PREFACE TO VOLUME 16
The Handbook series Magnetic Materials is a continuation of the Handbook series Ferromagnetic Materials. When Peter Wohlfarth started the latter series, his original aim was to
combine new developments in magnetism with the achievements of earlier compilations of
monographs, producing a worthy successor to Bozorth’s classical and monumental book
Ferromagnetism. This is the main reason that Ferromagnetic Materials was initially chosen as title for the Handbook series, although the latter aimed at giving a more complete
cross-section of magnetism than Bozorth’s book. In the last few decades magnetism has
seen an enormous expansion into a variety of different areas of research, comprising the
magnetism of several classes of novel materials that share with truly ferromagnetic materials only the presence of magnetic moments. For this reason the Editor and Publisher
of this Handbook series have carefully reconsidered the title of the Handbook series and
changed it into Magnetic Materials. It is with much pleasure that I can introduce to you
now Volume 16 of this Handbook series.
Magnetostrictive materials exhibit strains caused by the orientation of the magnetic moments when exposed to a magnetic field. A new class of magnetostrictive materials was
discovered in the 90th of last century by Ullakko and co-workers. These new materials
are called Magnetic Shape Memory (MSM) alloys or Ferromagnetic Shape Memory Alloys (FSMA). The thermoelastic martensitic phase transformation in MSM materials leads
to a low-symmetry phase characterized by a large magnetocrystalline anisotropy and by
highly mobile twin boundaries between the variants. These MSM materials can exhibit
giant magnetic field-induced strains (MFIS) that find their origin in the rearrangement of
the crystallographic domains (twin variants). When applying a magnetic field, the martensite variants that have their easy axis of magnetization along the field, start to grow due
to twin boundary motion and become predominant. This process lowers the magnetization
energy. The MFIS of MSM materials are unique as they can generate large strains with
fairly high frequencies without changes in the external temperature. The obtainable strains
are greater than those of magnetostrictive, piezoelectric or electrostrictive materials. Consequently, MSM materials are regarded as potentially important materials for actuator and
sensor applications. A comprehensive survey of this interesting novel class of materials is
presented in Chapter 1.
The development of modern magnetic materials and their application requires a deep
understanding of the magnetization processes that determine the magnetic properties. In
this respect, micromagnetics is of paramount importance since it relates the microscopic
distribution of the magnetization to the physical and chemical structure of a given material. In fact, micromagnetic modelling has become a most important tool in characterizing
v
vi
PREFACE TO VOLUME 16
the magnetic behaviour of many different materials as applied in thin film heads, various
types of recording media and nanostructured permanent magnets. The rapid progress of
nanotechnology has led to many new functional materials and devices. A prerequisite for
the development and application of structured materials is the detailed knowledge of the
correlation between the physical and magnetic structure of the system. So-called smart
magnetic materials require the possibility of predicting the response of the system to external fields, stress and temperature as a function of time. All these aspects are treated in
Chapter 2 where the basic theory of micromagnetism is explained for ferromagnetic and
antiferromagnetic materials. The authors introduce various methods for the solution of the
micromagnetic equation and deal with different numerical discretization schemes such as
the finite difference method and the finite element method. At the end of their chapter, the
authors present fully integrated recording simulations that allow to simulate a complete
recording process, including the relative movement of the externally magnetized head and
the recording medium.
A ferrofluid, or magnetic fluid as it is also frequently called, consists of a stable colloidal suspension of single-domain nano-sized particles of a ferrimagnet or ferromagnet in
a carrier liquid. A wide range of carrier liquids exists. Many ferrofluids are commercially
used to meet particular requirements. For instance in applications such as rotary vacuum
lead-troughs, it is essential that the carrier liquids have a very low vapor pressure. In other
applications, temperature may be a critical quantity. The first product based on ferrofluid,
introduced in the 60th of the last century, was a dynamic seal capable of operating under
pressure or vacuum. Since then quite a large number of new applications of the technology
have been discovered. Although ferrofluids may be present in many consumer products,
they are usually hidden from view. It is estimated that over 200 000 rotary vacuum seals,
350 million loudspeakers, 500 million computer disk drives, and 15 million DVD-ROM
drives have been built with ferrofluids. Chapter 3 of this Handbook Volume presents a
comprehensible overview of the production of ferrofluids and their colloidal and magnetic
stability, their basic properties and the possibility to magnetically control the properties
and flows of magnetic liquids by moderate magnetic fields. Moreover various examples
for the practical use of ferrofluids will be presented, including those in well established
fields but dealing also with novel and innovative applications currently in the focus of
actual research.
Antiferromagnetic exchange-coupled multilayer films are important ingredients for devices based on the giant magnetoresistance (GMR) and tunnel magnetoresistance (TMR)
effect as employed in many electronic and magnetic devices such as magnetic recording
heads and magnetic random access memories (MRAM). Furthermore, antiferromagnetic
exchange-coupled multilayer films are also employed in perpendicular magnetic recording
media where they serve as means that can reduce domain noise. Excellent high frequency
characteristics have been obtained by using antiferromagnetic/ferromagnetic/antiferromagnetic trilayers. In such devices, antiferromagnetic (AFM) systems based on Mn alloys
are useful as exchange biasing films for the ferromagnetic (FM) film and magnetic tunnel junctions. There are many systems having a complex (non-collinear) spin structure,
depending on the d-electron number and the crystal structure associated with geometrical
frustration of the magnetic moments. Experimental evaluation of their magnetic structures
is quite difficult, especially in γ -phase disordered alloys, and hence there still remain the
PREFACE TO VOLUME 16
vii
systems whose magnetic structures have not been established yet. Furthermore, the explanations of the mechanism of the exchange bias-field are still diverging, depending on the
proposed models. Chapter 4 presents an overview of the physical aspects of antiferromagnetism of Mn alloys both from theoretical and experimental view points, providing a basis
for future fundamental and practical developments. The authors discuss the electronic and
magnetic structures of Mn alloys, including results of first-principle calculations. Experimental data of antiferromagnetic transitions of various Mn alloys are reviewed in the light
of theoretical calculations. A discussion is also presented of the electrical resistivity and
its relation with the Néel transition and the concomitant changes in electronic states. Further topics addressed are the relations between lattice distortions, phase diagrams and spin
structures. Magnetovolume and magnetoelastic effects are discussed in terms of spin fluctuations. Finally, exchange-bias fields are reviewed and the mechanism of the exchange-bias
field between antiferromagnetic Mn alloy films and a ferromagnetic films are discussed
within the framework of the classical Heisenberg model.
Nanoparticulate materials have attracted increasing attention in the last decade. The
reason for this is the unique combination of small size, exotic properties and good processability, opening the possibility of their use in many technological and biomedical applications. For particle sizes falling into the nanometer range, materials can exhibit unusual
and interesting physical and mechanical properties. Many of the properties of magnetic
nanoparticles are described in Chapter 5. After a detailed description of the various methods used to synthesize magnetic nanoparticles, the authors give a comprehensible account
of the colloidal properties of nanoparticles which largely determine the applicability to
these materials. Special emphasis is given by them to several specific methodologies used
in the literature such as surface modification of nanoparticles and/or encapsulation which
render the magnetic nanoparticles useful in many areas of science and technology. The
magnetic behavior of magnetic nanoparticles represents a complex and challenging problem. It is clear that the potential applicability of the nanoparticulate systems requires a deep
knowledge of their magnetic properties and these properties are extensively discussed by
the authors. The last part of this chapter specifically deals with a detailed description of the
biomedical applications of magnetic nanoparticles such as NMR imaging, hyperthermia,
drug targeting, separation and selection.
Volume 16 of the Handbook on the Properties of Magnetic Materials, as the preceding
volumes, has a dual purpose. As a textbook it is intended to be of assistance to those who
wish to be introduced to a given topic in the field of magnetism without the need to read
the vast amount of literature published. As a work of reference it is intended for scientists active in magnetism research. To this dual purpose, Volume 16 of the Handbook is
composed of topical review articles written by leading authorities. In each of these articles
an extensive description is given in graphical as well as in tabular form, much emphasis
being placed on the discussion of the experimental material in the framework of physics,
chemistry and material science.
The task to provide the readership with novel trends and achievements in magnetism
would have been extremely difficult without the professionalism of the North Holland
viii
PREFACE TO VOLUME 16
Physics Division of Elsevier Science B.V., and I wish to thank Wim Spaans for his great
help and expertise.
K.H.J. B USCHOW
VAN DER WAALS -Z EEMAN I NSTITUTE
U NIVERSITY OF A MSTERDAM , N ETHERLANDS
CONTENTS
Preface to Volume 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
Contents of Volumes 1–15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
1. Giant Magnetostrictive Materials
O. SÖDERBERG, A. SOZINOV, Y. GE, S.-P. HANNULA
and V.K. LINDROOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Micromagnetic Simulation of Magnetic Materials
DIETER SUESS, JOSEF FIDLER and THOMAS SCHREFL . . . . . . . . . .
3. Ferrofluids
S. ODENBACH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4. Magnetic and Electrical Properties of Practical Antiferromagnetic Mn Alloys
K. FUKAMICHI, R.Y. UMETSU, A. SAKUMA and C. MITSUMATA . . . . .
5. Synthesis, Properties and Biomedical Applications of Magnetic Nanoparticles
PEDRO TARTAJ, MARIA P. MORALES, SABINO VEINTEMILLASVERDAGUER, TERESITA GONZALEZ-CARREÑO and
CARLOS J. SERNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
41
127
209
403
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
483
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
529
Materials Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
533
ix
CONTENTS OF VOLUMES 1–15
Volume 1
1. Iron, Cobalt and Nickel, by E. P. Wohlfarth . . . . . . . . . . . . . .
2. Dilute Transition Metal Alloys: Spin Glasses, by J. A. Mydosh and G. J. Nieuwenhuys
3. Rare Earth Metals and Alloys, by S. Legvold
. . . . . . . . . . . . .
4. Rare Earth Compounds, by K. H. J. Buschow
. . . . . . . . . . . . .
5. Actinide Elements and Compounds, by W. Trzebiatowski
. . . . . . . . .
6. Amorphous Ferromagnets, by F. E. Luborsky
. . . . . . . . . . . . .
7. Magnetostrictive Rare Earth–Fe2 Compounds, by A. E. Clark . . . . . . . .
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Volume 2
1. Ferromagnetic Insulators: Garnets, by M. A. Gilleo
. . . . .
2. Soft Magnetic Metallic Materials, by G. Y. Chin and J. H. Wernick
3. Ferrites for Non-Microwave Applications, by P. I. Slick . . . .
4. Microwave Ferrites, by J. Nicolas . . . . . . . . . . .
5. Crystalline Films for Bubbles, by A. H. Eschenfelder . . . . .
6. Amorphous Films for Bubbles, by A. H. Eschenfelder
. . . .
7. Recording Materials, by G. Bate
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8. Ferromagnetic Liquids, by S. W. Charles and J. Popplewell . . .
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Volume 3
1. Magnetism and Magnetic Materials: Historical Developments and Present Role in Industry
and Technology, by U. Enz
. . . . . . . . . . . . . . . . . . . . .
2. Permanent Magnets; Theory, by H. Zijlstra . . . . . . . . . . . . . . . .
3. The Structure and Properties of Alnico Permanent Magnet Alloys, by R. A. McCurrie . .
4. Oxide Spinels, by S. Krupička and P. Novák
. . . . . . . . . . . . . . .
5. Fundamental Properties of Hexagonal Ferrites with Magnetoplumbite Structure,
by H. Kojima . . . . . . . . . . . . . . . . . . . . . . . . . .
6. Properties of Ferroxplana-Type Hexagonal Ferrites, by M. Sugimoto
. . . . . . .
7. Hard Ferrites and Plastoferrites, by H. Stäblein . . . . . . . . . . . . . . .
8. Sulphospinels, by R. P. van Stapele . . . . . . . . . . . . . . . . . . .
9. Transport Properties of Ferromagnets, by I. A. Campbell and A. Fert
. . . . . . .
305
393
441
603
747
Volume 4
1. Permanent Magnet Materials Based on 3d-rich Ternary Compounds, by K. H. J. Buschow . . . . .
1
2. Rare Earth–Cobalt Permanent Magnets, by K. J. Strnat . . . . . . . . . . . . . . . . 131
3. Ferromagnetic Transition Metal Intermetallic Compounds, by J. G. Booth . . . . . . . . . . 211
xi
xii
CONTENTS OF VOLUMES 1–15
4. Intermetallic Compounds of Actinides, by V. Sechovský and L. Havela . . . . . . . . . . . 309
5. Magneto-Optical Properties of Alloys and Intermetallic Compounds, by K. H. J. Buschow
. . . . 493
Volume 5
1. Quadrupolar Interactions and Magneto-Elastic Effects in Rare-Earth Intermetallic Compounds,
by P. Morin and D. Schmitt . . . . . . . . . . . . . . . . . . . . . . .
2. Magneto-Optical Spectroscopy of f-Electron Systems, by W. Reim and J. Schoenes . . . . .
3. INVAR: Moment-Volume Instabilities in Transition Metals and Alloys, by E. F. Wasserman . .
4. Strongly Enhanced Itinerant Intermetallics and Alloys, by P. E. Brommer and J. J. M. Franse . .
5. First-Order Magnetic Processes, by G. Asti . . . . . . . . . . . . . . . . . .
6. Magnetic Superconductors, by Ø. Fischer
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511
1. Magnetism in Ultrathin Transition Metal Films, by U. Gradmann
. . . . . . . . . . .
2. Energy Band Theory of Metallic Magnetism in the Elements, by V. L. Moruzzi and
P. M. Marcus . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. Density Functional Theory of the Ground State Magnetic Properties of Rare Earths and Actinides,
by M. S. S. Brooks and B. Johansson . . . . . . . . . . . . . . . . . . . . .
4. Diluted Magnetic Semiconductors, by J. Kossut and W. Dobrowolski
. . . . . . . . . .
5. Magnetic Properties of Binary Rare-Earth 3d-Transition-Metal Intermetallic Compounds,
by J. J. M. Franse and R. J. Radwański
. . . . . . . . . . . . . . . . . . . .
6. Neutron Scattering on Heavy Fermion and Valence Fluctuation 4f-systems,
by M. Loewenhaupt and K. H. Fischer
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Volume 6
1. Magnetic Properties of Ternary Rare-Earth Transition-Metal Compounds, by H.-S. Li and
J. M. D. Coey . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Magnetic Properties of Ternary Intermetallic Rare-Earth Compounds, by A. Szytula
. .
3. Compounds of Transition Elements with Nonmetals, by O. Beckman and L. Lundgren . .
4. Magnetic Amorphous Alloys, by P. Hansen . . . . . . . . . . . . . . . .
5. Magnetism and Quasicrystals, by R. C. O’Handley, R. A. Dunlap and M. E. McHenry . .
6. Magnetism of Hydrides, by G. Wiesinger and G. Hilscher . . . . . . . . . . .
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Volume 7
. 139
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. 503
Volume 8
1. Magnetism in Artificial Metallic Superlattices of Rare Earth Metals, by J. J. Rhyne and
R. W. Erwin
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Thermal Expansion Anomalies and Spontaneous Magnetostriction in Rare-Earth Intermetallics
with Cobalt and Iron, by A. V. Andreev
. . . . . . . . . . . . . . . . . . .
3. Progress in Spinel Ferrite Research, by V. A. M. Brabers . . . . . . . . . . . . . .
4. Anisotropy in Iron-Based Soft Magnetic Materials, by M. Soinski and A. J. Moses . . . . .
5. Magnetic Properties of Rare Earth–Cu2 Compounds, by Nguyen Hoang Luong and
J. J. M. Franse
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. . 325
. . 415
Volume 9
1. Heavy Fermions and Related Compounds, by G.J. Nieuwenhuys . . . . . . . . . . . . .
2. Magnetic Materials Studied by Muon Spin Rotation Spectroscopy, by A. Schenck and F.N. Gygax . .
1
57
CONTENTS OF VOLUMES 1–15
xiii
3. Interstitially Modified Intermetallics of Rare Earth and 3d Elements, by H. Fujii and H. Sun . . . . 303
4. Field Induced Phase Transitions in Ferrimagnets, by A.K. Zvezdin
. . . . . . . . . . . . 405
5. Photon Beam Studies of Magnetic Materials, by S.W. Lovesey . . . . . . . . . . . . . . 545
Volume 10
1. Normal-State Magnetic Properties of Single-Layer Cuprate High-Temperature Superconductors and
Related Materials, by D.C. Johnston . . . . . . . . . . . . . . . . . . . . . .
1
2. Magnetism of Compounds of Rare Earths with Non-Magnetic Metals, by D. Gignoux and D. Schmitt
239
3. Nanocrystalline Soft Magnetic Alloys, by G. Herzer . . . . . . . . . . . . . . . . . 415
4. Magnetism and Processing of Permanent Magnet Materials, by K.H.J. Buschow . . . . . . . . 463
Volume 11
1. Magnetism of Ternary Intermetallic Compounds of Uranium, by V. Sechovský and L. Havela
. . .
1
2. Magnetic Recording Hard Disk Thin Film Media, by J.C. Lodder
. . . . . . . . . . . . 291
3. Magnetism of Permanent Magnet Materials and Related Compounds as Studied by NMR,
by Cz. Kapusta, P.C. Riedi and G.J. Tomka . . . . . . . . . . . . . . . . . . . . 407
4. Crystal Field Effects in Intermetallic Compounds Studied by Inelastic Neutron Scattering, by O. Moze 493
Volume 12
1. Giant Magnetoresistance in Magnetic Multilayers, by A. Barthélémy, A. Fert and F. Petroff
2. NMR of Thin Magnetic Films and Superlattices, by P.C. Riedi, T. Thomson and G.J. Tomka
3. Formation of 3d-Moments and Spin Fluctuations in Some Rare-Earth–Cobalt Compounds,
by N.H. Duc and P.E. Brommer . . . . . . . . . . . . . . . . . . . .
4. Magnetocaloric Effect in the Vicinity of Phase Transitions, by A.M. Tishin
. . . . .
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1
97
. . . . 259
. . . . 395
Volume 13
1. Interlayer Exchange Coupling in Layered Magnetic Structures, by D.E. Bürgler,
P. Grünberg, S.O. Demokritov and M.T. Johnson . . . . . . . . . . . . . . . . .
2. Density Functional Theory Applied to 4f and 5f Elements and Metallic Compounds, by M. Richter
3. Magneto-Optical Kerr Spectra, by P.M. Oppeneer . . . . . . . . . . . . . . . . .
4. Geometrical Frustration, by A.P. Ramirez
. . . . . . . . . . . . . . . . . . .
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1
. 87
. 229
. 423
Volume 14
1. III-V Ferromagnetic Semiconductors, by F. Matsukura, H. Ohno and T. Dietl
. . . . . . .
2. Magnetoelasticity in Nanoscale Heterogeneous Magnetic Materials, by N.H. Duc and P.E. Brommer
3. Magnetic and Superconducting Properties of Rare Earth Borocarbides of the Type RNi2 B2 C,
by K.-H. Müller, G. Fuchs, S.-L. Drechsler and V.N. Narozhnyi
. . . . . . . . . . . .
4. Spontaneous Magnetoelastic Effects in Gadolinium Compounds, by A. Lindbaum and M. Rotter .
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89
. 199
. 307
Volume 15
1. Giant Magnetoresistance and Magnetic Interactions in Exchange-Biased Spin-Valves, by R. Coehoorn
1
2. Electronic Structure Calculations of Low-dimensional Transition Metals, by A. Vega, J.C. Parlebas and
C. Demangeat
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
3. II–VI and IV–VI Diluted Magnetic Semiconductors – New Bulk Materials and Low-Dimensional
Quantum Structures, by W. Dobrowolski, J. Kossut and T. Story . . . . . . . . . . . . . 289
xiv
CONTENTS OF VOLUMES 1–15
4. Magnetic Ordering Phenomena and Dynamic Fluctuations in Cuprate Superconductors and Insulating
Nickelates, by H.B. Brom and J. Zaanen . . . . . . . . . . . . . . . . . . . . . 379
5. Giant Magnetoimpedance, by M. Knobel, M. Vázquez and L. Kraus . . . . . . . . . . . . 497
LIST OF CONTRIBUTORS
J. Fidler, Vienna University of Technology, Vienna, Austria
K. Fukamichi, Department of Materials Science, Graduate School of Engineering, Tohoku
University, Aoba 6-6-02, Sendai 980-8579, Japan
Y. Ge, Laboratory of Materials Science, a member laboratory of the Center for New Materials, Helsinki University of Technology, Helsinki, Finland
T. Gonzalez-Carreño, Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco,
28049, Madrid, Spain
S.-P. Hannula, Laboratory of Materials Science, a member laboratory of the Center for
New Materials, Helsinki University of Technology, Helsinki, Finland
V.K. Lindroos, Laboratory of Materials Science, a member laboratory of the Center for
New Materials, Helsinki University of Technology, Helsinki, Finland
C. Mitsumata, Advanced Electronics Research Lab., Hitachi Metals, Ltd., 5200 Mikajiri,
Kumagaya 360-0843, Japan
M.P. Morales, Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, 28049,
Madrid, Spain
S. Odenbach, Technische Universität Dresden, Chair for Magnetofluiddynamics, 01062
Dresden, Germany
A. Sakuma, Department of Applied Physics, Graduate School of Engineering, Tohoku University, Aoba 6-6-05, Sendai 980-8579, Japan
T. Schrefl, University of Sheffield, Sheffield, UK
C.J. Serna, Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, 28049,
Madrid, Spain
O. Söderberg, Laboratory of Materials Science, a member laboratory of the Center for New
Materials, Helsinki University of Technology, Helsinki, Finland
A. Sozinov, Laboratory of Materials Science, a member laboratory of the Center for New
Materials, Helsinki University of Technology, Helsinki, Finland
D. Suess, Vienna University of Technology, Vienna, Austria
xv
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LIST OF CONTRIBUTORS
P. Tartaj, Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, 28049,
Madrid, Spain
R.Y. Umetsu, Department of Materials Science, Graduate School of Engineering, Tohoku
University, Aoba 6-6-02, Sendai 980-8579, Japan
S. Veintemillas-Verdaguer, Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, 28049, Madrid, Spain
chapter 1
GIANT MAGNETOSTRICTIVE MATERIALS
O. SÖDERBERG, A. SOZINOV, Y. GE, S.-P. HANNULA
and V.K. LINDROOS
Laboratory of Materials Science, a member laboratory of the Center for New Materials,
Helsinki University of Technology, Helsinki, Finland
Handbook of Magnetic Materials, edited by K.H.J. Buschow
Volume 16 ISSN: 1567-2719 DOI 10.1016/S1567-2719(05)16001-6
1
© 2006 Elsevier B.V.
All rights reserved
CONTENTS
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.1. Magnetic shape memory effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2. Selected active materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2. Modelling the behaviour of MSM materials and the giant magnetic-field-induced strain (MFIS) . . . . .
5
3. Ni-Mn-Ga alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.1. Martensitic and reverse phase transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.2. Mechanical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
3.3. Magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
4. Martensite variant rearrangement in an applied magnetic field . . . . . . . . . . . . . . . . . . . . . . .
16
4.1. Effect of load on magnetic-field-induced strain (MFIS) . . . . . . . . . . . . . . . . . . . . . . . .
18
4.2. Dynamical actuation and fatigue of MFIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
4.3. Temperature dependence of MFIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
5. Selected properties of Ni-Mn-Ga alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
6. Promising MSM materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2
1. Introduction
When a magnetostrictive material is set in the magnetic field it strains while its magnetic
moments reorientate. The present review focuses on the ferromagnetic Magnetic Shape
Memory (MSM) alloys with a giant magnetic-field-induced strain (MFIS). MFIS is based
on the rearrangement of the martensite twin variants by the twin boundary motion. This is a
result of the interaction of the magnetic and crystal structure domains. The MSM alloys or
Ferromagnetic Shape Memory Alloys (FSMA) have a ferromagnetic thermoelastic twinned
martensite phase, high magnetocrystalline anisotropy and highly mobile twin boundaries.
These materials have potential for actuator and sensor applications since they combine
a large strain with rather high frequencies without a temperature change. The obtainable
strains are greater than those of magnetostrictive, piezoelectric or electrostrictive materials.
Ullakko (1995, 1996a) presented the idea of redistribution of twin variants by magnetic
field, i.e. MSM behaviour in martensitic phase. Experimental confirmation showed a 0.2%
MFIS in an unstressed single crystal of Ni2 MnGa in 800 kA/m magnetic field applied
along [001] at 265 K (Ullakko et al., 1996b, 1997). By applying the constantly increasing
knowledge of the Ni-Mn-Ga system, the obtainable strain and the service temperature is
raised and the applied magnetic field is reduced to some extent (Tickle and James, 1999;
Tickle et al., 1999; Murray et al., 2000a, 2000b; Heczko et al., 2000; Sozinov et al., 2001a,
2001b, 2001c, 2002a, 2002b, 2003a, 2003b, 2004a). To achieve the maximum MFIS, the
martensite should be in a single-variant state. In Ni-Mn-Ga alloys with the modulated fivelayered martensite structure (5M) a MFIS of 6% is obtained. In alloys with a modulated
seven-layered martensite structure (7M) the corresponding MFIS is 10%. The actuation
of the 5M alloys is demonstrated with 2.5% strains up to 500 Hz (Henry et al., 2001,
2002a, 2002b, 2003a, 2003b; Tellinen et al., 2002; Marioni et al., 2003a, 2003b). The
reverse application of the Ni-Mn-Ga MSM element as a sensor or a generator is studied by
Müllner et al. (2003a) and Suorsa et al. (2004).
The possibility for MFIS in other than Ni-Mn-Ga alloys is also studied in several alloy
systems with thermoelastic martensite transformation (James and Wuttig, 1998; Furuya et
al., 1998, 1999; Kakeshita et al., 2000, 2003; Yabe et al., 2000; Oikawa et al., 2001a, 2001b,
2002, 2003; Wuttig et al., 2001, 2002; Craciunescu et al., 2002a, 2002b, 2002c; Kakeshita
and Fukuda, 2002, 2003; Morito et al., 2002, 2003; Zhou et al., 2002a, 2003; Inoue et al.,
2003; Karaca et al., 2003; Sato et al., 2003; Li et al., 2004a, 2004b; Fukuda et al., 2004;
Efstathiou et al., 2004). Some of the alloys apply the actual MSM phenomenon, but in most
of them the idea is to control the stress-induced martensite (SIM) formation by the magnetic field. SIM based strain is also possible in Ni-Mn-Ga (Kokorin et al., 1992; Martynov
and Kokorin, 1992; Chernenko et al., 1995a, 1998a; Vasil’ev et al., 1999; Gonzàlez-Comas
et al., 1999; Inoue et al., 2000; Li et al., 2004a, 2004b; Heczko and Straka, 2004b).
3
4
O. SÖDERBERG et al.
Fig. 1.1. A MSM element in (1) a unidirectional and (2) a rotating magnetic field; and (3) in the actuator. a, b
and c are the crystallographic axes of martensite (c < a, b); c-axis is the easy axis of magnetization. (Courtesy
of Söderberg et al., 2005, reproduced by permission of Elsevier Science).
1.1. Magnetic shape memory effect
The magnetic shape memory effect (MSME) is demonstrated with a Ni-Mn-Ga MSM alloy
in the magnetic field (fig. 1.1). In the 5M and 7M martensite the shortest crystallographic
c-axis is also the easy axis of magnetization. Because the easy directions of magnetization
are different in the adjoining twins, the applied magnetic field creates a difference in energy
of the variants (O’Handley et al., 2000; O’Handley and Allen, 2001; Vasil’ev and Takagi,
2004; Kiang and Tong, 2005). This energy difference is a driving force for the growth of
those twin variants that are favourably oriented to the applied field. The structural change
occurs via twin boundary motion. Those martensite variants with c-axis along the magnetic
field become dominant and the material contracts in the direction of the applied field. The
shape change remains after removing the magnetic field. Actuation is achieved if the magnetic field is turned perpendicular to its original orientation—for example with rotating the
magnetic field—or with an external spring-back load. Then one single-variant martensite
structure changes to another and generates the cycling of the giant MFIS (Söderberg et al.,
2005).
1.2. Selected active materials
Active materials change their properties according to external impulses and they can work
as actuators, sensors or simultaneously both. Such materials in addition to the MSM alloys
are piezoelectrics (PZT), magnetostrictive materials (MS), shape memory alloys (SMA)
GIANT MAGNETOSTRICTIVE MATERIALS
5
and electroactive polymers (EAP). Piezoelectric materials are widely used in sensing
and actuation, especially with high frequencies. They change dimensions in an electric
field, or when loaded mechanically, they create an electric current (Muralt, 2001; Boller,
2001; Neurgaonkar, 2001; Quandt, 2001). Piezos can be polymers—such as PVDF—or
ceramics. The most utilized is Pb-Zr-Ti (PZT) with 0.16% strain. With single crystals
of Pb(Zn1/3 ,Nb2/3 )O3 (PZN), Pb(Mg1/3 ,Nb2/3 )O3 (PMN) and PbTiO3 (PT) the strain
can be much higher. Magnetostrictive materials change dimensions in a magnetic field
(Guruswamy et al., 2000; Quandt and Claeyssen, 2000; Peuzin, 2001; Quandt, 2002). The
new promising MS candidates are Fe-Ga-based alloys (Guruswamy et al., 2000; Clark
et al., 2003; Kumagai et al., 2004), but the most applied industrial MS material is Terfenol D (Tb0.3 Dy0.7 Fe2 ) operating at ambient temperature up to 20 kHz with 0.2%–0.24%
strain in 40 kA/m magnetic field (Clark, 1980; Clark et al., 1992; Clark, 1994; O’Handley
and Allen, 2001). With the shape memory alloys (SMAs) one can obtain large strains
and good stress output with changing temperature, but in actuation the frequencies are
smaller than with other active materials due to the lengthy cooling process (Duerig, 1990;
Otsuka and Wayman, 1998; Friend, 2001; Quandt, 2001). The most applied SMAs are
Ni-Ti-, Cu-Zn-Al- and Fe-Mn-Si-based alloys. Some of these materials have in austenitic
state a superelastic behaviour just above the martensite transformation temperature where
stress induced martensite can form. Also, electroactive polymers and ionic polymeric–
metal composites (IPMCs) have great potential as soft robotic actuators, artificial muscles,
and dynamic sensors in micro-to-macro size range (Shahinpoor, 2003).
2. Modelling the behaviour of MSM materials and the giant magnetic-field-induced
strain (MFIS)
Modelling the behaviour of MSM materials in simultaneously applied stress and magnetic field is complicated since these materials consist of multidomain crystallographic
and magnetic subsystems and their behaviour is affected by the interaction between these
subsystems. The first-principles numerical calculations have been applied to reveal the
microscopic origin of the lattice instabilities and the phase transformations in Ni-MnGa alloys (Ayuela et al., 1999, 2002; Godlevsky and Rabe, 2001; Bungaro et al., 2003;
Zayak et al., 2003; Zayak and Entel, 2004, 2005; Wan et al., 2005a, 2005b), their magnetocrystalline anisotropy (Enkovaara et al., 2002a), and some particularities in the magnetic
subsystem (Enkovaara et al., 2002b, 2003, 2004). The low temperature phase diagram for
Ni-Mn-Ga was calculated based on the phenomenological theory of phase transformations in ferromagnets (review by Vasil’ev et al., 2003). The premartensitic phenomenon
and the intermediate phase in it are reviewed by Planes and Mañosa (2001). The general
scheme considering the multistage structural transformation with the modulated phases is
suggested by Castán et al. (2003). Chernenko et al. (2004a) and Hirsinger et al. (2004)
have been modelling the stress-induced martensitic transformation.
According to the first theoretical considerations of the large magnetically induced strain,
the total energy density of the ferromagnetic martensite in a magnetic field contains the
Zeeman terms −H · mi (Ullakko et al., 1996b; James and Wuttig, 1998). Here H is
the magnetic field and mi the magnetic moment of the martensite variant i with changing value in the differently oriented variants. High magnetocrystalline anisotropy leads to
6
O. SÖDERBERG et al.
Fig. 2.1. Limitation of the magnetic field-induced driving force of the twin boundary motion. The difference in
the magnetic energy density of the twin variants is independent of the magnetic field value at high field, since the
magnetic moments of martensitic variants are oriented along the field. The deviation of a few degrees from the
90◦ between the c-axis inside the variants is neglected.
large differences in Zeeman energies and, thus, may promote the redistribution of martensite variants, when twin boundaries move easily. O’Handley’s model (1998) takes into
account the magnetization rotation. Due to it, the different martensite variants saturate at
a higher field than the anisotropy field, 2Ku /Ms (Ku —the uniaxial magnetic anisotropy
constant, Ms —saturation magnetization). Consequently, the driving force for the martensite variant redistribution by the magnetic field is restricted (fig. 2.1). James et al. (1999)
also acknowledged the importance of magnetization rotation. O’Handley’s model has been
the basis for the later models of the MSM materials in the alternating magnetic field (Henry
et al., 2002b; Marioni et al., 2003a).
Likhachev and Ullakko (2000a, 2000b) pointed out the importance of the hysteretic
mechanical behaviour in the stress-induced martensite variant rearrangement. The mechanical stress–strain relationship has been extended with the magnetic stress σmag (H )
when the magnetic field is applied. The Zeeman, magnetostatic and magnetic anisotropy
energies are all included. The saturation value is predicted to be σmag (H ) = Ku · ε0−1 ,
where ε0 = (1 − c/a), and a and c are the lattice parameters of the tetragonal phase.
With this model the magnetic stress σmag (H ) and deformation ε(σmec , H ) for the twovariant martensite structure can be calculated from the experimental mechanical testing
data εmec (σmec )H =0 and the field dependencies of magnetization along easy and hard directions in the single-variant state (Likhachev et al., 2004a). Thus, the large MFIS is possible
only in materials where the maximal magnetic stress exceeds the zero-field twinning stress
σtw , i.e. Ku · ε0−1 > σtw .
The approach of Hirsinger and Lexcellent (2003a, 2003b) and Creton and Hirsinger
(2005) is based on a thermodynamic model of irreversible process with internal variables. The magnetic subsystem is non-linear reversible, while the mechanical response
is decomposed to reversible and irreversible parts. Bogdanov et al. (2003) have applied
GIANT MAGNETOSTRICTIVE MATERIALS
7
the phenomenological theory of magnetoelastic interactions to describe the two-variant
tetragonal martensite in a magnetic field. The elastic energy for tetragonal lattice and the
magnetostatic together with the magnetic anisotropy energies are included. The field and
stress dependencies of strain are quantitative. Experimental data on elastic constants in the
martensitic phases is important for the model. This model can be improved by including
the mechanical hysteresis in it.
The Landau approach is widely used for modelling the structural and magnetic phase
transitions (Vasil’ev et al., 2003). Buchel’nikov et al. (2001, 2005) and L’vov et al. (2002)
apply it to the magnetic field-induced strain in MSM materials. The L’vov model was
developed further by taking into account the statistical nature of the obstacles for twin
boundary motion (Glavatska et al., 2003b). It was also applied to explain the different
Curie temperatures of cubic and martensitic phases as well as the magnetic anisotropy
properties of the tetragonal martensites in Ni-Mn-Ga and the mechanical behaviour of the
5M in a constant magnetic field (Chernenko et al., 2003a, 2003b, 2004b, 2004c). The main
difference between the others and the model of L’vov is that the normal magnetostriction
is considered responsible for the giant MFIS. All other above-mentioned models neglect
the normal MS deformation.
The microscopic details concerning the behaviour of MSM materials are proposed only
by Paul et al. (2003) and Müllner et al. (2003b). According to Paul et al. the twin boundary
motion is controlled by the motion of the magnetic domain walls and the model analyses
the region close to the twin boundary in the micro-magnetic scale. Müllner et al. explain the
MFIS of martensite by the effect of the magnetic force on the dislocations and, therefore,
the magneto-mechanical hysteresis is explained with a microscopic model dealing with
twinning dislocations, their mutual interaction and interaction with interfaces.
3. Ni-Mn-Ga alloys
The structural, magnetic and mechanical properties of the Ni-Mn-Ga alloys—the currently
best performing MSM materials—are highly sensitive to the chemical composition and
temperature. Their phase transformations, magnetic properties, behaviour in the magnetic
field and some selected physical and chemical properties are introduced here.
3.1. Martensitic and reverse phase transformations
The high temperature parent phase of near-stoichiometric Ni2 MnGa alloys has the highly
ordered L21 cubic structure with lattice parameter a = 0.576–0.597 nm depending on the
alloy composition and the external temperature (Webster et al., 1984; Chernenko et al.,
1995b; Brown et al., 1999; Inoue et al., 2000; Ge et al., 2003). The degree of atomic order
effects the structure and the magnetic properties. The ordering can be changed, for example, by fast cooling from homogenisation temperature resulting in 100 K lower martensitic
transformation temperature (Kreissl et al., 2004). The dendritic microstructure formed during casting of Ni-Mn-Ga alloys can be homogenized if the annealing temperature is high
enough and the duration of the heat treatment long enough—too low temperature and too
short time do not have the desired effect (Gupta et al., 2004; Pirge et al., 2004). A variety
8
O. SÖDERBERG et al.
of martensite crystal structures have been observed in Ni-Mn-Ga: non-modulated (marked
with T or NM in references), and modulated, so-called, 5-layered (5M), 7-layered (7M), 8layered (8M) or 10-layered (10M) (Martynov and Kokorin, 1992; Chernenko et al., 1998b;
Pons et al., 2000; Lanska et al., 2004). Modulation is observed as extra diffraction maxima
between the fundamental spots in 110∗ direction of reciprocal space and the crystal structure can be interpreted either as a long period stacking of closed-packed planes (110) (Pons
et al., 2000) or a periodic shuffling of basal planes (110) along [11̄0] (Martynov and Kokorin, 1992). When the crystal lattice is presented in the cubic parent phase co-ordinates the
ratio of the lattice parameters (of the basic or average lattice) in the 5M and 7M structures
is c/a < 1 and in the NM structure c/a > 1. The premartensitic or intermediate structure appearing in certain Ni-Mn-Ga alloys before the martensitic transformation (Cesari et
al., 1997; Kokorin et al., 1997; Planes et al., 1997; Khovailo et al., 2001; Vasil’ev et al.,
2003; Seguí et al., 2005) may have relevance in the formation of the modulated phases,
but it is not discussed in detail here, since this presentation focuses on the behaviour of the
martensitic phases.
The stoichiometric Ni2 MnGa transforms from the L21 phase (the parent phase P) into a
martensite structure approximately at 200 K (Webster et al., 1984; Ooiwa et al., 1992; Pons
et al., 2000; Brown et al., 2002). In the off-stoichiometric Ni-Mn-Ga alloys the martensitic
transformation is highly composition dependent and occurs at temperatures below 630 K
(Chernenko et al., 1995b, 1999; Mañosa et al., 1999; Lanska et al., 2004). The martensite
transformation temperatures Ms together with the Curie points Tc are mapped according to
the valence electron concentration in fig. 3.1 (see for example Chernenko, 1999; Tsuchiya
et al., 2001; Lanska et al., 2004). Ternary Ni-Mn-Ga alloys having their Curie points approximately at 370 K transform to the 5M structure at 343 K at the highest, while for
the 7M alloys the highest reported martensite start temperature (Ms ) is 356 K (Lanska
et al., 2004). Since the MSM occurs in certain martensite structures, the intermartensitic
reactions limit its lowest service temperature. Intermartensitic transformations may occur
thermally or stress induced (Martynov and Kokorin, 1992; Chernenko et al., 1995a, 1997;
Wang et al., 2001; Heczko et al., 2002a; Soolshenko et al., 2003; Seguí et al., 2003; Dai
et al., 2004; Khovailo et al., 2004a; Sozinov et al., 2004b; Söderberg, 2004). The final
structure is usually the NM phase and the transformation sequence depends on the alloy
composition. It occurs via modulated structures, for example P → 5M → 7M → NM
or P → 7M → NM. Attempts to increase the transformation temperatures and the Curie
point, i.e. rise the service temperature, as well as suppressing the intermartensitic reactions
are carried out by alloying with quaternary elements (Kokorin et al., 1989; Tsuchiya et al.,
2000a; Liu et al., 2002a, 2002b; Cherechukin et al., 2004; Khovailo et al., 2003, 2004b; Lu
et al., 2003; Yamaguchi et al., 2003; Kikuchi et al., 2004; Koho et al., 2004; Söderberg et
al., 2004a; Tsuchiya et al., 2004).
3.2. Mechanical properties
The crystal structure has a remarkable influence on the mechanical properties of the NiMn-Ga alloys. Mechanical behaviour is relevant for the possibility of MSM effect and
obtainable strains.
Three elastic constants of the Ni-Mn-Ga cubic parent phase C11 , C12 and C44 are
calculated from the stiffness constants measured in different crystal directions with the
GIANT MAGNETOSTRICTIVE MATERIALS
9
Fig. 3.1. The martensitic transformation temperatures and the Curie points vs. the valence electron concentration
in Ni-Mn-Ga. References: Albertini et al. (2001a, 2001b), Aliev et al. (2004), Brown et al. (1999, 2002, 2004),
Cesari et al. (1997), Chernenko (1999), Chernenko and Vitenko (1994), Chernenko et al. (1994, 1995a, 1995b,
1997, 1998a, 1998b), Elfazani et al. (1981), Ezer et al. (1999), Fritsch et al. (1994), Ge et al. (2002), Glavatska
et al. (2002, 2003a), Gonzàlez-Comas et al. (1999), Inoue et al. (2002), Heczko and Straka (2003), Heczko et al.
(2001a, 2002a, 2002b), James et al. (1999), Kokorin et al. (1992, 1996), Kudryavtsev et al. (2002), Lanska et al.
(2004), Ma et al. (2000), Mañosa et al. (1997), Martynov (1995), Martynov and Kokorin (1992), Matsumoto et
al. (1999), Mogylnyy et al. (2003), Murray et al. (1998), Obradó et al. (1998), Ooiwa et al. (1992), Pakhomov et
al. (2001), Park et al. (2003), Pasquale et al. (2002), Planes et al. (1997), Pons et al. (2000), Shanina et al. (2001),
Sozinov et al. (2001a), Stenger and Trivisonno (1998), Stuhr et al. (1997, 2000), Tickle and James (1999), Tickle
et al. (1999), Tsuchiya et al. (2000b, 2003a, 2003b), Ullakko et al. (1996b, 1997), Wang et al. (2001, 2002),
Webster et al. (1984), Wirth et al. (1997), Zasimchuk et al. (1990), Zheludev et al. (1995a, 1995b), Zuo et al.
(1998).
ultrasonic continuous-wave method and show softening of C = 1/2(C11 − C12 ) in the
vicinity of the structure phase transformation (Worgull et al., 1996; Planes and Mañosa,
2001; Stipcich et al., 2004). This indicates on the lattice instability to shear wave propagated in [110]. For the martensitic tetragonal phases due to the lower symmetry the amount
of independent elastic constants is six. Experimental studies of it in Ni-Mn-Ga have been
recently started (Dai et al., 2003, 2004). It is important to mention that a low shear elastic constant was found in the 5M phase (9 GPa) as well as in the cubic phase (7 GPa).
The rhombohedral constants C44 and C66 are close to each other, 51 GPa and 49 GPa
(Dai et al., 2003). The temperature dependence of the elastic constants C and C ∗ in a
magnetic field are plotted in fig. 3.2 from above the Curie point down to 200 K for the
10
O. SÖDERBERG et al.
Fig. 3.2. The elastic constants C and C ∗ = (a+b)−1 [cC44 +a(C33 −C44 −C13 )], where a = C11 −2C44 −C13 ,
b = C11 − 2C66 − C12 and c = C33 − 2C44 − C12 , of the alloy Ni50 Mn28.4 Ga21.6 in the temperature range of
200–380 K (above the Curie point). C was determined with magnetic field along [001] and C ∗ with the magnetic
field parallel to [100]. In the austenite (above 318 K) the two modes are degenerate. (Courtesy of Dai et al., 2004,
reproduced by permission of American Institute of Physics).
single-variant sample of the alloy Ni50 Mn28.4 Ga21.6 (Dai et al., 2004). Here, the upper
mode C describes the stiffness of the tetragonal distortion in the xy plane, while the lower
C ∗ represents physically the stiffness for the further tetragonal distortion in the xz or yz
plane.
In Ni-Mn-Ga alloys the hydrostatic pressure stabilizes the parent phase (Kanomata et al.,
1987), while the shear stress favours the martensite and causes the stress-induced martensite (SIM) formation (Kokorin et al., 1992; Martynov and Kokorin, 1992; Chernenko et al.,
1998a, 2003b; Gonzàlez-Comas et al., 1999). Stresses of 40–80 MPa or higher give approximately a 4% strain connected to the SIM formation in the parent phase; furthermore,
if the intermartensitic reactions are involved, strains may be 6–10% (Martynov and Kokorin, 1992). SIM formation is possible also in a high magnetic field, for example a field of
104 kA/m at 1 K above the martensite start temperature (Kakeshita et al., 1999; Vasil’ev et
al., 1999; Gonzàlez-Comas et al., 1999; Inoue et al., 2000; Cherechukin et al., 2001, 2003).
Training in the magnetic field increases the obtainable shape change in such a transformation (Cherechukin et al., 2003).
The thermally formed martensite has a multi-variant structure. The rearrangement of the
twins occurs when sufficient external stress is applied. This is observed as a stress plateau
in the stress–strain-curve (Otsuka and Wayman, 1998; Murray et al., 1998; O’Handley
and Allen, 2001). At the end of the stress plateau, the twin structure of the martensitic
phase is simpler. The single-variant state in the 5M structure can be obtained with a single compression and, the twinning stress, σtw , needed for variant reorientation can be less
than 1 MPa (fig. 3.3a). The twinning stress of the 7M structure can be decreased close to
1 MPa by the pre-straining including three compression-cycles to two different crystallographic orientations (fig. 3.3b, Sozinov et al., 2004a). The single variant state of the NM
structure can be obtained with three successive compressions to the three crystallographic
GIANT MAGNETOSTRICTIVE MATERIALS
11
(a)
(b)
Fig. 3.3. The martensite variant rearrangement in the Ni-Mn-Ga alloys in compression. (a) 5M-, (b) 7M- and
(c) NM-martensite. (Courtesy and compilation from Heczko and Straka, 2003 and Sozinov et al., 2003b, 2004a,
2004b, reproduced by permission of American Institute of Physics).
orientations or by tensile-compressive cycling. The lowest observed twinning stress for
NM martensite is about 6 MPa, which is much higher than the one required in MSME
(fig. 3.3c, Sozinov et al., 2004b; Söderberg et al., 2004b). The twinning stress decreases
with increasing temperature (Heczko et al., 2002b; Koho et al., 2003; Heczko and Straka,
2003; Sozinov et al., 2003a, 2004a, 2004b; Söderberg et al., 2004b). The maximum obtainable strain connected to a certain single variant martensite structure depends on the
crystallographic distortion of the structure and can be calculated as ε0 = 1 − c/a, where c
and a are given in the parent phase coordinates (O’Handley and Allen, 2001). This grants
for the maximum strains approximately 6% in the 5M-, 10% for the 7M- and 20% for the
NM-structure.
12
O. SÖDERBERG et al.
(c)
Fig. 3.3. (Continued.)
3.3. Magnetic properties
The magnetic anisotropy as well as the magnetization of the material are important factors
for the MSM effect. Also, it is important to notice the interaction of the twin domains and
the magnetic domains of the structure.
Magnetocrystalline anisotropy has been studied in the Ni-Mn-Ga system experimentally both for the cubic phase and for the different martensites (Tickle and James, 1999;
Likhachev and Ullakko, 2000b; Murray et al., 2000a; Heczko et al., 2000, 2002b; Sozinov et al., 2001b, 2002a; Straka et al., 2002, 2004; Straka and Heczko, 2003b; Heczko and
Straka, 2003, 2004a). Also, the summarised experimental results for Ni-Mn-Ga martensitic
phases are available (Sozinov et al., 2002c, 2003a; Heczko et al., 2003; Straka and Heczko,
2003a).
Magnetization curves at 300 K for the 5M, 7M and NM martensites measured in different crystallographic directions are shown in fig. 3.4. The magnetic anisotropy energy
density is calculated as the area between the easy and the hard magnetization directions.
For reliable results, the martensite must be in the single-variant state (Tickle and James,
1999). For the 5M martensite the easy axis of magnetization is its crystallographic short
c-axis. For the uniaxial anisotropy constant Ku at ambient temperature values in the range
of 1.2–2 × 105 J/m3 have been reported, while the second anisotropy constant is negligible (Heczko et al., 2002b; Enkovaara et al., 2004). The big difference in the Ku values
reflects probably the composition dependence (Albertini et al., 2002a; Heczko and Straka,
2004a, 2004b). Also, in the 7M martensite the shortest c-axis is the axis of easy magnetization (fig. 3.4c). The longest a-axis coincides with the axis of the hard magnetization and
the magnetization of the b-axis is the intermediate one. Therefore, two magnetic anisotropy
constants are needed to characterize this orthorhombic crystal structure. Their respective
GIANT MAGNETOSTRICTIVE MATERIALS
13
Fig. 3.4. Magnetization curves for different martensitic phases in the Ni-Mn-Ga system at 300 K. 1 Tesla corresponds with approximately 800 kA m−1 . (Courtesy and compilation from Sozinov et al., 2002c, reproduced by
permission of Institute of Electrical and Electronics Engineers (IEEE)).
values (Kb , Ka ) can be calculated from the magnetization data as the area cross-sections
between curves for the c-axis and for the b- or a-axis respectively. Kb = 0.7 × 105 J/m3
and Ka = 1.6 × 105 J/m3 for Ni48.8 Mn29.7 Ga21.5 and Kb = 1.02 × 105 J/m3 and
Ka = 1.74 × 105 J/m3 for Ni50.5 Mn29.4 Ga20.1 are established at ambient temperature
(Sozinov et al., 2002a; Heczko et al., 2003). In the NM martensite the c-axis is the hard
axis of magnetization and there is an easy plane of magnetization. This is connected to
the different kind of tetragonal distortions of the NM phase (c/a > 1) as compared with
the distortion of the 5M phase (c/a < 1). At ambient temperature the value of magnetic
anisotropy energy density for the NM martensite is close to −2 × 105 J/m3 . The magnetic
anisotropy increases, when the temperature decreases, as shown in fig. 3.5 (Heczko et al.,
2002b; Heczko and Straka, 2003; Straka and Heczko, 2003b; Straka et al., 2004).
The saturation magnetization in the Ni-Mn-Ga system depends on composition and temperature (Jin et al., 2002; Takeuchi et al., 2003; Heczko and Straka, 2004a). Upon cooling
the magnetization increases as shown in fig. 3.6, but during the structural phase transformations the magnetization shows an abrupt change (Webster et al., 1984; Marcos et al., 2002;
14
O. SÖDERBERG et al.
(a)
(b)
Fig. 3.5. Temperature dependences of magnetic anisotropy energy density of different martensitic phases in
Ni-Mn-Ga. (Courtesy and compilation from Straka and Heczko, 2003b, reproduced by permission of American
Institute of Physics).
Vasil’ev et al., 2003). In the Ni-Mn-Ga martensite the magnetic anisotropy is larger than
in the cubic phase; therefore, the magnetization of the martensite at low fields is smaller
than the one of the cubic phase (Tickle and James, 1999). In strong fields the saturation
magnetization of martensite is higher than that of the cubic phase and this can be applied
in the giant magnetocaloric effect (Marcos et al., 2002, 2003).
The martensitic transformation (cubic → 5M) of the single-crystalline Ni-Mn-Ga material produces a multi-variant structure, where the c-axes of the variants are close to the
[100]p , [010]p and [001]p directions of the parent cubic phase. According to Likhachev
and Ullakko (2000a) when the magnetic field is applied along the [100]p direction and the
saturation value in one martensite variant is reached, this variant is taken over by one large
magnetic domain. A rising magnetic field increases the total magnetization of the structure by rotation of local magnetic moments inside other martensite variants. This explains
GIANT MAGNETOSTRICTIVE MATERIALS
15
Fig. 3.6. Temperature dependencies of magnetization for Ni2 MnGa. The abrupt change at T = 202 K is connected with the structural cubic to tetragonal phase transformation. (Courtesy and compilation from Webster et
al., 1984, reproduced by permission of Elsevier Science).
the kink (fig. 3.7) in magnetization data observed at the field value corresponding to the
saturation field of the first variant (with c-axis along the field) reported by Ullakko et al.
(1996b) for a single-crystal sample.
When there is no constraint and the twinning stress is low enough, the MFIS connected
with twin boundary motion can be measured simultaneously with the magnetization data
(fig. 3.8). It is shown by a jump in the first quadrant of the magnetization curve. The critical
magnetic field for MFIS decreases when the magnetic anisotropy constant of the material
increases or when the twinning stress decreases.
The martensite of the MSM alloy Ni51.3 Mn24 Ga24.7 has a hierarchical magnetic domain
structure of herringbone patterns, where 180◦ magnetic domains form laminas within the
martensite variants (James et al., 1999). When magnetic fields up to 732 kA/m are applied
during the martensitic transformation the fir-tree magnetic domain structure changes to
one big domain (Pan and James, 2000). In the austenitic and premartensitic phase the 180◦
domain is dominant, while in the martensitic phase the magnetic domain coincides with the
twin boundaries and the microtwins of the structure (DeGraef et al., 2001, 2003; Heczko et
al., 2001a; Sozinov et al., 2001a; Park et al., 2003; Tsuchiya et al., 2003b; Grechishkin et
al., 2004). The magnetization of the main variant may distort the internal twins into a zigzag shape. (Ge et al., 2004). During the magnetic-field-induced twin boundary motion the
magnetic domain is observed to superimpose upon the martensite twin domains (Chopra et
al., 2000). In fig. 3.9 the 180◦ stripe domain follows the c-axis within one variant, the 90◦
domain wall coincides with the twin boundary when the c-axis is in plane in the adjoining
variants and a labyrinth domain forms if the c-axis is out of plane. Here the macroscopic
domain width is 5–40 μm. The Lorenz microscopy has given domain widths less than
0.1 μm, which may be due to the fact that magnetic domains might also have an internal
structure. Sullivan et al. (2004a, 2004b) investigated the reconfiguration of the magnetic
domains during the structural phase transformation of Ni-Mn-Ga by using a new magnetic
transition spectrum method (Chopra and Sullivan, 2005).
16
O. SÖDERBERG et al.
Fig. 3.7. Illustration of a simple additive model valid for the magnetization in the Ni-Mn-Ga 5M martensite. The
magnetization curves of the two-variant constrained sample are measured along the c-axis of one variant. The
c-axis of the second variant is perpendicular to the field direction within the accuracy of a few degrees. Due to
the constraint, x is constant in the applied magnetic field. 1 Tesla corresponds with approximately 800 kA m−1 .
(Courtesy and compilation from Likhachev et al., 2004a, reproduced by permission of Elsevier Science).
4. Martensite variant rearrangement in an applied magnetic field
Stress and magnetic field influence the MFIS of the MSM materials. Both the mechanical
behaviour in a constant magnetic field and the field-induced strain under a constant stress
are considered here.
The material parameters of the MSM alloys obtained in mechanical testing with a constant magnetic field can be applied for the device design. The mechanical behaviour of the
5M martensite in a constant magnetic field is studied by Jääskeläinen (2001), Straka and
Heczko (2003a, 2005), Müllner et al. (2003b), Chernenko et al. (2004a, 2004b, 2004c),
Suorsa and Pagounis (2004) as well as by Likhachev et al. (2004b). Figure 4.1 shows
schematically the mechanical behaviour of a 5M single-variant sample, when the compressive stress is applied along the [100] direction (a-axis) and the magnetic field to the
[001] direction (c-axis). Since the long a-axis is the hard axis of magnetization, growth
of the twin variant, which has the a-axis along the field direction, is retarded. Therefore,
GIANT MAGNETOSTRICTIVE MATERIALS
17
Fig. 3.8. The magnetization and straining of a single-variant 5M material as a function of the magnetic field.
Initially the field is applied along the a-axis. An abrupt change in magnetization and strain at a moderate field
is due to the twin variant redistribution. 1 Tesla corresponds with approximately 800 kA m−1 . (Courtesy and
compilation from Heczko et al., 2002b, reproduced by permission of American Institute of Physics).
Fig. 3.9. (a) The structure of a two-variant 5M martensite structure as seen in a magneto-optic image (MO) at left
and an optical image at right. The MO-images of a single variant 5M martensite structure obtained in a magnetic
field applied along (b) the horizontal direction, (c) the vertical direction and (d) perpendicular to the plane of
observation. The H—direction of the magnetic field, which was applied and removed after the reorientation of
the crystal (a, b and c—crystallographic axes of martensite lattice). (Courtesy of Sozinov et al., 2001a, reproduced
by permission of EPD Sciences).
18
O. SÖDERBERG et al.
Fig. 4.1. The magnetically assisted pseudo-elastic behaviour of a 5M martensite single-variant sample in compression at the constant magnetic field H = 800 kA m−1 . Inserts show the field dependence of σmag and the
crystallographic reorientation by the twin boundary motion (change between axes a ↔ c, b-axis is common for
both variants). The reorientation is illustrated schematically by the sample colour (black → gray during loading; gray → black during unloading). (Courtesy of Söderberg et al., 2005, reproduced by permission of Elsevier
Science).
more stress (shown as σmag in fig. 4.1) is needed in the magnetic field to obtain the
same deformation as in the mechanical testing without the field. This addition, σmag ,
is the magnetically induced stress. It increases with H and has the saturation value of
σmag = (1 − c/a)−1 Ku . If the mechanical deformation is carried out without the magnetic field, the 6% strain would remain in the 5M structure after unloading. However, in
the magnetic field, if the saturation value of σmag exceeds the twinning stress (σtw ), the
material behaves pseudo-elastically and the obtained shape change recovers totally in unloading. In alloys having this spring-like behaviour, the giant MFIS and the full size MSM
effect is possible. This magnetically assisted pseudo-elastic behaviour of the martensitic
Ni-Mn-Ga alloys exists in a broad temperature range. The 7M martensite shows a similar behaviour with the maximum magnetic stress of approximately 1.6 MPa at ambient
temperature and approximately 10% pseudoelastic strain (Sozinov et al., 2002a, 2004a;
Likhachev et al., 2004a). In the NM martensite the twinning stress exceeds the magnetic
stress and, consequently, the pseudo-elastic strain has not been observed in it (Sozinov et
al., 2004b).
4.1. Effect of load on magnetic-field-induced strain (MFIS)
The effect of the constant transversal stress on the magnetic-field-induced strain of the 5M
martensite has been studied by Murray et al. (2000b, 2001); Heczko et al. (2000, 2001b);
Jääskeläinen (2001); Jääskeläinen et al. (2003); Likhachev et al. (2001, 2002), O’Handley
and Allen (2001); O’Handley et al. (2003); Henry et al. (2001, 2002a, 2002b) and Tellinen
et al. (2002), Straka and Heczko (2005). If the martensitic single-variant sample is put into
GIANT MAGNETOSTRICTIVE MATERIALS
19
Fig. 4.2. The MFIS of a single-variant sample under a constant transversal stress. Insert shows the influence of
the stress on εMSM with H = 800 kA m−1 .
the magnetic field freely (σ = 0 MPa), the obtained MFIS (εMSM ) remains as a residual
stress (εres ) when the magnetic field is removed (fig. 4.2). The difference between εMSM
and εres is very small. If the compressive stress is applied together with the magnetic field,
the obtained MFIS (εMSM ) decreases and the saturation field value for the εMSM increases.
Also, the residual strain εres decreases, since a stress induced variant rearrangement occurs
when the field is removed. If the applied stress is tensile, the magnetic field required for the
twin boundary motion is less than in the case of compressive loading (Gans et al., 2004).
The giant MFIS is totally suppressed when the transversal stress exceeds 2.5–3 MPa. This
is explained by the competition between the applied stress (σ ) and the magnetically induced stress (σmag ) as the latter one can not exceed the value Ku · ε0−1 (see Chapter 2).
However, at compressive stresses higher than 3 MPa the conventional magnetostriction is
still observed (Tickle and James, 1999; Heczko, 2005).
In actuation the reversible strain (εrev ) is important (Likhachev et al., 2001) and εrev has
a maximum at a certain optimal value of the applied stress (fig. 4.3). With the correct stress,
the MSM-element is brought back to its initial state when the magnetic field is removed.
If the stress is too small, it cannot move the twin boundaries totally back and the shape
change is only partial. With too high stress the MFIS and the εrev are suppressed.
4.2. Dynamical actuation and fatigue of MFIS
For the dynamical actuation of Ni-Mn-Ga MSM elements, usually an axial movement is
applied, even though bending would also be possible (Suorsa et al., 2002; Tellinen et al.,
2002; Suorsa and Pagounis, 2004). The mechanical components of an ordinary actuator
are the MSM element, the moving mass and a spring for spring-back load. The MSM
20
O. SÖDERBERG et al.
Fig. 4.3. The actuation of a MSM element with the constant transversal stress in an alternating magnetic field. Insert shows the influence of the stress on εrev in H = 800 kA m−1 . (Courtesy of Söderberg et al., 2005, reproduced
by permission of Elsevier Science).
element is in the same magnetic circuit with the ferromagnetic core and the magnetic field
is generated by magnetising coils. The system is usually driven by a current, since with a
voltage drive there will be a delay in operation. During one full circle of the magnetic field
change the shape of the MSM element changes twice. The maximum change increases
with the growing magnetic field until the saturation value is reached (Aaltio and Ullakko,
2000). The magnetic-field-induced force of the actuator depends on the cross-sectional
area of the MSM element, while the stroke is related with the length of the element. At
high frequencies the eddy currents have to be reduced and the system may also reach the
resonance frequency.
The Ni-Mn-Ga single crystals are used for actuation of 2%–4% strains with frequencies decreasing with the generated stroke (Henry et al., 2001, 2003a; Suorsa et al., 2002;
Tellinen et al., 2002). The Ni-Mn-Ga element contracts along the field direction and extends in the plane normal to the magnetic field while conserving the volume during actuation. Figure 4.4 indicates that the Ni-Mn-Ga samples can be driven at least up to 320 Hz
(Henry et al., 2003b). Beyond 100 Hz the peak output strain seems to drop down sharply;
this is a response to the reduction in the applied field.
The fatigue of the MSM actuation is a less studied field. A demo-actuator driven by
a 1.8 A current showed unchanged behaviour after 40 000 cycles (Aaltio and Ullakko,
2000). A high-frequency actuator with 10 mm MSM element with about 2% strain and
1 MPa constant stress kept the stroke value approximately constant till 200 million cycles
of the alternating magnetic field (Tellinen et al., 2002). The evolution of the structure and
the MSM properties in the 5M specimens have been studied during testing in a rotational
magnetic field (6 Hz drive) with a 0.2 MPa bias stress (Heczko, unpublished). During the
first million cycles the initial 6% MSM strain decreased to approximately 3% and then
stayed approximately constant until the end of the test up to 36 million cycles. Müllner et
GIANT MAGNETOSTRICTIVE MATERIALS
21
Fig. 4.4. Dependence of the maximum field applied to the MSM element and the strain measured at the maximum
field as functions of the drive frequency and the actuation frequency for two different test systems (dot and solid
lines). 1 Tesla corresponds with approximately 800 kA m−1 . (Courtesy and compilation from Henry et al., 2003b,
reproduced by permission of SPIE—The International Society for Optical Engineering).
al. (2004) showed that the 5M material (in ref. 10M) showed a constant magnetic-fieldinduced strain in cycling, while in the 7M material (in ref. 14M) the MFIS decreased.
Müllner et al. (2004) and Xiong et al. (2005) have studied the fracture mechanism of NiMn-Ga martensites in thermal and magnetic cycling.
4.3. Temperature dependence of MFIS
The MSM effect is limited by the phase transformations and affected by the temperature
dependence of the twinning stress, the tetragonality of lattice and the magnetic anisotropy
(Heczko and Straka, 2003). In the 5M Ni-Mn-Ga martensite the lattice parameter a increases slightly, while the parameter c decreases with decreasing temperature (Ma et al.,
2000; Glavatska et al., 2002, 2003a) resulting in increasing of the lattice distortion (1−c/a)
and, consequently, in an increase of the largest possible MFIS. However, the temperature dependencies of the magnetic anisotropy of the material and the twinning stress are
more important. They both increase during cooling, but the twinning stress increases considerably (Heczko et al., 2002a, 2002b, 2003). Consequently, the lowest MSM service
temperature is reached when the magnetically induced stress is no longer able to exceed
the twinning stress. Another lowest limit is the possible intermartensitic transformation.
The highest service temperature is limited by the reverse transformation. The service temperature region of the MSM effect for the alloy Ni49.7 Mn29.1 Ga21.2 is shown in fig. 4.5
22
O. SÖDERBERG et al.
Fig. 4.5. The existence of the MSME in the alloy Ni49.7 Mn29.1 Ga21.2 is limited by the reverse phase transformation at a high temperature and by the temperature where magnetically induced stress (σmag = K1 /(1 − c/a),
see part 2) becomes lower than the twinning stress σTW . (Courtesy of Heczko and Straka, 2003, reproduced by
permission of American Institute of Physics).
(Heczko and Straka, 2003). Here, it was found that the obtainable MFIS increases with the
decreasing temperature, but it also has been reported that the MFIS decreases in cooling of
the alloy Ni52.6 Mn23.5 Ga23.9 (Pasquale et al., 2002).
5. Selected properties of Ni-Mn-Ga alloys
The unique transport properties (Zhou et al., 2002b; Lee et al., 2003) or the magnetocaloric
effect (Pakhomov et al., 2001; Marcos et al., 2003, 2004a, 2004b; Aliev et al., 2004;
Albertini et al., 2004; Zhou et al., 2004, 2005) may create possibilities for using some NiMn-Ga alloys in such applications as spin electronics and solid state refrigeneration. When
Ni-Mn-Ga alloys were grown by molecular beam epitaxy, the resulting heterostructures
showed in-plane magnetization and Curie temperatures of approximately 300 K (Dong et
al., 2000a; Palmstrom et al., 2002). In a magnetocaloric effect the applied magnetic field
may cause enhanced heat production or absorption in a material showing the simultaneous occurrence of structural and magnetic transitions. A large magneto-entropy change
has been observed in some Ni-Mn-Ga alloys (Hu et al., 2000, 2001a, 2001b; Pareti et al.,
2003) as well as the simultaneous occurrence of the structural and magnetic transition approximately at 370 K (Chernenko et al., 2002; Khovailo et al., 2002; Jiang et al., 2002;
Söderberg et al., 2004a).
In corrosion testing in NaCl-water-solution the non-modulated NM structure showed the
best chemical properties among the Ni-Mn-Ga martensites and the cubic parent phase was
at the same level (Liu et al., 2002a, 2002b, 2003a, 2003b). Ni-Mn-Ga behaved better than
the common low-alloyed steel, but not so well as the AISI316L stainless steel.
GIANT MAGNETOSTRICTIVE MATERIALS
23
6. Promising MSM materials
In addition to the bulk materials the research of the Ni-Mn-Ga MSM includes also thin
films (Suzuki et al., 1999; Dong et al., 2000a, 2000b, 2004; Ahn et al., 2001; Tello et al.,
2002; Wu et al., 2002a; Wu and Tseng, 2002b; Castaño et al., 2003; Kohl et al., 2003;
Rumpf et al., 2003; Chung et al., 2003; Hakola et al., 2004a, 2004b; Dubowik et al., 2004;
Golub et al., 2004; Kim et al., 2004), ribbons (Chernenko and Vitenko, 1994; Chernenko et
al., 1994; Kanada et al., 1998; Albertini et al., 2002b; Heczko et al., 2002c; Algarabel et al.,
2004) and composite structures (Feuchtwanger et al., 2003a, 2003b; Hosoda et al., 2004).
Also, the investigation of MSM powders has intensified, since they are important especially
while preparing composites (Berkowitz et al., 2004; Solomon et al., 2004). However, NiMn-Ga alloys are rather expensive, brittle and currently their MSME service temperature
range is below 340 K. The search for possible new MSM materials is targeted to the alloys
with a thermally formed or stress induced ferromagnetic thermoelastic martensite phase.
The martensite should have extremely mobile twin boundaries. The high magnetocrystalline anisotropy and the high saturation magnetization are also desirable properties. The
Fe-Pd and Fe-Pt alloys have shown MFIS, while the Fe-Ni-Co-Ti group has not been a
success. Co-Ni and Co-Ni-Ga have turned out to be promising candidates, while Co-Ni-Al
and Ni-Mn-Al have not been suitable.
Alloys close to Fe3 Pd show a martensite transformation up to 273 K and also a SIM
formation resulting in a tetragonal phase with a short c-axis (Sohmura et al., 1980; Matsui
et al., 1981; Sugiyama et al., 1984; Kato et al., 2002). The Curie point of the Fe70 Pd30 is
573 K (Matsui and Adachi, 1983; Koeda et al., 2001; Cui and James, 2001). The saturation
magnetization of the martensitic phase is 1400 kA/m and the magnetocrystalline anisotropy
constant Ku is approximately 1 MJ/m3 (Matsui and Adachi, 1983, 1989; Klemmer et al.,
1995; James and Wuttig, 1998). The single crystalline Fe-Pd has shown a free MFIS of
0.5% at 256 K in a 1 T cyclic field as well as a single strain of 3% and subsequent cycles with 0.1% MFIS at 77 K in a 4 T field (James and Wuttig, 1998; Koeda et al., 2001;
Yamamoto et al., 2004). At ambient temperature in a 1 T field, ribbons prepared by meltspinning showed 0.18% free strain and 0.08% strain with a 10 MPa tensile stress (Furuya
et al., 1998; Kubota et al., 2001, 2002a). The good MFIS of ribbons is based on the fine
columnar grains in which there was developed a 100 fiber texture and a 100 tilt grain
boundary (Kubota et al., 2002b; Yasuda et al., 2002). In Fe70.4 Pt29.6 melt-spun ribbons the
magnetostriction at 300 K has been 0.065% (Kubota et al., 2002b), while a larger shape
change of 1.5–2.3% has been observed in the single crystalline Fe3 Pt in a magnetic field of
4 T at 4.2 K (Kakeshita et al., 2000; Kakeshita and Fukuda, 2002; Sakamoto et al., 2004).
The Fe-Pt alloys have a martensitic transformation to a tetragonal phase up to 450 K, but
the thermoelastic structure is obtained only in annealed materials with a drop of the Tm
and an increase of TC (Wayman, 1971; Dunne and Wayman, 1973a, 1973b; Kajiwara and
Owen, 1974; Tadaki, 1977; Kakeshita et al., 1984). Close to the composition Fe3 Pt both
martensite and austenite are ferromagnetic and the Curie temperature is approximately
450 K (Wasserman, 1990). Fukuda et al. (2004) showed that the Fe68.8 Pd31.2 samples expand at 77 K in a 1.25 T field and at 4.2 K in a 4 T field the Fe3 Pt samples contract along the
field direction [001], since the easy axis of magnetization in the Fe-Pd alloy is the a-axis
and in the Fe-Pt alloy the c-axis.
24
O. SÖDERBERG et al.
The Co-Ni alloys show both the thermal and the SIM transformation when the Ni content
is less than 35 wt.%. In the single-crystalline Co–33 wt.% Ni approximately 3% reversible
MFIS due to SIM formation was obtained in a 2 T field below 180 K, and at ambient temperature Co–32 wt.% Ni showed a MFIS of 4.2% during the first 4–5 cycles in a 1.2 T field
along the [001] orientation (Jiang et al., 2001; Liu et al., 2001, 2003a; Zhou et al., 2003).
Also, Co-Ni-Ga alloys close to the Heusler composition show a thermoelastic martensite transformation and a shape memory effect (Oikawa et al., 2001a; Wuttig et al., 2001;
Craciunescu et al., 2002a, 2002c; Kishi et al., 2003). The higher Ni/Ga increases the transformation temperatures and decreases the Curie temperature—in the alloy Co47 Ni23 Ga30
Tm and TC merge at 370 K. The phase transformation temperatures increase also when Al
is added instead of Ga or by water quenching of the alloys having less than 26 at.% Ga. In
the [001] single crystalline sample of Co50 Ni22 Ga28 (TC 400 K, Tm 343 K) the two-way
shape memory can be continuously adjusted from −2.3% to 0 with a bias field of 0.8 T
and when increasing the bias field to 2.0 T a positive shape deformation of +2.2% occurs
(Li et al., 2004b). In Co-Ni-Al shape memory alloys the martensite transformation temperature increases and the Curie point decreases as the Ni content changes from 30 to 45
at.%, when the Al content is 30 at.% and the martensitic transition merges to the magnetic one at approximately 250 K when the Ni content is 35 at.% (Kainuma et al., 1996b;
Oikawa et al., 2001a, 2001b; Murakami et al., 2002). The magnetocrystalline anisotropy
of the Co37 Ni34 Al29 is 3.9 MJ/m3 and the reversible MFIS is limited to 0.06% (Morito et
al., 2002).
In the Ni-Mn-Al system the ordered L21 phase transforms to martensite the crystal
structure of which depends on the alloy composition (Inoue et al., 1994; Kainuma et al.,
1996a, 2000; Otsuka and Morito, 1996; Sutou et al., 1998). Ferromagnetic ordering with
a Curie temperature of approximately 330 K can be obtained by aging the quenched samples (Gejima et al., 1999). A magnetostrain of 1% in a single crystalline sample has been
obtained in a 7 T magnetic field (Fujita et al., 2000).
From other material groups Ni-Ga-Fe (Oikawa et al., 2002; Ota et al., 2002; Sutou et al.,
2004), Co2 NbSn (Garde and Ray, 2002; Neumann et al., 2002; Wolter et al., 2002), Fe-Rh
(Ibarra and Algarabel, 1994), Mn-As (Chernenko et al., 1999), La(Fex Si1−x )13 (Fujieda
et al., 2001), Gd5 (Six Ge1−x )4 (Morellon et al., 2000) and in a antiferromagnetic γ -MnFe(Cu) (Zhang et al., 2005) have been studied to some extent.
Acknowledgements
The present handbook chapter is initiated from the extensive research work carried out
at Helsinki University of Technology during the period of 1998–2004. Accordingly, the
support for this research work from The National Technology Agency of Finland (Tekes),
Outokumpu Research Oy, Metso Corporation Oy, Nokia Research Center, ABB Corporate Research Oy, AdaptaMat Ltd, Technology Industries of Finland (formerly Federation
of Finnish Metal, Engineering and Electrotechnical Industries (MET)), The Academy of
Finland and Helsinki University of Technology as well as the Outokumpu Foundation is
gratefully acknowledged. Furthermore, the authors would like to express their gratitude to
Ms. Marjatta Aav, B.Sc. (Econ.), and Ms. Arja Teramo, B.A., for their valuable assistance
in preparing this chapter.
GIANT MAGNETOSTRICTIVE MATERIALS
25
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chapter 2
MICROMAGNETIC SIMULATION OF MAGNETIC
MATERIALS
DIETER SUESS and JOSEF FIDLER
Vienna University of Technology, Vienna, Austria
THOMAS SCHREFL
University of Sheffield, Sheffield, UK
Handbook of Magnetic Materials, edited by K.H.J. Buschow
Volume 16 ISSN: 1567-2719 DOI 10.1016/S1567-2719(05)16002-8
41
© 2006 Elsevier B.V.
All rights reserved
CONTENTS
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
2. Basic micromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
2.1. Total Gibbs’ energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
2.2. Static micromagnetism—Browns equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
2.3. Dynamic micromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
2.4. Micromagnetism at finite temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
3. Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
3.1. Finite difference micromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
3.2. Finite element micromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
3.3. Numerical treatment of the stray field problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
3.4. Eddy currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
3.5. Fast integration methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
3.6. Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
3.7. Time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
3.8. Time integration at finite temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
3.9. Calculating energy barriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
3.10. Micromagnetic codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
4. Applied micromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
4.1. Perpendicular and tilted media for magnetic recording . . . . . . . . . . . . . . . . . . . . . . . .
91
4.2. Exchange spring media for perpendicular recording . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.3. Integrated simulations of recording devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Appendix A: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
A.1. Derivation of the interface condition for the eddy current field . . . . . . . . . . . . . . . . . . . . 117
A.2. Time integration of the Landau–Lifshitz equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
42
1. Introduction
The development and application of modern magnetic materials requires a basic understanding of the magnetization processes that determine the magnetic properties. Micromagnetics relates the microscopic distribution of the magnetization to the physical and
chemical microstructure of a material. Recently, micromagnetic modeling has become an
important tool to characterize the magnetic behavior of such different materials as thin
film heads, recording media, patterned magnetic elements, and nanocrystalline permanent
magnets.
The rapid progress of nanotechnology will lead to novel applications of magnetic materials in spin electronic devices, magnetic sensors, and functional materials within the next
years. A prerequisite for the application of structured magnetic materials is the detailed
knowledge of the correlation between the physical and magnetic structure of the system.
The design of smart materials requires to predict the response of the system to external
fields, stress and temperature as a function of time.
In magnetic recording the areal density (bits/in2 ) is almost doubled every year. This rally
is backed by a lot of research and development. Nowadays in commercial hard disks the bit
is stored parallel to the surface (longitudinal recording). Unfortunately, there is a physical
limit (superparamagnetic limit) which does not allow to reduce the size of a bit, respectively increase the areal density in the common way. Charap et al. (1997) determined an
upper bound for the areal density of 40 GBits/in2 . However, new methods have been developed that allow to overcome this limit like antiferromagnetically-coupled (AFC) media
(Fullerton et al., 2000; Abarra et al., 2000).
The spring 2005 world record in areal density is 230 GBits/in2 (Hitachi Global Storage,
2005) which was realized with perpendicular recording. This new level of storage density
translates to 140 hours of DVD-quality video on a single disk.
In perpendicular recording the magnetization of each bit is perpendicular to the surface.
To understand this method in detail, different aspects have to be investigated. Due to the
granular structure of the storage media, one has to consider its microstructure such as
the size and shape of the grains and intergranular phases which differ in their magnetic
properties. The design of the read–write head is a key problem in perpendicular recording
since the closing of the magnetic circuit is more difficult in this method (Khizroev, 2002).
Another new development in magnetic recording are magnetic random access memories (MRAM). MRAM technology makes use of an effect called giant magnetoresistance
(GMR), which was discovered in 1988 (Baibich et al., 1988). An MRAM element consists
of two magnetic layers with different coercivity. They are separated by a non-magnetic
interlayer. Depending on the angle of the magnetization between the two magnetic layers
the resistivity changes.
43
44
D. SUESS et al.
Numerical micromagnetics is a suitable method to treat magnetization dynamics on the
nanometer scale. A simple theory describing magnetic processes in magnetic particles was
developed by Stoner and Wohlfarth (Stoner and Wohlfarth, 1948). The theory describes
magnetization reversal of small magnetic particles. The magnetization within the particles
is assumed to be homogeneous. The Stoner–Wohlfarth theory successfully predicts the
critical fields and the dependence of the magnetization as a function of field strength and
field angle. Although the Stoner–Wohlfarth theory is widely applicable it breaks down
when domain processes dominate the reversal mechanism of the considered sample.
A theory that allows to resolve magnetization inhomogeneities is the theory of micromagnetics that goes back to the paper of Landau–Lifshitz (Landau and Lifshitz, 1935)
where the configuration of the magnetic moments within a domain wall was described using a continuum approach. This theory is suitable to calculate the magnetic configuration
on a mesoscopic length scale. In the continuum theory of micromagnetics the spontaneous
magnetic polarization is assumed to be a continuous function of space J (x). The magnetic
polarization is the magnetic moment per unit volume times μ0 . The magnetic moment
at a lattice point x a of the magnet is the value of the function J (x)/μ0 at the point x a divided by n, where n is the number of spins per volume (for a simple cubic lattice n = 1/a 3 ;
a = lattice constant). The stable equilibrium state of the specimen’s magnetization is found
by minimizing an expression for the total free energy. The total free energy is the sum of
the exchange energy, the Zeeman energy, the stray field energy, the crystalline anisotropy
energy, and the magnetostrictive energy. The magnetostrictive energy is usually not taken
into consideration. Magnetic domains, which originally where proposed by Weiss (1907)
in order to account for the observed magnetization and hysteresis curves, are formed due
to the interplay between the different energies.
Brown (1940, 1963a) used the continuum expression for the free energy of a ferromagnetic body to calculate the critical external field at which the magnetization of a uniform
magnetized body becomes unstable. The variation of the total Gibbs energy leads to a
system of partial differential equations, the so-called Brown’s equations. LaBonte (1969)
solved Browns equations iteratively in order to calculate the equilibrium configuration of
the magnetization in a domain wall in one and two dimensions. In equilibrium the magnetization is parallel to the total effective field, so in each iteration step the magnetization was
rotated a little bit towards the direction of the effective field. LaBonte’s method was used
widely to calculate equilibrium states in static micromagnetics (Nakatani et al., 1989).
In order to be able to deal with general magnetization distributions advanced numerical
tools are required. The finite difference method allows the simulation of magnetization
processes in regularly shaped elements (Schabes and Bertram, 1988). The calculation of
the strayfield of such regular elements can be performed using FFT (Yan and Della Torre,
1988). For more complex structures the finite element method (FEM) is more suitable
(Fredkin et al., 1992; Dahlberg and Zhu, 1995). FEM does not have any restrictions on
the shape of the element. In this framework the influence of the microstructure can be
investigated.
The length scales involved in these simulations vary widely and range from nanometers for the nucleation of reversed domains at material interfaces to micrometers for the
simulation of the magnetostatic interactions of the entire system. In the simulation of the
recording head we have to consider moving parts.
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
45
An adaptive mesh refinement algorithm can help to bridge these length scales, since a
fine mesh is only generated where it is needed. Fast boundary element methods can be
applied to accelerate the calculation of the long range magnetostatic interaction between
distinct parts. In addition, these methods make the simulation of moving parts or adaptive
grids very convenient.
The innovative combination of micromagnetic theory with advanced numerical techniques for the solution of partial differential equations gives rise to realistic simulations of
modern magnetic materials. The development of new magnetic materials used for permanent magnets, data storage, or magnetic sensors is linked with the ability to make accurate
predictions of their magnetic properties. Advanced numerical micromagnetic simulations
will have a great impact on the structural design of novel magnetic materials and devices.
In section 2 the basic theory of micromagnetics is explained for ferromagnetic and
antiferromagnetic materials. The stray field and the eddy current field are derived from
Maxwell equations. Basic theories for micromagnetics at finite temperature are given.
Various methods are introduced for the solution of the micromagnetic equation in section 3. Different numerical discretization schemes such as the finite difference method and
the finite element method are introduced. Furthermore a short overview of fast boundary
integration methods as well as comparisons between them are given. Efficient methods for
time integration for zero temperature and finite temperatures are explained in detail.
In section 4 the micromagnetic model is applied to magnetic recording applications.
Switching times and reversal modes of granular perpendicular media are investigated.
A new type of exchange spring media is introduced, in order to increase the areal density. In the last part of section 4 fully integrated recording simulations are presented that
allow to simulate the complete recording process. A current is applied in coils that magnetizes the recording head. The movement of the head across the media, the magnetization
dynamics in the data layer and in the soft underlayer are simulated simultaneously.
2. Basic micromagnetism
2.1. Total Gibbs’ energy
Thermodynamics shows that the most probable state of a magnetic system is described by
the minimum of the Gibbs’ free energy
Etot = U − H ext J s dV .
(2.1)
The inner energy U can be calculated as the sum of the stray field energy Es , the exchange
energy Eex , and the magnetocrystalline anisotropy energy Eani . So, the total Gibbs’ free
energy can be written as
Etot = Es + Eex + Eani + Eext ,
(2.2)
where Eext is the Zeeman energy and arises from the external field H ext . In the micromagnetic theory the Gibbs’ free energy is a function of the magnetization, where the vector field
M(r) = J (r)/μ0 is constrained by |M| = Ms (T ) = const, disregarding the Holstein–
Primakoff-effect, which says that |M| depends on the external field H ext (Holstein and
46
D. SUESS et al.
Primakoff, 1940). The magnetic polarization is given by Js = μ0 gnμB S, where n is the
number of spins per volume (for a simple cubic lattice n = 1/a 3 ), S the total spin quantum
number of one atom, g the Landé factor and μB the Bohr magneton.
2.1.1. Exchange energy
2.1.1.1. Ferromagnet The micromagnetic exchange energy term can be calculated using
quantum mechanics. We know the Heisenberg Hamiltonian of the exchange interaction
Eex = −
N
N Jij S i S j .
(2.3)
i=1 j =i
The exchange integral, J , decreases strongly with increasing distance between the atoms
and so we just consider the nearest neighbor interaction in the sum. By replacing the spins
S i with the continuous variable u = M/Ms = J /Js , using a Taylor expansion of the
vector product, and changing the summation to an integral over the ferromagnetic body,
we get
1 A(∇J )2 dV .
Eex =
(2.4)
2
Js
The exchange constant A depends on the exchange integral Jij . It can be determined by
experiments and the values are well known for many ferromagnetic materials.
The exchange energy becomes a minimum if all the spins are aligned parallel to each
other.
2.1.1.2. Ferrimagnets and antiferromagnets In contrast to ferromagnets the exchange
interaction between antiparallel spins in ferrimagnets and antiferromagnets is negative. As
a consequence neighboring spins point in opposite directions. In order to be able to apply
the concept of micromagnetism one can subdivide the magnet into two sublattices. Within
each sublattice the magnetic moments vary slowly in space. Figure 2.1 shows schematically
a domain wall in an antiferromagnet. The subdivision of the system into a sublattice A and
a sublattice B yields a distribution of the magnetic moments similar to that of a ferromagnet
for each sublattice.
Antiferromagnetic ordering on a simple cubic lattice If the nearest neighbor interaction is
responsible for the magnetic ordering, the six nearest neighbors (nN ) point antiparallel to
the spin in the middle as shown in fig. 2.2a. The local moments fall on two interpenetrating
face-centered cubic sublattices. Within each sublattice the spins vary only slowly with
space. If both sublattices have the same number of spins (N/2), equation (2.3) can be
written in the form
N/2
N/2
nN
nN
A
B
B
A
Eex,antiferro = −J
Si
S j + −J
Si
Sj
= −2J
i=1
j
N/2
nN
i=1
SA
i
j
i=1
SB
j.
j
(2.5)
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
47
Fig. 2.1. Schematic diagram showing the subdivision of an antiferromagnet into two sublattices. The antiferromagnetic wall structure can be represented in a continuum model by two walls, one in each sublattice.
= −2J
N/2
S A (x i )
= −2J
i=1
S B (x j )
j
i=1
N/2
nN
S A (x i )
6
S B (x i + δk ).
(2.6)
k=1
The displacement of neighboring atoms is given by the vector δk . In a continuum limit, we
want all our variables to be evaluated at the same point. Thus we expand S B (x j ) about x i .
For a cubic lattice with lattice constant a,
a
−a
0
0
0
0
, 0
, a
, −a
, 0
, 0
.
δk = 0
(2.7)
0 k=1
0 k=2
0 k=3
0 k=4
a k=5
−a k=6
The k-th component of the spin can be expanded as,
B
∂S B
∂S B
∂Sk
δx + k δy + k δz
SkB (x j ) ≈ SkB (x i ) +
∂x
∂x
∂x
2
B
B
2
2
1 ∂ S k 2 ∂ S k 2 ∂ Sk 2
δ + 2 δ y + 2 δy .
+
2 ∂ 2x x
∂ y
∂ z
(2.8)
The first-order terms and second order cross terms vanish in the sum over nearest neighbors
because of the symmetry of the cubic lattice. That is the reason why they are omitted in the
expansion of equation (2.8). Using equation (2.8),
nN
j
S B (x j ) = 6S B (x i ) + a 2 S B (x i ).
(2.9)
48
D. SUESS et al.
Fig. 2.2. (a) Antiferromagnetic ordering on a simple cubic lattice. Spins of the same kind form two interpenetrating face-centered cubic lattices. (b) Antiferromagnetic ordering on a body-centered cubic lattice. Spins of the
same kind form two interpenetrating simple cubic lattices. (c) Antiferromagnetic ordering on an fcc lattice for the
case of a negative next nearest neighbor interaction (e.g. CoO). The fcc lattice consists of four interpenetrating
simple cubic lattices. The vector δ connects next nearest neighbors.
To obtain the continuum expression (2.9), some assumptions have been used. In the summation over all position vectors δk it was assumed that all of them are inside the crystal.
However, on a surface the number of neighbors is reduced and the sum may come up different than at internal points. To compensate this error, an additional term to the surface
anisotropy can be added. Using equation (2.9), equation (2.6) can be written in the continuum limit as
Eex,antiferro = −
12J S 2 A
2J S 2 A
B
3
u
(x
)u
(x
)a
−
u (x i )uB (x i )a 3
i
i
a
a3
N/2
N/2
i=1
i=1
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
=−
12J S 2 1
a3 2
49
uA (x i )uB (x i )dV
V
2J S 2 1
−
a 2
uA (x i )uB (x i )dV ,
(2.10)
V
where S · uB (x i ) = S B (x i ). Applying Green’s first identity and the condition that the
normal derivative of the magnetic polarization at the surface vanishes, equation (2.10)
transforms to
A
B
Eex,antiferro = −6 2 uA (x i )uB (x i )dV + A
∇uA
x (x i )∇ux (x i )
a
V
V
B
A
B
+ ∇uA
y (x i )∇uy (x i ) + ∇uz (x i )∇uz (x i ) dV ,
(2.11)
where A = J S 2 /a.
In the case of a ferromagnet, where ∇uA = ∇uB
x , equation (2.11) transforms to the well
known expression for the exchange energy for ferromagnets (equation (2.4)).
Antiferromagnetic ordering on a body-centered cubic lattice For a dominating nearest
neighbor interaction all eight nearest neighbors of one atom point antiparallel. The derivation of the exchange energy is very similar to the previous derivation, only the number of
nearest neighbors
changes from 6 to 8, and the distance between nearest neighbors changes
√
from a to 3a/2. The sum over the nearest neighbors yields,
8
S B (x j ) = 8S B (x i ) + 3a 2 S B (x i ).
(2.12)
j =1
Hence we get for the exchange energy
A
A
B
B
∇uA
Eex,antiferro = −8 2 u (x i )u (x i )dV + 3A
x (x i )∇ux (x i )
a
V
V
B
A
B
+ ∇uA
y (x i )∇uy (x i ) + ∇uz (x i )∇uz (x i ) dV .
(2.13)
Antiferromagnetic ordering on an fcc lattice The face centered cubic lattice represents a
more complicated situation, since it is not possible to arrange all nearest neighbors of one
atom antiparallel. Neutron diffraction studies (Shull et al., 1951) revealed the structure of
MnO, as shown in fig. 2.2c. All the next nearest neighbors point antiparallel. Hence the
important interaction producing the antiferromagnetic ordering must be the next nearest
neighbor interaction (indirect exchange), which is along a cube edge in fig. 2.2c and denoted with the vector δ. In fig. 2.2c the spins of two interpenetrating cubic sublattices are
drawn separately. Four interpenetrating equivalent simple cubic lattices form an fcc lattice.
If nearest neighbor interaction is neglected, these four sublattices are not exchange coupled
in the bulk. However, they may be indirectly coupled, for example via the ferromagnet at
the interface of an antiferromagnetic/ferromagnetic bilayer. For all four sublattices equation (2.11) can be used for the calculation of the exchange energy.
50
D. SUESS et al.
2.1.2. Zeeman energy
The magnetostatic energy
Eext = − J H ext dV
(2.14)
represents the potential energy of the macroscopic ferromagnetic specimen in a fixed external field.
For the calculation of the Zeeman energy of ferro- and antiferromagnets again the system
is subdivided into two sublattices.
Eext,antiferro = − J A H ext dV − J B H ext dV .
(2.15)
Due to the strong antiparallel coupling of neighboring spins the Zeeman energy of antiferromagnets is usually negligible.
2.1.3. Magnetocrystalline anisotropy energy
In the continuum limit the exchange energy is usually assumed to be isotropic. The
most common type of anisotropy is the magnetocrystalline anisotropy, which is caused
by the spin–orbit interaction of the electrons. The electron orbits are linked to the crystallographic structure, and by their interaction with the spins they make the latter prefer to
align along well-defined crystallographic axes. Therefore, there are directions in space, in
which a magnetic material is easier to magnetize than in others. The spin–orbit interaction
can also be evaluated from basic principles. However, it is easier to use phenomenological expressions (power series expansions taking into account the crystal symmetry) and
experimentally obtained coefficients.
Depending on the symmetry requirements of the special point groups different expressions are derived for the case of cubic and hexagonal crystal structures. For a cubic crystal
the anisotropy energy can be written as
Eani,cubic = K1 (r)
(2.16)
γi2 (r)γj2 (r) + K2 (r)γ12 (r)γ22 (r)γ32 (r),
i=j
where γi (r) is the projection of the normalized magnetization on the cubic axis i.
For hexagonal crystals the anisotropy energy reads
K1
K2
2
(J
k)
dV
−
(J k)4 dV .
Eani,hexagonal = −
2
Js
Js4
(2.17)
The vector k points into the direction of the easy axis.
Considering only the first anisotropy constant K1 , equation (2.17) can be written in the
form
Eani,hexagonal = uMudV + const,
(2.18)
where M is a tensor with
Mlm = K1 k l k m
(2.19)
and u is the unit vector parallel to J . For antiferromagnets equation (2.17) or equation (2.18) can be applied for each sublattice separately.
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
51
2.1.4. Magnetic field
Maxwell derived equations for the averaged fields in the presence of material media.
They are:
∇D = ρ,
∇ ×H =I +
∇ ×E =
∂
D,
∂t
∂
B,
∂t
∇B = 0.
Coulomb’s law
(2.20)
Ampere–Maxwell equation
(2.21)
Faraday’s law
(2.22)
(2.23)
The relation between the dielectric flux density D, the electric field E and the polarization
P is given by:
D = ε0 E + P .
(2.24)
The material law gives a relation between the magnetic induction B, the magnetic field H
and the magnetization M
B = μ0 H + μ0 M.
(2.25)
By assuming Ohm’s law, the current density I depends linearly on the electric field via the
conductivity σ ,
I = σ E.
(2.26)
In the quasi-static case the second term in equation (2.21) can be neglected. Furthermore,
in the following no electric charges are assumed. Hence equation (2.20) transforms to
∇D = 0.
For the calculation of the magnetic field we split H into an irrotational field
∇H s = a,
(2.27)
∇ × Hs = 0
(2.28)
and the solenoidal field
∇H eddy = 0,
(2.29)
∇ × H eddy = b.
(2.30)
The total field is given by the sum of the stray field H s and the eddy current field H eddy
H = H eddy + H s .
(2.31)
The yet undetermined source terms, a and b, will be derived from Maxwell’s equations
and the material laws, equations (2.20)–(2.25).
52
D. SUESS et al.
2.1.4.1. Interface condition The four Maxwell equations can be used to calculate the
magnetic and electric field in a material and in free space. In order to calculate B and H
across an interface we start from Maxwell equations in integral form1
BdA = 0,
(2.32)
H ds =
I dA.
(2.33)
By applying the surface integral of equation (2.32) and the line integral of equation (2.33)
across the interface, the interface conditions follows as,
(B out − B in )n = 0,
(2.34)
(H out − H in ) × n = 0.
(2.35)
To obtain equation (2.35) we have assumed that the strength of the current density at the
surface is finite.
2.1.4.2. Strayfield Calculating the divergence of the magnetic induction (equation (2.25))
and considering that it vanishes, we get
∇H s = −∇M.
(2.36)
In order to guarantee that the curl of the magnetic field H s is zero, the stray field can be
represented using a scalar potential u,
H s = −∇u.
(2.37)
Substituting equation (2.37) into equation (2.36) leads to
u = ∇M.
(2.38)
The magnetic scalar potential u is regular at infinity
1
for r → ∞.
(2.39)
r
Inhomogeneities in the distribution of the spontaneous magnetic polarization within a particle causes a stray field H s in the interior and in the exterior of the particle. Additionally,
the divergence of M at the magnet’s boundary give rise to magnetic surface charges. Magnetic surface charges, which may occur at the grain boundaries and at the surfaces of the
particle, contribute to the stray field.
At the particle surface equation (2.36) leads to the boundary condition for the normal
component of the stray field. Starting from ∇(H s + M) = 0 and calculating the volume
u→
1 By integrating the Ampere Maxwell equation ∇ × H = I over a finite area and applying Stokes’ theorem the
integral form H ds = J dA follows. However, a prerequisite for applying Stokes’ theorem is that the vector
field H is continuously differentiable. Therefore by applying Stokes’ theorem a continuous transition of H is
implied a priory. To avoid this problem we start from Maxwell equations in integral form.
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
53
integral across the surface
in
H out
s − H s n = Mn,
(2.40)
follows. Equation (2.28) requires that the tangential component of the stray field is continuous across the interface. Therefore,
in
H out
s − H s × n = 0,
(2.41)
A scalar potential that is continuous at the interface implies a continuous tangential component of the gradient of the potential. Hence, the equation above are equivalent to,
(∇uout − ∇uin )n = −Mn,
(2.42)
uout = uin .
(2.43)
Equation (2.37) and equation (2.38) do not take care of the atomic structure of the material, since the magnetic moments are replaced by the continuum function of space J (x).
Applying the Lorentz approximation, the atomic structure can be incorporated into the formula for the demagnetizing field. For the Lorentz approximation a lattice made of magnetic
dipoles is assumed. For the calculation of the field at the lattice point xi , a small sphere
is drawn around the lattice point xi . The radius of the sphere has to be sufficient small,
so that the magnetization inside can be assumed constant, or at most as a linear function.
For the calculation of the demagnetizing field the region within the sphere and outside the
sphere are treated differently. Inside the sphere the dipoles on the lattice points are summed
up. Outside the sphere the dipoles are taken as a continuum. For a cubic crystal the dipole
fields inside the sphere sum up to zero for a linear and even for a quadratic function of the
magnetization (Adam and Corciovei, 1971). For other crystal structures the dipoles within
the sphere lead to a demagnetizing field of the form,
1 1
a
(2.44)
H stray =
J + CJ ,
μ0 3
where C is a tensor which depends on the crystalline symmetry. The total stray field is the
sum of the atomistic stray field produced by the dipoles within the small sphere a H stray
and the stray field produced by the magnetic polarization outside the sphere, m H stray .
H stray = a H stray + m H stray .
The atomic field leads to the additional energy
1
1 2
a
Estray = −
J + J CJ dV .
μ0
3
(2.45)
(2.46)
The first term in equation (2.46) is constant. Therefore it can be omitted since it only shifts
the zero level of energy. In equation (2.44) the first term does not change the dynamics of
the system, since the effective field in the Landau–Lifshitz–Gilbert equation only occurs
in terms of J × H eff . Hence all field contributions parallel to J vanish. The second term
in equation (2.46) is of the form of the anisotropy energy (2.18). Thus this additional energy term can be added to the anisotropy constant. If the anisotropy constant is measured
54
D. SUESS et al.
in experiments, the measurement can not distinguish between the different origins of the
anisotropy. So measured anisotropy constants already include this term.
Analogously, the atomic stray field for antiferromagnets with two sublattices follows as,
1 1 A
1 1 B
a
A A
B B
(2.47)
H s,antiferro =
J +C J
J +C J
+
μ0 3
μ0 3
and
m
H s,antiferro = −∇u
with u =
∇(J A + J B )
.
μ0
(2.48)
J A = JsA u and J B = JsB u denote the spontaneous polarization for sublattice A and
sublattice B, respectively. The atomic field a H s,antiferro leads to an additional contribution
to the stray field energy
1
1 A B
1 A A
a
A A A
A B B
Es,antiferro = −
J J + J C J + J J + J C J dV
μ0
3
3
1 B A
1
1
J J + J B CA J A + J B J B + J B CB J B dV . (2.49)
=−
μ0
3
3
The terms J A J A and J B J B can be omitted since they only shift the zero level of energy. The scalar product J A J B is a small correction to the exchange energy. The terms
J A CA J A and J B CB J B are already taken into account in the anisotropy constant. The
remaining terms are of the form J A CA J B and give rise to a small contribution to an
anisotropic exchange interaction.
2.1.4.3. Eddy current field In the following we assume that there are no external currents.
Thus the magnetic field arises from the divergence of the magnetization and from eddy
currents only. For the calculation of the eddy current field we start from the Ampere–
Maxwell equation (2.21). Considering that the curl of the stray field is zero, we get in the
quasi-static limit
∇ × H eddy = I .
(2.50)
The equation above shows that the divergence of the current I vanishes, since ∇ ·(∇ ×F ) =
0 for any vector field F . Therefore ∇ · I = 0, and we get at the boundary I in n − I out n =
I in n = 0, which implies that at the boundary the current flows parallel to the surface.
Applying the curl on both sides of equation (2.50) and considering Faraday’s law (equation (2.22)) and Ohm’s law we get
∂
(2.51)
B.
∂t
Taking into account that the divergence of the eddy current field is zero, we finally get
∇ × ∇ × H eddy = ∇ · (∇H eddy ) − H eddy = −σ
∂
(2.52)
B.
∂t
The boundary condition for the tangential component of the eddy current field follows
from equation (2.50). Assuming a finite current at the surface, we get for the tangential
H eddy = σ
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
55
component
in
H out
eddy − H eddy × n = 0.
(2.53)
The condition for the normal component of the eddy current field follows from ∇B = 0,
which leads to
B in · n = B out · n.
(2.54)
Using the material law for the magnetic induction B we get,
in
out
out
μ0 H out
s + μ0 H eddy + μ0 M · n = μ0 H s + μ0 H eddy · n.
(2.55)
Taking into account equation (2.40) that describes the jump of the normal component of
the stray field, the eddy current field reads
out
H in
eddy · n = H eddy · n.
(2.56)
In order to calculate the jump of the normal derivative of the eddy current field we apply
the curl on both sides of equation (2.50). We get
H eddy = −∇ × I .
(2.57)
For the component i of the eddy current field follows
∇ · ∇Heddy,i = (−∇ × I )i .
(2.58)
Calculating the volume integral
(∇ · ∇Heddy,i )dV = − (∇ × I )i dV ,
(2.59)
and applying Gauss’ theorem on both sides, the following expression is derived
∇Heddy,i ndA = − (n × I )i dA.
(2.60)
If we assume that equation (2.60) is also valid across the interface (∇ ·∇Heddy,i and (∇ ×I )
are not continuous across the interface and therefore in principle the requirement of Gauss’
theorem are not fulfilled) we get
out
in
∇Heddy,i
n − ∇Heddy,i
n = (n × I )in
i .
(2.61)
A rigorous derivation for the jump of the eddy current field without using Gauss’ theorem
is given in the appendix.
2.2. Static micromagnetism—Browns equation
In thermodynamic equilibrium a micromagnetic system tries to reach a state with minimum
total energy. Brown proposed a variational method, which is based on the calculation of the
variational derivative of the total energy with respect to the magnetic polarization (Brown,
1963a). The variation of the energy has to be performed subjected to the constraint that the
norm of the magnetic polarization remains constant. This can be realized introducing the
56
D. SUESS et al.
Lagrange Multipliers λ. The variation of the functional
2
2
2
2
˜
δE tot [J , λ] = δ Etot [J ] + λ Jx + Jy + Jz − Js dV = 0
leads to Brown’s equations
2A
2
u + K1 (uk)k + H S + H ext × J = 0,
Js
Js
for r ∈
∂J
= 0,
for r ∈ ∂ .
∂n
In other words, in equilibrium the magnetization within the magnet (r ∈
an effective field
H eff =
2A
2
u + K1 (uk)k + H S + H ext
Js
Js
(2.62)
(2.63)
(2.64)
) is parallel to
(2.65)
and the torque on the magnetization vanishes
H eff × J = 0.
(2.66)
Since J × J = 0 any arbitrary vector proportional to J may be added to H eff without
changing the result. It is therefore of no consequence if H or B = μ0 H + J is used for
the effective field.
Another approach to calculate the effective field is to calculate the functional derivative
of the total energy without the constraint of constant norm of the magnetization
−
δE
= H eff .
δJ
(2.67)
Brown’s equations have to be solved together with Maxwell’s equations for the magnetostatic field. Furthermore it is also necessary to check whether the solution is a minimum or
a maximum for which the variation vanishes.
2.3. Dynamic micromagnetics
Landau and Lifshitz (1935) calculated the equation of motion for the magnetic polarization
from the thermodynamical principle of irreversibility:
α |γ |
|γ |
∂J
J × H eff −
J × (J × H eff ),
=−
2
∂t
1+α
1 + α 2 Js
(2.68)
with the effective field defined in equation (2.65). This equation follows the physical path
of a magnetic system towards its equilibrium state. The first term in the equation describes
the precessional movement of the magnetization vector around an effective field H eff . The
second phenomenological term stands for the energy dissipation during the movement and
causes a movement into the direction of the effective field. The motion of the magnetization
can be seen in fig. 2.3.
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
57
Fig. 2.3. Movement of the magnetization vector according to the Landau–Lifshitz–Gilbert (LLG) equation
(Kirschner, 2003).
There are many processes which contribute to the damping in magnetic solid, like
magnon–magnon, magnon–phonon interactions and eddy currents (Berger, 1996; Ingvarsson et al., 2002; Capelle and Gyorffy, 2003).
γ is the gyromagnetic ratio, |γ | = 2.210175×105 m/(A s). The damping constant α can
be determined by ferromagnetic resonance measurements. The Landau–Lifshitz equation
is equivalent to the Gilbert equation (Gilbert, 1955)
∂J
α
∂J
(2.69)
= −|γ |J × H eff + J ×
.
∂t
Js
∂t
The intrinsic time scale is determined by the Larmor frequency ω = γ H eff which is usually
in the order of GHz.
It is interesting to note that even in a stable equilibrium state the effective field can be
oriented antiparallel to the magnetization. As an example let us consider a single domain
particle that can be described with one single spin vector. Initially the particle is saturated
in positive y-direction. The easy-axis direction is supposed to be aligned along the y-axis.
The effective field can thus be written as a function of the angle ϕ, which is measured
between the y-axis and the magnetization, as
2K1
J s sin ϕ
0
H eff =
(2.70)
cos(ϕ)
+ H ext − N
.
1
Js
μ0 cos ϕ
In both terms, the precession term and the damping term of the Landau–Lifshitz equation
(2.68), the effective field appears in the form J × H eff . If we split the effective field in a
component parallel and perpendicular to the polarization J , H eff = H eff,⊥ + H eff,|| , we
get J × H eff = J × (H eff,⊥ + H eff,|| ) = J × H eff,⊥ since J × J = 0.
Therefore the dynamics of the system does not change if we add a term proportional to
the polarization to the effective field. Hence, it also does not matter if we use the magnetic
induction B or the H field for the calculation of the time evolution of the magnetization,
since B = μ0 H + M.
The term Jᾱs J × (J × H eff ) rotates the magnetization into the direction of the effective
field. If the magnetization is exactly parallel to the external field the torque will be zero
and therefore the magnetization will not reverse even if the external field is greater than the
nucleation field, Hext > −2K1 /Js . The direction of the magnetization has to be perturbed
a little bit in order to generate a torque and to obtain realistic coercive fields.
58
D. SUESS et al.
Fig. 2.4. For a small particle with homogeneous magnetization the magnetization keeps the orientation aligned
up until the field becomes smaller than the critical value of −2K1 /Js . The bottom left figure shows that the
effective field can be antiparallel to the magnetization although the system is in a stable equilibrium state.
Figure 2.4 shows by simply adding the vectors of the demagnetizing field, the external
field and the anisotropy field that the magnetization is turned back into the easy axis direction as long as the field strength is smaller than the nucleation field. Hence, although the
effective field is oriented antiparallel to the magnetization, the magnetization can not be
reversed.
For fields larger than the critical field the magnetization of the particle will reverse, since
now the anisotropy field decreases with increasing angle.
2.4. Micromagnetism at finite temperature
2.4.1. Short time scales—stochastic dynamics
To introduce temperature into the dynamic equations, a randomly fluctuating field is
added to the total effective field. Equation (2.69) then transforms into
∂J
α
∂J
= −|γ |J × (H eff + H th ) + J ×
.
∂t
Js
∂t
(2.71)
We can transform (2.71) into the mathematical equivalent stochastic Landau–Lifshitz–
Gilbert (LLG) equation. The stochastic LLG equation is a system of 3N Langevin equations with multiplicative noise (Garcia-Palacios and Lazaro, 1998).
∂uli
l
l
Bik
(t)Hth,k
(t),
= Ali (t) +
∂t
(2.72)
k
Ali =
|γ |α
|γ |
ul × Hleff −
ul × ul × Hleff
−
2
1+α
1 + α2
,
i
(2.73)
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
l
Bik
= −|γ |
εij k ulj −
j
|γ |α
ul ul − δik ,
1 + α2 i k
59
(2.74)
where the indices i, j , k run over the three space directions, and the index l = 1, . . . , N
runs over the number of discrete magnetization vectors. The drift term, Ali , is the right
hand side of the deterministic Landau–Lifshitz–Gilbert equation. α is the Gilbert damping
constant and γ is the gyromagnetic ratio. Hlth is the random thermal field. The thermal field
is assumed to be a Gaussian random process with the following statistical properties:
l
l
k
and
Hth,i (t), Hth,j
(t ) = 2Dδij δlk δ(t − t ).
Hth,i (t) = 0
(2.75)
The average of the thermal field, taken over different realizations, vanishes in each direction i in space. The thermal field is uncorrelated in time and space. The strength of the
thermal fluctuations follows from the fluctuation-dissipation theorem (Garcia-Palacios and
Lazaro, 1998)
D=
αkB T
.
|γ |μ0 |ml |
(2.76)
2.4.2. Long time scales—energy barriers
An important problem in condensed matter physics and theoretical chemistry is the understanding of transition processes. In chemistry one is interested in reaction rates and
diffusion processes. In micromagnetics we are interested in thermally activated switching
events of submicron-sized magnetic elements. In magnetic storage applications these thermal switching events determine the long term stability of the stored information. The main
difficulty in the computation of transition processes is caused by the disparity of the time
scales.
If the thermal energy kB T is comparable to the energy barrier E separating two local
energy minima, direct simulations of the escape over the energy barrier using Langevin
equation(s) are possible (Scholz et al., 2000a, 2000b). Unfortunately, this is no longer the
case when kB T
E (either high energy barriers or low temperature limit) which is
the case for most of the interesting problems. An example where such events are crucial
are magnetic storage media. Here data bits are stored in small magnetic elements where
the magnetization can only occupy two stable states (energy minima). It is desired that
spontaneous switching events between these states occur as seldom as possible, since they
lead to a loss of stored data. The desired time scale for the stability is in the range of years
and decades. On the other hand the duration of the switching event itself takes only a few
nanoseconds. The dynamics proceeds by long waiting periods around the local minima
followed by sudden jumps from one state to another. In the simulation of the dynamics
of the system most of the computation time is spent in resolving unimportant fluctuations
around the minima. We would have to wait for very long time to observe the interesting
but rare switching events. This is what is meant with the “disparity of time scales”.
When kB T
E the task simplifies because it is sufficient to find the lowest saddle
point between the two energy minima. The height of the saddle point will then give the relevant energy barrier. Street and Woolley (1949) first discussed thermally activated reversal
processes by pointing out that the thermal energy kB T initiates magnetization reversal.
Starting from the classical Arrhenius formula of chemical reaction theory, Néel (1949)
60
D. SUESS et al.
Fig. 2.5. Thermal switching of the magnetization: fluctuations drive the magnetic system from the stable region A
to another stable region B in the energy landscape. The path with the smallest energy barrier is chosen since the
transition frequency decreases exponentially with the height of the energy barrier. To determine the crossing point
with the lowest energy the separating saddle points between the two regions needs to be calculated. An example
for two stable regions are the stable states in an MRAM (data bit “1” or “0”). Switching between “0” and “1”
passes a saddle point where the magnetization is in a two domain state.
expressed the probability of magnetization reversal as
E
f = f0 exp −
,
kB T
(2.77)
where f is the reversal rate between states separated by the energy barrier E . f0 is
the attempt frequency. The attempt frequency, f0 , depends on material parameters, like
anisotropy, particle shape, and damping (Braun, 1994). Its value ranges from f0 = 109 Hz
to f0 = 1012 Hz. The attempt frequency can be estimated by solving the Fokker–Planck
equation numerically or analytically (Brown, 1963b; Fukushima et al., 2002; Coffey et al.,
1998).
For single domain particles the energy barrier can be simply calculated by the model of
coherent rotation. If the particle has an uniaxial anisotropy the energy barrier is
E = K1 V ,
(2.78)
where K1 is the anisotropy constant and V the volume of the particle. Inspecting equation
(2.77) one finds that for an energy barrier E = 40kB T of the relaxation time, τ = 1/f ,
is about 10 years.
For larger particles the application of an external field greater or equal to the coercive
field initiates incoherent magnetization reversal. This indicates that also the thermally activated switching process may be non-uniform. In this case the energy barrier will differ from
equation (2.78). The calculation of energy barriers is a sophisticated problem. Depending
on whether the final state is known or not different methods were proposed. A short review about the method will be given in the section (3.9) “Calculating energy barriers”.
Figure 2.5 shows the crossing over an energy barrier for a MRAM cell. Thermal fluctua-
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
61
tions drives the micromagnetic system from the stable region A to another stable region B
in the energy landscape. The path with the smallest energy barrier is chosen since the population probability decreases exponentially with the energy of the system. To calculate the
crossing point with the lowest energy one needs to find the relevant saddle points between
the two stable regions. The example in fig. 2.5 shows that the thermal switching between
A and B passes a saddle point where the magnetization is in a two domain state.
3. Numerical methods
The numerical treatment of the micromagnetic equations requires a discrete representation of the continuous magnetization distribution on a computer. The discretization size is
usually much larger than the atomic lattice constant, which means that an equivalent physical system can be described with less variables. This is possible because the spins do not
change significantly from lattice point to lattice point owing to ferromagnetic exchange
coupling. The length over which the magnetic polarization changes from one domain to
the next depends on the material parameters. This length is called domain wall width. For
hard magnetic materials the domain wall width is in the order of some nanometers, for
soft magnetic materials it is even one or two orders of magnitude larger. Due to the slowly
varying spin directions on a length scale of the domain wall width the discretization error
of the numerical method remains small if the discretization is smaller than the domain wall
width.
The most important discretization techniques are based on the finite difference approximation or the finite element method. For example LaBonte (1969) used a 2-dimensional
square grid to represent the magnetization in the domain wall which resembles the finite
difference method. Schmidts and Kronmüller (1994) calculated magnetization processes
in cylindrical particles with square cross section by the numerical minimization of the total
energy. The total energy was discretized with finite difference schemes. The magnetization
direction of each computational cell was represented using polar coordinates. Seshan and
Cendres (1985) used triangular finite elements for the calculation of the domain pattern in
magnetic thin films. Fredkin and Koehler (1987) applied the finite element technique to calculate magnetization processes in small particles. With the use of the finite element method
arbitrarily shaped magnetic particles can be calculated. Schrefl et al. (1994a, 1994b) used
the finite element method for the calculation of nanocomposite magnets.
A different approach for the numerical treatment of the magnetization processes is
based on the solution of the dynamic equation of motion, the so-called Landau–Lifshitz
or Gilbert equation. In the late 80th different groups independently solved the Landau–
Lifshitz equation numerically. Magnetization processes in granular thin films were studied
by Victora (1987), Schabes and Bertram (1988). Nakatani et al. (1989) studied magnetization processes in small magnetic cubes addressing questions relevant to particulate
magnetic recording. Mansuripur (1988) investigated domain wall dynamics in thin films of
amorphous rare-earth-transition-metal alloys on a two dimensional square lattice. Based on
the early work of numerical micromagnetics, different algorithms where developed solving the micromagnetic equation for different problems. A compilation is given in table 3.1.
Generally a micromagnetic software package has to treat three tasks:
62
D. SUESS et al.
TABLE 3.1
Compilation of different micromagnetic models and discretization schemes
Discretization
scheme
Strayfield
calculation
Calculation of
equilibrium
Finite temperature
micromagnetism
Finite difference method
Brown (1963a, 1963b), LaBonte (1969), Schabes and
Bertram (1988), Nakatani et al. (1989), Scheinfein and
Blue (1991), Berkov et al. (1993), Oti (1993),
McMichael and Donahue (1997), Thiaville et al. (1998)
Finite element method
Seshan and Cendres (1985), Fredkin and Koehler
(1987), Bagneres-Viallix et al. (1991), Schrefl and
Fidler (1992), Ramstöck et al. (1994), Hertel and
Kronmüller (1999)
FFT methods
Mansuripur (1988), Yuan and Bertram (1992), Berkov
et al. (1993), Koch et al. (1998)
Magnetic scalar potential
(FEM)
Fredkin and Koehler (1990), Bagneres-Viallix et al.
(1991), Kinderlehrer and Ma (1994)
Magnetic vector potential
(FEM)
Fredkin and Koehler (1987), Schrefl et al. (1994a,
1994b)
Multipole expansion
Zhu and Bertram (1988), Miles and Middleton (1991),
Blue and Scheinfein (1991), Newns et al. (2004)
Matrix compression
Schrefl et al. (2000), Forster et al. (2003a, 2003b)
Energy minimization
Della Torre (1985), Bagneres-Viallix et al. (1991),
Berkov et al. (1993), Chen et al. (1993), Schmidts and
Kronmüller (1994)
Landau–Lifshitz–Gilbert
Victora (1987), Schabes and Bertram (1988), Nakatani
et al. (1989)
Langevin dynamics and
Monte Carlo simulations
Boerner and Bertram (1997), Nakatani et al. (1989),
Chantrell et al. (1998), Lyberatos and Chantrell (1993),
Zhang and Fredkin (1999), Nowak et al. (2000), Scholz
et al. (2000a, 2000b), Chubykalo et al. (2003), Grinstein
and Koch (2003), Kirschner et al. (2005)
1. The discretization of the total energy.
2. The calculation of the magnetostatic field.
3. The evolution of the magnetization toward equilibrium.
3.1. Finite difference micromagnetism
In order to solve the micromagnetic equation the total energy has to be evaluated for a
general distribution of the magnetization. Therefore the total energy has to be converted
into a form, which can be translated into an algorithm for a digital computer with finite
speed and memory. We have to reduce the problem of finding a continuous solution to one
with finite dimensionality. In the finite difference method (as later in the finite element
method) we replace the continuous solution domain by a discrete set of lattice points. In
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
63
each lattice point we replace any differential operators by finite difference operators. The
conditions on the boundary of the domain have to be replaced by their discrete counterparts.
For some differential equations, such as the wave equation in one dimension, it is even
possible to construct exact algorithms by nonstandard finite difference schemes (Cole,
1997). However, this is rarely the case, and so the finite difference method gives only
an approximate solution. In order to integrate the Landau–Lifshitz–Gilbert equation the
effective field has to be evaluated at every time step.
The calculation of the effective field contains two crucial parts. First, the calculation
of the long range interacting demagnetizing field, which will be discussed separately in a
following chapter. Secondly, the contribution of the exchange field that requires the calculation of the second derivative of the magnetic polarization. The easiest way to calculate
the exchange field is to evaluate the magnetic polarization on a regular grid. Approaches
where grids are used which do not necessarily form a regular array of points were also
proposed (Fredkin et al., 1992).
The expression of the exchange energy contains the second derivative of the magnetization. To express the second derivative of the magnetization with finite differences a Taylor
expansion of the magnetization is performed
u(x + x, y, z) = u(x, y, z) + x
(x)2 ∂ 2
∂
u(x, y, z),
u(x, y, z) +
∂x
2 ∂ 2x
∂
(x)2 ∂ 2
u(x, y, z).
u(x, y, z) +
∂x
2 ∂ 2x
Adding equation (3.1) and equation (3.2) leads to
(3.1)
u(x − x, y, z) = u(x, y, z) − x
(3.2)
u(x + x, y, z) + u(x − x, y, z)
∂2
u(x, y, z) = −2u(x, y, z) +
.
2
∂ x
(x)2
(3.3)
Using equation (3.3) to discretize the exchange energy (references see table 3.1) leads to
an exchange field at the lattice point i,
2A H exch,i = 2 2
(3.4)
J i,
Js x
i∈nN
where nN stands for the indices of the nearest neighbors of the lattice point i. The first term
in equation (3.3) could be omitted since it does not change the dynamics of the system. It
just gives a contribution parallel to the magnetization.
For a one-dimensional system Trouilloud and Miltat (1987) showed that a four-nearestneighbor approximation leads to an accuracy of the exchange field that is much higher than
the conventional two-nearest-neigbhor approach. In the four-nearest-neighbor approximation a Taylor expansion to the fourth order around the central mesh point is performed.
To obtain a system of four linear equations for the unknown derivatives of the magnetization, the Taylor expansion is evaluated at four lattice sites. Details for the calculation of
the exchange field with a Taylor expansion of the 4-th order can be found in (Berkov et
al., 1993; Labrune and Miltat, 1995; Fabian et al., 1996). Different representations of the
exchange energy on 2-D grids were investigated by Donahue and McMichael (1997). They
proposed a four-neighbor angle approach which has the advantage that the interpolation
64
D. SUESS et al.
preserves the norm in the interpolated region and a eight-neighbor dot product approach.
The comparison of these two representations with a conventional two-nearest-neighbor approach showed that all representations give accurate results for fine finite element meshes.
For coarser meshes the four angle representations is most suitable.
The contribution of the external field and the anisotropy field to the effective field are
straightforward using the finite difference method.
3.2. Finite element micromagnetism
The finite element method is a well established tool to solve partial differential equation
(PDE). It is successfully applied in structural analysis, fluid dynamics and electromagnetic
field calculation. The main advantage of the finite element method over the finite difference method is that arbitrarily shaped geometries can be calculated. The drawback is the
mathematical more complicated background. First the domain has to be discretized with
finite elements. Depending on the problem and the dimension of the system the used finite
elements range from triangles, squares and rectangles to tetrahedrons, cubes and hexahedra. Within these finite elements the solution of the problem (e.g. potential, temperature
field, stress, magnetic polarization) is interpolated with polynomials. The coefficients of
that expansion are determined in such a way that the distance of the approximated solution
to the exact solution is minimized.
Applying the finite element method to micromagnetism the magnetic polarization J (x)
is expanded with basis functions. In the finite element method the basis functions ϕi (x)
are chosen in such a way, that the support of each basis function (the region where ϕi (x)
is nonzero) is small. The support consists of only a few (connected) elements. The value
of every basis function is one on one node and zero at all the other nodes. Figure 3.1 gives
an example for linear basis function in two dimensions. Only a small number of basis
functions have a non-zero overlap.
The basis function obey
N
ϕi (x) = 1,
(3.5)
ϕi (x j ) = δ(x j − x i ),
(3.6)
i=1
where x i and x j denote the positions of the local node points i and j , respectively. N is the
number of node points of the finite element mesh that coincides with the number of basis
functions.
For k-th component of J we can write
k
(x) =
Japp
N
Js,i uki ϕi (x).
(3.7)
i=1
Js,i is the spontaneous polarization at the node point i. At a nodal point i the values of
the reduced magnetic polarization uk (k) are given by the coefficients uki . A different discretization scheme was used in the original work of Fredkin and Koehler (1987). In their
approach the magnetization was assumed to be constant within each finite element.
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
65
Fig. 3.1. Nodal linear basis functions and common support of two basis function.
To perform the time integration, the effective field on every node point of the finite
element mesh has to be calculated. Using the expansion according to equation (3.7) for the
magnetic polarization, the energy functional becomes a function of the coefficients uki
Et =
et (J )dV ∼
=
et
N
Js,i uki ϕi (x) dV .
(3.8)
i=1
As a consequence the energy functional has transformed to a function of the coefficients uki .
The k-th component of the effective field on node i is approximated using the box scheme
(Gardiner, 1985),
k
Hi,eff
≈−
1 ∂Et
,
mi ∂uki
(3.9)
where mi denotes the magnetic moment on the node point i. It follows from the integral
mi = Js (x)dV ,
(3.10)
Vi
where Vi , as shown in fig. 3.2, is the surrounding volume of the node i, such that
Vj = V
and
Vi ∩ Vj = 0 for i = j.
j
(3.11)
66
D. SUESS et al.
Fig. 3.2. The volume Vi surrounding the node i shown in a 2-dimensional example.
Usually, in a single phase magnetic material the spontaneous polarization, Js , is constant in
space. However, the spontaneous polarization becomes a function of space for composite
media or at grain boundaries. In our model the nodes of the finite elements are located at
grain boundaries. Regions with different values of the spontaneous polarization surround
these points. Thus we assume an average magnetic moment for these nodes as given by
equation (3.10).
For the calculation of the effective field the derivative of the total energy has to be calculated. Let us first consider only the contribution of the exchange energy to the total energy,
in order to give an example for the calculation. The exchange energy is given by equation
(2.4). Using the approximation for the magnetic polarization (3.7) for the energy follows
2
2
2
y
uxj ∇ϕj (x) + uj ∇ϕj (x) + uzj ∇ϕj (x) dV ,
Eex,e = A
(3.12)
j
e
where ukj denotes the k-th component normalized magnetization on node point j . Here we
just assumed the contribution
of the exchange energy from the finite element e to the total
exchange energy (Eex = e Eex,e , where the sum is over all finite elements).
For the gradient we obtain
∂
∂Eex
2
ukj ∇ϕj (x) dV .
=
A
(3.13)
k
k
∂ui
∂ui
j
e
Applying the chain rule gives
∂Eex
=
2A
ukj ∇ϕj (x)∇ϕi (x)dV .
∂uki
j
(3.14)
e
Introducing the element matrix Mlje ,
Mlj = ∇ϕl (x)∇ϕj (x)dV ,
(3.15)
e
the derivative of the exchange energy can be written as a matrix vector product,
∂Eex
= 2A(M · uk ).
∂uk
(3.16)
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
67
Due to the local support of the basis function, as shown in fig. 3.1, M is a sparse matrix.
Therefore the matrix vector product can be performed easily. The matrix M has to be
constructed only once since it only depends on the geometry of the finite element mesh.
In a similar way the anisotropy field can be calculated from (2.17) neglecting the term
with K2 . It follows
{x,y,z}
∂Eani
=
−2K
k
kl ukj ϕi (x)ϕj (x)dV .
(3.17)
1
k
k
∂ui
j
l
e
The external field can be easily added on every node point to the effective field. The calculation of the demagnetizing field is the most time consuming part in micromagnetic
simulation.
3.3. Numerical treatment of the stray field problem
The efficient calculation of the long range interaction of the magnetostatic field is the
most delicate problem in micromagnetic simulations. The origin of the magnetostatic field
are magnetic volume charges ρ(r ) = −∇J /μ0 within the sample and surface charges
σ (r ) = J · n/μ0 at grain boundaries and the surface of the sample with normal vector n.
The strayfield produced by these charges follows from the solution of equations (2.38)–
(2.40) as,
ρ(r )(r − r )
σ (r )(r − r )
1
1
dV
+
dS .
H (r) =
(3.18)
3
4πμ0
4πμ0
|r − r |
|r − r |3
V
S
To calculate the stray field numerically the sample can be subdivided into cubic cells. The
first numerical models assumed a constant magnetization within these elements (LaBonte,
1969; Schabes and Bertram, 1988; Zhu and Bertram, 1988; Yuan and Bertram, 1991).
The field follows from the superposition of the 6 magnetic charged surfaces for every
discretization cell. A model that also considers magnetic volume charges was developed by
Ramstöck et al. (1994) by interpolating the magnetization with linear or quadratic functions
within the discretization cells. Nakatani et al. (1989) calculated the demagnetizing field by
a dipole approximation. The field at the center point r of the observing cell is obtained by
superposing the dipole fields which are generated by the magnetic polarization J within
each source cell.
J (r )(r − r )
J (r )
1 −3
(r − r ) dx dy dz .
H (r) = −
4πμ0
|(r − r )|3
|(r − r )|5
cell
V
(3.19)
A less accurate method was applied by Boerner and Bertram (1997) who substituted the
constant magnetization within each cell with a magnetic dipole moment located at the
center of the cell. In the discrete form equation (3.18) and equation (3.19) can be written
as,
H (i, j, k)
Ny Nz
Nx 1 ⎛
Kxx
⎝ Kyx
=−
μ0
Kzx
i =1 j =1 k =1
Kxy
Kyy
Kzy
Kxz
Kyz
Kzz
(i −i,j −j,k −k)
Jx
Jy
Jz
⎞
⎠ , (3.20)
(i ,j ,k )
68
D. SUESS et al.
where Nx , Ny and Nz are the numbers of element in the x, y and z direction. Each cell is
specified by the indices i, j and k which are the discretized x-, y-, and z-coordinates of the
center point of the cell. The elements of the matrix K are called demagnetizing coefficients.
The calculation of the stray field is of the order of O(N × N ), when N is the number of
discretization points. Therefore more efficient methods are required.
3.3.1. Vector potential
The use of a magnetic scalar potential allows the calculation of the magnetostatic field
as a function of the magnetization state. It is applicable for dynamic calculation, where
the Landau–Lifshitz–Gilbert equation is integrated. At every time step the open boundary
problem has to be solved. An alternative way suitable for static micromagnetic simulations
is to treat the magnetostatic interaction as a variational problem as proposed by Brown
(1963a). The two variational formulations are,
1
Estray ≤ W1 [J , A] =
(3.21)
(∇ × A − J )2 dV ,
2μ0
Estray ≥ W2 [J , U ] =
(∇U )J dV −
μ0
2
(∇U )2 dV .
(3.22)
In both formulations the stray field energy only depends locally on J , U or A. Using W1 ,
the stray field can be calculated for a fixed magnetization configuration by minimizing W1
with respect to A. In order to calculate equilibrium states of the magnetization the functional W1 has to be minimized together with the total Gibbs’ free energy (Aharoni, 2000).
In order to satisfy the constraint of a constant magnetization, polar coordinates can be introduced for the representation of the magnetization vector. Numerically the minimization
problem of W1 has advantages over the maximization problem using the scalar potential U .
For the calculation of an equilibrium state of the magnetization, W2 has to be maximized,
while all other energy contributions have to be minimized. This saddle point problem may
lead to slow convergence.
The integration in W1 and W2 has to be performed over the entire space. To transform
the open boundary problem to a closed boundary problem the method of implicit mapping
can be applied (Brunotte et al., 1992; Imhoff et al., 1990).
3.3.2. Scalar potential
Instead of transforming the open boundary problem given by equations (2.38)–(2.40) to
an integral equation, it can be solved directly using numerical methods such as the finite
element (FE) or finite difference (FD) method. In principle to solve the Poisson equation by
taking into account the boundary condition at infinity the magnetic scalar potential u must
be determined everywhere in space. Practically the numerical discretization schemes (FE,
FD) can only be applied to a limited space. To overcome this problem different approaches
have been applied:
1. Finite elements are truncated in a limited area around the magnet and u is set to zero “far
away” of the magnet. To get reliable results the outer space should exceed five times the
length of the magnet as suggested by Chen and Konrad (1996).
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
69
2. Asymptotic boundary conditions. Hereby the asymptotic boundary condition at infinity
is transformed to Robin boundary conditions on the surface of the finite element domain
(Khebir et al., 1990). In Robin boundary conditions the solution can by expressed by its
derivative.
3. Coordinate transformation. The outer space of the magnet is limited to a bounded area
due to a parallelepipedic shell transformation (Brunotte et al., 1992).
In contrast to the first and second possibility the coordinate transformation method does
not bring any systematic error. But all described methods are very inflexible to different
geometries of the magnet. Another disadvantage is that additional finite elements outside
the magnets have to be used.
In contrast, the hybrid finite element/boundary element method (FE/BE) does not need
finite elements outside of the magnets. This method is an exact method to solve the equations (2.38)–(2.40).
3.3.2.1. Hybrid FE/BE method This method combines the advantages of the finite element method with those of the boundary element method. The boundary element method
takes into account the boundary condition at infinity (equation (2.39)), although no elements in the exterior are required. The volume divergences are considered by the finite
element method, which leads to sparse matrices.
Fredkin and Koehler (1990) suggested to split the magnetic scalar potential u into two
parts
u = u1 + u2 .
(3.23)
The potential u1 solves the Poisson equation with Neumann boundary conditions
uout
1 =0
uin
1 =
∇J s
μ0
outside the magnet,
(3.24)
within the magnet.
(3.25)
The potential u1 is zero outside the magnetic particles, uout
1 = 0. Therefore, for the normal
derivative of the solution at the boundary follows
∂uin
∂uin
∂uout
J
1
− 1 = − 1 = − n,
∂n
∂n
∂n
μ0
(3.26)
in
in
uout
1 − u1 = −u1 .
(3.27)
This problem can be solved with the standard finite element method as presented in the
following chapter.
Since the total potential u = u1 + u2 must fulfill equation (2.38), u2 has to satisfy the
Laplace equation
uout
2 =0
outside the magnet,
(3.28)
uin
2 =0
within the magnet.
(3.29)
70
D. SUESS et al.
The boundary conditions for the potential u2 follow from inserting equations (3.23), (3.26)
and (3.27) into equations (2.42) and (2.43) as,
∂uin
∂uout
2
− 2 = 0,
∂n
∂n
(3.30)
in
in
uout
2 − u2 = u1 .
(3.31)
Equations (3.28)–(3.31) are standing for the potential of a plane of dipoles with moment u1 .
Since we already know u1 , we can calculate u2 directly by using the analytical formula for
a dipole sheet with dipole density u1 .
in
u1 (y)(x − y)
1
ndy ,
u2 (x) =
(3.32)
4π
|x − y|3
where n is the normal vector at the boundary. One could calculate the potential u2 everywhere in space and superpose it with u1 receiving the potential u. To save memory, u2 is
only evaluated on the surface of the magnetic bodies. With these values as Dirichlet boundary conditions we solve the Laplace equation (3.29) inside the magnet by the standard finite
element method. Evaluating equation (3.32) on the boundary y , one has to integrate over
a singularity. The integral becomes
in
u1 (y)(x 0 − y)
1
(x 0 )
ndy +
− 1 uin
u2 (x 0 ) =
(3.33)
1 (x 0 ),
4π
4π
|x 0 − y|3
where (x 0 ) stands for the solid angle enclosing the boundary surface at x 0 . The dipole
in
moment uin
1 (y) as well as the potential u2 (y) can be approximated on N nodes of the
surface mesh by a linear combination of the local basis functions ϕ i (y)
j
uin
1 x0 =
N
j
j
ui1 ϕ i x 0 = u1 .
(3.34)
i=1
Inserting equation (3.34) into equation (3.33) we receive
j
u2 = Bij ui1 ,
(3.35)
where Bij is the interaction boundary matrix
Bij =
i
j
j
N ele
(x 0 )
ϕ (y)(x 0 − y)
1
nd
− 1 δij +
y .
j
4π
4π
(x 0 − y)3
i=1 e=1
(3.36)
e
The integral is split into a sum of integrals over each surface element e . Generally, the
N × N matrix B is fully populated. The advantage of the hybrid BE/FE method is that
the matrix B only depends on the geometry of the magnetic particles and the finite element
mesh and therefore has to be calculated only once. Furthermore the stray field between spatial separated particles can be calculated without requiring a finite element mesh between
the particles (Suess et al., 1999).
Summing up, to solve the stray field problem we have to perform the following steps:
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
71
1. At the beginning of the finite element micromagnetic simulation the boundary matrix B
has to be built according to equation (3.36).
2. At each time step the Poisson equation (3.25) is solved for a given distribution of the
magnetic moments to receive the potential u1 .
3. The matrix–vector multiplication of equation (3.35) is performed. One get the values of
the potential u2 on the N nodes of the surface mesh.
4. The Laplace equation (3.30) is solved for the potential u2 inside the magnetic particles
using the values of step 3 as Dirichlet boundary conditions.
5. Superpose the calculated potentials u = u1 + u2 .
6. Determine the stray field H s as the gradient of the scalar magnetic u according to equation (2.37).
As one can see steps two to six have to be performed at each time step. The advantage of
the hybrid BE/FE method is that the most time consuming step 1 has to be performed only
once at the beginning of the simulation. The boundary matrix B can be used at each time
step since it only depends on the geometry of the particle and the finite element mesh.
The matrix B can be computed directly by solving equation (3.36) for every entry of
the fully populated matrix. The memory requirements are N 2 and the multiplication of
the vector with the dense matrix in equation (3.35) needs a lot of CPU time. Therefore
different methods are used to reduce the dense matrix B to a sparse matrix. One possibility
is a wavelet transformation which transforms B into a sparse matrix (Goswami and Chan,
1999; Schrefl et al., 2000). The treecode (Barnes and Hut, 1986) interprets the integral
(equation (3.33)) as a sum of dipoles. Hierarchical matrices or so-called supermatrices
approximate equation (3.36) in a way that we receive data-sparse matrices (Börm et al.,
2002). The different methods are explained in the following chapters.
3.3.2.2. Finite element method—Galerkin approach In this chapter we demonstrate how
the partial differential equations (3.25)–(3.27) can be solved with the finite element
method. While the main idea of the finite difference method is to replace derivatives of
the partial differential equation by difference approximation, in the finite element method
the solution ũ(x) is approximated with basis functions ϕi (x)
ũ(x) ∼
= u(x) =
N
ui ϕi (x).
(3.37)
i=1
The ultimate aim is to construct a method for computing the coefficients ui such that the
error ũ(x) − u(x) is minimized. For most problems we don’t know the exact solution
ũ(x). Therefore we can not directly calculate the error ũ(x) − u(x). However, the error
can be measured by inserting u(x) in the PDE. Let us assume the exact solution fulfills the
Poisson equation with the source f (x),
ũ(x) − f (x) = L(ũ(x)) = 0,
(3.38)
where L is the differential operator. If we insert the approximation u(x) in the PDE, we
generally have
L(u(x)) = 0.
(3.39)
72
D. SUESS et al.
The error in the equation, R = L(u(x)), is called residual. In general we can expect that a
small residual leads to a small error ũ(x) − u(x).
The N equations needed to calculate the N parameters to u1 to uN , can be obtained by
forcing the residual to be small. In the Galerkin approach we multiply the residual with the
N test functions to obtain N equations
(3.40)
Rϕi (x) = 0, i = 1, . . . , N.
Equation (3.40) is also called weak formulation of the differential equation (3.38). The
approximated solution has to fulfill the PDE only in average. Let us now apply the Galerkin
approach to solve the partial differential equations (3.25)–(3.27). We multiply the Poisson
equation with the test function and obtain
∇J s
in
(3.41)
u1 ϕi (x)dV =
ϕi (x)dV .
μ0
Vm
Vm
Applying Greens’ theorem on both sides we get
in
− ∇u1 ∇ϕi (x)dV + n · ∇uin
1 ϕi (x)dA
Vm
=−
Vm
Am
J
∇ϕi (x)dV +
μ0
Am
ϕi (x)
nJ s dA.
μ0
(3.42)
Inserting the equation (3.26) for the normal derivative in equation (3.42), the surface integrals on the left-hand side and on the right-hand side cancel out, and one gets the following
N equation
J
(3.43)
∇uin
∇ϕ
(x)dV
=
∇ϕi (x)dV .
i
1
μ0
Vm
Vm
Substituting uin
1 (x) by the expansion
N
j =1
N
j =1 uj ϕj (x) we get
∇ϕi (x)∇ϕj (x)dV =
uj
Vm
Vm
Js
∇ϕi (x)dV .
μ0
(3.44)
The integrals over the derivatives of the shape functions only depend on the geometry and
have to be calculated only once. The integrals can be stored in the element matrix
Aij = ∇ϕi (x)∇ϕj (x)dV .
(3.45)
Vm
After discretization with the Galerkin approach the differential equation (3.25) with the
boundary condition (3.26) transforms to a set of N linear equations
N
Js
uj Aij =
∇ϕi (x)dV .
(3.46)
μ0
j =1
Vm
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
73
3.4. Eddy currents
For magnetic data storage (hard disk recording, MRAM) at high data rates and ultra fast
switching eddy currents cannot be neglected. Several eddy current methods have been presented in the past. These include the 2-dimensional quasi-static model by Della Torre and
Eicke (1997) and the 1-dimensional dynamic calculations by Sandler and Bertram (1997).
A hybrid method for three-dimensional eddy current problems based on the solution of the
differential equation for the current density and the magnetic field where the Biot–Savart
law is used to calculate the field intensity of the magnetic field on the surface was introduced by Kalimov et al. (1997). Serpico et al. (2001) developed a novel finite difference
scheme, that is applied to the analysis of eddy currents with the Landau–Lifshitz equation
as a constitutive relation.
Hrkac et al. (2005) developed a 3-dimensional finite element dynamic micromagnetic
model including eddy currents. This model simultaneously solves the Landau–Lifshitz–
Gilbert (LLG) equation and the quasi-static Maxwell equation using a hybrid FE/BE
method. The boundary element method is used to map the boundary conditions of the
magnetic field at infinity on equivalent boundary conditions on the surface of the conducting region, which reduces the necessity of a mesh outside the magnetic region. The
eddy current field is introduced as an additional part of the effective magnetic field in the
Landau–Lifshitz–Gilbert (LLG) equation, and is directly calculated from the space time
behavior of the magnetization rate of change.
3.4.1. Model
The dynamic magnetization process in the conducting region is described by the
Landau–Lifshitz–Gilbert equation of motion (2.68). The eddy current field is added to the
effective field. The solenoidal eddy current field has to fulfill the Poisson equation (2.52)
for each component inside the magnetic region and Laplace equation outside.
To solve the coupled Landau–Lifshitz–Gilbert equation and the quasi-static Maxwell
equations we use the hybrid FE/BE method as described by Fredkin and Koehler (Fredkin
and Koehler, 1990). For each component i of the eddy current field, we have to solve
∇ 2 Heddy,i = f,
inside and
(3.47)
∇ 2 Heddy,i = 0,
outside.
(3.48)
Both, the tangential and the normal component of the eddy current field are continuous as
given by equation (2.53) and equation (2.56). The normal derivative of the i-th component
of the eddy current field shows a jump of (n × j )i as described by equation (2.61).
In a similar manner like the calculation of the stray field, we split the eddy current field
into
1
2
Heddy,i = Heddy,i
+ Heddy,i
,
(3.49)
1,out
Heddy,i
=0
outside the magnet,
(3.50)
1,in
=f
Heddy,i
inside the magnet.
(3.51)
74
D. SUESS et al.
1
The normal derivative of the solution is defined on the boundary. The field Heddy,i
is zero
1
outside the magnetic particles, Heddy,i = 0,
1,out
∂Heddy,i
∂n
−
1,in
∂Heddy,i
∂n
=−
1,in
∂Heddy,i
∂n
= (n × J )i ,
1,out
1,in
1,in
− Heddy,i
= −Heddy,i
.
Heddy,i
(3.52)
(3.53)
The Poisson equation (3.51) with the Neumann boundary condition can be calculated using
standard finite element method. For the second part of the eddy current field follows
2,out
Heddy,i
= 0,
2,in
Heddy,i
= 0.
(3.54)
2
is given by
The boundary conditions for Heddy,i
2,out
∂Heddy,i
∂n
−
2,in
∂Heddy,i
∂n
= 0,
2,out
2,in
1,in
− Heddy,i
= Heddy,i
.
Heddy,i
(3.55)
(3.56)
Again equation (3.30) and equation (3.31) describe the potential of a dipole sheet and can
be calculated using a boundary integral method.
3.5. Fast integration methods
The direct computation of equation (3.18) scales with N 2 in storage and computation time,
where N is the number of grid points in the finite element or finite difference discretization. In order to save computational time and storage, various methods have been proposed
such as truncation of the interaction range (Zhu and Bertram, 1988), meanfield approximation for distant particles (Bertram, 1971) and hierarchical dipole interaction evaluation
schemes (Miles and Middleton, 1991). Mansuripur and Giles (1988) introduced the fast
Fourier transformation (FFT) to calculate the demagnetizing field on regular grids with periodic boundary conditions, with a computing time of O(N log2 N ). The FFT method was
extended to two-dimensional non-periodic systems by Yuan and Bertram (1992), Berkov et
al. (1993), and Hayashi et al. (1996). The method developed by Yuan and Bertram (1992)
allows the calculation of finite size magnetic systems with non-periodic boundary conditions in O(N log2 N )-computation time. The computation time could be decreased to
O(N log N ) by Blue and Scheinfein (1991) using multipole expansion (Greengard and
Rohklin, 1987). In the following the most common fast integration methods are introduced
in more detail.
3.5.1. Fast Fourier transformation
A very elegant and efficient method to calculate the integral in equation (3.18) is using
the convolution theorem for Fourier transformations. The convolution of the function f (t)
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
75
and g(t) is
∞
f ⊗g =
f (t)g(t − t )dt = I−1 [f˜(ν)g̃(ν)],
(3.57)
−∞
where f˜(ν) and g̃(ν) is the Fourier transform of f (t) and g(t), respectively. Equation
(3.18) may be easily understood as a convolution product and can be written in the form
H (r) = ∇r G(r − r ) ⊗ ρ(r ) + ∇r G(r − r ) ⊗ σ (r ),
(3.58)
with
G(r − r ) =
1
1
.
4πμ0 |r − r |
(3.59)
The convolution theorem provides an efficient way of calculating the convolution product
above.
In the discrete case the integral of equation (3.18) splits up into sums, and the demagnetizing field follows from equation (3.20). In the case of a translational invariant lattice the
demagnetizing matrix K only depends on the difference between the centres of cells i, j ,
k and i, j , k. Following Yuan and Bertram (1992) equation (3.20),
1 K(r − r )M(r ),
H (r) = −
(3.60)
Ms
r
can be recognized as a discrete convolution between the matrix K and function M. Such
a convolution can be changed into a scalar product in frequency space using the Fourier
transformation. The discrete convolution theorem requires two properties for the matrix K
and the vector M. It assumes that M and K are periodic with the same wavelength. For
magnetic samples of finite size these requirements can be fulfilled using the method of
zero-padding (Press et al., 1986). Applying zero-padding the length of the vector M and K
has to be extended to a power of two. Furthermore the number of cells in each dimension
after zero-padding has to be at least twice the number of physical cells. With these conditions non-periodic samples can be treated correctly since the cells in the extended periods
do not affect the cells in the simulated region.
Assuming that the number of cells after zero-padding is nx , ny , and nz , the discretized
Fourier transformation M̃ of the of magnetization M can be expressed as:
M̃(k1 , k2 , k3 )
=
n
y −1 nz −1
x −1 n
n1 =0 n2 =0 n3 =0
k1 n 1
k2 n 2
k3 n 3
M(n1 , n2 , n3 ) · exp 2πi
+
+
,
nx
ny
nz
(3.61)
and similarly for K.
Applying the convolution theorem equation (3.60) can be written in frequency space as
H̃ (k1 , k2 , k3 ) = −
1
K̃(k1 , k2 , k3 )J̃ (k1 , k2 , k3 ).
Js
(3.62)
76
D. SUESS et al.
The values of K in real space are in wrap around order to fulfill the requirement for the
convolution theorem (Press et al., 1986). Since the demagnetizing tensor K only depends
on the geometry, its Fourier transformation has to be calculated only once. The most time
consuming part is the calculation of the Fourier transform for every component of the magnetic polarization and the calculation of the inverse Fourier transform of the product K̃M̃.
Berkov et al. (1993) calculated the magnetic potential with the use of the convolution
theorem for Fourier transformations. In three-dimensional calculations they gained a factor
of three compared with the method proposed by Yuan and Bertram (1992).
3.5.2. Treecode
The first implementations of the so-called tree algorithms have been proposed by Appel
(1981). Tree algorithms arrange particles in a hierarchy of clusters, and compute the interaction at a given point by summing over multipole expansions of these groups. In this way
the computational requirements can be reduced to a O(N log N )-scaling. The clustering itself can be achieved in various ways, for example with nearest-neighbor pairings (Jernigan
and Porter, 1989), or with Eulerian subdivisions (Barnes and Hut, 1986).
The great advantage of tree codes is that they can be adapted flexibly to any clustering
state without significant losses in speed (Springel et al., 2001). Tree codes are basically
free from any geometrical restrictions.
When the force on a particular particle is computed, the interaction exerted by distant
groups is approximated by their lowest multipole moments. In this way, the computational
cost for a complete force evaluation can be reduced to an order of O(N log N ) (Appel,
1985). The forces become more accurate if the multipole expansion is carried out to higher
orders, but the increasing cost of evaluating higher order terms might make it more efficient
to terminate the multipole expansion and rather use a larger number of smaller tree nodes
to achieve a desired force accuracy (McMillian and Aarseth, 1993).
In the scheme of Barnes and Hut (1986) the domain, which surrounds all dipoles, is
hierarchically partitioned into a sequence of cubes, where each cube contains eight siblings,
each with half the side-length of the parent cube. These cubes form the nodes of an oct-tree
structure. The tree is constructed such that each cube contains either exactly one dipole, or
is parent to further cubes, in which case the parent cube carries the dipole moments of all
the dipoles that lie inside this cube. A schematic illustration of the Barnes and Hut tree is
shown in fig. 3.3.
The computation of the interaction between the dipoles proceeds by walking through the
tree, and summing up appropriate contributions from the tree elements. In the tree walk,
the dipole moment of a cell of size l is used only if
t
r> ,
(3.63)
θ
where r is the distance of the particular point to the center-of-mass of the cell and θ is an
accuracy parameter. If a cell fulfills this criterion the tree walk along this branch can be
terminated, otherwise it is “opened”, and the walk is continued with all its siblings. For
smaller values of the parameter θ the forces will in general become more accurate, but also
more CPU-time consuming.
The tree construction can be made by inserting the dipoles one after the other in the tree.
Once the grouping is completed, the multipole moments of each cube (if they are parent
cells) can be recursively computed from the moments of its daughter cubes.
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
77
Fig. 3.3. Schematic illustration of the Barnes and Hut oct-tree in two dimensions. The particles are first enclosed
in a square (root). This square is iteratively subdivided in four squares of half the size, until exactly one particle
is left in each final square (leaves of the tree). Note that empty squares need not to be stored. Image taken from
Springel et al. (2001).
3.5.3. H-matrices—adaptive cross approximation
The concept of H-matrices was introduced by Hackbusch (1999). It allows to transform
fully populated matrices arising from integral operators into sparse matrices. The matrices
are sparse in a sense that only few data are needed for their representation. The matrix–
vector multiplication is of almost linear complexity. The inverse of an H-matrix is again a
sparse H -matrix.
To give a short introduction to H -matrices we start from the discretization of boundary
integral equations which leads to large dense matrices. These matrices have in general no
explicit structure. However, by suitable renumbering and permuting the boundary nodes,
the dense matrix can be written in a form containing blocks which can be approximated
by low-rank matrices, which are explained later on. The renumbering of the nodes is done
by geometrical criterions. Consecutive boundary nodes should be located geometrically
close to each other. A simple example of the clustering is given in fig. 3.4. Boundary nodes
located close to each other are combined in clusters. A large distance between cluster
results in entries in the renumbered boundary matrix that are well separated from each
other. For example the cluster with the nodes {6,7,8,9,10} and the cluster {1,2,3,4,5} are
“admissible” cluster pairs. Admissible cluster pairs form a block in the boundary matrix
that can be approximated as described in the following paragraph with low rank matrices.
The boundary matrix corresponding to the clustering of fig. 3.4 is shown in fig. 3.5. The
off-diagonal blocks of the matrix describe the far field interactions. Generally a matrix with
the rank r can be written in the factorized form (singular value decomposition)
M = AB =
T
r
aν bνT
,
(3.64)
ν=1
where A ∈ Rn×r , B ∈ Rm×r . Varies methods can be used for the low rank approximation, such as Taylor expansion, multipole expansion or interpolation. In the following we
will focus on a method called adaptive cross approximation (ACA) technique (Bebendorf,
2000; Kurz et al., 2002). The ACA technique allows to construct a approximation of the
78
D. SUESS et al.
Fig. 3.4. Example of the clustering of 10 collocation points. A large distance between two points correspond to
entries in the boundary matrix that are well separated from each other.
Fig. 3.5. After renumbering the matrix in the way depicted in fig. 3.4 the boundary matrix has the given form.
The large off-diagonal blocks can be approximated by low-rank matrices. (Figure taken from Kurz et al., 2002).
form M ≈
k
T
ν=1 a ν bν up to an error
k
T
M −
aν bν ≈ εM2 .
ν=1
(3.65)
2
Therefore the rank k of the approximated matrix depends on the error criterion. With the
ACA algorithm the low-rank matrix can be approximated as following:
A LGORITHM 1 Adaptive cross approximation.
for μ = 1 to k do
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
79
Fig. 3.6. Successive computation of the approximation of a matrix block A. For a row (column) the maximum
error between the exact matrix and approximated matrix is located, which determines the position of the next
pivot element.
μ−1
1. Determine in the error matrix Eij := Mij − ν=1 (aν )i (bν )j the entry (i ∗ , j ∗ )
with the largest value δ = E(i ∗ , j ∗ ). This can be approximately done with partial
pivoting as described in the next algorithm.
2. Stop if δ = 0.
3. Calculate the entries of the vectors a μ and bμ
(aμ )i := Mij ∗ −
μ−1
(aν )i (bν )j ∗ ,
ν=1
μ−1
1
(bμ )j :=
(aν )i ∗ (bν )j .
Mi ∗ j −
δ
ν=1
4. Stop if aμ 2 bμ 2 ≤ εa1 2 b1 2 .
end do
The calculation of the largest matrix element in the error matrix would be of the order
(n × m). In order to decrease the cost of the algorithm in the ACA method partial pivoting
is applied.
A LGORITHM 2 Partial pivoting at step μ.
∗
from the last step (in the first step chose a column at random).
1. Take jμ−1
μ−1
∗
∗ | in this column. The
− ν=1 (aν )i (bν )jμ−1
2. Look for the maximum value of |Mijμ−1
∗
row of the maximum element gives iμ .
μ−1
3. Look for the maximum value of |Miμ∗ j − ν=1 (aν )iμ∗ (bν )j | in the row iμ∗ . This
gives the column jμ∗ of the pivot element.
The pivot elements are calculated successively as shown in fig. 3.6. Starting from the previous pivot element the next element follows. Both the memory costs and the computation
cost are of the order O(m + n). The ACA technique efficiently computes the matrix vector
product of boundary value problems and is well suited for problems with a large number
of iteration steps. The efficiency of the ACA is compared with the fast multipole method
of Buchau et al. (2003).
3.6. Time integration
Using the Landau–Lifshitz–Gilbert (LLG) equation the time evolution of the magnetization
can be calculated. The treatment of systems with realistic size leads to a system of ordinary
80
D. SUESS et al.
differential equations with up to one million unknowns. State of the art time integration
schemes provide an efficient numerical solution of the equations.
Traditionally, explicit time integration methods are used in numerical micromagnetics.
These methods are easy to apply since only the right hand side of the Landau–Lifshitz–
Gilbert equation has to be evaluated. Victora (1987) applied a Runge–Kutta method to
solve the Landau–Lifshitz–Gilbert equation in an array of columnar CoNi particles. Different types of Adams formulas are commonly used in micromagnetics. Mansuripur (1988)
applied an Adams method to calculate the time evolution of thin film recording media. The
Adams method used by Zhu and Bertram (1989) is a varying order and varying step-size
method and is well described by Gear (1971). Jones and Miles (1997) used the LLG equation to simulate the magnetic behavior of a metal evaporated tape. They performed the
integration by using a variable-order variable-step Adams method. More specifically, they
used the NAG D02CHF routine. Tako et al. (1996) used the Adams 4th-order predictorcorrector method with error control and step size adjustment described by Burden and
Faires (1997). McMichael and Donahue (1997) use a second order predictor-corrector technique of the Adams type to calculate the dynamic response of magnetic nanoelements.
However, for highly exchange coupled systems or complex microstructures the Adams
methods and explicit time integration schemes require an intolerably small time step to
maintain numerical stability. These problems are supposed to be stiff and can be more
suitably solved with backward differentiation formula methods. Originally, Hayashi and
Nakatani (1986) applied the backward Euler method which is a backward differentiation
method (BDF) of order 1, to solve the Landau–Lifshitz equation for magnetic bubble domain wall motion. Albuquerque et al. (2001) treat the exchange term implicitly applying
the Crank Nicolson method. This scheme applies a feedback mechanism from monitoring
the damping coefficient, maximum torque and total energy to obtain high accuracy. E and
Wang (2000) developed a projection method. The key point of this method is that they
relax the condition that the magnitude of the magnetization vector remains constant. After
each time step the magnetization vector is projected back to the unit sphere. The proposed
method is unconditionally stable.
The above methods are used together with a finite difference method for space discretization. The finite element method allows irregular grids suitable for complex microstructures
and adaptive refinement. Toussaint et al. (1996) showed that the time step required to obtain a stable solution of the Landau–Lifshitz–Gilbert (LLG) equation with an explicit time
integration scheme has to be proportional to h2g , where hg is the size of the spatial grid.
Thus a fine mesh which is required to resolve magnetization inhomogeneities near edges or
grain boundaries, limits the time step of explicit methods. Backward differentiation methods (BDF methods) are more suitable. However, the linear system which has to be solved
at every time step as part of the Newton method is ill conditioned. Proper preconditioning
decreases the number of iterations when the system is solved iteratively and thus speeds up
the computation considerably.
Yang and Fredkin (1998) developed a numerical procedure to study the dynamic behavior in micromagnetic systems using the finite element method. They solved the damped
Gilbert equation for a continuous magnetic medium, including all the interactions in standard micromagnetic theory in 3D regions of arbitrary geometry and physical properties.
The magnetization is linearly interpolated in each tetrahedral element in a finite element
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
81
mesh from its value on the nodes. They use the Galerkin method to discretize the dynamic
equation. The demagnetizing field is computed by solving Poisson’s equation, and they
treat the external region by means of an asymptotic boundary condition. They apply the
CVODE code (Cohen and Hindmarsh, 1994, 1996) to solve the stiff system of ordinary
differential equations. CVODE provides the option to solve the equations either with the
Adams or a backward differentiation formula (BDF). In order to apply a preconditioner
for the linear system, CVODE requires an approximate Jacobian, which is the first derivative of the right hand side of the Landau–Lifshitz–Gilbert equation. Yang and co-workers
cannot supply the Jacobian explicitly because they use the Galerkin method for space discretization. However, they are able to give a suitable approximation.
In the following we use the finite element method to compute the dynamic response of
thin film elements of different shape and structure. Starting from the finite element discretization of the total energy, the effective field can be evaluated using the box method.
Each node has its magnetic moment and its effective field. Thus the right hand side of the
LLG equation and the Jacobian can be calculated explicitly. For the preconditioning only
an approximation of the Jacobian is needed. Thus it is possible to keep the approximated
Jacobian sparse, omitting the stray field part. The Adams method, the BDF method and the
preconditioned BDF method are compared for μMAG Problem #4 and a thin film element
with granular microstructure in the paper of Suess et al. (2002). The results show a considerable decrease of the CPU time for the preconditioned BDF method for all investigated
samples.
3.7. Time discretization
In the following simulation the CVODE code (Cohen and Hindmarsh, 1994, 1996) for solving the LLG equation is used. The relative performance of the Adams method, the BDF
method with and without preconditioning changes, depending on the microstructure, material parameters, and the finite element mesh. The Adams method seems to be attractive
because of the low cost per time step. In contrast to the one-step methods (e.g. Runge–
Kutta), multistep methods make use of the past values of the solution to approximate the
solution in the future. At each time step a non-linear system of equations has to be solved.
The Adams method solves the non-linear system with functional iteration and thus requires
only the evaluation of the right hand side of equation (2.68). However, if the problem is stiff
the convergence of the functional iterations is slow. For a stiff problem it is advisable to use
an implicit method such as BDF. The nonlinear system is solved using a Newton method.
Normally only a few Newton steps are required. Within CVODE the linear system for each
Newton-step is solved either with a direct solver or with a Krylov subspace method. Krylov
subspace methods have been explored in micromagnetics by Tsiantos (2000), Tsiantos et
al. (2001). The solution is approximated iteratively by a linear combination of the basis
vectors of the Krylov subspace. At each iteration step one orthonormal basis vector is
added which increases the subspace dimension by one. If the Krylov subspace dimension
is equal to the number of unknowns the exact solution is found. For practical applications a
very good approximation is obtained for a Krylov subspace dimension much smaller than
the number of unknowns. The default value for the maximum Krylov subspace dimension
in CVODE is 5. As discussed in the next section the computation time drastically decreases
when this value is increased to about 300. An additional parameter in the CVODE package
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D. SUESS et al.
Fig. 3.7. Equilibrium states during the demagnetization process for a Co-grain with granular structure. The film
thickness is 25 nm. The letters identify the states marked on the demagnetization curve in fig. 3.8.
is the maximum order of the time integration method. Especially for small damping constants, we found that a maximum order, qumax = 2, improves the stability of the solution.
Further details of the time integration of the Landau–Lifshitz–Gilbert equation using the
CVODE package are given in the appendix.
3.7.1. Comparison of different time integration schemes
As a test case to compare the efficiency of different time integration methods we have
calculated the hysteresis loop for a Co element with a rough surface. The surface can be
seen in fig. 3.7. To create a rough surface the element was built up of 8 nm wide columnar
grains. The basal planes of the grains are irregular, constructed from Voronoi cells (Schrefl
and Fidler, 1992). These grains lead to edge irregularities of the same size. Each grain
consists in average of 26 finite elements. The grain structure leads to an inhomogeneous
finite element grid. Since the time step for explicit time integration scheme has to be proportional to h2 , where h is the size of the spatial grid, an inhomogeneous grid causes very
small time steps. The investigated element is 400 nm long, 100 nm wide and 25 nm thick.
No uniaxial anisotropy is assumed. The spontaneous polarization Js = 1.76 T, and the exchange constant A = 1.3 × 10−11 J/m. The damping constant α was set to 0.1. Figure 3.7
shows magnetization states during the demagnetizing process. The angle between the external field and the long axis of the Co element is 1◦ . The letters in fig. 3.7 identify states
marked on the hysteresis loop in fig. 3.8. The nucleation of reversed domains starts at edge
irregularities.
For the comparison of the integration schemes we simulated the switching process under the action of an external field. The external field is applied 1◦ off the easy axis and
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
83
Fig. 3.8. Hysteresis loop for the granular Co-particle. The field step is 4 kA/m.
the field strength is 100 kA/m. The field is applied instantaneously to the remanent state.
After 0.76 ns the normalized magnetization parallel to the external field direction becomes
smaller than −0.99. Then we regard the particle to be switched.
Figure 3.9 shows the efficiency of different time integration schemes. The CPU time
is plotted as a function of the simulated time. It clearly shows that the Adams method
(A) is not suitable to solve the LLG equation for samples with granular structure. In all
simulations with the BDF method the maximum integration order, qumax = 2, was used.
The BDF method shows a very bad performance if the dimension of the Krylov subspace
is restricted (lmax = 5 or lmax = 15). The maximum dimension of the Krylov subspace
restricts the number of iterations to solve the linear system (equation (A.12)), which has to
be solved for every Newton step. The linear iteration stops if the desired accuracy or lmax is
reached. If lmax is large enough so that the number of iterations does not reach lmax , the ratio
nli/nni (nli = number of linear iterations, nni = number of non-linear iterations) gives the
average number of linear iterations per Newton step. Figure 3.9 shows that the efficiency
of the time integration scheme increases with increasing lmax . With higher dimensions
of the Krylov subspace, the linear system can be solved more accurately which in turn
enables a larger time step. The maximum number of Krylov iterations is 162. So a further
increase of lmax > 162 has no influence on the simulation. It is interesting to note that the
CPU time decreases when the absolute tolerance of the time integration scheme abstol is
enhanced from 10−4 (curve D in fig. 3.9) to 10−5 (E). Smaller tolerances keep the norm
of the magnetization vector constant for longer time periods. Thus, less renormalization
steps are needed which also require a restart of the time integration algorithm. Restarting
the BDF method requires some precomputation steps like factorizing the matrices for the
linear systems. In addition the initial time step is small.
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D. SUESS et al.
Fig. 3.9. CPU time (Digital EV6—523 MHz) versus the simulated time for different time integration schemes for
a Co element with grain structure. (A) Adams method, (B–D) BDF method without preconditioning for different
dimensions of the Krylov subspace. (B) Krylov subspace dimension lmax = 5, (C) lmax = 15, (D) lmax = 400,
(E) lmax = 400 and high accuracy (abstol = 10−5 ). (F) BDF-method where an approximate Jacobian matrix is
provided analytically and lmax = 15.
The curve marked with (F) in fig. 3.9 shows that the CPU time is drastically decreased
if an approximation of the Jacobian matrix is provided. Then the linear system in each
Newton step can be solved more efficiently (see appendix).
3.8. Time integration at finite temperature
The previous sections showed the finite element treatment of micromagnetic dynamics
for the zero temperature limit. Powerful time integration methods can be applied since
we are dealing with an ordinary differential equation. The time is divided into a lattice
of discrete points tn . For micromagnetic simulations at finite temperatures the stochastic
Landau–Lifshitz–Gilbert equation (2.71) has to be solved. This was done numerically for
short time scales and small systems by Garcia-Palacios and Lazaro (1998), Zhang and
Fredkin (1999), Brown (1963a), and Scholz et al. (2000a, 2000b). Recently, numerical
solutions of the stochastic Landau–Lifshitz–Gilbert equation were reported for extended
micromagnetic systems (Zhu and Bertram, 1988, 1989; Scholz et al., 2000a, 2000b). Zhu
and Bertram analyzed the magnetization noise in submicron sized sensor elements based
on the numerical solution of the stochastic Landau–Lifshitz–Gilbert equation.
The Langevin equation is solved in the time interval tn to tn+1 with the initial condition
uli = uli (tn ). The right hand side of equation (2.72) is evaluated at the midpoint t¯n =
(tn + tn+1 )/2 = tn + t/2. The update of the direction cosines is given by
l+1
l
uli = Ali . . . , ūl−1
i , ūi , ūi , . . . t
l+1
l
l
l
+
. . . , ūl−1
Bik
i , ūi , ūi , . . . Hth,k
k
(3.66)
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
85
with the noise integral
l
Hth,k
=
tn+1
l
Hth,k
(t)dt.
(3.67)
tn
The magnetization direction at the midpoint of the time interval is
ūli = uli (tn ) + uli (tn+1 ) /2 = uli (tn ) + u/2,
(3.68)
assuming linearization of uli within the time interval. The midpoint value, ūli , is given
implicitly by the equations (3.66) and (3.68) which is solved by functional iteration. From
the midpoint value the magnetization at the time tn+1 is calculated from (3.68) and then
normalized:
ũli (tn+1 ) = 2ūli − uli (tn ),
ul = ũ/|ũ|.
(3.69)
The time integration of the stochastic Landau–Lifshitz–Gilbert equation is restricted to
short times and is thus a proper tool to analyze magnetization noise. Various methods
have been developed to speed up the computation and to overcome this problem (Xue and
Victora, 2000; Voter et al., 2002).
In computation micromagnetics, fluctuations of the magnetization at a spatial length
scale below the computation cell size cannot be taken into account. Various micromagnetic
simulations have shown that the straight forward solution of the stochastic Landau–Lifshitz
equation leads to wrong numerical results even at temperatures far below the Curie temperature TC . Various authors suggest to correct the system parameters like the magnetization
and the anisotropy constant depending on the computational cell size. Dobrovitsky et al.
(2000, 2003) and Grinstein and Koch (2003) suggested two different methods of statistical
coarse-graining and renormalization for magnetic systems. Contrary to their mathematical
treatment, Kirschner and coworkers propose an approximate and straight forward coarse
graining procedure (Kirschner et al., 2005). In a simulation where each atom corresponds
to a computational cell the zero temperature value of MS = MS,0 shall be used. As the cell
size is increased, the magnetization of a computational cell represents an average over the
atomic moments within the cell. Owing to thermal fluctuations the atomic moments within
a cell are not parallel to each other. Therefore the magnetization Ms,cell assigned to a cell
should be lower than MS,0 . For increasing cell size MS,cell will approach the experimentally measured value of Ms at a given temperature, MS,∞ (T ). Kirschner et al. computed
Ms,cell for a given temperature with Metropolis Monte Carlo simulations. The magnetic
moments are summed up over a cube of size x which is placed within the computational
domain of an atomistic Monte Carlo simulation (see fig. 3.10).
n3
MS,0 Mα = 3
mi,α ,
n
(3.70)
i=1
MS,cell =
3
α=1
1/2
Mα2
.
(3.71)
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D. SUESS et al.
Fig. 3.10. Schematic of the applied coarse graining procedure: Atomistic Monte Carlo simulations (left hand
side) yield the cell size dependent magnetization MS,cell as a function of the size of the averaging cube. MS,cell
is then used for non-atomistic simulations (right-hand side).
This procedure leads to the following Bloch-like scaling law for the magnetization
a 3/2
.
MS,cell (x, T ) = MS,∞ (T ) + (MS,0 − MS,∞ (T ))
(3.72)
x
In order to test this scaling law, the average magnetization parallel to an applied field was
calculated for different computational cell sizes. An external field of 0.1 T is applied parallel to the z-axis for a system with zero anisotropy. If the scaling works properly, Mz should
not depend on the cell size. Figure 3.11 shows that Mz decreases with decreasing computational cell size if the experimentally measured Ms values are used independent of the
cell size. The change of Mz with the computational cell size is less significant if the scaling law equation (3.72) is applied. The deviation of 3.2% at 1.5 nm for a constant Ms can
be reduced to 0.9% using cell size dependent MS,cell (x, T ). In addition fig. 3.11 compares results obtained with from the numerical solution of the stochastic Landau–Lifshitz
equation and Metropolis Monte Carlo simulations.
3.9. Calculating energy barriers
The previous section showed the numerical method for the simulation of thermal effects
on short time scales. However, transition events over large energy barriers (> 50 kB T )
as in magnetic recording media occur on very long time scales (e.g. decades). It is clear
that such events cannot be simulated directly by stochastic time integration methods. To
simulate such events we directly search for the most probable path through the energy
landscape for such an event.
The calculation of the energy barrier basically requires the calculation of the relevant
saddle point which is in general a sophisticated problem. Depending on whether the final
state is known or not, different methods were proposed.
Most schemes for finding the saddle point and the final configuration start from the
initial state and then trace stepwise a path of slowest ascent (Cerjan and Miller, 1981;
Quapp, 1996). However, these methods find paths that do not necessarily pass through the
saddle point.
If the initial and final states are given, more reliable methods can be applied. Berkov
(1998) calculated the transition path of interacting single domain particles by minimizing
the action along the path. He showed that a direct minimization of the action may also
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
87
Fig. 3.11. Mean value of the magnetization parallel to the field axis as a function of cell size x, resulting
from non-atomistic Monte Carlo (MC) and Landau–Lifshitz–Gilbert (LLG) simulations. No cell size corrections
were used for 1 (MC) and + (LLG). Simulations with cell size dependent MS; cell according to equation (3.72)
resulted in (MC) and × (LLG). Then, the error is less than 1% at 1.5 nm. (Kirschner et al., 2005).
give paths through local maxima which have to be excluded. Ren and Vanden-Eijnden
(2002) proposed the “string method”, a slightly modified version of the elastic band method
(Henkelman and Jónsson, 2000), to calculate a minimum energy path in micromagnetic
systems. Starting from an initial guess for the path which connects two local minima of
the system, a highly probable path is found by moving the points along the path according
to an algorithm which resembles tensioning an elastic band across a mountain. Variants of
elastic band methods are commonly used to calculate transition rates in chemical physics
(Henkelman et al., 2000a; Maragakis et al., 2002). Once the energy barrier is calculated,
the Néel–Brown theory can be used to calculate the transition probability for the energy
barrier crossing.
A common feature of path finding methods is the discrete representation of the path
connecting the initial state of the system with its final state. First we construct a sequence
of magnetic states from the initial magnetization state, M(i) , to the final magnetization
state, M(f ) . An optimization algorithm is then applied until at any point along the path the
gradient of the energy is only pointing along the path. This path is called minimum energy
path which means that the energy is stationary for any degree of freedom perpendicular to
the path. The minimum energy path typically represents the path with the greatest statistical
weight. From this path statistical quantities as the transition rate for the thermally induced
magnetization reversal can be estimated.
Various other methods exist which can calculate either the saddle points (transition state)
or the entire minimum energy path for the thermally activated barrier crossing. Most of
these methods come from chemical physics and condensed matter physics where thermally activated barrier crossing is of great interest for the calculation of reaction rates.
A nice overview of methods for finding saddle points and minimum energy paths in highdimensional systems is given by Henkelman et al. (2000b).
88
D. SUESS et al.
Fig. 3.12. Schematic representation of the path, which shows how Dk in equation (3.73) is calculated. A number
of images is chosen to discretize the path connecting the two states A and B.
Henkelman and Jónsson (2000) proposed the nudged elastic band method to calculate
minimum energy paths. The nudged elastic band method was successfully applied to micromagnetics by Dittrich et al. (2002a). Using the nudged elastic band method together
with the finite element method, energy barriers for complex micromagnetic systems such
as granular recording media can be calculated.
A path of the nudged elastic band method is represented by a sequence of images. An
initial path is assumed which connects the initial magnetization state M(i) = M(1) with the
final magnetization state M(f ) = M(m) . The index k runs from 1 to m. The path is optimal,
if for any image M(k) the gradient of the energy is only pointing along the path or in other
words the component of the energy gradient normal to the path, D, is zero. If t denotes the
unit tangent vector along the path, a minimum energy path has the following property
D(k) = ∇E(M(k) ) − [∇E(M(k) ) · t]t = 0,
for k = 1, . . . , m.
(3.73)
The optimal path can be found using an iterative scheme. In each iteration step the images
move towards lower energy in a direction perpendicular to the path. So the image M(k) is
moved into the direction, −D(k) . Figure 3.12 shows the calculation of Dk .
In order to keep an equal distance between successive images a spring force may be
introduced (Henkelman and Jónsson, 2000). A more detailed description of the method
and the implementation can be found in the paper of Dittrich et al. (2002a).
Figure 3.13 compares the transition between two stable minimum energy configurations
of a 150 × 100 × 5 nm3 thin film element (Dittrich et al., 2002b) calculated with stochastic
time integration and the elastic band method. In the sequence on the top the minimum
energy path is shown for the reversal between two S-states. The bottom sequence shows
the dynamic thermal switching between a C-state and a S-state. With both methods one
obtains the same transition state.
3.10. Micromagnetic codes
Various groups have established micromagnetic codes. Software available under the GNU
General Public License are MagPar developed by Scholz et al. (2003) and OOMMF developed by Donahue and Porter (1999). Both packages can be downloaded from the web
pages given in the references above.
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
89
Fig. 3.13. Thermal switching between the S-state and the C-state in NiFe platelets. The element size is
50 × 100 × 5 nm3 . Top: magnetization states at 300 K in a NiFe platelet at zero applied field. Bottom, left:
magnetization configurations along the minimum energy path: S-state, saddle point (not shown), C-state, saddle
point, inverted S-state. Bottom right: energy along the minimum energy path.
In the following a short overview of the program flow of the MagPar package is given. At
the beginning a finite element mesh of the desired geometry has to be created with adequate
software. Commonly used commercial software packages for the design of the geometry
and creating the finite element mesh are for example Patran2 and Gid3 . After discretizing
the geometry the initial magnetization has to be chosen. The following material parameters
have to be set for every grain:
• direction of the easy axes,
• type of magnetocrystalline anisotropy (uniaxial, cubic),
• magnetic anisotropy constants K1 and K2 ,
• saturation polarization Js ,
• exchange constant A, and
• damping constant α.
As shown in fig. 3.14 the simulation process starts up with reading the FE mesh and the
material parameters. If the hybrid finite element/boundary element method is used for the
calculation of the stray field, the boundary matrix which only depends on the geometry of
the system has to be calculated. This step can be skipped by using fast integration methods
such as the treecode or hierarchical matrices.
2 http://www.mscsoftware.com/products/products_detail.cfm?PI=6
3 http://gid.cimne.upc.es/
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D. SUESS et al.
Fig. 3.14. Schematic of the simulation cycle of a typical finite element micromagnetic package such as MAGPAR
(Speckmayer, 2003).
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
91
In the presented package for the time integration of the LLG equation the solver
“CVODE” is used (Cohen and Hindmarsh, 1994), which is distributed under the GNU
public license. Its capacity is solving initial-value problems (IVPs) for systems of ordinary
differential equations (ODE). Such as
∂Ji
= fi (t, J1 , J1 , . . . , JN−1 , JN ),
∂t
(3.74)
Ji (t0 ) = Ji,0 .
(3.75)
Equation (3.74) represents a system of ordinary differential equations and equation (3.75)
their initial conditions at some time t0 . In the case of micromagnetism the initial conditions
represent a desired magnetic state at the beginning of the simulation. J1 represents the x
component of the magnetization on the first node, J2 the y-component, J3 the z-component
and J4 the x-component of the magnetization of the second node and so on. For every
time step the right-hand side of equation (3.74) has to be calculated, which represents the
right-hand side of the Landau–Lifshitz–Gilbert equation (2.68). This is done by the routine,
which we will call in the following LLG calculation routine (LLG-CR). Within this routine
the following steps are executed:
• The solver posts a calculation request for a set of magnetic polarizations J1 , J2 , . . . , JN .
• The LLG-CR uses the FE method to determine Hani and Hex with equation (3.9), leaving
out the stray field term of the total energy.
• The stray field with the hybrid FE/BE method described in section 3.3.2.1 is calculated
(equations (3.24)–(3.33)).
• The LLG-CR superposes the acquired magnetic fields for each of the nodes of the FE
mesh.
• Using the set of magnetic polarizations J1 , J2 , . . . , JN given by the solver and the set of
magnetic fields H1 , H2 , . . . , HN the LLG-CR computes the right-hand side of the LLG
equation (2.68) considering the damping constant.
• The calculated results f1 , f2 , . . . , fN are transmitted to the solver which performs one
time step.
• These points are repeated until the final time is reached.
4. Applied micromagnetics
4.1. Perpendicular and tilted media for magnetic recording
A fundamental understanding for the underlying physical processes in magnetic recording
media is a key factor for further optimization. As the data rate reaches the GHz regime
precessional effects of the magnetization and ringing effects have to be understood and if
possible used to increase the data rate. In order to take care of realistic geometries, the
theory of micromagnetism is a suitable tool. In recent years there has been a renewed interest in perpendicular recording since an improvement of the areal density in longitudinal
recording is getting increasingly difficult. For longitudinal recording the demagnetization
field increases with increasing bit density, leading to a reduction of thermal stability. In
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D. SUESS et al.
contrast, the demagnetizing fields for perpendicular recording continuously reduces with
bit density.
Another advantage of perpendicular recording is that the areal density can be increased
without reducing the volume of the magnetic bit. A large volume of the magnetic domain
can be realized with an increased column length lc (film thickness). For coherent rotation
a higher grain volume leads to a higher energy barrier and hence to an improved thermal
stability. However as the column length is increased further the reversal mode changes from
quasi-coherent rotation to nucleation. As a consequence for thick films the energy barrier
does not depend to column length. The transition from rotation to nucleation does not
occur at a critical field but it is a smooth transition (Suess et al., 2001). We have performed
micromagnetic simulations to investigate the reversal modes in perpendicular recording
media in detail. The simulations at zero temperature are based on the integration of the
LLG equation. The energy barrier between two stable magnetization states are calculated
using the nudged elastic band method.
4.1.1. Reversal modes
The reversal process is calculated for one Co-Cr grain (Js = 0.5 T, A = 10−11 J/m,
K1 = 3 × 105 J/m3 ). The basal plane of the grain is an irregular pentagon with a diameter
of about 12 nm. The easy axis is perpendicular to the basal plane. The external field is
always applied 1◦ off the easy axis. The average element size of the finite element mesh is
2 nm, small enough to resolve domain walls which have a typical width of
dw = A/K1 ≈ 6 nm.
(4.1)
If the external field reaches the coercive field Hc , which increases linearly from 0.97
(2K1 /Js ) for lc = 20 nm to 1.03 (2K1 /Js ) for lc = 70 nm, the magnetization state becomes unstable and reversal starts. Figure 4.1 shows the maximum value of the exchange
energy during reversal for different column lengths. The exchange energy measures the
uniformity of the magnetization during reversal. The exchange energy remains zero if the
reversal mode is homogeneous rotation. It increases as the reversal mode becomes more
inhomogeneous. The exchange energy is normalized to the domain wall energy.
Ew = 4Aend AK1 .
(4.2)
Aend measures the area of the basal plane. It can be seen that nucleation does not occur at a
critical length, but the transition is smooth from coherent rotation to nucleation. Coherent
rotation dominates below 40 nm. Figure 4.2 shows non-equilibrium states of the magnetization during reversal for different column lengths at a time when Jz = 0. With increasing
column length (> 40 nm) the magnetization at the top and bottom of the grain becomes
increasingly misaligned to the magnetization in the middle. Above 80 nm the magnetization at the top and bottom points antiparallel to the magnetization in the middle, and the
magnetic exchange energy reaches its maximum. In this regime a domain wall is formed
on both ends. The maximum exchange energy during reversal is equal to the domain wall
energy Ew .
The transition from coherent rotation to nucleation is shifted to smaller column lengths
when the damping constant is reduced to α = 0.02. The magnetization becomes already
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
93
Fig. 4.1. Maximum value of the exchange energy during reversal as a function of the column length for two
different damping constants. A value of the exchange energy close to the domain wall energy Ew indicates
nucleation processes at the particle ends.
incoherent during reversal for lc > 20 nm. The transition from coherent rotation to nucleation is represented by the solid line in fig. 4.1. For a low damping constant the precession
term in the Landau–Lifshitz–Gilbert equation becomes important. Owing to gyromagnetic
precession a domain wall that is formed does not propagate with constant velocity through
the grain. An almost constant propagation of the domain wall can be observed for α = 1.
Because the magnetization in the center of the domain wall points perpendicular to the
long axis, a stray field perpendicular to the long axis occurs as shown in fig. 4.3. The magnetization starts to rotate around the stray field owing to gyromagnetic precession. As a
consequence the domain wall oscillates leading to oscillations in the z-component (easy
axis direction of the grain) of the magnetization as a function of time.
4.1.2. Energy barriers and thermally activated reversal modes
In this chapter the energy barriers between the state with magnetization up (initial state)
and the state with magnetization down (final state) are calculated using the nudged elastic
band method (Dittrich et al., 2002a). The column length is varied from 10 nm to 70 nm.
We start with a straight line in polar coordinates for the initial path. The angle of the magnetization and the long axis of the particle increases by a constant step from one image to
the next. It was found that above a certain critical length, lc , of the particle the minimum
energy path changes from coherent rotation to nucleation followed by domain wall motion
until the whole magnetization is reversed (see fig. 4.5). This can also be understood by
an analytical estimate. In the following, F denotes the
√ area of the basal plane of the elongated grain. The domain wall energy is Ewall = 4F AK1 while the energy for coherent
rotation is Erot = K1 V . Now if the particle length is increased, only V increases while F
remains constant. Thus, there will be a critical length where Ewall < Erot and the nucleation and expansion of a domain wall becomes energetically favorable. With A = 10 pJ/m
94
D. SUESS et al.
Fig. 4.2. Non-equilibrium states at Jz = 0 during reversal for different column lengths after the coercive field
was applied. The damping constant α = 1. The z-component of the magnetization is maped on the grey scale.
Fig. 4.3. When the magnetization rotates out of the long axis, a high stray field occurs. Owing to gyromagnetic
precession the magnetization rotates around the stray field direction.
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95
Fig. 4.4. Maximum exchange energy and effective volume for the thermal reversal of an elongated CoCr particle
as a function of the column length. The diameter is 13 nm. Above a size of 20 nm the reversal mode becomes inhomogeneous. Above ∼ 50 nm column length, the size is large enough to form a full domain wall. The maximum
exchange energy saturates as function of the length and takes the value of a domain wall exchange energy.
√
1)
and K = 300 kJ/m3 the critical length becomes lcrit = (4FK1 lexAK
= 19 nm. lex denotes the
F
length of the particle. This agrees very well with the numerical results. Figure 4.4 shows
the maximum exchange energy and the effective volume for the thermal reversal of an
elongated CoCr particle as a function of the column length. The diameter is 13 nm. Above
a size of 20 nm the reversal mode becomes inhomogeneous. Below 20 nm the maximum
exchange energy is almost zero. Here the particle reverses by coherent rotation. Above
40 nm the particle is large enough to form a full domain wall. A further increase of the
length does not change the maximum exchange energy any more. Once the wall is formed
it only moves through the particle. The same is true for the energy barrier, here expressed
as an effective switching volume. Above 40 nm the effective volume Veff saturates. It is
therefore clear that a further increase of the column length of the grains does not increase
the thermal stability in perpendicular recording material.
Figure 4.5 shows images along the initial path and the minimum energy path for a particle with a length l > lc . Clearly, the nudged elastic band method detects the minimum
energy path, which is given by the nucleation of a reversed domain at one end of the
particle. The wall moves through the particle, and the reversed domain expands. A nucleation of reversed domains at both ends would require twice the wall energy, Ewall , and
this is no minimum energy path. Forster and coworkers investigated this problem in detail
(Forster et al., 2003a, 2003b). For particles larger than the exchange length these barriers
represent non-uniform paths between the two approximately uniform equilibrium states.
Both the zero field energy barrier and the intrinsic switching field were determined over
a wide range of particle exchange and magnetocrystalline anisotropy parameters. Besides
the above mentioned path which involves a Bloch-type domain wall motion, also vortex
nucleations can occur when the anisotropy is smaller (fig. 4.6). Using the form of the energy barrier for a uniformly reversing particle an effective reversal volume was determined.
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D. SUESS et al.
Fig. 4.5. Magnetization states along the initial path and the minimum energy path for an elongated CoCr particle
with the easy axis parallel to the long axis. Grey: magnetization points up; black: magnetization points down.
This volume decreased with both increasing anisotropy and decreasing exchange. A simple
model by Forster et al. (2003a, 2003b) semi-quantitatively explained these effects corresponding to reversal by end nucleation of a cubic region and a domain wall.
4.1.3. Switching times
In the previous section the thermal stability which is important for the improvement of
the areal density was investigated. In addition to the areal density of magnetic recording
media the data rate becomes increasingly important. As the data rate approaches the GHz
regime a profound knowledge of the switching process is a prerequisite for further optimization. Kikuchi (1956) investigated the reversal time of a single domain sphere and a
single domain thin film. A minimum reversal time for a damping constant of α = 1 and
α = 0.01 for the sphere and the thin film was reported, respectively. He and Doyle (1996)
solved the Landau–Lifshitz equation numerically, in order to investigate switching with
very short field pulses. They conclude that switching times in the order of about 100 picoseconds are possible if the external field is applied 90◦ with respect to the anisotropy
axis. However, to guarantee switching, the pulse width has to be controlled extremely accurately. For too short or too long field pulses the particle may switch back to the initial
state. Recent experimental studies confirm that the conditions for deterministic precessional switching are hard to control (Tudosa et al., 2004).
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
97
Fig. 4.6. Minimum energy path for an elongated particle with zero anisotropy. A = 10 pJ/m, Js = 0.5 T. The
reversal starts with the nucleation of a vortex on one end. The diameter of the particle is ∼ 13 nm, the height is
70 nm (Forster et al., 2003a, 2003b).
He et al. (1996) found that if the rise time of the field pulse is less than a few nanosecond
and the Gilbert damping constant α < 1, switching can occur well below the Stoner–
Wohlfarth limit. Mallinson (1981, 2000) derived the switching time as a function of the
field strength for fields parallel to the anisotropy direction. The switching time decreases
with increasing external field. As the external field is applied at an angle of about 45◦
with respect to the anisotropy axis a higher torque acts on the magnetization. Therefore,
switching can be accelerated.
Tilted media are designed in a way that a high torque acts on the magnetization. Tilted
media consist of grains with an angle of the easy axes and the film plane of optimally 45◦ .
Therefore the switching time in tilted media is about 10 times smaller than in perpendicular
media.
Another advantage of tilted media compared to perpendicular recording is that the coercive field can be reduced by a factor of 2 as it can be derived by the Stoner–Wohlfarth
theory. As a consequence high anisotropy magnetic materials can be utilized leading to a
better thermal stability which in turn leads to a higher areal density.
Furthermore, in tilted media a high signal-to-noise ratio (SNR) can be achieved since
grains with slightly different easy axes have the same switching field in first order. The high
sensitivity of the switching field (and hence also on the SNR) on the easy axes distribution
is one of the main physical concerns in perpendicular recording.
In the following chapter magnetization dynamics of one grain of a tilted media is investigated. The presented simple model will be extended to a more realistic media, where the
exchange field and stray field is taken into account at the end of this chapter.
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D. SUESS et al.
4.1.3.1. Single domain particle The size of a magnetic particle, the material parameters
and the damping constant determine the reversal mode of magnetic particles. In this section
we will deal with a grain small enough to assume that the magnetization in the particle
reverses homogeneously. Thus the magnetization of the particle can be described with
only one single spin vector. A detailed study about the reversal mode of a perpendicular
recording media can be found in the paper of Suess et al. (2001).
The material parameter of the investigated grain are that of Co-Cr-Pt (Js = 0.5 T, A =
10−11 J/m, K1 = 3×105 J/m3 ) with a damping constant of α = 0.02. In all simulations the
angle between the easy axis and the external field is 45◦ . We will start our investigations
with a definition of the switching time.
We define the theoretical switching time as the minimum length of a field pulse to switch
the particle. However, for longer field pulses the magnetization may precess back and no
switching occurs. Slightly different parameters (e.g. in neighboring grains) may require
different field strengths and field pulses to guarantee switching. Therefore we call it theoretical switching time, since in practice it will be extremely difficult to tune the field pulse
to guarantee switching. This is confirmed by recent experiments using ultra short magnetic field pulses. It has been shown that precessional switching does not take place at
well-defined field strengths (Tudosa et al., 2004).
In our point of view a suitable definition of a practical switching time is the critical
duration of the field pulse, tp , so that all field pulses longer than tp will switch the particle.
Figure 4.7 shows the time evolution of the average magnetization parallel to the z-axis. The
magnetization oscillates owing to gyromagnetic precession. This effect is more pronounced
with larger angles between the external field and the easy axis. The arrows in fig. 4.7 show
the theoretical switching time tth and the practical switching time tp .
Figure 4.8 shows the switching time tp as a function of the strength of the external field
for different rise times, tr , of the external field. For t < tr the external field increases
linearly as function of time until the field strength reaches its maximum value at t = tr and
then stays constant.
Depending on the rise time of the external field we obtain different dependencies of
the switching time as a function of the external field strength. For field rise times larger
than 50 ps we observe the expected decrease of the switching time with increasing field
strength. Furthermore the simulations show a linear relationship between the switching
time and field rise time for tr > 50 ps.
The situation completely changes when the field rise time becomes smaller than tr <
50 ps. In that regime precessional effects dominate the switching behavior, and three interesting effects can be observed:
• Switching well below the Stoner–Wohlfarth limit is possible (He and Doyle, 1996).
• The switching time increases with shorter field rise times.
• The switching time does not decrease with increasing external field strength monotonically, but shows a maximum slightly above the Stoner–Wohlfarth switching field (Suess
et al., 2001; Gao et al., 2002; Uesaka et al., 2002).
The path the system follows through the energy landscape may elucidate these effects.
Figures 4.9 and 4.10 show the path of the magnetization through the energy landscape and
the locus of the magnetization just after the application of the field H ext = 0.9 (2K1 /Js )
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
99
Fig. 4.7. The average magnetization parallel to the long-axis as a function of time for an angle of 1◦ between the
applied field and the easy axis of a columnar grain of a CoCrPt perpendicular recording media. The theoretical
switching time tth and the practical switching time tp are marked.
Fig. 4.8. The switching time tp as a function of field strength for different rise times (from 0 ps to 100 ps) of the
external field for a single spin. The field is applied at 45◦ with respect to the easy axis.
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D. SUESS et al.
Fig. 4.9. Magnetization reversal in a single grain of a perpendicular media when an external field of 1.3 (2K1 /Js )
is applied at an angle of 1◦ with respect to the easy axis. The graphs show the energy landscape and paths of the
magnetic polarization toward the switched state. (A) and (B) show the energy landscape and the contour plot in
terms of its normalized magnetic polarization, respectively. (C) and (D): Locus of the polarization as a function
of time. The circles are plotted every 0.005 ns. (D) shows only the first 0.15 ns after applying the external field.
and H ext = 1.3 (2K1 /Js ), respectively. The angle between the external field and the easy
axis, β, is one degree. For β = 45◦ the switching times are smaller by about a factor of 10
than the switching times obtained for β = 1◦ .
Before the external field is applied, the system is in equilibrium at Jx = Jy = 0, marked
with dark dot in (A) and (B) in figs. 4.9 and 4.10. When a field of 1.3 (2K1 /Js ) is applied
the energy landscape suddenly changes. The system is no longer in equilibrium. A well
defined maximum is formed due to the Zeeman energy which contributes most to the total
energy. The current state is close to the maximum. Due to the precession term in the LLG
equation, the magnetization moves along a path with almost constant energy as shown in
fig. 4.9D with the bolded black line. If the damping constant is small the magnetization
will initially follow this path, indicated by the bold line in figs. 4.9A and 4.9B. During
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
101
Fig. 4.10. Magnetization reversal in a single grain of a perpendicular media when an external field of 0.9
(2K1 /Js ) is applied at an angle of 1◦ with respect to easy axis. The graphs show the energy landscape and
paths of the magnetic polarization toward the switched state. (A) and (B) show the energy landscape and the
contour plot in terms of its normalized magnetic polarization, respectively. (C) and (D): Locus of the polarization
as a function of time. The circles are plotted every 0.005 ns. (D) shows only the first 0.5 ns after applying the
external field.
this motion the angle between the magnetization and the effective field remains small,
and the magnetization precesses several times around the effective field, with Jz almost 1
(fig. 4.9C). Since the torque J × H is small for Jz almost 1, the relaxation towards the
reversed state is slow. A large switching time occurs. Only for fields exceeding 1.1 times
the anisotropy field the switching time decreases. This is due to the linear increase of the
torque with the strength of the external field.
If the external field is comparable with the anisotropy field a more complex energy
landscape is obtained, because anisotropy energy and Zeeman energy contribute in the
same order of magnitude to the total energy. The bold line in the contour plot in fig. 4.10B
gives the path of the magnetization for an external field of 0.9 (2K1 /Js ). The precession
drives the magnetization away from the effective field (fig. 4.10D) which leads to a high
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D. SUESS et al.
torque. Fast switching occurs when the magnetization does not relax back to the local
minimum as shown in fig. 4.10A but moves towards the local minimum which corresponds
to the reversed state. For an external field H ext ≥ 0.89 (2K1 /Js ) the initial motion of the
magnetization along a path with nearly constant energy leads to magnetization reversal. In
the highly damped case the magnetization would relax to the minimum that is still present,
and the magnetization will not switch. However in the low damping case as considered in
the simulation the magnetization precesses around the maximum. The particle can switch
well below the Stoner–Wohlfarth limit.
4.1.3.2. Granular media The calculations in the last section were performed for a very
idealized system. In the following we will extent the single domain particle to an exchange
and magnetostatically coupled granular media. Both, perpendicular and tilted recording
media with a weak exchange coupling between the grains, Ainter = 3 × 10−14 J/m, are
investigated. The material parameter within the grains are: Js = 0.5 T, A = 10−11 J/m,
K1 = 3.5 × 105 J/m3 . A damping constant of α = 0.02 is used. The thickness of the perpendicular media and the tilted media is 12 nm and 8.5 nm, respectively. The average grain
diameter is 10 nm for both media. More than 45 000 finite elements are used to provide a
finite element edge length smaller than the exchange length. A normal distribution of the
easy axes directions with a maximum opening angle of 5◦ is assumed for both media. For
the perpendicular media the averaged easy axis direction is parallel to the film normal. For
the tilted media the angle between the averaged easy axis direction and the film normal is
45◦ . Thus, for both samples the averaged easy axis direction points parallel to the edges
between the grains. The external field is applied 45◦ off the film normal and parallel to
the film normal for the perpendicular media and the tilted media, respectively. Thus the
averaged angle between the external field and the easy axis is 45◦ .
Figure 4.11 gives the switching time as a function of the field strength for the tilted media. The overall behavior is very similar to that of the single domain particle as depicted
in fig. 4.8. Thus the weak exchange interaction and stray field interaction does not significantly influence the switching process of the grains, the film is composed of. Figure 4.12
shows transient states after the application of the external field. The magnetization parallel
to the averaged easy axis is color coded. During the reversal process the magnetization
becomes inhomogeneous. Domain walls are formed between the grains. The magnetization within one grain remains almost homogeneous. Thus it seems reasonable to use a
macro-spin model, where the magnetization within one grain is described with a single
spin vector. The comparison of the switching time obtained from the full micromagnetic
approach and the macro-model showed that for external fields larger than the anisotropy
field the difference is smaller than 1%. However, for small external fields the system is
very sensitive to any changes of the conditions because of the complex energy landscape
that is formed. For small field strengths the switching times obtained from the macromodel and the finite element model differ by almost 50%. Hence, to guarantee accurate
results it is necessary to resolve the magnetization configuration within every grain of
a recording media even for grains with lateral dimensions in the order of the exchange
length.
Figure 4.13 compares the time evolution of the magnetization parallel to the average
easy axis for an external field of Hext = 0.5Ha and Hext = 0.8Ha . The field rise time
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
103
Fig. 4.11. The switching time as a function of field strengths for different rise times of the external field for a
tilted media.
Fig. 4.12. Transient states after application of an external field of Hext = 0.68 × Ha . The component of the
magnetization parallel to the long axes (45◦ degree with respect to the film normal) is color coded.
104
D. SUESS et al.
Fig. 4.13. Comparison of the time evolution of the magnetization parallel to the average easy axes of the tilted
media for two different field strengths. The field rise time is 10 ps.
is 10 ps. For Hext = 0.5Ha ringing of the magnetization can be reduced, leading to a
switching time that is about 4 times faster than for Hext = 0.8Ha .
The switching time of the tilted media in comparison with that of the perpendicular media is shown in fig. 4.14. For both media the angle between the averaged easy axis and
external field is 45◦ . So for the perpendicular media the field is applied at an angle of 45◦
with respect to the film normal. Similar to the calculation of a single spin the switching
time decreases with smaller magnitude of the external field. Owing to the difference in the
demagnetization fields for the two media, the switching times of the two investigated samples significantly differ. The demagnetizing field of perpendicular media is larger than that
of the tilted media. Due to the reversed behavior of the switching time with field strength
in the investigated field range (switching time increases with increasing field strength, see
fig. 4.11), the larger effective field of the perpendicular media leads to longer switching
times as compared to the tilted media. For external fields larger than the anisotropy field
the difference in the demagnetizing field of the two samples can be neglected, resulting in
almost identical switching times.
The finite element simulations also confirm that the switching time and the switching
fields are insensitive to small variations of the easy axes distribution for both samples.
A change of the opening angle of the easy axes distribution from 5◦ to 1◦ changes the
switching time by less than 0.1%. The small distribution of the switching field is an important factor to obtain high signal to noise ratios.
4.1.4. Conclusions
The investigation of the reversal mode of grains of perpendicular recording media show
that the exchange length is insufficient to determine whether switching occurs coherently
or inhomogeneously. For fixed exchange constant, anisotropy constant, and geometry the
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
105
Fig. 4.14. Comparison of the switching time for a perpendicular recording media and a tilted media. For the
simulation of the perpendicular media the angle between the external field and the film normal is 45◦ . The
external field is applied parallel to the film normal for the tilted media at an angle of 45◦ with respect to the film
plane for the perpendicular media. The external field is given in units of the anisotropy field.
reversal mode may change depending on the damping constant. For a constant damping
constant the reversal process changes slowly and steadily from coherent rotation to nucleation with increasing column length. The region between homogeneous rotation and
nucleation becomes smaller and is shifted to smaller column lengths if the damping constant is reduced from α = 1 to α = 0.02.
Calculations of the energy barriers for zero external field show that above a column
length of 20 nm the reversal mode becomes inhomogeneous. The transition from quasihomogeneous rotation to nucleation due to thermal agitation occurs at a similar column
length as for the field induced reversal and a damping constant of α = 0.02.
For field rise times below 50 ps the precessional motion determines the switching
process. Due to the precessional motion the field dependence of the switching time becomes a sophisticated problem. The switching time becomes larger when the field rise
time is decreased. Furthermore in a small field window for fields slightly larger than the
switching field the switching time decreases with decreasing field strength. That effect was
found by researchers of different groups for a single domain particle (Suess et al., 2001;
Gao et al., 2002; Uesaka et al., 2002). In the previous chapter it is shown that fast switching modes can also be found in weakly exchange and magnetostatic coupled thin films.
Depending on the field rise time the dependence of the switching time on the field strength
changes.
4.2. Exchange spring media for perpendicular recording
Further increase of areal density in magnetic recording is believed to be limited by the
superparamagnetic limit, as thermal fluctuations at room temperature (kB T300 ) may over-
106
D. SUESS et al.
come the energy barrier (E = KV ) separating both magnetization directions of an isolated
grain with a volume V and an uniaxial anisotropy constant K. According to the Stoner–
Wohlfarth theory (SW) an increase of anisotropy may compensate for the required decrease
of grain size, but results in an unfavorable increase of coercivity μ0 Hc . The maximum
fields achievable with perpendicular writing head are limited by about 1.7 T (Kanai et al.,
2003). The coercive field of a single domain particle can be decreased by 20% to 40%
while keeping the energy barrier constant by using magnetic materials with a combined
anisotropy (cubic and uniaxial) as shows by Usov et al. (2003).
Another possibility to overcome the writing problem is to employ thermally assisted
recording (Cumpson et al., 2000). Thiele et al. (2003) suggested to lower the coercive field
by the use of FePt/FeRh bilayer system. Recently magnetic multilayer structures being
composed of magnetically hard and magnetically soft layers were proposed, in order to
address the recording problem (Victora and Shen, 2005; Suess et al., 2005a; Suess et al.,
2005b).
In the model of Shen and Victora the soft and the hard part of each grain remain uniform.
In order to decrease the coercive field, the exchange coupling between these layers has to
be reduced.
In the following exchange spring media are presented which can be optimized for both,
high thermal stability and low coercivity. This differs significantly from the scope of pioneering work on exchange spring systems, aiming at an optimization of energy product of
bulk permanent magnets (Kneller and Hawig, 1991) and thin films (Fullerton et al., 1998).
In all calculations coercivity was fixed to 1.7 T and, if not stated otherwise, the grain size
of cylindrical grains was fixed to the dimensions expected for the next media generation
(diameter 6 nm, thickness 14 nm). After presenting an analytical model for single phase
media for comparison, micromagnetic calculations are used to analyze domain reversal in
bilayers.
4.2.1. Analytical model for single phase media
A simple analytical model for the optimization of a single phase data layer is given. With
a given head field one has to optimize the data layer properties such as K and Js . When
assuming that the film is composed of weakly coupled single domain grains, the energy
barrier for a single domain particle is given by
E = KV (1 − H /HSW )2 ,
(4.3)
where K is the anisotropy constant, V the volume and H the effective field acting on
the grain parallel to the easy axis. The Stoner–Wohlfarth switching field is given by
HSW = 2K/Js . The effective field H is composed of the demagnetizing field and the
exchange field from the neighboring grains. It acts parallel to the easy axis. The worst
case for thermal stability is when all neighboring grains are magnetized parallel, described by a demagnetizing factor of 1. However the exchange field from the neighboring
grains stabilizes the grain, opposing the demagnetizing field. The effective field can be
written as H = Js /μ0 − Bex μ0 . Here Js /μ0 is the demagnetizing field and Bex is the
exchange field in T. The maximum value of the anisotropy is limited by the maximum
head field, Bh . The head field has to be large enough in order to saturate the film. Assuming all grains except one to be reversed, the switching field of the last grain (in
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
107
the recording community also called saturation field or closer field) can be expressed as
HSW,last = 2K/Js − Bex μ0 + Js /μ0 . The exchange field of the neighboring grains supports the switching of the last grain whereas the demagnetizing field of the film hinders
the switching of the last grain. The minimum head field to record on the data layer is
Bh = μ0 2K/Js − Bex + Js . From this expression K can be expressed as a function of Js ,
Bex and Bh and substituted in equation (4.3). By setting the first derivative of equation (4.3)
with respect to K to zero the optimum value follows from an equation of third order. Thus
we obtain values for K and Js that maximize the energy barrier for a given head field. The
solution of interest is given by
K=
1
(3Bh + 3Bex − γ )(Bh + Bex + γ ).
32μ0
The optimum value of the magnetic polarization is given by
2 ,
γ = 5Bh2 + 6Bh Bex + Bex
Js = 1/4(3Bh + 3Bex − γ ),
(4.4)
(4.5)
(4.6)
and the energy barrier can be expressed as
E(Bh , Bex ) = V
(3Bh + 3Bex − γ )(−Bh + Bex + γ )2
.
8μ0 (Bh + Bex + γ )
(4.7)
Figure 4.15 shows a contour plot of the energy barrier as a function of the exchange field
and the maximum head field for a grain diameter of 8 nm and a layer thickness of 20 nm.
In a first approximation the isolines can be regarded as straight lines. Therefore the energy
barrier can be approximated to be a function of just one variable. E(Bh , Bex ) ≈ Ẽ(b)
for b = Bex + 0.36Bh .
4.2.2. Numerical results for single phase media and exchange spring media
The finite element method was used to optimize the magnetic properties of an exchange
coupled bilayer composed of a magnetically hard and soft layer for perpendicular magnetic
recording. In the limit of vanishing soft magnetic layer thickness, the numerical results can
be compared with the analytic model. The thickness of the hard layer and the thickness of
the soft layer are changed. Furthermore the values of K in the hard layer (Ksoft = 0) and
Js which is assumed as equal in the both layers are varied.
For each configuration the value of the anisotropy in the hard magnetic layer was
changed iteratively in order to obtain a coercive field of the bilayer structure equal to the
maximum possible head field of Bh = 1.7 T. Instead of using equation (4.3) we calculate the energy barrier numerically using the nudged elastic band method (Dittrich et al.,
2002a). For the calculation of the energy barrier both, the demagnetizing field and the exchange field are taken into account with an effective field, H = Js /μ0 − Bex /μ0 . This
effective field acts as an external field during the calculation of the saddle point configuration and saddle point energy. For each hysteresis loop calculation the Landau–Lifshitz
equation is integrated numerically while the external field is subsequently decreased. The
demagnetizing field is taken into account via a reduction of the external field by Js /μ0 . The
108
D. SUESS et al.
Fig. 4.15. Contour plot of the energy barrier for a single phase media as a function of the intergrain exchange
field and the head field. The grain diameter is 8 nm the film thickness 20 nm.
exchange field helps to reverse the last grain of a bit. Therefore it is added to the external
field.
The soft layer which is fully exchange coupled to the hard layer acts as a magnetic
spring that initiates the reversal of the hard layer similar to what is observed in composite
permanent magnets (Kneller and Hawig, 1991). A magnetic domain wall is created next
to the hard/soft interface. Compared to a single phase media with the magnetic properties of the hard layer the coercive field is significantly reduced (by a factor of 6). The
numerical results show two distinct reversal modes for switching by an external field
and for switching caused by thermal fluctuations. These reversal modes are compared in
fig. 4.16 for a single grain of an exchange spring recording media. The thickness of the
hard layer (A = 1 × 10−11 J/m, K = 9.6 × 105 J/m3 , Js = 0.26 T) and the soft layer
(A = 1 × 10−11 J/m, K = 0, Js = 0.9) is 8.7 nm and 5.3 nm, respectively. Figure 4.16a–d
shows the reversal of the grain under the influence of an external field. As the external
field reaches coercivity the domain wall formed at the hard/soft interface propagates into
the hard magnetic part. For zero external field, thermal activation helps the system to cross
the energy barrier. The right-hand side of fig. 4.16 depicts the magnetization distribution
at the saddle point. This quasi-uniform state is most probably chosen by thermal activated
reversal (fig. 4.16e) and differs significant from the reversal mode under the application of
the switching field (fig. 4.16d).
Due to different reversal mechanisms the ratio between the energy barrier and the coercive field is not constant, as predicted by the SW-theory, but allows an independent
optimization by changing media properties. First we investigate the influence of the fraction of the hard layer thickness to the total thickness (f = lhard / ltotal ) on the energy barrier
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
109
Fig. 4.16. Side view of a bilayer for perpendicular recording. (a–d): Magnetization states during the hysteresis
cycle for the bilayer with Js,hard = 0.26 T and Js,soft = 0.9 T. Configuration (a) shows the remanent state for the
fully magnetized disk. Configuration (e) gives the saddle point for thermally activated reversal at zero external
field.
where f = 1 describes a medium consisting of one complete hard magnetic layer. An exchange field produced from the neighboring grains of 0.2 T was assumed in the calculation.
This is about 12% of the coercive field. First both layers are set to an identical saturation
polarization (Js = Js,hard = Js,soft ). Figure 4.17 shows the calculated energy barrier as a
function of f . Under the constraint of a coercive field of 1.7 T the highest energy barrier
is obtained for f = 0.375 and for Js = 0.5 T. The angle between the external field and
the easy axis was assumed to be 0.7◦ . The energy barrier of this optimal bilayer is more
than two times larger than that of the single phase medium. The influence of Js on the
energy barrier for different film architectures is given in fig. 4.18. For a single layer the
finite element simulations results in an energy barriers with a maximum at Js = 0.42 T, in
excellent agreement with the analytic model predicting Js = 0.41 T.
The influence of an exchange breaking layer between the soft and the hard layer on the
energy barrier was investigated for Js,hard = Js,soft = 0.5 T for different values of the hard
layer thickness fraction f . In a 1 nm thick spacer layer separating the hard and soft layer,
the exchange constant was reduced by a factor in 2, 3, 4, 5, and 6. The results show that for
all those values the energy barrier decreases.
A small angular dependence of the coercive field is important in order to optimize the
signal to noise ratio. For the composite media, the angular dependence of the coercive field
can be fitted with the expression Hc (α) = Hc (0)/ cos(α). This expression is commonly
used for magnetic materials which are controlled by domain wall pinning. For an angle α
between 0◦ and 40◦ the deviation of the analytic formula and the finite element calculation is less than 5%. In contrast to the angular dependence of the coercive field for single
phase media which can be described by the Stoner–Wohlfarth theory the change of Hc (α)
is considerably weaker for composite systems. For a misalignment of 15◦ the coercive
110
D. SUESS et al.
Fig. 4.17. Energy barrier as a function of the fraction of the hard layer thickness to the total thickness for a
magnetic bilayer structure. Different curves are for different magnetic polarization which are set to be the identical
for the hard magnetic and soft magnetic layer.
Fig. 4.18. Energy barrier as a function of the magnetic polarization. For the bilayer the same value of Js was
assumed in the hard layer and the soft layer. The anisotropy constant in the hard layer was adjusted in order to
get a saturation field of 1.7 T.
field increases by a factor of 1.04 for the composite media, whereas it decreases by a factor of 0.60 for the single phase media. For perpendicular recording without intergranular
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
111
exchange (grain diameter is 10 nm) a change of the anisotropy distribution from σθ = 0
to σθ = 15◦ leads to an increase of the transition parameter a from about 10 nm to 18 nm
(Shimizu and Bertram, 2003). For the exchange spring media the same change in σθ increases a from about 10 nm to less than 11 nm. The signal to noise ratio is proportional to
SHR ∝ 1/a 2 (Gao and Bertram, 2002). Therefore for the given easy axis distribution the
SNR can be improved by more than a factor of 2.5 for the composite media.
4.2.3. Conclusion
It was shown that exchange spring media can improve thermal stability by a factor of two
compared to single phase media allowing at constant coercivity. The SNR can be increased
by more than a factor of 2.5. The different magnetization state during switching in the head
field compared to thermal switching allows to optimize thermal stability and coercivity
mostly independent from each other, resulting in a significant reduction of the stable grain
size. Recent experiments by Okamoto et al. (2004) showed that these properties indeed can
be optimized independently. They prepared epitaxial FePt L10 nanoparticles covered with
different Pt overlayer thicknesses. They observe a decrease of coercivity with Pt thickness
while the energy barrier remains unaltered. These experiments can be explained by an
induced magnetization within the Pt layer resulting in a kind of exchange coupled bilayer—
in qualitative agreement with the exchange spring media described in this chapter.
4.3. Integrated simulations of recording devices
The development of future recording systems with ultrahigh storage densities critically depends on a detailed understanding of the write process. The design of the magnetic write
head, the data layer, and in the case of perpendicular recording the soft underlayer requires
a joint optimization of these components based on their underlying dynamic magnetization processes. In order to calculate the time evolution of the magnetization, Maxwell’s
equations are solved simultaneously with an equation of motion for the magnetization vector. In perpendicular recording all parts of the recording system (current coil, write head,
data layer, and soft underlayer) are magnetostatically coupled. A precise simulation of the
recording process requires to solve the governing equations simultaneously for all these
parts while taking into account the movement of the write head (Schrefl et al., 2004).
Head field dynamics have been investigated previously using micromagnetic models.
Takano (2004) showed how the head field rise time is affected by the damping constant,
material, yoke length, and pole-tip dimension in ring heads for longitudinal recording.
Scholz and Batra (2005) solved the dynamic micromagnetic equations using a finite element/boundary element method for a single pole head. They conclude that an intermediate
value of the damping constant, short yoke length, and fast current rise time are needed for
maximum data rate. Senanan and Victora (2002) studied the effect of media permeability
on perpendicular recording. They showed that the presence of the medium makes changes
to the head field. However, their micromagnetic model used a truncated write poles driven
by magnetic charge sheets with explicit time dependence which represent unphysical dynamic constraints.
In this chapter the results are shown for integrated micromagnetic simulations. Details
for longitudinal recording can be found in papers of Schrefl et al. (2004) and Ertl et al.
112
D. SUESS et al.
(2005) and for perpendicular recording in papers of Schrefl et al. (2005) and Schabes et al.
(2005). The magnetization dynamics is calculated micromagnetically for the entire recording system, which consist of
(1) the current coil,
(2) the full three-dimensional head structure and the yoke,
(3) the data layer modeled by irregularly shaped Voronoi grains separated by a grain
boundary phase with reduced exchange, and
(4) in the case of perpendicular recording the soft underlayer.
This is a multi-scale simulation with the dimensions of the coil and the yoke in the range
of 10 000 nm to the grain boundary thickness of 0.5 nm. We compute the head field during
recording, taking into account both the magnetostatic field of the data layer onto the head
and the magnetostatic field of data layer onto the soft underlayer as well as the motion of
the head. Here we focus on the time dependence of the different contributions to the total
head field, which is composed of the field generated by the pole tip, the field generated
by the head, and the self-demagnetizing field of the data layer. Besides the magnetic properties, the only input for the simulations are the current profile for the coil excitation and
the head velocity. The head field dynamics is calculated from the induced magnetization
processes within the head, the data-layer, and the soft underlayer. Domain wall motion
within the yoke of the head, vortex motion within the pole tip of the head, and large-area
magnetization processes in the soft underlayer cause a lag of the head field behind the
current.
4.3.1. Numerical background
A preconditioned backward differentiation method is applied to solve the Landau–
Lifshitz–Gilbert equation for the recording system. At each time step the magnetostatic
interaction fields are calculated by using a magnetic scalar potential. We apply a hybrid
finite element/boundary element method to compute the magnetic scalar potential. Thus
no mesh is required outside the magnetic parts. All information concerning the mutual interaction of the different parts is stored in the boundary element matrix. In order to avoid
the matrix elements to be recomputed at each time step due to the movement of the head,
we compute the scalar potential at the surface of a so-called field box which moves together with the head. A fast Poisson solver is used to evaluate the potential within the
field box at high spatial resolution (Swarztrauber and Sweet, 1979). Numerical derivation
gives the head field which is then interpolated at the nodes of the data layer mesh. We apply the adaptive cross approximation method to construct so-called hierarchical matrices.
The above combination of the finite element method, the fast boundary element method,
and the finite difference method gives a highly efficient micromagnetics solver. A finite
element model consisting of 1.2 million tetrahedral elements can be solved on a single
CPU. It takes about 4 CPU hours to integrate the Landau–Lifshitz–Gilbert equation for
1 ns (1 GHz Alpha EV68 processor).
4.3.2. Longitudinal recording
For longitudinal recording we use the ring head which is shown in fig. 4.19. The gap
region is enlarged on the right-hand side. The gap distance is 40 nm. The intrinsic material
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
113
Fig. 4.19. (Up): Finite element model of the ring head showing the coil, the yoke, and the gap region.
parameters of the head are A = 1011 J/m, K1 = 800 J/m3 with the uniaxial anisotropy
direction parallel to the cross track directions and Js = 2.4 T in the pole tips. The intrinsic
damping constant of NiFeCo is low. In the ring head, eddy currents will give rise to dissipation, leading to an increase of the effective Gilbert damping constant (Chikazumi, 1997).
Since we are interested in the optimum value of the Gilbert damping constant for minimum
flux rise time, we varied the damping constant in the range from α = 0.02 to 1. The current
rise time was 0.1 ns. The maximum current was 120 mA. The frequency of the write pulse
was 286 MHz. In order to initialize the head magnetization the head was first exposed to a
field pulse sequence of 5 cycles. For a Gilbert damping constant of 0.1 the maximum head
field is reached 1.5 ns after the reversing the current. The time for the head field to reach
90% of its saturation value is 0.9 ns. This delay can be understood by the dynamic magnetization processes within the head. The damping value of α = 0.1 leads to the precession of
the magnetization. The local fields within the yoke strongly depend on the position within
the head. Thus, the motion of the head magnetization becomes non-uniform immediately
after reversing the current. The phase decoherence will lead to non-uniform magnetization
reversal within the yoke. Only after the full reversal of the yoke the pole tips can be saturated. This process implies that flux rise time increases with decreasing Gilbert damping
constant. The smaller the damping constant, the more dominant becomes the precession
term in the LLG equation. Small differences in the local field will create phase decoherence which in turns slows down the time to saturation. The results of fig. 4.20 clearly show
that large values of the Gilbert damping constant are preferable for fast flux rise times.
Figure 4.21 shows the influence of the head velocity on the read back signal. As the head
velocity is increased from 10 m/s to 20 m/s the read back signal decays by more than 25%.
The image on top in fig. 4.21 shows the written bit pattern after the head has moved with a
velocity of 20 m/s across the data-layer.
4.3.3. Perpendicular recording
Figure 4.22 shows the head with the coil used for the simulations. The yoke length of the
head is 12 μm. The pole width and thickness are 75 nm and 112 nm, respectively. We use a
graded mesh to bridge the length scales between the 3D coil structure, the yoke (Js = 2 T),
and the pole tips (Js = 2.4 T). The write pole is meshed using a mesh size of less than 5 nm.
The flying height is 10 nm, the thickness of the soft underlayer (Js = 1 T) is 100 nm. The
114
D. SUESS et al.
Fig. 4.20. Head field strength as a function of different Gilbert damping constants within the head. The dashed
line gives the current profile. The solid lines give the head field for α = 1, 0.5, 0.1, 0.05, and 0.02. The fasted
field rise time is achieved with intermediate damping (α = 0.5).
Fig. 4.21. (Top): The magnetization within the data layer is shown after writing with a head velocity of 20 m/s.
The small arrows indicate the strayfield produced by the magnetization. (Bottom): The read back signal is shown
for two different head velocities.
data layer model consists of grains based on modified Voronoi-cells (Schrefl and Fidler,
1992) with realistic shapes and a realistic grain size distribution. The grains (Js = 0.62 T,
average Ku = 2.3 × 105 J/m3 ) have an average diameter of 8 nm and a thickness of 14 nm.
The anisotropy axes are virtually perpendicular to the film plane and uniformly distributed
within a cone with an opening angle of 10 degrees. The Gilbert damping constant of the
head and soft underlayer was α = 0.1 and of the data layer α = 0.02. The grains are
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
115
Fig. 4.22. Model geometry for trailing shield pole head used for the perpendicular recording simulations.
Fig. 4.23. Dynamic writing of bit transitions in the data layer. The data layer magnetization is mapped as grey
scale at different times during the recording process. The head is moving from left to right.
separated by a grain boundary phase with zero anisotropy and reduced exchange. The
effective exchange field is C ∗ = 0.15 × 2K1 /Js .
We performed recording simulations using a head velocity of 20 m/s and a rise time
(zero-to-peak) of the recording current of 0.1 ns. The bit cell time is 2.5 ns. Figure 4.23
shows the data-layer magnetization at different times during the recording process.
During recording, the magnetization of the data-layer reverses. As a consequence, the
effective permeability of the data layer changes. As an example, we discuss the head field
dynamics when writing the first transition (see center image of fig. 4.23). In order to understand what limits transition sharpness in perpendicular magnetic recording we investigate
the time sequence of different field contributions as seen by the media grains. This field depends on the current profile, on the head and soft underlayer dynamics, and on the velocity
of the head. Initially, the media is in an ac-erased state. The current becomes positive and
the positive write field switches most of the grains on the left-hand side of the data layer.
In particular, grain 1 and grain 2 (as indicated in fig. 4.23) see a positive write field, which
monotonically increases for the first 2.5 ns. At a time t = 2.5 ns the current switches and
the write field starts to diminish. The zero crossing of the field occurs about 0.6 ns after
the zero crossing of the current which is a clear indication of the slow-down of the head
field caused by the magnetodynamics in the head (Scholz and Batra, 2005). Figure 4.24
gives the local write field at the position of grain 1 and grain 2. As negative write field
becomes reaches the switching threshold the previously positive magnetized grains switch
and a transition is written. Whether a grain can be switched or not depends on the total time
116
D. SUESS et al.
Fig. 4.24. Total write field as a function of time as seen by the media grains 1 and 2. The total write field is the
sum of the pole tip contribution and the SUL contribution.
Fig. 4.25. Contour plots of the write field contributions. Left: Pole tip field. Right: SUL field. The distance
between the lines is 0.04 T.
the grains sees a field above the switching threshold. The time dependence of the pole tip
field and the SUL field determine the strength of the local write field. In grain 1 (located
24 nm to the right from first transition), the contribution from the soft underlayer to the
head field reaches its maximum 1.3 ns after zero crossing of the field, whereas the field
generated by the pole tip reaches its maximum after 1.7 ns. For grain 2 (located 40 nm
from the transition) there is no significant time lag between the pole tip field and the SUL
field is less pronounced. At the position of grain 2, the pole field contribution and the SUL
contribution are in phase. Therefore maximum head field which is seen by grain 2 is 7%
larger then the maximum field acting on grain 1.
The time derivative of the SUL field is much smaller than the time derivative of the
dHpole /dt. This is partly due to the lower Js of the SUL
pole tip field, dHSUL /dt
assumed in these simulations. However, even if multiplied with Js,pole /Js,SUL the derivative of the SUL field is considerable smaller that the time derivative of the pole tip field
dHSUL /dt (Js,pole /Js,SUL ) < dHpole /dt. Similar, the head field gradient is smaller for the
dHpole /dx as shown in fig. 4.25. These
SUL field than for the pole tip field, dHSUL /dx
results underline the fundamental differences between the SUL and pole contributions to
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
117
the total write flux. While the contributions from the pole are generated by a lithographically defined object, the gradients for the SUL contributions are created along a moving
domain boundary in the SUL. Furthermore, vertical spacing and partial saturation of the
SUL may reduce the gradient of the SUL contributions.
4.3.4. Conclusion
The rise time of the head field generated by ring heads has been calculated using a
full micromagnetic description of the yoke, the pole tips, and the current coils. The flux
rise time increases with decreasing Gilbert damping constant. Small damping gives rise to
phase decoherence of the magnetization in different parts of the yoke which increase the
time required to saturate the pole tips. The time in ns for the head field to reach 90% of
its saturation value is 1, 0.8, 1.5 and 1.8 for a Gilbert damping constant of 1, 0.5, 0.1, and
0.05 respectively.
For perpendicular recording depending on the media position, a phase shift between the
SUL field and the pole tip field occurs. If both fields are not in phase the total write field
decreases and broadens in time and in space.
Acknowledgement
I want to thank my colleagues Markus Kirschner, Florian Dorfbauer, Gino Hrkac and
Otmar Ertl for all their help, inspiring and fruitful discussions about the details of micromagnetics.
The financial support of the Austrian Science Fund Grant No. Y-132-N02 is acknowledged.
Appendix A:
A.1: Derivation of the interface condition for the eddy current field
Here we show how equation (2.61) can be derived without the use of Gauss’ theorem. We
start from the Biot–Savart law,
j (r) × (r − r )
1
dV .
H eddy (r) =
(A.1)
4π
|r − r |3
Biot–Savart’s law can be written as the curl of a vector potential A,
j (r)
1
dV .
H eddy (r) = ∇ × A = ∇ ×
4π
|r − r |
For a constant current j we can write equation (A.2) as
1
j
H eddy (r) = ∇ × A =
∇×
dV
4π
|r − r |
1
j
=
∇ ×
dV .
4π
|r − r |
(A.2)
(A.3)
118
D. SUESS et al.
Fig. A1. (Left) A current density I (x) is assumed inside the body O. Close to the point x1 the homogeneous
current I h is assumed. (Right) The surface 1 can be regarded as an infinite plan. The gradient of the eddy current
field is evaluated at point x1 and point x2 .
We want to calculate the normal derivative of H eddy . Thus we apply the divergence on
both sides of equation (2.64), multiply with the normal vector n, and transform the volume
integral to a surface integral, which leads to
(r − r )n
1
(∇H eddy,i )n = −
(A.4)
(n × j )i dA .
4π
|r − r |3
Let us assume a closed body O, as shown in fig. A1. We evaluate the normal derivative
of one component of the eddy current field close the surface. The first time we evaluate
the gradient of the field at the location x1 which is inside the body. The second point x2
is located outside the body O. The surfaces seen from point x1 and x2 are 1 and 2 . In
the limit when x1 and x2 are located infinitesimal close to the surface of the body O, 1
can be regarded as an infinite plane. The current density around x1 can be assumed to be
homogeneous. Hence, the eddy current field can be calculated using equation (A.4). For
the calculation of the surface integral of the infinite plane we can set the normal vector n
parallel to the x-axis, without loss of generality. Therefore the integral has to be performed
over the y –z plane with x = 0. For the surface integral follows
A∞
(r − r )n
(n × j )i dA = (n × j )i
|r − r |3
∞ ∞
∞ −∞
−x
(x 2 + j 2 + z 2 )
= −(n × j )i 2π sgn(x).
dy dz
(A.5)
(A.6)
Therefore the integral over 1 is −2π(n × j )i for the point x2 and 2π(n × j )i for the point
x1 . The surface integral across the surface 2 is the same for point x2 and x1 . For the jump
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
119
of the normal derivative of the eddy current field across the surface follows
out
in
)n − (∇Heddy,i
)n = (n × I )in
(∇Heddy,i
i .
(A.7)
A.2: Time integration of the Landau–Lifshitz equation
A standard method for time integration of stiff ordinary differential equations is the backward differentiation method. The BDF method leads to the nonlinear system (Hindmarsh
and Petzold, 1995)
F (y n ) = y n − a n − hβ0 f (tn , y n ) = 0,
(A.8)
which is solved by Newton iterations. Here f is the right-hand side of the Landau–
Lifshitz–Gilbert equation (2.68). a n and β0 are constants which depend on the order of the
time integration method and on the previous time step size. Equation (A.8) is successively
solved by adding corrections y to the solution vector. y is the solution of the linear
system arising from the Newton method,
∂F (y n−1 )
y = Ay = −F (y n−1 ),
∂y
(A.9)
∂f
.
∂y
(A.10)
with
A = 1 − hβ0 J
and
J=
Equation (A.9) is solved with a Krylov method. J is the Jacobian matrix of the Landau–
Lifshitz–Gilbert equation. It starts with a guess x 0 for y, and corrects it successively
to get iterates x 1 , x 2 , . . . . After a few iterations a good approximation for y is found.
In every iteration only the product Ax i has to be calculated. So the large matrix A has
to be neither explicitly constructed nor stored. In micromagnetics A is fully populated
owing to the long range magnetostatic interaction. Ay is approximated using finite differences,
F (y n−1 + εy) − F (y n−1 )
.
(A.11)
ε
The number of Krylov iterations strongly depends on the matrix A. For some matrices A
the Krylov methods converge slowly.
A(y n−1 )y ≈
A.2.1: Preconditioning
The above problem of the slow convergence can be overcome by preconditioning. Instead of the system
Ay = b
(A.12)
the equivalent system
(AP−1 )(Py) = A x = b
(A.13)
−1
is solved. The matrix P should be an approximation to A. Then, AP is close to the identity matrix, and the system A x = b can be solved very efficiently with only a few Krylov
120
D. SUESS et al.
iterations. The matrix AP−1 cannot be calculated directly since generally the matrix A is
not explicitly constructed. To calculate the matrix vector product, s := AP−1 x, which is
needed in every Krylov iteration, the following procedure is applied. Instead of the calculation of s from
s = AP−1 x,
(A.14)
we multiply equation (A.14) by A−1 and calculate w from
A−1 s = P−1 x =: w.
(A.15)
w is calculated by solving the linear system
Pw = x
(A.16)
with an efficient iterative solver, since the inversion of a matrix is very time consuming.
Once w has been calculated, s follows from equation (A.15) simply by the matrix vector
product
s = Aw,
(A.17)
A = 1 − hβ0 J.
(A.18)
We found that the solution of equation (A.16) is the most time consuming part in our micromagnetic simulation. The linear system of equation (A.16) is sparse. As the number of
unknowns is large, the use of an iterative method is more appropriate than solving equation (A.16) with a direct method. Among different generalized minimum residual methods
(GMRES), the Bi-Conjugate Gradient method was found to be the most efficient one. For
some problems it is important to apply a preconditioner to the linear system (equation
(A.16)) to achieve good convergence. A good choice is the incomplete factorization technique RILU (Bruaset, 1997).
A.2.2: Error control
In the CVODE (Cohen and Hindmarsh, 1996) code the local truncation error ei is estimated. The tolerance parameters reltol and the vector abstol, which can be defined for
every solution component separately, can be used to control the error. The time step of the
integration is determined in such a way that the error satisfies the inequality
N
1
2
i=1 (ei reltol|ui |+abstoli )
(A.19)
≤ 1.
N
In micromagnetics a useful error indicator follows from the structure of the LLG equation.
The LLG equation maintains the magnitudes of the magnetic polarization. However the
discretization of the LLG equation using the BDF formula violates this relation. For example for the implicit Euler method with time step h (BDF method or Adams method of
order one) follows,
J i+1 − J i
|γ |
J i+1 × H eff (J i+1 )
=−
h
1 + α2
MICROMAGNETIC SIMULATION OF MAGNETIC MATERIALS
121
Fig. A2. The deviation of the magnitude of the magnetic polarization vector from Js indicates the quality of the
time integration method. Comparison of the deviation norm DN for the BDF method with and without preconditioning. The maximum order of the time integration was qumax = 2.
−
α |γ |
J i+1 × (J i+1 × H eff (J i+1 )).
1 + α 2 Js
If equation (A.20) is multiplied by J i+1 , the right-hand side vanishes,
J i+1 − J i
J i+1 = 0.
h
(A.20)
(A.21)
Substituting J i J i+1 by 12 (J 2i + J 2i+1 − (J i − J i+1 )2 ) in equation (A.21), it follows
(J i+1 )2 = (J i )2 + (J i+1 − J i )2 .
(A.22)
As a consequence (J i+1 =
> 0. To guarantee that the norm of the magnetic polarization does not drift away, we normalize the magnetic polarization on every
node if on at least one node the deviation becomes larger than the specified tolerance,
renormtol = max(abstol, reltol). To determine the accuracy of an integration method we
use the number of renormalization steps during the simulation as well as the deviation
norm of the spontaneous polarization,
)2
DN =
1
max(|J |i − Js ),
Js
J 20 for i
(A.23)
where |J |i is the magnitude of the polarization vector on node i.
Figure A2 shows the deviation norm DN (equation (A.23)) as a function of the simulated
time for a calculation with and without preconditioning. In both calculations lmax is 400.
122
D. SUESS et al.
A large deviation norm indicates a large error. Although for both simulations the same
values for the error tolerances, abstol = 10−4 and reltol = 0, were used, the accuracy is
much better if preconditioning is performed.
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chapter 3
FERROFLUIDS
S. ODENBACH
Technische Universität Dresden, Chair for Magnetofluiddynamics, 01062 Dresden, Germany
Handbook of Magnetic Materials, edited by K.H.J. Buschow
Volume 16 ISSN: 1567-2719 DOI 10.1016/S1567-2719(05)16003-X
127
© 2006 Elsevier B.V.
All rights reserved
CONTENTS
1. Synthesis of magnetic fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
1.1. Stability criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
1.2. Magnetite ferrofluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
1.3. Ferrofluids containing metallic particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
1.4. Ferrofluids for biomedical applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
2. Basic properties of ferrofluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
2.1. Magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
2.2. Viscous properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
3. Basic description for the field dependent behaviour of ferrofluids . . . . . . . . . . . . . . . . . . . . . . 147
3.1. Basic ferrohydrodynamic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
3.2. An overview on applications of magnetic fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4. Magnetic control of fluid properties and fluid flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
4.1. Field dependent viscous properties of ferrofluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
4.2. Thermal transport properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5. Medical applications of ferrofluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
128
Talking about “Magnetic Materials” usually implies a discussion of solid state materials
like magnets or recording media. The reason for this restriction is given by the fact, that all
known magnetic materials exhibiting interesting properties like ferro- or ferrimagnetism
loose these properties at a critical temperature well below their melting point. Usual liquid
materials do not show magnetic ordering and thus—in the best case—they are paramagnetic systems with susceptibilities in the order of 10−4 . These small susceptibilities imply
that the reaction of liquid systems to applied magnetic fields is weak even if very high
field strength is used. The exceptions from the rule that liquid materials do not show magnetic ordering—liquid helium at temperatures below 3 mK (Mermin and Lee, 1976) and
supercooled Co-alloys (Wilde et al., 1996a, 1996b)—do not represent magnetic liquids in
a sense that an efficient magnetic control of the properties of the material by means of
moderate magnetic fields can be obtained. Such a control, leading to applications of liquids controlled directly by magnetic fields, is also impossible for liquids like paramagnetic
salt solutions since, due to their low susceptibility, the magnetic fields strengths required
to control the fluids are to high for a practical use. Thus, defining the magnetic control of
a liquid medium as the employment of Kelvin forces to change the flow and the properties of the liquid, and calling a liquid a “Magnetic Material” only if an efficient control
by Kelvin forces can be obtained with magnetic fields requiring a technical effort for their
production which is small enough for practical use, one has to state that no homogeneous
magnetic liquid exists. In this context it has to be mentioned, that the well known use of
Lorentz forces controlling the flow of liquid materials especially in liquid metal magnetohydrodynamics (Davidson and Thess, 2000) is surely an efficient method to control the
flow of a liquid medium by means of a magnetic field, but since the forces employed are
not magnetic body forces, the liquid metal itself can not be defined as a magnetic material
in the above mentioned sense.
On the other hand the use of Kelvin forces for the control of liquid materials is a challenging idea which implies numerous possibilities in basic research as well as for technical
applications. Despite the fact that such a material was not available at all, these possibilities
forced theoretical considerations about the technical use of a hypothetic magnetic fluid in
the early 60’s of the twenties century. The predicted challenging properties of a liquid controlled by magnetic body forces led finally to the development of suspensions of nanosized
(Papell, 1964) magnetic particles in appropriate carrier liquids. Suspensions of micronsized
magnetic particles had already been investigated in the late 40’s (Rabinow, 1948) and had
shown the potential of magnetically influencing at least the viscosity of a liquid. While the
latter ones are usually called magnetorheological (MR) fluids, the suspensions of magnetic
nanoparticles have become known as magnetic fluids or ferrofluids.
In the following discussion we’ll mainly focus on ferrofluids, showing that they can be
called a magnetic material following the discussion given above. The following chapters
129
130
S. ODENBACH
will treat the production of ferrofluids, the question of their colloidal and magnetic stability, their basic properties and the possibility to magnetically control the properties and
flows of the magnetic liquids by moderate magnetic fields. Moreover various examples for
the practical use of ferrofluids—as well in well established fields as for new and innovative
applications currently in the focus of actual research—will be presented. Magnetorheological fluids will not be treated in detail in this contribution but similarities and differences
between ferrofluids and MR-fluids will be mentioned at distinct points.
1. Synthesis of magnetic fluids
Techniques for the synthesis of ferrofluids have to solve three major problems. First of
all magnetic nanoparticles with typical diameters around 10 nm have to be produced. Secondly it has to be ensured that the particles are magnetically stable. That means in particular
that no transition from a ferro- or ferrimagnetic state to an antiferromagnetic or diamagnetic state appears. E.g. ferromagnetic cobalt particles have to be prevented from oxidation
since CoO is a diamagnetic material and—as will be shown later on—a ferrofluid requires
suspended particles with a high overall magnetic moment to exhibit its unique properties.
Finally the magnetic particles have to be brought into a stable suspension.
1.1. Stability criteria
Since the last mentioned aspect, the stability of the suspension, requires particular effort in
the production process, we’ll now discuss the question of colloidal stability of suspensions
of magnetic nanoparticles before focussing on particular production routes.
To obtain a stable suspension of magnetic particles in a carrier liquid one has to solve the
problem that the particles have to stay homogeneously distributed over the sample even for
long times. The main hindrance for a long term homogeneous distribution is sedimentation
of the particles. The sedimentation can either be driven by the density difference between
the magnetic particles and the carrier liquid or by magnetic field gradients attracting the
magnetic particles and accelerating them relative to the carrier liquid. A stable suspension
is achieved if the thermal energy of the particles is able to keep them distributed. That
means that the thermal energy ET = kB T (kB : Boltzmann’s constant, T : absolute temperature) has to be larger than their energy in the gravitational field or in a magnetic field
gradient respectively. The energy of the particles in the gravitational field is given by
Eg = ρgV h,
(1.1)
where ρ denotes the density difference between the particles and the carrier liquid, g
is the gravitational acceleration, h a typical dimension of the sample and V = πd 3 /6 is
the volume of the magnetic particle with d being its diameter. Comparing thermal and
gravitational energy of the particles one obtains an upper limit for the particle size by
6kB T
.
d≤ 3
(1.2)
πρgh
Assuming magnetite particles in water (ρ ≈ 4 × 103 kg m−3 ) and a typical height
of the sample of h = 0.1 m (g = 9.81 m s−2 , T = 300 K) one finds that particles with
FERROFLUIDS
131
a diameter less than approximately 12 nm are stable against sedimentation in the earth’s
gravitational field.
Similarly one can check the stability against sedimentation in a magnetic field gradient.
For simplification we approximate the magnetic field gradient by a step in magnetic field
strength from 0 to a finite value H0 . That represents the extreme of an infinite field gradient
and the stability against sedimentation means that thermal energy has to be large enough
to allow the particles a free movement from the region with H0 to the region with H = 0.
With other words ET has to be larger than the energy of the particle in the field H0 .
Em = μ0 mH0 = μ0 M0 V H0 ,
(1.3)
where m = M0 V denotes the magnetic moment of the particle and M0 the spontaneous
magnetisation of the magnetic material (M0 = 4.5 × 105 A m−1 for magnetite, μ0 =
1.2566 × 10−6 V s A−1 m−1 the permeability of free space). As seen from equation (1.3),
the magnetic energy depends on the size of the particles and thus the stability criterion
leads again to an upper limit for the particle size
6kB T
.
d≤ 3
(1.4)
πμ0 M0 H0
For magnetite particles and a typical step in field strength of H0 = 104 A m−1 one
ends up again with a maximum particle size of about 10 nm. Thus the basic criterion for
stability of a suspension of magnetite nanoparticles requires that the particles have to have
diameters in the order of 10 nm and less.
This implies a second group of stability criteria since it requires that no agglomeration of the particles appears. Agglomeration would lead to a growth of the particles and
thus sedimentation would become efficient to destabilize the suspension. Agglomeration
processes in colloidal suspensions of magnetic particles can be either driven by magnetic
particle–particle interaction or by v.d.-Waals interaction. One can easily see, that the stability against magnetic agglomeration, which means that the thermal energy of two particles
has to be larger than their maximal magnetic interaction energy in contact
μ0 πM02 d 3
,
72
leads ones more to a size limit by
144kB T
d≤ 3
πμ0 M02
Edd =
(1.5)
(1.6)
which is fulfilled for magnetite particles with d ≤ 10 nm. In contrast the v.d.-Waals interaction energy
2
2
A
l + 4l
2
+
EvdW =
(1.7)
+
ln
6 l 2 + 4l
(l + 2)2
(l + 2)2
with the normalized distance l = 2δ/d between the particles (δ: surface–surface distance
of the particles; see fig. 1.1), diverges for particles in contact as long as the Hamaker
132
S. ODENBACH
Fig. 1.1. Schematic sketch for the explanation of the steric repulsion of the magnetic particles by surfactant
molecules. Details are given in the text.
constant A has a finite value. Therefore colloidal stability of a ferrofluid requires that the
particles are protected from coming into contact. This contact protection is usually provided by surfacting the particles with long chained molecules as it is shown in fig. 1.1. If
the particle surface distance δ falls below two times the thickness s of the surfactant layer
a steric repulsion with repulsion energy
kB T πd 2 ξ
l+2
l
1+t
2−
ln
−
ESteric =
(1.8)
2
t
1 + l/2
t
will appear. Here ξ denotes the surface density of the surfactant molecules and t = 2s/d
is the normalized thickness of the surfactant layer. The origin of the repulsive effect of
the surfactant is given by the reduction of the configuration space of the surfactant molecules appearing as soon as the surface distance of the magnetic particles becomes smaller
than 2s. Provided a sufficient thickness and density of the surfactant layer the steric interaction energy can create a potential wall large enough to prevent contact between the
magnetic particles and thus large enough to avoid their agglomeration due to v.d.-Waals
interaction.
Figure 1.2 shows in summary the various discussed contributions to the interaction
energy of the magnetic particles and their overall interaction potential. Here the steric
interaction is calculated with a surfactant thickness of s = 2 nm and a surfactant density ξ = 1 nm−2 . As seen, such a surfactant provides an energy barrier of about 20 kB T
and is thus sufficient to keep particles apart, allowing thermal energy to distribute them
homogeneously over the suspension.
For the production process the aspect of colloidal stability provides therefore the boundary condition that nanoparticles with a diameter of about 10 nm have to be produced which
must be covered with an appropriately chosen surfactant (see fig. 1.3). The choice of the
FERROFLUIDS
133
Fig. 1.2. The various contributions to the interaction potential between magnetic particles in a carrier liquid
and the resulting overall interaction potential. Calculations have been performed for magnetite particles with a
diameter of 10 nm and a 2 nm thick surfactant layer with a surfactant density of 1 nm−2 .
Fig. 1.3. Schematic sketch of the magnetic particles with surfactant. For clearness the surfactant is not shown in
scale.
134
S. ODENBACH
surfactant implies that the dielectric properties of the surfactant have to match the dielectric
properties of the carrier liquid. That means the Hamaker constant for the v.d.-Waals interaction of surfactant molecules in the carrier liquid becomes zero avoiding attractive forces
between the surfactant shells. During the production process an appropriate adding of the
surfactant has to be observed to prevent the particles from agglomeration during synthesis.
It should be mentioned here, that the question of colloidal stability is one of the central differences between ferrofluids and magnetorheological fluids. While ferrofluids are
synthesized in a way that their colloidal stability is maintained over decades, MR-fluids,
containing micronsized particles show rapid sedimentation. Therefore applications of MRfluids have to provide permanent mixing of the fluid to obtain a suspension with spatially
homogeneous properties. Moreover, since v.d.-Waals interaction does not play a significant
role for large particles no surfactants are usually used in MR-fluids. This causes problems
in applications since strong shear used to homogenize the fluids can lead to chipping of
nanosized fragments which agglomerate, leading to an increase of the suspensions viscosity known as “in use thickening”.
Keeping the stability requirements in mind one can now discuss various production
routes and types of ferrofluids.
1.2. Magnetite ferrofluids
Most ferrofluids produced nowadays and nearly all magnetic fluids used in practical applications contain magnetite (Fe3 O4 ) nanoparticles. The reason for this concentration on a
single material for the solid phase of the suspension is not given by unique magnetic properties of the material but by the fact that magnetite is a material which shows nearly no
magnetic degradation. Magnetite and the next and final oxidation state of iron, maghemite
(Fe2 O3 ), have nearly identical saturation magnetization and thus ferrofluids made of magnetite particles are magnetically stable since an oxidation of the particles does not severely
affect the magnetic properties of the fluid. The magnetic nanoparticles can either be produced by a grinding process from magnetite powders or by chemical precipitation.
The first mentioned process has been used in the original preparation of the first stable
ferrofluids synthesized by Steve Papell in 1964 (Papell, 1964). For this method micronsized magnetite powder is mixed with carrier liquid and surfactant and ground in a ball
mill for about 1000 hours. During the process centrifuging is used to extract large particles from the mixture. In this way the mean particle size is reduced stepwise towards
the limit of about 10 nm needed for a stable colloidal suspension. Due to the presence of
the surfactant the particles are covered by a protection layer during the grinding process
preventing agglomeration while particle size diminishes. An obvious disadvantage of this
process is the extremely long preparation time resulting from the necessary treatment in
the ball mill. Moreover the size distribution of particles produced this way is rather broad
and the particles’ shape is highly irregular.
A more convenient process is chemical co-precipitation of magnetite particles. The
original process has been established by Khalafalla and Reimers in 1973 (Khalafalla and
Reimers, 1973). In this process Fe2+ and Fe3+ salts are mixed in relation 1:2 and coprecipitated in NaOH
8NaOH + 2FeCl3 + FeCl2 → Fe3 O4 + 8NaCl + 4H2 O
FERROFLUIDS
135
Fig. 1.4. Production scheme for ferrofluids from chemical precipitation.
Adding the surfactant, e.g. oleic acid and heating the mixture allows the synthesis of
magnetite nanoparticles of appropriate size. After coagulation various washing processes
are needed to achieve a basic colloidal suspension (see fig. 1.4). This suspension is afterwards optimised concerning the particle size distribution by ultracentrifuging and magnetic
separation in strong field gradients. The resulting ferrofluid is a typical surfactant stabilized
magnetic fluid. Dilution or concentration processes can be used to achieve the necessary
volume content of magnetic particles. By appropriate choice of the surfactant material ferrofluids can be produced in various carrier liquids like water (Shimoiizaka et al., 1978),
kerosene (Khalafalla and Reimers, 1973), diesters (Wyman, 1984), polyphenyl ethers
(Bottenberg and Chagnon, 1982) or even silicone oil (Chagnon, 1982). The typical volume fraction of magnetic particles ranges from 2 to 15 vol%, but also high concentrated
fluids with up to 25 vol% have been synthesized (Scholten, 1983). As shown in fig. 1.5
such methods produce well separated magnetic nanoparticles with a size distribution ranging from about two to twenty nanometres.
Looking to the volume concentration of magnetic material φ in a ferrofluid containing
surfacted particles, one has to keep in mind that the surfactant significantly influences the
overall volume content of suspended material φ̃. First of all the volume content of the
136
S. ODENBACH
Fig. 1.5. Typical particle size distribution of a commercial ferrofluid.
particles including the surfactant φ , which can be calculated from
d + 2s 3
φ = φ
d
(1.9)
has to be considered. Additionally possibly added excess surfactant may be present leading
to values of φ̃ even larger than φ . Assuming a mean particle size of 10 nm and a surfactant layer thickness of 2 nm, equation (1.9) shows, that the total volume concentration of
suspended material for a ferrofluid without excess surfactant having a magnetic volume
content of φ = 0.25 is already near to 70 vol% which is close to the volume fraction of a
dense packing of spheres. Thus 25 vol% of magnetic material represents a clear maximum
for the volume fraction of magnetic particles in a surfactant stabilized ferrofluid leading to
a rather viscous suspension.
Besides the surfactant stabilized magnetite ferrofluids a second production route has
been established by Massart et al. (Massart, 1981). Instead of a surfactant they use surface charges on the magnetic particles produced by absorbed hydroxyl groups to generate
the repelling force needed to overcome the agglomeration due to v.d.-Waals attraction. In
acidic medium the particles have to be positively charged while an alkaline medium requires a negative particle charge. The advantage of this kind of fluids is the reduction of
the total suspended material at constant magnetic volume fraction compared with a surfactant stabilized fluid. Their great drawback for technical applications is their high sensitivity
against changes of the pH-value of the carrier medium which can easily lead to a complete
destabilization of ionically stabilized ferrofluids.
Various other production routes for magnetite and other ferrite containing ferrofluids
have been reported. Substitution of the Fe2+ ions by Co2+ , Ni2+ , Zn2+ or Mn2+ leads
to a variation of the magnetic properties of the particles (Lefebure et al., 1998; Auzans
et al., 1999). The use of micro-emulsion techniques employing inverse micelles allows the
FERROFLUIDS
137
production of particles with a narrow size distribution and with highly regular shape (Pileni,
1993, 1998). Details on such processes can be found e.g. in several reports (Rosensweig,
1985; Blums et al., 1997a, 1997b; Charles, 2002).
1.3. Ferrofluids containing metallic particles
From a magnetic point of view the use of ferromagnetic metallic particles like Co- or Feparticles would be highly preferable compared to the usually employed ferrite particles.
Incidentally—as mentioned above—oxidation of these particles to CoO or FeO leads to a
quick degradation of the magnetic properties of these fluids within a couple of days. While
such fluids are not of interest for technical use important basic results like the discovery
of rotational viscosity (McTague, 1969) which will be discussed later on, have been made
using experimental fluids containing metallic particles. Early production routes for such
fluids have been reported e.g. by Thomas (Thomas, 1966) or Hess and Parker (Hess and
Parker, 1966). Recently a new organo-metallic process has been established by Bönnemann
et al. (2003) which leads to Co particles covered with an aluminium oxide shell. This
shell has been shown to effectively suppress the oxidation of the Co particles allowing the
synthesis of long term stable ferrofluids. Meanwhile this process has also been employed
for the production of iron and iron-cobalt particles leading to ferrofluids with extremely
high magnetization values at low volume fraction of suspended magnetic material. An
additional advantage of this new method is the very narrow size distribution of the magnetic
particles achieved (see fig. 1.6). Moreover the mean particle size can be precisely adjusted
by well controlled changes in the production process. Thus a new class of ferrofluids has
become available which can enable new types of applications in the future.
Fig. 1.6. Electron microscopic picture of Co-particles and the corresponding size distribution (Bönnemann,
2005).
138
S. ODENBACH
Fig. 1.7. Typical agglomerate of particles in a biocompatible ferrofluid (Parak, 2005).
1.4. Ferrofluids for biomedical applications
As will be discussed later on, a field of recent research activities is the use of ferrofluids
for biomedical applications. It is obvious, that fluids employed for such applications have
to comply with the strong regulations applying for drugs in general. To achieve biocompatible fluids the carrier liquid has to be water. Moreover the choice of magnetic material
is also strictly limited and usually only magnetite is considered since magnetite particles
are approved as drugs. The major problem in the production of biocompatible ferrofluids
is the achievement of colloidal stability. Obviously the use of ionically stabilized fluids is
impossible since a fluid used in biomedical applications has to have a neutral pH. Thus
surfactant stabilization is needed, but the surfactant material has to be biocompatible too.
The usual way to stabilize magnetite particles in water is the use of starch as surfactant
(Bergemann et al., 1999). In contrast to the above discussed magnetite or cobalt particles
suspended for technical applications the particles stabilized by starch are not well separated but form agglomerates with a size of about 100 nm (see fig. 1.7). The low colloidal
stability of biocompatible ferrofluids represents a severe problem for the clinical use of the
fluids and is thus a centre point of actual research activities.
2. Basic properties of ferrofluids
In the last chapter the basic structure of ferrofluids has been discussed. The microscopic
make up of the fluids is of essential importance for their overall properties. In particular the
FERROFLUIDS
139
magnetic properties of ferrofluids, determining their reaction to magnetic fields, and their
basic viscous properties, describing their liquid behaviour, are of importance for all further
discussions on research and applications of magnetic fluids.
2.1. Magnetic properties
The magnetic properties of a ferrofluid are basically determined by the magnetic properties of the suspended particles. Concentrating on magnetite ferrofluids it is well known
that magnetite crystallizes in the inverse spinel structure leading to the fact that only the
Fe2+ ions contribute to the overall magnetic moment of the particles. The unit cell of
magnetite has a lattice constant of about 9 Å leading to an approximate volume of about
730 Å3 . Each unit cell contains 8 magnetite molecules with a magnetic moment of 4μB
(μB = 9.27 × 10−24 A m2 ; Bohr magneton). Assuming a particle size of 10 nm each particle contains about 5 × 103 molecules of magnetite. Since—due to their small size—the
magnetic nanoparticles are single domain particles, their overall magnetic moment equals
approximately m = 2 × 10−19 A m2 .
2.1.1. Equilibrium magnetization
If the concentration of the magnetic particles is not too high, effects of magnetic interparticle interaction can be neglected in the first step. In this case the magnetic particles—each
of them carrying a net magnetic moment m—can be treated as small thermally agitated permanent magnets in the carrier liquid. Therefore the magnetic behaviour of the whole liquid
will have paramagnetic character and can thus be described using Langevin’s equation for
the magnetisation of a paramagnetic system
1
= Ms L(α),
M = Ms ctgh α −
(2.1)
α
where Ms = φM0 denotes the saturation magnetisation of the ferrofluid given by the
product of the volume concentration φ of suspended magnetic material and its spontaneous
magnetisation (M0 = 4.5 × 105 A/m for magnetite). The character L denotes the Langevin
function with L(x) = ctgh x − 1/x. The so-called Langevin parameter
μ0 mH
(2.2)
kB T
gives the relation between the energy of a particle in the magnetic field H to its thermal
energy. The peculiarity of ferrofluids which distinguishes them from other paramagnetic
systems like paramagnetic salt solutions is given by the fact that the magnetic moment m
entering the Langevin parameter α (equation (2.2)) is not the magnetic moment of a single
molecule but the magnetic moment of the whole magnetic particle. Therefore it is about
four orders of magnitude larger than the values familiar from normal paramagnetic systems. If one recalls that the Langevin equation can be approximated by
α=
μ0 m
1 μ0 M02 V
1
φM0
H = χin H with χin = φ
(2.3)
3
kB T
3
kB T
for weak magnetic fields one can see immediately that the initial susceptibility χin of a
ferrofluid is also about four orders of magnitude larger than the one for common paramagnetic systems since χin is proportional to m. Using the values given above one can estimate
M=
140
S. ODENBACH
that the initial susceptibility of a ferrofluid containing 5 vol% of magnetite particles with a
mean diameter of 10 nm is in the order of χin ≈ 0.5. Due to this high initial susceptibility
even weak magnetic fields will induce strong magnetization in a ferrofluid and 90% of the
saturation magnetization can be easily reached with magnetic fields common for laboratory
use having a strength of about 50 kA/m as it is seen from fig. 2.1.
For normal magnetite ferrofluids the magnetization curve does not show any hysteresis
as it is expected for a pure paramagnetic system. The measure of the magnetization curve
provides fundamentally important data about the composition of the fluid. First of all the
saturation magnetization provides a direct measure of the volume concentration φ of the
suspended magnetic material in the fluid. The saturation magnetization is usually obtained
from the measured magnetization curve by plotting the magnetization versus 1/H since it
can be easily shown that equation (2.1) can be approximated by
1
kT 1
= Ms 1 −
M = Ms 1 −
(2.4)
α
μ0 m H
for large magnetic fields. Thus an extrapolation of M(1/H ) to 1/H → 0 leads to an immediate measure of Ms . If the saturation magnetization is known, equation (2.3) provides
an information about the mean particle size since the volume—and thus the mean particle
diameter—is the only unknown figure in the initial susceptibility and can thus be obtained
from the slope of the magnetization curve at low magnetic field strength.
Moreover, appropriate techniques can additionally provide an information about the size
distribution of the suspended particles by analysing the shape of the full magnetization
curve. Two different techniques have been established and proved to give reasonable results
on the size distribution. On the one hand Chantrell (1978) showed that the assumption of a
Fig. 2.1. Magnetization curve of the magnetite based ferrofluid APG513A. The inset shows the region of initial
susceptibility.
FERROFLUIDS
141
log-normal size distribution allows an easy approach by fitting
∞
L(α)f (y) dy,
M = Ms
(2.5)
0
with y = d/d̄ denoting a normalized particle diameter, f (y) representing the mentioned
log-normal size distribution in the form
(ln y)2
1
f (y) =
√ exp −
2σ 2
yσ 2π
and σ being the width of the distribution, to a measured magnetization curve. Another
method—established by Maiorov et al. (Maiorov, 1981)—is a multiparameter fit for superimposed Langevin functions
1
M = M0
(2.6)
φi ctgh αi −
,
αi
i
μ0 M0 Vi
αi =
H
kB T
with different particle volumes Vi and related volume concentration φi . By appropriate
numerical methods the ambiguity of the multiparameter fit can be well controlled and
the method called magnetogranulometry provides excellent size distributions without any
preconditioned shape of the distribution. The quality of size distributions obtained this
way can be estimated from the comparison between a distribution obtained from electron
microscopy with the one calculated with magnetogranulometry as shown in fig. 2.2.
Fig. 2.2. Size distributions of a ferrofluid as obtained by electron microscopy (full line) and magnetogranulometry
(circles) (after Wagener et al., 1999).
142
S. ODENBACH
Up to this point it has always been assumed that no interparticle interaction appears. But
for larger concentration of particles, greater particle diameter or a magnetic material with
higher spontaneous magnetization like e.g. Co, this assumption does not necessarily hold.
The easiest approach to account for the influence of interparticle interaction on the magnetization behaviour is a correction to the Langevin equation as it has been introduced e.g.
by Ivanov and Kuznetsova (2002) and which can be written for the weak field and the
saturation limit in the form
(4πχL )2
4πχL
+
H for H → 0
M = χL 1 +
3
144
(2.7)
φkB T
4π φkB T
M = Ms −
1+
for H → ∞
H
3 H2
using the Langevin susceptibility χL .
Besides this empiric approach various theoretical derivations have been made to account
for the dipolar interaction of the magnetic particles. In the Weiss model (O’Grady et al.,
1983) a virtual sphere around a particle is introduced in which no further particles exist.
Within the sphere the local field—needed to introduce the effect of dipolar interaction—is
calculated under the assumption that the magnetization and the inner field in the surrounding medium are constant. In an extended model (Huke and Lücke, 2000) this approach has
been modified in a way that the influence of the medium surrounding the particle is divided
into a part coming from larger distances and one from the nearer surrounding. The first
part is treated as in the Weiss model while the second part takes the influence of additional
dipoles inside the sphere surrounding the particle explicitly into account. The problem of
all these models is given by the fact that experiments are always made with polydisperse
suspensions, and the effect of polydispersity on the shape of the magnetization curve covers
the influences predicted by the models accounting for interparticle interaction.
If interparticle interaction becomes even stronger—i.e. if clusters or chains of magnetic
particles are formed even at vanishing magnetic field as it has been found for ferrofluids containing Co-particles with relatively large diameters—the correction of the initial
susceptibility as it is given in equation (2.7) will not be the only modification of the magnetization curve. In such fluids one finds even a small hysteresis of the magnetization curve
as shown in fig. 2.3. A comparison of the susceptibilities for the new curve and for the hysteresis loop at small fields provides here an estimate for the mean cluster size of particle
clusters formed in the system. For the example shown in fig. 2.3 this mean cluster size is
only about 1.6 particles/cluster for a ferrofluid containing Co-particles with a mean magnetic diameter of about 11 nm. It should be noted here that this estimate gives only a
reasonable value as long as the particles form straight aggregates. If rings are formed the
increase of initial susceptibility will only give the increased net magnetic moment of the
ring compared to a single particle but will not provide an information on the ring size.
For some effects in magnetic fluids which will be discussed afterwards the temperature
dependence of magnetization will be an important parameter. Starting from the Langevin
equation (equation (2.1)) one sees that temperature can influence the magnetization of a ferrofluid via two different means. On the one hand the spontaneous magnetization M0 of the
magnetic material is temperature dependent and on the other hand temperature enters M
FERROFLUIDS
143
Fig. 2.3. Region of initial susceptibility of a ferrofluid containing 10 nm Co-particles showing a slight hysteresis
(arrows indicate the direction of the change of magnetic field strength).
via the thermal energy of the particles. The first mentioned effect is usually negligible
for magnetite ferrofluids at temperatures around room temperature. Taking just the second
influence—the change of the thermal energy of the particles—into account, the change of
magnetization with temperature can be written as
∂M
1
μ0 mH
kT
(2.8)
= −Ms
.
−
∂T H
μ0 mH
kT 2 sinh2 α
Usually the absolute value of this change of magnetization with temperature, the pyromagnetic coefficient K = |∂M/∂T |H is taken as a characteristic parameter for the
temperature sensitivity of a ferrofluid. For small values of H the pyromagnetic coefficient
grows linear with field strength, reaches a maximum and decays as 1/H for large fields
as seen in fig. 2.4. For a normal magnetite ferrofluid the maximum of K is in the order of
50 A/mK−1 .
2.1.2. Relaxation of magnetization
Beside the general magnetic behaviour of a ferrofluid, expressed by its equilibrium
magnetization, the relaxation of magnetization, i.e. the reaction of the magnetization to
changes in field strength and direction is an important characteristic for a magnetic fluid.
In principle the relaxation of magnetization can take place by two fundamentally different
processes. On the one hand the magnetic moment of the particles can relax into magnetic
field direction by a rotation of the whole particle with the magnetic moment being fixed
relative to the particle. On the other hand a relaxation can also take place by a rotation of
the magnetic moment relative to the crystal structure of the particle without a rotation of
the particle itself. Both processes are characterized by typical relaxation times describing
how fast the respective relaxation leads to a reaction of magnetization on a change in field
direction.
144
S. ODENBACH
Fig. 2.4. The pyromagnetic coefficient of a ferrofluid containing 7 vol% magnetite particles with a diameter of
10 nm.
For the rotation of the whole particle in the carrier liquid—the so called Brownian
relaxation—the respective relaxation time is given by (Brown, 1963)
τB =
3Ṽ η
,
kB T
(2.9)
where η denotes the dynamic viscosity of the fluid and Ṽ is the hydrodynamic volume of
the particle, i.e. the volume including the surfactant layer Ṽ = π(d + 2s)3 /6. The second
process, called Néel-relaxation is characterized by the energy barrier the magnetic moment
has to overcome to flip relative to the crystal structure. The respective relaxation time reads
(Néel, 1949)
KV
τN = f0−1 exp
(2.10)
,
kB T
where K denotes the anisotropy constant of the particle and f0 is the Larmour frequency
of the magnetic moment in the anisotropy field of the particle. The relaxation process with
the smaller relaxation time will determine the magnetic relaxation behaviour of the fluid.
Thus for small particles relaxation will occur by the Néelian process. Such particles are
called magnetically soft. But while the Néel-relaxation time grows exponential with the
particle size, the respective time constant for the Brownian process scales only linear in Ṽ
and therefore a transition occurs at a critical particle size where the Brownian relaxation
becomes faster than the Néel process (see fig. 2.5). Particles relaxing by the Brownian
process are addressed as magnetically hard particles.
In the transition region around the critical diameter dcrit the actual relaxation time is
given by the geometric mean of the times for the single processes (Martsenyuk et al., 1974)
τN τ B
.
τeff =
(2.11)
τN + τB
FERROFLUIDS
145
Fig. 2.5. Brownian, Néelian and effective relaxation time of a ferrofluid calculated for 10 nm magnetite particles
in an oil with a viscosity of η = 0.05 kg/ms.
Fig. 2.6. The critical diameter for the transition from Néelian to Brownian relaxation for two different viscosities
(η = 0.05 kg/ms (full line) and η = 0.005 kg/ms (dashed line)) as a function of the anisotropy constant.
It is obvious that the critical diameter depends on the viscosity of the carrier liquid and in
particular on the anisotropy constant of the magnetic material employed. Figure 2.6 shows
the dependence of the critical diameter on the anisotropy constant for different viscosities.
Some typical magnetic materials used for ferrofluids are indicated.
146
S. ODENBACH
2.2. Viscous properties
For a liquid medium like a ferrofluid a second important basic characteristic has to be
discussed—its viscosity. For a magnetic liquid this discussion has two major aspects—on
one side the ferrofluid is a colloidal suspension and thus one has to look for the influence of
the suspended material on the fluid’s viscous properties. The second aspect is the influence
of the magnetic field on the viscosity and the general rheological behaviour of colloidal
suspensions of magnetic particles. The discussion of the second aspect will be postponed
to the chapter on magnetoviscous effects while here only the basic properties of the fluid,
i.e. its viscosity in the absence of a magnetic field will be in the focus.
The zero field viscosity of a suspension of magnetic particles, η0 , is mainly determined
by the viscosity of the carrier liquid ηc and the total volume concentration of suspended
material φ̃. It is well known, that the viscosity of a suspension can be described using
Einstein’s expression (Einstein, 1905)
5
η0 = ηc 1 + φ̃
(2.12)
2
as long as the volume fraction of suspended material is less than about 5%. As mentioned,
for a ferrofluid φ̃ accounts at least for the particles including their surfactant layer. Under
the typical assumption that the diameter of the particles is about 10 nm and the thickness of
the surfactant layer about 2 nm, equation (1.9) gives a content of magnetic material of less
than 2% for a fluid which could principally be described using equation (2.12). Moreover
it has to be taken into account, that the thickness s of the surfactant layer is just an estimate
and only few efforts have been undertaken to measure the surfactant layer structure in detail
(Kammel et al., 2001; Hoell et al., 2002). Since the value of s = 2 nm is a lower boundary
for the stability of a ferrofluid usually thicker layers may be employed and even a change
in the layer’s thickness of 10% makes a significant influence to the maximum of suspended
magnetic material allowed for the use of Einstein’s equation. In addition here it has only
been accounted for the particles and their surfactant layer when discussing the value of φ̃.
But usually ferrofluids contain excess surfactant necessary to keep the stabilization of the
fluid. Thus the nonmagnetic portion of the total volume fraction of suspended material
is even higher and it is obvious that the viscosity of a ferrofluid containing a reasonable
fraction of magnetic material will always require an extended description for its zero field
viscous behaviour.
Beyond numerous approaches to describe the dependence of the viscosity of a suspension for higher volume fraction of magnetic material e.g. by Batchelor (1970) or Chong
et al. (1971), Rosensweig (1985) developed a specialized approximation which is usually
used for ferrofluids. As common from approaches of other authors, Rosensweig makes a
correction to Einstein’s equation for higher volume fractions by introducing a quadratic
term in φ̃
−1
5
2
η0 = ηc 1 − φ̃ + bφ̃
(2.13)
,
2
5
b= 2
φ̃c − 1
φ̃c2
,
FERROFLUIDS
147
Fig. 2.7. Viscosity of a suspension normalized to the carrier liquid’s viscosity for low (left) and high (right)
volume concentration of suspended material.
where φ̃c denotes a critical concentration for which the suspension’s viscosity diverges.
Using φ̃c = 0.74 equation (2.13) gives reasonable results for ferrofluids with total volume
concentrations of magnetic material up to about 30% as shown in fig. 2.7 (left side).
For even higher concentrations up to φ̃ = 0.6 Pshenichnikov et al. (1998) used a modified Chong equation which over estimates the viscosity at low volume fraction up to
φ̃ = 0.3 but gives good results for higher values of φ̃ as seen in fig. 2.7 (right side).
It becomes clear that an increase of the amount of suspended magnetic material is not
necessarily an appropriate way to increase the magnetic reaction of a ferrofluid on a magnetic field since the increasing φ̃ leads to paste-like liquids, an effect which may have
serious drawbacks in technical use.
3. Basic description for the field dependent behaviour of ferrofluids
In the frame of the discussion of the equilibrium magnetization of ferrofluids it has been
seen, that a high initial susceptibility is the basic peculiarity of magnetic fluids. This high
value for χin leads to high magnetization of a ferrofluid in magnetic fields of the order of
some ten milli-Tesla. Recalling that the magnetic force density fm acting on a magnetised
body with magnetization M in a magnetic field gradient ∇H is given by the Kelvin force
fm = μ0 M∇H = μ0 χin H ∇H
(3.1)
one recognizes that it is proportional to χin for small magnetic field strength. Assuming
a standard ferrofluid with an initial susceptibility χin ≈ 1.6 and a density of about ρ =
1.3 × 103 kg/m3 one can calculate the magnetic force density and compare it with the
force density in the gravitational field. In a magnetic field with a strength of 20 kA/m, as it
can easily be produced with an electromagnet with a steel pole shoe as shown in fig. 3.1,
having a typical field gradient ∇H = 7 × 105 A/m2 , the magnetic force density equals
approximately 28 kN/m3 . For comparison, the force density in the gravitational field is only
about 13 kN/m3 for the mentioned standard ferrofluid. Thus the magnetic force generated
by a simple magnetic field is strong enough to compete with—and even overcome—the
action of gravitational forces on the fluid.
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S. ODENBACH
Fig. 3.1. Illustration for the action of a magnetic field gradient on a ferrofluid. The left side shows a sketch of the
experimental setup and the right side the behaviour of the ferrofluid in applied magnetic fields.
Figure 3.1 shows this by the attraction of a ferrofluid towards the pole piece of an electromagnet. The series of photographs illustrates how the fluid is attracted and flows under
the influence of the magnetic field towards the pole shoe. The characteristic spike pattern
is a result of the interaction of magnetic and gravitational forces with the fluid’s surface
tension. As mentioned the fluid flows under the influence of the magnetic field against the
direction of the gravitational force towards the pole shoe. That means that the magnetic
force is able to drive a flow in the fluid while the fluid remains liquid under the influence of
the magnetic field. Thus a direct magnetic control of the flow of a ferrofluid by moderate
magnetic fields is possible.
3.1. Basic ferrohydrodynamic equations
From the hydrodynamic point of view this force, driving a flow in a magnetic fluid, gives
an additional term in the Navier–Stokes equation. For a ferrofluid this basic equation for
fluid mechanic description has to be written in the form
dv
= −∇p + ν∇ 2 v + μ0 M∇H,
(3.2)
dt
where v denotes the flow velocity, p the pressure in the fluid, ν its kinematic viscosity and
dv/dt = ∂ v/∂t + v∇ v is the total derivative of the velocity field.
Due to the non-linearity the Navier–Stokes equation is usually not a suitable tool to
illustrate the action of a magnetic field on the flow of a ferrofluid. For steady flow it is much
easier to make use of Bernoulli’s equation which is nothing else than the energy equation
for a fluid in flow. For a magnetic liquid this equation has to include a term describing the
magnetic energy density of a ferrofluid in a magnetic field
EH = −μ0 MH.
(3.3)
FERROFLUIDS
149
Fig. 3.2. An example for the use of the ferrohydrodynamic Bernoulli equation. A ferrofluid subjected to the field
of a horse shoe magnet. Explanations are given in the text.
Therefore the Bernoulli equation for a ferrofluid reads
ρ
p + v 2 + ρgh − μ0 MH = const,
(3.4)
2
where ρgh denotes the potential energy density of a fluid with density ρ in the gravitational
field with a height h measured relative to the reference level for the potential energy.
The magnetic extensions for the basic hydrodynamic equations of a ferrofluid have been
first formulated by Rosensweig (1985) and have been—together with the equations for
relaxation of magnetization (Shliomis, 1974)—the basis of the classical description of the
dynamics of ferrofluids.
To get a first insight to the importance of the Bernoulli equation for the description of
the behaviour of ferrofluids in a magnetic field one can have a look at two simple examples
showing a ferrofluid in a static situation as well as the action of a field on a simple steady
flow.
In the first example (see fig. 3.2) equation (3.4) will be used to determine the height
of rise of a ferrofluid between the poles of a horse shoe magnet. In between the poles a
homogeneous magnetic field H is assumed and the poles are far enough from each other
to allow the neglect of capillary effects. Outside the magnet the magnetic field vanishes
(H = 0). The pressure at the surface of the fluid, in particular at the two positions marked
1 and !
2 is the ambient air pressure p0 . The velocity of the fluid is zero everywhere. For
!
1 and !
2 gives
this situation equation (3.4) applied to positions !
p1 + ρgh1 = p2 + ρgh2 − μ0 M̄H,
(3.5)
where the indices indicate the respective values at the marked positions and M̄ is the mean
magnetization in the field H
M̄ =
1
H
H
M(H ) dH,
(3.6)
0
an approach which accounts for the non-linearity of M(H ) which becomes important for
strong magnetic fields. With p1 = p2 = p0 and using h = h2 − h1 one obtains from
150
S. ODENBACH
Fig. 3.3. The magnetic nozzle. Explanations are given in the text.
equation (3.5)
h =
μ0 M̄H
.
ρg
(3.7)
In the second example a free ferrofluid jet entering a homogeneous magnetic field parallel to the flow direction as shown in fig. 3.3 is considered. In this situation the magnetic
field will induce a reduction of the cross section of the jet, an effect commonly called mag1 and !
2 , one inside the
netic nozzle (Rosensweig, 1985). Again we chose two positions !
magnetic field and the other one outside with related fluid velocities v1 and v2 and the
2 only. The magnetic Bernoulli equation
magnetic field of strength H acting on position !
reads in this situation
ρ
ρ
p1 + v12 = p2 + v22 − μ0 M̄H,
(3.8)
2
2
where it is assumed that the height h in the gravitational field is equal at both points. Again
the pressures at both positions equals the ambient air pressure p0 and thus the change in
velocity can be written as
v22 − v12 =
2μ0 M̄H
.
ρ
(3.9)
Employing the equation of continuity one obtains a relation between the velocities at
1 and !
2 and the respective jet diameters a1 and a2
positions !
a2
v2
= 12 .
v1
a2
(3.10)
The last two equations together allow the determination of the change in the jet diameter
induced by the magnetic nozzle to be
a1
2μ0 M̄H 1/4
(3.11)
= 1+
.
a2
ρv12
For a standard ferrofluid, a magnetic field strength of 20 kA/m and an initial flow velocity
of v1 = 0.1 m/s one obtains with equation (3.11) a reduction of the jet diameter by a factor
of about 2.
FERROFLUIDS
151
3.2. An overview on applications of magnetic fluids
It is obvious, that a magnetic force, which can be created by simple technical means and
which gives rise to significant changes in the flow behaviour of a fluid will force the development of related applications. Astonishingly most of these applications—and in particular
those which gained the highest commercial value—make only use of the possibility to position a ferrofluid by means of a magnetic field. The change of the properties of the fluid and
a direct magnetic driving of a flow are yet not used to design practical applications. Thus
the discussion of applications will be subdivided in two major parts—those applications
which are already realized will be discussed here while future applications, being in the focus of recent research will be addressed in the context of the related basic research fields.
Due to the large number of applications proposed for ferrofluids we’ll here only be able to
give representative examples for various classes of applications. A good overview on the
possibilities resulting from the magnetic control of suspensions of magnetic nanoparticles
can be found e.g. in articles of Berkovsky et al. (1993), Berkovsky and Bashtovoy (1996)
and Rosensweig (1985).
3.2.1. Sealing applications
The most famous mechanical application is the sealing of a rotary feed through using
a liquid seal given by a ferrofluid in a strong magnetic field. In this application a rotating
axis is surrounded by permanent magnets as indicated in fig. 3.4. The arrangement of the
magnets is chosen in a way that a strong magnetic field gradient is generated in the small
gap between the axis and the magnet ring close to the permanent magnets. A small amount
of ferrofluid put into this gap will be fixed by the field gradient and form a kind of liquid
O-ring. The advantage of this arrangement compared with e.g. oil seals is obvious. Since
the ferrofluid remains liquid, the rotating axis experiences only the viscous friction instead
of solid body friction in conventional seals. This reduces the power consumption for the
rotation as well as the generation of heat in the system. Such seals are nowadays standard
for sealing of hard disk drives and vacuum feed throughs e.g. in the production lines for
compact discs.
An important characteristic of a ferrofluid seal is the burst pressure, i.e. the maximum
pressure difference the seal can keep. This pressure difference can easily be calculated
Fig. 3.4. Schematic sketch of a ferrofluid seal.
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S. ODENBACH
Fig. 3.5. The sealing region of a ferrofluid seal enlarged to indicate the points needed for the calculation of the
burst pressure. Explanations are given in the text.
using the Bernoulli equation (equation (3.4)) for magnetic fluids. For this fig. 3.5 gives a
1 –!
4 needed to set up equation (3.4). In
closer look to the ferrofluid O-ring with positions !
1 ,!
2 ) and low pressure on
the figure high pressure is assumed on the left side positions (!
3 , !
4 ). The Bernoulli equation reads for this situation for the positions !
2
the right side (!
3
and !
p2 − μ0 M̄(H2 ) H2 = p3 − μ0 M̄(H3 ) H3
or
p2 − p3 = μ0 M̄(H2 ) H2 − M̄(H3 ) H3 .
(3.12)
Here it has been assumed that the magnetic field is homogeneous and tangential to the
1 –!
2 and !
3 –!
4 , gravitational effects are neglected and that the rotating axis is at
surfaces !
rest providing the static burst pressure in this approach.
1 –!
2 and !
3 –!
4 —the pressures inside and
Since—for stability reason of the interfaces !
outside the liquid at the interfaces have to be equal
p1 = p2 ;
p 3 = p4 .
Equation (3.12) provides directly the pressure difference between the high and low pressure
region separated by the seal
H2
p1 − p4 = μ0
M dH.
(3.13)
H3
This pressure difference becomes maximal if the design is chosen in a way that the field
3 is as low as possible, i.e. in an optimised situation H3 → 0, and it
strength at point !
2 to saturate the fluid’s magnetization. In this optimised
should be strong enough at point !
case the maximum pressure difference the seal can stand can be written as
p = p1 − p4 = μ0 M̄H,
FERROFLUIDS
153
Fig. 3.6. Schematic sketch of the levitation of a nonmagnetic body in a ferrofluid under influence of a field
gradient. Explanations are given in the text.
with H being the field strength in the centre of the gap and in the middle of the magnet
arrangement. For a typical one stage seal with a highly concentrated ferrofluid with a saturation magnetization of Ms = 56 kA/m and with a magnet arrangement providing about
1.8 T in the gap the burst pressure in the seal can easily reach 1 bar.
3.2.2. Ferrofluids for separation purposes
Another interesting application of ferrofluids is the density separation of materials. Here
use is made of the fact that a nonmagnetic body in a magnetic fluid exposed to a magnetic
field gradient will experience a force antiparallel to the magnetic field gradient. If the field
gradient is directed parallel to gravitational acceleration as depicted in fig. 3.6 this force
will counteract the gravitational force acting on the body (Rosensweig, 1966; Odenbach,
1998a, 1998b). Depending on the body’s density it will float in a certain depth in the fluid.
In separation technique this phenomenon is commonly described by an apparent density of
the ferrofluid induced by the field gradient
ρapp = ρ +
μ0 M∇H
.
g
(3.14)
If the field gradient is inhomogeneous over the height of the fluid sample, e.g. if it
increases towards the bottom of the fluid beaker like it will happen if simply a permanent
magnet is placed below the beaker as in fig. 3.7, the apparent density will vary over the
height too. Therefore nonmagnetic objects of different density immersed in the fluid will
float in different depth’ allowing their separation.
Despite the fact that this technique has been proposed quite early (Rosensweig, 1966)
the high price of ferrofluids has hindered its use for the separation of materials with not too
high value as proposed by Gogosov et al. (1994). Only in situations where the separation
of high value material like ores (Smolkin et al., 1992) or diamonds (Svoboda, 2000) from
low value stuff like excavation residues has to be done automatically with high separation
154
S. ODENBACH
Fig. 3.7. Illustration of the levitation of a nonmagnetic body in a ferrofluid. Left side—no magnet below the
ferrofluid beaker, right side—magnet below the beaker creating a levitation field gradient. Levitation is indicated
by the position of the pointer.
Fig. 3.8. Schematic sketch of a tilt sensor employing ferrofluid. Explanations are given in the text.
efficiency the magnetic fluid separation technique can be a suitable tool. In diamond mining
it is nowadays used on a commercial basis.
3.2.3. Ferrofluids for sensor applications
Numerous applications have been proposed using ferrofluids as active medium in sensor
devices. As an example which principally represents the basic idea of such sensors fig. 3.8
shows a tilt sensor (Kubasov, 1994) using a ferrofluid. The fluid partly fills a cylindrical
container surrounded by two different groups of coils. One of the coils magnetizes the fluid.
The second set of coils consists of two separate sensor coils covering the right and left end
of the container. If the sensor is tilted the fluid level will be different on both sides leading
to different fluid volumes in the two sensor coils. Since the impedance of the sensor coils
FERROFLUIDS
155
Fig. 3.9. Schematic sketch of the cooling of a loudspeaker with ferrofluid (left) and the power dependence of the
speaker’s temperature with and without ferrofluid (right, after Berkovsky and Bashtovoy, 1996).
depends on the amount of magnetic fluid inside the coil they will show different impedance
values and the difference can be used as a measure for the tilt of the sensor.
The great backdraw of sensors employing ferrofluids as active medium is the nonlinearity of the sensor signal induced by the non-linearity of the magnetization curve. This
makes the use of such sensors in feedback loops more difficult than the use of conventional sensor types. Moreover ageing of the fluids can change the sensor properties in an
uncontrolled way leading to high maintenance efforts. Due to these reasons ferrofluid sensors have—in contrast to other applications of magnetic fluids—not reached commercial
importance.
3.2.4. Ferrofluids as heat transfer medium
The field of application with highest amount of ferrofluid used nowadays is the cooling
of loudspeakers. A loudspeaker, as shown schematically in fig. 3.9, consists mainly of a
membrane driven by the voice coil which oscillates in a permanent magnet system when
subjected to a time modulated current through the coil. The amplitude of the membrane and
thus the power of the acoustic output is determined by the strength of the current through
the coil. Therefore high power loudspeakers or loudspeakers with small dimensions will
always face thermal problems due to the ohmic heat produced in the coil. If the coil oscillates in air the generated heat can not efficiently be transferred to the outer structure which
can be cooled by external fans or other cooling means. The use of a normal liquid as cooling medium with significantly enhanced thermal conductivity is impossible since there is
no way to locate the fluid in the gap between the voice coil and the permanent magnet.
The magnet system of a loudspeaker creates on the other hand a strong magnetic field in
the vicinity of the voice coil and especially in the gap between coil and magnet. Thus a
ferrofluid can be located there being held by the magnetic forces. The voice coil will now
oscillate in the ferrofluid which only slightly damps the oscillation by viscous friction—an
156
S. ODENBACH
Fig. 3.10. Schematic sketch of a ferrofluid bearing.
effect which is even favourable to reduce parasitic oscillations and improve the sound quality. The central advantage of the use of a ferrofluid in this application is given by the fact
that it increases the heat transfer from the coil towards the structure leading to a significant
enhancement of the maximum power achievable with a certain speaker system. It should be
noted, that this application requires specialized ferrofluids which withstand temperatures
up to 150◦ C like they are common close to the thin wires of loudspeakers used in car HiFi
systems.
3.2.5. Ferrofluids for positioning applications
As last example for a class of applications of magnetic fluids the positioning of objects
in different application fields should be mentioned. Within the discussion of separation
applications the phenomenon of levitation of non-magnetic bodies in a ferrofluid subjected
to a field gradient has already been addressed. This phenomenon can not only be used to
separate materials by density but also to create bearings with low friction or positioning
systems.
Figure 3.10 shows schematically the principle of a ferrofluid bearing (Berkovsky and
Bashtovoy, 1996). The rotating rod, made of non-magnetic material, is surrounded by a
set of ring magnets creating strong magnetic field gradients in the gap between rod and
magnet system. If a ferrofluid is filled into the gap the rod will levitate in the fluid. Since
the magnetic field gradient increases radially from the centre of the system to the surface
of the magnets, the rod will experience a radial force directed to the centre of the bearing.
Since this force acts equally from all sides the rod will be kept in the centre. By proper
design of the magnet system keeping the gap between rod and magnets thin, such bearings
can withstand strong radial forces without displacement of the rod.
Another kind of application of the phenomenon of levitation is its use in micropositioning systems. In such systems as shown in fig. 3.11 the levitation in an electromagnetically
controlled field gradient against gravitational action is employed (Uhlmann and Bayat,
2003). Using modern control algorithms the positioning can be as precise as a few microns
with setting times in the order of a millisecond for static loads up to about 100 g. The
advantage of such a micro positioning system compared with e.g. piezzo positioners is on
the one hand the large adjustment travel of the positioner and in addition the fact that a
“soft positioning” takes place. The softness results from the fact, that the positioner is levitated on a liquid medium and not held by a stiff actuator. Thus an additional force acting
FERROFLUIDS
157
Fig. 3.11. 3D-drawing of a high precision positioning system using the levitation effect (Uhlmann and Bayat,
2003).
on the positioner will result in a soft answer of the system. That makes it favourable for
the handling of sensitive devices like glass capillaries etc. Using stronger fields and larger
dimensions of the system even fine positioning systems for machine tools providing loads
of more than 10 kg can be designed that way (Hesselbach and Abel-Keilhack, 2003).
4. Magnetic control of fluid properties and fluid flow
4.1. Field dependent viscous properties of ferrofluids
In the preceding chapters it has been seen, that the magnetic forces acting on ferrofluids
are strong enough to enable a certain magnetic control of the fluids. Nonetheless it has not
been addressed whether the properties of magnetic fluids in a magnetic field will change
or whether it is practically possible to drive and control the flow of a ferrofluid by the
means of magnetic forces. Thus it is the scope of the present and the following chapter to
address these problems and to show that magnetic fields can have significant influence on
the properties as well as the flow of ferrofluids.
4.1.1. The magnetoviscous effect
In the frame of the discussion of relaxation of magnetization of a ferrofluid it has been
shown that one has to distinguish between the Néelian and the Brownian relaxation type.
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S. ODENBACH
Fig. 4.1. Sketch for the explanation of the rotation of the magnetic particles in a shear flow.
Concentrating on particles following the Brownian relaxation it is obvious that the process
of relaxation of magnetization will influence the flow of a fluid since a rotation of a particle in the carrier liquid will cause a motion of the liquid itself due to the viscous friction
between the particle and the surrounding fluid. The other way around a flow can also influence the actual magnetization of a magnetic liquid containing Brownian particles. This
can easily be understood if one considers a ferrofluid under the influence of a shear flow.
The velocity gradient in the shear flow will induce a rotation of the magnetic particles (see
fig. 4.1).
If the magnetic moment is spatially fixed in the particle, that means the relaxation time
for the Néel process has to be larger than the Brownian relaxation time, the rotation of the
particle will also cause a temporal change of the direction of the magnetic moment of the
particle. If a magnetic field is subjected to the fluid the magnetic moments of the particles
will tend to align with the magnetic field direction.
Looking to the particular situation where the magnetic field direction is parallel to the
vorticity = 1/2rotv of the flow (fig. 4.2), i.e. parallel to the axis of rotation of the
particles in the flow, the magnetic moment can easily align with the field direction while
the particle is rotating freely within the flow. In this situation the particle’s rotation will
not affect the alignment of the magnetic moment with the field direction which is only
determined by the relation between the thermal energy of the particle and the energy of the
magnetic moment in the field. Macroscopically the magnetization of the fluid will equal
the equilibrium value given by the magnetization curve.
The situation changes significantly if the magnetic field is directed perpendicular to the
vorticity of the flow (fig. 4.3). Now the magnetic field tries to align the magnetic moment
in a direction perpendicular to the vorticity. The viscous friction exerted by the fluid on
the particle causes the particle’s rotation leading to a finite angle between the magnetic
moment and the field direction. This finite angle gives rise to the appearance of a magnetic
torque
D m = μ0 m × H
(4.1)
which counteracts the mechanic torque exerted by viscous friction. Thus, the action of the
magnetic field leads here to a hindrance of the free rotation of the magnetic particle in
FERROFLUIDS
159
Fig. 4.2. On the explanation of rotational viscosity—a ferrofluid in a shear flow with a magnetic field aligned
parallel to the vorticity of the flow. Explanations are given in the text.
Fig. 4.3. On the explanation of rotational viscosity—a ferrofluid in a shear flow with a magnetic field aligned
perpendicular to the vorticity of the flow. Explanations are given in the text.
the flow. Macroscopically this means that an increase of the viscosity of the fluid will be
observed which should be accompanied by a deviation of the magnetization of the fluid
from its equilibrium value.
It should be noted, that the above qualitative argument bases on two major assumptions.
First it has been stated that the magnetic particles are assumed to be of Brownian type.
Particles relaxing by the Néel process would simply rotate around the magnetic moment
aligned with the field direction and in the given picture this would not cause a change of the
viscosity of the fluid. Nonetheless the rotation of the particle around the magnetic moment
would cause a certain energy dissipation giving an increase of measured viscosity too,
but the respective effect can be assumed to be significantly smaller than the phenomenon
discussed above and thus the requirement of magnetically hard particles has to be kept as
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S. ODENBACH
a basic assumption for the further discussion. The second fundamental constraint made
above is given by the fact that only single particles have been considered. That means
that interparticle interaction leading to formation of any kind of clusters is excluded at
this point. Practically this means that the discussion concerning the increase of viscosity
caused by the hindrance of rotation of single magnetic particles has to concentrate on
highly diluted systems.
4.1.2. The magnetoviscous effect in diluted ferrofluids
The first discovery of an enhancement of viscosity in a highly diluted suspension of
magnetic particles has been published in 1969 by McTague (McTague, 1969). In his experiments a ferrofluid containing as less as 0.05 vol% of Co-particles with a mean diameter
of at least 6 nm in a low viscous carrier has been used. This is clearly a highly diluted suspension in which interparticle interaction can be neglected and—as seen from fig. 2.6—the
particles are magnetically hard. Thus both basic assumptions made above are fulfilled. The
investigations have been carried out using a capillary viscosimeter placed in the gap of an
electromagnet. Laminar tube flow exhibits an azimuthal vorticity of the form
v0z r
(4.2)
,
a2
where v0z is the maximum flow velocity in the centre of the tube, a is the tube’s diameter
and r is the radial coordinate. Since the vorticity is directed azimuthally a magnetic field
parallel to the flow direction will cause perpendicular arrangement of vorticity and field
while a field perpendicular to the flow will result in an angle between H and which varies
over the cross section of the tube. As seen above, the angle between field and vorticity has
significant influence on the viscosity changes which are to be expected and thus the second
arrangement requires an averaging of the angle over the cross section of the tube. This
averaging leads to the conclusion, that the increase of viscosity for the arrangement H ⊥v
should be half as strong as for the situation with H ||v. Figure 4.4 shows the experimental
results of McTague’s investigations, clearly verifying the factor 2 in the magnitude of the
effect of the magnetic field on viscosity.
McTague’s paper has been accompanied by a first approach for a theoretical explanation
based on the discussed microscopic picture and trying to treat the additional portion of
viscosity as a correction term to Einstein’s formula for the viscosity of suspensions (equation (2.12)) (Hall and Busenberg, 1969)
3
5
η(H ) = η0 1 + φ̃ + φ sin2 ε ,
(4.3)
2
2
tube =
where φ denotes the volume fraction of the particles with the surfactant while φ̃ is as
before the total volume fraction of all suspended material, i.e. including e.g. additional
excess surfactant. The parameter ε describes the influence of the magnetic field using the
angle β between H and and the ratio between the magnetic and the viscous torque acting
on the fluid
ξ −1 =
μ0 mH
4πη0 d 3 γ̇
(4.4)
FERROFLUIDS
161
Fig. 4.4. The change of viscosity in a Co-ferrofluid measured in a pipe flow with the field directed in flow
direction (full circles) and perpendicular to flow direction (stars) (after McTague, 1969).
with the velocity gradient γ̇ —commonly called shear rate—leading to
1
sin ε = 1 + ξ −2 −
2
2
1
2
1 + ξ −2 − ξ −2 sin2 β
4
1/2
.
(4.5)
For strong magnetic fields equation (4.3) describes well the factor 2 in the magnitude
of the effect observed by a change of the field direction since equation (4.5) reduces to
sin2 ε = sin2 β for large values of H . But it fails to predict the behaviour of the fluid at low
field strengths since it does not account for the thermal energy of the particles which is of
severe importance for nanosized suspended particles. In particular equation (4.3) predicts
saturation of the effect for magnetic field strengths well below the values observed in the
experiment.
The problem has been overcome by Shliomis (Shliomis, 1972) introducing the internal
angular momentum density
S = I ωp
(4.6)
with I being the sum of the moments of inertia of the particles per volume and ωp their
averaged angular velocity as a new variable.
This leads to a new formulation of the Navier–Stokes equation for magnetic fluids which
accounts not only for the magnetic force acting on the fluid (as in equation (3.2)) but also
for the relative rotational movement between particles and carrier liquid
dv
1
S
ρ
= −∇ p + (MH ) + (S − I ) + η∇ 2 v + (M∇)H
dt
2
I
1
+
(4.7)
rot(S − I ).
2τs
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S. ODENBACH
Here τs denotes a typical relaxation time for rotational movement, i.e. a typical time
in which the angular velocity of the particle relaxes to the equilibrium value given by the
vorticity of the flow.
The concept of internal angular momentum has then been used to calculate the extra
portion of viscosity due to the hindrance of the rotation of the particles, an effect which is
called rotational viscosity ηr (Shliomis, 1972). The result of the calculation reads as
3 α − tanh α 2 φ η0
sin β
(4.8)
2
α + tanh α
with η(H ) = η(H =0) + ηr , α being again the ratio of magnetic and thermal energy of the
particles (equation (2.2)) and . . . denoting the spatial average over a certain quantity.
The viscosity η0 is the viscosity of the suspension in the absence of a magnetic field as
it has been discussed in the context of equations (2.12), (2.13). Obviously equation (4.8)
accounts for the effect of relative alignment of H and and since tanh α vanishes for
H → ∞ it provides the already proven change of magnitude of rotational viscosity with
β (see fig. 4.4).
Denoting the relative change of viscosity with field strength as
η = ηr =
R=
η(H ) − η0
ηr
η(H ) − η(H =0)
=
=
η(H =0)
η0
η0
(4.9)
one can make predictions for this measurable quantity for the special cases of low field
strength and saturation. For saturation it has already been mentioned that tanh α vanishes
and thus equation (4.8) with equation (4.9) becomes
3 2 φ sin β .
(4.10)
2
This allows to give an upper border for the effect of rotational viscosity which can be
achieved for β = π/2, i.e. perpendicular arrangement of H and .
R(H →∞) =
3 φ.
(4.11)
2
That means that for a standard ferrofluid—neglecting any interparticle interaction—a
maximum change of viscosity of about 30% should be the upper limit for the effect. For
the situation of low field strength equation (4.8) can be approximated by
Rmax =
R=
1 2
φα
4
(4.12)
again assuming optimal alignment of H and , i.e. β = π/2. This gives a quadratic
increase of the change of viscosity with field strength for weak magnetic fields as it is
proven in fig. 4.5 for a commercial ferrofluid with weak interparticle interaction at low
field strength.
4.1.3. Negative viscosity
In the introduction of the present chapter is has shortly been mentioned, that the rotation
of a magnetically hard particle caused by the relaxation process of magnetization will
force a movement of the surrounding fluid due to the viscous friction between the particle
FERROFLUIDS
163
Fig. 4.5. The change of viscosity in a ferrofluid for weak magnetic field as a function of the square of H .
and the liquid. Using appropriate alternating magnetic fields a permanent rotation of the
magnetic particles may be induced leading to a spin up flow in the fluid. The rotation
speed ωp of the particles in a high frequency magnetic field can be calculated on the basis
of the fundamental ferrohydrodynamic equations (Shliomis, 1972). Shliomis and Morozov
carried out such calculations leading to (Shliomis and Morozov, 1994)
α 2 1 − ω2 τB2
ωp = 1 −
(4.13)
,
6 (1 + ω2 τB2 )2
where ω denotes the frequency of the alternating magnetic field, τB the Brownian relaxation time of the particles and the vorticity of the flow.
While a static field leads to a hindrance of rotation of the particles and results in an
increase of viscosity, an alternating field is able to drive the particles and this externally
driven rotation can be expected to lead to a macroscopically observable decrease of viscosity. Using an intuitive description for rotational viscosity (Bacri et al., 1995a, 1995b)
3 − ωp
φ η0
,
(4.14)
2
relating the change of viscosity to the difference between the angular velocity of the particles and the vorticity of the flow one sees immediately that magnetic particles rotating
slower than the flow will give rise to a positive value of ηr while particles with ωp > will force a negative rotational viscosity and thus lead to a decrease of viscosity in the field
compared with the zero field value. Using the angular velocity of the particles in an alternating field as given in equation (4.13), one obtains for the change of viscosity (Shliomis
and Morozov, 1994)
ηr =
ηr =
1 − ω2 τB2
1
.
η0 φ α 2
4
(1 + ω2 τB2 )2
(4.15)
164
S. ODENBACH
Fig. 4.6. Frequency dependence of the change of viscosity of a ferrofluid with field strength in an alternating
magnetic field (after Bacri and Perzynski, 1995).
For a static field (ω = 0) equation (4.15) reduces to the low field approximation of the rotational viscosity given in equation (4.12). But with increasing values of the field frequency
equation (4.15) predicts a decrease of ηr with ηr = 0 for ωτB = 1. If the field frequency
exceeds the value of ω = 1/τB the viscosity change caused by the field becomes negative
leading to a reduction of the viscosity of the fluid below its zero field value. This possibility
to reduce the viscosity of a ferrofluid by means of a magnetic field lead to the expression
“negative viscosity” for this phenomenon (Shliomis and Morozov, 1994). Looking to the
relaxation times given in fig. 2.5 it becomes clear that an experimental proof with magnetite
ferrofluids would require relatively large particles and high magnetic field frequencies in
the order of 10 kHz. Therefore Bacri et al. (1995a, 1995b) used Co-ferrite particles with a
diameter of about 10 nm. Due to the high crystal anisotropy of Co ferrite the Néel process
is effectively blocked for 10 nm particles and the relaxation times are enhanced due to
slight particle interaction leading to reasonable frequencies in the order of 1 kHz to reach
the resonance condition ωτB = 1. Figure 4.6 shows the experimental results of these experiments, well proving the reduction of the viscosity increase with increasing field frequency
and even showing the appearance of negative values of ηr for high frequency fields.
4.1.4. Magnetoviscous effects in concentrated ferrofluids
Up to this point we’ve discussed the change of viscosity of a ferrofluid in a magnetic
field just for relatively idealized fluids which are diluted and in which interparticle interaction does not give a significant contribution to the properties of the fluids. This leads to
interesting effects but the magnitude of the field dependent changes was relatively low not
favouring a technical use of these phenomena. In contrast it is known that magnetorheological fluids containing micronsized magnetic particles exhibit tremendous changes in their
viscous behaviour which can be accounted for by strong interparticle interaction leading to
formation of complex particle networks (Bossis et al., 2002).
FERROFLUIDS
165
The question that has to be risen at this point is whether commercial ferrofluids, containing high volume fraction of magnetic particles, will behave like normal ferrofluids or
whether they will exhibit a more complex rheological behaviour due to non vanishing interparticle interaction. Very early experiments by Rosensweig et al. (1966) showed that a
strong increase of viscosity can be observed in a commercial ferrofluid. But since the paper
by McTague (1969) published about eight weeks after Rosensweig’s work focused on an
aspect of field dependent viscosity changes providing an easier theoretical approach due
to the exclusion of complex interparticle interaction, the further research in that field focussed on rotational viscosity while the changes in concentrated fluids weren’t investigated
further.
This changed more than 20 years later when experiments were undertaken which measured the change of viscosity in a commercial ferrofluid at high shear rate (Ambacher et al.,
1992). The results, shown in fig. 4.7, gave a strong increase of viscosity even at low field
strength. Testing, whether these experimental results can be explained with Shliomis’ theory (equation (4.8)) one has to recall the basic assumptions made for the derivation of the
expression for rotational viscosity. This has mainly been the restriction that only magnetically hard particles can contribute to the viscosity changes. From fig. 2.6 it is seen that this
is valid for magnetite particles with diameters of more than about 13 nm only. Taking into
account the size distribution of magnetic particles shown for the fluid used in the paper of
Ambacher et al. (1992) as an inset in fig. 4.7 one finds that only a small portion of particles has to be considered for the comparison of the experimental data with equation (4.8).
Taking this fluid data, the theory provides the full line in fig. 4.7, which obviously disagrees significantly with the experimental results. On the other hand fitting equation (4.8)
to the experimental values, leaving the size of the particles as a fitting parameter one gets
an excellent agreement with a mean particle size of 16 nm. This result leads to two major
conclusions: first, the possibility to fit equation (4.8) to the data obtained with a concentrated fluid shows, that the concept of hindrance of rotation of magnetic objects by the
action of a magnetic fields holds even in the present experimental situation. Secondly the
large particle size obtained from the fit leads to the assumption that interparticle interaction
forces the formation of particle structures like chains or other small agglomerates with a
mean size of 2–3 particles, dominating the change of viscosity in such fluids.
Since the microscopic background of the effects observed here is basically different
from the microscopic assumption for the derivation of rotational viscosity (Shliomis, 1972)
the term “magnetoviscous effect” has been introduced (Odenbach, 2002) to distinguish
the phenomena in concentrated ferrofluids with significant interparticle interaction from
those effects appearing in highly diluted fluids with non interacting particles. In the further
process of the investigations of magnetic field dependent changes of the rheological properties of ferrofluids three major goals had to be addressed. Numerous experiments were
devoted to the investigation of the microscopic reasons for the effects observed, a second
line of interest was devoted to the clarification of the overall rheological behaviour of the
ferrofluids with interparticle interaction while the most recent efforts have the goal to optimise the effects for technical use on the basis of the fundamental understanding of the
phenomena obtained with the first two research lines.
166
S. ODENBACH
Fig. 4.7. The change of viscosity in a commercial ferrofluid (APG513A). The full line indicates the calculated
change using Shliomis’ theory and the fluid data (particle size distribution shown in the inset). The dashed line is
a fit of the theory with the particle diameter as free fitting parameter (after Ambacher et al., 1992).
4.1.5. The rheological behaviour of ferrofluids
To get a deeper insight in the rheological behaviour of magnetic fluids a rheometer is required allowing the determination of shear rate dependent changes of the magnetoviscous
behaviour. Since the effects expected are significantly smaller than in magnetorheological
fluids, commercial rheometers designed for MR-investigations are not suitable here. Therefore the results discussed in the following have been obtained with a specialized equipment
dedicated to the investigation of ferrofluids (Odenbach et al., 1999a). Figure 4.8 shows the
ferrofluid-rheometer which consists of a cone plate measuring cell with the plate connected
to a driving unit allowing rotational as well as oscillating movement of the plate. The cone
is connected via an air bearing, reducing parasitic friction, with a torque sensor. The fluid
cell is subjected to a magnetic field parallel to the symmetry axis of the system which is
generated by a pair of Helmholtz coils. The fluid cell is equipped with a water cooling
system keeping the temperature of the fluid constant with an accuracy better than 0.05 K
avoiding any effect of temperature changes on the results obtained. Figure 4.9 shows a
typical set of flow curves for different magnetic field strengths for the ferrofluid APG513A
which will be used as a standard for the following descriptions of various rheological effects in ferrofluids. The influence of different microscopic make up will be discussed within
the frame of the analysis of the general microscopic description of the effects.
It is seen, that the flow curves change significantly with the field strength, but a better
comparison with the formerly shown data can be achieved by plotting the magnetoviscous
effect as a function of field strength for different shear rates as shown in fig. 4.10. The
main feature observed here is the obvious reduction of the magnetoviscous effect with
increasing shear rate—a field dependent shear thinning shown in fig. 4.11 by plotting the
magnetoviscous effect vs. shear rate for various magnetic field strengths. Moreover the left
FERROFLUIDS
167
Fig. 4.8. A specialized rheometer for the investigation of ferrofluids (Odenbach et al., 1999a, 1999b).
side of fig. 4.10, showing the effects for very small shear rates illustrates, that extremely
strong changes of viscosity can appear. This leads to the conclusion, that large structures
of magnetic particles can be created as long as the disturbance by shear is not too high.
Since these structures are only weakly bound, they degrade quickly with increasing shear
rate leading to the observed strong shear thinning. The strong increase of viscosity with
decreasing shear rates raises the question whether ferrofluids show a yield stress as it is
known from MR-fluids. Since the rheometer used for the experiments is a controlled shear
setup, an extrapolation of the flow curves (fig. 4.9) to zero shear rate is needed to obtain
the yield stress σ y . The field dependent result of this extrapolation is shown in fig. 4.12.
168
S. ODENBACH
Fig. 4.9. Flow curves of the ferrofluid APG513A for various magnetic field strengths.
Fig. 4.10. Shear rate dependence of the magnetoviscous effect for low shear rate (left) and higher shear rate
(right).
The yield stress found for this fluid grows quadratically with magnetic field strength but
its absolute value is about four orders of magnitude smaller than for common MR-fluids.
Nonetheless, the result shows that ferrofluids are able to build up large particle networks
even if they are only weakly bound.
The results shown up here indicate that structures are formed in the fluid by interparticle
interaction. Thus it is reasonable to check whether these structures give rise to the appearance of viscoelastic effects (Odenbach, 2004). Therefore the fluid has to be subjected to an
oscillating load created by an oscillating motion of the plate (for detailed information on
rheological techniques see e.g. Whorlow, 1992). In such oscillating measurements the fluid
shows a reaction to the load which is phase shifted by δp = π/2 for a Newtonian liquid
and which equals 0 for a pure elastic material. Figure 4.13 shows a typical experimental
run plotting the shear γ and the stress σ in the fluid as a function of time. The shear is
modulated with a sinoidal temporal modulation leading to a sinoidal answer in the stress in
the fluid. The comparison of the left side of fig. 4.13 displaying the behaviour for vanishing
FERROFLUIDS
169
Fig. 4.11. Shear thinning of ferrofluids for different magnetic field strengths.
Fig. 4.12. Yield stress in APG513A as a function of magnetic field strength. The values are extrapolated from
the flow curves.
field and the right side taken for a finite value of H shows immediately that the phase shift
is reduced from π/2 found for H = 0 to a value of 0 < δp < π/2 for finite field indicating
viscoelastic behaviour. In fig. 4.14 the field dependent results of δp are shown as a function
of the frequency of the load. For vanishing field δp = π/2 over the whole investigated
frequency range, while increasing field leads to significant viscoelastic properties.
The oscillating measurement allows additionally to determine the real and imaginary
part of the complex viscosity and the complex storage module. Figure 4.15 shows the
imaginary part of the complex viscosity η for various field strengths as a function of the
load frequency. For H = 0 η vanishes as it is expected for a pure Newtonian fluid indicating that the ferrofluid does not show any internal structures for vanishing field. Increasing
170
S. ODENBACH
Fig. 4.13. Shear γ and stress σ in APG513A under oscillating load for vanishing field (left) and H > 0 (right).
A significant change of the phase shift between the two signals due to the applied field is observed.
Fig. 4.14. The phase shift in APG513A as a function of the frequency of the oscillating load for vanishing and
finite field strength.
H leads to a growing imaginary part of the complex viscosity showing a pronounced maximum which is related to a typical rheological relaxation time of the fluid by τR = 1/ωmax .
For H = 30 kA/m this relaxation time is in the order of 10 s indicating large structures in
the system. With decreasing field strength the maximum of η shifts towards larger values
of ω which corresponds to shorter relaxation times as it is expected if the structures in the
fluid reduce in size with decreasing field. From the imaginary part of the complex viscosity
one can derive the real part of the storage module G by the relation
G = ωη .
(4.16)
FERROFLUIDS
171
Fig. 4.15. The imaginary part of viscosity of a ferrofluid as a function of the frequency of the oscillating load for
various magnetic field strengths.
Fig. 4.16. The storage module for the same fluid as in fig. 4.16 as a function of the frequency of the oscillating
load in a logarithmic plot showing typical Maxwell behaviour of the fluid.
A plot of G against ω allows a simple check whether the fluid behaves like a Maxwell
fluid (Whorlow, 1992). Figure 4.16 shows the result for the ferrofluid used here. For
vanishing field G is more than an order of magnitude smaller than for the lowest field
investigated, indicating that the fluid shows mainly Newtonian behaviour. But with in-
172
S. ODENBACH
creasing field one finds the typical quadratic increase of G with ω for small values of ω
and the plateau at higher oscillation frequencies familiar from Maxwell type fluids.
Finally oscillating rheological investigations allow the determination of the zero shear
limit of the first normal stress coefficient ν10 from the relation
G
ν10 = lim 2 .
(4.17)
ω→0 ω
Together with investigations of the Weissenberg-effect, i.e. the climb of a free surface
of a fluid at a rotating rod due to normal stress effects, which provide the combination
ν10 + 4ν20 of the normal stress coefficients one can determine ν20 too. For the magnetite
ferrofluid APG513A investigations of the Weissenberg effect have been undertaken for a
shear rate of γ̇ = 0.57 s−1 under microgravity conditions in a parabolic flight using the
1/g dependence of the height of climb of the liquid at the rod to enhance the measurable
effect (Odenbach et al., 1999b). In combination with the rheologically taken estimates for
ν10 one obtains for this liquid ν10 = 0.75 Pa s2 and ν20 = −0.18 Pa s2 for H = 25 kA/m
and γ̇ = 0.57. As known from polymers ν20 is negative and its absolute value is smaller
than ν10 . But while polymer melts usually show a relation of 10:1 for ν10 :ν20 one finds
for the ferrofluid under the given conditions a relation of 4:1. Interestingly this is in good
agreement with actual molecular dynamic simulations of the behaviour of ferrofluids under
shear in a magnetic field (Ilg, 2004).
Taking all the discussed results together, it is clear that a magnetic field influences the
overall rheological behaviour of suspensions of magnetic nanoparticles in a much more
complex way than the basic theoretical approach of rotational viscosity suggests. Moreover
the experimental findings favour a model for the origin of the observed phenomena which
is based on the formation of preferably chain like structures build by the magnetic particles
due to magnetic interparticle interaction. Since numerous of the described phenomena like
e.g. the viscosity enhancement or the appearance of a yield stress might have technical
importance if their strength could be improved, a detailed microscopic model is strongly
required to enable a specific modification of the fluids.
4.1.6. Microscopic models for the magnetoviscous behaviour
As mentioned above, chain formation of magnetic particles seem to be a reasonable
approach to explain the experimental findings for the rheological behaviour of ferrofluids. Chain formation requires that the interparticle interaction of the magnetic particles is
strong enough to keep the structures together. As shown in the discussion on the stability of
magnetic fluids the magnetic dipole interaction energy of 10 nm particles is smaller than
their thermal energy avoiding agglomeration and thus chain formation of such particles.
Therefore a discussion about the reversible formation of chains of magnetic particles due
to magnetic interparticle interaction requires at first information about the minimum particle size of magnetic particles having the ability to form structures. Usually this information
is taken from the interaction parameter
μ0 M02 V
(4.18)
24kT
relating the dipole–dipole interaction energy of two particles in contact to their thermal
energy. Since suspensions of magnetic particles in fluids being stable enough for applications and thus being of interest for rheological investigations are usually coated with a
λ=
FERROFLUIDS
173
Fig. 4.17. The coupling parameter λ∗ for cobalt and magnetite particles as a function of the particle size.
surfactant layer of a certain thickness a direct contact of the particles is prohibited and a
modified interaction parameter (Thurm and Odenbach, 2003)
3
d
λ∗ = λ
(4.19)
d + 2s
should be used. If this interaction parameter exceeds a value of λ∗ ≥ 1 the dipole interaction of the particles at closest possible distance exceeds their thermal energy and chain
formation becomes possible. Figure 4.17 shows the dependence of λ∗ on the particle diameter calculated for a surfactant layer thickness of s = 2 nm for different magnetic
materials used for the synthesis of ferrofluids. As seen here, ferrofluids with magnetite particles require a minimum particle diameter of about 13 nm for efficient chain formation
while other materials with higher spontaneous magnetization will show agglomeration already at smaller particle diameters. To enable the calculation of the influence of chains in
a ferrofluid on its viscous properties the length of the chains or—more realistic—the chain
length distribution function is required to calculate the stress tensor.
An early approach for the calculation of the length of chains in a suspension of magnetically interacting particles has been made by Jordan (1979) on the basis of empirical
assumptions and been proven by Hayes (1975) showing good agreement for sufficiently
large particles. Since Jordan’s calculations do not allow to calculate analytically the field
dependence of the chain length, Rosensweig (1985) gave approximations for the extreme
situations of zero and infinite field strength
2 φ 2λ −1
n(H =0) = 1 −
e
3 λ3
(4.20)
2 φ 2λ −1
n(H →∞) = 1 −
e
3 λ2
174
S. ODENBACH
Fig. 4.18. The length of particle chains for magnetite particles for vanishing and infinite field strength as calculated from equation (4.20).
where n denotes the number of particles in a chain. Calculating the chain length using equation (4.20) as a function of the particle diameter for a magnetite ferrofluid with 5 vol% of
suspended magnetic material assuming again a surfactant layer thickness of 2 nm, i.e. replacing λ in equation (4.20) with λ∗ , one finds, that the value of n remains more or less
equal to 1 up to a particle diameter of 12 nm in zero field (see fig. 4.18). For infinite field
strength chain formation is also negligible for d < 10 nm and still weak for d ≈ 11 nm but
it increases dramatically for d ≥ 12 nm. For larger diameters above 12 nm slight agglomeration is seen for the zero field situation too. This calculation gives a first insight in chain
formation in ferrofluids but since it neither gives a chain length distribution nor accounts
for shear rate it is not suitable for the further development of an explanation of the rheological behaviour of ferrofluids. Nonetheless, the discussion of the coupling parameter as
well as the chain length estimates from equation (4.20) make clear that only relatively large
particles with diameters about 13 nm and above will contribute significantly to chain formation and related changes in the viscous behaviour of magnetite ferrofluids. This forced
the suggestion of a model for the microscopically relevant structure of a ferrofluid which
describes the magnetic fraction of the fluid as a bidisperse system (Odenbach and Stoerk,
1998). In this approach a large fraction of small particles with volume content φs and mean
particle diameter ds determines the overall magnetic properties of the fluid but does not
significantly contribute to the magnetoviscous phenomena. A second fraction with small
volume content φl and a large particle diameter dl is able to form chains and thus important for the magnetic effects in rheology. In this simple model the maximum chain length
in a shear flow can be estimated by evaluating the equilibrium of the magnetic interaction
force, keeping two parts of a chain with a length of n particles in the middle together and
the Stokes force appearing due to viscous friction in a shear flow (see fig. 4.19). The magnetic dipole force keeping the chain together is assumed to be given by the dipole–dipole
FERROFLUIDS
175
Fig. 4.19. Schematic sketch of a chain of n particles for the explanation of the maximal stable chain length in a
shear flow. Explanations are given in the text.
interaction of the two particles in the centre only
μ0 M02 πd 6
(4.21)
24(d + 2s)4
while the force breaking the chain up results from the different viscous forces acting on the
two parts of the chain in the shear flow
2
1 n
Fvis = 6πη0 γ̇
(4.22)
(d + 2s) .
2 2
Fm =
Assuming that the chain breaks for Fvis > Fm one can calculate (Odenbach and Stoerk,
1998) the maximum number of particles to be
μ0
d3
nmax =
(4.23)
.
M0
18η0 γ̇
(d + 2s)3
Figure 4.20 shows the dependence of the maximum number of particles in a chain on
shear rate. Comparing the characteristic decrease of nmax with the decrease of the magnetoviscous effect with γ̇ also shown in fig. 4.20 one finds an interesting qualitative agreement
which encourages a further investigation of the model approach.
To check more in detail whether the small fraction of large particles in ferrofluids is
responsible for the experimentally found effects in their rheology, various experiments
(Odenbach and Raj, 2000; Thurm and Odenbach, 2003) have been performed with fluids
with varied content of large particles. As an example fig. 4.21 shows the magnetoviscous
effect for five fluids which differ only in the content of large particles. Obviously a reduction of φl leads to a diminishing of the magnetoviscous effect clearly strengthening the
above model.
To concretise the microscopic description of magnetic field induced effects in rheology
of ferrofluids Zubarev et al. (2002) gave a mathematical formulation of the two fraction
176
S. ODENBACH
Fig. 4.20. The maximal stable chain length calculated from equation (4.23) for magnetite particles and the magnetoviscous effect in APG513A as a function of shear rate.
Fig. 4.21. The variation of the magnetoviscous effect with the content of large particles in the fluid for 5 different
concentrations of large particles.
model. Under the assumption that the large particles show significant interparticle interaction with other large particles while interaction between large and small particles and
between small particles themselves is negligible, the free energy EF of the chains in the
FERROFLUIDS
applied magnetic field H
gn
1
E F = kB T
gn ln
−
μ0 mchain H ,
e
2kB T
n
177
(4.24)
with the chain length distribution function gn and the total magnetic moment of the chain
mchain , can be calculated. Here it is furthermore assumed that the chains formed by large
particles only are rigid and straight and do not interact with each other. The assumption of
rigid and straight chains is reasonable as long as the chains are short—a point which has
to be clarified from comparison of calculations and experiments. It should be noted, that
an extended approach for flexible chains has been made (Morozov and Shliomis, 2002)
which has not yet been experimentally proven. The neglect of interchain interaction takes
its basis from numerical simulations of Satoh et al. (1996a, 1996b) which showed that
the formation of bundles of chains—so-called rods—appears only for highly interacting
particles, i.e. particles with even significantly larger diameters than they are considered in
a discussion concerning ferrofluids. With all these assumptions the chains can be treated
as an ideal gas of chains leading to the free energy expression in equation (4.24) which
shows in the brackets the normal logarithmic term representing the entropic part of the free
energy together with the energy of the chain in the magnetic field. Equation (4.24) allows
the calculation of the chain length distribution function gn by a minimization of the free
energy following the methods in (Zubarev and Iskakova, 1995a, 1995b). With gn one can
e.g. calculate the mean number of particles in a chain
n =
φl
Vl
n gn
(4.25)
using the volume fraction φl and the volume Vl of the large particles including their surfactant. With the help of the methods of statistical hydrodynamics of diluted suspensions of
rigid ellipsoids (Brenner and Condiff, 1980) one can calculate the stress tensor of the fluid
and thus the resulting effect of the chains in the field on the fluid’s viscosity (Zubarev et al.,
2002). The resulting equation has to be fitted to the experimental data numerically observing, that the theoretical approach holds strictly for vanishing shear rate only. This requires
either shear free experiments like they’ve been performed by Embs et al. (2000) or experiments with shear rates low enough to ensure that the measurement is undertaken in the
lower Newtonian region of the flow curve of the fluid investigated. Since Embs et al. (2000)
did not carry out extensive parameter studies of the magnetoviscous behaviour of ferrofluids and since the development of a specialized stress controlled rheometer for ferrofluids
properly allowing measurement at ultra low shear rate is a matter of current research, the
only tests of the theory (Zubarev et al., 2002) undertaken till now had to be made with
the lowest shear rate available with a shear controlled rheometer (Odenbach et al., 1999a).
Since this measurement can not ensure that the fluid is in the lower Newtonian region, the
absolute values taken from the fits are a first step only and may have to be modified in
future investigations. In the fits the mean particle size dl of the large particles and their
volume fraction including surfactant φl are the free fit parameters. Figure 4.22 shows the
result of such a fit together with the experimental data obtained for γ̇ = 0.1 s−1 . The fit
provides dl = 16.5 nm and φl = 0.017 which corresponds to a magnetic volume fraction
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S. ODENBACH
Fig. 4.22. The magnetoviscous effect in APG513A for three low shear rates with fits using Zubarev’s theoretical
approach (Zubarev et al., 2002). The inset shows the size distribution of the particles in the fluid.
of the large particles φl = 0.007 assuming a surfactant layer thickness of s = 2 nm. If
one compares this result with the size distribution of the particles in the fluid given in the
inset one finds excellent agreement suggesting that the experimental conditions—beside
all restrictions—fit well to the theoretical constraints and that the assumptions of the theory are quite reasonable for the fluid investigated. The latter statement can additionally be
proven calculating the mean chain length of the chains in the fluid using the fit parameters.
As fig. 4.23 shows, the mean length of the chains is between 2 and 4 particles which is
short enough to keep the assumption of straight and rigid chains. Chain lengths of this size
have additionally been found in various numerical studies of the bidispersed model (Wang
et al., 2002; Ilg and Hess, 2003; Wang and Holm, 2003). The theory can even be expanded
to describe the experimentally observed shear thinning by cutting all summations over the
chain length at a maximum value given by equation (4.23). As seen from the values for
higher shear rates in fig. 4.22 this simple approach holds for not too high shear rates.
As a final point in the discussion of microscopic explanations for the magnetoviscous
behaviour in ferrofluids a series of new experiments using small angle neutron scattering
(SANS) for a microstructure evaluation in a sheared ferrofluid under the influence of a magnetic field should be mentioned (Pop et al., 2004). Neutron scattering investigations have
been shown already earlier to be a powerful tool for structural investigations in ferrofluids
regarding as well the microstructure of the particles and their surfactant layer (Kammel
et al., 2001; Hoell et al., 2002) as structures formed by the particles (Cebula et al., 1983,
Rosman et al., 1990; Itoh et al., 1992). But all these investigations have been undertaken in
unsheared ferrofluids, thus not providing information about the relation of microstructure
and rheological properties.
For the experiments discussed here a specialised rheometer has been developed which allows normal rheological investigations and SANS experiments under the same experimen-
FERROFLUIDS
179
Fig. 4.23. The mean chain length of particle chains calculated from Zubarev’s theory.
Fig. 4.24. Schematic sketch of the experimental setup for SANS experiments with sheared ferrofluids.
tal conditions especially concerning shear and magnetic field (Pop et al., 2003). Figure 4.24
shows the shear cell of the rheometer in the arrangement used for SANS investigations.
The peculiarity of the system are the niobium beam windows, allowing the neutron beam
180
S. ODENBACH
to pass through the ferrofluid. The fluid is sheared in a cone plate arrangement as used for
the rheological investigations discussed above. The magnetic field is generated by two coils
and directed towards the measuring region via steel bars and pole shoes. The holes in the
pole shoes, which allow the neutron beam to pass through, do not affect the homogeneity
of the field by more than 1% (Pop et al., 2003).
Since the investigations of the microstructure should preferably be carried out with
normal ferrofluids as used in the rheological investigations one has to observe that the
scattering by the magnetic particles is superposed by scattering from the carrier liquid
as well as from the surfactant shell. Since ferrofluids are moreover polydisperse systems
the overall scattering pattern will be composed in a complex way by numerous contributions from different sources which are not well known. Since this severely reduces the
probability to perform an unambiguous analysis, it is reasonable not to discuss the direct
scattering patterns taken from the whole sample for a certain shear rate and field strength
but the changes appearing relative to an appropriately chosen reference. These changes
in the scattering pattern will be only determined by changes in the microstructure of the
fluid generated by formation and rupture of chains or agglomerates of magnetic particles
while the influence of parasitic effects, like the scattering from the carrier liquid, remains
unchanged and will thus disappear in the difference scattering patterns.
Looking at the assumptions made in the chain model for the microstructure of magnetic
fluids one can make some estimates concerning the expected scattering behaviour. If the
magnetic field is aligned parallel to the neutron beam, chains of particles will be formed
which align with the field direction for γ̇ = 0 s−1 (see fig. 4.25a). The scattering pattern
will show the effect of the projection of the chains on the detector plane which is nearly
equal to the projection of single particles for well aligned chains but with a slightly reduced
number of scattering centres compared to a real single particle situation of the same fluid.
This single particle situation is achieved for the fluid APG513A, which has been discussed
extensively concerning its rheological behaviour above, for shear rates as high as 100 s−1
where a status is reached in which the magnetoviscous effect becomes independent from
γ̇ and is nearly zero as seen in fig. 4.20. Thus, choosing the single particle situation, i.e.
the scattering pattern for γ̇ = 200 s−1 , as a reference the difference scattering pattern for
γ̇ = 0 s−1 should not show any significant changes since the projections of the structures
are more or less equivalent and the change of the number of scattering centres due to
the formation of the chains does not make an impact since the number of chain forming
particles is small and the change of scatterers due to chain formation is also small since the
chains are short (see fig. 4.23). The expectation of a minimal change between γ̇ = 0 s−1
and high shear rate is approved experimentally for all investigated magnetic field strengths.
For the experiments carried out at the beam line V4 at the Hahn Meitner Institute
in Berlin a neutron wavelength of 6 Å has been chosen which yields together with a
sample–detector distance of 4 m a range of scattering vectors from 0.1 nm−1 to 1 nm−1
corresponding to detectable structure sizes between 6 nm and 60 nm and with a sample–
detector distance of 12 m to a range of detectable structures between 18 nm and 180 nm.
For other shear rates, as shown schematically in fig. 4.25, the chains should be deviated from the field direction and—for increasing shear rate—should break up into smaller
pieces. The deviation and length of the chains depends both on shear rate and magnetic
field strength applied, resulting in a change of the scattering pattern with shear rate and
field strength as is it given in fig. 4.26.
FERROFLUIDS
181
Fig. 4.25. Sketch for the explanation of the behaviour of particle chains in a sheared ferrofluid and the influence
on the scattering pattern. Explanations are given in the text.
To ensure that the changes of the scattering pattern are definitely due to changes of
the fluid’s microstructure a reference experiment has been undertaken using a purified
ferrofluid containing small particles only and showing thus a negligible magnetoviscous
effect. As seen in fig. 4.27 the changes in the scattering patterns vanish for this fluid proving that the effects seen in fig. 4.26 are a real footprint of the structural changes in the
fluid giving rise to the rheological effects. SANS experiments do in principle not only provide a qualitative image of structural changes in a system but allow the direct comparison
with e.g. numerical simulations. Since the difference patterns contain information about
the large particles and structures formed by these particles only, a very clear relation to
molecular dynamic simulations investigating a set of particles with properties corresponding to the large particles in a ferrofluid becomes possible (Ilg and Kröger, 2002; Wang
et al., 2002; Wang and Holm, 2003).
Due to the fact that the chains formed in APG513A are rather short a fluid with stronger
interparticle interaction has to be chosen to make the comparison with simulations more
reliable. Rheological measurements of such a fluid, based on Co-particles with a diameter
of about 9 nm providing an interaction parameter of λ∗ = 6 are shown in fig. 4.28. Obviously strong effects are observed here despite the fact that the concentration of magnetic
particles is only φ = 0.0035. In the microscopic picture this can directly be related to
large structures formed by the particles which can all contribute to chain formation. In the
182
S. ODENBACH
Fig. 4.26. SANS patterns for different shear rates and magnetic field strength’ for APG513A.
Fig. 4.27. SANS patterns for different shear rates and magnetic field strength’ for a ferrofluid with negligible
magnetoviscous effect.
FERROFLUIDS
183
Fig. 4.28. The magnetoviscous effect for a ferrofluid containing 0.35 vol% of Co-particles with a mean diameter
of 9 nm.
Fig. 4.29. SANS patterns for different shear rates and magnetic field strength’ for the Co-ferrofluid shown in
fig. 4.28.
scattering patterns these larger structures yield much stronger changes as seen in fig. 4.29.
In particular an anisotropy of the scattering pattern is observed which results from the
magnetic scattering of the neutrons by the magnetic moment of the chains. The increasing
anisotropy is thus a direct measure for the deviation of the chains from the field direction.
184
S. ODENBACH
The great difference between the effects in figs. 4.26 and 4.29 can be accounted for by
the fact that the Co particles form stronger chains due to the higher value of λ∗ than the
magnetite particles.
Further SANS investigations for different rheological states of the ferrofluids and a data
evaluation comparing the experimental results with numerical findings are the subject of
actually ongoing research.
4.1.7. Consequences of the microstructural picture for the magnetoviscous effects
As a result of the above discussions on the rheological properties of ferrofluids one can
fix three major points:
1. Magnetic fields can give rise to significant changes in the rheological behaviour of
ferrofluids but the effects observed in commercial magnetite based fluids degrade to
strongly with shear rate to make them suitable for technical applications.
2. The phenomena observed can be related to changes in the microstructure of ferrofluids,
i.e. the formation and rupture of chains formed by the magnetic particles.
3. The formation of the mentioned structures depends on the magnetic interparticle interaction which has to be strong enough to enable efficient formation of chains.
From these three points some major consequences for magnetic fluid research have to
be drawn. First of all an improvement of the magnetoviscous effects, as it is needed for
technical applications, requires a modification of the interaction parameter λ∗ . Additionally
the formation of internal structures in a ferrofluid influencing its rheological properties as
strong as shown above raises the question whether the formulation of ferrohydrodynamics
(Shliomis, 1972; Rosensweig, 1985) basing on a single particle model gives a complete
description of the behaviour of ferrofluids.
Concerning the first point two generally possible ways to an improvement of the magnetoviscous effects can be used. On the one hand—as shown above—larger magnetite
particles will exhibit stronger interparticle-interaction. Following this path one ends up
with particles with a size of about 20–30 nm as they’ve been used in the synthesis of so
called nano-MR fluids (Kormann et al., 1996a, 1996b). A further increase of particle size
would result in two problems, sedimentation of particles would appear and the particles
would become polydomain with still high remanence magnetization. The high remanence
magnetization would cause a strong interaction parameter which would give rise to structure formation in zero field too resulting in a small effect of an applied magnetic field which
would only weakly enhance the structure formation. This problem is also known from the
nano-MR fluids and it reduces significantly the usable change of the viscous properties in
a magnetic field. An approach using the increase of particle size will thus, reasonably, lead
to micronsized particles, i.e. to MR-fluids, which have small remanence and therefore no
significant structure formation in the absence of a field.
The second path to enhance the interparticle-interaction is the change of the magnetic
material of the particles to materials with higher spontaneous magnetization. As mentioned
in the chapter on synthesis of ferrofluids this path has meanwhile successfully been used in
the production of aluminium coated Co-particles (Bönnemann et al., 2003). As seen from
fig. 4.28 these fluids show a strong change of viscosity with magnetic field strength even
if the concentration of the particles is rather small. The small concentration results in the
FERROFLUIDS
185
Fig. 4.30. Comparison of the shear thinning in the Co-ferrofluid from figs. 4.28 and 4.29 with the fluid APG513A
at H = 60 kA/m.
fact that the structures which are formed in zero field in the fluid due to the high coupling
parameter will not span the fluid volume but will form ring like structures (Charles, 2000).
The small concentration of these rings does not significantly influence the zero field viscosity of these fluids. In presence of a magnetic field the rings will transform to chains
which are strongly bound due to the high value of λ and will thus be able to withstand
even stronger shear loads. This is illustrated in fig. 4.30 comparing the shear stability of
the magnetite based ferrofluid APG513A with that of a Co based ferrofluid containing
0.35 vol% of 10 nm Co particles. As seen the viscosity change for low shear rate is comparable for both fluids which can easily be understood by the fact that the amount of structure
forming particles is nearly identical in both fluids. But while the magnetoviscous effect decreases strongly with γ̇ for the magnetite fluid, it remains more or less constant at a change
of about 200% for the cobalt fluid even at the highest shear rates investigated here. This
shear stability of a significant change of viscosity is a new feature of these fluids which
distinguishes them as well from conventional ferrofluids as from MR-fluids. Future development will show how this feature of a shear stable adjustable change of viscosity in weak
magnetic fields can be used for technical purposes.
The second mentioned consequence of the structure formation is of more fundamental
nature. The classical formulation of ferrohydrodynamics bases on the microscopic picture
of single magnetic particles subjected to magnetic and viscous torques. As seen, this approach describes well the behaviour of diluted fluids with non interacting particles, while
it does not include any of the viscoelastic effects or even the shear thinning due to the rupture of the chains. Thus it is reasonable to check whether other effects may appear due to
the structure formation, which are not predicted by the classical approach. As known, the
basis of ferrohydrodynamics is the equation for the relaxation of magnetization in the fluid
which assumes the above mentioned microscopic picture (Shliomis, 1972)
1
dM
= − (M − Meq ) + ( × M),
dt
τ
(4.26)
186
S. ODENBACH
where M denotes the actual magnetization of the fluid and Meq its equilibrium value. The
relaxation time τ of magnetization will usually be assumed to be τ B for magnetically hard
particles. Since this basic equation does not account for structure formation one can either
try to extend the model to include internal structures and their influence on the magnetization behaviour and therefore on the behaviour of the fluid as a whole, or one can try to find
a completely new way to formulate an appropriate basis for the macroscopic description of
the behaviour of magnetic fluids. With the first approach one runs into serious problems in
the formulation of basic equations since it would require an extension of the microscopic
picture providing the basis of the formulation. A more complex microscopic model would
increase the complexity of the formulation of the equations and—even more problematic—
a change of the properties of the fluid resulting in another microstructure would require a
new formulation of the basic equations again. Such an approach is obviously not satisfying
since it will not lead to a real generally valid basic formulation. A more general approach
would require a relief from the concept of a founding microscopic model, i.e. it would have
to focus on a macroscopic description employing material specific parameters reflecting the
characteristics of a special ferrofluid used. These parameters would have to be determined
empirically. Such an approach is generally well known in fluid mechanics where e.g. the
basic equation of motion, the Navier Stokes equation, contains the fluid characteristics in
form of the viscosity which has to be known from measurements.
The formulation of a basic equation for the relaxation of magnetization in a ferrofluid has
been undertaken by Müller and Liu (2001) on the basis of a thermodynamic approach. As
result this formulation provides an extended equation for the relaxation of magnetization
in the form
d
1
Mi + λ1 Mi (∇ v) + λ2 (∇i vj + ∇j vi ) − ( × M)i = (H − Heq ).
dt
2
(4.27)
As seen, this equation includes as a special case, for the situation where the parameters λ1 and λ2 vanish, the equation of relaxation of magnetization as it has been given
by Shliomis (equation (4.26)). In this situation the fluid characteristic parameter is τ , the
relaxation time for magnetization, which would have to be measured. But in addition to
this formulation equation (4.27) contains two further couplings of the flow to the magnetization dynamics. The first one, employing the fluid characteristic parameter λ1 , describes
the influence of the divergence of the velocity field, an effect which may be important for
questions of sound propagation in ferrofluids. The second additional term accounts for the
coupling of the symmetric velocity gradient vij = 1/2(∇i vj + ∇j vi ) to the magnetization
dynamics.
This coupling is described by the parameter λ2 . The open question at this point—and
in fact the proof for the new theoretical approach for the description of the dynamics of
ferrofluids—is the point whether one of the additional coupling parameters takes a finite
value for a ferrofluid with internal structure formation. A first check for additional influences of internal structures on the relaxation of magnetization has been undertaken using
the ferrofluid APG513A in an attempt to measure the value of λ2 for this fluid (Odenbach
and Muller, 2002). The experimental setup for this investigation employs the flow between
two concentric rotating cylinders with radii R1 and R2 as depicted in fig. 4.31. The pecu-
FERROFLUIDS
187
Fig. 4.31. Schematic sketch of the experimental arrangement for the measurement of the parameter λ2 .
liarity of this flow is the independent variation of the vorticity
=
ω2 R22 − ω1 R12
(4.28)
R22 − R12
and the symmetric velocity gradient which contains the shear rate given in a dimensionless
form as
R2
1
R1
ω1 − ω2
γ̇
=
s= 2 1 2
(4.29)
R2 − R1
R 2 − R1
with
⎛
0
vij = −s ⎝1
0
1
0
0
⎞
0
0⎠
0
by appropriate adjustment of the angular velocities ω1 and ω2 . Measuring the field component H⊥ generated by the magnetization component M⊥ perpendicular to the applied
magnetic field in the centre of the cylinder system provides the requested information on
relaxation of magnetization. If both cylinders are rotated at the same angular velocity, vij
vanishes and the relaxation of magnetization is only determined by the vorticity which
means that Shliomis’ theory and equation (4.27) yield the same result
H⊥0 = τ Hext F(Ri ,χ) ,
(4.30)
where the function F depends on the geometry of the cylinders and the susceptibility of
the fluid and enters the equation due to demagnetisation effects. The measurement of the
linear relation of the perpendicular component of H in the shear free situation H⊥0 as a
function of vorticity (see fig. 4.32) will thus provide the first characteristic parameter of
the fluid, the relaxation time for magnetization τ .
If now shear is applied at constant vorticity by appropriate change of ω1 and ω2 a different perpendicular component of the magnetic field H⊥ will be observed which is given
by
H⊥
H⊥0
= 1 + γ̇ (G(Ri ) − λ2 F̃(Ri ,χ) ).
(4.31)
188
S. ODENBACH
Fig. 4.32. The dependence of the nonequilibrium component of the magnetic field on the vorticity of the flow for
the shear free situation for the ferrofluid APG513A.
Fig. 4.33. The dependence the nonequilibrium component of the magnetic field on the shear parameter for the
ferrofluid APG513A. The full line is a fit of the theory providing λ2 = 0.2 while the dashed line shows the
corresponding behaviour for λ2 = 0 (Shliomis theory).
Again, a linear relation of the perpendicular component of H on γ̇ is found, but the
slope of this relation depends on λ2 allowing to determine its value. The functions G and
F̃ are again known functions of the geometry and the susceptibility of the fluid which
appear due to demagnetisation effects (Odenbach and Muller, 2002). Figure 4.33 shows
the measured values of H⊥ /H⊥0 as a function of s with a fit of equation (4.31) providing
a value of λ2 = 0.2. The dashed line gives the result that would have been expected for
FERROFLUIDS
189
Fig. 4.34. Sketch for the explanation of the microscopic origin of the coupling of the symmetric velocity gradient
to the relaxation of magnetization in a ferrofluid.
λ2 = 0. The investigations have been carried out at an external field of Hext = 20 kA/m
corresponding to an inner field in the fluid of H = 14 kA/m.
Microscopically the appearance of a finite value of λ2 and thus a coupling of vij to
the magnetization dynamics can be related to the formation of chain like structures in
the fluid. This is easily understood looking to fig. 4.34 with = 0 and vij = 0. In
such a flow a spherical particle would not be affected leading to λ2 = 0 in accordance
with Shliomis’ theory while an ellipsoidal particle will experience a torque leading to a
coupling of vij to its orientation and thus to a change of the magnetization dynamics which
is described by λ2 = 0. It is obvious that this interpretation suggests that λ2 will depend
on the axis ratio of the ellipsoid and thus for the particular case of a ferrofluid on the
length of the chains built up by the magnetic particles. An estimate for the chain length
can be made using a relation given by Brenner (1974) for the orientational distribution of
ellipsoids in a pure elongational flow. As a result one obtains for λ2 = 0.2 an axis ratio of
2 corresponding to a mean number of particles in the chain of n = 2, which is in good
agreement with the result found from rheological measurements using Zubarev’s theory
(see fig. 4.23), providing n = 2.4. As seen in the discussion of the magnetoviscous
effects, the chain formation, and thus the influence of a magnetic field on the viscous
properties of the fluid, depends strongly on the microscopic composition of the fluid. Thus
it is finally an interesting point to compare the above measured values of λ2 for APG513A
with those obtained for a fluid with stronger magnetoviscous effects. Figure 4.35 shows the
shear thinning behaviour of APG513A at H = 14 kA/m together with another magnetite
based ferrofluid—TTR630—which contains a larger amount of large particles being able
to form chains (Odenbach and Müller, 2005). This larger number of chain forming particles
gives rise to a much stronger increase of viscosity at small shear rates up to approximately
γ̇ = 30 s−1 while the weak coupling of the magnetite particles leads to weakly bound
chains only, resulting in a significant decay of the magnetoviscous effect for higher shear
rates (γ̇ > 30 s−1 ) approaching the values found for APG513A. The rheological analysis
following Zubarev et al. (2002) provides a mean chain length of n = 4.8 for γ̇ < 30 s−1
and n = 2.4 for γ̇ > 30 s−1 . In fig. 4.36 the relation of H⊥ /H⊥0 on s is depicted for
the fluid TTR630. In contrast to the results shown for APG513A (fig. 4.33) a significant
dependence of the slope of the curve on shear rate is found. For larger shear rates |γ̇ | >
30 s−1 a value of λ2 = 0.12 is found which is close to the value for APG513A and provides
190
S. ODENBACH
Fig. 4.35. The magnetoviscous effect for the magnetite based ferrofluids APG513A and TTR630.
Fig. 4.36. The dependence the nonequilibrium component of the magnetic field on the shear parameter for the
ferrofluid TTR630. The full line gives the fits of the theory providing different λ2 -values for the different regions
of shear rate as given in the text.
via Brenner’s relation again a value of n = 2. For small shear rates |γ̇ | < 30 s−1 the
slope gets much smaller corresponding to λ2 = 0.88. This leads to a mean chain length of
n = 5 which corresponds again well with the rheological findings.
As a general result one can fix at this point, that internal structure formation leads to a
more complex coupling of the flow to the dynamics of magnetization of a ferrofluid. The
FERROFLUIDS
191
coupling constant between vij and the relaxation of magnetization λ2 can be interpreted
by a chain model leading to good agreement with rheologically obtained results. Since
this additional coupling obviously exists it will be worth in the future to check the further
consequences of the new approach for ferrofluid dynamics (Müller and Liu, 2001) e.g. by
measuring the sound propagation in ferrofluids to determine λ1 (Müller et al., 2003) or
by investigating more complex phenomena predicted by the theory like the appearance of
shear induced sound in a ferrofluid under influence of a magnetic field (Müller and Liu,
2003).
4.2. Thermal transport properties
4.2.1. The Soret effect in ferrofluids
The viscosity of a ferrofluid is not the only characteristic parameter which can be
changed by magnetic fields. A second example for a strong—and still not in detail
understood—influence of magnetic fields is the change of the Soret effect in ferrofluids.
The Soret effect—thermodiffusion—describes the demixing of a two component system
under the influence of a temperature gradient.
Looking at thermal transport phenomena in a fluid in general one has to describe heat
and mass transfer under the influence of a temperature gradient ∇T . As well known the
temperature gradient drives a heat flux
jQ = −λT ∇T ,
(4.32)
where λT , the thermal conductivity of the material, is the related transport coefficient. If
the driving force, given by ∇T , overcomes a certain critical value the pure diffusive heat
transfer will be replaced by a convective transport which will be the subject of the next
section.
Beside the direct coupling of a temperature gradient to heat transfer, a second phenomenon, thermodiffusion, given by a mass flux driven by ∇T may appear in a two component
system like a ferrofluid. The mass flux is given by
jc = −ρ(cDT ∇T + D0 ∇c),
(4.33)
where the left term in the parentheses accounts for thermodiffusion with the thermodiffusion coefficient DT while the second term describes conventional diffusion with the
diffusion coefficient D0 which counteracts the thermodiffusion effect by equilibrating the
concentration gradient ∇c of the species diffusing in the fluid. Usually the phenomenon of
thermodiffusion is characterized by the relation of the thermodiffusion coefficient and the
conventional diffusion coefficient which is called Soret coefficient ST = DT /D0 . In the
stationary state the Soret coefficient provides the maximal concentration difference cmax
that may appear due to thermodiffusion by
1 cmax
,
(4.34)
c0 T
where c0 denotes the initial concentration of the diffusing species. A positive value of ST
means a thermodiffusion antiparallel to the temperature gradient.
Focussing on ferrofluids the diffusion phenomena are related to the relative movement
of the particles with respect to the carrier liquid, i.e. a demixing of the fluid leading to
ST = −
192
S. ODENBACH
Fig. 4.37. Schematic sketch on the determination of the Soret coefficient using a thermodiffusion column.
an inhomogeneous distribution of the particles. Since the diffusion coefficient for nanosized particles is in the order of D0 = 10−11 m2 /s, the characteristic times to reach an
equilibrium state in a fluid volume of a reasonable macroscopic size of about 5 mm is in
the order of several month’, making experiments on such phenomena time consuming and
difficult. On the other hand colloidal suspensions are known (Schaertl and Roos, 1999) to
be characterized by large values of ST in the order of 0.1 providing cmax /c0 = 1 for a
driving temperature difference of T = 10 K, i.e. colloids can be completely demixed
by thermodiffusion. Thus the effect itself, and for ferrofluids in particular its dependence
on magnetic fields, is an interesting research phenomenon but connected to unpractical
long time scales. To overcome this problem experiments on thermodiffusion employ either
very small length scales in thermal grating experiments (Bacri et al., 1995a, 1995b) or they
make use of thermodiffusion columns. In these columns, shown schematically in fig. 4.37
the thermodiffusion between two walls at different temperatures fixed at a small distance of
typically 0.5 mm is combined with a buoyancy driven convective flow which moves fluid
with reduced concentration at the hot wall—assuming positive Soret coefficient—upwards
and vice versa fluid with enhanced particle concentration downwards at the cooled wall.
Attaching large reservoirs to the ends of this thermodiffusion channel will therefore give
rise to an increase of concentration in the lower and a decrease of the volume fraction
of magnetic material in the upper reservoir. In ferrofluids this change of concentration of
magnetic material can be detected by means of the change of the impedance of sensor coils
in the reservoirs (Blums et al., 1997a, 1997b; Völker et al., 2000). Using modern measuring
techniques this detection concept allows the observation of concentration changes down to
10−6 vol% of magnetic material (Völker et al., 2000). This high accuracy is necessary
since an analytic theory for the demixing behaviour can only be formulated for the initial
part of the separation curve, i.e. the change of concentration as a function of time, and for
saturation (Blums, 1995; Blums et al., 1998; Blums and Odenbach, 2000). As mentioned,
saturation will take place on long time scales only making the investigation of the short
FERROFLUIDS
193
Fig. 4.38. The separation curve for a ferrofluid in the initial time regime of the separation process. The full line
is a fit using Blums’ theory providing ST = 0.16.
initial period of the diffusion process more suitable provided that the small concentration
differences appearing in this time span can be detected reliably. Figure 4.38 shows the
separation curve for a ferrofluid in the initial time span using a dimensionless time scale
τc = tD0 /d 2 normalized with the characteristic diffusion time for the colloid. The whole
time span shown corresponds to a real time of 8300 s. For the initial time span, i.e. for
0 < τc < 1/48 Blums (1999) gave an analytic expression for the separation curve showing
a t 5/2 behaviour of c/c0 . A fit of this theory to the experimental data, shown as solid line
in fig. 4.38, provides the value of ST = 0.16 K−1 .
While the thermodiffusion column is obviously an excellent tool for the investigation of
the Soret coefficient in a zero field situation it faces serious problems as soon as a magnetic
field is applied (Völker and Odenbach, 2003). In the presence of a field the thermal flow is
disturbed due to magnetic body forces leading to experimental results that can not be interpreted using the theory for the thermodiffusion column. Since a highly precise detection
system for the concentration of magnetic particles had been developed for the investigation
of the initial changes of the volume fraction of magnetic material in the thermodiffusion
column, a new experiment could be set up providing a convection free investigation of the
Soret effect.
Figure 4.39 shows a sketch of this setup which consists of two reservoirs, containing
the detection coils, which are connected via 1 mm thick horizontal fluid layer. Heating the
system from above and cooling from below excludes any thermal convection phenomena
and—as will be seen in the discussion of thermomagnetic convection in the next chapter—
magnetically driven thermal flows are also suppressed in this thin layer. Fine grids at the
contact between the reservoirs and the diffusion layer act as flow breaks avoiding any
influence of flows in the reservoirs on the diffusion layer (Völker and Odenbach, 2005).
Using this apparatus the separation curves for field alignment parallel and perpendicular
to the temperature gradient have been measured over a field strength range from 0 kA/m
194
S. ODENBACH
Fig. 4.39. Sketch of the experiment for the determination of the magnetic influence on the Soret effect.
Fig. 4.40. The effective Soret coefficient in a ferrofluid in the presence of a magnetic field.
to 320 kA/m. Determining from these curves an effective Soret coefficient leads to the
values shown in fig. 4.40. Obviously a perpendicular arrangement of H and ∇T leads to
significant increase of the thermodiffusion while a parallel arrangement forces a reduction and leads even to a change of the direction of the diffusion for magnetic fields with
H > 40 kA/m. The change in the strength of the effect is even stronger than the conventional Soret effect for both alignments. Assuming that the effective Soret coefficient can be
splitted into the conventional part ST and STm , being a magnetic Soret coefficient, (Blums
FERROFLUIDS
195
Fig. 4.41. Schematic sketch for the explanation of the driving force of thermal convection. Explanations are given
in the text.
et al., 1998) the experiments provide thus values of STm in the same order as ST . This result is in strong contradiction with the classical theory of the magnetic Soret effect which
predicts qualitatively also an increase of STeff for perpendicular alignment and a decrease
for the parallel arrangement, but which gives a quantitative estimate for STm in the order of
10−3 ST . This discrepancy is yet unexplained even if various theoretical approaches have
been undertaken trying to either explain the strength of the magnetic Soret effect (Lange,
2004) or by introducing new transport mechanisms (Blums, 2005). It will be a challenge
for future research efforts to clarify this problem.
4.2.2. Thermal convection in ferrofluids
As mentioned above the second transfer phenomenon driven by temperature differences
in a magnetic fluid—heat transfer—may become unstable if the driving force becomes
to large leading to a convective flow, enhancing the thermal transport in the system. The
general problem of thermal convection is well known since more than 100 years (Benard,
1900). The driving force for thermal convection in a flat fluid layer heated from below can
be understood by a simple buoyancy argument (see fig. 4.41). The temperature gradient,
aligned parallel to gravitational acceleration, causes a density gradient in the fluid antiparallel to ∇T . If a volume element V is displaced adiabatically by thermal fluctuations
from the hot towards the cold wall it will be surrounded by colder fluid with higher density
and will thus experience a buoyant force in the direction of the original displacement. This
force amplifies the disturbance of the fluid layer and can thus drive a convective flow if
it is strong enough to overcome the stabilizing mechanisms given by viscous friction and
thermal conductivity. The stabilizing influence results from the fact, that thermal conductivity equilibrates the temperature of the volume element and its surrounding diminishing
the driving buoyancy force and the viscous friction provides the necessary time for this
equilibration by keeping the velocity of fluid motion finite. The actual status of the system
is usually described by a dimensionless parameter, the Rayleigh number
Ra =
βT ρgT d 3
κη
(4.35)
196
S. ODENBACH
which gives the relation of destabilizing and stabilizing effects. Here βT denotes the coefficient of volume expansion of the fluid, ρ its density and κ its thermometric constant. If the
Rayleigh number exceeds a certain critical value Ra∗ determined by the thermal boundary
conditions (Ra∗ = 1708 for ideal thermal conductivity of the upper and lower bounding
wall) the convective flow will set in.
If such an experiment in an enclosed fluid layer would be performed with the fluid heated
from above, a stable stratification of the fluid layer would appear and no convective motion
would improve the heat transfer.
This can be changed if the fluid used is a magnetic fluid under the influence of a magnetic
field collinear with the temperature gradient (see fig. 4.42). Since the magnetization of the
fluid is temperature dependent (see fig. 2.4; equation (2.8)) the temperature gradient will
give rise to a magnetization gradient in the fluid antiparallel to ∇T . Due to the fact that the
inner field inside the fluid Hi is given by
Hi = H − DM,
(4.36)
with D denoting the demagnetisation factor of the fluid layer, the magnetization gradient
will give rise to a gradient of the inner magnetic field ∇Hi antiparallel to ∇M. If it is
now—as above in the discussion of thermal convection—assumed that a volume element is
displaced adiabatically from the lower, cold wall towards the hot wall, this volume element
will have higher magnetization than its surrounding. Therefore the inner field gradient
∇Hi will exert a net volume force on this volume element in the direction of the field
gradient, i.e. in the direction of the initial displacement. The same argument could be given
for a volume element displaced from the top to the bottom, again leading to a magnetic
force in the direction of the initial disturbance. Since the magnetic force, as the buoyant
force in thermal convection, acts always in the direction of the temporal disturbance of the
fluid layer it has destabilizing character and is able to drive a convective flow in the fluid
(Schwab et al., 1983). The stabilizing effects are again thermal conductivity and viscous
friction in the fluid and the actual status of the system can be described by a dimensionless
Fig. 4.42. Schematic sketch for the explanation of the driving force of thermomagnetic convection. Explanations
are given in the text.
FERROFLUIDS
197
parameter called magnetic Rayleigh number Rm with
Rm =
μ0 |∂M/∂T |∇H T d 3
.
κη
(4.37)
If the experiment is carried out under terrestrial laboratory conditions the effect of thermomagnetic convection will always be accompanied by buoyancy driven effects leading to
the fact that the sum of Rayleigh number and magnetic Rayleigh number has to exceed a
certain critical value. If the fluid layer is heated from above, buoyancy will have stabilizing
character expressed by a negative value of Ra while it will be destabilizing for a fluid layer
heated from below. In contrast the magnetic force will always have destabilizing character independent from the direction of the temperature gradient, since, due to the nature of
their origin, magnetization gradient and the gradient of the inner magnetic field are always
antiparallel.
Due to the coupling of magnetic and buoyant effects the flow profile of thermomagnetic convection in a terrestrial experiment is usually quite complex (Schwab et al., 1983;
Schwab and Stierstadt, 1987). This can be overcome by performing the experiments under
conditions of reduced gravitational acceleration, e.g. in a drop tower, a sounding rocket
or on the International Space Station. As seen from the argument for the appearance of
the magnetic driving force, gravitational acceleration has no importance for this force, and
thus experiments in an environment with g ≈ 0 will not affect thermomagnetic convection
but sufficiently suppress buoyancy driven effects.
Moreover, the absence of a distinct direction given by gravitational acceleration enables
the performance of experiments with periodic boundary conditions using a cylindrical fluid
layer as shown in fig. 4.43. If the inner cylinder is heated and the outer one cooled, a radial
temperature gradient from the outer to the inner cylinder will be established. If the fluid
Fig. 4.43. Schematic sketch of the arrangement for the investigation of thermomagnetic convection in a cylindrical fluid layer under conditions of reduced gravitational acceleration.
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S. ODENBACH
layer is additionally subjected to an azimuthal magnetic field produced by a current leading wire along the symmetry axis, having a gradient directed radially inward one will again
achieve a situation in which a ferrofluid layer is influenced by the interaction of antiparallel directed field and magnetization gradients. Thus the described situation is capable to
lead to a thermomagnetic convective flow with the advantage compared to the situation
discussed above, that the magnetic field gradient is not a result of a second order effect
produced by the thermally induced magnetization gradient, but is created directly by the
magnetic field arrangement leading to much higher field gradients achievable. The main
problem in experiments dealing with flow detection in ferrofluids appears from the fact
that ferrofluids are optically opaque disabling all common fluid mechanic flow detection
techniques. For the example of the magnetic convection this problem can be overcome
by measuring the temperature distribution along the azimuthal circumference of the outer
cylinder bounding the fluid layer. If the flow profile is assumed to consist of counter rotating vortices along the cylinder axis as seen in fig. 4.43, positions where fluid flows from the
hot inner cylinder towards the outer wall will be characterized by a high temperature while
the counter directed flow towards the inner cylinder can be detected by a lower temperature
at the outer cylinder. Measuring the temperature distribution along the circumference will
thus lead to a sinoidal temperature profile which reflects not only the flow structure by a
series of maxima and minima but which provides additionally the convection amplitude A
by the amplitude of the temperature differences. The amplitude of convection is connected
with the magnetic Rayleigh number by
Rm − Rm∗
A≈
(4.38)
,
Rm∗
where Rm∗ denotes the critical value of the magnetic Rayleigh number, at which convection occurs.
A measurement of the amplitude provides thus an access to the determination of Rm∗ by
plotting A2 against Rm and extrapolating the resulting linear relation to A2 = 0. Respective experiments have been carried out using sounding rockets and drop tower experiments
(Odenbach, 1995a, 1995b). Figure 4.44 shows a typical temperature distribution obtained
by microthermistors placed in the outer cylinder. The distribution shows a series of maxima
and minima which corresponds to a set of counter rotating flow vortices with a diameter
equal to the fluid layers’ thickness as shown in the inset of fig. 4.44. Taking these temperature profiles for various magnetic Rayleigh numbers above Rm∗ and plotting the square
of the amplitude vs. Rm leads to the linear relation expected from equation (4.38) as seen
in fig. 4.45. The extrapolation to A2 = 0 provides the critical magnetic Rayleigh number
as Rm∗ = 1820. This result, as well as the flow profile detected are in excellent agreement
with independently performed numerical studies.
This agreement forced various theoretical studies of the problem e.g. (Zebib, 1996;
Morimoto et al., 1999) which dealt with three dimensional flow structures and the appearance of higher instabilities at stronger magnetic driving forces. Actually efforts are undertaken to enable an experimental proof of these calculations performing experiments on
thermomagnetic convection on the International Space Station (Odenbach, 1998a, 1998b).
A deeper understanding of the process of thermomagnetic convection could lead to the
possibility of a technical use of this magnetically driven flow. An example for the advantages of thermomagnetic convection in a technical device is the cooling of high power
FERROFLUIDS
199
Fig. 4.44. The temperature distribution along the azimuthal circumference of the cylinder shown in fig. 4.43
indicating the flow structure of thermomagnetic convection as shown in the inset.
Fig. 4.45. The linear relation between the square of the amplitude of convection and the magnetic Rayleigh
number.
electric transformers. In such systems—as depicted schematically in fig. 4.46—magnetic
gradients and temperature gradients appear naturally in parallel arrangement. This situation
is, as shown above, preferable for the appearance of thermomagnetic convection. Thus, if
the process would be well understood, a ferrofluid could be employed as a cooling agent
pumped by magnetic forces without the need for maintenance intensive external pumps.
But the process of thermomagnetic convection is not the only effect influencing thermal
convective flow in ferrofluids. Even in the absence of magnetic fields thermal convection in
a ferrofluid will have different characteristics than in ordinary one component fluids since
the temperature gradient does not only cause the buoyancy driven flow but will additionally
lead to a significant demixing of the fluid due to the Soret effect as discussed above. Taking
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S. ODENBACH
Fig. 4.46. Schematic sketch of the temperature and magnetic field gradients in a transformer.
Fig. 4.47. Schematic sketch for the explanation of the driving force of convection in the presence of Soret effect
in a binary mixture containing colloidal particles. Explanations are given in the text.
this into account thermodiffusion with a positive Soret coefficient will cause an additional
density gradient in the fluid due to the redistribution of the magnetic particles. As shown in
fig. 4.47 this density gradient will add to the temperature driven density variation leading
to a critical density difference at temperature differences well below the values found for a
one component fluid. This effect is known e.g. for water–alcohol mixtures but in contrast to
these molecular mixtures the characteristic parameter for this phenomenon, the separation
ratio
βc
ST
=
(4.39)
βT
will not be in the order of 0.1 but can reach values of about 1000 and more due to the
high Soret coefficient and the large density difference between the particles and the carrier liquid. Here βc denotes the volume expansion coefficient due to concentration changes
while βT is as above the volume expansion coefficient due to thermal effects. In a numerical study Ryskin et al. (2003) showed, that, despite the fact that thermodiffusion is a slow
process, the onset of convection even well below the critical Rayleigh number for a one
component system Ra∗0 takes place in reasonably short times not much different from the
FERROFLUIDS
201
Fig. 4.48. Schematic sketch of the experimental setup used for the investigation of the influence of the Soret
effect on thermal convection in ferrofluids.
Fig. 4.49. The temperature distribution along one of the legs of the cross of thermistors in the experimental setup
shown in fig. 4.48.
behaviour of simple fluids and that the amplitudes of convection for Ra < Ra∗0 are expected
to be large enough for an experimental investigation already for values of ψ = 100. A corresponding experiment (Odenbach and Völker, 2005) using the same ferrofluid as it has
been employed for the investigations of the Soret effect was performed in a flat layer system detecting the flow profile and convection amplitude by means of 13 microthermistors
arranged in a cross-like structure in the centre of the upper cooled plate (see fig. 4.48). The
fluid layer thickness equalled 5 mm with a relation between diameter and thickness of the
layer of 30. Figure 4.49 shows the temperature distribution measured along one of the legs
of the cross of microthermistors showing the typical sinoidal trend. The fact that the wavelength of the distribution is significantly larger than 10 mm results from the case that the
convection rolls are inclined relative to the direction of the microthermistors arrangement.
Taking the wavelength results of both legs of the thermistor cross, the typical wavelength
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S. ODENBACH
Fig. 4.50. The amplitude of convection in a ferrofluid as a function of the reduced Rayleigh number. The dashed
line indicates the behaviour expected for a one component system while the full line is a fit of the theory in
(Hollinger et al., 1998).
of the convective flow can be determined to be (10.1 ± 0.3) nm leading to the typical structure of flow vortices with a diameter equal to the thickness of the fluid layer. In fig. 4.48 the
amplitude of this convective flow is shown as a function of the reduced Rayleigh number
ε=
Ra − Ra∗0
.
Ra∗0
(4.40)
The dotted line gives the square root behaviour for a one component system in accordance with equation (4.38). Obviously the convection amplitude comes close to the
behaviour of a one component system for large Rayleigh numbers (ε 0) while it is
increased close to the critical point and in particular significantly strong convection is observed at low Rayleigh numbers well below Ra∗0 as it is expected due to the influence of
the Soret effect. The full line in fig. 4.50 is a fit of the theory for double diffusive convection by Hollinger et al. (1998) leading to a separation ratio for the fluid employed of
ψ = 1850 ± 100.
While the above mentioned experimental and theoretical investigations on thermal transport phenomena are in the first instance basic research one should bear in mind that the
phenomena described here can have serious impact on the performance of applications
of ferrofluids. Taking just as a single example the sealing of rotary feed throughs as it
was discussed in the context of the general application overview (see fig. 3.4). In such an
arrangement strong magnetic field gradients are accompanied by strong thermal gradients
especially if high rotation rates are employed. Thus thermodiffusion, thermal convection,
thermomagnetic convection and the respective couplings of these effects—which can be of
highly complex nature (Ryskin and Pleiner, 2004)—can influence the flow in the fluid as
well as its properties. Altogether an accurate understanding of the underlying fundamental
processes can help to improve the performance of such application.
FERROFLUIDS
203
5. Medical applications of ferrofluids
The final chapter of this contribution will be devoted to an actually quickly developing field
in magnetic fluid research. The use of magnetic nanoparticles in biomedical applications
has been discussed for a long time already but only recently some of the approaches became
more concrete.
The major problem of biomedical applications of magnetic fluids is the question of stability of a biocompatible ferrofluid as it has been discussed in the chapter on synthesis of
ferrofluids. Therefore the very first approaches for the use of magnetic fluids in medicine
employed the fluid in closed vessel systems only. The classical example for such an application idea has been the so called magnetic muscle (Rosensweig, 1985). Here a magnetic
fluid is enclosed in an elastic membrane and the counteraction of elastic and magnetostrictive forces were supposed to be used to generate a magnetically controlled contraction
leading to a muscle-like movement. Another example for such a kind of mechanic device
for medical use is the ferrofluid supported artificial heart. Usual artificial hearts being used
for patients waiting for a heart transplantation have a size which does not allow their use
inside the patient’s body leading to serious restrictions for the patient and limiting the time
of use to a maximum of a few month. So the idea of ferrofluid supported artificial heart
(Nethe et al., 1999) has been to reduce the size of the system in a way that an implantation
in the abdominal cavity would be possible. Therefore a small electromagnetic driving unit
has been developed in which a ferrofluid with high saturation magnetization was used to
enhance the magnetic force in the electromagnetic system. The major problems appearing
in the design of such a device are the biomedical boundary conditions, especially the short
pumping pulse, the high life time requirement and—most crucial—the problem of blood
heating due to the power needed to drive the pump. In particular the last boundary condition forced the employment of a ferrofluid in such systems since the force enhancement
yields a significant reduction of power required for the operation of the system leading
to less heat release and therefore to a more acceptable heating level of the blood pumped
by the device. The central difficulty appearing in the design of these pumps has been the
contradiction of a high saturation magnetization of the fluid—needed to keep the force
enhancement high—and a low viscosity required to enable short switching times for the
pump. Since—at the time of these developments—no suitable ferrofluids were available,
other approaches, basing on fluid mechanic principles were preferred for the ongoing design studies.
For the above mentioned devices the particular requirements for the ferrofluids are given
by electromagnetic aspects only. This changes as soon as the fluids are used in vivo leading
to the mentioned strong requirement of biocompatibility. Despite all actually unsolved
problems appearing from this demand the expected advantages in certain medical areas are
so large that strong efforts are actually undertaken to force the respective developments.
The use of magnetic nanoparticles as a contrast agent for X-ray examinations (Papisov
et al., 1993) is a meanwhile well established technique being approved for clinical use. The
direct injection of a suspension of magnetic particles into the patient’s vein is particularly
required for applications devoted to cancer treatment. Here two different basic approaches
are actually investigated. On the one hand the particles can be used as a carrier for drugs,
e.g. chemotherapeutic agents, transporting them to the region in the body where a certain
204
S. ODENBACH
medical treatment is needed. The second approach uses the particles themselves for the
medical treatment by subjecting them to an AC magnetic field leading to a localized heat
generation. It is obvious, that both techniques require a targeted delivery of the magnetic
particles towards a certain position in the patient’s body, usually a tumour.
Due to the possibility to attract the magnetic particles by means of magnetic field gradients, one possibility to guide them and locate them inside the tumour tissue is the so called
magnetic drug targeting (Ruuge and Rusetski, 1993; Alexiou et al., 2000, 2001; Alexiou
et al., 2001). For this technique, usually employed for tumours located directly beneath
the skin, the surfactant layer is functionalized with the drug to be delivered to the tumour.
After injecting the suspension of these functionalized nanoparticles into a vein leading to
the tumour the flow of particles in the blood is guided towards the target area by means of a
magnetic field gradient. This gradient can be either produced by a permanent magnet or by
the pole of an electromagnet positioned directly on the tumour. Since tumour tissue is usually highly vascularized an efficient delivery of the particles into the tumour tissue can be
achieved. Due to the action of the field gradient the particles are now captured in the tumour
region and the magnetic influence is even strong enough to force them to cross the vascular
wall leading to their immobilization in the tissue. This immobilization allows the removal
of the magnetic field after a certain time while the particles remain in the targeted area. In
this way the effect of the chemotherapeutic agent attached to the particles can be concentrated in the tumour region reducing side effects usually accompanying a chemotherapy. In
animal experiments Alexiou et al. (Alexiou et al., 2000; Alexiou, 2001) have shown, that
magnetic drug targeting can significantly enhance the efficiency of a chemotherapy reducing side effects below the limit of detection. But it was also shown that the lack of stability
of biocompatible ferrofluids leading to an agglomeration of the particles to larger structures
with diameters comparable to the size of capillary vessels bears the risk of embolism—a
problem that obviously has to be treated seriously.
While magnetic drug targeting is usually employed in chemotherapeutic treatment, the
also mentioned magnetic hyperthermia, using the heat release of magnetic particles in an
AC-field, utilizes usually tissue specific markers to locate the particles in the tumour. This
therapeutic approach makes use of the fact, that heating of tissue even slightly above normal blood heat will lead to a damage weakening the tumour and making it more sensitive to
other therapeutic means. For the normal hyperthermia approach temperatures about 42◦ C
are the target for the heating process while temperatures about 55◦ C are employed for the
so called thermoablation leading to a irreversible damage of the tumour tissue (see e.g.
Jordan et al., 1993; Hergt et al., 1998; Hiergeist et al., 1999; Hilger et al., 2002).
Like the magnetic drug targeting, magnetic hyperthermia is also actually in the state
of animal experiments exploring the dependence of the efficiency of heat release on field
strength and frequency as well as on the composition of the fluids utilized. While the field
strength is usually limited by technical boundary conditions, its frequency and the fluid
composition can be optimised. It is obvious that the relaxation behaviour of magnetization
will strongly influence the heat release since particles relaxing by the Brownian mechanism
will have to turn inside the tissue leading to strong heat generation by viscous friction.
Recently it has been shown that high heat release with a small amount of magnetic material
can be achieved if the size of the particles—as mentioned earlier only magnetite can be
used due to the necessary medical approval—is about 20 nm in diameter (Müller et al.,
2004).
FERROFLUIDS
205
The frequency of he magnetic field has to be adjusted as good as possible to the relaxation time of the particles to meet the resonance condition ωτB = 1. For magnetite particles
of the mentioned size this means that the frequency of the magnetic field should be in the
range between 0.2 MHz and 1 MHz taking into account the fact that particles always have
certain size distribution.
Both approaches to new therapeutic techniques for cancer treatment are quite challenging due to the low risk of side effects, but it should not be concealed, that numerous open
questions will have to be clarified before a standardized clinical application can be established. This may take a couple of years for basic research but keeping in mind the possible
advantages it is obviously worth to take the effort.
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chapter 4
MAGNETIC AND ELECTRICAL PROPERTIES
OF PRACTICAL ANTIFERROMAGNETIC Mn
ALLOYS
K. FUKAMICHI and R.Y. UMETSU
Department of Materials Science, Graduate School of Engineering, Tohoku University, Aoba 6-6-02,
Sendai 980-8579, Japan
A. SAKUMA
Department of Applied Physics, Graduate School of Engineering, Tohoku University, Aoba 6-6-05,
Sendai 980-8579, Japan
C. MITSUMATA
Advanced Electronics Research Lab., Hitachi Metals, Ltd., 5200 Mikajiri, Kumagaya 360-0843, Japan
Handbook of Magnetic Materials, edited by K.H.J. Buschow
Volume 16 ISSN: 1567-2719 DOI 10.1016/S1567-2719(05)16004-1
209
© 2006 Elsevier B.V.
All rights reserved
CONTENTS
Main list of abbreviations and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
2. Morphologies of Mn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
3. Electronic and magnetic structures of Mn alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
3.1. Model for the itinerant electron system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
3.2. Minimal models of itinerant electron magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
3.3. Path-integral approach for itinerant electron magnetism . . . . . . . . . . . . . . . . . . . . . . . . 222
3.4. Saddle point (molecular filed) approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
3.5. Helical magnetic structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
3.6. Rotation of local spin axes for the complex magnetic structures . . . . . . . . . . . . . . . . . . . . 231
3.7. Magnetic excitation energies and the exchange constants . . . . . . . . . . . . . . . . . . . . . . . 234
4. First principles approach for the magnetic structures of transition metal systems . . . . . . . . . . . . . 241
4.1. Tight-binding (TB)—LMTO method for the complex magnetic structures . . . . . . . . . . . . . . 242
4.2. Coherent potential approximation (CPA) for disordered alloys . . . . . . . . . . . . . . . . . . . . 245
4.3. Effective exchange constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
5. Electronic and magnetic structures of γ -Mn, L12 -type (≡ γ -phase) and L10 -type alloys . . . . . . . . 246
5.1. Pure γ -Mn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
5.2. L12 -type (≡ γ -phase) ordered and γ -phase disordered Mn3 Ir and Mn3 Rh alloys . . . . . . . . . 251
5.3. L10 -type ordered MnRh, MnIr, MnPt, MnPd and MnNi alloys . . . . . . . . . . . . . . . . . . . . 256
5.4. Comparison between the ordered and disordered phases of equiatomic MnPt alloys . . . . . . . . . 261
5.5. Stability of B2 phase of MnRh equiatomic alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
6. Experimental observations of antiferromagnetic transition and magnetic moments of Mn alloys . . . . . 267
6.1. The Néel temperature and the magnetic moment of L12 -type (≡ γ -phase) ordered (O) alloy systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
6.2. The Néel temperature and the magnetic moment of γ -phase disordered (DO) alloy systems . . . . 269
6.3. Magnetic structures of L12 -type (≡ γ -phase) Mn3 Pt and Mn3 Ir ordered (O) alloy systems . . . . 273
6.4. Crystallographic phase transformation between B2 and L10 -type phase . . . . . . . . . . . . . . . 279
6.5. The Néel temperature of L10 -type alloy systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
6.6. Strong bonding state correlated with magnetic moment and Pauling valence . . . . . . . . . . . . . 284
7. Fundamental properties of practical Mn alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
7.1. Low-temperature specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
7.2. Electrical resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
7.3. Reflectivity and thermoelectric power of L10 -type MnPt alloys . . . . . . . . . . . . . . . . . . . . 302
7.4. Magnetic anisotropy in γ -phase disordered alloys and L10 -type ordered alloys . . . . . . . . . . . 304
210
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
211
7.5. Spin structures and lattice distortions of γ -phase disordered alloys . . . . . . . . . . . . . . . . . . 310
7.6. Appearance of the β-phase and stabilization of the γ -phase . . . . . . . . . . . . . . . . . . . . . . 320
7.7. Spin fluctuation effects in β-Mn alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
7.8. Concentration and temperature dependences of magnetic properties of the L10 -type alloy systems
328
7.9. Magnetovolume effects and elastic properties of Mn alloys . . . . . . . . . . . . . . . . . . . . . . 340
8. Mechanism of exchange bias-field in spin valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
8.1. Survey proposed models of exchange coupling bias . . . . . . . . . . . . . . . . . . . . . . . . . . 354
8.2. Generalized model based on classical Heisenberg model . . . . . . . . . . . . . . . . . . . . . . . . 365
8.3. Blocking temperature and training effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
9. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
MAIN LIST OF ABBREVIATIONS AND SYMBOLS
AF-I
AFM
ASA
ASW
CIP
CPA
CPP
DLM
DO
DOS
FM
FS
GGA
GF
GMR
H-S
LSDA
LMTO
KKR
MAE
MDW
MFA
ML
MQSDW
MRAM
MT
O
PM
PV
SCR
SDW
SF
SP
TB
T1
collinear antiferromagnetism
antiferromagnetic, antiferromagnetism
atomic sphere approximation
augmented-spherical-wave
current in the plane
coherent potential approximation
current perpendicular to the plane
disordered local moment
disordered state
density of states
ferromagnetic, ferromagnetism
frustrated spin
generalized gradient approximation
Green function
giant magnetoresistance
Hubbard–Stratonovich
local spin density functional approximation
linear muffin-tin orbital
Korringa–Kohn–Rostoker
magnetocrystalline anisotropy energy
magnetic domain wall
molecular field approximation
monolayer
multiple-Q spin density wave
magnetic random access memory
muffin tin
ordered state
paramagnetic, paramagnetism
Pauling valence
self-consistent renormalization
spin density wave
spin fluctuation
saddle point
tight-binding
triangular magnetic structure
212
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
TMR
tunnel magnetoresistance
AAFM
B0
C
Ccor
Cmv
Cp
C, γ , +
)
ciσ (ciσ
EF
EEX
f (ε)
G(ω)
Hex
Jij
J0
KAFM
Kd
kB
l, m
M
m
mi , mi N(EF )
n
ni , ni P
p(ω)
S
SL,L
T
TB
TC
tij
Tm
TN
Tt
U
U (θ, φ)
Um
V
α
δd
γe
exchange stiffness
bulk modulus at ambient pressure
integral path
correlation effective Coulomb energy
magnetovolume coupling constant
specific heat
potential parameters in the LMTO method
annihilation (creation) operator of σ spin at the i-th site
Fermi level
exchange-correlation energy in the density functional theory
Fermi distribution function
Green function
exchange bias-field
pair exchange constant between the moments at the i-th and the j -th sites
effective exchange constant
magnetocrystalline anisotropy constant of antiferromagnet
unidirectional anisotropy constant
Boltzmann constant
orbital index
magnetization
local magnetic moment
operator of magnetic moment at the i-th site, expectation value
density of states at the Fermi level
electron density
number operator of electron at the i-th site, expectation value
pressure
potential function in the LMTO method
action
structure constant in the LMTO method
temperature
blocking temperature
Curie temperature
hopping integral of electron between the i-th and the j -th sites
transformation temperature of magnetic structure
Néel temperature
transformation temperature of crystal structure
Coulomb interaction energy between electrons of opposite spins
rotation matrix of quantization axis of spin 1/2 to the direction of (θ, φ)
exchange splitting
volume
thermal expansion coefficient
magnetic domain wall width
electronic specific heat coefficient
213
214
κ
νσ (r)
θ, φ
θD
ρ
σ
ω
ωs
ξ 2 (T )
K. FUKAMICHI et al.
compressibility
exchange-correlation potential for the electron of σ spin
angles defining the direction of the magnetic moment
Debye temperature
density matrix, electrical resistivity
spin index
energy of electron
spontaneous volume magnetostriction
mean square amplitude of thermal spin fluctuations
1. Introduction
Practically, the giant magnetoresistance (GMR) and tunnel magnetoresistance (TMR) are
very important for electronic and magnetic devices such as magnetic recording heads and
magnetic random access memory (MRAM) cells (Hirota et al., 2002). Furthermore, antiferromagnetic exchange-coupled multilayer films are necessary in order to reduce domain
noise in perpendicular magnetic recording media (Takahashi et al., 1999; Jung and Doyle,
2001). Excellent high frequency characteristics have been obtained by using antiferromagnetic/ferromagnetic/antiferromagnetic trilayers (Viala et al., 2004). In these devices, the
antiferromagnetic (AFM) Mn alloy systems are useful as exchange biasing films for the
ferromagnetic (FM) film and magnetic tunnel junctions. There are many systems having a
complex (non-collinear) spin structure, depending on the d-electron number and the crystal
structure associated with the geometrical frustration of magnetic moments. Experimental
evaluation of their magnetic structures is quite difficult, especially in γ -phase disordered
alloys, and hence there still remain the systems whose magnetic structures have not been
established yet. Furthermore, the explanations of the mechanism of the exchange bias-field
are still divergent, depending on the proposed models.
Several kinds of excellent overviews on the exchange bias-field are available (Berkowitz
and Takano, 1999; Nogués and Schller, 1999; Stamps, 2000; Kiwi, 2001). Review articles
(Barthélémy et al., 1999; Daughton, 1999; Coehoorn, 2003; Nogués et al., 2005a, 2005b)
and the extensive treatise (Hirota et al., 2002) of basic researches and applications for GMR
and TMR have been published.
In the present article, we look over physical aspects of antiferromagnetism of Mn alloys both from theoretical and experimental view points in order to provide a base for
the future fundamental and practical developments. The construction of this article is as
follows. In the next section, the morphologies of Mn are outlined. We briefly explain the
theoretical frame applied to the first principles version in order to deal with the actual Mn
alloy systems in section 4. The calculated results of electronic and magnetic structures,
the amplitude of the exchange field of Mn alloys in γ -phase disordered and L10 -type ordered states will be discussed in the same section 5. In section 6, the experimental data
of antiferromagnetic transitions of various Mn alloys are reviewed,referring the theoretical calculations. The electrical resistivity is associated with the Néel temperature and the
electronic state, the relations among the lattice distortion, phase diagram and spin structure
are discussed, and the appearance of β-Mn phase also presented. The magnetovolume and
magnetoelastic effects are discussed in terms of spin fluctuations in section 7. In section 8,
the exchange-bias field is reviewed, and the mechanism of the exchange-bias field between
the Mn alloy antiferromagnetic (AFM) film and the ferromagnetic (FM) film are discussed
within the framework of the classical Heisenberg model. Finally, the concluding remarks
are given in the last section 9.
215
216
K. FUKAMICHI et al.
2. Morphologies of Mn
Structural and magnetic properties Mn are the most complex in all metallic elements. According to the general structural trends in the series of 4d and 5d transition metals, Mn
in the group VII would be expect to crystallize in a hexagonal closed-packed (hcp) A3
structure. However, appearance of magnetism in 3d elements disturbs this trend and Mn
has several kinds of morphologies. α-Mn is the most stable polymorph under normal conditions of temperature and pressure, and forms an exotic complex structure containing 8
atoms in a cubic unit cell (Hobbs et al., 2003). β-Mn exists in the temperature range between 1000 and 1368 K, and is simple cubic with 20 atoms per unit cell (Hafner and
Hobbs, 2003). γ -Mn of fcc phase is formed in the high-temperature region between 1368
and 1406 K. At higher temperatures up to the melting point of 1517 K, the δ-phase has
a body-centered cubic (bcc) structure. By quenching γ -Mn to room temperature, γ -Mn
is stabilized in a face-centered tetragonal (fct) structure. By addition of several percent of
Cu, the quenched γ -Mn phase in a face-centered cubic (fcc) structure is stabilized (Endoh
and Ishikawa, 1971). By applying high pressures up to about 160 GPa, α-Mn transforms
to ε-Mn in an hcp structure phase (Zheng-Johansson et al., 1998) in analogy with α-Fe
(Bundy, 1964; Shimizu et al., 2001). Morphology, crystal structure, space group, stabilization temperature and pressure for Mn element are listed in table 2.1.
The Néel temperature TN of α-Mn is 95 K and the magnetic transition is accompanied by
a tetragonal distortion. In the complex crystal structure, the different atomic sites are also
magnetically inequivalent. Neutron diffraction data (Yamada et al., 1970; Lawson et al.,
1994), magnetic torque measurements (Yamada and Tazawa, 1970) and nuclear magnetic
resonance (NMR) investigations (Yamagata and Asayama, 1972) have been explained in
terms of a non-collinear antiferromagnetic structure. The magnitudes of the magnetic moments calculated by the local spin density approximation (LSDA) (Sliwko et al., 1994)
and the augmented-spherical-wave (ASW) method (Williams et al., 1979) for four kinds
of atomic sites are in reasonable agreement with those experimental values (Yamada et al.,
1970).
In the cubic cell of β-Mn, there are two inequivalent crystallographic sites, site 1 (8c
sites) and site 2 (12d sites). The longitudinal nuclear spin–lattice relaxation rate 1/T1 at
site 2 is about 20 times larger than that at site 1 (Kohori et al., 1993), namely, the electronic
TABLE 2.1
Morphology, crystal structure, space group and stabilization temperature and pressure for the Mn element
Phase
Structure
Space group
Temperature (K)
α-Mn
β-Mn
γ -Mn
δ-Mn
ε-Mn
A12
A13
A1 (fcc)
A2 (bcc)
A3 (hcp)
I 4̄3m
P 41 32
F m3̄m
I m3̄m
P 63 /mmc
1000–1368
1368–1406
1517–m.p.a
a m.p.: melting point.
b Zheng-Johansson et al. (1998).
Pressure (GPa)
above 165b
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
217
state of the Mn site 2 is more magnetic than that at the site 1 (see fig. 7.53). No magnetic
order with strong spin fluctuations in β-Mn has been confirmed down to the lowest temperatures (1.4 K) (Nakamura and Shiga, 1997), showing an enhanced Pauli paramagnetic
behavior. It has been pointed out that magnetic properties of β-Mn are very similar to those
of materials with a fully frustrated lattice such as kagomé or pyrochlore lattices where
the frustration overcomes any magnetic ordering, and hence the magnetic state should be
considered as a spin liquid (Canals and Lacroix, 2000). The β-Mn lattices are fully frustrated and the frustration cannot be eliminated by non-collinear spin arrangement. The
magnetic state of β-Mn is affected by additional elements substituted into site 1 or site 2.
The elements substituted into site 1 such as Fe and Ni have been considered to induce the
antiferromagnetic state (Katayama and Asayama, 1978), although no direct observations
of the long-range antiferromagnetic ordering have been carried out. Recently, however,
a definite long-range antiferromagnetic order has been confirmed by neutron diffractions
for β-MnOs alloys (Miyakawa et al., 2005) (see fig. 7.55). On the other hand, the elements
substituted into site 2 such as Al results in a spin-glass-like behavior (Nakamura and Shiga,
1997). The antiferromagnetic magnetic order in fct γ -Mn and quenched γ -Mn takes place
at 450–500 K. γ -Mn phase may be stabilized by an epitaxial growth on Cu (100) (Wuttig
et al., 1993) and Ni (100) substrates (Rader et al., 1997). In higher concentration ranges,
the γ -Mn phase is stabilized and lattice distortion and a magnetic structure change occur,
depending on the kind and concentration of solute elements. Especially, some γ -Mn phase
alloys exhibit a very high Néel temperature and are promising as spintronic devices. The
magnetic state, equilibrium atomic volume V , enthalpy difference H at ambient pressure
relative to α-Mn, bulk modulus B0 at ambient pressure for five polymorphs of Mn are tabulated in table 2.2 (Hafner and Hobbs, 2003), together with available related data. In the
TABLE 2.2
Magnetic state, equilibrium atomic volume V , enthalpy difference H at ambient pressure relative to α-Mn, bulk
modulus B0 , pressure derivative B0 of five polymorphs of Mn (Hafner and Hobbs, 2003)
B0
(theor)
(GPa)
B0
(exp)
(GPa)
B0
(theor)
B0
(exp)
158b
6.0
4.6b
146
188
269 (ferri)
300 (non-mag)
⎧
⎨ 144 (GGA)
107 (GGA)e
⎩
110 (ASW)f
166
61
246
Phase
Magnetic
state
V
(theor)
(Å3 )
V
(exp)
(Å3 )
H
(meV)
α-Mn
AFM-NCL
11.08
12.05a
0
α-Mn
FIM
10.84
12.44c
63
γ -Mn
AFM
11.16
12.95d
67
AFM
AFM,
PMg
11.12
10.72
δ-Mn
ε-Mn
4.7
7.4
6.8
5.2
NCL: non-collinear, FIM: ferrimagnetic.
a Lawson et al. (1994), b Fujihisa and Takemura (1995), c Preston (1928), d Wyckoff (1971). The extrapolated value from high-temperatures to room temperature. e Asada and Terakura (1993), f Moruzzi et al. (1989),
g Zheng-Johansson et al. (1998).
218
K. FUKAMICHI et al.
equation of state associated with finite strain theory, Murnaghan equation is given by
B B0
V0 0
P = (2.1)
−1 ,
B0
V
where B0 is the pressure derivative of B0 (Murnaghan, 1951). The value of B0 for β-Mn is
the highest in the morphologies of Mn, being comparable to that of non-magnetic 4d and
5d metals. That is, the values calculated by the Birch–Murnaghan equation (Birch, 1952)
and the universal equation of state (Vinet et al., 1987) are 269 GPa in the ferromagnetic
state and about 300 GPa in the non-magnetic state as given in table 2.2. On the other hand,
the outstanding softness of α-Mn is closely correlated with the strong volume dependence
of the magnetic moments around the equilibrium volume, and the value of γ -Mn and δ-Mn
is also low. Especially, the bulk modulus of 144 GPa of γ -Mn is about one half the value
found for β-Mn. Various fundamental properties of α- and β-Mn are reviewed by Hobbs
et al. (2003), and Hafner and Hobbs (2003).
3. Electronic and magnetic structures of Mn alloys
Most Mn-based alloys become antiferromagnetic due to their occupation of 3d orbitals
close to the half-filled state. Among them, a number of face centered cubic (fcc) alloys such
as γ -phase Mn-based alloys are considered to have complex non-collinear spin structures
in which the relative angle between the neighboring spins deviates from 0 to 180 degrees.
However, detailed discussion on their spin structures has been avoided because of difficulties both of experimental and theoretical techniques, and some scatter of data due to
different experimental methods.
Several kinds of equiatomic Mn alloys with Ir, Rh, Pt, Pd and Ni have a B2 (≡ CsCl)type cubic phase at high temperatures and transform to an L10 (≡ CuAu-I)-type tetragonal
phase with a diffusionless martensitic transformation process at low temperatures. The
L10 -type ordered alloys, except for MnRh (Selte et al., 1972), have a quite high Néel
temperature TN .
Practically, the application of exchange bias-field has been proposed for recording heads
based on anisotropic magnetoresistance. In this case, an AFM layer is deposited on the
edges of the FM layer to avoid closure domains, and thus reduce the Barkhausen noise of
devices (Tang, 1984; Tsang, 1989; Cain and Kryder, 1990; Miyamoto et al., 1995; Lin et
al., 1996). Recently, the antiferromagnetic Mn-based alloy systems have become important as an exchange biasing film in the GMR and TMR devices (Barthélémy et al., 1999;
Berkowitz and Takano, 1999; Daughton, 1999; Nogués and Schller, 1999; Hirota et al.,
2002). As is well-known, the exchange biasing characteristics and the blocking temperature TB are closely interrelated with the spin structures and also the magnitude of the Néel
temperature TN . Therefore, the knowledge of the microscopic origin such as the spin structures of these systems is strongly desirable to develop excellent properties for spintronic
devices including GMR and TMR spin valves. In this section, the magnetic and electronic
structures of Mn alloys are calculated by the tight-binding (TB)-linear muffin-tin orbital
(LMTO) method based on the local spin density functional approximation (LSDA). Noncollinear spin structures are treated as the local rotation of the spin quantization axis on
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
219
each site. In both the disordered alloy and paramagnetic (PM) state, the coherent potential
approximation (CPA) is adopted for each sublattice within the local frame of the spin axis.
The effective exchange constant (Liechtenstein et al., 1987; Sakuma, 2000) acting on each
Mn moment is also calculated to examine the magnetic stability and to give an estimation of the Néel temperature. Before going into the results and discussion, we give a brief
description of the methodology in the following several subsections.
3.1. Model for the itinerant electron system
Let us first start with the single band Hubbard model (Hubbard, 1963, 1964a, 1964b) given
below
+
tij ciσ
cj σ + U
H =−
ij
σ
ni↑ ni↓ ,
(3.1)
i
+
) denotes the electron annihilation (creation) operator of σ (= + or −) spin at
where ciσ (ciσ
+
ciσ is the number operator. As shown in fig. 3.1, the first term reprei-th site and niσ ≡ ciσ
sents electron hopping between i-th site and j -th site with probability amplitude of tij , the
second term means Coulomb interaction between electrons of opposite spin encountered at
a same site. Though, boldly dropped multi-orbital and intra-atomic Coulomb interactions
between them play an important role especially for ferromagnetism, we restrict ourselves
to this simple model to avoid the confusion from plural effects. Even simple, the model can
hardly be solved exactly except for a few cases, since the second term reflecting particle
character of the electrons puts the treatment into typical many body problems. Here, let us
dare to give an intuitive interpretation on the role of the two terms in equation (3.1). The
first term reflecting wave character of the electrons provides band effects which supply the
bonding (collective) energy with electron hopping between the sites. If all electrons located
at each site have a same spin direction (ferromagnetic alignment), the electron hopping is
restricted due to the Pauli principle and then the ferromagnetic state is unfavorable for this
term. In short, the collective force to form solids or molecules acts as a resistance for ferromagnetic ordering. The second term, on the other hand, insists that the energy increases
when the two electrons encounter at a same site. Since, two electrons of the same spin direction can not meet at the same site; this term favors the ferromagnetic alignment. In other
words, since the Coulomb repulsion does not work for electrons of parallel spin (within the
frame of the Hubbard model), this term helps the ferromagnetic alignment. As a result, it
can be said that the magnetic ordering can be realized as a consequence of the competition
between the wave character and the particle character of electrons. In the real systems, the
Fig. 3.1. Schematic illustration of the Hubbard model. The circles represent the atomic positions and the arrows
the spin direction of electrons. The electrons can hop to the other site with a probability amplitude of t and interact
with the other electron with the opposite spin direction when they encounter at the same site.
220
K. FUKAMICHI et al.
influence of these two terms on the magnetic structure is strongly dependent on the lattice
structure and the number of electrons. The details of this situation will be discussed in the
following sections.
3.2. Minimal models of itinerant electron magnetism
We consider about the condition of occurrence of magnetic ordering with using minimal
systems. In this analysis, one can see the importance of lattice structure and carrier density
for the spin structure. The simplest system in which both two terms in equation (3.1) can
work is a two-electron in the two-site system as shown in fig. 3.2(a). In this case, the oneelectron eigen-states realized by the first term are located at E = −|t| and |t|. When the
two electrons have the same spin direction, they must occupy different state not to commit
the Pauli principle. In this case, the second term does not work and then the total energy
is exactly Et = |t| − |t| = 0. When the two spins are anti-parallel, on the other hand, two
Fig. 3.2. Minimal models of the itinerant electron magnetism in the two electron system. In (a) and (b) the
energy of the singlet state (Es ) is always lower than the triplet state (Et ), while for the case (c) where two
levels degenerate at E = t (<0) realizing the triplet state. Ecor is the correlation energy due to the Coulomb
interaction U .
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
221
electrons can occupy the lower level at E = −|t|. If the Coulomb energy U is zero, the
total energy turns to be Es = −|t| − |t| = −2|t|. When the second term U is switched on,
the total energy increases from −2|t| and is given exactly by
1
U − U 2 + 16t 2 .
(3.2)
2
It can be easily seen that Es is lower than Et for any U and thus the ground state is always
spin singlet (S = 0) state. Especially for the strong coupling limit of U/|t|, Es can be
approximated to be
Es =
4t 2
.
U
This gives an effective (antiferromagnetic) exchange interaction of
Es ≈ −
(3.3)
t2
(3.4)
U
for two spins, which is the so-called super-exchange interaction. Physically, this result
means that the bonding of two sites through the electron hopping with opposite spins is
more advantageous in energy than the parallel spin (triplet S = 1) state in which the
Coulomb interaction is free but the electron hopping is tightly restricted. Generally, this
situation can work also for N -site N -electron system (half-filled system) in which the antiferromagnetic ordered state is realized in two- and three-dimensional systems. In the large
U/|t| case, this system belongs to insulator and can be seen in some transition metal oxides where the occupation of the valence orbital dominating the magnetism is just unity.
Then, as a metallic system, we consider two-electron in three-site system. When the three
sites are√arranged√linearly as shown in fig. 3.2(b), one-electron eigen-values are given by
E = − 2|t|, 0, 2|t|. Since the situation is essentially the same as the case mentioned
above, the ground sate is always spin singlet (S = 0) when two electrons are involved in
these eigen-states. That is, for any√
value of U , the singlet state where two electrons occupy
the one-electron level at E = − 2|t| with the Coulomb interaction U is always lower
in energy
√ than the triplet state where two electrons are located separately in each level of
E = − 2|t|, 0 (Kanamori, 1963). On the other hand, when the three sites form a triangular as shown in fig. 3.2(c), the one-electron eigen-states have doubly degenerate states
at E = t and single level at E = −2t. If t < 0, the degenerate states at E = t are lower
in energy and it is possible for both electrons to occupy the level at E = t, regardless of
the spin configuration, because, for the parallel spin case, two electrons are to be sited in
the different state at E = t in order not to commit the Pauli principle. Since two electrons
with parallel spin configuration is free from the Coulomb interaction, the total energy of
the triplet state is exactly Et = 2t. The singlet state, on the other hand, has an additional
positive energy due to the Coulomb interaction U . Thus the triplet state is always more
stable than the singlet state and we can understand there is a possibility that the ferromagnetic ordering is realized in the system as fig. 3.2(c). To extend this situation, one can
infer that the ferromagnetism is realized in the case where the one-electron energy interval
(E) around the highest level in the occupied states (Fermi level in the solids, or HOMO
level in the molecular system) is smaller than the correlation (effective Coulomb) energy
J ∝−
222
K. FUKAMICHI et al.
(Ecor ). Intuitively, the ferromagnetic state can be established, if the energy consumption of
the kinetic (one-electron) energy to align the spin direction is smaller than the correlation
energy acting on the electrons with anti-parallel spin. Since, generally in solids, E can
be approximated to be 1/N(EF ) (N (EF ): the density of states at the Fermi level), the ferromagnetic condition leads to 1/N(EF ) < Ecor (N (EF )Ecor > 1). This is a generalized
form of the Stoner criterion.
3.3. Path-integral approach for itinerant electron magnetism
In the preceding section, a qualitative consideration has been made on the mechanism of
magnetic ordering with using finite cluster models. In the following sections, we will give
an explanation from the microscopic viewpoint on the theoretical approaches to describe
the magnetism of solids.
A powerful tool to describe the ordered state and the (thermal or quantum) fluctuations in the many body systems is the path-integral approach. In this method, using the
Hubbard–Stratonovich (H-S) transformation for the second term in equation (3.1), the partition function can be expressed as (Hubbard, 1959; Wang et al., 1969; Korenman et al.,
1977; Schulz, 1990)
Z = Dξ Dη exp −S(ξ , η) ,
(3.5)
where Dx = (Uβ/4π)1/2 i dxi with β = 1/kB T , S is the action given by
S(ξ , η) = − Tr ln(∂τ − tˆ + ε̂) +
U
β
4
2
ξi + ηi2
(3.6)
i
with  ≡ Aij and
U
(iηi + ξ i · σ )δij .
(3.7)
2
In equation (3.6), Tr denotes trace over site (i), spin (σ ) and time (τ ), and σ in equation (3.7) is the Pauli matrix. Here, we take a static approximation for the auxiliary fields
{ξ , η}, regarding the fact that the spin excitation energies are a few ten meV at most, while
the band energies of the electrons are order of eV. The virtue of the H-S transformation is
to transform the many body problems to a one-electron model under the potential given by
equation (3.7).
Before going into the elaborate inspections, let us first make a brief analysis with a
preliminary approach. The simplest way to express the magnetic interaction is a second
order perturbation expansion in terms of the effective magnetic potential in equation (3.7).
εi = −
U
ξ · σ.
2 i
Disregarding the terms concerning the virtual electric field η, the action is written as
−
S0
S
=
−
β
β
J (q)ξ q · ξ −q ,
q
(3.8)
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
223
where S0 is the unperturbed term
and the second term represents the exchange term for the
auxiliary filed ξ q = (1/N)1/2 i ξ i eiq·R i (R i : position of the i-th site, N : the number of
sites) with the following exchange energy
U
U χ(q) − 1 .
4
χ(q) in the above equation is expressed by the Fermi distribution function f (ε) as
J (q) =
χ(q) =
1
N
k
f (εk ) − f (εk+q )
εk+q − εk
(3.9)
(3.10)
with εk = − i ti0 eik·R i .
The above expression for J (q) takes a part of the Murata and Doniach (1972) formulation, which was extended to a more general form by Moriya and Takahashi (1978). Also,
this can be regarded as a crude version of formalism given by Prange and Korenman (1979)
where the spin polarization due to U is included in the single particle eigen-states. From
equations (3.8) and (3.9), the ferromagnetic ground state is found to require
J (q = 0) ∝ U χ(0) − 1 = U N (EF ) − 1 ≥ 0.
(3.11)
Thus, we have reached the Stoner criterion again. Note, however, that this is no more
than a necessary condition for the ferromagnetism, since there remains a possibility that
a larger positive J (q) exists at finite q, which gives RKKY (Ruderman–Kittel–Kasuya–
Yoshida) (Ruderman and Kittel, 1954; Kasuya, 1956; Yoshida, 1957) like spin density
wave with the wave length of 1/q. Here, let us consider the physical meaning that the
condition J (0) > 0 does not necessarily lead to the ferromagnetism. The exchange term
in equation (3.9) contains not only Jij (i = j ) but also Jii in the real space representation.
The local exchange part Jii is given by
1
N
J (q)
(3.12)
q
from which we understand that this term takes a maximum value at the electron number per
orbital (n) of unity (half-filled state). Thus the local moment would also exhibit a maximum
value at this point. However, this does not mean the ferromagnetic alignment of moments,
and rather leads to an opposite result. To see this, we examine the following quantity
1
U2
1
J0 = J (0) −
(3.13)
J (q) =
χ(q) .
χ(0) −
N q
N
N q
J0 is found to be i Ji0 , which corresponds to J (q = 0) in the Heisenberg model (in the
Heisenberg model, J (q) does not contain the local exchange term Jii because Jii = 0 in
the Hamiltonian). Also, this is formally equal to
1
N
L(q)
q
224
K. FUKAMICHI et al.
Fig. 3.3. Exchange constant calculated from equation (3.13) as a function of the Fermi level EF . The inset shows
the assumed density of states (DOS) with a band width of 2D (Sakuma, 1999a).
with
Jij 1 − eiq·R ij .
L(q) =
ij
Figure 3.3 shows the calculated result of J0 as a function of the Fermi level EF (or band
filling), assuming the semi-elliptic density of state (DOS) for the electronic system (f (ε)
is treated as a step function). Obviously, J0 takes a negative value at around EF = 0, and
positive when EF lies near the edge of the band (Sakuma, 1999a). This aspect is consistent with well-known results discussed by Asano and Yamashita (1972), Hamada (1981)
and Terakura et al. (1982) with spin susceptibilities of 3d transition metals. However, the
present result implies that this trend is not ascribed to the nature of electronic structures of
the transition metals. Furthermore, one can see from equation (3.13) that the aspect does
not depend on the strength of U provided U > 0, and is mainly attributed to the constraint
of the integral range of χ(q) by the Fermi distribution function. In this sense, the behavior
can be regarded as a general property of electronic systems reflecting the Pauli principle.
Note, however, that the occurrence of the local moment is brought about by U > 0.
3.4. Saddle point (molecular filed) approximation
As seen from above discussion, the magnetic interaction derived in the itinerant electron
system involves not only inter-site interaction of spins but also intra-site interaction giving rise to the local spin polarization. The exchange interaction defined in the Heisenberg
model corresponds to the former term. Therefore, next, we take a remedy to treat these
two magnetic interactions separately from the beginning. The basic idea was first proposed
by Hubbard (1979). The scenario is as follows. At the first step, a magnetic solution of the
Hubbard model is provided by means of the saddle point approximation in the path-integral
approach. In the second step, the slight deviation from the saddle point is expressed as a
change of the direction of obtained magnetic moment at a given site. In the last step, the
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
225
Fig. 3.4. Schematic illustration of the action as a function of auxiliary fields ξ and η. The SPs with respect to ξ and
η correspond to the expectation values of the magnetic moment m and the electron number in, respectively.
Heisenberg type exchange constants are derived from the energy change due to the deviation from the saddle point solution. This can be easily linked to the LSDA version of
the formulation by Liechtenstein et al. (1987) through the following correspondences. The
saddle point (SP) solutions turn to be molecular field solutions which can be given by usual
self-consistent band calculations and the deviation from it on a certain site can be treated
within the single-site multiple scattering theory. This relation leads us to the first principles
calculations for the exchange constants as given afterward (section 4.3).
As is well known, the saddle points for the action S in equation (3.6) are obtained from
∂S/∂ηi = ∂S/∂ξ i = 0 as given in fig. 3.4. This gives
1
ηi = i Trσ,τ (−∂τ + tˆ − ε̂)−1
ii = ini ,
β
1
ξ i = Trσ,τ (−∂τ + tˆ − ε̂)−1
ii σ = mi .
β
(3.14)
(3.15)
At the saddle points, the action is written as
U
SS = SSP − Tr ln −G−1 + β
4
mi 2 − ni 2 ,
(3.16)
i
where G is the Green function (GF) given by
−1 U
U
G ij = −∂τ δij + tij −
ni + mi · σ δij .
2
2
(3.17)
The action in this case acts as a thermodynamic potential and the effective Hamiltonian
can be expressed as
+
tij ciσ
cj σ +
HMF = −
ij
+
U
4
σ
vi ni −
i
mi 2 − ni 2
i
B i · mi
i
(3.18)
226
K. FUKAMICHI et al.
+
+
with vi = U/2ni , B i = U/2mi , mi = σ,σ ciσ
σ σ,σ ciσ , ni = σ ciσ
ciσ , and ni ,
mi are
ni ≡ ni↑ + ni↓ ,
mi = mi (cos φi sin θi , sin φi sin φi , cos θi ),
1/2
2
mi = σix
+ σiy2 + σiz2
,
+ + σix = 2 Re ci↑ ci↓ ,
σiy = 2 Im ci↑
ci↓ ,
−1 σiz
−1 σiy
θi = cos
,
φi = tan
.
mi
σix
σiz = ni↑ − ni↓ ,
It can easily be understood that equation (3.18) is nothing but the molecular field Hamiltonian derived from an approximation for the second term in equation (3.1)
n↑ n↓ → n↑ n↓ + n↓ n↑ − c↑+ c↓ c↓+ c↑ − c↓+ c↑ c↑+ c↓ − n↑ n↓ + c↑+ c↓ c↓+ c↑ .
+
+
ci↓ and ci↓
ci↑ should be solved self-consistently, from which
Here, ni↑ , ni↓ , ci↑
the amplitude mi and the direction (θi , φi ) of the magnetic moment at the i-th site can be
obtained. The vector B i in the third term in equation (3.18) corresponds to the exchange
(molecular) field acting on the electron at the i-th site and is proportional to the magnetic
moment mi brought about by itself. The last term in equation (3.18) is the correction
term for total energy due to double count in the molecular field approximation. The matrix
form of equation (3.18) can be given by
−1 0
1 0
tij i+
vi i+
j +
i
HMF =
0 −1
0 1
ij
i
cos θi
e−iφi sin θi
+
+
Bi i
(3.19)
i
eiφi sin θi
− cos θi
j
ci↑
and Bi = U mi /2.
ci↓
From equation (3.19), the collinear magnetic structure (relative angles between two moments are 0 or π) can be calculated with the up spin and the down spin states, separately,
since the off-diagonal elements vanish due to sin θ = 0. For non-collinear structure, on the
other hand, it is necessary to treat the (2 × 2) matrices of the Hamiltonian.
with i =
3.5. Helical magnetic structures
Here, let us try actual calculations of the electronic structures by using equation (3.19).
To see how the magnetic arrangements influence the electronic structures, we consider the
helical structures specified by wave vector Q in the two-dimensional system by putting
mi = m, θi = θ , φi = Q · R i in equation (3.19) as illustrated in fig. 3.5. The ferromagnetic structure can be obtained with θ = 0 or Q = (0, 0), the collinear antiferromagnetic
alignment to the x-direction corresponds to θ = π/2, Q = (π/a, 0) (a: the lattice constant), and the arbitrary helical structure with the wave length of (a/p, a/q) is expressed
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
227
Fig. 3.5. Schematic diagram of the spin spiral structure in two-dimensional lattice. All spins cant from the z-axis
with the angle θ and the azimuthal angle of each spin is defined by φ.
by Q = (pπ/a, qπ/a) (0 < p < 1, 0 < q < 1). After some algebra, the Hamiltonian of
equation (3.19) can be rewritten as
+
+
ck+Q/2↑ ck−Q/2↓
HMF =
k
×
1
ε(k + Q
2 ) − 2 U m cos θ
− 12 U m sin θ
− 12 U m sin θ
1
ε(k − Q
2 ) + 2 U m cos θ
ck+Q/2↑
ck−Q/2↓
(3.20)
with ε(k) = v − t eik· = v − 2t (cos kx a + cos ky a). Equation (3.20) can easily be
diagonalized and the eigen-energies are given by
1 +
Ek± =
(3.21)
εk + εk− ± (εk+ − εk− )2 + g 2
2
with εk± = ε(k ∓ Q/2) ± 1/2U m cos θ and g = U m sin θ. The amplitude of the moment
m is determined by m = 2 Rec↑+ c↓ , the x-component of the moment at the origin of
position (R i = 0), which leads to the following equation:
m sin θ =
f (ε − ) − f (εk+ )
g k
.
(εk+ − εk− )2 + g 2
k
1
N
(3.22)
Inserting g = U m sin θ , one obtains
1=
U
N
k
f (εk− ) − f (εk+ )
(εk+ − εk− )2 + (U m sin θ )2
.
(3.23)
This is analogous to the gap equation of the BCS theory and gives the order parameter,
m in this case. The critical condition for m = 0 is found by putting m = 0 in the above
equation and we have
1
1
≤
U
N
k
f {ε(k − Q/2)} − f {ε(k + Q/2)}
= χ(Q)
ε(k + Q/2) − ε(k − Q/2)
(3.24)
228
K. FUKAMICHI et al.
Fig. 3.6. Band structure of ferromagnetic state. The energy splitting between up- and down-spin states is given
by U m.
which gives U χ(Q) ≥ 1. For the ferromagnetic case, we obtain the Stoner criterion
N(EF ) ≥ 1 from χ(Q = 0) = N (EF ). The ferromagnetic state can be realized by putting
θ = 0 in equation (3.20) and the energies of up- and down-spin states are respectively
expressed by
1
1
εk− = ε(k) − U m
and
εk+ = ε(k) + U m.
2
2
The splitting is given by U m as shown in fig. 3.6. When θ = π/2, on the other hand,
the energy splitting of U m takes place at an intersection of εk+ and εk− , and there appears
a finite gap in the direction of Qx or Qy when either of them has a value of π/a. Figure 3.7(a)–(d) shows the energy dispersions for the magnetic structures of Q = (π/2, 0),
Q = (π, 0), Q = (π/2, π/2) and Q = (π, π), respectively (Sakuma, 2005). Below each
dispersion, the corresponding magnetic structure is drawn schematically. In each case, the
splitting has a maximum value of g = U m at the intersection of εk+ = ε(k − Q/2) and
εk− = ε(k +Q/2). However, it does not give a forbidden gap in the DOS for Q = (π/2, 0),
Q = (π/2, π/2) and Q = (π, 0). In the case of Q = (π, π), on the other hand, where
the moments are antiferromagnetically aligned both in the x- and y-directions, and the
splitting of g = U m appears as a forbidden gap in the DOS. Here, let us consider what
dominates the wave vector Q. Generally, it is determined by the strength of U and the
electron concentration n. In most cases, the magnetic structure of finite Q (= 0) is realized
as the ground state if the Fermi level EF is located in the gap formed by the staggered
field specified by Q in the k-space. Therefore, for finite Q, the electrical resistivity is
expected to be larger in the direction of the Q vector. Actually, in Cr metal and Cr-Mn
alloys the resistivity increases rapidly below the Néel temperature and the change is most
remarkable when the current direction coincides with the direction of the wave vector
Q (Muir and Ström-Olsen, 1971). Especially, when n = 1 (half-filled state), EF is located at E = 0 in fig. 3.7, and so if the bands split at E = 0 in all the directions, the
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
229
Fig. 3.7. Band structures of spin spiral states of several Q vectors. The corresponding spin structure of each state
is drawn below by the arrows (Sakuma, 2005).
230
K. FUKAMICHI et al.
total energy is lowered. Thus for the wave vector of Q = (π, π) which brings about
the forbidden gap at E = 0 is realized in the ground state. For U/|t| 1 in the onedimensional system the forbidden gap can be given approximately from equation (3.23)
as g ≈ D exp[−1/{U N (EF )}] (D denotes the effective band width), which is analogous
with the weak coupling scheme in the BCS theory. This shows that the gap g increases
rapidly with U and the behavior can also be seen in the two-dimensional system. Since,
under the condition of U/|t| 1, the critical temperature (the Néel temperature TN in the
present case) is of the order of g, it is expected that TN behaves like exp[−1/{U N (EF )}]
against U . Note here that the finite gap due to Q = 0 formed in the ground state (insulator gap if n = 1) vanishes above TN and the system becomes metallic. This is because,
in the case of U/|t| 1, the origin of the gap lies in the magnetic long-range order.
In U/|t| 1, on the other hand, the electrons can hardly move in the lattice due to
the strong Coulomb repulsion with other electrons, and the system becomes insulator regardless of the magnetic ordering. This corresponds to a Mott insulator which should be
distinguished from the former case in the sense that the gap remains obstinately above TN .
In this region, as mentioned before, the exchange interactions between spins is shot-range
and can be expressed approximately as t 2 /U which is just the super-exchange interaction. Therefore TN is expected to behave like 1/U in this region. Figure 3.8 shows the
phase diagram in the U –T plane (Hasegawa, 1980; Moriya and Hasegawa, 1980) in the
half-filled state (n = 1) calculated with a path integral method under the static approximation combined with the coherent potential approximation (CPA) (Soven, 1967; Tayor,
1967; Velicky et al., 1968). One can see that TN behaves like exp[−1/{U N (EF )}] for
U/|t| 1 and like 1/U for U/|t| 1. Qualitatively, the behavior does not change so
much even when the electron concentration n slightly deviates from unity. Note that, in
this case, the system becomes metallic in any region and there is a possibility that the ferromagnetic ordering occurs in the large U region through a point of TN = 0. This can be
understood by considering the situation that the metallic system has a finite N (EF ) and
has a possibility to satisfy the Stoner criterion U N (EF ) ≥ 1 at a certain value of U . According to Ichimura et al. (1992), in the two-dimensional square lattice the parameter q
Fig. 3.8. Schematic view of the phase diagram of the half-filled Hubbard model. PM, PI and AI represent the paramagnetic metal, the paramagnetic insulator and the antiferromagnetic insulator, respectively (Hasegawa, 1980;
Moriya and Hasegawa, 1980).
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
231
in Q = (q, q) of the helical structure approaches to 0 continuously as n increases and
the transition to the ferromagnetic phase takes place by the second order. Furthermore,
it was shown by many investigators (see for example, Nolting and Borgiel, 1989) that
the Curie temperature in the ferromagnetic region increases monotonically with U and is
saturated to a certain value, depending on the hopping integral t and the electron concentration n.
3.6. Rotation of local spin axes for the complex magnetic structures
In what follows, we go to arbitrary or more complicated structures than the helical structures discussed above. In principle, the direct diagonalization of equation (3.18) or equation (3.19) can give any magnetic structure. In the practical calculation, however, it is
useful to rotate, at each position, the spin quantization axis to the direction of the moment
as shown in fig. 3.9, which enables us to avoid treating the off-diagonal elements of the exchange potential in the spin space which is unadvisable in the conventional LSDA scheme.
Mathematically, this can be performed by the transformation of the basis operators as
ci↑
d
(3.25)
= U (θi , φi ) i↑ ,
ci↓
di↓
where U is the rotation matrix of spin 1/2 given by
−iφ /2
e i cos θ2i −e−iφi /2 sin θ2i
U (θi , φi ) =
.
eiφi /2 sin θ2i
eiφi /2 cos θ2i
(3.26)
By using the matrix U , the last term in equation (3.19) can be expressed as U σz U + . By
the rotation, the effective Hamiltonian can be rewritten as
H̄MF = −
ij σ σ
−
i
+
dj σ +
t¯ijσ σ diσ
+
vi diσ
diσ
i
σ
U +
+
di↓ .
mi di↑ di↑ − di↓
2
(3.27)
One can see that the exchange potential is expressed along each spin quantization axis
with no off-diagonal elements. In place of the exchange potential, the hopping integrals
take charge of the non-collinearity of the moments through the transformation
+
1 ± cos θij
σ
σ
t¯ij = tij U (θi , φi )U (θj , φj ) σ σ = tij
(3.28a)
exp iαijσ σ ,
2
Fig. 3.9. Rotation of the local spin axis to the direction of the exchange (molecular) field on each site.
232
K. FUKAMICHI et al.
which connects the local spin axis between the i-th site and the j -th site. In equation (3.28a)
the sign + is taken for σ = σ , and − for σ = σ . θij is the relative angle between the
moments at the i-th and the j -th sites, defined by
cos θij = cos θi cos θj + sin θi sin θj cos(φi − φj ).
(3.28b)
The phase αijσ σ for σ = σ is given explicitly by
αijσ σ = ± tan−1
cos{(θi + θj )/2} sin{(φi − φj )/2}
cos{(θi − θj )/2} cos{(φi − φj )/2}
(+ for ↑↑, − for ↓↓), (3.29)
sin{(θi + θj )/2} sin{(φi − φj )/2}
sin{(θi − θj )/2} cos{(φi − φj )/2}
(+ for ↓↑, − for ↑↓). (3.30)
and for σ = σ αijσ σ = ± tan−1
We see that the phase plays an important role especially in the strong coupling case, U m ≥
↑↑
|t| in which only αij remains and the system reduces to that of spinless fermions. When the
azimuthal angle φ varies as the site, the phases added to electrons are different depending
on the path in the lattice. Therefore a finite phase remains in the wavefunction when the
electron takes a closed path in the lattice. This implies that, if one defines a vector α by
Ri − Rj
αij = (R i − R j ) · α
(3.31)
,
2
the electrons move in the magnetic field of B = rot α or in the vector potential of α, and
the quantity
a · dr = rot a · dS = B · dS = (3.32)
remains finite. Here, is called spin chirality which can be regarded as a magnetic flux
running through the area enclosed by the path.
The quantities of mi and (θi , φi ) in equations (3.28a), (3.28b) should be solved selfconsistently by using mi = Tr ρi σ which provides mi and (θi , φi ) to be used again in
equations (3.28a), (3.28b). Here, ρ is the density matrices defined in the global frame of
the spin space. Generally, this can be obtained with the GF of equation (3.17) by
1
i
Gii (iωn ) =
dεf (ε){Gii (ε + iδ) − Gii (ε − iδ)}
ρi =
(3.33)
β n
2π
with
ωn = (2n + 1)/β,
where f (ε) is the Fermi distribution function and δ is an infinitesimal. The GF G(ω) in
the above equation can be given through the transformation G = U ḠU + of the GF Ḡ(ω)
defined under the Hamiltonian of equation (3.27) by
Ḡ(ω) = (ω − H̄MF )−1 .
Once the density matrices are obtained, mi and (θi , φi ) are given explicitly as
(3.34)
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
↑↑
↓↓ − ρi
mi
↑↓ Im ρi
φi = tan−1
,
↑↓
Re ρi
mi = |mi |.
θi = cos−1
ρi
233
,
(3.35)
When these quantities coincide with the input values in equations (3.28a), (3.28b), one can
reach the self-consistent results.
The form of equation (3.27) is available not only for the first principles calculations to be
mentioned afterward but also for understanding the condition of the magnetic ordering of
ferromagnetism and antiferromagnetism. Letting U/|t| 1, for brevity, let us consider the
electronic structures of ferromagnetic and the antiferromagnetic states by putting θij = 0
and θij = π (i, j denote adjacent sites), respectively. The schematic DOS of each state
is given in fig. 3.10. In ferromagnetic case (fig. 3.10(a)), the exchange splitting of ↑ spin
and ↓ spin bands is given by U m and the band width is 4d|t| (for simple lattice of ddimension). In the antiferromagnetic case, on the other hand (fig. 3.10(b)), each spin state
has upper and lower bands separated by 2{t 2 + (U m/2)2 }1/2 ∼ U m + 2t 2 /(U m) which is
always larger than U m in the ferromagnetic state. The weight of the lower band is larger
in the ↑ spin state at a site where the moment m = n↑ − n↓ is positive, while in a site
where the moment m = n↑ − n↓ is negative the lower band is larger in the ↓ spin state.
It should be marked that the band widths are smaller than that in the ferromagnetic case,
which are of the order of dt 2 /(U m). Keeping above situation in mind, let us first consider
the case in which the electron density (n) is far from the half-filled state (n = 1), for
example n = 0.5. The electrons in this case occupy the ↑ spin band in the ferromagnetic
case and the lower band in the antiferromagnetic case. As seen from fig. 3.10, the gravity
center of the occupied states is lower in the ferromagnetic case than the antiferromagnetic
case, since the band width is larger in the ferromagnetic case. Physically this is because
Fig. 3.10. Schematic diagrams of the density of states of (a) ferromagnetic and (b) antiferromagnetic states in the
large splitting case (U m/t 1). The band width of the ferromagnetic state is larger than the antiferromagnetic
state and the splitting of up spin state and the down spin state is larger in the antiferromagnetic state.
234
K. FUKAMICHI et al.
the flat potential realized in the ferromagnetic case is more advantageous in energy when
the electrons can move freely in the lattice as in the case of n < 1. Therefore, it is possible
that the ferromagnetic state becomes stable when the electron concentration is far from
the half-filled state. On the other hand, when the electron concentration equals or nearly
closes to the half-filled state, the gravity center of the occupied states is almost same as the
gravity center of the whole lower band in the antiferromagnetic state. Consequently, the
antiferromagnetic state becomes stable because the energy of its lower band is lower than
that in the ferromagnetic state. The energy difference from the ferromagnetic state is almost
equals to the difference of the band splitting. This is approximately 2t 2 /(U m) ∼ 2t 2 /U ,
and then we obtain the order of the super-exchange interaction energy.
3.7. Magnetic excitation energies and the exchange constants
We provide a method to calculate the magnetic excitation energy in the ground state of the
magnetically ordered states including non-collinear structures. Roughly speaking, the strategy is to evaluate the energy change due to a slight deviation of the direction of a moment
at the ground state (saddle point). With this approach, one can derive the effective exchange
constants acting on a given moment, and according to a generalized molecular field theory
by Liechtenstein et al. (1985) magnetic transition temperatures (Néel temperature in our
case) can be estimated from this value.
Following the local force theorem, the variation of the action due to a slight change of
the moment direction at the 0-th site from the saddle point (equation (3.16)) can be given
by
δS = − Tr ln(1 − GδV0 ),
(3.36)
where G has been defined by equation (3.17) and δV0 is a perturbation of the exchange
potential due to the deviation of the moment direction as shown fig. 3.11.
δV0 = −
U
· σ δi,0 .
2
(3.37)
Here, for the calculation of equation (3.36), it is available to rotate the local spin axis to
the moment direction at each site. The action in this case is given by
δS = − Tr ln(1 − U + GδV0 U ) = − Tr ln(1 − Ḡδ V̄0 ),
Fig. 3.11. Magnetic excitation due to a slight deviation of the moment direction at the 0-th site.
(3.38)
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
235
where the GF Ḡ is defined by equation (3.34) with the time (τ ) representation and δ V̄0 is
given by
U
M −1
N
+
δ V̄0 = U δV0 U = − m0
(3.39)
−M + 1
N∗
2
with
M = cos(δθ ) = cos θ0∗ cos θ0 + sin θ0∗ sin θ0 cos φ0∗ − φ0 ,
N = − cos θ0∗ sin θ0 + sin θ0∗ cos θ0 cos φ0∗ − φ0 − i sin θ0∗ sin φ0∗ − φ0 .
(3.40)
(3.41)
Here, (θ0 , φ0 ) and (θ0∗ , φ0∗ ) are the angles of the moment before and after the deviation,
respectively, and δθ is the relative angle between them. From equations (3.39) and (3.40),
the trace of equation (3.38) in terms of the spin space gives
− Trσ ln(1 − Ḡδ V̄0 )
U 2 2 ↑↑ ↓↓ U ↑↑
↓↓ m0 Ḡ00 − Ḡ00 +
m Ḡ Ḡ
1 − cos(δθ ) ,
= − ln 1 −
2
2 0 00 00
(3.42)
where we have used the identity Tr ln A = ln det A and exploited the fact that the GFs
Ḡ are diagonal with respect to the spin space in each local spin axis. Further, when the
rotation angle is small enough {1 − cos(δθ )} 1, the approximation ln(1 − A) ≈ −A
leads to
− Trσ ln(1 − Ḡδ V̄0 )
U 2 2 ↑↑ ↓↓ U ↑↑
↓↓ m0 Ḡ00 − Ḡ00 +
m0 Ḡ00 Ḡ00 1 − cos(δθ ) .
=
2
2
Performing the trace over the time (τ ) for equation (3.43), we have
↑↑
U m0
↓↓
↑↑
↓↓
Im dεf (ε) Ḡ00 (ε) − Ḡ00 (ε) + U m0 Ḡ00 (ε)Ḡ00 (ε)
δS = −β
2π
× 1 − cos(δθ ) .
(3.43)
(3.44)
The quantity δS/β can be regarded as a variation of the thermodynamic potential due to
a change of the moment direction (δθ ) at the 0-th site. By comparing with the following
classical Heisenberg model
H =−
Jij ei · ej
(e: the unit vector)
(3.45)
i,j
one can notice that the coefficient of (1 − cos δθ ) in equation (3.44) can be related to
themagnitude of total sum of the exchange fields acting on the moment at the 0-th site
(| i =0 Ji0 ei |). Defining this coefficient by J0 (= | i =0 Ji0 ei |), we obtain
↑↑
U m0
↓↓
↑↑
↓↓
Im dεf (ε) Ḡ00 (ε) − Ḡ00 (ε) + U m0 Ḡ00 (ε)Ḡ00 (ε)
J0 = −
(3.46)
4π
as an effective exchange constant. This is the same form as the equation obtained for the
collinear case by Liechtenstein et al. (1987). Note, however, that the GF used here is defined in the local frame of the spin space rotated to each direction of the moment (Sakuma,
236
K. FUKAMICHI et al.
2000). Especially, in the collinear case, using a sum rule for the following GF
↑↑
↑↑
↓↓
Ḡ − Ḡ↓↓ 00 = −
Ḡ0j (U mj )Ḡj 0 ,
j
one can obtain the pair exchange constants between i-th site and j -th site as
1
↑↑
↓↓
Jij =
Im dεf (ε) U mi Ḡij (ε)U mj Ḡj i (ε) .
4π
(3.47)
At this stage, let us make a brief consideration about the physical nature of the exchange
constants derived from equation (3.47). For the sake of simplicity, we assume a ferromagnetic structure in the simple lattice as an ordered state, in which the GF can be written
simply as
−1 Un
Um
ˆ
Ḡij (ε) = ε l̂ + t −
(3.48)
.
l̂ +
σ̂z
2
2
ij
When U/|t| 1, at first, the Coulomb repulsion term U n/2 and the exchange term U m/2
in the denominator of equation (3.48) can be neglected and then
↑↑
↓↓
Ḡij (ε)Ḡj i (ε) =
=
where εk = −
i t0i e
1
N2
1
N2
k
q
1
1
ei(k−k )·(Ri −R j )
ε − εk ε − εk k
iq·R ij
1
1
e
−
,
ε − εk
ε − εk−q εk − εk−q
(3.49)
k
ik·R i0 . In this case, equation (3.47) turns to be
↑↑
(U m)2
↓↓
Jij =
Im dεf (ε) Ḡij (ε)Ḡj i (ε)
4π
(U m)2
χ(q)eiq·R ij ,
=
4N
q
(3.50)
where R ij ≡ R i − R j and χ(q) is the spin susceptibility defined by equations (3.10) and
(3.24). One finds that equation (3.50) reproduces the form of equation (3.9) subtracted by
the correction term for double counting of the energy (−U/4) which automatically provides the necessary Stoner criterion for the ferromagnetism. This difference comes from
the situation that the present case starts from the magnetically ordered state including ferromagnetic state, while in the previous case (equation (3.9) in subsection 3.3) the unperturbed
state is non-interacting (non-magnetic) system. Therefore, in the present case, it is necessary to take into account the Stoner criterion carefully, when χ(q) has a peak at q = 0
which gives ferromagnetic character to Jij . Since, in the present case of the weak coupling
system, however, the ferromagnetic condition U χ(0) = U N (EF ) > 1 is hard to be realized, one can understand that the exchange interaction exhibits an oscillatory behavior,
similar to the RKKY-type interaction.
Next, in the strong coupling case U/|t| 1, it is useful, in equation (3.47), to shift the
integral variable by ε → ε + U n/2 − U m/2 in order to expand only the ↓ spin GF up to
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
237
the third order of 1/U m as (Solovyev and Terakura, 1999)
−1
Un Um
↓↓
−
= ε l̂ + tˆ − U ml̂
ε +
Ḡ
2
2
≈−
1
(ε l̂ + tˆ) (ε l̂ + tˆ)2
−
.
−
Um
(U m)2
(U m)3
Inserting the above equation and G↑↑ (ε +U n/2−U m/2) = (ε l̂+tˆ)−1 into equation (3.47),
we have
tj l tlj
1
n−m
Jij =
f εk + U
cos k · R ij tj i +
4N
2
Um
l
k
n−m
1
f εk + U
tj l cos k · R j l .
cos k · R ij tj i
−
(3.51)
U mN
2
k
l
The first term in equation (3.51) is mainly dominated by the term tij in the parenthesis ( )
rather than l tj l tli /(U m), since U/|t| 1. Then the first term can be approximated to
be
tj i
n−m
f εk + U
cos k · R ij
(3.52)
4N
2
k
which behaves like the double exchange type (ferromagnetic) interaction (Anderson and
Hasegawa, 1955; de Gennes, 1960) in the sense that the linear term of the electron hopping
tij promotes the ferromagnetic alignment and this is always positive, regardless of the sign
of tij . Actually, this is the same form as just obtained for the double exchange ferromagnet
with the spin wave analysis (Furukawa, 1996; Wang, 1998). One can find that generally
this term starts from zero at n = 0 and approaches zero again at n = 1 (half-filled state)
through a maximum value around n = 0.5. In the region near n = 1, the second term in
equation (3.51) can take the place of the first term. The leading term in the summation of
the second term has a form of
−
tij2
U mN
kF
cos2 k · R ij ,
(3.53)
k
whose magnitude increases rapidly with n or the Fermi momentum k F near the half-filled
state and reaches to the super-exchange interaction at the half-filed state. The superexchange form just at the half-filled state can also be obtained by adopting another approximation for Ḡσijσ (ε) as
Ḡσijσ =
1
1
tij
ε + (U/2)σ ε + (U/2)σ
(3.54)
to the first order in t. Here we assume m = n = 1. From this approximation, we have
238
K. FUKAMICHI et al.
Fig. 3.12. Integral path in the complex plane in equation (3.55).
Jij =
U 2 tij2
4π
εF
dε
Im
1
1
2
(ε + U/2) (ε − U/2)2
U 2 tij2 1 1
1
=
dz
4 2πi
(z + U/2)2 (z − U/2)2
C
=
U 2 tij2
=−
4
tij2
Res
1
1
z=U/2 (z + U/2)2 (z − U/2)2
(3.55)
,
2U
where the integral path C is shown in fig. 3.12 by assuming U kB T . Thus in the strong
coupling case, we have seen that eq. (3.47) takes the form of double-exchange type ferromagnetic interaction in the low electron concentration region and changes gradually into
the super-exchange type antiferromagnetic interaction when n approaches to unity. The
schematic behaviors of these terms as a function of n are drawn in fig. 3.13 in which the
line denoted by JijDE corresponds to the first term and the line JijSE to the second term in
equation (3.51) (Sakuma, 2005). Finally, in this subsection, let us look into the specification of the effective exchange constants given by equation (3.46). The contribution from
the first term in [ ] is 1/4U m2 . This corresponds to the exchange energy in the Stoner
model, while the second term is found, from the form of equation (3.47), to be the local
exchange energy, −J00 . Therefore equation (3.46) has a form of the magnetic total energy
from which the local exchange energy is subtracted. This confirms us that equation (3.46)
has the same physical meaning as equation (3.13) and corresponds to the exchange constant J (q = 0) in the classical Heisenberg model. The virtue of the quantity J0 rather than
Jij is availability for the estimation of the magnetic transition temperatures in the frame
of the molecular field approximation (MFA). In three dimensional spin systems, the magnetic critical temperatures can be given by 2J0 /3kB in the MFA. Especially in the collinear
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
239
Fig. 3.13. Schematic behaviors of the exchange constants describing the double exchange character and the super
exchange character denoted by JijDE and JijSE , respectively, each of which corresponds to the first and the second
term in equation (3.51). The line denoted by Jij is the summation of JijDE and JijSE (Sakuma, 2005).
Fig. 3.14. Local magnetic moment m and the effective exchange constant J0 calculated from equation (3.46) as
a function of the number of electrons per orbital. The inset shows the assumed DOS with a band width of 2D.
The area of DOS per spin is set at unity (Sakuma, 1999b).
antiferromagnetic case, J0 calculated from equation (3.46) corresponds to J (q = Q) (Q:
wave vector characterizing the antiferromagnetic alignment) and then the Néel temperature
TN is also given by 2J0 /3kB .
Figure 3.14 shows the calculated results of J0 as a function of the electron density n for
the semi-elliptic DOS as depicted in fig. 3.3 (Sakuma, 1999b). The local magnetic moment
240
K. FUKAMICHI et al.
Fig. 3.15. Exchange constant J0 for U = 2D and 3D as a function of the number of electrons per orbital
(Sakuma, 2005).
m0 in equation (3.46) is self-consistently derived from
↑↑
1
↓↓
m0 = − Im dεf (ε) Ḡ00 (ε) − Ḡ00 (ε) .
(3.56)
π
Finite m starts at around n = 0.25. One can easily find that this is Stoner critical point for
the ferromagnetism, U N (EF ) = 1. Though the moment m increases with increasing n to
m = n = 1, J0 turns to negative value at around n = 0.65 and makes the ferromagnetic
state unstable in the range 0.65 ≤ n ≤ 1.35. This is qualitatively the same behavior as in
fig. 3.3 and then is ascribed to the Pauli principle which restricts the electron hopping in the
ferromagnetic configuration. In fig. 3.15, we show the behavior of J0 in the region n 1
for U = 2D and 3D. It is seen that the increase of U raises J0 toward the positive side
and stabilizes the FM state (Sakuma, 2005). Based upon these results, the magnetic phase
diagram of the Hubbard model in the (U –n) plane can be schematically drawn as shown in
fig. 3.16. In the vicinity of n ∼ 2 (or n ∼ 0), the system is non-magnetic, because in this
region N (EF ) is too small to satisfy the Stoner criterion U N (EF ) = 1 even though U is
large. For U/|t| 1, the system is also non-magnetic because of the shortage of U N (EF ).
With increasing U , the ferromagnetic phase starts to appear in the region between n = 1
and 2, and the ferromagnetic region tends to spread as U/|t| further increases. In the strong
coupling limit (U/|t| 1), the ferromagnetic exchange interaction is dominated by the
hopping amplitude |t|, where the scenario for appearance of the ferromagnetism is quite
similar to that of the double exchange ferromagnet. When the carrier density n approaches
to unity in this large U/|t| region, the super exchange interaction grows up competing
with the double exchange interaction, and the system goes to the antiferromagnetic Mott
insulator.
In figs. 3.15 and 3.16, the variation of the electron density (number of electrons per
orbital) in the region (1 ≤ n ≤ 2) corresponds to the series of 3d transition metals from Cr
or Mn to Cu metals (Sakuma, 2005). One can find that J0 is negative in the region around
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
241
Fig. 3.16. Phase diagram in the (U –n) plane deduced from the results in fig. 3.15 FM: ferromagnetic phase,
AFM: antiferromagnetic phase, PM: paramagnetic (Sakuma, 2005).
Mn metal whose electron density per d-orbitals is close to n ∼ 1, and turns to positive
value around n ∼ 1.3 corresponding to Fe metal and reaches to the maximum value at
around the region of Co metal (n ∼ 1.5), then finally decreases to 0 through Ni metal
(n ∼ 1.7). Although the electronic structures of these materials are so intricate and different
from each other, the curves in figs. 3.15 and 3.16 obtained from a quite simple model with
varying only the electron density reproduce well the relative difference among the magnetic
properties of these metals. In this sense, it is possible to consider that the electron density
of d-orbital plays a decisive role to dominate the exchange constants of the 3d transition
metals. Accepting this premise, we can infer the following from figs. 3.15 and 3.16. The
electron densities of d-orbitals of Mn and Fe metals may be located around or across
the point for J0 = 0, so the magnetic structures (ferro- or antiferromagnetism) of these
metals strongly depend on the circumstances such as crystal structure, lattice constants,
alloying and so on. As for Co metals, on the other hand, the electron density may be
located around the stationary maximum point. Therefore, the value of J0 is stable against
the circumstances and it may keep a high Curie temperature TC . From above the qualitative
consideration, the transition metal elements are plotted in the (U –n) plane in fig. 3.16. To
see more quantitative natures, we proceed to evaluate the magnetic properties of the actual
transition metal systems with the first principles calculations in the following section.
4. First principles approach for the magnetic structures of transition metal systems
In this section, the theoretical framework discussed above will be mapped into the linear
muffin-tin orbital (LMTO) (Andersen, 1975; Skriver, 1984) method based upon the density
functional theory in order to make a quantitative evaluation of the magnetic structures and
its stability of the actual magnetic systems including substitutional disordered alloys. The
treatment given below can also be applied to the Korringa–Kohn–Rostoker (KKR) method
in a similar manner.
242
K. FUKAMICHI et al.
4.1. Tight-binding (TB)—LMTO method for the complex magnetic structures
As mentioned in subsection 3.4, the Hamiltonian for the spin polarized band calculations
has a form of equation (3.18). This equation has a correspondence with the LSDA Hamiltonian through the relation
−t → −∇ 2 + νHartree + νnuclei ,
1
νi → νi↑ (r) + νi↓ (r) ,
2
1
B i → − νi↑ (r) − νi↓ (r) ei .
2
Here, the term νHartree is the Hartree potential and νnuclei the potential from the nuclei. The
potential νiσ (r) which depends on the spin σ is given by the functional derivative of the
exchange correlation energy EXC with respect to the electron density niσ (r) as
δEXC
νiσ (r) =
(4.1)
.
δniσ (r)
The last term in equation (3.18), the correction term for double counting of the interaction
energies, is given by
n(r)n(r )
1
drdr −
dr n↑ (r)ν↑ (r) + n↓ (r)ν↓ (r) + EXC , (4.2)
Edc = −
2
|r − r |
n(r) =
nσ (r),
σ
nσ (r) =
niσ (r).
i
The resultant form of the LSDA Hamiltonian can be given as
HMF = −∇ 2 l̂ +
{ν̄i (r)l̂ − B i (r) · σ },
(4.3)
i
1
(4.4)
νi↑ (r) + νi↓ (r) + νHartree + νnuclei ,
2
1
B i (r) = − νi↑ (r) − νi↓ (r) ei ,
(4.5)
2
where ei = (cos ϕi sin θi , sin ϕi sin θi , cos θi ) indicates the direction of the molecular (exchange) field acting on a moment at the i-th site. Although, generally, the direction of the
molecular field varies as a function of position r, we adopt the rigid spin approximation
assuming that the direction is constant within each muffin tin (MT) sphere. In a situation
where each moment points to a different direction on the individual sites, it is convenient
to rotate each spin axis within the MT sphere to the direction of the moment (molecular
field). Upon this concept, we define the MT orbital of the σ spin as (Kübler et al., 1988;
Sticht et al., 1989)
ν̄i (r) =
−1/2
(−)
χLσ (r)ησ = Lσ Lσ (r)(Ui ησ )
−
L ,σ −1/2 (+)
L σ L σ (r) Ui+ ησ Ui+ SL ,L Ui σ σ ,
(4.6)
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
243
where L = (i, l, m) (i denotes site, l, m are orbital index), ησ means bispinor basis functions
1
0
, η↓ =
,
η↑ =
0
1
and the matrices U are the rotation matrices of spin 1/2 defined by equation (3.26). In
(±)
equation (4.6), Lσ (r) is a trial function for a single MT sphere in the atomic sphere
approximation (ASA), and , are the normalization constants. The SL ,L is the structure constants which connect the orbitals on L and L . In equation (4.6), the relation
between the spin axes on the i-th and i -th sites are taken into account through the matrices
(Ui+ SL ,L Ui )σ σ which have off-diagonal elements in terms of the spin. The resultant form
of the LMTO Hamiltonian (H LMTO ) and the overlap matrix (O) in the secular equation
det(H LMTO − EO) = 0 are
H LMTO = AT Eν A + AT B + B T Eν pB,
(4.7)
O = A A + B pB,
A = −1/2 δL ,L δσ ,σ − −1/2 S̄ ,
B = −1/2 ω(−)δL ,L δσ ,σ − −1/2 ω(+)S̄ ,
(4.8)
T
T
(4.9)
where Eν , ω(±) and p combined with , are the potential parameters determined within
each atomic sphere with use of the LSDA. One can find in equations (4.7)–(4.9) that the
different point from the collinear case is only the generalized structure constant S̄ defined
by
S̄L σ ,Lσ = Ui+ SL ,L Ui σ ,σ .
(4.10)
Thus the calculations for non-collinear spin structure can be performed through the transformation of the structure constants S to S̄. Note, here, that the quantities as potential
parameters are defined in the local frame of the spin axis rotated to the molecular field
direction on each MT sphere.
For the later preparation of the CPA to deal with disordered alloys, we need to go into
the tight-binding scheme where the overlap matrix is unity as det(H orth − E l̂) = 0. By
putting pLσ = 0 in equations (4.7) and (4.8), one obtains the Hamiltonian H orth in the
orthogonal basis set
H orth = Eν + BA−1 = C + 1/2 S̄(1 − γ S̄)−1 1/2
(4.11)
with
C = Eν + ω(−),
γ = 1/2 −1/2 .
(4.12)
Generally, the form of equation (4.11) is called the γ representation. The Green function
(GF) in the γ representation is given by
Ḡ(ω) = −1/2 g γ (ω)−1/2 ,
−1
g γ (ω) = p γ (ω) − S γ
,
(4.13)
(4.14)
244
K. FUKAMICHI et al.
where g γ (ω) is the so-called auxiliary GF constituted of the potential function p γ (ω) and
the screened structure constants S γ given, respectively, by
p γ (ω) =
ω−C
δL ,L δσ ,σ ,
S γ = S̄(1 − γ S̄)−1 .
(4.15)
(4.16)
Since at this stage the site-off-diagonal elements in the screened structure constants depend
on the species of atoms through the values of γ , a further step is required for the CPA. To
avoid this, we follow the way proposed by Kudrnovský and Drchal (1990), where the auxiliary GF of the γ representation is transformed into the most localized (β) representation
through a relation which is valid for arbitrarily representation
g γ = (p γ )−1 − (p γ )−1 p β (p γ )−1 + (p γ )−1 p β g β p β (p γ )−1 ,
(4.17)
where the matrices in the β representation are
−1
,
g β (ω) = p β (ω) − S β
−1
p β = p γ 1 − (β − γ )p γ
,
−1
S β = S γ 1 − (β − γ )S γ
= S̄(1 − β S̄)−1 .
(4.18)
(4.19)
(4.20)
It has been found that for closed-packed lattices the optimum values of β are given by
βl=0 = 0.3485, βl=1 = 0.0530, βl=2 = 0.0107 which give the fastest and essentially
monotonic decay of S β in the real space. It is of importance that S β in these values becomes
free from species of atoms.
With the same manner as in the section 3.6, the moment mi defined in the MT sphere at
the i-th site is expressed by
↑↓
↑↓
↑↑
↓↓ mi = Tr ρi σ = 2 Re ρi , 2 Im ρi , ρi − ρi ,
(4.21)
where the density matrices ρi should be calculated with the GF, G(ω) defined in the global
frame as
ρiσ,σ =
i
2π
εF
dε Gσ,σ
(ω + iδ) − Gσ,σ
(ω − iδ) .
i
i
(4.22)
The G can be calculated by the unitary transformation of Ḡ
Gi (ω) = U + (θi , ϕi )Ḡi (ω)U (θi , ϕi ).
(4.23)
The direction of the moment at the i-th site can be expressed in the global frame by equation (3.35). If one wants a self-consistent solution of the moments, the band calculation
should be repeated until the newly obtained (θi , ϕi ) coincides with the input angle.
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
245
4.2. Coherent potential approximation (CPA) for disordered alloys
For non-collinear magnetic structures in the substitutional disordered alloys having several
sublattices, we apply the CPA individually in each sublattice. The CPA condition in the
i-th sublattice is given by
β
i (ω) =
cia gia (ω)
(4.24)
a
or equivalently
β
β
−1
cia pia (ω) − p̃i (ω) 1 + pia (ω) − p̃i (ω) i (ω)
= 0,
Ti =
(4.25)
a
where cia denotes the density (probability) of atom a occupying the i-th sublattice. The
quantity i (ω) is the coherent propagator of i-th sublattice, which is given by
i (ω) =
1
N
β −1 p̃(ω) − S̄k
,
ii
(4.26)
k
β
where p̃i (ω) and S̄k are the coherent potential function and the screened structure conβ
stants, respectively, in the β representation in the k-space. The quantity gia (ω) is the local
(auxiliary) GF of atom a in the i-th sublattice which is given with i (ω) based on the
multiple scattering theory as
β
−1
β
gia (ω) = i (ω) 1 + pia (ω) − p̃i (ω) i (ω) ,
(4.27)
β
where pia (ω) is the potential function in the β representation. As seen in equation (4.19),
β
the pia (ω) is diagonal with respect not only to (l, m) but also to spin σ since it is defined
β
in the locally rotated spin axis. Note, however, that the quantities i (ω), gia (ω) and p̃i (ω)
β
generally have off-diagonal elements with respect to (l, m) and σ . Once gia (ω) is solved
γ
self-consistently, the GF Ḡ, defined in the local frame, can be obtained by gia (ω) through
γ
β
the transformation of gia (ω) to gia (ω) with equation (4.17). The concrete form is
Ḡia (ω) MM =
(γM − βM )
δMM M + (γM − βM )(ω − CM )
+
1/2
β
M
gia (ω) MM M + (γM − βM )(ω − CM )
×
M ,
M + (γM − βM )(ω − CM )
1/2
(4.28)
where M = (l, m, σ ). The Ḡ is further transformed into G defined in the global frame
through equation (4.23) to get the density matrices. Calculating mia , the moment of atom
a at the i-th sublattice by equations (4.21) and (4.22), one can obtain the average moment
at the i-th sublattice by
m̄i ≡
cia mia .
a
246
K. FUKAMICHI et al.
When the direction of this moment coincides with the input angle (θi , ϕi ) through the S̄ in
equation (4.10), the self-consistent solution is achieved.
4.3. Effective exchange constant
Based upon the scenario discussed in the subsection 3.7, the effective exchange constant
can be calculated in the LSDA scheme. As mentioned in the subsection 3.7, the magnetic
transition temperatures can be estimated to be 2J0 /3kB by the values obtained within the
MFA scheme.
From the expressions of J0 and Jij given by equations (3.46) and (3.47), respectively,
the matrices (U m)Ḡ(ω) can be rewritten in the TB-LMTO method as
(U mi )Ḡ(ω) → (ω)g γ (ω)
γ
γ
with (ω) ≡ p↑ (ω) − p↓ (ω).
Then, the expressions for J0 and Jij in the LSDA scheme are given by
γ ↑↑
1
γ ↓↓
J0 = − Im dεf (ε)Trlm 0 (ε) g00 (ε) − g00 (ε)
4π
γ ↑↑
γ ↓↓
+ 0 (ε)g00 (ε)0 (ε)g00 (ε)
(4.29)
and
1
γ ↑↑
γ ↓↓
(4.30)
Im dεf (ε) Trlm i (ε)gij (ε)j (ε)gj i (ε) ,
4π
where Trlm represents the trace over the orbital part (l, m). Note here again that J0 of
equation (4.29) can work both for collinear and non-collinear magnetic structures, while
Jij of equation (4.30) can be used only for the collinear case.
In disordered alloys, Ḡ should be calculated in the CPA process by using the form of
equation (4.28). When the number of magnetic components in alloying system is greater
than two, the Néel temperature TN is considered to be determined by the largest value of
J0 , because the magnetism of the system should be dominated by the component with the
largest J at around the critical temperature. According to a generalized molecular field
theory by Liechtenstein et al. (1985), the magnetic transition temperatures such as the
Curie temperature TC or the Néel temperature TN can be estimated from the following
expression.
Jij =
2J0
(4.31)
,
3kB
where kB is the Boltzmann constant. The detailed description for this subsection was given
in the literature (Sakuma, 2000).
TC or TN =
5. Electronic and magnetic structures of γ -Mn, L12 -type (≡ γ -phase) and L10 -type
alloys
The calculated results based on the first principles approach are given for magnetic structures of pure γ -Mn, several kinds of L12 -type (≡ γ -phase), γ -phase and L10 -type Mn
alloys discussed above because their magnetic structures are practically important and
closely associated with the exchange biasing behavior as given in section 8.
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
247
5.1. Pure γ -Mn
The phase stability of pure γ -Mn is obtained in the temperature range 1368–1406 K. One
may notice that the pure γ -Mn does not exist metallurgically at room temperature and
the actual system contains Cu or C atoms of a few atomic percents, accompanied by a
lattice distortion of c/a ∼ 0.95 (Endoh and Ishikawa, 1971). Experimentally, the magnetic
structures and the lattice constants of pure γ -Mn are discussed by extrapolating to zero
additive concentration.
5.1.1. Magnetic structure of pure γ -Mn
As mentioned in the introduction, almost all Mn alloys exhibit antiferromagnetism due
to their occupation of 3d orbitals close to the half-filled state. Among them, a number of
face centered cubic (fcc) alloys, such as FeMn (Endoh and Ishikawa, 1971) and Mn3 Pt
(Krén et al., 1968a), have a complex spin structure other than a collinear or simple helical structure. Experimentally, on the other hand, γ -Mn is considered to be a collinear
antiferromagnet, though it also has a face centered structure.
Theoretically, so far, several attempts have been made to clarify the magnetic structure
of pure γ -Mn. Asano and Yamashita (1971) have first carried out the band calculation
for pure γ -Mn, assuming the collinear AFM spin arrangement. Cade and Young (1980)
proposed the 3Q state (non-collinear) as the ground state, and the result was followed
by Hirai and Jo (1985) with a Hartree–Fock calculation. Based on LSDA approach for
non-collinear magnetism developed by Kübler et al. (1988), more recently, Fujii et al.
(1991) showed that the most stable magnetic structure is a collinear AFM as proposed
experimentally by Meneghetti and Sidhu (1957). Unfortunately, all of them, except Asano
and Yamashita (1971), neglected the lattice distortion of pure γ -Mn for the calculations.
In this subsection, based on the LSDA calculations by Sakuma (1998a), let us focus our
attention mainly on the effect of the lattice distortion on the magnetic structure of γ -Mn.
The exchange-correlation term is adopted as the form of von Barth and Hedin (1972) with
the parameters given by Janak (1978). The core charge density is used as the so-called
frozen core. For the valence states, the s-, p- and d-basis functions are adopted, neglecting
the spin–orbit interaction. The atomic sphere radius is settled at a = 0.375 nm for the fcc
structure.
The spin structure examined here is shown in fig. 5.1, illustrating a magnetic primitive
cell constituted of four atoms in the face-centered structure. One cubic cell indicates 1/8 of
the chemical unit cell having an fcc crystal structure (Krén et al., 1966, 1968a). Note that
the angles θ and φ are defined individually
for each site. As shown in fig. 5.2, with φ settled
√
at 45 degrees, θ = 0, θ = cos−1 (1/ 3) = 54.7 degrees and θ = 90 degrees correspond
to the so-called 1Q, 3Q and 2Q spin density wave (SDW) structures, respectively, which
are generally referred to as the multiple-Q SDW (MQSDW) structures (Fishman and Liu,
1998).
Figure 5.3 shows the θ dependence of the total energy of γ -Mn with φ = 45 degrees at
c/a = 1 (Sakuma, 1998a). Starting from θ = 0, the energy reaches the minimum value
at around θ = 20 degrees. This state, however, is not a self-consistent solution since the
output angle (θS ) of the magnetic moments deviates from the input angle θ by δθ = θS − θ
degrees, whereas δθ = 0 at θ = 0. With increasing θ , on the other hand, the energy
248
K. FUKAMICHI et al.
Fig. 5.1. Magnetic primitive cell constituted of four atoms in an fcc lattice. The solid circles represent disordered
atoms.
Fig. 5.2. Multiple-Q spin density wave (MQSDW) structures in an fcc lattice. MQSDW structures, 1Q, 2Q and
3Q states are realized by putting θ = 0, 90 and 54.7 degrees, respectively (Fishman and Liu, 1998).
√
increases and has a maximum value at around θ = cos−1 (1/ 3) = 54.7 degrees. At
this point, δθ vanishes and the solution becomes self-consistent. Actually, δθ changes its
sign from minus to plus at this point, getting toward zero again at θ = 90 degrees. The
results reveal that the 3Q SDW structure (θ = 54.7 degrees) is remarkably unstable, and
within the other structures the 2Q structure (θ = 90 degrees) is more stable than the 1Q
structure (θ = 0) by 0.08 mRy/atom. The results for γ -Fe are also shown in the same
figure for comparison. The lattice constant is chosen at 0.36 nm. In contrast to γ -Mn,
the 3Q structure (θ = 54.7 degrees) has the lowest energy among the given magnetic
structures. Note, however, that the clarification of the magnetic ground state of γ -Fe may
require super cell structures. More sophisticated analysis for γ -Fe has been performed by
Antropov et al. (1996).
Shown in fig. 5.4 is the θ dependence of the total energy of γ -Mn for several values of
c/a under the constant cell volume (Sakuma, 1998a). Obviously the 2Q structure is stable
at c/a = 1.05. Across c/a = 1, the total energy for the 2Q structure goes up abruptly
and that for the 1Q structure is getting lower with decreasing c/a from unity. It is worth
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
249
Fig. 5.3. The θ dependence of the electronic total energy of γ -Mn, together with that of γ -Fe (Sakuma, 1998a).
Fig. 5.4. The angle θ dependence of the electronic total energy of γ -Mn for several values of the axial ratio c/a
(Sakuma, 1998a).
noting that the self-consistent solutions are restricted only to θ = 0 and θ = 90 degrees,
except for c/a = 1 where the state of θ = 54.7 degrees is also self-consistent. Therefore,
the stable state for c/a < 1 can be regarded as the 1Q structure. Furthermore, the value
of the total energy of the 1Q structure decreases with decreasing c/a. The total energy
and the magnetic moment at the 1Q structure are plotted against c/a in fig. 5.5 (Sakuma,
1998a). The 1Q structure is found to have a minimum energy at c/a ∼ 0.9. Although this
value is considerably smaller than that expected from experiments (c/a ∼ 0.95), both the
magnetic structure accompanied by the lattice distortion of c/a < 1 and the moment of
250
K. FUKAMICHI et al.
Fig. 5.5. Electronic total energy and the magnetic moment of γ -Mn as a function of the axial ratio c/a. The angle
θ is settled at 0 (Sakuma, 1998a).
around 2.3 μB are in good agreement with the experiments. From these results, it can be
concluded that the lattice distortion to c/a < 1 is strongly responsible for the collinear
structure in γ -Mn and concurrently the system is stabilized by this distortion.
5.1.2. Exchange constant of pure γ -Mn
An intuitive interpretation of the mechanism that the magnetic structure favors 1Q for
c/a < 1 and 2Q for c/a > 1 may be ascribed to a change of the magnitude of the AFM
exchange interactions between the Mn moments depending on the interatomic distance.
That is, when c/a < 1, the atomic distances between the inter-c-plane Mn atoms are
shorter, and the exchange interactions between them are expected to be stronger than that
for intra-c-plane Mn atoms and vice versa. Actually, applying the Heisenberg model in
fig. 5.6, one can reproduce both the 1Q and 2Q structures by changing only the relative
strengths of J12 and J13 . In order to examine the validity of this feature, the exchange
constants are calculated by the first principles approach using equation (4.30). The results
calculated for the collinear structure of θ = 0 with different ratios of c/a are listed in
table 5.1 (Sakuma and Fukamichi, 2005). Contradictory to the above-mentioned inference,
no reversal of the strength between J12 and J13 can be confirmed in the change from c/a <
1 to c/a > 1, and both values at c/a = 0.95 are smaller than those at c/a = 1. Especially, it
is noticeable that J13 remarkably decreases with decreasing c/a. These results suggest that
the pair exchange constants are not locally dominated by the interatomic distance only,
and the relative strength of J12 to J13 is not responsible for determining the 1Q and 2Q
structures. The effective exchange constant J0 calculated by equation (4.29) is also shown
in table 5.1 (Sakuma and Fukamichi, 2005). The Néel temperatures TN estimated from
these values are in the range of about 420 to 500 K, corresponding to the experimentally
estimated value of about 480 K (Endoh and Ishikawa, 1971). However, it should be noted
that J0 for c/a = 0.95 is smaller than others even though it is energetically most stable.
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
251
Fig. 5.6. Pair exchange constants J12 and J13 between the nearest neighboring Mn moments in the magnetic unit
cell of γ -Mn.
TABLE 5.1
Pair exchange constants J12 and J13 between the nearest neighbor Mn atoms and
the effective exchange constant J0 in meV for c/a = 0.95, 1.00 and 1.05 for γ -Mn
(Sakuma and Fukamichi, 2005)
c/a
J12
J13
J0
0.95
1.00
1.05
−18
−18
−19
12
16
17
54
68
64
This trend can also be seen in J12 and J13 . One can infer from these results that the most
stable magnetic structure does not always have a higher magnetic stiffness than others
against the transverse fluctuation of the moment, because the magnetic stiffness reflects the
curvature of the action around the SP, whereas the stable magnetic structure is determined
by the global minimum of the action. For γ -phase disordered Mn alloys, such as MnNi
(Honda et al., 1976), MnGa (Hori et al., 1995) and FeMn (Endoh and Ishikawa, 1971), the
1Q, 2Q and 3Q spin density wave (SDW) structures are observed by changing temperature
and/or composition. Multiple domains in a 1Q or 2Q structure exhibit the same neutron
diffraction fingerprint as a 3Q structure. Therefore, γ -ray emission from spin-polarized
nuclei (Kawarazaki et al., 1988) and Mössbauer spectra (Kennedy and Hicks, 1987) have
been used to identify the MQSDW structures in γ -MnNi and FeMn alloys.
5.2. L12 -type (≡ γ -phase) ordered and γ -phase disordered Mn3 Ir and Mn3 Rh alloys
Figure 5.7 illustrates the T1 (triangular-type) structure in the fcc chemical unit cell in which
the solid and open circles represent Mn and additional elements, respectively. This structure appears in L12 -type ordered Mn alloys. The Mn moments lie in the (111) plane and
point to the 211 direction (Krén et al., 1966).
From a practical viewpoint, the alloys having a high TN without any heat treatments are
desired for the complex multi-layered structures in GMR and TMR devices. Mn-rich systems such as γ -phase disordered Mn1−x Irx alloys with 0.2 ≤ x ≤ 0.3 are promising
candidates satisfying such condition mentioned above. Experimentally, the lattice constants and TN of γ -phase Mn1−x Irx alloys were first investigated by Yamaoka (1974). He
252
K. FUKAMICHI et al.
Fig. 5.7. T1 (triangular) spin structure of L12 -type ordered Mn alloys. The solid circles stand for Mn atoms
(Krén et al., 1966).
Fig. 5.8. Density of states (DOS) of L12 -type ordered Mn3 Ir alloy with the T1 structure (Sakuma et al., 2003).
demonstrated that TN of the disordered (DO) alloy at x ∼ 0.25 was around 730 K, very
high among Mn-rich alloy systems. A recent study (Tomeno et al., 1999) reconfirmed that
L12 -type ordered Mn3 Ir alloy also has a high TN of about 950 K. On the other hand, no
systematic theoretical investigations have been performed yet for Mn-rich alloys, except
for L12 -type ordered Mn3 Pt and Mn3 Rh alloys. The magnetic structure of these two ordered alloys was suggested to be the so-called triangular structure T1 as given in fig. 5.7
from both by experimental (Kouvel and Kasper, 1964; Krén et al., 1968a) and theoretical
(Kübler et al., 1988) works. In this subsection, the electronic and magnetic structures of
Mn3 Ir in both the ordered and disordered states are presented, and the magnetic stability is
discussed.
Shown in fig. 5.8 is the local DOS of L12 -type ordered Mn3 Ir alloy with the triangular
magnetic structure (T1) (Sakuma et al., 2003). The upper and lower curves refer to the
majority and minority spin states, respectively, for the Mn and Ir sites. The lattice constant
is chosen as 0.3785 nm from the experimental data (Yamaoka, 1974). The result is drawn
from the imaginary part of the GF in equation (4.13), not from the usual technique with
secular equations, in order to make a comparison with the later results by the CPA for
the disordered alloy. It has been confirmed that this procedure well reproduces both the
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
253
Fig. 5.9. Density of states (DOS) of L12 -type ordered (O) Mn3 Ir alloy in the paramagnetic (DLM) state (Sakuma
et al., 2003).
Fig. 5.10. Effective exchange constant J0 of L12 -type ordered Mn3 Ir alloy with the T1 structure as a function of
the Fermi level EF (band filling) (Umetsu et al., 2003b).
DOS and the moment obtained by the usual secular equation. The moments of Mn and
Ir atom are 2.62 μB and 0, respectively. Figure 5.9 shows the DOS in a disordered local
moment (DLM) state in the PM state for L12 -type ordered Mn3 Ir alloy (Sakuma et al.,
2003). The Mn moment is reduced to 2.20 μB , whereas the behavior of the local DOS
below −0.1 Ry is retained in both the T1 and the PM states, and the dip around the Fermi
level EF found in the T1 state is swept out in the PM state. This situation is similar to the
case of L10 -type MnPt alloy in which the pseudo-gap realized in the AFM state vanishes
completely in the PM state (see fig. 5.17), which means that the gap is closely connected
to the AFM long-range order. The energy in this PM state is higher than that in the T1 state
by as much as 6 mRy/atom, giving the same order of TN . To see the magnetic stability in
the T1 state, the effective exchange constant J0 is calculated. Presented in fig. 5.10 is the
value of J0 of the Mn moment as a function of band-filling in the rigid band scheme by
254
K. FUKAMICHI et al.
Fig. 5.11. Density of states of γ -phase disordered (DO) Mn3 Ir alloys for the 1Q, 2Q and 3Q structures (Sakuma
et al., 2003).
scanning EF for L12 -type ordered Mn3 Ir alloy (Umetsu et al., 2003b). It is found that J0
has a maximum value of about 160 meV at EF (= 0). This result gives TN of 1250 K,
larger by about 30% than the observed TN . Such a high value of TN , compared with the
experimental value, would be permissible in the generalized molecular field approximation
for three dimensional systems.
Figure 5.11 sets out the DOS of the 1Q, 2Q and 3Q structures for disordered (DO) Mn3 Ir
alloy (Sakuma et al., 2003). The lattice parameter is settled at a = 0.3785 nm. Contrary to
the case of L10 -type MnPt alloy considered in the following subsection 5.3, the local DOSs
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
255
Fig. 5.12. Difference in the total energy relative to the value of the T1 structure between L12 -type ordered and
γ -phase disordered Mn3 Ir alloys with each magnetic structure (Umetsu et al., 2003c; Sakuma and Fukamichi,
2005).
Fig. 5.13. Difference in the total energy relative to the value of the T1 structure between L12 -type ordered
and γ -phase disordered Mn3 Rh alloys with each magnetic structure (Umetsu et al., 2003a, 2003c; Sakuma and
Fukamichi, 2005).
of Mn and Ir sites maintain well the characteristic features of the DOS of the T1 state in
the ordered alloy. Especially, it should be marked that the DOS of the 3Q structure is most
close to that of the T1 state and exhibits a large dip around EF , which is comparable to that
of the ordered alloy. Very similar DOSs of Mn3 Rh alloy have been reported by Sakuma
et al. (2002). Therefore, it is considered that the 3Q structure in the disordered systems
of Mn3 Ir and Mn3 Rh has a lower energy, compared with the PM state in their ordered
systems. Actually, as shown in figs. 5.12 and 5.13, the 3Q state in the γ -phase disordered
256
K. FUKAMICHI et al.
Mn3 Ir and Mn3 Rh alloys is more stable than the PM state (Umetsu et al., 2003a, 2003c;
Sakuma and Fukamichi, 2005). In these figures, we plot the energy in the PM state of the
disordered systems by using the CPA for the random alloys of the compositions Mn (↑)
0.375 Mn (↓) 0.375 Ir (↑) 0.125 Ir (↓) 0.125 and Mn (↑) 0.375 Mn (↓) 0.375 Rh (↑) 0.125
Rh (↓) 0.125. For convenience, the abscissa is divided into the equal spacing for each
magnetic structure. The energy in the PM state is higher by about 110–120 meV/atom than
that in the MQSDW states, which means that the MQSDW states in the disordered system
are stable, comparable with the paramagnetic state in the ordered systems. The effective
exchange constant J0 for each MQSDW structure as a function of the Fermi level EF for
disordered Mn3 Ir alloy is set out in fig. 5.14 (Umetsu et al., 2003b). All exhibit a similar
behavior to that in the T1 state in fig. 5.10, though the magnitude of the ordinate is slightly
reduced. Especially, the actual value of J0 at EF = 0 becomes small with a slight peak shift
position from EF = 0 for the 1Q structure. The order of magnitude of J0 in these magnetic
structures is consistent with the relative positions of the energy shown in fig. 5.14 (Umetsu
et al., 2003b), and thus the results imply that the 3Q structure is most stable in the γ -phase
disordered system. The value of TN estimated from J0 for disordered Mn3 Ir alloy is about
735 K, in agreement with the observed value of about 730 K (Yamaoka, 1974), though
the result for the stable 3Q spin structure is different from the 1Q spin structure suggested
by Yamaoka (1974). It will be useful to keep in mind that the distinction between the
3Q and 1Q structures is experimentally difficult (see subsection 5.1.2), in addition, Ir is a
high absorber of neutrons, hindering detailed analysis. The calculated total energy (E in
meV/atom) relative to the value of the T1 structure, magnetic moments (m in μB ), effective
exchange constant (J0 in meV) and the Néel temperature (TNcal in K) for both L12 -type
γ -phase ordered (O) and the γ -phase disordered (DO) Mn3 Ir alloys are summarized in
table 5.2 (Sakuma et al., 2003). In addition, J0 of a disordered alloy with a smaller lattice
constant of a = 0.375 nm, the same value as γ -Mn, does not change within the numerical
accuracy and is still larger than that of γ -Mn (J0 = 64 meV). On the contrary, J0 of γ -Mn
with the same lattice constant as that of the γ -phase disordered Mn3 Ir alloy is confirmed
to be almost the same as that of γ -Mn with a = 0.375 nm (Sakuma, 1998a). These results
disclose that the enhancement of TN by the addition of Ir to γ -Mn is not attributed to the
lattice expansion but to the effect of the electronic structure change induced by Ir atoms.
Note that the band calculations in the same methods for L12 -type γ - and γ -phases Mn3 Rh
alloys give almost the same results as those for Mn3 Ir (Umetsu et al., 2003c) as given in
table 5.3 (Sakuma et al., 2002).
5.3. L10 -type ordered MnRh, MnIr, MnPt, MnPd and MnNi alloys
Ordered MnRh, MnIr, MnPt, MnNi and MnPd alloys form the L10 (≡ CuAu-I)-type structure. Four kinds of possible spin configurations (I–IV) in the L10 -type MnPt alloy were
discussed by Krén et al. (1968a). The structure I is ferromagnetic with magnetic moments
pointing in either [100] or [110] direction of the basal plane or the direction of the tetragonal axis. The Pt atoms may have a magnetic moment, too. Experimental results do not
support this spin structure because L10 -type MnPt alloy is antiferromagnetic. The spin
structures II–IV are antiferromagnetic with the same possible directions of the magnetic
moment as those in the ferromagnetic state. It is noted that the Pt atoms in the structure
II carry no magnetic moment (see fig. 7.73). At low temperatures, L10 -type MnPt alloy
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
257
Fig. 5.14. Effective exchange constant J0 of γ -phase disordered (DO) Mn3 Ir alloy as a function of the Fermi
level EF (band filling) for each magnetic structure (Umetsu et al., 2003b).
with the equiatomic composition exhibit the spin structure II as shown in fig. 5.15 and
this structure has been called the AF-I-type structure (Sakuma, 2000). The spin structure
III will be given in subsection 7.8.3 (see fig. 7.73). They, except for MnRh, have a quite
high Néel temperature TN of about 1145 for MnIr (Umetsu et al., 2004a, 2004b), 970 K
for MnPt (Krén et al., 1968a), 1070 K for MnNi (Pál et al., 1968) and 810 K for MnPd
258
K. FUKAMICHI et al.
TABLE 5.2
Calculated total energy (E in meV/atom) relative to the value of the T1 structure, magnetic moments (m in
μB ), effective exchange constant (J0 in meV) and the Néel temperature (TNcal in K) estimated from 2J0 /3kB for
ordered (O) and disordered (DO) Mn3 Ir alloys (Sakuma et al., 2003). The experimental values of the magnetic
exp
moment (mexp ) and the Néel temperature TN are also shown, for comparison
exp
Phase
Spin
E
mMn
mIr
J0
TNcal
TN
L12 -type
(O)
γ -phase
(DO)
T1
PM
1Q
2Q
3Q
PM
0
77.5
93.8
81.6
76.2
201.3
2.62
2.20
2.46
2.47
2.51
2.27
0.00
0.00
0.09
0.11
0.12
0.00
162
1253
960a
41
85
95
317
660
735
730b
a Tomeno et al. (1999), b Yamaoka (1974).
TABLE 5.3
Calculated total energy (E in meV/atom) relative to the value of the T1 structure, magnetic moments (m in
μB ), effective exchange constant (J0 in meV) and the Néel temperature (TNcal in K) estimated from 2J0 /3kB for
ordered (O) and disordered (DO) Mn3 Rh alloys (Sakuma et al., 2002), together with the experimental values of
exp
the magnetic moment (mexp ) and the Néel temperature TN
exp
Phase
Spin
E
mMn
mRh
J0
TNcal
TN
L12 -type
(O)
γ -phase
(DO)
T1
PM
1Q
2Q
3Q
PM
0
123.8
117.0
103.4
99.3
205.4
2.78
2.57
2.57
2.62
2.66
2.48
0.00
0.00
0.04
0.07
0.08
0.00
163
1250
853a
39
80
88
300
620
680
668b
a Krén et al. (1966), b Yamauchi et al. (1999).
Fig. 5.15. Crystal structure with the AF-I-type spin structure of L10 -type MnPt ordered alloy (Sakuma, 2000).
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
259
Fig. 5.16. Density of states of L10 -type ordered MnRh alloy. (a) Antiferromagnetic state, (b) paramagnetic (PM)
state (Umetsu et al., 2004d, 2005a).
Fig. 5.17. Density of states of L10 -type ordered MnIr alloy. (a) Antiferromagnetic state, (b) paramagnetic (PM)
state (Umetsu et al., 2004a, 2004b).
(Kasper and Kouvel, 1959), and these alloys are expected to have a high potential of application to the exchange biasing film for GMR and TMR devices. Although the ordered
phase is favorable in these bulk systems, the disordered phase can be easily realized by
means of the thin film deposition process. It has been considered that the disordered phase
is practically undesirable because it is paramagnetic at room temperature, and hence the
exchange bias-field does not work when such a film is used for GMR and TMR devices.
Accordingly, little investigation has been made on the magnetism not only in the bulk state
but also in the film structure for these disordered alloys.
Figures 5.16–5.20 show the DOSs of L10 -type ordered phase of MnRh (Umetsu et al.,
2004d, 2005a), MnIr (Umetsu et al., 2004a, 2004b), MnPt (Sakuma, 2000), MnPd (Umetsu
et al., 2002a) and MnNi (Sakuma, 1998b), respectively. In each figure, (a) is for the L10 type tetragonal structure with c/a = 0.9033, 0.9430, 0.9175, 0.9412 and 0.8796 for MnRh,
MnIr, MnPt, MnNi and MnPd, respectively, and (b) is for the PM state of the corresponding
system. The AFM magnetic structure of (a) in each figure is collinear of the AF-I type, and
the PM states in (b) are calculated based on the DLM model where the Mn moments
260
K. FUKAMICHI et al.
Fig. 5.18. Density of states of L10 -type ordered MnPt alloy. (a) Antiferromagnetic state, (b) paramagnetic (PM)
state (Sakuma, 2000).
Fig. 5.19. Density of states of L10 -type ordered MnPd alloy. (a) Antiferromagnetic state, (b) paramagnetic (PM)
state (Umetsu et al., 2002a).
Fig. 5.20. Density of states of L10 -type ordered MnNi alloy. (a) Antiferromagnetic state, (b) paramagnetic (PM)
state (Sakuma, 1998b).
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
261
pointing upward and downward are distributed randomly in the Mn sublattices in fig. 5.15.
It is found that the DOS in the AF-I state of each alloy reveals similar behavior and all have
a prominent dip at EF in both the spin states. This characteristic feature can be connected
with the AF-I staggered field since the dip completely vanishes in the PM state as shown
in (b) in each corresponding figure. These are closely related to the gap caused by the AFM
long-range order. What is important is that the non-magnetic layers such as Pt and Pd layers
between the Mn layers interfere opening the full gap and give a small but a finite intensity
at EF . Besides the shapes of DOS, the magnetic moments of the AF-I and the PM states
are close in each system. In L10 -type MnPt alloy, for example, the moments are 3.18 μB
and 2.94 μB in the AF-I and the PM states, respectively, showing that the reduction of
the moment in the PM state is less than 10%. As far as it concerns the total energy, the
energy in the PM state (b) is about 88 meV/atom larger than that in the AF-I state (a) in
MnPt, by 119 meV/atom in MnNi and by 68 meV/atom in MnPd. The calculated results
of J0 in the L10 -type ordered structure for each system appear in fig. 5.21 (Umetsu et
al., 2003b). The value of J0 at EF (= 0) for these alloys is around 150 meV, comparable
to the energy difference between the AF-I and the PM states given above. The value of
TN estimated from J0 is about 1000 K, which can be compared quantitatively with the
experimental values. On these grounds, it seems reasonable to consider that the AF-I state
of these ordered alloys is stabilized through the formation of a pseudo-gap located at EF ,
resulting in the increase of TN at the same time.
5.4. Comparison between the ordered and disordered phases of equiatomic MnPt alloys
As an example, we discuss the magnetic structure in the disordered equiatomic MnPt phase
(Sakuma, 2000). The lattice is fixed at an fcc structure with the same volume of the ordered
phase. The considered magnetic structures are the three kinds of multiple-Q states given in
fig. 5.2 and the PM state is treated by the CPA as the DLM state. Figure 5.22 shows the
DOSs of each magnetic structure and the paramagnetic state (Sakuma, 2000). Two magnetic and two non-magnetic atoms are randomly distributed in the disordered state. It is
clear that the pseudo-gap formed in the ordered phase vanishes and the whole structure
changes drastically compared with that of the AF-I ordered phase. A highly important aspect is that the DOS in the PM state does not differ so much from those of the multiple-Q
structures. As shown in fig. 5.23, this is a distinctive feature in the ordered phase, reflected in the total energy given by the difference relative to the value of the AF-I structure
(Sakuma, 2000). The energy change in the disordered phase is much smaller than that in
the L10 -type ordered phase. We can infer the reason for that from the DOSs as follows. The
random arrangement of Mn and Pt atoms, whose characteristics are much different from
each other, disturbs the AFM staggered field and prevents the formation of the pseudogap which stabilizes the AFM state. Among them, the 3Q structure has the lowest energy,
though the difference from the 2Q structure is negligibly small. In addition, the value of the
1Q structure is about 5 meV/atom higher than that of the 3Q structure, and hence considered to be unstable. Looking at fig. 5.24, in fact, J0 of the 1Q structure is slightly negative,
indicating that the 1Q structure is not stable in the ground state (Sakuma, 2000). This
could be attributed to magnetic frustration of the Mn atoms, which is caused by the random arrangement with the non-magnetic Pt atoms. On the other hand, the values of J0 for
the 3Q and 2Q structures are around 40 meV, being in the region of the energy difference
262
K. FUKAMICHI et al.
Fig. 5.21. Effective exchange constant J0 of L10 -type ordered alloys as a function of the Fermi level EF (band
filling); (a) MnRh (Umetsu et al., 2006), (b) MnIr, (c) MnPt, (d) MnPd and (e) MnNi (Umetsu et al., 2003b).
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
263
Fig. 5.22. Density of states of disordered (DO) MnPt alloy for the 1Q, 2Q and 3Q spin structures and the paramagnetic (PM) state (Sakuma, 2000).
Fig. 5.23. Difference in the total energy relative to the value of the AF-1 structure of ordered and disordered
MnPt alloys for each magnetic structure relative to the value of the AF-1 structure (Sakuma, 2000).
264
K. FUKAMICHI et al.
Fig. 5.24. Effective exchange constant J0 of disordered (DO) MnPt alloys as a function of the Fermi level EF
(band filling) for each magnetic structure (Sakuma, 2000).
from the PM state. The estimated TN from these J0 values value is about 300 K. The magnetic moment of Mn atoms in the 3Q structure is of about 2.7 μB , smaller than that in the
ordered phase, as much as about 30%. In the Mn alloys with non-magnetic atoms, such as
Pt, the results mentioned above lead us to conclude that the random distribution remarkably
weakens the effects of the staggered field and makes the collinear spin structure unstable
because of magnetic frustration, which brings about a non-collinear spin structure close to
the 3Q state as well as a low TN decreased to around room temperature. The situation is
qualitatively the same for MnIr, MnPd and MnNi alloys in the disordered state. Therefore,
magnetic measurements below room temperature of the disordered alloys should be carried out for profound understanding. Figure 5.25 shows the effective exchange constant J0
of L10 -type (c/a < 1) and cubic (c/a = 1) MnPt ordered alloys with the AF-I structure
as a function of the Fermi level EF (band filling) (Sakuma, 2000). The values of J0 at the
actual EF are almost the same for L10 and cubic structures. From the above results, we can
infer that the AF-I state of the MnPt ordered alloy is stabilized through the formation of
a pseudo-gap located at EF , which increases TN at the same time. Additionally, the lattice
distortion to c/a < 1 lowers the electronic energy but does not have a strong influence on
the antiferromagnetism. The obtained data for the L10 -type and cubic ordered and disordered phases are summarized in table 5.4 (Sakuma, 2000). We may go on from these results
to the conclusion that the random distributions in Mn alloys with non-magnetic atoms as
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
265
Fig. 5.25. Effective exchange constant J0 as a function of the Fermi level EF for L10 -type and cubic ordered
MnPt alloys (Sakuma, 2000).
TABLE 5.4
Calculated total energy (E in meV/atom) relative to the value of the AF-I structure, magnetic moments (m in
μB ), effective exchange constant (J0 in meV) and the Néel temperature (TNcal in K) estimated from 2J0 /3kB for
ordered (O) and disordered (DO) MnPt alloys (Sakuma, 2000). The experimental value of the Néel temperature
exp
TN (Krén et al., 1968a) is also shown, for comparison
Ordered (O)
Disordered
(DO)
L10
PM
cubic
1Q
2Q
3Q
PM
exp
E
mMn
J0
TNcal
TN
0.0
107.5
7.5
300.5
283.8
280.5
301.0
3.22
2.99
3.11
2.75
2.76
2.76
2.72
98.8
764
975
101.0
−6.3
38.6
47.1
0.0
781
298
364
0
Pt weaken the effects of the staggered field and make the collinear magnetic structure unstable due to magnetic frustration, bringing about non-collinear structures close to the 3Q
state with TN lowered to around room temperature.
5.5. Stability of B2 phase of MnRh equiatomic alloy
No high Néel temperature of L10 -type MnRh alloy has been reported. The phase transformation between the B2 and the L10 -type phases is impractically low, compared with other
four kinds of L10 -type alloys (see fig. 6.17). Figure 5.26(a) and (b) shows the DOS curves
of MnRh equiatomic alloy in the AFM and PM states, respectively (Umetsu et al., 2004d,
2005a). No remarkable difference is observed between the B2 and the L10 -type phases
and the feature in the PM state is very similar to that of the other four kinds of L10 -type
alloys in the PM state (see figs. 5.16–5.20).
The effective exchange constant J0 as a function of the Fermi level EF for the B2type MnRh alloy is given in fig. 5.27 (Umetsu et al., 2005a). Note that the curve goes
266
K. FUKAMICHI et al.
Fig. 5.26. Density of states of the B2-type MnRh alloy, (a) antiferromagnetic (AFM) state, (b) paramagnetic
(PM) state (Umetsu et al., 2004d, 2005a).
Fig. 5.27. Effective exchange constant J0 as a function of the Fermi level EF for the B2-type MnRh alloy
(Umetsu et al., 2005a).
through the origin at EF , indicating that the B2-type alloy is not liable to establish the
antiferromagnetic state with a high Néel temperature. Figure 5.28 indicates the difference
in the total energy relative to the value of the L10 -type structure in the AFM state for
L10 -type alloy in the paramagnetic state and B2- type alloys in the AFM and PM states
(Umetsu et al., 2005a). The important point to note is that the value in the PM state of
the B2 phase is lower than that in the PM state of the L10 -phase, consistent with the
experimental results such as the phase transformation (see fig. 6.17) and magnetic susceptibility (see fig. 6.18). Recently, it has been pointed out that the total energy obtained
by using the LMTO-ASA method with the generalized gradient approximation (GGA) in
the antiferromagnetic and ferromagnetic states is very close to each other (Yamada et al.,
2005).
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
267
Fig. 5.28. Difference in the total energy relative to the value of the L10 -type structure in the AFM state for
L10 -type alloy in the paramagnetic state and B2-type alloys in the AFM and PM states (Umetsu et al., 2005a).
6. Experimental observations of antiferromagnetic transition and magnetic
moments of Mn alloys
Investigations of magnetic properties of Mn alloys became active in the 1960s because
neutron diffractions were used as very powerful methods for studies of antiferromagnetic
materials. In the following decades, the relationship between the martensitic phase transformation and the antiferromagnetic transition in the γ -phase Mn alloys was investigated.
Main studies of antiferromagnetic properties and spin structures of L10 -type Mn alloys
also began in the 1960s. Especially, detailed investigations were carried out for the MnPt
alloy system. Furthermore, the martensitic transformation from the B2 (≡ CsCl) structure
to the L10 -type phase was investigated from metallurgical viewpoints for several kinds of
alloy systems.
As mentioned in the introduction, γ -phase, L12 -type (≡ γ -phase) alloys and equiatomic
L10 (≡ CuAu-I)-type Mn alloys have been extensively investigated from practical viewpoints because these alloys exhibit a high TN . In this section, antiferromagnetic transitions
and magnetic moments of L12 -type, γ - and L10 -type phases are described.
6.1. The Néel temperature and the magnetic moment of L12 -type (≡ γ -phase) ordered
(O) alloy systems
It has been reported that several kinds of additional elements stabilize γ -Mn phase, and
γ -MnIr, γ -MnRh and γ -MnPt disordered alloys easily transform into an L12 -type ordered
phase (≡ γ -phase) (Raub and Mahler, 1955). In these investigations, it has been demonstrated that Ir significantly increases TN in γ -phase ordered and γ -phase disordered phases
(Yamaoka et al., 1971, 1974; Yamaoka, 1974). Therefore, a number of studies have been
carried out to find novel exchange biasing films for spin valves. Especially, γ -MnIr alloys
are identified as promising antiferromagnets due to their high exchange bias-field, high
blocking temperature and low critical thickness for exchange bias-field.
268
K. FUKAMICHI et al.
Fig. 6.1. Temperature dependence of the magnetic susceptibility for L12 -type (≡ γ ) ordered (O) MnRh alloys
(Yamauchi et al., 1998).
Magnetic properties of Mn and its alloys have often been discussed in terms of the
magnitude of the Mn–Mn distance within the framework of the localized moment model
(Yamada et al., 1970; Yamada and Tazawa, 1970). However, systematic trends of the concentration dependence of TN are not explained in terms of the model mentioned above. In
addition, the magnetic susceptibility of L12 -type ordered MnRh alloys does not follow a
Curie–Weiss-type temperature dependence as given in fig. 6.1, showing no decrease with
temperature in the PM temperature ranges (Yamauchi et al., 1998). The susceptibility and
specific heat peaks at the Néel temperature TN become broad with decreasing Rh concentration because of the decrease in the degree of order. In addition, the spin fluctuation
contribution would become strong with decreasing Rh concentration, or decreasing TN ,
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
269
Fig. 6.2. Exchange constant J0 as a function of the Fermi level EF for γ -Mn with a collinear spin structure
(Sakuma, 1998c).
accompanied by a rather dullness in peak shape (Makoshi and Moriya, 1975; Takahashi,
1999).
In the band theory, the calculations for γ -Mn (Sakuma, 1998a) under the conditions of
the collinear spin structure with the lattice constant a = 0.375 nm and the axial ratio c/a =
1.0 provide the exchange energy J0 as a function of the Fermi level EF as shown in fig. 6.2
(Sakuma, 1998c). As discussed in the preceding subsection 4.3, the Néel temperature TN is
given by equation (4.31). The magnitude of J0 becomes larger with decreasing d-electron
number at the Mn site, and hence this figure shows that TN becomes higher with decreasing
d-electron number at the Mn site. In fact, the calculations by the linear muffin-tin orbital
(LMTO)-atomic sphere approximation (ASA) method indicate that the order of d-electron
number at the Mn site is Mn3 Rh < γ -Mn < Mn3 Pt (Sakuma, 1998c). The concentration
dependence of TN shown in fig. 6.3 (Krén et al., 1968a; Yamaoka, 1974; Yamauchi et al.,
1998; Tomeno et al., 1999) is consistent with the theoretical calculations. In other words,
the order of magnitude of TN is Mn3 Rh > γ -Mn > Mn3 Pt (Yamauchi et al., 1998). In
addition, on the basis of equation (4.31), it is expected that not only Ir but also Rh and
Ru increase TN . With increasing Rh concentration, as seen from fig. 6.3, TN increases in
a similar manner as Ir, though the value is different in magnitude (Yamauchi et al., 1998).
This difference will be discussed in the subsection 7.8. L12 -type (≡ γ -phase) ordered
MnRu alloys are not obtainable as discussed in the following subsection 6.3.
6.2. The Néel temperature and the magnetic moment of γ -phase disordered (DO) alloy
systems
The γ -phase disordered (DO) Mn-based alloys are stabilized by several kinds of additional
elements in analogy with γ -phase Fe-based alloys. The temperature dependence of magnetic susceptibility χ(T ) for γ -phase disordered (DO) Mn100−x Rhx alloys is set out in
fig. 6.4 (Yamauchi et al., 1998). Instead of a clear peak observed in the ordered state (see
fig. 6.1), a shoulder associated with the magnetic transition is found. Furthermore, irregular
270
K. FUKAMICHI et al.
Fig. 6.3. Concentration dependence of the Néel temperature TN for L12 -type-phase (≡ γ ) ordered (O) alloy
systems; Mn100−x Irx (: Yamaoka, 1974, : Tomeno et al., 1999), Mn100−x Rhx (Yamauchi et al., 1998) and
Mn100−x Ptx (Krén et al., 1968a).
dependences above the Néel temperature are caused by the crystallographic phase transformation. The concentration dependence of TN of γ -phase disordered Mn100−x TMx alloy
systems is gathered in fig. 6.5 (Krén et al., 1966; Hicks et al., 1968; Endoh et al., 1969;
Uchishiba, 1971; Yamaoka, 1974; Sasao et al., 1999; Yamauchi et al., 1999). The value of
TN exhibits a similar trend with that of the ordered phase given in fig. 6.1. In contrast to
L10 -type ordered MnPt alloy, Mn-based systems such as Mn100−x Rhx (Raub and Mahler,
1954; Kasper and Kouvel, 1959) and Mn100−x Irx (Raub and Mahler, 1955; Andresen et al.,
1965; Kjekshus et al., 1967; Krén et al., 1968a; Pál et al., 1968) γ -phase disordered alloys
require no post-heat-treatments in industrial processes, and hence Mn100−x Irx (Hoshino
et al., 1996; Nakatani et al., 1997; Devasahayam and Kryder, 1999a, 1999b; Mao et al.,
1999; Takahashi et al., 2000; Li et al., 2001; Childress et al., 2002; Tsunoda et al., 2002b)
and Mn100−x Rhx (Araki et al., 1998a, 1998b; Veloso et al., 1998) γ -phase alloys are also
considered as promising candidates for spintronic devices. What has to be noticed is that
the magnitude of TN for the γ -phase disordered alloys is lower by 150–200 K than that for
the L12 -type ordered counterparts, depending on the alloying element and its composition.
The curve in fig. 6.6 illustrates the concentration dependence of the room temperature
lattice constant a of Mn100−x Rhx (Yamauchi et al., 1999) and Mn100−x Irx (Yamaoka,
1974) alloy systems in the ordered and disordered states. The open and solid symbols stand
for the ordered and disordered alloys, respectively. The lattice constant increases with the
Rh and Ir concentrations, but no distinct evident difference between the disordered and
ordered states are found, although a slight difference is found at high concentration ranges
in the Mn-Ir alloy system. Consequently, the difference between TN in the disordered and
ordered states has no relation with the magnitude of the lattice constant, or the Mn–Mn
distance. This difference is explained by taking randomness into consideration (Sakuma et
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
271
Fig. 6.4. Temperature dependence of the magnetic susceptibility for γ -phase disordered (DO) MnRh alloys
(Yamauchi et al., 1998).
al., 2002, 2003). Namely, careful comparison of fig. 5.10 with fig. 5.14 shows the magnitude of the ordinate is different from each other. Such a smaller exchange constant J0 of
the disordered (DO) alloy is obtained by using the CPA (see subsections 4.2 and 4.3). It is
important to note that TN of γ -phase disordered MnRu alloy system is the highest because
the d-electron number at the Mn site in MnRu is smaller than that in MnRh, in good agreement with equation (4.31). Therefore, the smaller the d-electron number at the Mn site, the
higher the Néel temperature. The concentration dependence of the magnetic moment mMn
per Mn atom extrapolated to 0 K for integrated neutron diffraction intensity of γ -phase
disordered (DO) Mn100−x Irx alloy system is shown in fig. 6.7 (Yamaoka et al., 1974). The
theoretical value of Mn75 Ir25 disordered (DO) alloy is given in the same figure (Sakuma
272
K. FUKAMICHI et al.
Fig. 6.5. Concentration dependence of the Néel temperature TN for γ -phase disordered (DO) alloy systems;
Mn100−x Irx (Yamaoka, 1974), Mn100−x Rhx (Yamauchi et al., 1999), Mn100−x Rux (Sasao et al., 1999),
Mn100−x Ptx (Krén et al., 1966), Mn100−x Pdx , Mn100−x Nix (Hicks et al., 1968), Mn100−x Fex (Endoh et al.,
1969) and Mn100−x Cux (Uchishiba, 1971).
Fig. 6.6. Concentration dependence of the room temperature lattice constant for γ -phase disordered (DO) and
L12 (≡ γ ) ordered (O) Mn100−x Rhx (Yamauchi et al., 1999) and Mn100−x Irx (Yamaoka, 1974) alloy systems.
et al., 2003). The experimental value (•) of mMn increases linearly with Ir concentration,
being consistent with the theoretically calculated value (◦) in magnitude. Furthermore, the
value of mMn extrapolated to x = 0 is about 2.2 μB , being slightly smaller than the estimated value of 2.4 μB in Mn-Cu alloys with an fct structure (Bacon et al., 1957). However,
in view of the experimental accuracy associated with diffuse scattering and so on, the magnetic moment should be said to be almost the same as that of fct Mn.
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
273
Fig. 6.7. Concentration dependence of the magnetic moment mMn per Mn atom of γ -phase disordered (DO)
Mn-Ir alloy system (•: Yamaoka et al., 1974; ◦: Sakuma et al., 2003).
6.3. Magnetic structures of L12 -type (≡ γ -phase) Mn3 Pt and Mn3 Ir ordered (O) alloy
systems
L12 -type (≡ γ -phase) alloys are observed in Mn-Pt, Mn-Ir and Mn-Rh alloy systems.
With increasing temperature, only Mn3 Pt exhibits a first-order transition, accompanied by
the change of the spin structure and the lattice constant.
6.3.1. D- and F-phases of L12 -type (≡ γ -phase) ordered Mn3 Pt
In ordered Mn3 Pt alloy, a first-kind of magnetic transition from the D-phase to the Fphase takes place with increasing temperature (Krén et al., 1968a). In the former state,
all of the Mn sites have magnetic moments, which lie in the (111) plane and point to the
211 direction, exhibiting the so-called T1 structure. On the other hand, the magnetic
unit cell in the latter state has a double atomic cell and the magnetic moments are parallel to the 001 crystal axis, showing a collinear structure as illustrated in fig. 6.8(Krén et
al., 1966, 1968a). It has been reported that one third of the Mn atoms have no magnetic
moments in the F-phase state (Krén et al., 1968a). However, recent inelastic neutron scattering data have revealed that the F-phase is understood as a new type of 3D geometrical
spin frustration system (Ikeda and Tsunoda, 2003a, 2004). Above the Néel temperature,
the crystal lattice has a cubic structure. Therefore, Mn atoms at the face center position
form the equilateral octahedral structure which is composed of eight equilateral triangles.
Since the first-neighbor Mn spins have antiferromagnetic coupling among the vertices of
the equilateral triangles, the Mn atoms of Mn3 Pt satisfy the conditions of the geometrical
spin frustration above the Néel temperature. As a result, strong diffuse inelastic neutron
scatterings of ordered Mn3 Pt above TN are attributed to this geometrical spin frustration.
The 400 Bragg reflection lines in the F-phase (T /TN = 0.897) and in the paramagnetic
phase (T /TN = 1.225) were examined. The diffraction line width of the former is wider
than that of the latter. Therefore, the line broadening in the F-phase is attributed to the
lattice distortion. Consequently, the X-ray diffraction line of the F-phase in fig. 6.9 is fitted by two Gaussian functions with the same line width as the cubic phase and with the
274
K. FUKAMICHI et al.
Fig. 6.8. Magnetic structures of ordered Mn3 Pt in the D- and F-phases (Krén et al., 1966, 1968a). The open and
closed circles stand for Pt and Mn atoms, respectively.
Fig. 6.9. X-ray diffraction line of the F-phase fitted by two Gaussian functions with the same line as the cubic
phase and with the intensity ratio of 2 to 1 (Ikeda and Tsunoda, 2003a).
intensity ratio of 2 to 1 (Ikeda and Tsunoda, 2003a). The X-ray diffraction line above the
Néel temperature was first fitted by a Gaussian function as an experimental line width of
the Bragg peak. Next, the X-ray diffraction line of the F-phase was fitted by two Gaussian
functions with the same line as the cubic phase and with intensity rations of 2 to 1. As seen
from the line shape fitting, the lattice parameter c is smaller than the lattice parameter a.
The lattice parameters a and c are determined to be 0.3882 and 0.3876 nm, respectively, at
T /TN = 0.897. Thus, the tetragonality c/a in the F-phase is estimated as 0.9985. Due to
this lattice deformation, the condition of 3D geometrical spin frustration with the octahedral symmetry is broken, resulting in the F-phase magnetic structure. It should be pointed
out that one-third of Mn spins are not paramagnetic (see fig. 6.8) but are still dynamically
fluctuating. In other words, the F-phase is stabilized by suppressing the inconsistency of
the triangle antiferromagnetic coupling through the dynamical motion of one third of Mn
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
275
Fig. 6.10. The magnetic phase diagram of Mn-Pt alloy system in the concentration range from 17 to 28% Pt
for ordered and disordered states. The temperatures TN (O) and TN (DO) stand for the Néel temperatures in the
ordered and disordered states, respectively (Krén et al., 1968a).
Fig. 6.11. Magnetic phase diagram under hydrostatic pressure for L12 -type ordered Mn3 Pt alloy (Yasui et al.,
1987).
spins. The tetragonal lattice distortion supports the suppression of the octahedral frustration and determines which Mn spins should be non-magnetic (Ikeda and Tsunoda, 2003a,
2004).
Figure 6.10 shows the magnetic phase diagram of the Mn-Pt alloy system in the concentration range from 17 to 28% Pt in the ordered and disordered states (Krén et al., 1968a). In
the ordered state, the F-phase range decreases with decreasing Pt concentration and disappears in the vicinity of 21% Pt, accompanied by a linear increase of the Néel temperature
TN . It should be pointed out again that TN in the disordered state is much lower than that in
the ordered state, in analogy with the Mn-Ir alloy system. The concentration dependence
curve is slightly convex upward, extrapolating to 480 K at x = 0 (see fig. 6.5). The magnetic phase diagram under hydrostatic pressure for ordered Mn3 Pt alloy is given in fig. 6.11
(Yasui et al., 1987). The Néel temperature TN in the F-phase state decreases on applying
pressure and the first-order magnetic transition temperature Tt increases linearly. Because
276
K. FUKAMICHI et al.
TABLE 6.1
The lattice parameter a, transformation temperature Tt (K) and the Néel temperature TN , magnetic moment per
Mn atom mMn (μB ) for several kinds of L12 -type (≡ γ -phase) ordered (O) alloys (Krén et al., 1967b)
Alloy
a (nm)
Tt (K)
TN (K)
mMn (μB )
Mn3 Pt
Mn3 Pt0.95 Rh0.05
Mn3 Pt0.90 Rh0.10
Mn3 Pt0.80 Rh0.20
Mn3 Pt0.50 Rh0.50
Mn3 Rh
0.3833
0.3833
0.3832
0.3828
0.3820
0.3813
365
415
∼ 507
475
465
505
555
685
855
3.0
3.5
2.8
3.4
3.5
3.6
Fig. 6.12. Magnetic phase diagram of Mn3 Pt1−x Rhx system in the ordered and disordered states (Krén et al.,
1966, 1968a).
of the first-order transition, we observe a clear hysteresis between heating and cooling
curves for the transition temperature Tt . The decrease in TN is associated with the so-called
invar effect (see fig. 7.82). The F-phase fades out at around 3 kbar, and then the Néel temperature TN increases with increasing pressure. These behaviors are closely related with
the negative or positive spontaneous volume magnetostriction (see figs. 7.80 and 7.82). It
means that the F-phase disappears with decreasing lattice constant. It should be pointed
out that similar alloys with a smaller lattice constant such as Mn3 Ir (Tomeno et al., 1999)
and Mn3 Rh have no collinear magnetic structure of the F-phase. In table 6.1, we compare
the lattice parameter, the Néel temperature TN and excrescence of the F-phase for several
kinds of L12 -type (≡ γ -phase) ordered (O) alloys (Krén et al., 1967b). With increasing
Rh concentration, the lattice constant a decreases, whereas the transformation temperature
Tt and the Néel temperature TN increase. The magnetic moment mMn also exhibits a tendency to increase with increasing Rh concentration. These data tell us that the smaller the
lattice parameters, the higher the Néel temperature TN with increasing Rh concentration,
and then the F-phase is pushed out by addition of a small amount of Rh. Figure 6.12 illustrates the magnetic phase diagram of Mn3 Pt1−x Rhx system (Krén et al., 1966, 1968a)
obtained from systematic neutron diffraction studies. With increasing Rh concentration x,
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
277
Fig. 6.13. Magnetic phase diagram of Mn3 PtNx system in the ordered state (Krén et al., 1971).
Fig. 6.14. Temperature dependence the magnetic moment of the L12 -type ordered Mn3 Pt alloy (Yasui et al.,
1992).
both TN (O) and Tt increase. With increasing x, TN (DO) also increases with the same rate,
though TN (DO) is much lower than TN (O). Figure 6.13 shows the magnetic phase diagram of ordered MnPtNx alloy system (Krén et al., 1971). The nitrogen atoms occupy the
body-centered interstitial position in the L12 -type (Cu3 Au) unit cell. With increasing x, or
the lattice constant, the F-phase region becomes wider, that is, the Néel temperature TN increases, whereas the first-order transition temperature Tt decreases. In the figure, the solid
and dashed lines represent the transition temperature Tt obtained on heating and cooling
processes, respectively. By assuming the presence of a chemical pressure caused by expansion of the lattice constant due to the nitrogen atoms, a comparison of figs. 6.11 and 6.12
with fig. 6.13 is interesting. That is, both the magnetic transition temperature TN and the
crystallographic transformation temperature Tt markedly depend on the lattice constant.
The temperature dependence the magnetic moment of L12 -type ordered Mn3 Pt is presented in fig. 6.14 (Yasui et al., 1992). There is a jump of the magnetic moment of about
30% and a fast decrease of the moment takes place when approaching to the Néel tem-
278
K. FUKAMICHI et al.
perature TN , accompanied by an abrupt lattice expansion of about 0.9%. This behavior is
associated with the drastic change in the thermal expansion curve (see fig. 7.82). In addition, the lattice distortion to the tetragonal lattice occurs in the F-phase (see fig. 6.8).
6.3.2. Magnetic reflection of L12 -type (≡ γ -phase) ordered Mn3 Ir
The (111) surface magnetic structure of the T1 structure in an ordered single crystal of
Mn3 Ir (Tomeno et al., 1999) is given in fig. 6.15. The Mn magnetic moments lie in the
(111) plane and align in the 112 direction. From a neutron scattering study, the absence
of magnetic diffuse scattering has been observed in the ordered Mn3 Ir, showing that each
Mn moment points exactly along the 112 direction. The temperature dependence of the
magnetic part of the integrated intensity of the 100 reflection I100M in L12 -type ordered
Mn3 Ir is given in fig. 6.16. The solid line is the least-squares fit to the data with a power
Fig. 6.15. The (111) surface magnetic structure of the T1 magnetic structure for the L12 -type ordered Mn3 Ir
alloy (Tomeno et al., 1999). The solid and open circles represent Mn and Ir atoms, respectively.
Fig. 6.16. The magnetic part of the integrated intensity of the 100 reflection I100M of L12 -type ordered Mn3 Ir.
The solid curve is fitted to the experimental data by using the power law function with βc = 0.17 (Tomeno et al.,
1999).
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
law function given by
T 2βc
I100M ∝ 1 −
TN
279
(6.1)
with βc = 0.17 (Tomeno et al., 1999). On the other hand, the intensity of the 110 magnetic
reflection I110 for the disordered Mn83 Ir17 alloy is found to obey the same relation with
βc = 0.38 (Yamaoka et al., 1974). The difference between βc in the ordered and disordered
alloys suggests that the ordering in the L12 -type (Cu3 Au) structure reduces the temperature
dependence of the ordered magnetic moment.
6.4. Crystallographic phase transformation between B2 and L10 -type phase
Figure 6.17 draws the contour of the transformation temperature between the B2 and
L10 -type phases of five kinds of alloy systems (Raub and Mahler, 1954, 1955; Hahn and
Kneller, 1958; Hellawell, 1959; Brun et al., 1965; Tsiuplakis and Kneller, 1969; Gendo,
1999). Excepting the MnRh alloy system (Raub and Mahler, 1955; Gendo, 1999), the
transformation temperature is much higher than room temperature. In the vicinity of the
equiatomic composition, the value of TN of several kinds of L12 -type alloy systems is
high enough to use for GMR and TMR devices. In these alloy systems, a high temperature
phase has B2 (≡ CsCl)-type structure and the martensitic phase transformation induces
a low temperature phase with L10 (≡ CuAu-I)-type structure. The transformation in the
MnRh alloy system takes place around 200 K, accompanied by a large thermal hysteresises
of the magnetic susceptibility χ(T ) as given in fig. 6.18 (Umetsu et al., 2005b). Similar
thermal hysteresis has been observed in the electrical resistivity ρ–T curves of Mn50 Rh50
and Mn66 Rh34 alloys (Kouvel et al., 1963; Zavadskii and Medvedeva, 1973). The low temperature phase of L10 -type structure has been shown to be antiferromagnetic (Selte et al.,
Fig. 6.17. The contour of the transformation temperature between the B2-type high temperature phase and
L10 -type low temperature phase for five kinds of alloy systems. Rh: Raub and Mahler (1955), Hellawell (1959),
Gendo (1999), Pd: Raub and Mahler (1954), Pt: Raub and Mahler (1955), Brun et al. (1965), Ir: Raub and Mahler
(1955), Ni: Hahn and Kneller (1958), Tsiuplakis and Kneller (1969).
280
K. FUKAMICHI et al.
Fig. 6.18. Thermal hysteresis of magnetic susceptibility χ(T ) for Mn50 Rh50 alloy (Umetsu et al., 2005b).
Fig. 6.19. Crystallographic relation between the unit cells of the tetragonal L10 -type with the lattice constants a
and c and the cubic B2-type structures (Krén et al., 1968b).
1972). The curve χ(T ) in fig. 6.18 has a negative temperature dependence in a wide range
of temperature, suggesting that its Néel temperature is very high.
Figure 6.19 represents the crystallographic relation between the unit cells of the L10 type tetragonal structure with the lattice constants a and c and the cubic B2 (≡ CsCl)-type
cubic structure with c0 /a0 = 1 (Krén et al., 1968b). This phase transformation occurs
without atomic diffusion, and hence this transformation is regarded as a martensitic transformation.
6.5. The Néel temperature of L10 -type alloy systems
Plotted in fig. 6.20 is the concentration dependence of TN of L10 -type ordered Mn100−x Irx
(Umetsu et al., 2004a, 2004b), Mn100−x Nix (Kjekshus et al., 1967), Mn100−x Ptx (Krén et
al., 1968a) and Mn100−x Pdx (Pál et al., 1968) alloys. A glance at this figure reveals that the
Mn100−x Irx alloy system exhibits the highest value of TN in the whole concentration range.
In addition, the concentration dependence of TN for the Mn100−x Irx alloy system is not so
sensitive, compared with that of the other L10 -type ordered alloy systems. It is noted that
there have been no reports until recent years on the Néel temperature of MnIr, although an
antiferromagnetic state was confirmed by room temperature neutron diffractions (Selte et
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
281
Fig. 6.20. Concentration dependence of TN of L10 -type ordered Mn100−x TMx alloy systems; Mn100−x Irx
(Umestu et al., 2004a, 2004b), Mn100−x Nix (Kjekshus et al., 1967), Mn100−x Ptx (Krén et al., 1968a) and
Mn100−x Pdx (Pál et al., 1968).
al., 1968). The theoretical calculations are similar to the experimental results (see figs. 7.72
and 7.77). Furthermore, what needs to be noted is that the maximum value of TN does not
necessarily occur at the equiatomic composition. Especially, for the Mn100−x Pdx alloy
system, the highest TN is observed at 53% Pd as seen from the figure, consistent with
theoretical calculations (see fig. 7.77). A similar concentration dependence is displayed by
the specific heat coefficient γe curve (see fig. 7.7).
L10 -type MnPt alloys are widely used as exchange biasing films in spin valves
(Childress et al., 1997; Farrow et al., 1997; Saito et al., 1997, 1999). Generally speaking, MnPt sputtered films are in a disordered state and non-magnetic at room temperature,
and hence a heat-treatment for ordering is necessary to obtain the AFM state. In order to reduce the annealing temperature, the MnPd (Pt) alloy system has been investigated (Tanaka
et al., 1997).
Shown in fig. 6.21 is the effective exchange constant J0 of the equiatomic L10 -type
MnIr alloy as a function of the Fermi level EF calculated by using the experimental lattice
constants of MnIr (a) (c/a = 0.945) and the J0 curve obtained by using the lattice constant
of the equiatomic L10 -type MnPt alloy (b) (c/a = 0.918) instead of the lattice constants
of the equiatomic L10 -type MnIr, together with the J0 curve for the equiatomic L10 -type
MnPt alloy itself using the experimental lattice constants (c) (c/a = 0.918) (Umetsu et
al., 2004a). The actual Fermi level EF is located at the origin of the abscissa and the value
of J0 at EF = 0 is 193 meV in (a). The calculated Néel temperature TNcal obtained from
equation (4.31) is 1495 K, permissible as explained in connection with fig. 5.10. The values
of TNcal of L10 -type equiatomic ordered MnPt (Umetsu et al., 2002a, 2002b) and MnNi
(Sakuma and Fukamichi, 2005) alloys obtained from J0 in the same way are 973, 1087
and 1387 K, respectively (see table 7.1). Therefore, TNcal of MnIr is also the highest among
these systems. The calculated magnetic moment of Mn (mcal
Mn in μB ) in the AF and PM
states, the effective exchange constant (J0 in meV) and the calculated Néel temperature
282
K. FUKAMICHI et al.
Fig. 6.21. The effective exchange constant J0 as a function of the Fermi level EF for L10 -type alloys.
(a) Equiatomic MnIr alloy calculated by using the experimental lattice constants (c/a = 0.945), (b) equiatomic
MnIr alloy obtained by using the lattice constants of the isotypic MnPt alloy (c/a = 0.918), (c) equiatomic MnPt
alloy itself calculated by using the experimental lattice constants (c/a = 0.918) (Umetsu et al., 2004a).
TABLE 6.2
cal in μ ) in the antiferromagnetic (AFM) and the paramagnetic
The calculated magnetic moments of Mn (MMn
B
(PM) states, the effective exchange constant (J0 in meV) and the calculated Néel temperature (TNcal in K) of
L10 -type equiatomic ordered MnIr and MnPt alloys. (a) The values calculated by using the experimental lattice
constants of MnIr, (b) the values of MnIr alloy, using the lattice constants of isotypic MnPt instead of those of
MnIr and (c) the values of MnPt, using the lattice constants of MnPt (Umetsu et al., 2004a)
L10 -type alloy
MnIr (a)
MnIr (b)
MnPt (c)
cal in AF
MMn
cal in PM
MMn
J0
TNcal
2.74
3.12
3.22
2.18
2.86
2.99
193
179
126
1495
1387
973
(TNcal in K) obtained from the curves (a), (b) and (c) are given in table 6.2 (Umetsu et al.,
2004a). The shape of the J0 curves (a) and (b) is very similar to each other, although the
magnitude of J0 at EF = 0 is slightly different. The difference between J0 of two curves
is 14 meV, corresponding to be about 100 K in TN . Moreover, by comparing the J0 curve
of the MnIr (a) with that of the MnPt (c), the former shifts to a higher energy side, as a
result, the position of EF = 0 comes close to the peak. This causes a large difference in
J0 of 70 meV between two curves (a) and (c), corresponding to be about 540 K in TN .
It is evident that the difference between the electron number of Ir and Pt gives a larger
contribution to J0 rather than the difference between the lattice constants and/or the c/a
ratio. Consequently, the magnitude of TN for the equiatomic Mn alloy system is dominantly
governed by the d-electron number in analogy with the γ -Mn system alloys (Yamauchi et
al., 1998, 1999).
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
283
Fig. 6.22. The Néel temperature TN as a function of d-electron number for L10 -type Mn alloys, together with that
of γ -Mn, γ -phase (DO) and L12 -type (O) alloys. The solid and open squares, and , stand for the experimental
and theoretical values, respectively (Sakuma, 1999a; Fukamichi, 2000; Umetsu et al., 2003b, 2003c).
Fig. 6.23. The calculated Néel temperature TNcal as a function of the c/a ratio for the equiatomic MnAu alloy (Wang et al., 2001, 2003), together with the experimental TN of the equiatomic L10 -type ordered MnTM
(TM = Ir, Pd, Pt and Ni) alloys (Umetsu et al., 2004a).
Figure 6.22 shows the Néel temperature TN vs. the d-electron number at the Mn site
for L10 -type Mn alloys, together with that of L12 -type ordered (O) and γ -phase disordered (DO) Mn alloys (Sakuma, 1999a; Fukamichi, 2000; Umetsu et al., 2003b, 2003c).
The solid and open squares stand for the experimental and theoretical values, respectively.
A glance at this figure reveals that the magnitude of TN tends to increase with decrease in
the d-electron number calculated by the LMTO-ASA method without regard to the crystal
structures. Recently, Wang et al. (2001, 2003) have carried out first principles calculations
on MnAu superlattices and pointed out that TN increases with increasing c/a ratio. Their
284
K. FUKAMICHI et al.
results are represented in fig. 6.23, together with the experimental value of TN for L10 type Mn50 TM50 alloys with the c/a ratio for MnIr, 0.94 for MnNi, 0.92 for MnPt and 0.88
for MnPd (Umetsu et al., 2004a). In the figure, the AF2, AF3 and AF4 phases indicate
a collinear-type antiferromagnetic spin structure determined by calculating the total energies. The detailed collinear spin structure in each phase is given in the literature (Wang
et al., 2001, 2003). Since the equiatomic MnAu has a crystal structure slightly deformed
from the B2 (≡ CsCl)-type structure, the calculations were performed as a function of the
c/a ratio in the L10 -type structure with the fixed volume of the experimental data (Bacon
and Street, 1958). The calculated magnetic structure and TNcal for the equiatomic MnAu in
which the c/a ratio corresponds to 0.7 in the L10 -type are in accord with the experimental
one (Bacon and Street, 1958). It is worth noting that TNcal linearly increases with the value
of the c/a ratio in the AF2 phase region, in which the c/a ratio varies from 0.81 to 1.00.
The relation between the values of TNcal for the equiatomic MnAu alloys and the c/a ratio
is qualitatively in agreement with the experimental data for other L10 -type MnTM alloys.
Strictly speaking, however, the experimental results of TN for the equiatomic MnTM alloys deviate from the theoretical line as shown by the dashed line in the same figure. This
could be caused by the fact that the collinear spin structure in MnAu is not AF2 but AF1
as given in their literature (Wang et al., 2001), and the d-electron number is different between Au and the TM elements. Consequently, the data given in fig. 6.23 tell us that the
lattice constant and/or the c/a ratio also contribute to the magnitude of TN to a certain
extent, although the d-electron number dominates TN in the equiatomic Mn alloy systems
as discussed in relation to fig. 6.22.
6.6. Strong bonding state correlated with magnetic moment and Pauling valence
The half-filled d band character of Mn is associated with a strong bonding state, reflected
in the magnitude of the magnetic moment. These behaviors are explained by the empirical Pauling valence model which gives a straight line for various kinds of Mn alloys and
compounds.
6.6.1. Volume contraction of L10 -type alloy systems
Figure 6.24 shows the concentration dependence of the room temperature ratio c/a for
the L10 -type Mn100−x TMx (TM = Ir, Pt, Pd and Ni) alloy systems (Pearson et al., 1965;
Umetsu et al., 2003b). For comparison, the data for the non-antiferromagnetic Cu100−x Aux
alloy system is also shown in the same figure (Bjerkelund et al., 1967). The dotted line
(Brun et al., 1965) for the MnIr alloy system substantially coincides with the recent experimental data (◦) obtained by Umetsu et al. (2003b). In all the alloy systems, the value
of c/a tends to become the smallest in the vicinity of the equiatomic composition. In detail, the value of c/a is insensitive to the concentration x in L10 -type MnIr and MnNi
alloy systems. That is, especially, the case for x < 60 at.% Ni. The concentration dependence of the volume V obtained from the room temperature lattice constants a and c for
the L10 -type Mn100−x TMx alloy systems is plotted in fig. 6.25, together with that of the
L10 -type Cu100−x Aux alloys which exhibit a monotonic increase (Bjerkelund et al., 1967;
Umetsu et al., 2003b). In the case of the MnNi alloy system, V decreases with increasing
Ni concentration because the atomic size of Ni is smaller than that of Mn. For L10 -type
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
285
Fig. 6.24. Concentration dependence of the room temperature lattice constant ratio c/a for L10 -type alloys (Brun
et al., 1964, 1965; Pearson et al., 1965; Bjerkelund et al., 1967; Umetsu et al., 2003b).
Fig. 6.25. Concentration dependence of the unit cell volume V for L10 -type alloys (Bjerkelund et al., 1967;
Umetsu et al., 2003b).
MnIr alloy system, the concentration dependence of V is insensitive to the Ir concentration because the atomic size is very close to each other. However, for the MnPt and MnPd
alloy systems, the value of V exhibits the smallest value at the equiatomic composition,
regardless of the larger atomic size of TM. In contrast, the value of CuAu alloy exhibits a
monotonic increase (Bjerkelund et al., 1967). The shrinkage at the equiatomic composition
implies that the covalent bonding state becomes strong at the equiatomic composition. In
Mn and its alloys, the strange properties arise from the conflicting tendencies to maximize
simultaneously according to Hund’s rule the magnetic moment and the bond strength, as
expected from a half-filled d-band character. That is, the short interatomic distances produced by strong bonding tend to quench magnetism (Hafner and Hobbs, 2003). The outer
electrons in transition metal alloys and compounds contribute to chemical bondings, electrical conduction and the magnetic moment. Under such circumstances, the d electrons in
286
K. FUKAMICHI et al.
Mn atoms in L10 -type alloys tend to be localized, giving rise to large spin polarization
within each Mn atom (Sakuma, 1998b). Such a strong bonding state would be correlated
with the Pauling valence and the characteristic electronic structures. As seen from figs. 6.24
and 6.25, the concentration dependences of the lattice constants and volume of the L10 type Mn100−x TMx alloy systems exhibit an anomalous behavior, suggesting that strong
bonding character becomes significant in the vicinity of the equiatomic composition.
6.6.2. Relation between magnetic moment and Pauling valence
The calculated and experimental values of the magnetic moment for L10 -type and L12 type alloys are summarized in table 6.3. It is noted that the calculated values (Sakuma,
1998a; Sakuma et al., 2002; Umetsu et al., 2003b) of L10 -type alloys are larger than those
of the experimental values (Krén et al., 1968a, 1968b).
In the empirical Pauling valence model (Pauling, 1960), the bond length D(n)ij for i
and j atoms in metals and alloys is given by
D(n)ij = D(1)ij − 0.60 log10 n,
(6.2)
where D(1)ij is the bond length for a single bond between the i and j atoms and n is the
bond number. Note that the factor of the second term is 0.71 for organic compounds. The
value of n is defined by the following relation;
PV
(6.3)
,
CN
where CN is the coordination number and PV is the Pauling valence. This empirical rule
was derived from the bonding states in organic materials. Therefore, in magnetic materials,
this rule was first applied to localized magnetic moment systems, and demonstrated that
the magnetic moment of various Mn alloys and compounds are fitted to a universal straight
line (Mori and Mitsui, 1968). The d electrons of the Mn atoms in L10 -type MnNi, MnPt,
MnPd and MnIr are rather localized, which gives rise to large spin polarization within each
Mn atom. To realize the interatomic coupling by keeping both the large spin polarization
and satisfying the Pauli principle, the magnetic moments of the Mn atoms are forced to be
n=
TABLE 6.3
The calculated and experimental magnetic moments of Mn (mcal
Mn in μB ) in the antiferromagnetic (AFM) of L10 type equiatomic ordered MnIr, MnRh, MnPt, MnPd and MnNi alloys, together with those of L12 -type Mn3 Ir,
Mn3 Pt and Mn3 Rh ordered alloys
Alloy
L10 -type
L12 -type
MnIr
MnRh
MnPt
MnPd
MnNi
Mn3 Ir
Mn3 Pt
Mn3 Rh
mcal
Mn (μB )
2.71,2
3.13
3.24
3.85
3.36
2.67
2.938
2.889
exp
mMn (μB )
3.410
3.411
4.312
4.413
3.813
–
3.014
3.614
1 Umetsu et al. (2004a), 2 Umetsu et al. (2004b), 3 Umetsu et al. (2006), 4 Sakuma (1998a), 5 Umetsu et al.
(2002a, 2002b), 6 Sakuma (2000), 7 Sakuma et al. (2003), 8 Kübler et al. (1988), 9 Sakuma et al. (2002), 10 Selte
et al. (1968), 11 Selte et al. (1972), 12 Pál et al. (1968), 13 Krén et al. (1968a), 14 Krén et al. (1967b).
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
287
arranged antiferromagnetically (Sakuma, 1998a). The anti-parallel arrangement of the moments lowers the electronic energy by forming a pseudo-gap in the energy bands. On the
other hand, recent detailed theoretical and experimental investigations have disclosed that
several kinds of Mn alloys and compounds exhibit itinerant-electron characters (Edwards
and Bartel, 1972; Motizuki and Katoh, 1984, 1986; Katoh et al., 1986; Motizuki, 1987).
As a result, the Pauling valence plots of various data for Mn alloys and compounds exhibit
a nice straight line for not only localized moment systems but also itinerant-electron systems now. On the concept of the Pauling valence, let us try to plot the relation between the
experimental value of the magnetic moment and the Pauling valence. In the plots, the room
temperature values of the magnetic moments are used. Note that the Néel temperature is
high enough to ignore the temperature dependence of their magnitude (Yamaoka et al.,
1974), that is, the room temperature values of the magnetic moment could be considered
as the values at 0 K. From equations (6.2) and (6.3), the resultant data are added to the figure previously obtained by Mori and Mitsui (1968), together with the data of Mn3 Ir, Mn3 Pt
and Mn3 Rh, as given in fig. 6.26 (Umetsu et al., 2003b, 2004c). For Mn3 Pt and Mn3 Rh
alloys, the values of the magnetic moment are taken from neutron diffraction data (Krén
et al., 1967b; Yamaoka et al., 1974) and the lattice constants from Yamauchi et al. (1999)
and Sasao et al. (2003). In the case of Mn3 Ir, the calculated value of m was used. These
plots by using equations (6.2) and (6.3) and the experimental data in table 6.3 are fitted to
the universal line as given by several solid circles, though there some scatter is observed.
Fig. 6.26. Relation between the magnetic moment (m) and the Pauling valence (P V ) by using equations (6.2)
and (6.3) for several kinds of Mn alloys (Umetsu et al., 2003b, 2004c), together with various Mn alloys and
compounds (Mori and Mitsui, 1968). The close circles indicate the data of L12 -type Mn3 Ir (Yamaoka et al.,
1974), Mn3 Pt (Krén et al., 1966), Mn3 Rh (Krén et al., 1967b) and the L10 -type MnTM equiatomic alloys. The
magnetic moment and the lattice constant for Mn3 Ir, Mn3 Pt and Mn3 Rh alloys are taken from neutron diffraction
(Krén et al., 1967b; Yamaoka et al., 1974) and recent results (Yamauchi et al., 1999; Sasao et al., 2003). Note
that the values of PV were calculated by using the first nearest neighbor distance and coordination number in the
same manner as Mori and Mitsui (1968).
288
K. FUKAMICHI et al.
Inevitably, it is considered that the anomalous concentration dependence of the lattice constants, the electronic structure with the pseudo-gap and the Pauling valence associated to
the covalent bonding state are closely interrelated in the L10 -type MnTM equiatomic alloy
systems. Regarding to the magnitude of the magnetic moment, the theoretical values are
small, compared to the experimental values (see table 6.3). Therefore, there is some room
for further theoretical investigation.
7. Fundamental properties of practical Mn alloys
Being associated with the electronic structure, the stability of spin structures and the magnitude of the Néel temperature have been discussed in the preceding sections 5 and 6. In this
section, the low temperature specific heat and electrical resistivity are presented in order
to verify experimentally the calculated electronic structures. The electrical properties are
also related to the electronic structure and its magnitude is concerned with the electronic
characteristics in the antiferromagnetic layer of spin valves. Next, the relation between
the spin structures and the lattice distortion in the γ -phase Mn-Ir alloy and the magnetic
phase diagram of L10 -type MnPt alloy system are discussed from theoretical and practical
viewpoints. Finally, the magnetovolume effects and thermal strains are discussed because
multilayer films in spin-valves are influenced by a high level of strains. In other words,
the magnetic properties are sensitive to thermal strains which would be related with the
durability of spin-valves.
7.1. Low-temperature specific heat
The specific heat is expressed by the sum of the band, spin fluctuation and phonon terms
in the temperature region much lower than the Debye temperature θD as
C = Cband + Csf + Cph = γe T + AT 3 ,
(7.1)
where γe = γband + γsf is the electronic specific heat coefficient for the band and spin
fluctuation terms, respectively. The coefficient A is given by
A=
12π 4 R
5θD3
,
where R is the gas constant. Since the Néel temperature of γ -phase, L12 (γ )-phase, L10 phase alloys is very high and their antiferromagnetic interactions are strong, the value of
γsf becomes negligibly small. Therefore, γe is given by
(πkB )2
N (EF )
(7.2)
3
in units of states/eV · atom. The low-temperature specific heat coefficient is directly connected with the density of states which reflect the phase stability and magnetic state. In
addition, the electronic structure governs the electrical properties. Accordingly, the experimental data are useful for comparison with the theoretical electronic structures.
γe =
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
289
7.1.1. L12 -type (≡ γ -phase) ordered and γ -phase disordered alloys
The theoretical electronic states in γ -phase and L10 -type Mn alloy systems have been
discussed in the preceding section 5. It is characteristics of L10 -type ordered MnIr, MnPt,
MnPd and MnNi alloys to have a pseudo-gap in the electronic states. Therefore, the lowtemperature specific heat data would provide useful information on the electronic states.
Figure 7.1 shows the low-temperature specific heat Cp in the conventional form of Cp /T
vs. T 2 for γ -phase (DO) and L12 -type ordered (O) MnRh alloys (Umetsu et al., 2003a).
The value of γe in transition metals and alloys is usually ranging from 4 to 7 mJ/mol · K2 .
The specific heat coefficient γe proportional to the DOS at EF is obtained by a linear
extrapolation to T = 0. Similar results are presented for Mn100−x Rux , Mn75 Ir25 and Mn3 Ir
alloys in fig. 7.2 (Umetsu et al., 2005c).
Fig. 7.1. Temperature dependence of the specific heat Cp in the form of Cp /T − T 2 . The closed and open
circles indicate Cp of ordered (O) and disordered (DO) Mn3 Rh alloys, respectively. The closed and open triangles
indicate Cp of the ordered (O) and the disordered (DO) Mn3.20 Rh0.80 alloys, respectively (Umetsu et al., 2003a).
Fig. 7.2. Temperature dependence of the specific heat Cp in the form of Cp /T − T 2 for Mn100−x Rux , Mn75 Ir25
and Mn3 Ir alloys (Umetsu et al., 2005c).
290
K. FUKAMICHI et al.
7.1.2. L10 -type alloys
The plots for L10 -type Mn100−x TMx alloys are presented in figs. 7.3–7.6 (Umetsu et
al., 2003b, 2004a, 2005c, 2006). The linearly extrapolated value of γe is much smaller
than the data for γ -phase and L12 -type ordered MnRh alloys (see figs. 7.1, 7.2), consistent
with the pseudo-gap obtained by the band calculations given in figs. 5.17–5.20. A direct
confirmation of the pseudo-gap of L10 -type ordered MnPt alloy has been made by an
optical conductivity measurement (Kubota et al., 2006). The concentration dependence of
the value of γe of four kinds of L10 -type ordered alloy systems is given in fig. 7.7 (Umetsu
et al., 2003b). The values of γe for L10 -type ordered MnPt, MnPd and MnNi are very
small in the vicinity of the equiatomic composition. Strictly speaking, the minimum point
of the curve for MnPd alloy system is observed around 53% Pd, corresponding to the
maximum point of TN (see fig. 7.77). The experimental electronic specific heat coefficient
(γe in mJ/mol · K2 ), the calculated electronic specific heat coefficient (γecal in mJ/mol · K2 )
obtained by the DOS appear in table 7.1, together with the lattice constant ratio c/a, the
Debye temperature θD and the experimental TN (TN in K) and the calculated TN (TNcal in K)
Fig. 7.3. Temperature dependence of the specific heat Cp in the form of Cp /T − T 2 for L10 -type ordered Mn-Ir
alloys (Umetsu et al., 2004a).
Fig. 7.4. Temperature dependence of the specific heat Cp in the form of Cp /T − T 2 for L10 -type ordered MnPt
alloys (Umetsu et al., 2006).
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
291
Fig. 7.5. Temperature dependence of the specific heat Cp in the form of Cp /T − T 2 for L10 -type ordered MnPd
alloys (Umetsu et al., 2005c).
Fig. 7.6. Temperature dependence of the specific heat Cp in the form of Cp /T − T 2 for L10 -type ordered MnNi
alloys (Umetsu et al., 2005c).
Fig. 7.7. Concentration dependence of the low-temperature specific heat coefficient γe for L10 -type ordered
MnTM (TM = Ir, Pt, Pd and Ni) alloy systems (Umetsu et al., 2003b).
292
K. FUKAMICHI et al.
TABLE 7.1
The experimental electronic specific heat coefficient (γe in mJ/mol · K2 ), the calculated electronic specific heat
coefficient (γecal in mJ/mol · K2 ) obtained by the total density of states (DOS), the lattice ratio (c/a), the Debye
temperature (θD in K), the experimental Néel temperature (TN in K) and the calculated Néel temperature (TNcal
in K) given by the effective exchange constant J0 for L10 -type equiatomic MnTM (TM = Rh, Ir, Ni, Pd and Pt)
alloy systems (Umetsu et al., 2003b)
L10 -type alloy
γe
γecal
c/a
θD
TN
TNcal
MnRh
MnIr
MnPd
MnPt
MnNi
5.0
2.1
0.3
0.2
0.3
∼1.5
∼1.0
∼0.3
∼0.3
∼0.3
0.903
0.945
0.880
0.918
0.942
440
400
340
300
420
–
1145
810
970
∼1070
1330
1495
1087
973
1352
obtained from the effective exchange constant J0 for the equiatomic MnTM (TM = Ir, Pt,
Pd and Ni ) alloy systems (Umetsu et al., 2003b). Although γe of the MnIr alloy is twice
as large as γecal , these values are very small, compared with the data of ordinary transition
metals and alloys. As shown in fig. 7.8(a), the concentration dependences of γe and γecal
accord with each other, however, γe deviates from γecal in the concentration range less
than 50 at.% Pt (Umetsu et al., 2003b). The concentration dependence of the experimental
γe is sensitive in the concentration range less than 50 at.% Pt, compared with that above
50 at.% Pt, in analogy with the concentration dependence of TN (see fig. 7.72). It has been
pointed out that the contributions of spin fluctuations and the magnetic frustration often
enhance the electronic specific heat coefficient γe . The theoretical investigations for the
spin magnetic moment of the L10 -type MnPt alloys indicate that Mn atoms substituted
at the Pt site have comparatively large magnetic moments (see fig. 7.62). This situation
would induce magnetic fluctuations and results in some enhancement of γe value in the
concentration range less than 50 at.% Pt. For L10 -type MnPd, the minimum points in
both the experimental and theoretical curves in fig. 7.8(b) do not occur at the equiatomic
composition (Umetsu et al., 2003b) in analogy with the concentration dependence of the
Néel temperature (see fig. 7.77).
7.2. Electrical resistivity
The electrical resistivity ρ is closely associated with the electronic structure, magnetic state
as well as crystallographic state. Therefore, the electrical resistivity sensitively changes
against the concentration and temperature. In this subsection, the temperature and concentration dependences of ρ of L12 -type, γ -phase and L10 -type alloys are presented.
Practically, the magnitude of ρ is associated with the CIP and CPP currents in spin valves.
7.2.1. L12 -type (≡ γ -phase) ordered and γ -phase disordered alloy systems
The temperature dependence of electrical resistivity of L12 (≡ γ -phase) ordered MnRh
alloys (Umetsu et al., 2003a) is presented in fig. 7.9. The curves exhibit a monotonic
decrease below TN , which is associated with a usual magnetic order–disorder scattering.
Figures 7.10 and 7.11 show the temperature dependence of electrical resistivity of Mn-Rh
(Umetsu et al., 2003a) and Mn-Ir (Yamaoka, 1974) alloys, respectively, in the disordered
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
293
Fig. 7.8. Comparison the theoretical values with experimental values of low-temperature specific heat coefficient
for L10 -type alloy systems. (a) Mn100−x Ptx , (b) Mn100−x Pd (Umetsu et al., 2003b).
Fig. 7.9. Temperature dependence of the electrical resistivity ρ of L12 -type ordered (O) MnRh alloys (Umetsu
et al., 2003a).
294
K. FUKAMICHI et al.
Fig. 7.10. Temperature dependence of the electrical resistivity ρ of γ -phase disordered (DO) alloy MnRh alloys
(Umetsu et al., 2003a).
Fig. 7.11. Temperature dependence of the electrical resistivity ρ of γ -phase disordered (DO) Mn100−x Irx alloys
(Yamaoka, 1974).
state. Comparing the curves with those of in the ordered state in fig. 7.9, the residual resistivity is much larger than that of the ordered alloys and the change at TN is not so clear,
though TN is defined by confronting with the temperature dependence of the magnetic susceptibility. From figs. 7.9 and 7.10, the Néel temperature TN in the ordered state is higher
than that in the disordered state. Figure 7.12 shows the temperature dependences of magnetic susceptibility χ and electrical resistivity ρ for Mn74.4 Ir25.6 alloys in the disordered
(DO) and ordered (O) states (Yamaoka, 1974). The susceptibility curves in both states exhibit a clear anomaly associated with TN . On the other hand, the resistivity curve with
the degree of order S = 0.83 exhibits a clear bend at TN . In addition, the value of ρ the
disordered state exhibits no appreciable change at TN , though that in the disordered state
is much larger than that in the ordered state. Figure 7.13 shows the concentration dependence of electrical resistivity at room temperature (RT) for Mn100−x Irx and Mn100−x Rhx
(Yamaoka, 1974; Umetsu et al., 2003b, 2003c) in the ordered and disordered states. In the
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
295
Fig. 7.12. Temperature dependences of the magnetic susceptibility χ and the electrical resistivity ρ of
Mn74.4 Ir25.6 alloys in the γ -phase disordered (DO) and L12 -type ordered (O) states (Yamaoka, 1974).
Fig. 7.13. Concentration dependence of the electrical resistivity at room temperature (RT) for Mn100−x Irx and
Mn100−x Rhx (Yamaoka, 1974; Umetsu et al., 2003b, 2003c) in the L12 -type ordered and γ -phase disordered
states.
disordered state, ρ increases with increasing x due to the disruption of periodic potential
of Mn. On the other hand, the value of ρ in the ordered state takes a minimum at x = 0.25
because of existence of Mn3 Ir and Mn3 Rh ordered phases. It should be noted that the value
of ρ in the ordered state is much lower than that in the disordered state.
296
K. FUKAMICHI et al.
7.2.2. L10 -type alloy systems
The resistivity data of L10 -type ordered MnIr, MnPt, MnPd and MnNi alloy systems
are also shown in figs. 7.14–7.17 (Umetsu et al., 2003b, 2003c, 2004a, 2004b). As is wellknown, antiferromagnetic materials exhibit two types of the electrical resistivity ρ curve,
depending on the band structure at the Fermi level EF . One is a gap-type, showing a hump
below TN and the other is gapless-type which exhibits a steeper decrease below TN . In
general, the nature of ρ in antiferromagnets at TN depends on the periodicity and dimensionality of their magnetic structure. Using the following relation with the scaling of critical
exponents, αc , βc and γc , αc +2βc +γc = 2, the temperature derivative of resistivity dρ/dT
is expressed as (Suezaki and Mori, 1969; Rossiter, 1987)
dρ
= −Aε −(αc +γc )/2 + Bε −(αc +γc −1)
dT
(T < TN )
(7.3)
N
with ε = | T −T
TN |. Thus, the sign of dρ/dT at TN is determined by the magnitudes of the
Fig. 7.14. Temperature dependence of the electrical resistivity ρ of L10 -type ordered MnIr alloys (Umetsu et al.,
2003c, 2004a).
Fig. 7.15. Temperature dependence of the electrical resistivity ρ of L10 -type ordered MnPt alloys (Umetsu et
al., 2003b, 2003c, 2004b).
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
297
coefficients, A and B. In the case of the pseudo-gap type antiferromagnets, the first term
in equation (7.3) becomes dominant. In consequence, dρ/dT exhibits a minimum at TN
and the ρ–T curves is accompanied by a hump just below TN . In figs. 7.14–7.16, the arrows indicate TN defined from the minimum point in the dρ/dT –T curves (Umetsu et al.,
2002a, 2003b, 2003c, 2004a, 2004b, 2004d). Accordingly, the indicated Néel temperature
TN in figs. 7.14–7.16 for L10 -type Mn ordered alloys is in support of theoretical discussions. Additional anomalous behavior above TN observed in figs. 7.14 and 7.16 is caused
by the phase transformation from the L10 -type to the B2 structure. Furthermore, for the
MnNi alloy system, the transformation temperature is located below or above TN , depending on the concentration (Krén et al., 1968b). Therefore, the temperature dependence of
the curves is complicated as shown in fig. 7.17 (Umetsu et al., 2003b, 2005c). The anomalous temperature dependence of the solid curves is connected with the martensitic phase
transformation. The temperature dependence of electrical resistivity ρ and its derivative
Fig. 7.16. Temperature dependence of the electrical resistivity ρ of L10 -type ordered MnPd alloys (Umetsu et
al., 2002a, 2004d).
Fig. 7.17. Temperature dependence of the electrical resistivity ρ of L10 -type ordered MnNi alloys (Umetsu et al.,
2003b, 2005c).
298
K. FUKAMICHI et al.
Fig. 7.18. Temperature dependence of the electrical resistivity ρ and its derivative dρ/dT for L10 -type ordered
Mn45.5 Ni54.5 alloy (Umetsu et al., 2003b).
Fig. 7.19. Concentration dependence of the electrical resistivity ρ at 300 K for L10 -type ordered MnIr, MnPt,
MnPd and MnNi alloy systems, together with that at 4.2 K () for L10 -type ordered MnPt alloys (Umetsu et al.,
2003c).
dρ/dT for Mn45.5 Ni54.5 alloy are shown in fig. 7.18 as a representative curve (Umetsu et
al., 2003b). In this composition, TN is easily observed below the transformation temperature Tt , showing a hump just below TN in analogy with other L10 -type alloy systems.
Figure 7.19 shows the concentration dependence of electrical resistivity ρ at 300 K for
L10 -type MnIr, MnPt, MnPd and MnNi alloy systems (Umetsu et al., 2003c). It should be
pointed out that the data () at 4.2 K are similar to those () at 300 K as given in the same
figure for the MnPt alloy system.
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
299
Within the Drude scheme, the resistivity ρ0 at 0 K is given by
ρ0 ∝
Nσ (EF )τσ
−1
σ
∝
−
σ
Nσ (EF )
Im σ (EF )
−1
,
(7.4)
where N(EF ) and τ are the DOS at EF and the relaxation time, respectively, and σ denotes
the spin state. Here, σ (EF ) is the coherent potential at EF . Accordingly, the steep increase
in ρ0 in the off-equiatomic composition ranges can be connected with the decrease in
Nσ (EF ) and/or τσ when deviating from the equiatomic composition. The theoretical value
of Nσ (EF ) increases when deviating from the equiatomic composition as seen from fig. 7.7.
This leads us to infer that the increase of ρ0 is mainly attributed to a significant decrease
of the relaxation time τσ . Theoretically, − Im σ (EF ) is proportional to (τσ )−1 as given
in equation (7.4), and hence it is also proportional to ρ0 . Although it is difficult to obtain
the absolute value of ρ0 , − Im σ (ω) is obtainable from the CPA calculations, thus, it will
provide us the understanding of the qualitative concentration dependence of ρ4.2 . Strictly
speaking, if we define the coherent potential function as
p CPA (ω) =
ω − (ω)
,
(7.5)
where ω means the energy of electron. Under the CPA calculations, the following relation
is given as
− Im
(ω)
≈ Im p CPA (ω).
(7.6)
Figure 7.20(a) and (b) shows the representative results of the CPA calculations of
Im p CPA (ω) for the magnetic quantum number m = −1 in the Mn d-state for L10 -type
Mn45 Pt55 and Mn40 Pt60 alloys, respectively (Umetsu et al., 2006). In these figures, the
solid and dotted lines represent the results for the down- and up-spin states, respectively.
It is clear that Im p CPA (ω) at EF increases when deviating from the equiatomic composition. The CPA calculations were also performed for the magnetic quantum number m = 0
and −2 in the Mn d-state. For the compositions with 40 and 45% Pt, similar trend has been
confirmed. That is, the value of ρ0 relative to that of the equiatomic alloy of MnPt increases
when deviating from the equiatomic composition. It is well-known that the electrical resistivity of CuAu alloy system depends on the concentration and the degree of atomic order
(Rossiter, 1987).
In the disordered alloy system, the residual electronic resistivity ρ0 exhibits a broad
maximum at the equiatomic composition. On the other hand, it exhibits a minimum at Au
concentrations of 25 at.% and 50 at.% in the ordered alloy system (Johansson and Linde,
1936; Rossiter, 1987). In the L10 -type MnPt alloy system, the concentration dependence
of ρ0 is similar to that of CuAu ordered alloy, although the variation of ρ0 against to the Pt
concentration is significantly larger. The substitution effect of Mn element on the electronic
scattering would be more significant. In order to discuss the absolute value of ρ0 for this
alloy system, the effects on the electronic scattering of s- and p-states and s–d exchange
interactions should be taken into consideration.
300
K. FUKAMICHI et al.
Fig. 7.20. Imaginary part of the potential function from the CPA calculations ImpCPA as a function of energy ω
for the magnetic quantum number m = −1 in the Mn d-state. (a) Mn45 Pt55 and Mn40 Pt60 alloys (Umetsu et al.,
2006).
7.2.3. Spin valves
Consider now implication of resistivity in practical applications. Schematic spin-valve
structures, having the sense current in the plane (CIP-type) and the sense current perpendicular to the plane (CPP-type) are illustrated in fig. 7.21. The resistance R of CIP and
CPP-type elements is proportional to w/t and t/w, respectively, because the sense current
direction is given by the arrow. In spin valve devices, the electrical resistivity characteristic is important because a relatively thick antiferromagnetic film layer is necessary in
order to obtain the exchange bias-field, Hex , dominating the electrical properties of spinvalves (see fig. 7.22). Antiferromagnetic films with a high resistivity are favorable for the
CIP-type spin valves, whereas low resistivity is preferable for the CPP-type spin valves
(Seyama et al., 1999). In the light of these facts, the off-equiatomic composition films
would be useful for the CIP-type and the equiatomic composition films for the CPP-type
spin valves. To be more precise, we should settle the alloy composition after verifications
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
301
Fig. 7.21. Relation between the layer structure and the sense current for CIP and CPP-type spin valves.
Fig. 7.22. Layer stack of a synthetic spin valve (Lenssen et al., 1999).
of the film thickness and temperature dependences of ρ because the value of ρ in the thin
film state is markedly different from that in the bulk state. It is important to bear in mind
that this difference becomes more significant when reducing film thickness and for lower
temperature.
A layer stack of a synthetic spin valve is given in fig. 7.22 (Lenssen et al., 1999). A cap
layer is used to protect the film, a Cu spacer is for an enhanced MR ratio and thermal
stability, the FM/AFM part is replaced by an FM/X/FM trilayer called “artificial antiferromagnet” (AAF), which is named the synthetic antiferromagnet (Sy-AF). It should be
302
K. FUKAMICHI et al.
Fig. 7.23. Usable area map for the relation among the resistance-area product (RA), the MR ratio and the position
of actual CPP-GMR materials (Takagishi et al., 2002).
noted that the antiferromagnetic bias layer γ -MnIr is relatively thick in the layer stack.
For a replacement of γ -MnIr by L10 -type MnPt, we need a thicker layer film to induce
the exchange bias-field (see fig. 7.27). Therefore, it is considered that the magnitude of
the resistivity of the antiferromagnetic layer dominantly influences the GMR effect for the
CPP-type spin valve. Figure 7.23 shows the usable area map for the relation among the
resistance–area product (RA), the MR ratio and the position of the actual CPP-GMR materials (Takagishi et al., 2002). This figure obtained by micromagnetic simulations tells
the fact that we need to reduce the RA value to reach a high MR ratio for high density
recordings as indicated by the arrows from 100 Gb/in2 to 1 Tb/in2 .
7.3. Reflectivity and thermoelectric power of L10 -type MnPt alloys
The photon energy dependence of the reflectivity spectra at room temperature as a function
of Pt concentration for L10 -type Mnx Pt100−x alloys is shown in fig. 7.24 (Kubota et al.,
2006). The decrease of reflectivity in the vicinity of 400 meV becomes more significant
with coming close to the equiatomic composition. In general, the phonon frequency is
approximately less than 100 meV, and hence this depression of reflectivity is associated
with the pseudo-gap, consistent with the results given in fig. 7.7.
Figure 7.25 shows the temperature dependence of thermal conductivity σth for L10 -type
MnPt alloy (Umetsu and Fukamichi, 2004). The value of σth is expressed as
σth = σe + σph ,
(7.7)
where σe and σph are the electron and the phonon thermal conductivities, respectively. At
low temperatures, σth significantly increases with temperature in analogy with the heat
capacity. At high temperatures, however, the heat capacity becomes almost constant, following the Dulong–Petit law, and the relaxation time decreases because of the increase of
phonons. As a result, σth decreases at high temperatures. The contribution of σe is given by
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
303
Fig. 7.24. Photon energy dependence of the reflectivity spectra at room temperature as a function of the Pt
concentration for L10 -type Mnx Pt100−x alloys (Kubota et al., 2006).
Fig. 7.25. Temperature dependence of the thermal conductivity σth for L10 -type MnPt alloy (Umetsu and
Fukamichi, 2004).
the following expression associated with the Wiedemann–Franz law in high temperatures:
Ln T
(7.8)
,
ρ(T )
where Ln is the Lorenz number, and ρ is the electrical resistively. Above the Debye temperature θD , Ln has a constant value of 2.45 × 10−8 W /K2 because the elastic scattering of
electrons dominates in this temperature (Parrott and Stuckes, 1975). From equation (7.7),
the value of σe is estimated to be 42.8 W/m K by using ρ = 0.2 × 10−6 m in fig. 7.15.
This value is about 95% of σth . Accordingly, σe in equation (7.8) for L10 -type MnPt alloy is
dominant in the vicinity of 350 K. Shown in fig. 7.26 is the temperature dependence of the
σe =
304
K. FUKAMICHI et al.
Fig. 7.26. Temperature dependence of the Seebeck coefficient S for L10 -type MnPt alloy (Umetsu and
Fukamichi, 2004).
Seebeck coefficient, or the thermoelectric power, S for L10 -type MnPt alloy (Umetsu and
Fukamichi, 2004). Since the Debye temperature θD is 300 K from table 7.1, the anomalous
behavior around 0.2θD (∼60 K) in the S curve would be associated with the phonon drag.
At low temperatures, the lattice specific heat varies as T 3 below about 20 K, and hence
the phonon drag effect is ineffective. On the other hand, the mean free path of phonons
becomes so short that the phonon drag effect is unimportant at high temperatures. Generally, therefore, we can expect the peak to occur somewhere in the temperature range
0.1θD ≤ T ≤ 0.2θD . Another minimum around 150 K would also be originated from the
pseudo-gap structure in analogy with Fe2 VAl compound system (Nishino et al., 2001).
7.4. Magnetic anisotropy in γ -phase disordered alloys and L10 -type ordered alloys
Practically, a reduction of the antiferromagnetic layer is necessary for spin valve devices.
As discussed in subsection 7.2.3, the magnetocrystalline anisotropy should be large for
such purpose (see equation (7.11)). In this subsection, the magnetic anisotropy in γ -phase
disordered alloys and L10 -type ordered alloys is discussed.
7.4.1. Magnetic anisotropy in γ -phase disordered alloys
Figure 7.27 shows the unidirectional anisotropy constant Kd as a function of the AFM
layer thickness for exchange coupled bilayers with various AFM γ -phase disordered alloys, together with that of L10 -type ordered alloys (Tsunoda et al., 2002a, 2002b; Tsunoda
and Takahashi, 2004). The unidirectional anisotropy constant Kd given in this figure indicates that the critical thickness dc of γ -Mn-Ir AFM layer is the thinnest, being associated
with the magnitude of the magnetic anisotropy (see table 7.2). For L10 -type alloys, dc of
MnPt is smaller than that of MnNi, being consistent with the magnitude of the magnetocrystalline anisotropy constant (see table 7.3). The substrate and annealing temperature
dependences of the unidirectional anisotropy constant Kd of Mn73 Ir27 /Co70 Fe30 bilayers
is presented in fig. 7.28 (Imakita et al., 2004). The marked increase of Kd is closely associated with the deposited fiber structure and the appearance of Mn3 Ir ordered phase with a
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
305
Fig. 7.27. Unidirectional anisotropy constant Kd as a function of the AFM layer thickness for exchange coupled
bilayers with various AFM γ -phase disordered alloys, together with that of L10 -type ordered alloys (Tsunoda et
al., 2002b; Tsunoda and Takahashi, 2004).
TABLE 7.2
Symmetry, crystal plane and magnetic anisotropy energy of γ -phase Mn82 Ni18 and Mn75 Ir25
films, determined from magnetic torque analysis of exchange coupled NiFe/Mn-Ni and MnIr/NiFe (Tsunoda and Takahashi, 2004)
Symmetry
2θ
KAF
Anisotropy energy (erg/cm3 )
γ -Mn82 Ni18
γ -Mn75 Ir25
(110)
3.2 × 105
8.5 × 105
Four-fold
(001)
1.4 × 104
5.0 × 104
Six-fold
(111)
7 × 103
103 –104
Two-fold
4θ
KAF
6θ
KAF
Crystal
plane
TABLE 7.3
The electron number q, the magnetocrystalline anisotropy energy (MAE) and
the magnetocrystalline anisotropy constant KAFM of L10 -type MnPt, MnPd,
MnNi, MnRh and MnIr equiatomic alloys (Sakuma, 1998b; Umetsu et al.,
2005a)
L10 -type alloy
MnPt
MnPd
MnNi
MnRh
MnIr
Electron number
q/cell
MAE
(meV/cell)
KAFM
(106 J/m3 )
34
34
34
32
32
0.51
−0.57
−0.29
−0.63
−7.05
1.39
−1.53
−0.97
−1.85
−20.77
306
K. FUKAMICHI et al.
Fig. 7.28. The substrate temperature Tsub during the depositions dependence of the unidirectional anisotropy
constant Kd of Mn73 Ir27 /Co70 Fe30 bilayers as a function of annealing temperature Ta in an in-plane magnetic
field of 1 kOe (Imakita et al., 2004).
Fig. 7.29. Torque curves as a function of the applied in-plane field H and Mn74 Ir26 AFM layer thickness 3 and
4 nm (Takahashi and Tsunoda, 2002; Tsunoda and Takahashi, 2004).
high Néel temperature. Figure 7.29(a) and (b) shows the torque curves for below and above
the critical thickness, 3.7 nm, in the bilayer system of Ni79 Fe21 /Mn74 Ir26 as a function of
the applied in-plane field H and the AFM layer thickness dAFM = 3 and 4 nm (Takahashi
and Tsunoda, 2002). The ordinate represents the product tL, where t is the total film thick-
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
307
ness and L is the torque per unit film area. The abscissa gives the in-plane field angle
relative to the deposition field direction. When dAFM = 3 nm, the torque curves exhibit
sin θ characteristics in low fields below H = 250 Oe, but they display sin 2θ characteristics with increasing magnetic field, whereas a small sin θ component remains up to high
fields. In the case of dAFM = 4 nm, the torque curves exhibit sin θ characteristics in all
applied fields. The magnitude of the sin θ component gradually increases with increasing
field up to the critical field of 400 Oe, being saturated at a value of 0.12 dyn · cm/cm2 . This
value corresponds to the unidirectional anisotropy constant of the same film. The important
relation between the saturation amplitude of the torque and the anisotropy energy is given
as
⎫
2θ d
for sin 2θ ⎪
(tL)sat = 2KAFM
AFM
⎪
⎬
4θ d
.
(7.9)
for
sin
4θ
= 4KAFM
AFM
⎪
⎪
⎭
6θ d
= 6KAFM
for sin 6θ
AFM
The symmetry, crystal plane and the magnetic anisotropy energy determined from magnetic torque analyses of exchange coupled NiFe/γ -Mn82 Ni18 and NiFe/γ -Mn75 Ir25 alloys
are tabulated in table 7.2 (Tsunoda et al., 2002b). It should be noted that these values
obtained from the single spin model (Meiklejohn and Bean, 1956, 1957) do not refer to
the microscopic intrinsic magnetic anisotropy energy in the domain wall model (Mauri
et al., 1987). Furthermore, it is important to bear in mind that the moments in the AFM
are locked together under the magnetization process of the exchange-coupled bilayers because the evaluation mentioned above is based on the single spin model. In the domain
wall model which assumes the relative rotation of the AFM moments, and which would
be more appropriate than the single spin model for the exchange-biasing phenomena, the
nθ
do not exactly represent the magnetic anisotropy energy of the
obtained values of KAFM
nθ values mentioned above, they are pracAFM. However, the experimentally obtained KAFM
tically useful as the effective magnetic anisotropy energy of AFM.
Figure 7.30 shows the concentration dependence of the room temperature amplitude of
the 1θ, 2θ and 4θ components of the torque curves for γ -Mn100−x Ptx /NiFe bilayers grown
by applying a magnetic field along the γ -Mn100−x Ptx [100] or [110] direction (Kume et
al., 2004). The 1θ , 2θ and 4θ components correspond to the unidirectional anisotropy,
the uniaxial anisotropy and the cubic anisotropy, respectively. The 1θ component shows
a maximum value at x = 0.11, whereas the 2θ component is small and does not show a
strong concentration dependence. A large value of the 4θ component appears in the Pt concentration range where the θ component is relatively large. Additional experiments show
that the 4θ component is considerably small for NiFe/CrMgO. These facts strongly suggest that the 4-fold symmetry anisotropy originates from the antiferromagnetic γ -Mn89 Pt11
spin structure and/or from the interface between NiFe and antiferromagnetic γ -Mn89 Pt11
(Kume et al., 2004). On the other hand, the torque curves of γ -Mn80 Ir20 /NiFe bilayers are
dominated by only the 1θ component unlike γ -Mn89 Pt11 /NiFe bilayers, which could be associated with the difference between the magnetocrystalline anisotropy in γ -Mn80 Ir20 and
γ -Mn89 Pt11 . Shown in fig. 7.31 is the temperature dependence of the 1θ component (unidirectional anisotropy) and 4θ component (cubic anisotropy) obtained from torque curves
for γ -Mn80 Ir20 /NiFe bilayers grown epitaxially on MgO (001) (Kume et al., 2005). The
308
K. FUKAMICHI et al.
Fig. 7.30. Concentration dependence of the room temperature amplitude of the θ, 2θ and 4θ components of the
torque curves for γ -Mn100−x Ptx /NiFe bilayers grown by applying a magnetic field along the γ -Mn100−x Ptx
[100] or [110] direction (Kume et al., 2004).
Fig. 7.31. Temperature dependence of the 1θ component (unidirectional anisotropy) and 4θ component (cubic
anisotropy) obtained from torque curves for γ -Mn80 Ir20 /NiFe bilayers grown epitaxially on MgO (001) applying
magnetic field Hdep parallel to [100] and [110] directions (Kume et al., 2005).
squares and circles correspond to the 1θ and 4θ components, and the closed and open
symbols are for films grown by applying a magnetic field Hdep parallel to the [100] and
[110] directions, respectively. Both the layers show a similar temperature dependence. The
blocking temperature TB is about 570 K, in agreement with the reported result (van Driel
et al., 1999).
7.4.2. Magnetic anisotropy in L10 -type alloys
The magnetocrystalline anisotropy energy of L10 -type alloys has been calculated theoretically. For the calculations of the magnetocrystalline anisotropy energy (MAE), the
spin–orbit interaction is introduced in the LMTO Hamiltonian, following the method given
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
309
Fig. 7.32. The magnetocrystalline anisotropy energy (MAE) defined by E = E[100] − E[001] of the L10 -type
MnPt (Umetsu et al., 2005a), MnPd (Umetsu et al., 2005c) and MnNi (Sakuma, 1998b) alloys as a function of
electron number of q. The solid vertical line stands for the actual valence electron number in the formula unit.
by Andersen (1975) for the spin–orbit coupling and the spin–orbit matrix in the LMTO
Hamiltonian. By using the force theorem (Mackintosh and Andersen, 1980), the MAE is
obtained from the following expression.
E = E[100] − E[001],
(7.10)
where E[n] is the sum of the eigenvalues of the Kohn–Sham equation (Kohn and Sham,
1965) when the magnetic moment is parallel to the crystal axis [n] in the L10 -type structure. Figure 7.32 depicts the calculated results of magnetocrystalline anisotropy energy
(MAE) defined by equation (7.10) as a function of electron number q/cell of the L10 -type
MnPt alloy (Umetsu et al., 2005a), together with that of the L10 -type MnNi (Sakuma,
1998b) and MnPd (Umetsu et al., 2005c) alloys. The solid and dotted lines give the results
of MnPt and MnNi, respectively. The solid vertical line stands for the actual valence electron number in the cell. Although the MAE is quite sensitive to the calculation conditions,
the MAE curves as a function of q are meaningful for a comparison with other calculated
results. At the actual value of q (= 34), the MAE in MnPt exhibits a positive value, suggesting that the direction of the magnetic moments of Mn is parallel to the c-axis of the
L10 -type structure. On the contrary, the MAE of MnNi shows a negative value, indicating
that the magnetic moments of Mn lie in the (001) plane (Sakuma, 1998b). These results
are consistent with the spin structure determined from neutron diffractions (Kasper and
Kouvel, 1959; Pál et al., 1968). Furthermore, we should not overlook that the variations of
the MAE against q for MnPt is more sensitive than that of MnNi. Especially, in the vicinity
of the actual value of q (= 34), the sign of the MAE changes from negative to positive.
This fact implies that the sign of the MAE, in other words, the direction of the magnetic
moment can be easily changed, depending upon the changes of the temperature and/or the
concentration (see fig. 7.72). From fig. 7.32, the value of MAE at the actual value of q
(= 34) is estimated to be about 0.51 meV/cell. This leads about 1.39 × 106 J/m3 of the
magnetocrystalline anisotropy constant KAFM , and this value is larger in magnitude but
with opposite sign than that of MnNi of about −0.97 × 106 J/m3 (Sakuma, 1998b). The
310
K. FUKAMICHI et al.
Fig. 7.33. The magnetocrystalline anisotropy energy (MAE) defined by E = E[100] − E[001] of the L10 -type
MnRh (Umetsu et al., 2005a) and MnIr (Umetsu et al., 2005c) alloys as a function of electron number of q. The
solid vertical line stands for the actual valence electron number per formula unit.
value of MnPd is also negative and is larger than that of MnNi. The difference could originate from the large spin–orbit coupling of the Pd atom. In fig. 7.33, the calculated results
using the same procedure are illustrated for L10 -type MnIr and MnRh alloys (Umetsu et
al., 2005a, 2005c). These two alloy systems also exhibit a negative value, consistent with
their spin structure. Note that the ordinate is different in magnitude, compared with that of
in fig. 7.32. The value of MnRh is about twice that of MnNi with the same sign. It will be
useful to note that the value of MnIr is the largest in the five alloy systems.
In table 7.3, we summarize the anisotropy constant KAFM for these five L10 -type
equiatomic alloys (Sakuma, 1998b; Umetsu et al., 2005a), together with the electron number, the magnetocrystalline anisotropy energy (MAE) defined by E = E[100] − E[001]
and the magnetocrystalline anisotropy constant KAFM . It is very meaningful to investigate
the MAE of antiferromagnetic material in the field of spintronics. The exchange-bias field
which is one of the characteristic properties of the spin-valve-type multilayers is thought
to be closely correlated with the MAE of the antiferromagnetic layer, although the mechanism of the exchange coupling between the ferromagnetic and the antiferromagnetic layer
is controversial. Some models assume that the exchange-bias field Hex is due to the formation of an antiferromagnetic domain-wall (Mauri et al., 1987; Koon, 1997). For some of
the exchange bias-field models, Hex is given as follows.
Hex ∝
AAFM KAFM ,
(7.11)
where KAFM and AAFM are the magnetocrystalline anisotropy constant and the exchange
stiffness of the antiferromagnets, respectively. Accordingly, it can be said that the antiferromagnets with a large magnetocrystalline anisotropy bring about a large Hex .
7.5. Spin structures and lattice distortions of γ -phase disordered alloys
Mn-based γ -phase disordered alloys exhibit various lattice distortions, depending on the
concentration and temperature. The lattice distortions seem to be connected with the spin
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
311
Fig. 7.34. Phase diagram of γ -phase disordered Mn-Ni alloy system (Honda et al., 1976).
structures. However, detailed investigations disclose that such distortions are not always
accompanied by the spin structure change.
7.5.1. Spin structure and magnetic susceptibility
In the Mn100−x Nix alloy system, lattice distortions of the γ -phase, depending on the
concentration and temperature, were demonstrated by several authors (Hicks et al., 1968;
Uchishiba et al., 1969; Honda et al., 1976). As seen from fig. 7.34, the cubic phase is retained down to low temperatures in the concentration range x > 22% Ni, although the Néel
temperature is practically not so high. With decreasing Ni concentration, lattice distortions
take place as given in the same figure. In the orthorhombic phase, the order of size of the
lattice constants becomes opposite, depending on the Ni concentration, that is, a < b < c
in the concentration close to the T2 line and a > b > c in the concentration range close to
the T1 line (Honda et al., 1976). Jo and Hirai (1986) gave a phenomenological model for
the relation between these lattice distortions and the spin configurations. Extending their
discussion, Fishman and Liu (1998, 1999) have theoretically calculated the temperature
dependence of magnetic susceptibility by using the random phase approximation (RPA)
for the three kinds of spin structures illustrated in fig. 5.2, namely, the single (1Q), double (2Q), and triple (3Q)-spin density wave (SDW) structures. Following their theory, we
can determine the spin structure change. That is, the Hamiltonian composed of kinetic energy of the electrons, the Coulomb interaction between the N spins and the lattice, and the
interaction with the external magnetic field can predict the spin excitations in each SDW
structure. In addition, the RPA has been used to solve the equation for the GFs in zero field.
When T → 0, the averaged magnetic susceptibilities χav are given as
⎫
V
5
χav → μ2B ρeh
(for 1Q)⎪
⎪
⎪
⎪
6 N
⎪
⎪
⎪
⎪
2
⎬
V
1
z
3
0
2
+
(for 2Q) ,
χav → μB ρeh
(7.12)
⎪
N
4 48 ⎪
⎪
⎪
⎪
⎪
⎪
V
1 z0 2
13
2
⎭
χav → μB ρeh
+
(for 3Q)⎪
N
18 36 312
K. FUKAMICHI et al.
Fig. 7.35. Average χav versus T /TN for 1Q, 2Q and 3Q states with (a) Z0 = 0 and (b) Z0 = 4TN∗ (Fishman and
Liu, 1999).
where V , ρeh , z0 and are the volume, the density of states of the electron and the holes,
the energy mismatch parameter of the Fermi momentum and the energy gap, respectively.
Although the relative magnitude of χav depends on the size mismatch between the Fermi
surfaces, one of the most useful results is that the value of χav is different among three
configurations even when averaged over all possible spin domains. Therefore, the magnetic
susceptibility exhibits a clear change at the temperature where the spin structure changes.
Figure 7.35(a) and (b) presents numerical results for the temperature dependence of the
magnetic susceptibility of average χav values normalized to TN for 1Q, 2Q and 3Q spin
structures under the condition of z0 = 0 and 4TN∗ , respectively. When a perfect nesting
z0 = 0, and then the electron Fermi surface is larger (smaller) than the hole Fermi surface,
z0 is negative (positive). The Néel temperature of a perfectly nested alloy is given by
2γ
ε0 e−8/Uρeh ,
(7.13)
π
where ln γ ∼ 0.577 is Euler’s constant, ε0 is the cutoff in the quasiparticle energy spectrum and U is the Coulomb interaction. The actual TN will be much lower than because
the surface nesting is imperfect with z0 = 0. At the triple point where the commensurate
SDW structures become unstable to an incommensurate SDW structure at TN , the value
of z0 is given by z0 = 4.29TN∗ (Fishman and Liu, 1999). As shown in fig. 7.35(b), the
TN∗ =
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
313
Fig. 7.36. Temperature dependence of magnetic susceptibility χ of disordered (DO) Mn86 Rh14 and Mn90 Rh10
alloys (Yamauchi et al., 2000a).
Fig. 7.37. Phase diagram of γ -phase disordered (DO) MnRh alloy system (Yamauchi et al., 2000a).
relative magnitudes of χav for the larger Fermi surface mismatch are reversed. In addition, the values of χav merge into one curve close to TN and the temperature range over
which χav is approximately linear near TN is narrower for z0 = 4TN∗ than for z0 = 0.
The temperature dependence of magnetic susceptibility χ of disordered (DO) Mn86 Rh14
and Mn90 Rh10 alloys is given in fig. 7.36 (Yamauchi et al., 2000a). In these curves, two
anomalies are observed, that is, the temperatures TN and Tt are the Néel temperature and
the phase transformation temperature, respectively.
7.5.2. Phase diagram and lattice distortion of Mn-Rh, Mn-Ga and Mn-Ru alloy systems
The phase diagram of Mn-Rh alloy system in fig. 7.37 (Yamauchi et al., 2000a) is similar to that of the Mn-Ir alloy system (see fig. 7.46), but an additional phase with a > c
appears in the Mn-Rh system. The spin structure is 3Q in the case of a = c, and the other
structures depend on the lattice distortion. It is important to stress that the 2Q spin structure
of Mn90 Rh10 with a < c, the 1Q spin structure with a > c are in good consistency with
314
K. FUKAMICHI et al.
Fig. 7.38. Relation among the lattice distortion, the magnitude of magnetic moment and multiple spin structures
1Q, 2Q and 3Q in the Mn-based alloy system (Fishman and Liu, 1998, 1999; Yamauchi et al., 2000a).
Fig. 7.39. Temperature dependence of neutron diffraction intensities of I110 and I101 for γ -Mn89 Rh11 alloy
(Hori et al., 2001).
the neutron diffraction data (Hori et al., 2001) and also with theoretical calculations (Jo
and Hirai, 1986). In addition, another type of 2Q magnetic structure 2Q given in fig. 7.38
is formed in the concentration range of a > b > c (Yamauchi et al., 2000a). The relation
among the lattice distortion, the magnitude of magnetic moment and spin structures are
illustrated in fig. 7.38 (Fishman and Lui, 1998, 1999; Yamauchi et al., 2000a). Figure 7.39
shows the temperature dependence of the neutron diffraction intensities I110 and I101 for
γ -Mn89 Rh11 alloy. Note that the given nominal composition Mn90 Rh10 (Hori et al., 2001)
corresponds to Mn89 Rh11 (Umetsu and Fukamichi, 2000a). The total magnetic intensity
I110 + I101 exhibits a Brillouin function-like decrease without any disturbance at the distortion temperature T2 and vanishes at the Néel temperature TN . A significant 110 reflection
below T2 means that the magnetic moment lies in the (001) plane with μC = 0 and the
absence above T2 indicates that μC = 0 in the (001) plane (Hori et al., 2001). The magnetic phase diagram in fig. 7.40 is correlated with the lattice distortion for the Mn-Ga alloy
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
315
Fig. 7.40. Lattice distortion and the magnetic phase diagram of the Mn-Ga alloy system (Hori et al., 1995).
Fig. 7.41. Neutron diffraction intensities of I110 , I101 and I110 + I101 as a function of temperature for the
Mn81 Ru14 Cu5 alloy system (Hori et al., 2001).
system (Hori et al., 1995). The distortion sequence is very similar to that in fig. 7.37. In
addition, it is important from a practical point of view that the Néel temperature is scarcely
affected by the addition of Ga. The temperature dependence of neutron diffraction intensities I110 , I101 and I110 + I101 is shown in fig. 7.41 for Mn81 Ru14 Cu5 alloy (Hori et al.,
2001). A small amount of Cu is added to stabilize the γ -phase. The total intensity shows a
monotonic decrease without a disturbance at the crystallographic transition temperature T2
and then becomes zero at TN . These behaviors are very similar to those of Mn89 Rh11 alloy
(see fig. 7.39). The magnetic phase diagram of the Mn96−x Rux Cu4 alloy system given in
fig. 7.42 is associated with the lattice distortion (Hori et al., 2001). The lattice distortion
sequence is very similar to that of the Mn100−x Rhx alloy system (see fig. 7.37).
316
K. FUKAMICHI et al.
Fig. 7.42. Lattice distortion and the magnetic phase diagram of the Mn96−x Rux Cu4 alloy system (Hori et al.,
2001).
Fig. 7.43. Concentration dependence of the lattice constants of the Mn-Ir alloy system (Sasao et al., 2003),
together with the data obtained by Yamaoka (1974).
7.5.3. Phase diagram and spin structures of Mn-Ir alloy system
The concentration dependence of the lattice constants of the Mn-Ir alloy system is plotted in fig. 7.43. Recent results (Sasao et al., 2003) are slightly different from those reported
by Yamaoka (1974). One may notice that chemical analyses tend to cause a slight underestimation of the Ir concentration because the alloys including Ir atoms are not so easy to
dissolve completely (Sasao et al., 2003). In high concentration ranges, right of the dashed
line, disordered alloys form an fcc structure. In contrast, the lattice distorts to the fct (c > a)
structure below about 15 at.% Ir. Figure 7.44 shows the representative X-ray diffraction
data measured at various temperatures for γ -phase disordered Mn85.2 Ir14.8 alloy (Sasao et
al., 2003). The 002 and 200 peaks identified as the fct (c > a) structure are observed at
low temperatures. With increasing temperature, the 002 and 200 peaks are progressively
close together, indicating that the c/a ratio becomes gradually close to 1, as expected for
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
317
Fig. 7.44. X-ray diffraction data measured at various temperatures for γ -phase disordered (DO) Mn85.2 Ir14.8
alloy (Sasao et al., 2003).
the fcc structure. The fct–fcc structural transition temperature Tfct/fcc (≡ Tt ) is determined
to be located between 275 and 285 K by Rietveld analysis of the c/a value.
The temperature dependence of magnetic susceptibility χ of the disordered
γ -Mn100−x Irx alloys with x = 14.8 and 15.5 is shown in fig. 7.45(a) and (b) (Sasao et
al., 2003). At the transition temperature from the high-temperature 3Q spin structure to
the low-temperature 2Q spin structure, T2Q/3Q , the susceptibility suddenly increases or
decreases by a few percent, depending on the size difference between the electron and
hole Fermi surfaces (Fishman and Liu, 1999). The point to observe is that a clear decrease
takes place below T2Q/3Q as seen from fig. 7.45(a), whereas no anomaly in the temperature
dependence of magnetic susceptibility is observed at the crystallographic transition temperature Tt from the fcc to the fct phase, Tfct/fcc (≡ Tt ) determined by X-ray diffraction,
as given in fig. 7.45(a) and (b). Accordingly, we ascertain that the lattice distortion is not
simultaneously accompanied by a spin structure change. Figure 7.46 constitutes the phase
diagram of the disordered Mn-Ir alloy system quenched from 1273 K (Sasao et al., 2003),
together with the data obtained by Yamaoka (1974). As given by the dashed line, the Néel
temperature TN of γ -Mn is extrapolated to be 480 K (see fig. 6.5). On the whole, the composition studied by Yamaoka (1974) for the γ -phase is slightly lower than in recent data
for the reason explained in connection with fig. 7.43. It is meaningful to note that epitaxial
growth characteristics of Mn-Ir films drastically change around 20% Ir (Fuke et al., 1997),
being very close to the phase transformation. The stabilization of the γ -phase below several percent is very difficult (see fig. 7.52), accompanied by the transformation to a β-phase
(Miyakawa et al., 2003a, 2003b). Therefore, the concentration range of a > b > c in the
Mn-Ir alloy system is strictly restricted to be narrow. For γ -phase Mn-Ir alloys, the relation
between the electronic total energy and the spin structure has been investigated (Sakuma
et al., 2003). To investigate systematically the variation of energy with the change of the
318
K. FUKAMICHI et al.
Fig. 7.45. Temperature dependence of the magnetic susceptibility χ of disordered (DO) γ -Mn100−x Irx alloys
with (a) x = 14.8 and (b) 15.5 (Sasao et al., 2003).
Fig. 7.46. Phase diagram of the disordered (DO) Mn-Ir alloy system (Sasao et al., 2003), together with the data
reported by Yamaoka (1974).
magnetic structure, the electronic total energy for several compositions of x in the γ -phase
disordered Mn100−x Irx alloys is plotted in fig. 7.47 as a function of angle θ defined in
fig. 5.1 (Sakuma et al., 2003). Note that θ = 0◦ , 90◦ and 54.7◦ correspond to the 1Q, the
2Q and the 3Q structures, respectively, as explained in connection with fig. 5.2. The lattice
constant is fixed at a = 0.3785 nm and the axial ratio c/a is also fixed at unity for simplicity. The curve of x = 0 indicates that the 2Q structure (θ = 90◦ ) is the most stable and the
3Q structure (θ = 54.7◦ ) has a maximum energy in analogy with previous results for pure
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
319
Fig. 7.47. The θ dependence of the electronic total energy for x = 0, 5, 15 and 25 in the γ -phase disordered
(DO) Mn100−x Irx alloys. The lattice constant is fixed at a = 0.3785 nm (Sakuma et al., 2003).
Fig. 7.48. Concentration dependence of the electronic total energy of the γ -phase disordered (DO) Mn100−x Irx
alloys for the 2Q and the 3Q structures. The total energy is given by the value relative to that of the 1Q structure
(Sakuma et al., 2003).
γ -Mn metal (Sakuma, 1998a). Noteworthy is that only 5% addition of Ir atoms makes the
energy of the 3Q structure lower than that of the 1Q structure. The energies both of the 2Q
and the 3Q structures are lowered relative to that of the 1Q structure with increasing x.
Figure 7.48 represents the concentration dependence of the total energy of the 2Q and the
3Q structures relative to that of the 1Q structure (Sakuma et al., 2003). The total energies
both of the 2Q and the 3Q structures exhibit a monotonic decrease with x. The slope is
steeper in the 3Q structure than that in the 2Q structure, intersecting at the concentration
x between 10 and 15. Thus, the 2Q structure is stable until x of about 13 and the 3Q
320
K. FUKAMICHI et al.
structure is found to be realized for x > 13. Returning to fig. 7.47, we can confirm that
there is no intermediate state between the 2Q and the 3Q structure as the ground state,
that is, the curves exhibit no minimum in the range 54.7◦ < θ < 90◦ , and hence the
transition is of first-order. The spin structure changes from the 2Q to 3Q state around
13% Ir under the condition of c/a = 1. Emphasis to be made in fig. 7.48 is that the Ir
concentration where the magnetic transition from the 2Q to 3Q structure takes place is
quite close to the concentration at which the axial ratio changes from c/a > 1 to c/a = 1.
It has been predicted that even in pure γ -Mn metal, the lattice distortion of c/a > 1
further lowers the total energy of the 2Q structure (Sakuma, 1998a). In a similar manner as
above, the total energy of the 2Q structure in γ -phase disordered Mn85 Ir15 alloy becomes
lower than that of the 3Q structure when the lattice is distorted to c/a = 1.05, whereas
the 3Q structure in γ -phase disordered Mn80 Ir20 alloy is still stable under the condition
of c/a = 1. Thus, it can be inferred that the critical concentration defined with xm for
the magnetic transition is pushed forward to a higher concentration of a few percent if
the constraint for the lattice distortion is free. It should be remarked, however, that the
concentration xm does not necessarily coincide with the concentration defined by xt for
the lattice distortion. This can be understood by noting a situation that γ -phase disordered
Mn80 Ir20 alloy having the 3Q structure is found to prefer c/a > 1 energetically rather than
c/a = 1, while for γ -phase disordered Mn75 Ir25 alloy the lattice keeps c/a = 1 with the
3Q structure. From these relations, one can expect that xt is higher than xm . This is because
the lattice distortion of c/a > 1 lowers not only the total energy of the 2Q structure but
also that of the 3Q structure. When the total energy of the 3Q structure is low enough
for c/a = 1, any degree of lattice distortion cannot always cause a reversal from the 3Q
to the 2Q magnetic structure (Sakuma et al., 2003). From the discussions based on the
experimental data and theoretical calculations, it is evident that the lattice distortions do
not always result in a change of the spin structures.
It is meaningful to note that excellent spin valve characteristics of γ -phase disordered
Mn-Ir alloys are not obtainable below about 20% Ir (Hoshino et al., 1996; Fuke et al.,
1997), very close to the composition in which the lattice distortion and also the spin structure change take place as seen from fig. 7.48 (Sasao et al., 2003), implying that exchange
biasing characteristics are sensitive to the spin structure (see fig. 8.9). It is readily expected
that the spin structure of γ -phase disordered alloys is easily modified by lattice defects,
magnetovolume effects and so on, because the energy difference between 1Q and 2Q, 2Q
and 3Q states is not so significant (see figs. 5.12 and 5.13).
7.6. Appearance of the β-phase and stabilization of the γ -phase
The smaller the d-electron number at the Mn site, the higher the Néel temperature as given
in subsection. However, it is metallurgically worth noting that the stable γ -phase range
becomes narrower with decreasing outer electron number. For example, γ -phase MnRu
alloys are unstable and easily bring about a phase separation with a β-phase having a low
Néel temperature below room temperature after low-temperature annealing (Sasao et al.,
2001). As seen from fig. 6.5, the γ -phase MnRu is obtainable only in the concentration
range from 30 to 36% Ru by quenching from 1373 K, in contrast to the Mn–Ru phase diagram (Raub and Mahler, 1955). For example, γ -phase Mn69 Ru31 decomposes into β + γ
phases by annealing at 900 K. However, the more stable γ -phase is obtained by addition
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
321
Fig. 7.49. Temperature dependence of electrical resistivity of γ -phase MnRhRu ternary alloys (Sasao et al.,
2005).
Fig. 7.50. Temperature dependence of the magnetic susceptibility for γ -phase Mn71.2 Ru22.4 Ga6.4 with a = c
(Umetsu et al., 2005d).
of Rh. Figure 7.49 shows the heating and cooling resistivity curves of γ -phase MnRuRh
alloys with a = c (Sasao et al., 2005). The γ -phase Mn70 Ru20 Rh10 is stabilized, even
though the Rh concentration is much lower than 30% Ru. It should be noted that γ -phase
Mn80 Ru15 Rh5 and Mn70 Ru25 Rh5 alloys are also stabilized up to about 1100 K without
the phase decomposition into β + γ phases (Sasao et al., 2005). From the standpoint of
practical applications, the stabilization of γ -phase MnRu alloys should be accomplished
in order to obtain a high Néel temperature. For this alloy system, therefore, no excellent
exchange bias-field characteristics are obtained without the addition of Rh (Araki et al.,
1998a, 1998b; Shimazawa et al., 1999; Tsuchiya et al., 2000). In order to obtain a relatively high Néel temperature TN by using economically cheap elements, the Mn-Ru-Ga
alloy system has been investigated. Figure 7.50 shows the temperature dependence of the
322
K. FUKAMICHI et al.
Fig. 7.51. Mn-Os phase diagram in the Mn-rich range (Yamauchi et al., 2000b).
magnetic susceptibility for a stabilized γ -phase Mn71.2 Ru22.4 Ga6.4 with a = c (Umetsu et
al., 2005d). The Néel temperature TN is estimated to be about 530 K, comparable with that
of FeMn (Endoh and Ishikawa, 1971) which has been investigated as an exchange biasing
film (Kamiguchi et al., 1996; Gurney et al., 1997; Schwartz et al., 1998). The value of TN
of Mn-Ru-Ga alloy system is higher than that of pure γ -Mn metal (see fig. 6.5) and MnGa binary alloy system (see fig. 7.40), comparable that of Mn-Ru-Cu alloy system (see
fig. 7.42). In the case of further smaller outer electron number, say, in the Mn-Os binary
alloy system, the β-phase concentration range becomes wide, thus no γ -phase is formed
as seen from fig. 7.51 (Yamauchi et al., 2000b), although it has been pointed out that γ Mn77 Os23 alloy used for magnetic tunnel junctions is stable up to temperatures exceeding
670 K (Parkin and Samant, 1999). In the figure, the symbols, ◦, and represent the β,
α and α + β phases, respectively. It should be emphasized that the β-phase is stable in a
wide range of Os concentration. Figure 7.52 is the phase diagram of the Mn-Ir alloy system
(Raub and Mahler, 1955; Miyakawa et al., 2004). Several kinds of symbols •, , and
respectively stand for the β, β + γ , γ (DO) and γ (O) phases. The β-phase is formed
in a wide concentration range, expelling the γ -phase (DO) which has been investigated
actively because the exchange bias-field is obtained by the thinnest film among antiferromagnetic films (see fig. 7.27). In this alloy system, the concentration range of the β-phase is
much wider than that in the previously reported phase diagram (Raub and Mahler, 1955).
It should be recalled that the excellent biasing characteristics are obtained above about
20 at.% Ir (Fuke et al., 1997) without the phase decomposition from the γ -phase into the
mixed phases of γ + β after annealing.
Pure β-Mn does not order in the antiferromagnetic state down to the lowest temperatures (Nakamura and Shiga, 1997). The crystal structure of β-Mn is illustrated in fig. 7.53.
The unit cell has a simple cubic structure with 20 atoms, and there are two kinds of Mn
sites. The Mn atoms at the site 1 are stabilized in the antiferromagnetic state by additional
elements. On the other hand, the Mn atoms at site 2 form corner sharing triangles lying in
the (111) plane, resulting in the magnetic instability (Nakamura and Shiga, 1997). That is,
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
323
Fig. 7.52. Phase diagram of Mn-Ir alloy system (Raub and Mahler, 1955; Miyakawa et al., 2004).
Fig. 7.53. Crystal structure of β-Mn. Note that there are two kinds of Mn sites of 8c and 12d.
each triangle is perpendicular to one of the [111] axes such that neighbor triangles from an
angle φ satisfy cos φ = 1/3, and accordingly the sublattice of Mn atoms at site 2 becomes
a three-dimensional Kagomé lattice (Hafner and Hobbs, 2003). The site preference of additional elements is tabulated in table 7.4. Note that the 8c site is occupied with transition
and noble metals. A number of papers have been published for β-Mn alloys. The Néel
temperature TN has been determined from conventional methods such as the temperature
dependence of magnetic susceptibility. However, careful determinations are necessary because the magnetic properties of β-Mn alloys are conceivably complicated by the presence
of spin frustrations.
Shown in fig. 7.54 is the temperature dependence of the magnetic susceptibility χ of
β-Mn100−x Osx (Miyakawa et al., 2001). The anomaly in the curves shifts to higher temperature ranges with increasing Os concentration. Since magnetic properties of β-Mn alloys
are really complicated, careful direct observations are necessary to confirm the long range
magnetic order, that is, whether this anomaly corresponds to the Néel temperature or not.
Figure 7.55(a) and (b) shows the neutron diffraction profiles at 295 and 10 K, respectively,
for β-Mn75 Os25 alloy (Miyakawa et al., 2005). Comparing both the figures, the diffraction
324
K. FUKAMICHI et al.
TABLE 7.4
Site preference of substitutional elements in various β-Mn alloy systems
Site
Element
Reference
Site 1
(8c)
Fe
Co
Ni
Ru
Os
Ir
(1)–(5)
(1), (2), (6), (7)
(1), (2), (8)
(9)
(10)–(13)
(14)
Site 2
(12d)
Al
Sn
In
(15)–(21)
(3), (15), (22)–(24)
(6), (15)
(1) Shinkoda et al. (1979), (2) Oyamatsu et al. (1989), (3) Nishihara et al. (1977), (4) Kimball
et al. (1967), (5) Kimball and Sill (1970), (6) Funahashi and Kohara (1984), (7) Hori
(1975), (8) Whittaker et al. (1971), (9) Sasao et al. (2001), (10) Yamauchi et al. (2000b),
(11) Miyakawa et al. (2001), (12) Miyakawa et al. (2003b), (13) Hama et al. (2004),
(14) Miyakawa et al. (2003a), (15) Kohori et al. (1996), (16) Nakamura et al. (1997),
(17) Nakamura and Shiga (1997), (18) Stewart et al. (1998), (19) Stewart and Cywinski
(1999), (20) Mekata et al. (1999), (21) Stewart and Cywinski (2004), (22) Dunlop et al. (1977),
(23) Nakai (1994), (24) Nakai (1998).
Fig. 7.54. Temperature dependence of magnetic susceptibility of χ of β-Mn100−x Osx (Miyakawa et al., 2001).
profile at 10 K in (b) evidently exhibits additional peaks of magnetic reflections due to the
antiferromagnetic long-range order. Therefore, the anomalous temperature dependence of
magnetic susceptibility in fig. 7.54 is associated with the Néel temperature, and hence it is
decisive that β-Mn alloys in which site 1 is occupied by transition metals such as Ir, Ru
and, Os as well as Fe, Co and Ni are antiferromagnetic. Similar neutron diffraction data
have been reported for β-Mn type Mn3 IrSi in which the 8c site is split into two sites of Ir
and Si and the Néel temperature was determined to be 210 K from the temperature dependence of the magnetic susceptibility (Eriksson et al., 2004). The concentration dependence
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
325
Fig. 7.55. The neutron diffraction profiles at 295 (a) and 10 K (b) for β-Mn75 Os25 alloy (Miyakawa et al., 2005).
of the Néel temperature TN is given in fig. 7.56 for several kinds of β-Mn100−x TMx alloy
systems (Shinkoda et al., 1979; Miyakawa et al., 2003b). The values of TN increase with
increasing concentration of additional elements of Ir, Ru and Os. Unfortunately, however,
such β-Mn alloys are not practical because their Néel temperature is about 200 K at most,
much lower than room temperature.
7.7. Spin fluctuation effects in β-Mn alloys
According to the self-consistent renormalization (SCR) theory by Hasegawa (1975) for
itinerant-electron weak antiferromagnets, the low-temperature specific heat coefficient γe
is given by
γe = γband + γsf
3/4
with γsf = A − BTN ,
(7.14)
where A and B are the constant coefficients. The specific heat coefficient enhanced spin
fluctuations γsf is expressed as
3N0
1
γsf =
(7.15)
1+ 2 ,
4T0
K0
326
K. FUKAMICHI et al.
Fig. 7.56. Concentration dependence of the Néel temperature TN for several kinds of β-Mn100−x TMx alloys
(Miyakawa et al., 2003b; Shinkoda et al., 1979).
where N0 is the number of magnetic atom, T0 and K0 are spin fluctuation parameters. In
itinerant-electron weak antiferromagnets, the spin fluctuation modes have strong intensities
around the antiferromagnetic vector Q. As a result, both 1/T0 and 1/K0 become large
(Konno and Moriya, 1987), accompanied by a large value of γsf . The specific coefficient
of the band term is given by
1 2 2
1
(7.16)
π kB (1 + λ)N(EF ) ∼ π 2 kB2 N (EF ),
3
3
where λ is the electron–phonon coupling constant and N (EF ) is the density of states
at the Fermi level. As seen from figs. 7.2–7.7, the value of γband is generally less than
3/4
10 mJ/mol · K2 (Mori et al., 1980). The relation between γe and TN is given in fig. 7.57
for several kinds of β-Mn100−x TMx alloy systems in which the additional atoms preferentially occupy site 1 (Shinkoda et al., 1979; Sasao et al., 2001; Miyakawa et al., 2001;
Miyakawa et al., 2005). Mössbauer spectroscopy results for β-MnFe and β-MnCo alloys
tell us that Fe and Co atoms have no localized moment (Nishihara et al., 1977). Therefore,
Fe and Co atoms are regarded as non-magnetic atoms in β-Mn alloys. The plots in the form
3/4
of γe –TN for several kinds of β-Mn alloy systems are in accord with equation (7.14),
showing that the value of γsf is insensitive to the kind of additional elements. For low concentrations, the value of γe is large for all kinds of additional elements. On the other hand,
for higher concentrations, the value of γe deviates from the straight line, coming close to
γband . Therefore, these alloys are no longer itinerant-electron weak antiferromagnets. From
these data, it is concluded that the magnetic state of β-Mn alloys varies from the itinerantelectron weak antiferromagnetic state to the intermediate state with increasing additional
element concentration x.
Figures 7.58 and 7.59 show the temperature dependence of the specific heat capacity Cp
for β-MnOs (Miyakawa et al., 2001) and MnRu alloys (Sasao et al., 2001). The Cp curves
also changes remarkably with the increase of Os or Ru concentration. To take an example,
γband =
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
327
Fig. 7.57. The relationship between the electronic specific coefficient γe and the Néel temperature TN in the form
3/4
of γe versus TN for various β-Mn100−x TMx alloy systems (Fe, Co, Ni: Shinkoda et al., 1979, Ru: Sasao et al.,
2001, Os: Miyakawa et al., 2001, Ir: Miyakawa et al., 2005).
Fig. 7.58. Temperature dependence of the specific heat capacity Cp for β-Mn100−x Osx alloys (Miyakawa et al.,
2001).
in the high Ru concentration range, a clear peak is observed in the temperature dependence
of Cp as given in fig. 7.59. The peak corresponds to the inflexion point of the susceptibility
curve shown in fig. 7.54, being defined as the Néel temperature. The entropy change due
to the magnetic transition, S, estimated from fig. 7.58 increases with increasing Ru concentration, implying that the magnetic behavior approaches an intermediate state between
the itinerant-electron and localized-electron magnetic states. On the other hand, the peak at
the Néel temperature TN becomes vague in low Ru concentration ranges. The magnetism
of β-Mn alloys in which additional elements mainly occupy site 1 has been successfully
explained in terms of weak antiferromagnetism in itinerant-electron systems (Sasao et al.,
2001; Miyakawa et al., 2001, 2004). In itinerant-electron weak antiferromagnetic alloys,
the effect of spin fluctuations is widely enhanced around TN , that is, the spin fluctuation
328
K. FUKAMICHI et al.
Fig. 7.59. Temperature dependence of the specific heat capacity Cp for β-MnRu alloys (Sasao et al., 2001).
modes have strong intensities around Q, and hence the magnetic correlation length becomes significantly large, therefore, these alloys show no distinct anomaly at TN in the
specific heat curve (Makoshi and Moriya, 1975; Takahashi, 1999; Takahashi and Nakano,
2004). Consequently, β-MnRu alloys in this concentration range are regarded as itinerantelectron weak antiferromagnetic alloys (Sasao et al., 2001). The peak associated with the
magnetic transition at TN also becomes smaller with decreasing Os content in a similar
manner as β-MnRu alloys. The peak is hardly confirmed below x = 0.13, exhibiting a
characteristic itinerant-electron weak antiferromagnetic property.
7.8. Concentration and temperature dependences of magnetic properties of the L10 -type
alloy systems
In L10 -type equiatomic alloys, the spin structure of MnPt is different from that of other
equiatomic alloys, depending on the concentration and temperature (see fig. 7.72). Therefore, the change in the spin structure against temperature and composition is highly interesting. In this subsection, the concentration dependence of the magnetic moment and
the temperature dependence of magnetic susceptibility of L10 -type MnPt are discussed,
comparing with those of other alloy systems.
7.8.1. Magnetic moment and the Néel temperature of MnPt and MnPd alloy systems
Figure 7.60 shows the concentration dependence of magnetic susceptibility at 4.2 K,
χ4.2 for three kinds of L10 -type alloy systems (Umetsu et al., 2003b). These alloys exhibit a minimum value in the vicinity of the equiatomic composition, and the value of χ4.2
increases for concentrations deviating from the equiatomic composition. This behavior
strongly implies that a characteristic electronic structure exists at the equiatomic composition. Panels (a) through (d) in fig. 7.61 set out the calculated density of states (DOS) of
Mn and Pt of MnPt alloys with the L10 type tetragonal structure with c/a = 0.9175 in
the collinear antiferromagnetic state with the composition of 45, 50, 55 and 60% Pt, respectively (Umetsu et al., 2003b, 2006). It is clear that a pseudo-gap formed around the
Fermi level EF is most prominent at the equiatomic composition and becomes indistinct
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
329
Fig. 7.60. Concentration dependence of the magnetic susceptibility at 4.2 K, χ4.2 for three kinds of L10 -type
alloy systems (Umetsu et al., 2003b).
in the off-equiatomic compositions. These results are in good agreement with the concentration dependence of γe , that is, the smallest value of γe is observed in the vicinity of the
equiatomic composition (see fig. 7.7). Especially, the excess Mn atoms substituted at the Pt
site effectively increase the DOS at EF as shown in fig. 7.61(a), making these alloys more
sensitive to the γe value in the concentration range less than 50 at.% Pt.
Shown in fig. 7.62 is the concentration dependence of the calculated magnetic moments
of Mn and Pt for the L10 -type ordered MnPt alloy system (Umetsu et al., 2003b; 2006).
The magnetic moment of Mn for the equiatomic composition is about 3.6 μB , being smaller
than the reported value of about 4.3 μB from neutron diffraction studies (Krén et al., 1968a;
Pál et al., 1968). In the off-equiatomic compositions, the Mn atoms substituted at the Pt
site have a comparatively large magnetic moment of about 2.5 μB , whereas Pt atoms substituted at the Mn site also have a small magnetic moment, less than 0.3 μB . Similar panels
(a) through (c) in fig. 7.63 also set out the calculated density of states (DOS) of Mn and Pd
of MnPd alloys with the L10 -type tetragonal structure with c/a = 0.8796 in the collinear
antiferromagnetic state with the composition of 40, 50 and 60% Pd, respectively (Umetsu
et al., 2003b, 2005c). In a similar manner as the MnPt alloy system, a distinct pseudo-gap
is formed around the Fermi level EF . The excess Mn atoms substituted at the Pd site effectively also increase the DOS at EF as shown in fig. 7.64 in the concentration range less than
50 at.% Pd. Figure 7.64 shows the concentration dependence of the calculated magnetic
moments of four sites for Mn and Pd in the L10 -type ordered MnPd alloy system (Umetsu
et al., 2005c). In analogy with the MnPt alloy system, the magnetic moment of Pd is almost
zero in contrast to the proposed spin structure III (see fig. 7.73).
7.8.2. Temperature dependence of the magnetic susceptibility and magnetic cooling effect
The difference in the magnetic moment discussed in connection with figs. 7.62 and 7.64
causes a difference in the susceptibility between two antiferromagnetic sublattices. As it
turned out, the magnitude of magnetic susceptibility χ becomes larger for off-equiatomic
compositions (Umetsu et al., 2003b). The temperature dependence of magnetic susceptibility χ is given in figs. 7.65 and 7.66 for L10 -type MnPt alloys with Pt concentrations
330
K. FUKAMICHI et al.
Fig. 7.61. The calculated density of states (DOSs) of Mn and Pt of MnPt ordered alloys with the L10 -type
tetragonal structure with c/a = 0.9175 in the collinear antiferromagnetic state with compositions of 45, 50, 55
and 60 at.% Pt. Note that the left and right four panels show the local DOS at the Mn and Pt sites, respectively
(Umetsu et al., 2003b, 2006).
lower and higher than 50 at.%, respectively (Umetsu et al. 2003b, 2006). As mentioned
above, the value of χ increases when deviating from the equiatomic composition. The arrows indicate the Néel temperature TN determined from the temperature dependence of
the electrical resistivity (see fig. 7.15). From neutron diffraction, it has been reported that
another additional magnetic transition below TN can be observed only in the MnPt alloy
system (Krén et al., 1968a) among the equiatomic MnTM (TM = Pt, Pd and Ni) alloy sys-
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
331
Fig. 7.62. Concentration dependence of the calculated magnetic moments of Mn and Pt for the L10 -type MnPt
alloy system (Umetsu et al., 2003b; Umetsu et al., 2006).
tems (Krén et al., 1968a; Andresen et al., 1966). However, no clear anomaly corresponding
to the transition is confirmed in the resistivity curve in fig. 7.15 and the magnetic susceptibility curves in figs. 7.65 and 7.66. Details are given in subsection 7.8.3. Figure 7.67 shows
the magnetic cooling effect for Mn40.7 Pt59.3 alloy which exhibits hysteresis between the
field cooling (FC) and zero field cooling (ZFC) in the magnetic susceptibility χ curves in
1 kOe (Umetsu et al., 2005c). Recent detailed theoretical calculations speculate the following possibility; the Mn moment at the Pt site couples perpendicular to the Mn moment at
the Mn site as the spin structures (a) or (c) illustrated in fig. 3.7, in contrast to the collinear
structure. However, the difference between the magnetic energy in the perpendicular and
collinear states is comparably small. This situation could result in such magnetic cooling
effect (Umetsu et al., 2005c). For MnPd alloy system, a similar magnetic cooling effect is
observed in the concentration range less than 50 at.% Pd as seen from fig. 7.68 (Umetsu et
al., 2005c).
More significant magnetic cooling effect is observed in the Mn100−x Irx and Mn100−x Nix
alloy systems. The representative curves for Mn100−x Irx and Mn100−x Nix alloy systems
are drawn in fig. 7.69 and figs. 7.70(a) and (b), respectively (Umetsu et al., 2005c). The
cooling effect becomes significant with decreasing Ir concentration in the range less than
50 at.% (Umetsu et al., 2005c). The important point to note is that such cooling effect is
observed in the concentration range below 50 at.% where the Mn moment at the Pt, Pd or
Ir site is relatively large (see figs. 7.62 and 7.64). In the case of Mn100−x Nix alloy system,
the cooling effect is also observed more than 50% Ni, which may be associated with its
very complicated phase diagram (Tsiuplakis and Kneller, 1969).
7.8.3. Magnetic phase diagram and spin structures
In order to determine the Néel temperature TN , following equation (4.31), based on the
results of the DOSs shown in fig. 7.61, the effective exchange constant J0 of L10 -type
Mn100−x Ptx alloys with c/a = 0.9175 were calculated. The J0 curves as a function of the
332
K. FUKAMICHI et al.
Fig. 7.63. The calculated density of states (DOSs) of Mn and Pd of MnPd ordered alloys with the L10 -type
tetragonal structure with c/a = 0.8796 in the collinear antiferromagnetic state with compositions of 40, 50 and
60 at.% Pd. Note that the left and right four panels show the local DOS at the Mn and Pd sites, respectively
(Umetsu et al., 2003b, 2005c).
Fermi level EF for the L10 -type Mn100−x Ptx alloys with c/a = 0.9175 in the collinear
antiferromagnetic state with the composition of x = 45, 50 and 55 are represented in
fig. 7.71. The actual value of J0 corresponds to the value at the origin of the abscissa, EF
(= 0), and each value is about 112 for x = 0.45, 126 for x = 0.50 and 120 meV, for
x = 0.55 (Umetsu et al., 2006).
The magnetic phase diagram and the spin structures of the MnPt alloy system are illustrated in fig. 7.72, together with the calculated theoretical values of TN (Umetsu et al.,
2006), and the types II and III of the spin structures (Krén et al., 1968a). The proposed
spin structures are illustrated in fig. 7.73 (Krén et al., 1968a). The results of TN are in
agreement with each other. Similar magnetic structures were reported by Andresen et al.
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
333
Fig. 7.64. Concentration dependence of the calculated magnetic moments of Mn and Pd for the L10 -type MnPd
alloy system (Umetsu et al., 2005c).
Fig. 7.65. Temperature dependence of the magnetic susceptibility of L10 -type ordered MnPt alloys with a Pt
concentration lower than 50 at.% (Umetsu et al., 2003b; Umetsu et al., 2006).
(1965). As shown in fig. 7.72, below and above the magnetic transition temperature Tm , the
direction of the magnetic moment changes from perpendicular to parallel to the c-axis. In
other words, the magnetocrystalline anisotropy changes at Tm as given in fig. 7.73 (Krén et
al., 1968a), in which the Pt atoms carry a magnetic moment in spin structure III. The magnetic transition has been recognized from the observation of 100 and 101 magnetic peaks
in the neutron diffraction diagrams, although no changes were observed in the temperature
dependences of the magnetic susceptibility χ and the lattice constant (Krén et al., 1968a).
Figure 7.74 shows the neutron diffraction patterns measured at 293 and 803 K, respectively, for MnPt alloy (Krén et al., 1968a). The magnetic structure factors for a collinear
334
K. FUKAMICHI et al.
Fig. 7.66. Temperature dependence of the magnetic susceptibility of L10 -type ordered Mn100−x Ptx alloys with
a Pt concentration higher than 50 at.% (Umetsu et al., 2003b; Umetsu et al., 2006).
Fig. 7.67. Magnetic cooling effect of L10 -type ordered Mn40.7 Pt59.3 alloy (Umetsu et al., 2005c).
Fig. 7.68. Magnetic cooling effect of the L10 -type ordered MnPd alloys (Umetsu et al., 2005c).
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
335
Fig. 7.69. Magnetic cooling effect of the L10 -type ordered MnIr alloys with the concentration below 50 at.% Ir,
together with that of Mn49.9 Pt50.1 (Umetsu et al., 2005c).
Fig. 7.70. Magnetic cooling effect of the L10 -type ordered MnNi alloys with Ni concentrations (a) higher and
(b) lower than 50 at.% (Umetsu et al., 2005c).
336
K. FUKAMICHI et al.
Fig. 7.71. The effective exchange constant J0 as a function of the Fermi level EF for the L10 -type Mn100−x Ptx
alloys with c/a = 0.9175 in the collinear antiferromagnetic state with the compositions of x = 45, 50 and 55
(Umetsu et al., 2006).
structure are given by the following expression:
$
$2
$
2 $
2 2
2
$
|Fhkl | = qhkl $
|Fhkl | ,
pj exp(2πik · r j )$$ = qhkl
(7.17)
j
where pj , k, and r are the magnetic scattering amplitude, the reciprocal lattice vector, and
2 = sin2 α, where α is the angle of the magnetic
the atomic position vector, respectively. qhkl
moment with respect to the [hkl] direction. For tetragonal symmetry,
2 l 2 + [ 12 (c/a)2 (h2 + k 2 ) − l 2 ] sin2 ϕc
qhkl = 1 −
,
(c/a)2 (h2 + k 2 ) + l 2
(7.18)
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
337
Fig. 7.72. The magnetic phase diagram of the L10 -type MnPt ordered alloy system (Krén et al., 1968a), together
with recent experimental ( ) and theoretical () results (Umetsu et al., 2006).
•
Fig. 7.73. Proposed spin structures of L10 -type MnPt alloys in the vicinity of the equiatomic composition (Krén
et al., 1968a).
where a and c are the lattice constants of the tetragonal unit cell and φc is the angle of the
magnetic moment with respect to the c-axis. Model II corresponds to φc = 0, and models
|2 are given in table 7.5 for MnPt, and the
III and IV imply φc = 90◦ . The values of |Fhkl
2
values of qhkl can be obtained from equation (7.18) (Krén et al., 1968a). The structure
factors for model III and IV coincide and are different from those for model II only in the
338
K. FUKAMICHI et al.
Fig. 7.74. Neutron diffraction patterns at 293 and 803 K for MnPt equiatomic alloy (Krén et al., 1968a).
TABLE 7.5
|2 for the models in L1 -type lattice, b , b are the nuclear and P , P the
Values of |Fhkl
Mn Pt
Mn Pt
0
magnetic scattering amplitudes (Krén et al., 1968a)
Model
II
III
h, k, l unmixed
4(bMn + bPt )2
4(bMn + bPt )2
h, k, l mixed
h − k = 2n
h − k = 2n + 1
4(bMn − bPt )2
4(bMn − bPt )2
2
4PMn
2
2)
4(pMn + pPt
2 when the Pt atoms have no magnetic moment. Accordingly, a distinction
values of qhkl
between models III and IV is impossible by powder diffraction methods (Pál et al., 1968).
The relative magnetic reflection intensities depend on the angle φc . That is, the integrated
intensity ratio of reflections 100 to 101 is 4:1 for structure II, and 1:1 for structures III and
IV. From the change in neutron diffraction intensities at 293 and 803 K for L10 -type MnPt
equiatomic alloy in fig. 7.74 reveals that structure II at 293 K, and III (or IV) at 803 K
are formed (Krén et al., 1967a, 1968a; Pál et al., 1968). A recent neutron diffraction study
using a single crystal of MnPt has arrived at a distinction of model III (Ikeda and Tsunoda,
2003b). The temperature dependence of the 100 and 101 peak intensities of L10 -type MnPt
alloy is presented in fig. 7.75 (Krén et al., 1967a, 1968a; Pál et al., 1968). The spin structure
transformation from model II to III clearly takes place in a wide temperature range, and
hence the center of which is taken to be the transformation temperature Tm .
The effective exchange constant J0 curve as a function of the Fermi level EF for the
L10 -type Mn100−x Pdx alloys with c/a = 0.8796 in the collinear antiferromagnetic state
with the composition of x = 45, 50 and 55 is presented in fig. 7.76 (Umetsu et al., 2003b,
2005c). The values of J0 at the origin of the abscissa EF (= 0) for x = 45, 50 and 55
are 131, 140 and 143 meV, respectively. It should be noted that the origin of the abscissa
EF (= 0) is located at the maximum of the peak in 55% Pd, resulting in a larger value
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
339
Fig. 7.75. Temperature dependence of the 100 and 101 magnetic reflections for L10 -type MnPt alloy with the
equiatomic composition (Krén et al., 1968a; Pál et al., 1968).
of J0 than that of 50% Pd, consistent with the data of low-temperature specific coefficient
(see fig. 7.7). The difference in the J0 value, about 3 meV, corresponds to about 20 K in
TN . The concentration dependence of the experimental and calculated TN of the L10 -type
Mn100−x Pdx alloy system is shown in fig. 7.77 (Umetsu et al., 2003b), together with the
obtained TN from neutron diffraction studies by Pál et al. (1968). The difference would be
permissible on the basis of the generalized molecular field approximation (Liechtenstein
et al., 1985), that is, the calculated values of TN are higher than those of the experimental
data. However, the trend of the concentration dependence of TN well accords with each
other. The figure highlights that the highest TN is observed around 53% Pd concentration,
being different from that of MnPt alloy system in which the equiatomic composition shows
the highest TN (see figs. 6.20 and 7.72).
The temperature variation of the 100 magnetic reflections for L10 -type MnNi, MnPt
and MnPd with equiatomic composition is shown in fig. 7.78 (Pál et al., 1968). For MnNi,
the curve exhibits an abrupt decrease in the vicinity of Tt = 980 K because of the phase
transformation to B2 (≡ CsCl) structure (see fig. 6.17). The curve for MnPd exhibits a
Brillouin function-like dependence up to TN . Therefore, no anomalous dependence due to
Tt is observed in MnPd alloy in contrast to MnPt alloy. In recent investigations, no distinct
anomaly associated with the magnetic transformation can be confirmed (see figs. 7.15,
7.65 and 7.66). Powder neutron diffraction data detected no change in the spin direction
as a function of Pt composition (Severin et al., 1979). On the other hand, more recently,
the direction change of the magnetic moment from perpendicular to parallel to the c-axis
of MnPt has been reconfirmed by neutron diffraction for a single crystal (Ikeda and Tsunoda, 2003b). In this connection, the calculated Pt moment in fig. 7.62 is almost zero. On
the other hand, we may recall that MnPt3 becomes ferromagnetic and both Mn and Pt
atoms carry magnetic moments (Pickart and Nathans, 1962; Krén et al., 1968a). Therefore,
precise experiments on type III are necessary for detailed discussion.
340
K. FUKAMICHI et al.
Fig. 7.76. The effective exchange constant J0 as a function of the Fermi level EF for the L10 -type Mn100−x Pdx
alloys with c/a = 0.8796 in the collinear antiferromagnetic state with the compositions of x = 45, 50 and 55
(Umetsu et al., 2003b, 2005c).
7.9. Magnetovolume effects and elastic properties of Mn alloys
Elastic properties of Mn and its alloys are significantly different from each other, depending on the crystal structure and the magnetic state as given in table 2.2. A marked thermal
expansion anomaly and a pressure effect on the Néel temperature associated with magnetovolume effects have been observed in α-Mn (Petrenko et al., 1974; Kagayama et al.,
1999). On the other hand, there are strong elastic strains in multilayers and they interact
with the magnetic properties of the antiferromagnetic and ferromagnetic layers. In making
processes of spin valve devices with multilayers, thermal strains are inevitably induced
at high level in the multilayers due to the difference in the thermal expansion coefficient
among the film layers. In addition, the magnetoelastic stresses related with the spontaneous
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
341
Fig. 7.77. Concentration dependence of TN obtained from the experiments and calculations of the L10 -type
Mn100−x Pdx alloy system (Umetsu et al., 2003b), together with the data of TN determined by neutron diffraction
studies by Pál et al. (1968). The solid circles and triangles indicate TN defined from dρ/dT and J0 , respectively.
Fig. 7.78. Temperature dependence of the 100 magnetic reflections for L10 -type MnNi, MnPt and MnPd with
equiatomic composition (Pál et al., 1968).
volume magnetostriction ωs are induced. From practical viewpoints, ωs and thermal strains
are not ignored because both of them are closely correlated with durability and stability of
spin valves. Therefore, it is important to discuss the elastic properties of antiferromagnets
used for exchange biasing.
7.9.1. L12 -type (≡ γ -phase) ordered and γ -phase disordered alloys
Kakehashi et al. (1999) have applied a molecular dynamics (MD) theory to γ -Mn with
108 atoms in a MD unit cell introducing a site-dependent non-collinear effective medium.
The temperature dependence of magnetic moments calculated by the improved moleculardynamics for γ -Mn is given in fig. 7.79 (Kakehashi et al., 1999). The local magnetization
mi smoothly decreases with increasing temperature, accompanied by the second-order
342
K. FUKAMICHI et al.
Fig. 7.79. Calculated temperature dependence of the local magnetization |mi | and the amplitude of local magnetic moment (LM) |m2i |1/2 for γ -Mn (Kakehashi et al., 1999).
Fig. 7.80. Thermal expansion curves of the Mn100−x Rhx ordered (O) alloys (Umetsu et al., 2003a), together with
that of ordered (O) Mn82 Pt18 (Krén et al., 1968a) and Mn3 Ir (Tomeno et al., 1999) alloys. The Néel temperature
TN is given by the arrows.
transition at TN = 510 K. Although the calculated TN depends on the system size in the
molecular-dynamic unit cell, the value is comparable to the experimental values of 480–
510 K (see fig. 6.3). On the other hand, the amplitude of local moment (LM) shows a weak
temperature dependence; m2i 1/2 = 3.77 μB at T = 0 and 3.24 μB at T = TN . Such
temperature dependence deviates from the dashed line for localized moment systems.
Several kinds of thermal expansion curves of L12 -type ordered alloys are drawn in
fig. 7.80 in which L0 is the length of the specimen at room temperature (Krén et al., 1968a;
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
343
Fig. 7.81. The temperature dependence of the thermal expansion coefficient α of L12 -type ordered (O)
Mn80 Rh20 and Mn3 Rh alloys (Umetsu et al., 2003a).
Tomeno et al., 1999; Umetsu et al., 2003a). These curves exhibit a clear shrinkage below TN , indicating a negative spontaneous volume magnetostriction ωs . According to this
theory, the spontaneous volume magnetostriction ωs at T is expressed by
ωs (T ) = κCmv M 2 (T ) + ξ 2 (T ) ,
(7.19)
where κ, Cmv , M and ξ 2 (T ) are the compressibility, the magnetovolume coupling constant,
the magnetization and the mean square amplitude of thermal SFs, respectively. The curve
of L12 -type ordered (O) Mn3 Rh alloy in fig. 7.80 exhibits a marked SF effect over a wide
temperature range even above TN . The temperature dependence of the thermal expansion
coefficient α of L12 -type Mn3 Rh and Mn80 Rh20 ordered alloy is presented in fig. 7.81
(Umetsu et al., 2003a). A large positive peak is observed at the Néel temperature TN . The
thermal expansion coefficient α is given by
α = αph + αe + αm ,
(7.20)
where αph , αe and αm are the phonon, the electron and the magnetic terms, respectively.
The solid lines in the figure stand for αph + αe , which are estimated from the lowtemperature specific heat data within the Debye model. Note that the thermal expansion
coefficient in the PM temperature range is very large of about 30 × 10−6 /K, compared
344
K. FUKAMICHI et al.
with the value of conventional 3d metals and their alloys, deviating from the solid curve.
The enhancement of the thermal expansion above the magnetic transition temperature is
explained by the spin fluctuation (SF) theory (Moriya, 1985). In most Mn-based alloys,
the occupation of 3d orbitals is close to the half-filled state, and hence the intra-atomic
Coulomb interaction related to Hund’s rule makes the repulsive energy small, accompanied by a large local moment which induces remarkable SFs. It should be emphasized that
α above TN shows a large value of about 30 × 10−6 /K. The volume thermal expansion
coefficient ωs = 3α and specific heat Cp are related with the following Grüneisen relation
(Gopal, 1966)
1 κCp
1
ωs =
,
(7.21)
3
3 V
where is the Grüneisen parameter, κ the compressibility and V the molar volume. In
general, the total thermal expansion coefficient is expressed as the sum of the lattice, electronic and magnetic contributions. The sum of the lattice and electronic contributions to
the thermal expansion coefficient is defined as
α=
αph + αe =
1 κ(Cph + Ce )
3
V
(7.22)
with
Cph = 9N kB
T
θD
3 θD /T
dx
0
x 4 ex
(ex − 1)2
(7.23)
and
Cph = γe T ,
(7.24)
where Cph and Ce stand for the phonon and the electronic contributions to the specific
heat, respectively. N , kB and γe are the number of atoms, the Boltzmann constant and the
electronic specific heat coefficient, respectively. The phonon contribution Cph (αph ) can be
described by the Debye theory, independent of temperature above the Debye temperature
θD . The electronic contribution Ce (αe ) is proportional to temperature. The solid line in the
figure stands for the sum of contributions of αph and αe to the thermal expansion coefficient
calculated by equations (7.22)–(7.24). In the calculations, the proportional coefficient value is defined as to meet the calculated value (αph + αe ) with the observed curve around
room temperature and becomes an appropriate value of about 1.4 (Barron et al., 1980),
using the compressibility κ obtained from the experiment (see fig. 7.90). The observed
α in paramagnetic regions is about 30 × 10−6 /K, much larger than the calculated value
(αph + αe ) given by the solid line in fig. 7.81. It is difficult to explain such a large α of the
Mn3 Rh ordered alloy as the sum of the phonon and band terms.
According to the SF theory (Moriya, 1985), M is zero in the paramagnetic state, but
ξ shows an effective value. As a result, the thermal expansion coefficient α of itinerantelectron materials increases with temperature even in the paramagnetic state. The thermal
expansion curves of ordered Mn3+x Pt1−x and Mn3 Pt1−x Rhx alloys are drown in fig. 7.82
(Krén et al., 1968a). The discontinuous change accompanied by thermal hysteresis at the
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
345
Fig. 7.82. Thermal expansion curves of L12 -type ordered Mn3+x Pt1−x and Mn3 Pt1−x Rhx alloys (Krén et al.,
1968a).
transition temperature Tm from the D-phase to F-phase (see fig. 6.8), depending on the
value of x and y, corresponds to the abrupt increase in the magnetic moment (see fig. 6.14).
The so-called invar effect is observed between the first-order transition temperature Tm and
the Néel temperature TN , that is, in the F-phase range. The thermal expansion coefficient
α above TN is larger than that of ordinary transition metals and alloys because of the spin
fluctuation effect in analogy with Mn3 Rh and Mn80 Rh20 alloys (see fig. 7.81). Similar
discontinuous change behavior of the ordered Mn3 Pt has been confirmed by Yasui et al.
(1992). It should be pointed out that the thermal expansion anomaly below TN is positive
in the F-phase and negative in the D-phase, resulting in a lower and higher thermal expansion coefficients α, respectively, compared with the value in the paramagnetic temperature
ranges.
7.9.2. β-Mn alloys
For paramagnetic temperatures, very similar behavior is also observed in β-Mn100−x Osx
alloys (Miyakawa et al., 2001, 2003a) and β-MnIr alloys (Miyakawa et al., 2003b). We
may recall that a strong SF effect persists up to paramagnetic temperature ranges of about
3TN for the YMn2 compound, which exhibits a remarkably large α of about 55 × 10−6 /K
above TN (Shiga et al., 1987). Figure 7.83 provides the thermal expansion curves of
β-Mn100−x Osx alloys (Miyakawa et al., 2001). The hypothetic paramagnetic curves estimated from the Debye model are given by the dashed lines. The experimental thermal
expansivity is small, that is, the value of α is smaller than that of the dashed curves below TN . Therefore, a positive volume effect is observed below TN , in contrast to the data
in fig. 7.80. What should be noted is the thermal expansion characteristics in the paramagnetic temperature range, that is, the thermal expansivity L/L0 becomes larger with
decreasing Os content. In order to make such behavior clear, we also show the temperature
dependence of the thermal expansion coefficient α is shown in fig. 7.84 for Mn100−x Osx
paramagnetic alloys (Miyakawa et al., 2005). The Néel temperature TN of those alloys is
much lower than room temperature (see fig. 7.56). With increasing x, the value of α is
346
K. FUKAMICHI et al.
Fig. 7.83. The thermal expansion curves of paramagnetic β-Mn100−x Osx alloys (Miyakawa et al., 2001).
Fig. 7.84. Temperature dependence of the thermal expansion coefficient α for paramagnetic β-Mn100−x Osx
alloys (Miyakawa et al., 2005).
reduced from about 30 × 10−6 /T to 15 × 10−6 /T which is the same magnitude of ordinary
transition metals and alloys. In other words, the value of α for the alloys with low concentrations x is extremely large and associated with the spin fluctuation effect. This tendency
is common to β-Mn alloys (Miyakawa et al., 2005).
7.9.3. L10 -type alloys
The temperature dependence of the lattice constants a and c, and the volume V for
L10 -type ordered MnPt alloy is given in fig. 7.85 (Krén et al., 1968a; Pál et al., 1968).
Remarkable shrinkages in a and also V occur below TN . A similar temperature dependence
is also found in MnPd as shown in fig. 7.86 (Kjekshus et al., 1967; Pál et al., 1968). For
MnNi alloy given in fig. 7.87 (Krén et al., 1968b; Pál et al., 1968), the crystallographic
transformation takes place below TN (see fig. 6.17), and then no clear shrinkage below TN
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
347
Fig. 7.85. Temperature dependence of the lattice constants ac, the ratio c/a and the volume V for L10 -type
ordered (O) MnPt alloy (Pál et al., 1968; Krén et al., 1968a).
is observed in the measuring temperature range, although the temperature dependence is
very similar to that of MnPt and MnPd alloys. Therefore, it is expected that the shift of TN
against pressure for L10 -type ordered Mn alloys is in analogy with that of γ - and γ -phase
Mn alloys (see fig. 7.89). Accordingly, the magnetovolume effects in L10 -type ordered
Mn alloys should be explained in a similar way as those in the γ -phase disordered and
L12 -type ordered alloys.
7.9.4. Compressibility and bulk modulus of Mn alloys
The thermal expansion anomaly given in the preceding section is associated with the
pressure effects on TN as described below. The thermodynamic Eherenfest equation is
given by
V TN αV
dTN
,
=
dP
Cm
(7.25)
where V and P are the volume and pressure, respectively, αV and Cm are the difference
between the thermal expansion coefficient and the specific heat before and after of the magnetic phase transition, respectively. The equation indicates that the negative spontaneous
volume magnetostriction ωs increases TN on applying hydrostatic pressure. Figure 7.88
shows the temperature dependence of electrical resistivity under various pressures for the
ordered (O) Mn3 Rh alloy (Umetsu et al., 2003a). Because the measurements were made
by a quasi-four-probe method using a multianvil-type system, the background noises are
348
K. FUKAMICHI et al.
Fig. 7.86. Temperature dependence of the lattice constants a, c, the ratio c/a and the volume V for L10 -type
ordered (O) MnPd alloy (Kjekshus et al., 1967; Pál et al., 1968).
Fig. 7.87. Temperature dependence of the lattice constants ac, the ratio c/a and the volume V for L10 -type
ordered (O) MnNi alloy (Krén et al., 1968b; Pál et al., 1968).
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
349
Fig. 7.88. Temperature dependence of electrical resistivity under various pressures for L12 -type ordered (O)
Mn3 Rh alloy (Umetsu et al., 2003a).
Fig. 7.89. Pressure effect on TN for L12 -type ordered (O) Mn3 Rh and Mn80 Rh20 alloys (Umetsu et al., 2003a).
somewhat high compared with the data in fig. 7.9. Thus, the measurements were repeated
two times to confirm the reproducibility. A clear change in electrical resistivity at TN is
observed when using the fine solid line as guides to the eyes. The Néel temperature TN
indicated by the arrows increases on applying pressure.
The pressure effect on TN for L12 -type ordered MnRh alloys is presented in fig. 7.89
(Umetsu et al., 2003a). With increasing pressure, or decreasing lattice constant, TN increases with a rate of dTN /dP = 7–9 K/GPa in accordance with equation (7.25). The
relation between the pressure effect on the Néel temperature and the thermal expansion
characteristics in figs. 7.85–7.87 is described in a similar way. From the pressure effect
on the volume of L12 -type Mn3 Rh ordered alloys in fig. 7.90 (Umetsu et al., 2003a),
the compressibility κ of L12 -type (≡ γ -phase) ordered Mn3 Rh alloy is estimated to be
350
K. FUKAMICHI et al.
Fig. 7.90. Pressure dependence of V /V0 of α-Mn (Takemura et al., 1988), L12 -type ordered Mn3 Rh (Umetsu
et al., 2003a) and Mn78 Pt22 (Saunders et al., 1993) alloys. The value of V indicates the difference between the
volume under ambient pressure V0 and the volume under applied pressures V at room temperature.
1.4 × 10−2 /GPa by using the following standard form of the equation of state of solids.
V
= −P + ϑP 2
(7.26)
V0
with ≡ κ = −d ln V /d ln P . The determined value is more compressive than that of
α-Mn of about 0.8 × 10−2 /GPa at room temperature (Takemura et al., 1988) and also much
larger than that of non-magnetic elements such as Ta, Mo and Ru. For example, the value
of κ of the L12 -type alloy is about five times larger than that of Ru. The pressure effect on
the volume change of ordered (O) Mn78 Pt22 alloy (Saunders et al., 1993) is given in the
same figure and the value of κ is much larger than that of Mn3 Rh. It is well known that the
bulk modulus B (≡ κ −1 ) is generally lowered by the appearance of magnetism, especially
in antiferromagnets (Janak and Williams, 1976; Moruzzi et al., 1977; Sliwko et al., 1994).
It has been reported that the theoretical bulk modulus of magnetic elements in 3d transition
metals is deviant from the experimental results in the early theoretical calculations by the
local density spin functional approximation (LSDA) (Janak and Williams, 1976; Moruzzi
et al., 1977). However, the lattice constant and the magnetic moment of Mn calculated
by the generalized gradient approximation (GGA) (Perdew and Wang, 1986) are nearly in
agreement with the experimental data (Asada and Terakura, 1992, 1993). It is interesting to
note that the value of ordered Mn3 Rh alloy is close to the theoretical value of γ -Mn (fcc)
(Asada and Terakura, 1992). Available data of the lattice constant and bulk modulus B0 of
several kinds of materials are listed in table 7.6. For MnPt alloys, the value in the D-phase
seems more compressive than in the F-phase. The value of B0 of α-Mn is relatively low,
even though the available data were obtained in the paramagnetic state.
The low temperature specific heat coefficient γe , the Debye temperature θD and the bulk
modulus at room temperature B0 as a function of the electron concentration e/a for antiferromagnetic alloys are plotted in fig. 7.91(a), (b) and (c) (Umetsu et al., 2005c), together
with those of various materials (Lenkkeri, 1981; Wassermann, 1990). Large values in the
range e/a = 8.2–8.6 in (a) are caused by superposition of two linear terms in the low temperature specific heat; the band term and a spin-glass or re-entrant spin glass term from the
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
351
TABLE 7.6
Lattice constant a and bulk modulus B0 at room temperature for several kinds of metals and Mn-based alloys
a (nm)
B0 (N m−2 ) × 1011
Mn78 Pt22
D-phasea
Mn3 Pt
F-phaseb
Mn3 Rh
(O)c
Mn75 Ni25 d
Mn40 Fe60 e
α-Mnf,g
0.3842
0.55
0.3833
1.1
0.3813
0.71
0.3698
0.75
0.3620
1.23
0.8913
1.25
(1.58)
a Saunders et al. (1993), b Yasui et al. (1987), c Umetsu et al. (2003a), d Honda et al. (1976), e Cankurtaran et al.
(1993), f Asada and Terakura (1993), g Moruzzi et al. (1989).
low-energy excitations. On the other hand, e/a of L10 -type MnNi, MnPt and MnPd alloys
have a pseudo-gap, showing a very small value of γe . The Debye temperature θD is given
by the following relation with the compressibility or bulk modulus (James, 1954; Asada
and Terakura, 1993).
√ −1
κ
h 9N 1/3
,
θD =
(7.27)
√
k 4π
(MfD (μ) ρ)1/3
with
fD (μ) =
1 + μ 3/2
2(1 + μ) 3/2
+2
,
3(1 − μ)
3(1 − 2μ)
where h, k, κ, M, ρ, N and μ are Plank constant, Boltzmann constant, compressibility,
atomic weight, density, Avogadro number and Poisson ratio, respectively (James, 1954).
The free energy F as a function of rWS and temperature T is expressed by the following
equation under the quasiharmonic approximation (Asada and Terakura, 1993);
9
F (rWS , T ) = Etot (rWS ) − kB T D(θD /T ) − 3 ln 1 − e−θD /T + kB θD ,
(7.28)
8
where θD is the Debye temperature and D(x) is the Debye function. The last term in
equation (7.21) represents the zero-point energy. The Debye temperature (θD )0 for an equilibrium Wigner–Seitz sphere radius (rWS )0 is given by
B0 1/2
,
θD = c (rWS )0
(7.29)
Ma
where B0 is the bulk modulus for (rWS )0 , or P = 0, Ma is the atomic mass and c is a
constant. As seen from fig. 7.91(c), significant softening of antiferromagnetic and ferromagnetic alloys respectively take place around e/a = 7.7 and 8.7, where the value of θD
becomes small as displayed in fig. 7.91(b).
The concentration dependence of the lattice constant of γ -phase disordered alloys is
given in fig. 7.92 (Krén et al., 1966, 1968a; Yamaoka, 1974; Sasao et al., 1999; Yamauchi
et al., 1999). For the additives Ir and Rh, belonging to the same group in the periodic table with the same outer electrons, the lattice constant of the Mn100−x Irx alloys is smaller
than that of γ -Mn100−x Rhx alloys. On the other hand, TN of γ -MnIr is higher than that
352
K. FUKAMICHI et al.
Fig. 7.91. The low temperature specific heat coefficient γe , the Debye temperature θD and the bulk modulus B0
as a function of the electron concentration for antiferromagnetic alloys (Umetsu et al., 2005c), together with those
of various materials (Rosen, 1968 Lenkkeri, 1981; Yasui et al., 1987; Takemura et al., 1988; Moruzzi et al., 1989;
Wassermann, 1990; Saunders et al., 1993; Umetsu et al., 2003a).
of γ -MnRh in fig. 6.3. At first glance, this difference could be explained from the data in
figs. 7.89 and 7.92. That is to say, the shrinkage of the lattice constant brings about the
increase of TN . Presuming the same compressibility for the ordered Mn3 Ir alloy, the room
temperature lattice constant estimated by using the compressibility in fig. 7.90 is comparable with the value of the ordered Mn3 Rh alloy under applied pressure of about 2.0 GPa. On
the other hand, the Néel temperature of 960 K for the ordered Mn3 Ir alloy is much higher
than the value of 855 K for the ordered Mn3 Rh alloy (see fig. 6.3). However, fig. 7.89 tells
us that TN of the latter does not go up to TN of the former under the corresponding pressure
of 2.0 GPa. Therefore, the strength of the exchange interaction of Mn systems cannot be
explained by the Mn–Mn distance alone, or by the localized magnetic moment model. In
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
353
Fig. 7.92. The concentration dependence of the room temperature lattice constant of γ -phase disordered (DO)
Mn100−x TMx alloy systems (Krén et al., 1966, 1968a; Yamaoka, 1974; Sasao et al., 1999; Yamauchi et al.,
1999).
the discussion mentioned above, we pay no attention to the difference between the lattice
constants in the ordered and disordered states because no distinct difference between the
two states is observed in the MnRh alloy system (see fig. 6.6).
Spin valve devices are constructed with multilayers. In manufacturing processes of such
multilayers, the substrate temperature could become high above room temperature. Under
such circumstances, it is expected that thermal strains are inevitably induced at high level
in the multilayers due to the difference in the thermal expansion coefficient among the
film layers. In addition, magnetoelastic stresses related with ωs are induced. From practical viewpoints, ωs and thermal strains are not ignored because they are closely correlated
with durability and stability of spin valves. Namely, these stresses and strains easily accelerate diffusion associated with electromigration and stressmigration in spin valves, causing
deterioration of spin valve characteristics. Being associated with deterioration, we should
call attention from the standpoint of metallurgy that the diffusion constant of Mn is much
larger than that of other transition metals. For example, the diffusion constant of Mn is
several times larger than that of Pt in Co (Le Claire and Neumann, 1990).
8. Mechanism of exchange bias-field in spin valves
Exchange biasing characteristics have been investigated for almost 50 years after the
discovery in CoO (Meiklejohn and Bean, 1956, 1957; Jacobs and Bean, 1966). The phenomenological exchange bias-field, its applications and the leading theoretical models have
been reviewed. However, no clear-cut consensus has emerged regarding its origin or possible mechanisms. In spin valve devices, many materials to produce an exchange bias-field
have been developed, and also several models such as the domain wall model (Mauri et
al., 1987), random field model (Malozemoff, 1987), spin flop model (Koon, 1997) and
polycrystalline models (Nishioka et al., 1996; Tsunoda and Takahashi, 2000) have been
proposed to explain the mechanism of the exchange bias-field. In fact, it has been an active
354
K. FUKAMICHI et al.
subject of controversy. Exchange bias-field and related properties have been reviewed by
Nogués and Schller (1999), Berkowitz and Takano (1999), Stamps (2000), Kiwi (2001),
Coehoorn (2003) and Nogués et al. (2005a, 2005b).
After surveying theoretical models proposed by many researchers, in this section, the
AFM spin structures at the interface between the AFM and FM bilayers are modeled by
taking the atomic configuration into consideration in order to discuss the origin of the
exchange bias-field in spin valves (Mitsumata and Sakuma, 2001, 2003; Mitsumata et al.,
2003a, 2003b).
8.1. Survey proposed models of exchange coupling bias
Due to the complex interplay between the microstructure of the AFM layer and interfaces,
as well as the magnetic structures in the AFM layer, the microscopic mechanisms that bring
about the exchange biasing effect are not fully understood yet. A comprehensive overview
for the exchange coupling models is given in this subsection.
8.1.1. Single spin model
Currently, the exchange coupling bias becomes very important to achieve high performance of magnetic devices such as spin valve heads, perpendicular recording media and
magnetic memories. The exchange coupling bias phenomenon at the interface between Co
and CoO as shown in fig. 8.1 was originally found by Meiklejohn and Bean (1956). The
curves give a unidirectional anisotropy in the ferromagnetic (FM) material that contacts
with the antiferromagnetic (AFM) material. Since the first discovery of exchange coupling
bias, many of FM/AFM systems have been investigated experimentally and theoretically.
In the early works of those investigations, a primitive explanation was given (Meiklejohn,
Fig. 8.1. Hysteresis loops of fine oxide-coated particles of cobalt taken at 77 K. The dashed lines stand for the
hysteresis loop when the material is cooled in the absence of a magnetic field. The solid lines show the hysteresis
loop when the material is cooled in a saturating magnetic field (Meiklejohn and Bean, 1956).
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
355
1962), and the free energy of the FM/AFM system is expressed by equation (8.1);
EF = H Ms cos θ − Kd cos θ + Ka cos2 θ,
(8.1)
where Ms is the magnetization of the system, θ is the angle between the magnetic easy
axis and the direction of magnetization, Kd and Ka are the unidirectional and uniaxial
anisotropy energy, respectively. The effective field Heff = H − Kd /Ms is obtained from
equation (8.1). However, they declared no origin of the unidirectional anisotropy energy
Kd . Thus, the microscopic models to explain the mechanism of the exchange coupling bias
are expected at the atomic scale level. The original single spin model was based on the rigid
spin structure for AFM materials. Later on, Néel (1967) included the thermal fluctuation of
spins in the single spin model regarded as the molecular field approximation (MFA). The
perturbation of the fluctuated spins influences the coupling energy at the interface between
FM/AFM. The shrinkage of averaged spin in AFM materials corresponds to the exchange
bias-field. This idea is similar to the spin-wave theory by Suhl and Schuller (1998) (see
section 8.1.9).
8.1.2. Antiferromagnetic (AFM) domain wall model
The exchange energy of the magnetic atoms at the interface of FM/AFM systems is expressed by JI SFM SAFM , where JI is the exchange constant at the interface. SFM and SAFM
are the spins in FM and AFM materials, respectively. The exchange energy becomes of the
order of 10 to 100 meV (100 to 1000 K in temperature), when JI is of the same order of the
exchange energy in FM or AFM material. However, the energies of the exchange bias-field
for many experimental data only reach the order of 1 to 10 µeV. If JI is small enough to
explain the experimental data, the exchange biasing phenomena should appear only near
absolute zero temperature. Mauri et al. (1987) proposed the AFM domain wall model to
solve the quantitative discrepancy between the simple exchange model and the experimental data. The AFM domain wall model brings about an interfacial domain wall, which runs
parallel to the FM/AFM interface. The model assumes a perfectly uncompensated interface with a collinear spin structure in the AFM material given the one-dimensional array
of magnetic atoms. The magnetic structure of the interfacial domain wall is illustrated in
fig. 8.2 (Mauri et al., 1987). The arrows represent the spin vector, and the magnetic easy
axes lie in the z-direction in both AFM and FM layers. The rotation of FM spins forms
the AFM domain wall near the interface due to the exchange coupling of spins at the interface. The energy difference between the magnetic states with and without the domain wall
corresponds to the energy of the exchange bias-field. The magnetic energy of such domain
wall is given by the following equation:
Ewall = 4 AAFM KAFM ,
(8.2)
where AAFM and KAFM are the exchange stiffness and the anisotropy constants, respectively, in the AFM material. The exchange stiffness is AAFM ∼ JAFM /a, and the anisotropy
constant KAFM ∼ DAFM /a 3 . JAFM is the exchange constant, DAFM is the single spin
anisotropy energy, and a is the lattice constant in the AFM material, respectively. The exchange constant JAFM and the anisotropy constant DAFM for FeMn alloy were reported
to be 10 meV and 0.2 µeV, respectively. Thus, the estimated energy becomes the order of
10 µeV, consistent with the experimental data.
356
K. FUKAMICHI et al.
Fig. 8.2. Magnetic model for the interface between a thin ferromagnetic film and a thick antiferromagnetic substrate. The uniaxial anisotropy of the antiferromagnet is along the z axis. The figure depicts a situation in which
an external magnetic field is applied opposite to z and in which the exchange coupling across the interface with
thickness ξ is positive. For the sake of simplicity, the spins of only one sublattice of the antiferromagnet are
shown (Mauri et al., 1987).
Fig. 8.3. Schematic side view of possible atomic moment configurations in the ferromagnetic–antiferromagnetic
sandwich with a non-planar interface. The bump should be visualized at a two-dimensional interface. The configuration (c) represents a lower energy state of (a). The shifted bump of (b) is energetically equivalent to flipping
the ferromagnetic spins of (a) (Malozemoff, 1987).
8.1.3. Random-field model
The assumption of the interfacial domain wall model is not realistic in some cases. Especially for industrial applications of sputtered metallic bilayers, the interface is expected
to be compensated. The summation for AFM sublattice magnetizations notes zero at a
certain direction of the atomic plane. If the interface of FM/AFM system corresponds to
such atomic plane, which is called the compensated interface. Malozemoff introduced the
random-field model to explain the exchange phenomena at the compensated interface. The
physical aspect of the random-field model is illustrated in fig. 8.3 (Malozemoff, 1987). He
assumed the interface roughness due to the existence of atomic steps or defects. The mixture of the positive and negative exchange coupling is associated with a spin frustration at
the rough interface. The spin frustration reduces the exchange energy, and the AF domainlike structure is stabilized in such a frustrated spin state. The following calculated energy
Ebias for the exchange bias-field coincidentally agrees with the order of the energy in the
AFM domain model:
√
2f AAFM KAFM
Ebias =
(8.3)
,
π2
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
357
Fig. 8.4. Frozen domain states in the AFM. Shown are staggered spin configuration (gray and black) of the AFM
interface layer after the initial cooling procedure for dilution p = 0.3. AFM thickness t = 1 (left figure) and
t = 10 (right figure). Vacancies are left white (Nowak et al., 2001).
where f is the structure factor to count the frustrated spins in the unit of coupled spins.
The calculated results within the Ising spin model with the random-field approximation
(Imry and Ma, 1975) were limited within the simple cubic (sc) structure for the atomic
configuration.
Malozemoff tried to extend the model to a Heisenberg spin system. However, the structure factor f becomes visionary (Malozemoff, 1988a, 1988b). The idea of the interface
roughness is necessary to create the localized uncompensation. The physical aspect was
examined for the case of NiFe/CoO system (Takano et al., 1997, 1998), and concluded that
only 1% of interfacial spins contributed to the uncompensation and such a small amount
of unbalanced spin in the collinear spin system results in a quantitative agreement between
the models and the experiments.
8.1.4. Domain state model
The mechanism of the local uncompensation due to interfacial roughness was extended
to the domain state model (Miltényi et al., 2000; Nowak et al., 2001). This model examines the influence of the interfacial roughness by using an ion dosage, and shows an AFM
domain structure where the domain wall runs perpendicular to the interface as shown in
fig. 8.4 (Nowak et al., 2001). The model assumes Ising spins in the AFM layer and Heisenberg spins in the FM layer in Co/CoO bilayer. The roughness is associated with the dilution
by inserting non-magnetic substitutions or defects of the magnetic atoms in the AFM layer,
and the model requires a dilution ratio of p = 30–40% of atomic vacancies at the interface
to obtain the exchange bias-field. Unlikely, the number of required vacancies in the domain
state model is quite different from the interface properties reported by Takano et al. (1997,
1998). In addition, the domain state model was unexplainable for the critical thickness of
the AFM layer. In fact, a numerical simulation shows that exchange bias-field is obtained
in a thickness range from 1 atomic monolayer (ML) to 10 MLs for the AFM layer. Thus,
no critical thickness is found in the FM/AFM system.
358
K. FUKAMICHI et al.
Fig. 8.5. Spin configuration near the interface plane for a 15/15 ML FM/AFM film with lowest energy orientation
(ϕ = 90◦ ). Two interface planes are L15 and L16. The angles are approximately to scale (Koon, 1997).
8.1.5. Spin flop model
The exchange bias-field was also found in the Fe/FeF2 system, and the influence of
interfacial roughness was examined (Nogués et al., 1996). Their report points out that
the smoother the interface, the stronger the exchange bias-field at the fully compensated
interface. This experimental result motivated to propose the spin flop model (Koon, 1997).
The spin flop model assumes that both AFM and FM layers are a body centered cubic (bcc)
structure, and the interface extends along the 110 direction where the AF spins are fully
compensated. The spin configuration near the interface is shown in fig. 8.5 (Koon, 1997).
The spins in the FM layer roughly make an angle of 90 degrees with the AFM spins. The
90 degrees configuration is called the spin flop state. The basic spin structure in the AFM
layer shows an AF-I type collinear configuration, but the AFM spins near the interface cant
away from the Néel axis. The canting effect is most significant in the spin flop model. The
exchange interaction between the FM and AFM spins realizes the canting of spins, and
the resultant spin vector from AFM sublattices becomes finite. The resultant spin vector
is oriented perpendicular to the Néel axis, and the FM spins couple with such resultant
spins. In other words, the canted spins bring about the local uncompensation instead of
the interface roughness. The exchange bias-field was calculated by the rotation of the FM
spins in the spin flop model. The AFM spins near the interface provokes the motion of
the FM spins. However, the AFM spins away from the interface hardly rotate against the
rotation of FM spins. Those behaviors of AFM spins forms the interfacial domain wall,
which is similar to the model by Mauri et al. (1987). Koon (1997) also suggested that the
width of the AFM domain wall dominates the critical thickness of the AFM layer to obtain
the exchange bias-field.
8.1.6. Defect models
The spin flop model seems to succeed the explanation of the microscopic origin for
the exchange bias-field. However, Schulthess and Butler (1998a, 1999) reported another
solution for the spin flop configuration in the FM/AFM bilayer system. Unlikely, the calculation of the exchange bias-field in the spin flop model is based on the rotation of FM
spins parallel to the interface, which is hereinafter described 2D reversal. They pointed out
that the restriction for such 2D spin rotation was responsible for the exchange bias-field.
Their reports note the solution of the dynamic motion equation for Heisenberg spin. The
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
359
Fig. 8.6. Magnetic configuration of a CoO-FM (111) interface, where one FM site is replaced by an AFM site.
The open and filled arrows indicate the relaxed and unrelaxed moment directions, respectively, in the AFM layer.
The FM directions are given by the triangles, where the open triangles in configuration A indicate that the FM
moments are slightly canted away from the ideal spin-flop coupling axis. In configuration A the FM spins are
simply inverted. The dashed lines highlight the next nearest neighbor interactions between the defect site and one
of two AFM sublattices (Schulthess and Butler, 1999).
calculation assumes that the face centered cubic (fcc) structure in the AFM layer, and the
interface is oriented in the 111 direction, where the AFM spins are compensated. The
obtained spin configuration in the AFM layer shows the single Q (≡ 1Q) spin structure,
which is typically collinear. In the FM layer, the spins are perpendicular to the Néel axis
of the AFM layer. The spin structure of this model is similar to the spin flop configuration shown in fig. 8.6 (Schulthess and Butler, 1999). However, the switching path of the
FM spins runs perpendicular to the interface, which is called 3D reversal, in contrast to
the spin flop model by Koon (1997). In addition, that perpendicular magnetization reversal forces a reversible switching at AFM spins. Thus, the magnetization process shows a
symmetric loop without any evidence of unidirectional exchange bias-field. A similar solution was also reported by Xi and White (2000) and Mitsumata et al. (2003a). On the other
hand, the calculated magnetization curve shows a large coercive force. As solution of the
dynamic motion equation, the spin flop state is only responsible for the enhancement of
coercivity.
Based on those discussions, Schulthess and Butler (1999) examined the influence of a
defect at the compensated interface. The defect was simulated by the substitution of the
AFM atom into the FM site at the interface including 4 × 4 atoms. It is regarded as the
“Heisenberg spin version” of the random-field model. The substituted AF atom plays the
role of anchor to fix the FM spin in the given direction. The AFM spins near the interface
are canted away from the original 1Q spin structure with a small angle, and the canting
direction reverses due to the switching of the applied field. The canting angle gradually
decreases and reaches the original 1Q spin structure from the interface to the bottom of
the AFM layer. The calculated magnetization process shows a displaced loop due to the
exchange bias-field, but no remarkable domain wall is observed in the AFM layer.
It should be noted that the spin structure in the AFM layer is important for the case of
the fcc lattice. The reported spin configuration shows the 1Q spin structure in the model by
Schulthess and Butler (1998a, 1998b). However, 1Q, 2Q and 3Q spin structures degenerate
in the ground state for the fcc AF model (Villain et al., 1980; Henley, 1987). In order to
lift such degeneracy, spin fluctuations are required somehow (Long, 1989). The reason to
stabilize the 1Q spin structure should be discussed by Henley (1987). Later on, Kim et al.
(1999) proposed another defect model. The model assumes a simple cubic (sc) structure,
360
K. FUKAMICHI et al.
and the defect was simulated by an atomic vacancy (Kim et al., 2000). A strong anisotropy
is developed in the direction perpendicular to the interface. It forces the spins to lie in the
interface plane. Accordingly, this model corresponds to the model by Koon (1997). In this
case, both the compensated and uncompensated interfaces generate exchange bias-field.
For the compensated interface, the density of defects changes the relative angle between
FM and AFM spins. For the uncompensated interface, the exchange bias-field reduces with
increasing defect density. At the same time, the coercive force increases with increasing
defect density, thus the exchange bias-field and the coercive force behave compensatory.
The temperature dependence of the exchange model was also investigated by Kim et al.
(1999) and Wee et al. (2001). In the low temperature phase, the exchange bias-field is able
to generate. The exchange bias-field suddenly disappears above a certain temperature, and
the compensatory coercive force appears above such a critical temperature. Their defect
model extended to general distributions for vacancies (Kim and Stamps, 2001). The vacancies are distributed in a given atomic plane and such modified atomic plane is located
either at the interface or away from the interface. The interfacial domain wall is formed in
the AFM layer, and it gives rise to the exchange bias-field. The density of defects affects
the exchange bias-field. It is noted that the change in the exchange bias-field is not stable
when the defect-modified plane is located away from the interface.
Sakurai and Fujiwara (2003) discussed planar defects. Their calculation followed the
model proposed by Kim et al. (1999), and it contains planar defects in a simple cubic
lattice system instead of atomic vacancies. The planar defects cause stacking faults in the
crystal, and lead to a reduction of the exchange constant JAFM at a certain depth of the
atomic plane. In their model, the defect plane is located four MLs away from the interface.
The exchange bias-field is realized, even in the case of a fully compensated interface. The
AFM domain wall is formed near the interface, and the shape of the AFM domain wall is
equivalent to the model by Mauri et al. (1987). No defect is necessary to be located at the
interface for the generation of exchange bias-field. Accordingly, the exchange bias-field
is independently realized either by a fully compensated interface or by a uncompensated
interface due to the interfacial roughness in that model. However, it is noted that the energy
calculation for the exchange bias-field is done by rotation of the FM spins in the model. It
is the same method as the spin flop model proposed by Koon (1997). As mentioned above,
in this subsection, the in-plane rotation of the FM spins potentially leads to the generation
of exchange bias-field. To avoid such argument, therefore, the calculation in an external
applied field would be expected to ensure the planar defect model.
8.1.7. Analytical models
A few analytical formulas have been proposed to discuss the AFM domain wall model.
In the early works of the AFM domain wall model, the exchange bias-field was calculated
numerically. This interfacial domain wall model assumes a collinear spin configuration in
the AFM layer with uncompensated interface. These assumptions provide the following
simple expression of the magnetic energy.
E = −Ewall cos α − JI cos(α − β) − KFM t cos2 β − H MFM t cos β,
(8.4)
where α and β are the angles of the AFM and FM magnetization
with respect to the mag√
netic easy axis. The AFM domain wall energy Ewall = AAFM KAFM is given by the
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
361
exchange contrast constant AAFM and the magnetic anisotropy constant KAFM in the AFM
layer. JI represents the exchange contrast at the interface between the AFM and FM layers.
KFM , MFM and t represent the anisotropy constant, the magnetization and the thickness
of the FM layer, respectively. The external field H is applied along the easy axis direction. Based on equation (8.4), analytical solutions were reported by using the continuum
approximation (Xi and White, 1999; Geshev, 2000). Equation (8.4) can be extended to describe the local uncompensation of the interface, corresponding to the defect model with
interfacial roughness. The second term in equation (8.4) was modified by Matsuyama et al.
(2000) as
JI υM FM · M AFM1 + (1 − υ)M FM · M AFM2 ,
(8.5)
where M FM is the magnetization vector in the FM layer. The unit vectors M AFM1 and
M AFM2 represent the magnetizations in the AFM sublattices at the interface with respect
to the collinear spin structure. The coefficient υ stands for the population of defects at
the interface, and υ = 0.5 for a fully compensated interface. In this case, the exchange
bias-field is not realized. On the other hand, the imbalance of the sublattice magnetization
(υ = 0.5) generates the exchange bias-field.
8.1.8. Polycrystalline models
The polycrystalline models do not mention the microscopic origin of the exchange biasfield. However, it is useful to discuss the influence of the magnetic anisotropy in the AFM
layer for the phenomenological analysis of FM/AFM bilayer systems. The basic idea of a
polycrystalline model is illustrated in fig. 8.7. The interface between the FM layer and the
AFM grain follows the model of Meiklejohn and Bean (1956). In other words, the interface
consists of the uncompensated plane. The difference between the analytical model and the
polycrystalline model is the state of frustration for the magnetization. The polycrystalline
model includes the frustration due to the difference between magnetic anisotropy energy
in the individual grains. The grain size and the easy axis direction are defined in each
individual AFM grain, which is in contrast with the uniform FM layer.
Tsunoda and Takahashi (2000) applied the polycrystalline model to investigate the critical thickness of the exchange bias-field. In their report, the model of the torque measurement was discussed to evaluate the magnetic anisotropy energy in the AFM layer. Fulcomer
Fig. 8.7. Schematic model of the FM/AFM bilayer. The spin configuration in the FM layer and in the AFM grain
and the angular relations are indicated (right) (Tsunoda and Takahashi, 2000).
362
K. FUKAMICHI et al.
and Charap (1972) explained the temperature dependence of the exchange coupling for the
oxidized Co particles based on the thermal fluctuation theory within the molecular field approximation. Nishioka et al. (1996) and Fujiwara et al. (1999) applied the polycrystalline
model to the calculation of the blocking temperature in the NiFe/FeMn metallic bilayer.
The blocking temperature TB of the exchange bias-field is expressed by the thermal excitation against the magnetic anisotropy energy in the AFM grains. The calculation shows a
close relation between the grain size and the blocking temperature (see subsection 8.6).
Another polycrystalline model was proposed by Stiles and McMichael (1999a). Basic
configuration for this model was similar to other polycrystalline models. Namely, the uncompensated spin consists of the interface, and the easy direction randomly orients in each
AFM grain. In addition, this model presumes the AFM domain wall, which was originally
introduced by Mauri et al. (1987). The consideration of the AFM domain wall enables the
evaluation for the critical thickness of AFM layers. The critical angle of magnetization
rotation in each AFM grain and the frustration between easy directions of AFM grains influence the exchange bias-field and the enhanced coercivity for the FM layer. In addition,
the temperature dependence of the exchange bias-field is evaluated by considering the
thermally activated spin switching. Accordingly, the blocking temperature TB of bilayer
systems agrees with the Néel temperature TN of the AFM layer (Stiles and McMichael,
1999b).
8.1.9. Other models
The difference between the exchange energy of up and down spins of FM atoms at the
interface may cause the displacement of the magnetization loop. The model for the asymmetric exchange energy at the AFM and FM interface was proposed within the framework
of the spin-wave theory (Suhl and Schuller, 1998). Emission and absorption of spin waves
due to quantum fluctuations in the AFM spin state provides a shift of the ground state energy. The effective field for the energy shift is roughly given by the following expression:
gμB Heff ≈
2SAFM JI2
,
JAFM N
(8.6)
where SAFM is the size of the AFM spin, and N is a normalizing factor. Quantitative
approaches to reveal the exchange phenomena were also investigated in terms of first principles calculations. The exchange constants in the Co/MnNi bilayer system were evaluated
within the Korringa–Kohn–Rostoker (KKR) scheme (Schulthess and Butler, 1998b). The
magnetic structure in the NiFe/MnNi bilayer system was discussed within the full-potential
linearized augmented-plane-wave (FLAPW) scheme (Nakamura and Freeman, 2001).
A summary for the exchange bias-field models is listed in tables 8.1a, 8.1b. The name
of the model is listed in the first column. The lattice system, the spin model used in the
calculations, the spin structure in the AFM layer and the modeled interface are given in the
second column. “One-dimensional array” means that the crystal structure is not specified.
The degree of freedom for FM and AFM spins is specified in term of spin models. “Ising”
notes the Ising spin model, which considers ±S spins. “XY” notes the XY spin model,
which allows the in-plane rotation. “Vector” stands for the classical Heisenberg spin model.
The assumed or calculated spin configuration in the model of the AFM layer is classified
as “Collinear” or “Non-collinear” spin structure. The mechanism for the exchange biasfield is listed in the third column. The origin of the exchange bias-field is expressed in key
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
363
TABLE 8.1a
Models for the exchange bias-field between ferromagnetic and antiferromagnetic system
Model
Lattice system
Spin model
AFM spin
structure
Interface type
(A)
(B)
(C)
Interface only
Interface only
One-dimensional
array
Simple cubic
Simple cubic
XY
XY
XY
Not defined
Not defined
Collinear
Uncompensated
Uncompensated
Uncompensated
(D)
(E)
Single spin
Spin fluctuation
AFM domain
wall
Random-field
Domain state
Collinear
Collinear
Compensated
Compensated
(F)
Spin flop
Collinear
Compensated
(G)
Defect A
(roughness)
Defect B
(vacancy)
Defect C
(planar defect)
Analytical
Body-centered
cubic
Face-centered
cubic
Simple cubic
Ising
Vector (FM)
+ Ising (AFM)
XY
Vector
Collinear
Compensated
Vector
Collinear
Uncompensated
Vector
Collinear
Compensated
XY
Not defined
Uncompensated
XY
Not defined
Uncompensated
(H)
(I)
(J)
Simple cubic
One-dimensional
array
(K) Polycrystalline A
Two-dimensional
array
(L) Polycrystalline B
Simple cubic
(M) Thermal fluctuation Thermal average
aftereffect
at interface
(N) Spin wave
One-dimensional
array
(O) Disordered alloy
Face-centered
cubic
Both XY and vector Collinear
XY
Collinear
Uncompensated
Uncompensated
Spin-wave
Collinear
Compensated
Vector
Non-collinear,
Collinear
Compensated
(A) Meiklejohn and Bean (1956), (B) Néel (1967), (C) Mauri et al. (1987), (D) Malozemoff (1987, 1988a, 1988b),
(E) Nowak et al. (2001), Miltényi et al. (2000), (F) Koon (1997), (G) Schulthess and Butler (1999), (H) Kim et
al. (1999), (I) Sakurai and Fujiwara (2003), (J) Xi and White (2000), Geshev (2000), Matsuyama et al. (2000),
(K) Nishioka et al. (1996), Tsunoda and Takahashi (2000), (L) Stiles and McMichael (1999a), (M) Fulcomer and
Charap (1972), (N) Suhl and Schuller (1998), (O) Mitsumata et al. (2003a).
words. “Interfacial wall” means an AFM domain wall, which runs parallel to the interface.
Relating to the interfacial domain wall, the way of definition for the critical thickness of
AFM layer is indicated for realizing exchange bias-field. It is noted that the critical thickness naturally becomes zero in cases of “Random-field model” and “Domain state model”,
because the magnetic anisotropy energy is regarded as infinite within the framework of the
Ising spin model in the AFM layer. Additional remarks are described in the last column.
8.1.10. Main problems in the proposed models
For almost all available models (A)–(N) listed in tables 8.1a, 8.1b, two major problems
should be pointed out. First, the quantitative disagreement with experiment of the magnetic
energy which is much smaller than the interatomic exchange interaction. Second, the can-
364
K. FUKAMICHI et al.
TABLE 8.1b
Models for the exchange bias-field between ferromagnetic and antiferromagnetic system
Model
Mechanism
Critical
thickness
Remarks
No origin of the unidirectional
anisotropy is given
Molecular field approximation
AFM domain wall is induced
by the rotation of FM spins
Vector spin model can be considered
AFM domain size depends on
the AFM layer thickness
Energy calculation is made by
FM spin rotation
(A)
Single spin
Direct exchange
Unable
(B)
(C)
Spin fluctuation
AFM domain wall
Unable
Wall width
(D)
Random-field
Spin fluctuation
Interfacial planar
wall
Spin frustration
(E)
Domain state
Spin frustration
(F)
Spin flop
(G)
Defect 1
(roughness)
Defect 2
(vacancy)
Spin frustration +
Interfacial domain
wall
Spin frustration
(H)
Interfacial domain
wall
Anisotropy
energy
Anisotropy
energy
Wall width
Anisotropy
energy
Wall width
Interface roughness is required
(I)
Defect 3
(planar defect)
Interfacial domain
wall
Wall width
(J)
Analytical
(K)
Polycrystalline A
Interfacial domain
wall
Spin frustration
Anisotropy
energy
Anisotropy
energy
(L)
Polycrystalline B
Wall width
(M)
Unable
Molecular field approximation
(N)
Thermal fluctuation aftereffect
Spin wave
Frustration + Interfacial wall
Spin fluctuation
Temperature dependences of
coercive force and exchange
bias-field can be calculated
Not required interface roughness
Energy calculation done by rotation of FM spins
Continuum approximation is
applied
Temperature dependence of
magnetic anisotropy can be
discussed
Spin flop coupling is included
Spin-wave excitation
Unable
(O)
Disordered alloy
Spin frustration +
Interfacial domain
wall
Wall width
Self energy shift due to emission and reabsorption spinwaves
Magnetic phase transition is allowed. AFM domains also realize exchange bias-field
(A) Meiklejohn and Bean (1956), (B) Néel (1967), (C) Mauri et al. (1987), (D) Malozemoff (1987, 1988a, 1988b),
(E) Nowak et al. (2001), Miltényi et al. (2000), (F) Koon (1997), (G) Schulthess and Butler (1999), (H) Kim et
al. (1999), (I) Sakurai and Fujiwara (2003), (J) Xi and White (2000), Geshev (2000), Matsuyama et al. (2000),
(K) Nishioka et al. (1996), Tsunoda and Takahashi (2000), (L) Stiles and McMichael (1999a), (M) Fulcomer and
Charap (1972), (N) Suhl and Schuller (1998), (O) Mitsumata et al. (2003a).
cellation of spins ±S at the interface of the AFM layer, that is, the so-called compensated
interface. Introducing the magnetic domain wall structure near the interface, quantitative
agreement with experiment was first achieved by Mauri et al. (1987). Both the quantitative
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
365
agreement and the problem of the compensated interface were explained within the framework of the random field model by Malozemoff (1987). In this model, both the roughness
at the interface and the AFM domain are required. On the other hand, Koon (1997) proposed a spin flop model having a smooth interface, and explained relaxed spin structures
at the interface. Later on, however, different solutions in the spin flop arrangement for the
magnetization reversal process were pointed out (Schulthess and Butler, 1998a, 1998b; Xi
and White, 2000). Those models are based on the collinear spin structure which shows antiparallel arrangement of AFM spins in the AFM layer. However, it should be emphasized
that various types of Mn alloys are used in practical applications and their spin structures
are not necessarily collinear.
8.2. Generalized model based on classical Heisenberg model
Antiferromagnetic γ -phase alloys with a high Néel temperature have intensively been
investigated as exchange biasing films. The spin structure of antiferromagnetic γ -phase
alloys with the 3Q spin structure is non-collinear as discussed in section 5. Therefore, we
need a generalized model to explain the exchange bias-field properties for AFM films with
not only collinear but also non-collinear spin structures. In this subsection, several kinds
of exchange bias-field characteristics are discussed by using a generalized model based on
the classical Heisenberg model listed in the last column in tables 8.1a, 8.1b (Mitsumata et
al., 2003a).
8.2.1. Total magnetic energy and equation of motion
The atomic configuration in the generalized model is illustrated in fig. 8.8 (Mitsumata
and Sakuma, 2002; Mitsumata et al., 2003a). The triangular net planes are stacked in the
111 direction, forming a compensated interface between the FM and the AFM layers. The
closed and open circles respectively represent the magnetic and non-magnetic atoms in the
AFM layer, composing a binary alloy. The atomic configuration in fig. 8.8 corresponds
to the case of an L10 -type ordered AFM layer. For the γ -phase disordered AFM layer,
the lattice becomes an fcc structure. The double circles denote the magnetic atoms in the
FM layer. The layer thickness of the FM layer is 9 monolayers (MLs), corresponding to
about 1.5 nm for typical FM layers, e.g., Ni, Fe and Co. On the other hand, the AFM
layer thickness ranges form 10 to 120 MLs. One may notice that the interface between the
FM and the AFM layers is clearly defined without any roughness. The periodic boundary
condition is applied to the in-plane direction of the layered film, and the calculated region
includes 24 by 24 atoms in each monolayer (ML).
The spin structures and the magnetization process of the FM/AFM bilayer have been
investigated by using the classical Heisenberg model. Within the framework of this model,
the interactions up to the second nearest neighbors are discussed, because the interaction
between the second nearest neighbors plays an important role to establish the long-range
order; the antiferromagnetic spin arrangement in frustrated spin systems in the present case.
The thick arrows stand for the exchange interaction J1 between the first nearest neighboring spins, and the dashed arrows represent the FM exchange interaction. The thin arrows
indicate the exchange interaction J2 between the second nearest neighboring spins and the
dashed arrows represent the FM exchange interaction as well.
In the classical Heisenberg model, the interlayer exchange constant is assumed to be
equal to the exchange constant in the AFM layer. The total magnetic energy is given by the
366
K. FUKAMICHI et al.
Fig. 8.8. The model lattice of the FM and AFM bilayer (Mitsumata and Sakuma, 2002; Mitsumata et al., 2003a).
following Hamiltonian (Mitsumata and Sakuma, 2001; Mitsumata et al., 2003a).
H =−
J1ij S i · S j −
i,j −
J2ik S i · S k
i,k
A(S i ) − gμB
i
(H di + H app ) · Si ,
(8.7)
i
where S i is the unit spin vector of the i-th magnetic atom, J1 and J2 are the exchange constants for the first and the second nearest neighbors, respectively. The summations i, j and i, k are carried out over all possible pairs of the first and the second nearest neighbors. The third term in equation (8.7) represents the magnetic anisotropy energy. When the
system shows the uniaxial anisotropy, the function A(S i ) is given by
A(S i ) = Di S i · n2 ,
where Di and n mean the single spin anisotropy energy and the unit vector along the easy
direction of the magnetic anisotropy, respectively. On the other hand, when the system
shows the cubic anisotropy, the function A(S i ) is given as
A(S i ) = D1 αb2 αc2 + αc2 αa2 + αa2 αb2 + D2 αa2 αb2 αc2 ,
(8.8)
where D1 and D2 denote the second and fourth orders of magnetic anisotropy constants in
the AFM layer, respectively. Also, αa , αb , and αc , respectively, denote the elements of the
spin vector S i in the directions of the a-, b-, and c-axes in the γ -phase alloy (Mitsumata
et al., 2003a). The last term in equation (8.7) describes the Zeeman energy, H d is the demagnetized field due to the dipole interaction between spins, and H app is the applied field.
Typically, the exchange constants of magnetic alloys composed of Mn, Ni, Fe and Co are
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
367
of 1–30 meV, which can be estimated from the Néel temperature or the Curie temperature.
For the sake of simplicity, the first nearest exchange constant J1 is assumed to be 20 meV
for the FM layer and to be −20 meV for the AFM layer and the interface, and the second
nearest exchange constant J2 is given by J2 = |J1 |/2 (Schulthess and Butler, 1998a). For
the calculation of the demagnetized field H d , the magnetic moment of the atom is assumed
to be 1.7 μB like that of Co. To take a well-known example, Ni-Fe or Co-Fe alloy is used
for the FM layer in the FM/AFM bilayer system, and a typical anisotropy field of such
alloys shows the order of 1–10 Oe. Accordingly, the anisotropy energy of 0.116 µeV corresponds to 10 Oe. The anisotropy energy of the AFM layer becomes 103 –104 times larger
than that of the FM layer (Sakuma, 1998a), and hence we can assume it to be 0.116 meV.
For the Monte Carlo calculations of the spin structures in the AFM layer system, the
single spin flip Metropolis algorithm is employed. First, 5 × 103 Monte Carlo steps are
spent for the relaxation of the system, and next 2 × 105 Monte Carlo steps are adopted as
the Monte Carlo average for the physical observable. Furthermore, the following Landau–
Lifshitz motion equation is used to solve the magnetization process in the external applied
field to the FM/AFM bilayer.
dS i
= −νS i × H effi − α S i × (S i × H effi ) ,
(8.9)
dt
where S i is the unit vector of the i-th spin in the system, and α is the damping constant
which is assumed to be 0.1 in the calculations. The effective field at the i-th site H effi is
given by the following derivative of the Hamiltonian in equation (8.7):
∂H
.
(8.10)
∂S i
This motion equation is solved by using a forward difference method. The time step t
in the difference equation is defined by t = 0.02/(νHapp ), where ν and Happ are the
gyromagnetic constant and the applied field, respectively (Mitsumata et al., 2003a).
H effi = −
8.2.2. Unidirectional exchange bias-field
The spin structure of the disordered AFM layer including 75% of magnetic atoms and
25% of non-magnetic atoms in analogy with the γ -phase disordered Mn alloys is calculated
by employing the Monte Carlo method for the 3Q structure in fig. 5.2. Also, the spin
structure of the disordered alloy with 97.5% of magnetic atoms and 2.5% of non-magnetic
atoms is calculated for the 1Q structure in fig. 5.2. The atomic arrangements and the spin
configurations in the AFM layers are illustrated in fig. 8.8 (Mitsumata and Sakuma, 2002;
Mitsumata et al., 2003a). The temperature used in the calculations is T /J1 = 0.08617,
giving the condition of T TN . In fact, the value of TN /J1 is 1.896±0.038, 2.887±0.053
and 3.567 ± 0.061 for the 3Q, 1Q and AF-I structures in figs. 5.2 and 5.15, respectively.
This is the reason why we use such a value of T /J1 is for the representation of the spin
structures in this model. Later on, as given in fig. 8.9 and the following discussion, we
evaluate the exchange bias-field by using the Landau–Lifshitz motion equation which is
generally solved at 0 K. The Néel temperatures TN in figs. 5.2 and 5.15 are obtained from
the following equation:
−1
T − TN
∂
=
,
lnS̄I (8.11)
∂T
βc
368
K. FUKAMICHI et al.
Fig. 8.9. The magnetization loops of the FM/AFM bilayer with the L10 -type (AF-I) ordered and the γ -phase
(1Q, 2Q and 3Q) disordered AFM alloys (Mitsumata et al., 2003b). AF-I (thin solid line), 1Q (dashed line), 2Q
(dotted line) and 3Q (thick solid line).
where S̄I is the average size of the classical spin. The spin vector is obtained by averaging
the spins on equivalent sites:
S̄I =
1
N
Si ,
cos θ = S̄I · S̄K ,
(8.12)
i
where N is the number of magnetic atoms on each sublattice I. The correlation angle θ of
the spin structures is calculated by a linear combination of averaged spins. In fig. 5.2, it
averages over four spins on the equivalent sites I (I = 1, 2, 3, 4), it averages over two spins
on the equivalent site I (I = 1, 2, where 1 and 2 consist of magnetic atoms). βc is the critical
exponent in the expansion equation of m ∝ (T − TN )βc near the critical temperature. In
the magnetic unit cell of a γ -phase disordered AFM layer in the 3Q structure in fig. 5.2,
four equivalent atoms are included, and the spins of those atoms point into the [111], [1̄1̄1],
[11̄1̄] and [1̄11̄] directions, respectively. This spin configuration corresponds to the 3Q spin
structure, which is a typical non-collinear spin structure in the fcc lattice, resulting from
the frustrated spins due to the atomic geometry.
Within the framework of the classical Heisenberg model, in principle, the ground state
energy of the collinear spin structure, or the 1Q structure, is not distinguishable from the
non-collinear spin structure of the 3Q structure, in the fcc lattice (Villain et al., 1980;
Henley, 1987) where all equivalent atoms are purely magnetic like γ -Mn. However, the
presence of non-magnetic atoms changes the number of interacting spins at the nearest
neighboring atomic sites, causing spin frustration in the γ -phase disordered alloy system.
This spin frustration lowers the energy of the 3Q spin structure, compared with that of the
collinear spin structure (Long, 1989). The validity of this behavior when using the motion
equation of spins is confirmed to assume the initial spin configuration is the 1Q spin structure, and hence the final solution is obtained to be the 3Q spin structure in the case of the
disordered alloy with 75% magnetic atoms and 25% non-magnetic atoms. The magnetic
energy decreases from −120 to −126 meV/spin by changing the spin structure from 1Q
to 3Q. In addition, the spin configuration of the ordered AFM layer having the L10 -type
lattice structure is calculated by using the Monte Carlo method. Two magnetic and two
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
369
Fig. 8.10. The magnetization switching process in the FM layer of an FM/AFM bilayer consisting of L10 -type
ordered and γ -phase disordered AFM alloys. (a) and (b) are schematic images of the magnetization reversal for
the L10 -type ordered and the γ -phase disordered AFM alloy, respectively, (c) the three magnetization components
against time steps for the bilayer consisting of L10 -type ordered AFM alloy having the collinear spin structure
arrangement, (d) the same plots for the γ -phase disordered AFM alloy having a non-collinear spin structure
arrangement (Mitsumata et al., 2003a).
non-magnetic atoms are included in the magnetic unit cell as seen from fig. 8.8. The spin
correlation angle between the nearest neighboring spins indicates 180 degrees and it corresponds to the AF-I spin structure, being a typical collinear spin structure of AFM alloys.
Then, the magnetization process in an external field is calculated by adding the FM layer to
such AFM layers. The calculated magnetization loops are presented in fig. 8.9, where ms
is the saturation magnetization (Mitsumata et al., 2003b). In these cases, the magnetic easy
directions in the AFM and FM layer are [112̄] and [11̄0], respectively. As seen from the
figure, the FM/AFM bilayer with the 1Q, 2Q or AF-I structures only shows coercivity in
the magnetization loop without any shift. That is, only the 3Q structure can realize a loop
shift caused by the unidirectional exchange bias-field. These distinct different magnetization loops are explained in the following way. Each component of the magnetization vector
m in the FM layer during the magnetization reversal process is given in fig. 8.10(a) and
(b) for the γ -phase disordered AFM and the L10 -type ordered AFM layer, respectively
(Mitsumata et al., 2003a). The x- and y-components lie in the layered film, and the z-
370
K. FUKAMICHI et al.
component is perpendicular to the film plane. The initial magnetization points into the +y
direction, and the switching field is applied along the −y direction. While the y-component
my decreases from 1 to −1, mx and mz oscillate and converge to zero, exhibiting Larmor
precession during the magnetization process. Let us attempt to focus on the mz component
for an ordered AFM layer having the AF-I spin structure in fig. 8.10(a), and the case of a
γ -phase disordered AFM layer having the 3Q spin structure in fig. 8.10(b). In fig. 8.10(c)
and (d), the vertical axis denotes the magnetization of the x, y and z components, and
the horizontal axis denotes the time step in the calculations. The mz component in the
former increases and makes the first peak of the oscillation before my crossing the zero
line in fig. 8.10(c), while mz in the latter is restricted to a small value and makes the first
peak after my crossing the zero lines in fig. 8.10(d). The former corresponds to the 3D
reversal and the latter to the 2D reversal in the first half of the magnetization process. As
a result, the magnetization loop shift is developed only in the bilayer system having the
non-collinear spin structure, namely the 3Q structure (Mitsumata et al., 2003a, 2003b). As
pointed out above, those switching modes are influenced by the spin structure of the AFM
layer. Therefore, the investigation of the spin structure at the interface is important. The
correlation angle between the FM and the AFM spins indicates about 90 degrees for the
ordered AFM layer, and the correlation angle of all the FM and AFM spins is likely to keep
90 degrees during the whole magnetization reversal process. In fig. 8.11(a), the FM spins
always lie in the film plane and the AFM spin structure is called the spin flop structure, and
the spins show a 3D reversal with the component perpendicular to the interface, because
the AFM spins remain in the film plane. This is because the correlation angle between
the FM and AFM spins corresponds to the bisection of the correlation angle between the
AFM spins in a collinear configuration, and the switching path along such bisection gives
the minimum state in the exchange energy. Next, we show the spin configuration at the
interface with a γ -phase disordered AFM layer as shown in fig. 8.11(b). The FM spins
lie in the film plane, but the spin component perpendicular to the interface is due to the
γ -phase disordered AFM layer where the AFM spin makes angles of +40 or −40 degrees
with the interface. The correlation angle between the FM and the AFM spins agrees with
the bisection of the correlation angle between AFM spins, and it is also equivalent to the
Fig. 8.11. The spin configuration at the interface of bilayer system. (a) L10 -type ordered AFM layer, (b) γ -phase
disordered AFM layer (Mitsumata and Sakuma, 2001, 2002; Mitsumata et al., 2003a).
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
371
angle between the AFM spins and the interface. As a consequence, the direction of the FM
spins is restricted near the parallel plane to the interface during the switching process. In
other words, the FM spins with the γ -phase disordered AFM layer exhibit a 2D reversal
(Mitsumata and Sakuma, 2001, 2002; Mitsumata et al., 2003a).
8.2.3. Formation of magnetic domain wall in antiferromagnetic layer
As seen from tables 8.1a, 8.1b, the formation of domain walls in the antiferromagnetic
layer plays an important role in the exchange bias-field. In this subsection, therefore, the
relation between the formation of domain walls in the γ -phase and L10 -type antiferromagnetic layers and the exchange bias-field is discussed.
8.2.3.1. Magnetic domain wall in 3Q spin structure The thickness dependence of the
exchange bias-field in the bilayer system is shown in fig. 8.12 (Mitsumata et al., 2003a).
Below about 25 MLs, the exchange bias-field Hu decreases when the AFM layer thickness
ML decreases, and it becomes zero below a certain critical thickness. The coercive force
Hc takes a maximum value around such a critical thickness. The systematic error in the
calculations of Hu and Hc is 0.005 in the normalized units in fig. 8.12, and the statistical
error is about ±0.15. These behaviors are very similar to the measured results for an FeMn
film as the AFM layer (Jungblut et al., 1994), which has been reported to have the 3Q spin
structure (Endoh and Ishikawa, 1971; Sakuma, 1998a). The unidirectional exchange biasfield is influenced not only by the spin structure at the interface but also by the macroscopic
spin arrangement in the AFM layer. The magnetization components m/ms for the four
equivalent atoms along the applied field are plotted as a depth profile of the stacking films
in fig. 8.13 (Mitsumata et al., 2003a). The magnetization of an equivalent atom is averaged
over the net plane for each depth. The stacked planes for depths from 1 to 9 ML correspond
to the FM layer, and the stacked planes for depths from 10 to 49 ML correspond to the
AFM layer. The magnetization in the FM layer equals +1 and −1 in positive and negative
applied fields, respectively. The value of m/ms at each equivalent atomic site, I, II, III and
IV, in the AFM layer is about +0.5 or −0.5, reflecting the correlation angle between the
FM and the AFM spins. Approaching the interface, the value of the magnetization |m|/ms
Fig. 8.12. The thickness dependence of the unidirectional exchange bias-field Hu and coercivity Hc with the
γ -phase disordered FM/AFM bilayer. The unidirectional exchange bias-field and coercivity are given by Hu and
Hc , respectively (Mitsumata et al., 2003a).
372
K. FUKAMICHI et al.
becomes small and a magnetic domain wall is formed parallel to the interface, similar to
the spin configuration presented by Mauri et al. (1987), and the wall shape is changed,
depending on the direction of the applied field. The magnetic energy difference between
positive and negative applied magnetic fields in the 3Q structure agrees with the order
of measured energy values for the exchange bias-field. Such magnetic energy difference
is due to the magnetic energy difference resulting from the formation of the magnetic
domain wall (MDW) parallel to the interface. The wall structures in the bilayer with 3Q
structure are illustrated in fig. 8.14(a) and (b) (Mitsumata et al., 2003b). The directions
of the spins on the equivalent site B3 in fig. 8.11(b) are plotted as the depth profile. The
direction of depth corresponds to the z-axis, and the AFM and FM layers are at z < 0
and z > 0, respectively. In the AFM layer, a twisted state of spins is observed, forming
a MDW parallel to the interface. Near the interface, the rotation angle of the spin in the
Fig. 8.13. The depth profile of the magnetizations in the stacked planes of the γ -phase disordered FM/AFM
bilayer. Four types of lines; dots, thin, cross and thick, indicate the averaged magnetization of the equivalent
atomic sites, I, II, III and IV in each stacked plane. The depth from 1 to 9 corresponds to the FM layer, and that
from 10 to 49 corresponds to the AFM layer (Mitsumata et al., 2003a).
Fig. 8.14. The magnetic domain wall structures in a bilayer with 3Q structure in the AFM layer. The bold lines
denote the direction of spins on the atomic site B3 in fig. 8.11. (a) For a positive applied field, (b) for a negative
applied field (Mitsumata et al., 2003b).
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
373
MDW is changed under the applied field +H in fig. 8.14(a) and the applied field −H in
fig. 8.14(b). This small change of the MDW provides the magnetic energy difference in the
3Q structure. Apparently, the change of the MDW shape gives rise to the difference in the
magnetic energy equivalent to the energy of the exchange bias-field.
8.2.3.2. Introduction of multi-domains into L10 -type antiferromagnetic layer The apparent magnetization loop shift in fig. 8.9 was explained to be caused by the non-collinear spin
structure, which is due to the random arrangement of magnetic and non-magnetic atoms
in the γ -phase disordered AFM alloys. In practical applications, however, the L10 -type
ordered AFM alloys such as MnPt (Farrow et al., 1997; Saito et al., 1997), MnNi (Lin
et al., 1994; Wong et al., 1996a) and MnPdPt (Tanaka et al., 1997) have excellent properties for the unidirectional exchange bias-field even though they have the collinear spin
structure. Accordingly, at first glance, this model seems to be invalid for the formation of
unidirectional exchange bias-field in the collinear spin structure systems. We shall attempt
to include the spin frustration effects in the L10 -type ordered AFM layer. For this purpose,
we introduce multi-domains as illustrated in fig. 8.15 (Mitsumata et al., 2003a). In this
connection, it should be noted that multi-variants exist with keeping the energy of shear
strains minimum in the phase transformation from cubic structures to L10 -type structure,
showing various crystallographic orientations (Tanaka et al., 2000; Fujita et al., 2003). In
addition, there are many twin boundaries (Newkirk et al., 1950; Vlasova and Shur, 1987;
Wong et al., 1996a, 1996b) and antiphase boundaries in the transformed microstructures
(Klemmer et al., 1995; Frommen and Rösner, 2003). Accordingly, the introduced domain
structures are regarded as the multi-variants in single crystals and grains in polycrystals.
Two magnetic atoms and two non-magnetic atoms are included in the magnetic unit cell in
Fig. 8.15. Schematic drawing of the multi-domains model (Mitsumata et al., 2003a).
374
K. FUKAMICHI et al.
Fig. 8.16. The magnetization loop of the FM layer stacked on the L10 -type ordered AFM layer when
multi-domains are introduced (Mitsumata et al., 2003a).
Fig. 8.17. The thickness dependence of the unidirectional exchange bias-field Hu (dotted line) and coercivity Hc
(solid line) of the FM / AFM bilayer of the L10 -type ordered alloy (Mitsumata et al., 2003a).
fig. 8.15. These magnetic atoms are arranged along the [01̄1̄], [1̄01̄] or the [1̄1̄0] direction
in the 111 oriented stacking sequence of planes. The easy axis of magnetic anisotropy
in the AFM layer is along the same direction in each unit cell, and the easy axis in the
FM layer is [1̄1̄0]. The configurations of atoms are equal in all of these three patterns,
forming the L10 -type ordered lattices. The single domain is composed of one of those
three patterns, which includes 12 × 12 atoms in the stacking plane. The entire calculating model includes 3 × 3 domains, and the outer edge of the calculating region in the
111 stacking plane is defined as the periodic boundary condition. The frustrated spins
are caused by the geometry of the magnetic atoms at the domain boundaries. As a result,
in contrast to the loop in fig. 8.9, the calculated magnetization loop is actually shifted by
a unidirectional exchange bias-field even in the L10 -type ordered alloy systems as shown
in fig. 8.16 (Mitsumata et al., 2003a). Therefore, it is demonstrated that the introduction of
frustrated spins causes a unidirectional exchange bias-field. The thickness dependence of
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
375
Fig. 8.18. The depth profile of magnetizations on stacked planes after introducing multi-domains in the FM/AFM
bilayer of the L10 type ordered alloy. (a) The depth from 1 to 9 corresponds to the FM layer with 9 MLs, and
the depth from 10 to 29 corresponds to the AFM layer with 20 MLs, (b) the depth from 1 to 9 corresponds to the
FM layer with 9 MLs, and the depth from 10 to 49 corresponds to the AFM layer with 40 MLs. Note that the
AFM layer thicknesses correspond to values below and above the critical thickness for the exchange bias-field
phenomenon (Mitsumata et al., 2003a).
the exchange bias-field is given in fig. 8.17 (Mitsumata et al., 2003a). The unidirectional
exchange bias-field decreases with decreasing AFM layer thickness, coming close to zero
below a certain critical thickness. The maximum value of coercivity Hc is obtained near
such a critical thickness, accompanied by a sharp peak. These critical behaviors are very
similar to the results in fig. 8.17. The depth profile of the spin configuration is shown in
fig. 8.18(a) and (b), where the AFM layer thickness is 20 MLs in fig. 8.18(a) and 40 MLs in
fig. 8.18(b) (Mitsumata et al., 2003a). The magnetizations are measured along the applied
field direction, and they are averaged over the net plane in each depth. The stacked planes
in depth from 1 to 9 MLs correspond to the FM layer. The stacked planes in depth from
10 to 29 MLs correspond to the AFM layer in fig. 8.18(a), and the stacked planes in depth
from 10 to 49 MLs correspond to the AFM layer in fig. 8.18(b). The thicknesses of 20 and
40 MLs of the AFM layer respectively represent the conditions below and above the critical thickness where the magnetization loop shift appears. In the former case in fig. 8.18(a),
there is a small rotation of spins in the AFM layer. Switching the external field from +Happ
to −Happ , the magnetization at the site I changes from positive to negative and vice versa
at the site II. As a result, the magnetic structure shows a symmetric configuration in the
applied field. In the latter case in fig. 8.18(b), on the other hand, the spins at the sites I and
II retain the same direction without switching, e.g., the magnetization at the site I always
indicates a positive value in the AFM layer regardless the positive +Happ and negative
−Happ applied fields. Then, a MDW structure is formed in the AFM layer, and the wall
width changes from 10 to 30 MLs in alternative applied fields. Therefore, this spin structure
shows an asymmetric configuration in the applied field. Obviously, the symmetric change
376
K. FUKAMICHI et al.
Fig. 8.19. The magnetic energy of the FM/AFM bilayer as a function of applied field. (a) 20 ML thickness of the
AFM layer, (b) 40 ML thickness of the AFM layer. Note that these thicknesses correspond to values below and
above the critical thickness for the exchange bias-field phenomenon (Mitsumata et al., 2003a).
of the magnetic configuration in the applied field represents the absence of a unidirectional
exchange bias-field Hu , and this asymmetric configuration in the applied field results in
the magnetization loop shift. Also, this domain wall motion induces a difference between
the magnetic energy in the bilayer under positive and negative fields. The magnetic energies are plotted as a function of applied field in fig. 8.19(a) and (b) for 20 and 40 MLs of
the AFM layers, respectively (Mitsumata et al., 2003a). Symmetric and asymmetric curves
of the magnetic energies are obtained for the thinner and the thicker AFM layers, respectively. Assuming a bilayer constructed by the 111 stacking of triangular net planes which
forms a compensated interface between the FM and AFM layers, Monte Carlo calculations
within the framework of the classical Heisenberg model have been carried out (Mitsumata
and Sakuma, 2001, 2003; Mitsumata et al., 2003a, 2003b). For the magnetic structures in
the AFM layer, a MSDW for γ -phase disordered alloys and an AF-I structure for L10 type ordered alloys were considered by taking appropriate concentrations of non-magnetic
atoms in the fcc lattice. In the classical Heisenberg model, it should be noted that the 3Q
structure is essentially preferred among the MSDW structures because of the existence of
randomly arranged non-magnetic atoms in the AFM fcc lattice. The 1Q and 2Q and AF-I
structures can be realized by making the exchange interactions anisotropic; different values
for the interactions within and perpendicular to the (001) plane.
As presented in fig. 8.9, the point to observe is that only the 3Q structure can create exchange coupling bias instead of the 1, 2Q and AF-I structures. These results tell us that the
non-collinear spin structure caused by the geometric frustration in the AFM layer is respon-
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
377
sible for the magnetization loop shift. Experimentally, it is well known that the exchange
bias-field is drastically reduced with decreasing Ir concentration below about 20 at.% Ir
(Hoshino et al., 1996; Fuke and Kamiguchi, 1998), very close to the concentration where
the spin structure change from the 3Q to 2Q spin structure takes paces (see fig. 7.46). On
the other hand, a collinear spin structure formed in the L10 -type ordered AFM alloys results only in an enhancement of coercivity of the FM layer. On introducing multi-domains
into the L10 -type ordered AFM layer, however, the magnetization loop shift of the FM
layer is evidently developed by the geometrically frustrated spins induced at the magnetic
domain boundaries as given in fig. 8.16.
It is important to note that the magnitude of the exchange bias-field corresponds to the
gain of the exchange energy due to the canting of spins caused by the frustration. This
well explains that L10 -type ordered alloys having a large canting angle of spins exhibit a
large exchange bias-field (Lin et al., 1994; Saito et al., 1997), compared with that of the
γ -phase disordered alloys. In this frustration model, the introduction of the frustrated spins
configuration in the AFM layer and the formation of the magnetic domain wall parallel to
the interface of the FM and AFM bilayer are necessary for the development of the unidirectional exchange bias-field. These prerequisites, the frustrated spins and the domain
wall, are common to both the disordered γ -phase and L10 -type ordered AFM systems.
In addition, the multi-domain structures to create frustrated spins are also required for the
collinear spin structure in the L10 -type ordered AFM layer. We see from fig. 8.15 that the
AFM domain structures are in part similar to the random field model (Malozemoff, 1987)
which stands within the framework of the Ising spin system. In a wide sense, however,
the AFM domain in the random field model can be regarded as the frustration model. The
domains also correspond to the grains in the polycrystalline models (Nishioka et al., 1996;
Tsunoda and Takahashi, 2000). Those two kinds of the polycrystalline models include no
influences of the frustrated spins at the interface. As it turned out, those polycrystalline
models are naturally extended to the generalized model of the compensated interface by
taking the influence of frustrated spins into consideration. The role of the spin frustration
caused by the competition of exchange interactions due to geometry of the magnetic atoms
in the AFM disordered layer is significant as discussed above. From the generalized frustration model, it is possible to explain the mechanism for the dependence of the exchange
coupling bias on the composition in AFM fcc alloys (Mitsumata et al., 2003b). If the origin
of the exchange bias-field is the interface roughness, the interface roughness should change
with the composition in the AFM layer, or else the dependence of the exchange bias-field
on the composition in the AFM layer is hardly explained.
8.2.4. Cubic anisotropy and torque curves
The exchange bias-field models are based on the collinear spin structure of AFM materials. However, the single Q (1Q), double Q (2Q), and triple Q (3Q) spin structures are
obtained in a disordered AFM alloy (see fig. 5.2). In these structures, the 3Q spin structure
without interface roughness is necessary to form the exchange bias-field, and neither the
1Q nor the 2Q spin structure in the AFM layer shows the exchange bias-field. In addition, when the exchange bias-field is obtained, a magnetic domain wall is formed in the
AFM layer, running parallel to the interface in the bilayer. In both ordered and disordered
systems, the formation of a domain wall is closely related to the generation of exchange
378
K. FUKAMICHI et al.
bias-field (Mauri et al., 1987; Kim and Stamps, 2001; Mitsumata et al., 2003a, 2003b;
Sakurai and Fujiwara, 2003), and affects the critical thickness of the AFM layer.
In the preceding subsection, the unidirectional magnetic anisotropy in the AFM layer
was discussed. On the other hand, also cubic anisotropy is expected in the disordered γ phase AFM alloy. In this subsection, therefore, the influence of cubic anisotropy in the
AFM layer is discussed. First, the dependence of exchange bias-field on the anisotropy
energy in the disordered γ -phase alloy is explained. Next, the torque curves of AFM/FM
bilayers are calculated with various anisotropy energies, because the magnetic structure
under a rotational applied magnetic field is an important aspect of the domain wall model
(Mauri et al., 1987; Koon, 1997; Kim and Stamps, 2001; Zhang et al., 2002; Sakurai and
Fujiwara, 2003).
8.2.4.1. Cubic anisotropy and exchange bias-field The magnetic structure of the
AFM/FM bilayer was investigated within the framework of the classical Heisenberg model
(Mitsumata et al., 2004a). The model Hamiltonian for the calculation of the magnetic energy is given by equation (8.7). In the third term, the function A(S i ) denotes the magnetic
anisotropy energy which is expressed by the following equations (8.13) and (8.14). When
Si belongs in the FM layer,
A(S i ) = Du S i · ni 2 ,
(8.13)
where Du denotes the uniaxial anisotropy constant and ni is the unit vector pointing in the
direction of the magnetic easy axis in the FM layer. On the other hand, when S i belongs in
the AFM layer,
A(S i ) = D1 αb2 αc2 + αc2 αa2 + αa2 αb2 + D2 αa2 αb2 αc2 ,
(8.14)
where D1 and D2 denote the second and fourth order magnetic anisotropy constants in
the AFM layer, respectively. The quantities, αa , αb , and αc denote the direction cosines
between the spin vector Si and the a-, b-, and c-axes in the γ -phase alloy, respectively.
The magnetization process is calculated to solve the Landau–Lifshitz–Gilbert equation
given by equation (8.9). The effective magnetic field Heff is given by the derivative of the
Hamiltonian in equation (8.7) as expressed by equation (8.10). The atomic configuration
of the FM/AFM bilayer is the same in fig. 8.8. The lattice site of fcc is randomly occupied
by 75% magnetic atoms and 25% non-magnetic atoms, and then these atoms compose
the disordered γ -phase alloy. The orientation of the bilayer is the 111 stacking as in
exchange coupling layers in industrial applications. In the 111 directions, the thickness
of the FM layer is 9 monolayers (MLs), and that of the AFM layer is 100 MLs in the
model. For simplicity, the figure shows only the atoms near the interface. In the [1̄10] and
[011̄] directions, 24 atoms are aligned, and the periodic boundary condition is applied. The
magnetic structure of the FM/AFM bilayer is shown in fig. 8.20 (Mitsumata et al., 2006).
The gray and black arrows represent the FM and AFM spins, respectively. In the model, the
magnetic easy axis of the AFM layer is one of the 100 directions, and is selected [1̄10]
as the easy axis of the FM layer. The exchange constant J1 is set to be consistent with the
Néel temperature of the AFM layer (Mitsumata et al., 2003a, 2003b), and J2 is defined
as J2 = |J1 |/2 (Schulthess and Butler, 1998a; Mitsumata et al., 2003a). The anisotropy
energy is set as D1 /J1 = 5.8 × 10−3 . Disordered AFM alloy shows the 3Q spin structure,
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
379
Fig. 8.20. Magnetic structure of the AFM/FM bilayer with the disordered alloy model (Mitsumata et al., 2006).
(a) 3Q spin structure in a magnetic unit cell, (b) spin configuration in the vicinity of the interface.
and such non-collinear spin structure is associated with spin frustration due to the atomic
geometry. The 3Q spin structure in a magnetic unit cell is shown in fig. 8.20(a). The spins
of equivalent atoms I, II, III, and IV point to the [1̄11], [11̄1], [1̄1̄1̄] and [111̄] directions,
respectively. The spin configuration near the interface is illustrated in fig. 8.20(b). The
external magnetic field is zero in this calculation of the spin structures. The exchange
interaction between the AFM and FM layers changes the AFM spin direction, resulting in
that AFM spins at the interface are canted from the exact 3Q spin structure. The canting
angle becomes small when the AFM atoms are distant from the interface, ranging in the
vicinity from π/30 to π/12. The exchange bias-field is calculated as a function of the cubic
anisotropy energy in fig. 8.21 (Mitsumata et al., 2006). The anisotropy constant in the FM
layer is assumed to be Du /J1 = 5.8 × 10−6 , and the anisotropy constants in the AFM
layer range from D1 /J1 = D2 /J1 = 5.8 × 10−6 to D1 /J1 = D2 /J1 = 5.8 × 10−3 .
The exchange bias-field is determined by the displacement of the magnetization loop. An
external magnetic field is applied in the direction of the magnetic easy axis of the FM layer.
The magnetic field (H1 + H2 )/2 denotes the displacement of the magnetization loop. The
values of H1 and H2 are external magnetic fields where the magnetization is zero. In the
case of D1 /J1 ≤ 1.0 × 10−3 , the exchange bias-field becomes zero. On the other hand,
a finite value for the exchange bias-field is obtained in the case of D1 /J1 ≥ 2.9 × 10−3 .
Accordingly, the magnetic anisotropy needs to be larger than a certain critical value to yield
exchange bias-field. In the magnetic domain wall model (Mauri et al., 1987), the critical
thickness of the AFM layer is determined by the magnetic domain wall width δd . In the
present case, the magnetic domain wall in the AFM layer forms parallel to the interface.
Therefore, the rotation of spins in the domain wall is restricted to the (111) plane. As a
380
K. FUKAMICHI et al.
Fig. 8.21. Dependence of the exchange bias-field on the magnetic anisotropy energy in the AFM layer. The critical anisotropy is presumed to be D1 /J1 = 1.0 × 10−3 for a thickness of the AFM layer of 100 MLs (Mitsumata
et al., 2006).
result, the anisotropy energy in equation (8.14) can be expressed as
2D2
D1
+
sin2 3θ,
(8.15)
a3
27a 3
where θ is the rotation angle of spins in the (111) plane, and a represents the lattice constant
of the AFM layer. Using equation (8.15), the profile of the magnetic domain wall z(θ ) is
given by
J1 S · S
dθ
z(θ ) =
(8.16)
× √
.
a
A(θ )
A(θ ) =
From equation (8.16), the magnetic domain wall width δd is obtained, and the critical
thickness dc is approximately by
%
J1 S · S
a.
dc = δd = π
(8.17)
D1
Taking the critical thickness dc equal to the thickness of the AFM layer, the critical
anisotropy energy becomes D1 /J1 = 1.0 × 10−3 , reaching agreement with the calculation
in fig. 8.21. Therefore, the formation of a domain wall strongly influences the exchange biasing phenomena in analogy with previous results (Mitsumata et al., 2003a, 2003b; Umetsu
et al., 2003c). The calculated magnetization loops are shown in fig. 8.22 (Mitsumata et al.,
2006). The magnetization process with the anisotropy energy D1 /J1 = 5.8 × 10−4 shows
a symmetric loop without any exchange bias-field as given by the solid lines. On the other
hand, the magnetization process with the anisotropy energy D1 /J1 = 2.9 × 10−3 shows a
clear loop shift due to the exchange bias-field as shown by the dashed curves.
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
381
Fig. 8.22. Magnetization loops of the AFM/FM bilayer. The dashed lines denote the magnetization process when
the anisotropy energy is D1 /J1 = 5.8 × 10−4 , which is larger than the critical anisotropy energy. The solid
lines denote the magnetization process when the anisotropy energy is D1 /J1 = 2.9 × 10−3 , less than the critical
anisotropy energy (Mitsumata et al., 2006).
8.2.4.2. Torque curves of FM/AFM bilayer The torque measurement is important to evaluate the magnetic anisotropy of the AFM layer in a bilayer system. For γ -phase AFM alloy,
Tsunoda and Takahashi (2000) reported the results of a disordered γ -phase MnIr system.
However, those measurements were carried out under the condition where the AFM thickness is thinner than the critical thickness. Therefore, the motion of a magnetic domain wall
which gives birth to the exchange bias-field in an AFM layer is not yet clear because a
polycrystalline bilayer was used. In this connection, the torque of the FM/AFM bilayer
with various anisotropies has been investigated. The torque of the FM/AFM bilayer is obtained from the following derivative of magnetic energy.
∂E
(8.18)
,
∂θ
where θ represents the angle of the external magnetic field. The external magnetic field
which rotates in the (111) plane is set as gμB Hext /J1 = 5.8 × 10−2 . The torque curves
of the FM/AFM bilayer are shown in fig. 8.23(a)–(d) (Mitsumata et al., 2006). The solid
and dotted lines represent the curves from the AFM layer and the FM layer, respectively.
The anisotropy energies of the AFM layer in these figures are (a) D1 /J1 = 5.8 × 10−6 , (b)
D1 /J1 = 5.8 × 10−5 , (c) D1 /J1 = 5.8 × 10−4 , and (d) D1 /J1 = 2.9 × 10−3 , respectively.
The direction θ = 0 corresponds to the [011̄] direction. The torque curves of the FM layer
show a two-fold symmetry, reflecting the unidirectional anisotropy in the FM layer. For
the same reason, six-fold symmetry is expected in the torque curves of the AFM layer,
because of the cubic anisotropy. However, the torque curve of the AFM layer shows a
two-fold symmetry in the (111) plane as shown in fig. 8.23(a). When the anisotropy is
small, no magnetic domain wall appears in the AFM layer. Therefore, spins in an AFM
layer follow the motion of spins in the FM layer. The torque curve of the AFM layer is
L=−
382
K. FUKAMICHI et al.
Fig. 8.23. Calculated torque curves in AFM/FM bilayer systems with various anisotropy energies. The solid
and dotted lines represent the torque curves for the AFM and FM layers, respectively. The anisotropy energies
of the AFM layer are (a) D1 /J1 = 5.8 × 10−6 , (b) D1 /J1 = 5.8 × 10−5 , (c) D1 /J1 = 5.8 × 10−4 , and
(d) D1 /J1 = 2.9 × 10−3 (Mitsumata et al., 2006).
slightly deformed from a sinusoidal curve. This is due to the influence of the small cubic
anisotropy in the AFM layer. The spins in the AFM layer have a close relation with the
spins in the FM layer because of the exchange interaction between the FM and AFM layers.
The same phase rotation of spins in the AFM and FM layers provides a four-fold symmetry
in the torque curve for the AFM layer. When the anisotropy in the AFM layer increases
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
383
to D1 /J1 = 5.8 × 10−5 , six-fold symmetry appears in the torque curve of the AFM layer
as shown in fig. 8.23(b). Since the anisotropy energy is still smaller than the critical value,
no domain wall is expected in the AFM layer. In this case, all spins of both the AFM
and FM layers rotate in the same phase in the external magnetic field. Accordingly, the
torque curves show symmetry of unidirectional and cubic anisotropy at the same time. In
addition, the spins in the AFM layer are closely connected to spins in the FM layer. That is,
the six-fold symmetric curve is slightly modulated by the two-fold symmetric curve from
the FM layer. When the anisotropy energy increases to D1 /J1 = 5.8 × 10−4 , the six-fold
symmetric torque curve in the AFM layer is deformed from a sinusoidal curve, as shown
in fig. 8.23(c). The monotonic rotation of spins in the FM layer results in a sinusoidal
torque curve. Since the anisotropy energy is still less than the critical value, no domain
wall forms in the AFM layer. In addition, the close connection between the AFM and FM
spins forces to monotonic rotation of all the spins. However, the magnetic easy directions in
the AFM and FM layers are different from each other. Furthermore, the anisotropy energy
in the AFM layer is larger than that in the FM layer, and consequently the torque in the
AFM layer is different at points near and distant from the interface. The deformation of
the torque curve could originate from the deviation of the relative angle between the AFM
spins and the direction of the external magnetic field. It is considered that the deformation
of the torque curve is caused by the domain wall formed in the AFM layer. The influence
of the cubic anisotropy appears in the torque curve of the FM layer as well: the curve
is slightly modulated by the three-fold symmetry (see the FM torque curve near π/4).
In the case of D1 /J1 = 2.9 × 10−3 , where the displaced magnetization curve appears
(see fig. 8.22), the torque of the AFM layer no longer exhibits a sinusoidal curve as seen
from fig. 8.23(d). In this model, the anisotropy energy is larger than the critical anisotropy,
allowing the formation of an AFM domain wall parallel to the interface. Therefore, the
deformation of the torque curve is considered to be caused by the incoherent rotation of
spins in the AFM layer. The exchange interaction connects the spins at the interface in the
AFM and FM layers. The AFM spins near the interface follow the rotation of the FM spins,
causing the six-fold symmetry of the torque in the AFM layer. Away from the interface,
the AFM spins retain their initial direction because of the incoherent rotation, resulting in
little change in the anisotropy energy. In other words, the spin torque disappears far away
from the interface in the AFM layer. That is, the total torque is suppressed, and hence
the shape of the torque curve is significantly deformed. Moreover, modulation appears in
the torque curve for the FM layer as in fig. 8.23(c). The torque curves of the AFM and
FM layers in fig. 8.23 are calculated separately. On the other hand, the thickness of the
AFM layer is eleven times larger than that of the FM layer. Therefore, in the cases shown
in fig.8.23(c) and (d), the anisotropy energy of the AFM layer is dominant in the total
anisotropy energy of the bilayer. For example, Du tFM /J1 = 3.0×10−5 a and D1 tAFM /J1 =
0.17a in fig. 8.23(d), where tAFM and tFM are the thicknesses of the AFM and FM layers,
and a is the lattice constant. Thus, it is expected from fig. 8.23(c) and (d) that the torque
curve of the entire bilayer follows the curve of the AFM layer. In order to confirm the
motion of the magnetic domain wall in the AFM layer, the depth profile of spins in the
bilayer is discussed. The element of the averaged spin is plotted in fig. 8.24(a) and (b)
(Mitsumata et al., 2006). The averaged spin S̄ I,R is obtained on each net plane at a certain
384
K. FUKAMICHI et al.
Fig. 8.24. Depth profiles obtained for the magnetic anisotropy energies in the AFM/FM bilayer system. The
depth profiles are. (a) D1 /J1 = 5.8 × 10−6 and (b) D1 /J1 = 2.9 × 10−3 . The solid and dotted lines show the
profiles at angle θ = π/3 and θ = π/3 + π/2 in the external magnetic field, respectively. The depth profile shows
the element of spin on the equivalent atom IV in fig. 8.20 (Mitsumata et al., 2006). The element of spin represents
the projection of the spin vector onto [1̄10] direction.
depth of the bilayer, and is defined by the following equation
S I,R =
S I,p,q,R
.
NI
p,q
(8.19)
The indices I = I, II, III, and IV denote the equivalent atomic site as shown in fig. 8.20. The
index R represents the depth of a net plane in the [111] direction. The net plane consists of
a triangular fcc lattice in the (111) plane (see fig. 8.8). The indices p and q represent the
positions of the magnetic atoms in the [1̄10] and [011̄] directions, respectively. NI is the
number of magnetic atoms at the equivalent site I at the depth R. Thus, the averaging is
carried out at each depth of the net plane. The element of spins is given by the projection
of the spin vector onto the [1̄10] direction. The depth conditions of R < 0 and R ≥ 0
correspond to the FM and AFM layers, respectively. The depth profile shows the spin
element of the equivalent atom IV. The profile in fig. 8.24(a) represents a condition in
which the anisotropy (D1 /J1 = 5.8 × 10−6 ) is less than the critical anisotropy, and the
profile in fig. 8.24(b) represents the case in which the anisotropy (D1 /J1 = 2.9 × 10−3 ) is
larger than the critical value. The solid and dotted lines show the profiles at θ = π/3 and
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
385
θ = π/3+π/2 of the external magnetic field, respectively. The angle θ = π/3 corresponds
to the easy direction of the FM layer. The rotated angle θ = π/3 + π/2 corresponds to
the hard direction of the FM layer. In both fig. 8.24(a) and (b), the spin elements change
in a range between about 1 to 0 in the FM layer, when the external magnetic field rotates
from θ = π/3 to π/3 + π/2. These changes denote the π/2 rotation of the FM spins from
the easy to hard directions. When the external magnetic field is at θ = π/3, in fig. 8.24(a),
the spin element of the AFM atom at the equivalent site IV is substantially directed in the
[111̄] direction. The angle between [1̄10] and [111̄] is π/2, and thus the spin element in
the AFM layer is zero. The depth profile from 0 to 99 is nearly 0, and no domain wall
appears in the AFM layer. The π/2 rotation of the external magnetic field changes the
spins in the AFM layer. The projection of the rotated AFM spin on site IV becomes 0.8–
0.9 in the [1̄10] direction as shown in fig. 8.24(a). To summarize, the rotational external
magnetic field changes the FM and AFM spins without the formation of a magnetic domain
wall when the cubic anisotropy energy is less than the critical anisotropy. The relative angle
between the AFM and FM spins remains unchanged during such rotation. The depth profile
in fig. 8.24(b) represents the case in which the cubic anisotropy energy is larger than the
critical value. The depth profile at θ = π/3 shows a spin configuration similar to that in
fig. 8.24(a). The elements of the FM and AFM spins are nearly 1 and 0, respectively. When
the external magnetic field rotates from θ = π/3 to π/3 + π/2, the element of FM spins
changes from 1 to 0. The AFM spins near the interface follow the motion of FM spins, and
the element of AFM spins becomes 0.8–0.9. These values correspond to the π/2 rotation
of spins at site IV in the AFM layer. The rotation of AFM spins can be observed at a depth
from 0 to 30. In contrast, the element of AFM spins remains near 0 at a depth from 30 to 99.
This part of the depth profile shows no rotation of AFM spins in response to the application
of an external magnetic field. As a result, the motion of spins shows an incoherent rotation
in the AFM layer, and forms a magnetic domain wall parallel to the interface. The relative
angle between the FM and AFM spins near the interface remains unchanged during the
rotation of the external magnetic field. As explained above, all the spins in the AFM layer
are monotonically rotated in fig. 8.24(a). Thus, all the AFM spins change the magnetic
anisotropy energy in the same phase in the rotational external magnetic field, accompanied
by a six-fold sinusoidal torque curve. On the other hand, the incoherent rotation of AFM
spins occurs due to the formation of a domain wall, resulting in deformation of the torque
curve.
8.2.5. Appearance of 2Q in 3Q spin structure
The spin structure in the FM/AFM bilayer is also influenced by the stacking direction.
The spin correlation S i · S j in each atomic plane is shown in fig. 8.20. For the 3Q spin
structure, the angle between each spin equals 0.61π, therefore, the correlation becomes
S i · S j = −1/3. In fig. 8.25 (Mitsumata et al., 2004a), for the 111 stacking of the
AFM/FM bilayer, the 3Q spin structure is stabilized in the AFM layer. For the 100 stacking, however, the 3Q spin structure is disturbed near the surface (depth = 39 monolayer,
MLs) and the middle of the AFM layer (depth = 15–20 MLs). At those atomic planes, the
spin correlations are S i · S j = 0 and S i · S j = −1, corresponding to the angles of π/2
and π. Consequently, the 2Q spin structure is realized at such atomic planes. In the 100
stacking of the bilayer, therefore, the 2Q spin arrangement appears in the 3Q structure.
386
K. FUKAMICHI et al.
Fig. 8.25. Spin correlation in the atomic plane for the 111 and 100 directions of bilayers. The depth 0 represents the position of the interface, the depths > 0 and < 0 correspond to the AFM and FM layers, respectively
(Mitsumata et al., 2004a).
The appearance of the 2Q spin structure in the γ -phase AFM layer is able to deteriorate
the strength of the exchange bias-field in the FM/AFM bilayer.
8.2.6. Uncompensated spin element in AFM layer at the interface
In this subsection, we examine the exchange bias of L12 -type ordered AFM alloy. As
discussed in the preceding subsections 6.2 and 7.4, this ordered alloy system such as Mn3 Ir
has a large exchange bias-field and a high Néel temperature (Imakita et al., 2004).
In fig. 8.8, the bilayer is assumed to be 111 orientation, and the thickness of FM and
AFM layer are 9 monolayers (MLs) and 80 MLs, respectively. The thickness of 80 MLs
corresponds to 17–18 nm in typical Mn-based alloys. As shown in fig. 5.7, such AFM
alloy realizes the triangular spin structure where an angle between AFM spins indicates
2/3π, showing the typical non-collinear spin configuration. The triangular spin structure
can be generated due to the frustration of spins, where the spin frustration is caused by
the triangle lattice configuration for magnetic atoms in the L12 -type ordered structure. For
AFM spins, three equivalent atoms are contained in the unit cell, and the summation of
those spins should become zero. Therefore, in the case of 111 stacking of the FM/AFM
bilayer, the interface
consists of the compensated interface where the summation of AFM
spins becomes
S i = 0. Strictly speaking, uncompensated spin element of the AFM
layer remains near the interface. The exchange coupling between FM and AFM layer influences to relax the direction of spins. Consequently, the resultant spin element becomes
finite in the AFM layer. The spin configuration at the interface in fig. 8.26 is shown when
the triangular spin structure is realized in the L12 -type ordered AFM layer. The indices
of AFM spins represent the equivalent atomic sites. The angle between AFM spins deviates from the value of 2/3π, which corresponds to ground state of the spin correlation.
Such deviation angle is listed in table 8.2, depending on the depth of the atomic layer
(Mitsumata et al., 2004b). The large deviation from 2/3π is observed at the interface. At
the center or bottom of the AFM layer, however, the deviation angle is close to zero. The
similar behavior of the resultant AFM spin was reported by Nowak et al. (2002) within the
scheme of Ising spin model. Their model is based on the interfacial roughness to remain
the uncompensated spin component. However, in this model, no assumption of the interfacial roughness is also possible to create the uncompensated spin element. Accordingly,
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
387
Fig. 8.26. Spin configuration at the interface between FM and AFM layer having the triangular spin structure in
L12 -type ordered AFM layer (Mitsumata et al., 2005).
TABLE 8.2
Deviation form 2/3π for the angle between AFM spins having the triangular spin structure in the
bilayer system (Mitsumata et al., 2004b)
Atomic layer
Interface
Center
Bottom
Between 1 and 2
Between 2 and 3
Between 3 and 1
−0.011π
−0.001π
−0.001π
−0.016π
0.001π
0.002π
−0.014π
0.000π
0.000π
Indices 1, 2 and 3 respectively correspond to the equivalent atomic sites shown in fig. 8.26.
the frustration of spins at the interface would be able to realize the finite value of the resultant magnetization in the AFM layer. Such uncompensated spin element is extraordinary
small and can be observed by X-ray magnetic circular dichroism (XMCD) spectroscopy
(Ohldag et al., 2003). The characteristics of magnetization in the AFM layer showed the
shifted loop both in field axis and magnetization axis of M–H loop in XMCD measurement. The calculated magnetization loops are plotted in fig. 8.27 (Mitsumata et al., 2005).
The magnetization m is obtained by the average of all FM and AFM spins, respectively.
The resultant magnetization in the AFM layer is obtained from the summation of all AFM
spins including equivalent atomic site 1, 2 and 3 as shown in fig. 8.26. The FM spins show
the shifted magnetization loop under the influence of the exchange bias-field in fig. 8.27(a).
On the other hand, in the AFM layer, the resultant spin vector becomes a finite value, forming an uncompensated spin element. The uncompensated spin element in the AFM layer
shows the hysteresis loop as well as the FM layer, and it also indicates the biasing shift
in the direction of the abscissa (field axis) in fig. 8.27(b). That biasing field in the AFM
layer is equivalent to the bias in the FM layer. Consequently, the existence of such biasing shift in both FM and AFM layers, and the agreement of bias values between FM and
AFM layer could reproduce the measured results in XMCD spectroscopy (Ohldag et al.,
2003). In fig. 8.27(b), the asymmetry h between positive and negative elements of uncompensated spin is calculated by h = (mpos + mneg )/(mpos − mneg ). Here, mpos and mneg
are determined from the magnetizations at positive and negative signs of switching fields,
respectively. Unfortunately, the calculated data show scattered value near the switching
388
K. FUKAMICHI et al.
Fig. 8.27. Magnetization loops in (a) FM and (b) AFM layers, respectively. The resultant magnetization in the
AFM layer is obtained by the average of all AFM spins including equivalent atomic site 1, 2 and 3 as shown in
fig. 8.26. The asymmetry in positive and negative magnetization in the AFM layer is h = 0.035, corresponding
to the loop shift toward up in the ordinate (Mitsumata et al., 2005).
field in fig. 8.27(b). Therefore, ten data points before the switching are taking account of
the averaged values for mpos and mneg . The calculation of asymmetry h in the AFM layer
becomes h(AFM) = 0.035. At the same time, the asymmetry in the FM layer becomes
h(FM) = 0.001. Namely, the asymmetry h(AFM) is significantly larger than h(FM) in the
calculation of magnetization process. The finite value of asymmetry h(AFM) describes the
loop shift upward in the ordinate (magnetization axis). Those behaviors of magnetization
loop of AFM layer also agree with the results of XMCD measurements.
8.3. Blocking temperature and training effect
Practically, the thickness of antiferromagnetic bias film should be as thin as possible in
spintronic devices. However, to make the AFM layer film thin risks lowering the blocking
temperature TB . Therefore, we should select the AFM having a large value of the magnetic anisotropy constant. In several consecutive hysteresis loop measurements for many
exchange biasing films, the loop shift of consecutive loops often becomes small, and this
behavior is called the training effect (Tsang and Lee, 1982; Schlenker et al., 1986; Chen et
al., 1991; Lai et al., 1997a).
8.3.1. Blocking temperature
In the preceding subsection 8.2, several exchange bias-field properties are discussed on
the basis of the calculated results in the ground state. At finite temperatures, the exchange
bias-field does not work above a temperature denoted as the blocking temperature TB which
is much lower than the Néel temperature TN of its bulk counterpart in some cases (Hoshino
et al., 1996; Soeya et al., 1996; van der Zaag et al., 1996; Soeya et al., 1997). In other cases,
however, TB is very close to TN (Moran et al., 1995; Nogués et al., 1996; van der Zaag et al.,
1996; Choe and Gupta, 1997). Those behaviors could be related to the size effects, that is,
the grain size (or the layer thickness), is smaller (or thinner) than the critical dimension of
the AFM, TN is significantly reduced. In addition, being associated with the size effects, the
following condition is necessary for the observation of exchange anisotropy (Meiklejohn,
1962).
KAFM tAFM > JINT ,
(8.20)
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
389
Fig. 8.28. Thickness dependence of the blocking temperature TB of Ta/CoFe/Cu/CoFe/MnIr/Ta with FWHM
∼3.5◦ and Ta/CoNbZr/NiFe/CoFe/Cu/CoFe/MnIr/Ta with FWHM ∼5◦ (Fuke et al., 1998).
Fig. 8.29. Grain size dependence of the blocking temperature TB of Ta/CoFe/Cu/CoFe/MnIr/Ta with FWHM
∼3.5◦ and Ta/CoNbZr/NiFe/CoFe/Cu/CoFe/MnIr/Ta with FWHM ∼5◦ (Fuke, 2002).
where KAFM and tAFM are the anisotropy constant and the thickness of the AFM layer,
respectively, JINT is the interface coupling constant. Accordingly, the value of TB would
be reduced by assuming that the AFM anisotropy decreases as its size is reduced. A smaller
anisotropy would also bring about a smaller exchange bias-field, resulting in the relation of
TB < TN . Other reasons decreasing TB could be attributed to metallurgical problems such
as non-stoichiometric compositions and presence of mixed phases, and also the degree in
ordered phases.
Figure 8.28 (Fuke et al., 1998) and fig. 8.29 (Fuke, 2002) show the thickness and the
grain size dependences of the blocking temperature TB for Ta/CoFe/Cu/CoFe/Mn78 Ir22 /Ta
with the full width at half-maximum (FWHM) ∼3.5◦ and Ta/CoNbZr/NiFe/CoFe/Cu/
CoFe/Mn78 Ir22 /Ta with FWHM ∼5◦ , respectively. Both figures clearly exhibit that
TB is markedly affected by the thickness and the grain size of the γ -Mn78 Ir22 film,
as well as by its crystallinity. Similar results were obtained for NiFe/FeMn system
(Nishioka et al., 1996). Ta/NiFe/MnIr/CoFe/Cu/Ta films exhibit such thickness depen-
390
K. FUKAMICHI et al.
Fig. 8.30. Temperature dependence of the unidirectional anisotropy constant Kd of Mn73 Ir27 /Co70 Fe30 bilayers
prepared at 443 K and room temperature of the substrates (Imakita et al., 2004).
dence of TB (Tsunoda et al., 2002b). Figure 8.30 shows the temperature dependence
of the unidirectional anisotropy constant Kd of Mn73 Ir27 /Co70 Fe30 bilayers prepared at
443 K and room temperature of the substrates (Imakita et al., 2004). The layer structure is the substrate/Ni27 Fe7 Cr66 5 nm/Cu 50 nm/Mn73 Ir27 10 nm/Co70 Fe30 4 nm/Cu
1 nm/Ni27 Fe7 Cr66 2 nm. The value of TB of the layer prepared at room temperature is
comparable with that in fig. 8.28 at the same thickness of Mn-Ir films. Marked increases
of Kd and TB for the layer prepared at 443 K are attributed to the increase of TN because
of appearance of L12 -type Mn3 Ir ordered phase (see figs. 6.3, 6.5 and 7.28). Note that
the value of TB is much lower than TN of Mn3 Ir ordered alloy (see fig. 6.3). The value
of TB of Mnx Pt1−x -Cr/NiFe has been investigated by Dai et al. (2004) and found that a
small amount of Cr promotes the ordering of L10 phase and increases in TB , although the
grain size becomes small. It should be noted that TB will be distributed because of different
grain sizes and roughness in films (Gangopadhyay et al., 1993; Soeya et al., 1994; Lai et
al., 1997b). Practically, the thickness of the antiferromagnetic bias film should be as thin
as possible in spintronic devices. However, making the AFM layer film thin leads to the
risk of lowering the blocking temperature TB . Therefore, we should select an AFM having
a large value of the magnetic anisotropy constant.
8.3.2. Training effect
In several consecutive hysteresis loop measurements of many exchange-biasing films,
the loop shift of consecutive loops often becomes small, and this behavior is called the
training effect (Tsang and Lee, 1982; Schlenker et al., 1986; Chen et al., 1991; Lai et al.,
1997a, 1997b). The training effect was reported several decades ago (Paccard et al., 1966;
Schlenker, 1968; Schlenker and Paccard, 1967). The origin of this effect may come from
a growth induced metastable spin configuration (Malozemoff 1987, 1988a, 1988b) and
could be related to partial reorientation of the AFM domains with each FM magnetization
reversal and then the AFM spins try to find energetically favorable configurations after each
cycle. The training effect is significant in polycrystalline AFM, and very small or negligible
in layers based on single crystals of bulk and thin films (Moran et al., 1995; Michl et al.,
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
391
Fig. 8.31. (a) Hysteresis loops of training effect of a Si (100)/Cu (15 nm)/NiFe (12 nm)/MnIr (3.2 nm) Ta
(10 nm) film starting from the positive (pinned direction) saturation. The loop cycles 1, 2, 5, 10 are shown.
−
+
(b) |HSW
|, HSW
, Heb and Hc (in Oe) as a function of the number of the cycles (Zhang et al., 2002).
1996; Nogués et al., 1996; Gökemeijer et al., 1997). Detailed studies of the training effect
have been carried by Fujiwara et al. (2001) and Zhang et al. (2002). In fig. 8.31(a), the
hysteresis loops of the training effect as a function of the number of the loop cycles are
shown for a Si (100)/Cu (15 nm)/NiFe (12 nm)/MnIr (3.2 nm) Ta (10 nm) film starting
−
+
from the positive (pinned direction) saturation (Zhang et al., 2002). HSW
and HSW
are
the switching fields of the descending (antiparallel to the pinned direction) and ascending
−
(parallel to the pinned direction) fields. The magnitude of |HSW
| decreases drastically,
+
−
while HSW slightly changes. The value of the exchange bias-field Hex is defined as |HSW
+
+
−
+
−
+
HSW |/2 and coercivity as Hc = |HSW − HSW |/2. Figure 8.31(b) shows |HSW |, HSW , Hex
and Hc (in Oe) as a function of the antiferromagnetic layer thickness tAF (Zhang et al.,
2002). Both the values of Hex and Hc decrease with increasing number of loop cycles, and
the training effect is more prominent in the descending curve than in the ascending curve.
392
K. FUKAMICHI et al.
−
+
−
Fig. 8.32. The training effect ratio given by [HSW
(0)−HSW
(20)]/HSW
(0) as a function of the antiferromagnetic
MnIr layer thickness tAFM (Zhang et al., 2002).
In fig. 8.32, the training effect ratio defined by
$
$ −
$H (0)$ − H + (20) /H − (0)
SW
SW
SW
(8.21)
−
(n) denoting
is plotted as a function of thickness tAF of the AFM MnIr layer, with HSW
−
the HSW of the nth cycle (Zhang et al., 2002). With increasing tAF , the training effect
starts to appear by affecting Hex and Hc , then the ratio defined in equation (8.21) increases
drastically to form a peak and then decreases quickly, eventually to almost zero. That is,
the training effect behavior changes from the type I to the type II at around tAFM = 5 nm.
According to the extended Fulcomer and Charap model, in which planar wall formation
and the distribution of not only the ferromagnetic–antiferromagnetic coupling strength and
the anisotropy constant of the antiferromagnetic grains but also the random distribution of
the antiferromagnetic easy axes are taken into consideration (Fujiwara et al., 1999), the
mean
mean
type mentioned above is classified by the strength ratio of |JAFM
-AFM |/JFM-AFM , where
mean
mean
JAFM-AFM and JFM-AFM are the mean exchange strength between the antiferromagnetic–
antiferromagnetic layers and antiferromagnetic–ferromagnetic layers, respectively. For the
ratios 0.4 and 0.04, the type I and type II are expected to occur, respectively (Fujiwara et
al., 2001). On the other hand, it should be noted that the ratio is not so small in Co/MnNi
layers (Schulthess and Butler, 1998a).
9. Concluding remarks
In the case of ferromagnets, magnetic domains are easily formed in order to reduce the
magnetostatic energy. On the contrary, generally antiferromagnets have no magnetostatic
energy as expected from a simple AFM spin structure, and hence no magnetic domains
would exist in ideal antiferromagnets. However, there are many defects and strains in real
crystals, and then domains with finite sizes are formed in antiferromagnets. We pointed
out in section 8 that the magnetic domain wall plays an important role in the formation
PROPERTIES OF PRACTICAL ANTIFERROMAGNETIC Mn ALLOYS
393
of exchange bias-fields. Usually, the observation of AFM domains is not easy by conventional methods because the observations have been carried out through a very small
lattice distortion caused by the exchange strain below the magnetic transition temperature.
It has been considered that the AFM domain size is smaller than the crystal grain size. In
line with such assumption, many published papers have discussed the relation between the
grain size and the strength of exchange coupling and also the blocking temperature TB . At
present stage, however, the discussions are divergent on this point. Recently, a powerful
method for the observation of the AFM domains has been developed. By using X-ray magnetic linear dichroism (XMLD) spectroscopy, it has been demonstrated that the domain
size in NiO film grown on Mg (100) is smaller by 1 or 2 orders of magnitude than that in
bulk NiO (Stöhr et al., 1999). On the other hand, the correspondence between the blocking
temperature TB and the Néel temperature TN has often been discussed by many researchers.
Associated with these discussions, the AFM domain observation of LaFeO3 is meaningful.
What has to be noticed is that the domains observed by XMLD spectroscopy for LaFeO3
grown epitaxially on SrTiO3 (100) thoroughly disappears at a temperature lower by about
70 K than TN of the bulk specimen (Scholl et al., 2000). Thermal strains bring about this
behavior by changing the angle of the Fe-O-Fe superexchange interaction, reducing TN .
This data clearly shows that the Néel temperature is sensitively affected by thermal strains.
Similar discussions have been presented for ferroelectric PbTiO3 thin films on MgO (111)
(Bater et al., 1996). The energy difference between the spin structures in the γ -phase is
not very large (see figs. 5.12 and 5.13) and hence stresses as well as strains could influence
the AFM properties. The magnetovolume effects and also crystallographic defects result
in stresses and strains. These facts allow us to speculate that the spin structure in AFM
films of spin valves is locally different, depending on the magnitude of induced strains and
defects. Accordingly, direct observations of the domain structures in Mn alloy systems are
very important to develop excellent spintronic devices because the thermal strains, elastic
stress and the magnetovolume effects presented in section 7 are liable to change TN as well
as the spin structure.
In L10 -type ordered alloys, the AFM state is stabilized when the staggered field caused
by the Mn atomic plane makes a pseudo-gap at the Fermi level EF . The experimental
magnetic and electrical properties are in harmony with the theoretical calculations. The
characteristic behaviors of the concentration dependence of specific heat coefficient and
the temperature dependence of electrical resistivity are closely related with the pseudo-gap
in the electronic structures. The magnitude of electrical resistivity ρ which is sensitive to
the composition is important for practical applications to CIP- and CPP-type spintronic
devices. In electrical properties of thin films, when the grain size less than the mean free
path of electrons, grain boundary scattering becomes striking, and the quantum size effects
and localization behavior could appear in very thin films. These properties are sensitive to
temperature, especially low temperatures. Therefore, the thickness and temperature dependences of ρ of L10 -type alloy films should be examined. Exchange bias-field and related
properties have been calculated in zero-temperature ground state. For more detailed discussions, the analyses at finite temperatures are necessary.
Turning now to fundamental researches on the magnetism of Mn alloys, much exciting
progress has been made. For instance, as seen from fig. 5.7, Mn atoms at the face center
positions form an octahedral structure, which is composed of eight equilateral triangles.
394
K. FUKAMICHI et al.
Then, if the nearest neighbor Mn spins couple antiferromagnetically, L12 -type ordered
Mn3 Pt alloy would be a good candidate of a new-type of three-dimensional frustration
system (Ikeda and Tsunoda, 2003a, 2004). As to the cohesive energy, it will be useful to
keep the fact in mind that the bulk modulus of the γ -phase alloys is about several times
lower than that of ordinary non-magnetic elements such as Mo and Ru. This behavior
could be attractive to discuss in terms of the generalized gradient approximation (GGA1) (Perdew and Wang, 1986) and GGA-2 (Becke, 1988). Another significant aspect is the
study of β-Mn alloys. In the Mn100−x Alx alloy system, antiferromagnetic SFs become
remarkable and result in a quantum spin-liquid state (Nakamura and Shiga, 1997). We
may recall that the β-phase is easily formed in many Mn-based alloys having a disordered
γ -phase (see subsection 7.6).
Finally, we should stress that magnetic and other fundamental data for antiferromagnets
are a little amount, compared with those for practical ferromagnets. Even the equilibrium
phase diagrams, almost all Mn alloy systems are uncertain and depicted by dotted lines. For
further progresses of advanced spintronic devices, active basic researches and the corpora
of the data are earnestly desired for antiferromagnets not only in the bulk state but also in
the thin film state.
Acknowledgements
The study of Mn alloy systems given in this article was supported by a Grant-in-Aid for
Scientific Research (B) (2), No. 13450255, from the Japan Society for the Promotion of
Science. One of the authors (R.Y. Umetsu) has been supported by the Research Fellowships
of the Japan Society for the Promotion of Science for Young Scientists.
We wish to thank Professor J. Echigoya of Iwate University, Professor T. Hori of Sibaura
Institute of Technology and Professor K. Ono of High Energy Accelerator Research Organization for cooperation throughout the research of antiferromagnetic Mn alloys. We are
indebted to Professor U. Mizutani of Nagoya University for his useful discussion on the
thermal conductivity of Mn alloys.
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chapter 5
SYNTHESIS, PROPERTIES AND BIOMEDICAL
APPLICATIONS OF MAGNETIC
NANOPARTICLES
PEDRO TARTAJ, MARIA P. MORALES,
SABINO VEINTEMILLAS-VERDAGUER,
TERESITA GONZALEZ-CARREÑO AND CARLOS J. SERNA
Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, 28049, Madrid, Spain
Handbook of Magnetic Materials, edited by K.H.J. Buschow
Volume 16 ISSN: 1567-2719 DOI 10.1016/S1567-2719(05)16005-3
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© 2006 Elsevier B.V.
All rights reserved
CONTENTS
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
2. Synthesis of magnetic nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
2.1. Solution routes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
2.2. Gas-phase routes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
2.3. Solid routes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
2.4. Size selection methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
3. Colloidal properties of magnetic nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
4. Surface modification or encapsulation of nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
4.1. Organic hydrophilic coatings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
4.2. Inorganic hydrophilic coatings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
4.3. Hybrid hydrophilic coatings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
4.4. Hydrophobic coatings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
4.5. Transformation of hydrophobic into hydrophilic particles . . . . . . . . . . . . . . . . . . . . . . . 448
5. Magnetic properties of nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
5.1. Particle size effects (finite size and/or surface effects) . . . . . . . . . . . . . . . . . . . . . . . . . 450
5.2. Other microstructural effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
5.3. Coating and encapsulation effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
6. Applications of magnetic nanoparticles in biomedicine . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
6.1. Nuclear magnetic resonance (NMR) imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
6.2. Hyperthermia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
6.3. Magnetic drug-targeting and gene delivery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
6.4. Separation and selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
6.5. Other biomedical applications of magnetic nanoparticles . . . . . . . . . . . . . . . . . . . . . . . 470
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
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1. Introduction
Nanoparticulate materials are experiencing increasing attention mainly because of the
unique combination of small size, exotic properties and processability, opening the possibility of their use in many technological applications (e.g. Whitesides, 2005). As the size
reduces into the nanometer range, the materials exhibit peculiar and interesting physical
and mechanical properties. For example, when the particle dimensions of a semiconductor
become comparable to, or smaller than, the Bohr radius, the space in which the charge
carriers move decreases, and thus additional quantum confinement is imposed on their motion. This leads to an increase in the band gap energy, the electron and hole kinetic energy,
and the density of the charge carriers within and at the nanoparticle surface (e.g. El-Sayed,
2001). Moreover, charge transfer processes that arise from the attachment of single or small
groups of (usually organic) molecules to semiconductor and metal nanoparticles are also
strongly affected at the nanoscale (e.g. Adams et al., 2003).
In noble metals, the decrease in size below the electron mean free path (the distance
the electron travels between scattering collisions with the lattice centers) gives rise to an
intense absorption in the visible-near-UV associated with surface plasmons (Bohren and
Huffman, 1983). In fact, gold colloidal nanoparticles were used to give stained glass its brilliant ruby color used in cathedrals at 17th century (Kerker, 1985). Another series of unique
properties of metal nanoparticles are derived from their large number of grain boundaries
compared to coarse-grained polycrystalline counterparts. In nanocrystalline solids, a large
fraction of atoms are boundary atoms. Thus the interface structure plays an important role
in determining their physical and mechanical properties. Nanocrystalline metals have been
found to exhibit creep and superplasticity with high strain rates at lower temperatures than
their micro-grained counterparts. High strain-rate superplasticity at lower temperatures is
of practical interest because it can offer an efficiently near-net-shape forming technique to
industrial sectors (Tjong and Chen, 2004).
Ceramic materials composed of powders with a particle size in the nanometric range
are also receiving attention because they may significantly enhance sintering rates or dramatically lower sintering temperatures (Hahn et al., 1990; Zhou and Rahaman, 1993;
Tartaj et al., 2002a; Tartaj and Tartaj, 2002a). Also, ceramic matrix composites with
dispersed nanoparticles have better mechanical properties (Niihara, 1991). Special interest has generated the preparation and study of ferroelectric ceramics in the nanoscale
regime (Ahn et al., 2004). Their switchable electric polarization combined with their
small size make these materials ideal for use in memory storage devices and integrated
microelectronics (Ramesh et al., 2001). From fundamental perspectives to technological
applications, the issue of the size dependence of the ferroelectric properties, and their
possible disappearance at a critical size becomes crucial (Junquera and Ghosez, 2003;
Dan et al., 2005).
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In magnetic nanoparticles finite-size effects along with surface effects, collective phenomena arising from interparticle interactions and unusual transport properties, cause
the nanomagnets to exhibit a number of outstanding physical properties such as superparamagnetism, giant magnetoresistance, quantum tunneling of the magnetization, large
coercivities and the development of a net magnetic moment and ferrimagnetism on antiferromagnetic nanoparticles (Morup et al., 1995; Mehn et al., 1996; Gu et al., 1997;
Martinez et al., 1998; Kodama, 1999; Battle and Labarta, 2002; Lazaro et al., 2003;
Serrate et al., 2005; Yakushiji et al., 2005). Examples of exotic magnetic phenomena at
the nanoscale are observed in zero dimensional magnets (small molecular clusters that
bear large spins, which do not exist naturally in any of the elements in the periodic table).
They tend to have large Ising-type anisotropy, and the magnetization relaxes very slowly
at low temperature, being adequate candidate for the observation of quantum tunneling
of magnetization (e.g. Blundell and Pratt, 2004). A small magnetic particle with uniaxial
anisotropy will have its magnetization parallel to an easy axis. Switching between one easy
axis direction and another will take place via thermal activation. Thermal activation is not
the only way to switch the magnetization; another possibility is quantum tunneling. The
exchange interaction constrains all the electronic spins inside the particle to remain parallel
with one another, but the ‘macro-spin’ can coherently tunnel from one state to the other.
Another interesting example of exotic magnetic phenomena at the nanoscale is observed
in Pd and Au nanoparticles. The possible ferromagnetism of Pd and late 4d transition
metals has been intensively investigated giving rise to a continuous issue of theoretical and experimental studies (Sampedro et al., 2003). Pd metal exhibits enhanced Pauli
paramagnetism with large susceptibility. However, in bulk Pd no spontaneous ferromagnetic order has been observed. Although the density of states shows a sharp peak just
below the Fermi level, the Stoner criterion for ferromagnetism is not satisfied. Nevertheless, Pd lies close to a ferromagnetic instability. The factors that can affect the onset
of ferromagnetism are those increasing the density of states at the Fermi level (confinement effects associated with a reduced coordination number, local symmetry changes
and lattice expansion that induces a narrowing of the d band) (Sampedro et al., 2003;
Shinohara et al., 2003). On the other hand, Au particles show ferromagnetism at the
nanoscale (Teranishi et al., 1997; Hori et al., 1999, 2004; Crespo et al., 2004; Yamamoto et
al., 2004). Such ferromagnetic spin polarization is surprisingly unique and the polarization
mechanism is quite interesting. Because Au metal is a typical diamagnetic material, its
ferromagnetic polarization mechanism is thought to be quite different from the ferromagnetism observed in transition metals.
Similar to other nanoparticulate systems the unique combination of small size, exotic
properties and processability makes magnetic nanoparticles attractive for a variety of technological applications from data recording to magnetic seals in motors and magnetic inks
for bank checks, magnetooptical and biotechnology applications (Charles and Popplewell,
1986; Ziolo et al., 1992; Berkovsky et al., 1993; Moser et al., 2002; Tartaj et al., 2003a;
Terris and Thomson, 2005). Magnetic storage has played a key role in audio, video and
computer development since its invention more than hundred years ago by Valdemar
Poulsen (Moser et al., 2002). In 1956 IBM built the first magnetic hard disk drive featuring a total storage capacity of 5 MB at a recording density of 2 kbit in−2 . In the quest to
improve the performance, the areal density, i.e. the number of bits/unit area on a disk surface, has increased more than 20 million-fold in modern disk drives and currently doubles
SYNTHESIS, PROPERTIES, APPLICATIONS OF MAGNETIC NANOPARTICLES
407
every year. An obvious advantage of using nanomagnets for data storage applications is to
increase the areal density. However, enhanced thermal effects limit the use of nanomagnets
(McDaniell, 2005).
One of the experimental approaches to overcome thermal fluctuations is to prepare
needle-like FeCo nanoparticulate metallic alloys of about 100 × 15 nm because of their
high magnetic anisotropy, a consequence of their elongated shape. In fact, these particles
deposited longitudinally on a film, are nowadays routinely used for the storage of digital and analogue signals in the area of advanced flexible media (Onodera et al., 1996;
Hisano et al., 1997; Hisano and Saito, 1998; O’Grady et al., 1998; O’Grady and Laidler, 1999; Sharrock, 2000; Nuñez et al., 2003, 2004). Another approach is to prepare
directly metallic nanoparticles with compositions that intrinsically have a high crystalline
anisotropy, say for example FePt alloys that crystallize in the tetragonal phase (Sun et al.,
2000a, 2000c, 2000b) or taking advantage of the increase anisotropy as a result of the
unique interaction between ferromagnetic Co nanoparticles and antiferromagnetic CoO
matrices (Skumryev et al., 2003).
Ferromagnetic nanowires and their arrays are also attracting a great deal of interest for
magnetic storage. The magnetic nanowire arrays have high-density recording ability in
excess of 100 Gb in−2 (Jo et al., 2005). In particular, Fe-rich nanowire alloys with high
coercivity are widely used to control the magnetic properties of recording media by varying the concentration (Zhan et al., 2002). Moreover, ferromagnetic nanowires and their
arrays are also attracting interest because of their potential applications in magnetic sensors, memory devices, spintronic nanodevices, and energy storage (Chou et al., 1994;
Prinz, 1998; Guo et al., 2003). They are also good candidates for giant magnetoresistance (GMR) materials without the occurrence of structural defects in multilayers. Magnetic nanorings are also excellent candidates for high-density storage devices because
of the existence of vortex or flux-closed states in which the magnetization is oriented
circularly and for which stray magnetic fields are essentially absent (Li et al., 2001;
Castaño et al., 2003; Wang et al., 2005a, 2005b).
Another application in which the use of magnetic nanoparticles is relevant is in magnetooptics. For example, transparent magnetic materials, which potentially could be used
as sensors based on the Faraday effect, require the use of γ -Fe2 O3 nanoparticles embedded in transparent matrices such as transparent polymers, sol-gel derived silica or Vycor
glasses (Ziolo et al., 1992; Guerrero et al., 1997; Bentivegna et al., 1999; Zayat et al., 2003;
Rosa et al., 2005). Incorporation of photochromic compounds into magnetic systems is
one of the new strategies being implemented to realize the control of magnetic properties
by photoillumination. Recently, Taguchi et al. (2003) have designed composite materials
comprising iron oxide particles and photochromic spiropyran vesicles as cast films on substrates. Photofunctional magnetic vesicles of amphiphilic spiropyran (SP1822) containing
iron oxide ferrite nanoparticles exhibit superparamagnetic properties at room temperature.
Both photoisomerization and photoinduced aggregation of the photoresponsive magnetic
vesicles were observed, even without any supporting media such as solvents or polymer
matrices. As a result of the assembly of the vesicles, magnetic dipolar interactions among
adjacent iron oxide nanoparticles were induced by photoillumination at room temperature,
even in the solid state. The same group has also recently reported the reversible photoswitching of the magnetization at room temperature in γ -Fe2 O3 nanoparticles coated with
azobenzene-containing amphiphilic compounds (Mikami et al., 2004).
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Even magnetic nanoparticles can be used to manipulate microfluidic devices. Reactions
in microfluidic devices require minimal quantities of reagents, produce little waste, and can
be very fast through integration with robotics and automation (Buriak, 2005). Dorvee et al.
(2004) describe a clever and adaptable self-assembly approach to manipulating microlitresized liquid droplets. Using the same principles that govern the self-assembly of surfactant
molecules at interfaces, they made flat crystals of porous silicon, with a hydrophobic and
hydrophilic side, that align themselves at the interface of an aqueous droplet immersed in
an organic solvent. If the quantity of two-sided porous silicon crystals is sufficient, an entire
droplet can be coated in a self-assembled porous silicon wrapper. The two-sided porous
silicon was rendered magnetic through soaking in a basic solution of superparamagnetic
Fe3 O4 nanoparticles, making the drops sensitive to external magnetic fields.
Having so far briefly illustrated the importance of the magnetic nanoparticles both in
fundamental studies and technological applications, clearly there is a need to present a
comprehensible and structured state-of-the-art description on the synthesis, magnetic properties and applications of nanomagnets. Our approach to reach this goal has been starting
from a detailed description of some of the most relevant preparation routes to obtain
magnetic nanoparticles to finally cover a broad range of biomedical applications of these
nanoparticles that to our understanding nowadays can be considered as one of the most
steaming fields in the nanomagnets’ world. Only to mention that nowadays articles about
this topic are frequently situated among the top 10 most accessed articles in physics, chemistry and materials related journals. This seems not surprising since it is generally accepted
in the scientific community that the 21st century is the “biocentury” (see for example an
essay of George M. Whitesides in Angewandte Chemie on some of the possible new directions in chemistry, Whitesides, 2004).
Magnetic nanoparticles offer abundant attractive possibilities in biotechnology (Häfeli et
al., 1997; Safarik and Safarikova, 2002; Berry and Curtis, 2003; Pankhurst et al., 2003; Tartaj et al., 2003a, 2005; Mornet et al., 2004; Tartaj, 2004, Gupta and Gupta, 2005a, 2005b).
First, they have sizes that place them at dimensions comparable to those of a virus (20–
500 nm), a protein (5–50 nm) or a gene (2 nm wide and 10–100 nm long). Second, the
nanoparticles are magnetic, which means that they obey Coulomb’s law, and can be manipulated by an external magnetic field gradient. Third, nanoparticles have a large surface
that can be properly modified to attach biological agents. Special interest deserves the use
of dispersions of magnetic nanoparticles as contrast enhancement agents in magnetic resonance imaging (MRI) (Weinmann et al., 2003). Because of their larger magnetic moment
in comparison to paramagnetic ions, particulate contrast agents produce enhanced proton
relaxation rates at significantly lower doses (in the range μmol/kg body weight (B.W.))
(Merbach and Tóth, 2001). Finally, magnetic nanoparticles exposed to an alternating magnetic field might act as localized heat sources at certain target regions inside the human
body.
Therefore we firstly deal with the different methods described in the bibliography capable of producing magnetic nanoparticles of different characteristics. In almost all applications the preparation method of the nanomaterials represents one of the most important
challenges that will determine the particle size and shape, the size distribution, the surface
chemistry of the particles and consequently their magnetic properties. Ferri- and ferromagnetic materials such as Fe3 O4 , SrFe12 O19 , Fe-C and some alloys like SmCo5 , have irregular
SYNTHESIS, PROPERTIES, APPLICATIONS OF MAGNETIC NANOPARTICLES
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particle shape when obtained by grinding bulk materials but can have a defined shape when
prepared by wet chemistry, plasma atomization or from the aerosol and gas phases. Also,
depending on the mechanism of formation, the particles obtained in solution can be amorphous or crystalline if they result from a disordered or ordered aggregation of crystallites,
respectively. In addition, the preparation method determines to a great extent the degree of
structural defects or impurities in the particle, as well as the distribution of such defects
within the particle and therefore its magnetic behavior (Morales et al., 1999a, 1999b; Battle
and Labarta, 2002).
After a detailed description of the different methods used to prepare magnetic nanoparticles the following section deals with the colloidal properties of nanoparticles. Having
information on the colloidal properties of magnetic nanoparticles is essential to impart applicability to these materials. Applications from magnetic inks for bank checks to biotechnology require the preparation of liquid dispersions containing magnetic nanoparticles.
Once described the colloidal properties the next section deals with the different protocols
used in the literature (surface modification of nanoparticles and/or encapsulation) to render the magnetic nanoparticles useful in biomedical applications though some protocols
intended for improving their stability in non-polar solvents will also be described.
The next section of this chapter deals with the magnetic properties of nanoparticles.
The magnetic behavior of magnetic nanoparticles represents a complex and challenging
problem. Thus, it is clear that the potential applicability of these nanoparticulate systems
require a deep knowledge of their magnetic properties. Finally, the last part of this chapter
specifically deals with a detailed description of the biomedical applications of magnetic
nanoparticles (NMR imaging, hyperthermia, drug targeting, separation and selection).
2. Synthesis of magnetic nanoparticles
We discuss in this section recent advances made in the synthesis of magnetic nanoparticles
using different synthetic approaches. Some general reviews on the synthesis and properties of nanoparticulate magnetic materials have been recently published. Sugimoto (2000)
have edited an excellent book on the synthesis of fine particles with the contribution of
several leading scientific authorities on this field. Jolivet (2000) has provided a comprehensive explanation on the synthesis in aqueous solution of several metal oxides by using
the partial charge model. Cushing et al. (2004) have reviewed the liquid-phase synthesis of
inorganic particles including many examples on magnetic nanoparticles. Recently, Tartaj
et al. (2003a, 2005) have described some synthetic routes for the preparation of magnetic
nanoparticles for biomedical applications. Gutsch et al. (2005) have also recently reviewed
the gas-phase synthesis of nanoparticles including examples on magnetic nanoparticles.
It is generally accepted that the next generation of synthetic routes must be able to
produce materials with a rigorous control in the microstructure and crystallochemical characteristics. This will allow to better tune the properties of the resulting materials. Moreover,
the ability to make large quantities of stable materials could be extremely useful. Thus, in
this review particular attention should be paid to the preparation methods that allow the
synthesis of nanoparticles with control in microstructure. This goal can be achieved by precipitation from a homogeneous solution under controlled conditions or by controlling the
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particle growth in a process where a precursor in aerosol or vapour form is decomposed. We
have grouped the different synthetic routes into three general categories mainly solution,
gas and solid routes. We have firstly described the solution routes because these techniques
are by far the most developed for the production of nanoparticulate materials with control in microstructure. Secondly, we have described the gas-phase routes. Although these
routes are not as developed as the solution routes in controlling the microstructure, they
allow scalability for mass production. Thirdly, we have briefly described some of the solid
routes that in some cases can lead to the efficient production of nanoparticulate materials
with the required microstructure. Finally, though strictly size selection methods cannot be
considered synthetic routes, their importance, have prompted us to briefly describe some
of these methods.
2.1. Solution routes
In the liquid phase, uniform particles are usually prepared via homogeneous precipitation
reactions, a process that involves the separation of the nucleation and growth of the nuclei.
A schematic representation of the different mechanisms proposed to explain the formation
of uniform particles is shown in fig. 2.1.
In a homogeneous precipitation, a short single burst of nucleation occurs when the
concentration of constituent species reaches critical supersaturation. Then, the nuclei so
obtained are allowed to grow uniformly by diffusion of solutes from the solution to their
surface until the final size is attained. To achieve monodispersity, these two stages must
be separated and nucleation should be avoided during the period of growth. This is the
classical model proposed first by LaMer and Dinegar (1950) to explain the mechanism of
formation of sulphur colloids and also for a limited number of cases (curve I of fig. 2.1).
However, uniform particles have also been obtained after multiple nucleation events. Den
Ouden and Thompson (1991) have explained the uniformity of the final product in this case
through a self-sharpening growth process (Ostwald ripening, curve III of fig. 2.1). Another
Fig. 2.1. Formation mechanism of uniform particles in solution: curve I: single nucleation and uniform growth
by diffusion (classical model of LaMer and Dinegar); curve II: nucleation, growth and aggregation of smaller
subunits; curve III: multiple nucleation events and Ostwald ripening growth.
SYNTHESIS, PROPERTIES, APPLICATIONS OF MAGNETIC NANOPARTICLES
411
mechanism has been supplied by Morales et al. (1992) and later by Ocaña et al. (1995)
illustrating that uniform particles can also been obtained as a result of aggregation of much
smaller subunits rather than continuous growth by diffusion (curve II of fig. 2.1).
On the other hand, an artificial separation between nucleation and growth processes
may be achieved by seeding, a process in which foreign particles are introduced into the
solution of monomers below the critical supersaturation. A study specifically addressing
the theory and thermodynamics of coprecipitation has been recently published by Cushing
et al. (2004).
2.1.1. Precipitation from homogenous solutions
2.1.1.1. Precipitation from aqueous solutions Precipitation routes from aqueous solutions are widely used for the synthesis of oxides in general and magnetic particles in
particular (Ozaki, 2000; Tartaj et al., 2003a). For example, two main methods based on
this synthetic approach have been reported for the production in aqueous solution of iron
oxide ferrite spherical particles in the nanometer range. In the first one, due to Sugimoto
and Matijevic (1980), ferrous hydroxide suspensions are partially oxidized with different
oxidizing agents. In this procedure, spherical magnetite particles of narrow size distribution with mean diameters above 30 nm can be obtained from an Fe(II) salt, a base and a
mild oxidant (nitrate ions).
According to the above method, an amorphous ferrous hydroxide is first precipitated and
then the aqueous gel is aged at 90◦ C for various periods of time in the presence of nitrate
anions. By studying the evolution of the precipitation with time it has been shown that very
small primary particles nucleated in the Fe(OH)2 gel, first aggregate and then larger particles are formed by a contact-recrystallization mechanism. The aggregation of the primary
magnetite particles results from the Van der Waals and magnetic forces acting under the
conditions of week repulsion, closely to the isoelectric point. This is an interesting case of
particle formation by ordered aggregation, as suggested by Ocaña et al. (1995), that gives
rise to spherical crystalline particles.
The most widely used method to obtain iron oxide ferrite nanoparticles in aqueous solution is that of Massart (1981), Massart and Cabuil (1987). By aging stoichiometric mixtures
of ferrous and ferric hydroxides in aqueous media, homogenous spherical ferrite particles
with sizes from 13 to 2 nm can be prepared. In addition, it was found that by adjusting the
pH and the ionic strength of the precipitation medium, it is possible to control the mean
size of the particles over one order of magnitude (Jolivet et al., 1997). The size decreases
as the pH and the ionic strength in the medium increases, since both parameters determine
the chemical composition of the surface and consequently, the electrostatic surface charge
of the particles (fig. 2.2). Jolivet et al. (2004) have recently overviewed condensation phenomena of iron cationic species in aqueous solution. A typical microstructure of magnetic
nanoparticles produced by this method is shown in fig. 2.3.
Iron oxide ferrite nanoparticles can also be generated after adding surfactants or polymers to the iron salts during the precipitation (Lee et al., 1996; Tartaj et al., 2005).
Magnetite nanoparticles of nearly uniform size have been prepared by Si et al. (2004)
precipitating ferrous ions in the presence of different polyelectrolytes at basic pH. The size
of the magnetite nanoparticles can be controlled from 5 to 15 nm by varying the concentration of the polyelectrolyte in the medium. Very fine magnetite particles have also been
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Fig. 2.2. Mean diameter of magnetite nanoparticles (D) formed at different pH (ionic strength constant at 1 M,
NaNO3 ) and different ionic strengths (NaNO3 , pH = 12). Adapted from Jolivet et al. (1997).
© 1997 Materials Research Society.
Fig. 2.3. Typical TEM microstructure of particles produced by precipitation from aqueous solutions (Massart
method). The visual differences in microstructure are the result of the process followed for TEM observation.
The particles shown in the left part are dispersed in H2 O and deposited onto carbon grids while the ones in the
right part were stabilized with oleic acid and dispersed in hexane prior to deposition onto carbon grids.
SYNTHESIS, PROPERTIES, APPLICATIONS OF MAGNETIC NANOPARTICLES
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Fig. 2.4. Typical TEM microstructure of particles produced by precipitation from organic solutions. In this particular case correspond to a) 6-, b) 7-, c) 8-, d) 9-, e) 10-, f) 11-, g) 12-, and h) 13-nm-sized air-oxidized iron oxide
nanoparticles showing the one nanometer level increments in diameter. Reprinted with permission from Park et
al. (2005).
© 2005 Wiley-VCH Verlag.
obtained by Bee et al. (1995) by coprecipitation of Fe(II) and Fe(III) salts in the presence
of citrate ions. Modifications of the Massart method by Molday and Mackenze (1982) have
allowed the synthesis of iron oxide ferrites in the presence of dextran that renders the magnetic nanoparticles biocompatible. A detailed description of these methods will be given
in the section dedicated to the surface modification and/or encapsulation protocols used to
render the magnetic nanoparticles useful in biomedical applications.
2.1.1.2. Precipitation from organic solutions The decomposition of iron complexes in
the presence of surfactants using organic solvents with high boiling temperatures has yield
markedly improved samples with good size control, narrow size distribution and good
crystallinity of individual and dispersible magnetic iron oxide nanoparticles (fig. 2.4). The
unique microstructural control achieved by this method is a contribution of several factors such as temperature of synthesis, concentration, nature of the solvent and precursors,
complexing strength, addition of seeds and even the speed at which the reagents are added
over the seeds (Hyeon, 2003; Caruntu et al., 2004; Jana et al., 2004; Park et al., 2005).
Particularly, the high boiling temperature of the organic solvents used in this method allows producing nanoparticles with high crystallinity. On the other hand, the fact that the
iron oxide precursors contain an organic component assure after decomposition a reducing
atmosphere able, if necessary, to reduce Fe(III) to Fe(II) species. Finally, the reversible adsorption of surfactant onto the surfaces of nanoparticles provides a dynamic organic shell
that mediates growth, stabilizes the nanoparticles in solution and limits oxidation after
synthesis.
Numerous examples exist in the bibliography on the preparation of magnetic iron oxide
nanoparticles in organic solutions. For example, Rockenberger et al. (1999) have demon-
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strated that injecting solutions of FeCup3 (Cup: N -nitrosophenylhydroxylamine) in octylamine into long-chain amines at 250–300◦ C yields nanocrystals of maghemite. These
nanocrystals range from 4 to 10 nm in diameter and are dispersable in organic solvents.
Sun and Zeng (2002) have been able to prepare monodispersed magnetite nanoparticles
with sizes from 3 to 20 nm by a high-temperature (265◦ C) reaction of iron(III) acetylacetonate in phenyl ether in the presence of alcohol, oleic acid and oleylamine. In particular,
magnetite nanoparticles around 4 nm were obtained by the thermal decomposition of the
iron precursor but to obtain diameters up to 20 nm a seed-mediated growth method was
required. Hyeon et al. (2001) prepared 13 nm monodisperse maghemite nanoparticles by
a non-hydrolytic method consisting in the injection of Fe(CO)5 into a solution containing
surfactants and a mild oxidant (trimethylamine oxide). The same group (Park et al., 2004)
and parallely the group of Peng at Arkansas University (Jana et al., 2004) have reported
the synthesis of magnetite nanocrystals using non-toxic iron chloride salts as reactants that
form iron oleate precursors. The group of Peng also used commercial iron stearate precursors to check for the influence of the fatty acid chain length. Also at the same time the
group of Colvin at Rice University used a similar approach to obtain magnetite nanocrystals from 6 to 30 nm though in this case they started from hydrated iron oxides instead iron
chlorides (Yu et al., 2004).
Li et al. (2004a, 2004b) have recently reported the preparation of water-dispersable iron
oxide nanocrystals by thermal decomposition of Fe(acac)3 in 2-pyrrolidone. 2-pyrrolidone
was chosen as solvent and stabilizer because it has strong polarity, a high boiling point,
and coordination capacity with transition metal ions. Also recently Caruntu et al. (2004)
have described a new method for synthesis of magnetite nanoparticles that is based on the
hydrolysis of chelate metal alkoxide complexes at elevated temperature in solutions of the
parent chelating alcohols (fig. 2.5). Particularly, these authors use diethylene glycol and
diethanolamine as solvents because both their chelating abilities inhibit the formation of
metal oxides/hydroxides and their high boiling temperature is suitable for variable temperature synthesis (these two solvents have similar physical properties but the complexing
strength of diethanolamine is higher). This method allows the preparation of nonaggregated nanocrystals with variable size and composition and high crystallinity. The surface
of the isolated nanoparticles is coated with a labile layer of the solvent and remains
chemically active and available for further derivatization. Nanocrystals in this state are capable of forming stable aqueous colloids without using capping ligands or surfactants (by
adding oleic acid during synthesis these authors are able to prepare particles dispersable
in nonpolar solvents). A summary of the precursors and experimental conditions used to
obtain monodisperse magnetite nanocrystals from organic solutions has been included in
table 2.1. We have also included in this table the Sugimoto/Matijevic and Massart methods
(precipitation from aqueous solutions) for comparison.
The formation of metallic nanoparticles in organic solvents can be carried out by decomposition of metal carbonyl precursors or by reduction of metallic salts. Metallic carbonyls
are metastable organometallic compound whose facile decomposition has made it an extremely useful reagent to obtain metallic nanoparticles not only in solution but also (as
we will show below with detail) for gas-phase synthesis (e.g. Huber, 2005). Previous reviews on the topic can be found in Bates (1980), Toshima (2000) and Cushing et al. (2004).
In theory, one should be able to develop surface ligands that nucleate the seeding of the
SYNTHESIS, PROPERTIES, APPLICATIONS OF MAGNETIC NANOPARTICLES
415
Fig. 2.5. Formation of metal chelated complexes and their decomposition yielding colloidal transition metal
ferrites. Reprinted with permission from Caruntu et al. (2004).
© 2004 American Chemical Society.
TABLE 2.1
Summary of the precursors and experimental conditions used to obtain monodisperse iron oxide ferrite nanocrystals from organic solutions. The Sugimoto/Matijevic and Massart methods (precipitation from aqueous solutions)
are included in the first two rows for comparison
Conditions
(T a /solvent)
Particle size
(nm)
Source
Fe(II) + base + nitrates
90◦ C/water
30–1000
Fe(II) + Fe(III) + base
FeCup3
25◦ C/water
250–300◦ C/octylamine
15–2
4–10
265◦ C/oleylamine + 1,2hexadecanediol
100◦ C/octyl ether
180◦ C/dodecylamine
Reflux/2-pyrrolidone
300◦ C/octadecene
3–20
4–16
12, 6
5–11
3–50
Sugimito and Matijevic
(1980)
Massart (1981)
Rockenberger et al.
(1999)
Sun and Zeng (2002)
Sun et al. (2004)
Hyeon et al. (2001)
Cheon et al. (2004)
Li et al. (2004a, 2004b)
Jana et al. (2004)
5–18
Caruntu et al. (2004)
13
6–30
Park et al. (2004)
Yu et al. (2004)
Precursors
Fe(Acac)3
Surfactants
Oleic acid
Fe(CO)5
Oleic acid
TOPO
Fe(CO)5
Fe(acac)3
2-pyrrolidone
Fe(II) stearate or oleate
Fatty acids
(from Fe(II, III) chloride) (oleic or stearic)
Fe(II, III) chlorides
Oleic acid
200◦ C/diethylene glycol
and/or diethanolamine
FeCl3
Oleic acid
320◦ C/octadecene
FeOOH
Oleic acid
300◦ C
Cup = C6 H5 N(NO)O; Acac = acetylacetonate; TOPO = trioctylphosphine oxide.
416
P. TARTAJ et al.
magnetic nanocrystals in the solvent and passivate the surface of the nanoparticles against
unwanted aggregation. Although the metal particles usually oxidize quickly under normal
handling, the selected surfactant both favors the crystallization of metal over metal oxide
and slows down the oxidation of nanoparticles surface after growth.
The reduction of some salts to metals requires reducing agents whose reducing ability
can be afforded by some organic compounds such as amines, hydroxycarboxylic acids, or
alcohols. Thus, direct metal precipitation of fine particles can be obtained by reduction
of dissolved metallic salts in polyols and heating at moderately high temperatures. This
process was first used by Fievet et al. (1989) and by Viau et al. (1996) to prepare noble
metals such as Ru, Pd, Pt, Au and others such as Co, Ni or Cu. Later, the process has
been extended to the synthesis of other materials such as Fe-based alloys that could be
used for biomedical applications (Fievet, 2000). Furthermore, iron alloys can be obtained
by coprecipitation of Fe, Ni and/or Co in ethylene glycol and poly(ethylene glycol) but
the polyols are not able to directly reduce Fe(II) and Fe(III) ions to the zero-valence state.
In the polyol process, the liquid polyol acts as the solvent of the metallic precursor, the
reducing agent and in some cases as a complexing agent for the metallic cations. The metal
precursor can be highly or only slightly soluble in the polyol. The solution is stirred and
heated to a given temperature reaching the boiling point of the polyol for less reducible
metals. By controlling the kinetic of the precipitation, non-agglomerated metal particles
with well-defined shape and size can be obtained.
A better control of the average size of the metal particles can be obtained by seeding the
reactive medium with foreign particles (heterogeneous nucleation). In this way, nucleation
and growth steps can be completely separated and uniform particles result. Thus, larger
particles are usually obtained without seeding (homogeneous nucleation) while particles
between 50 and 100 nm have been obtained using Pt as the nucleating agent (heterogeneous
nucleation). A typical microstructure of magnetic nanoparticles produced by the polyol
process is shown in fig. 2.6. A summary of the metals prepared including their particle size
and the relevant reaction conditions are included in table 2.2.
In case where the reduction necessary to produce metallic nanoparticles exhibits a substantially negative E 0 potential, the use of an aqueous solvent is precluded and aprotic
solvents such as dimethyl ether or tetrahydrofuran (THF) are required. Bönnemann et al.
(1990) have demonstrated that trialkylborohydride, ABEt3 H (A = Li, Na or K), reduces an
impressive number of electropositive metals from their various salts in non-polar organic
solvents, such as toluene, dioctyl ether or THF. The reduction of ions such as Ni2+ , Co2+
and Fe2+ was typically performed with borohydride salts (table 2.2).
Sun et al. (1999, 2001) and Murray et al. (2001a, 2001b) developed an analogous
elevated-temperature method of synthesizing nanocrystalline cobalt and nickel and their
alloys with narrow size distribution. This method involved the reduction of metal salts,
chlorides or acetates, in the presence of two types of capping ligands: one strongly bound
(oleic acid) and other weakly bound (trialkyl phosphine and phosphine oxide). Injecting a
solution of reducing agents, long chain 1,2 diol or LiBEt3 H, into a preheated solution of
metal salts and capping agents in a high-boiling solvent resulted in a fast nucleation followed by slower nanocrystal growth. The reaction products formed stable dispersions in
nonpolar solvents due to the presence of carboxylic (oleic) acid groups covalently attached
to the nanocrystal surface (table 2.2).
SYNTHESIS, PROPERTIES, APPLICATIONS OF MAGNETIC NANOPARTICLES
417
Fig. 2.6. Typical TEM microstructure of particles (Co in this case) produced by the polyol process.
The decomposition of metal carbonyl precursors has also been used as a source of metal
nanoparticles. Thus, a study of the thermolysis of Fe(CO)5 has been reported by Smith and
Wychick (1980) in which they were able to prepare 5–15 nm colloidal Fe nanoparticles in
dilute solutions of butadiene containing polymers. For this purpose the metal carbonyl in
the reaction system was heated to temperatures between 130–160◦ C while Ar was purged
thoroughly. Further development of the metal decomposition method led to the use of
capping ligands instead of polymers for controlling the particle size and protecting the
products from oxidation and agglomeration. Thus Murray et al. (2001a, 2001b) have been
able to prepare very uniform 6 nm Fe nanoparticles by decomposition of Fe(CO)5 at 250◦ C
in the presence of two surfactants, oleic acid and tributylphosphine (TBP). The reason for
using two surfactants is because one binds tightly to the metal surface (oleic acid), favoring
slow growth and the other (TBP) binds weakly, permitting rapid growth. The ratio of both
can be adjusted to control the particle size.
One interesting case from the crystallochemical point of view is that of Co nanocrystals
since they posses multiple crystal structures (hcp, fcc and epsilon) that are close in energy.
Consequently, differences in surfactants adhesion or slight variations in temperature may
be used to control the crystal phase, size and shape. Many studies have been carried out
by decomposition of Co2 (CO)8 in a mixture of hot organic solvents and surfactants. Thus,
according to Puntes et al. (2001, 2002) the presence of TOPO resulted initially in the formation of disks, which later dissolve and a very narrow size distribution of spherical shaped
microcrystals of the thermodynamically stable ε-Co phase are formed. The same reaction
performed in the absence of TOPO resulted in the formation of a pure fcc phase. However,
the presence of long-chain aliphatic amines surfactants yields hcp-Co disks in large proportion and persisting for a long period of time. By using cobalt salts (Co(CH3 COO)2 ) Murray
et al. (2001a, 2001b) have been able to obtain Co nanoparticles with a predominantly hcp
418
TABLE 2.2
Summary of the precursors and experimental conditions used to obtain monodisperse metallic nanocrystals from organic solutions
Compound and precursors
Reducing agents
Surfactants
Fe, Fe(CO)5
Oleic acid, TBP
NaBEt3 H
EG
NaBEt3 H
1,2-dodecanediol
Co mt-fcc, Co2 (CO)8
ε-Co, Co2 (CO)8
Oleic acid, TOP
Oleic acid,
octadecylamine
Oleic acid, TBP
Oleic acid, TOPO
Fe45 Co55 , Fe(CO)5 + Co2 (CO)8
Oleic acid, TOPO
FePt, Pt(acac)2 + Fe(CO)5
1,2-hexadecanediol
FePt, Pt(acac)2 + FeCl2
1,2-hexadecanediol,
LiBEt3 H
TEG
FePt, Pt(acac)2 + Fe(acac)2
Oleic acid,
oleyl amine
Oleic acid,
oleyl amine
Particle size
(nm)
Reference
130–160◦ C/
butadiene
250◦ C /DPE
67◦ C/THF
180–220◦ C
23◦ C/THF
200–250◦ C/DPE
180◦ C/o-dichlorobenzene
200◦ C/DPE
180◦ C/o-dichlorobenzene
180◦ C/o-dichlorobenzene
180◦ C/1,2hexadecanediol
200, 260◦ C/1,2hexadecanediol
300◦ C/TEG
5
Smith and Wychick (1980)
6
10–100
50
10–100
2–20
Murray et al. (2001a, 2001b)
Bönnemann et al. (1990)
Fievet (2000)
Bönnemann et al. (1990)
Murray et al. (2001a, 2001b)
Puntes et al. (2002)
8–10
5–9
Murray et al. (2001a, 2001b)
Puntes et al. (2001)
1–11
Hütten et al. (2004)
3–10
Sun et al. (2000a, 2000c, 2000b, 2001)
4
Sun et al. (2004)
5–10
Jeyadevan et al. (2003)
THF = tetrahydrofuran; EG = ethylene glycol; OAc = acetate; NaBEt3 H = sodium trialkylborohydride; acac = acetylacetonate; TEG = tetraethylene glycol;
DPE = diphenylether; TOP = trioctylphosphine; TBP = tributylphosphine; TOPO = trioctylphosphine oxide; DDAB = didodecyldimethylammonium bromide.
P. TARTAJ et al.
Fe, Fe(CO)5
Fe, Fe(OEt)2
Co, Co(OH)2
Co, Co(OH)2
hcp-Co, Co(OAc)2
hcp-Co disks, Co2 (CO)8
Conditions
(T a /solvent)
SYNTHESIS, PROPERTIES, APPLICATIONS OF MAGNETIC NANOPARTICLES
419
internal structure. Staking faults along the c axis were the main source of disorder in the
hcp-Co nanoparticles. According to the above authors, carbonyl-based synthesis typically
do not yield single-crystal particles but they are comprised of multiple fragments of the
bulk fcc lattice with several radial twin planes in each particle. Hütten et al. (2004) have
prepared FeCo-alloyed nanoparticles from the simultaneous decomposition of two precursors, Co2 (CO)8 and Fe(CO)5 in the presence of oleic acid. Nanoparticles from 1 to 11 nm
were obtained with an extremely narrow particle size distribution.
Based also in the high-temperature decomposition of complexes, Sun et al. (2000a,
2000c, 2000b, 2001) have reported the synthesis of monodisperse FePt nanoparticles by
the simultaneous reduction of platinum acetylacetonate (Pt(acad)2 ) and the thermal decomposition of iron pentacarbonyl (Fe(CO)5 ) in the presence of oleic acid and oleylamine.
The composition of Fe and Pt is readily controlled from Fe30 Pt70 to Fe80 Pt20 and the size
can be tuned from 3 to 10 nm by adjusting the molar ratio of Fe(CO)5 to Pt(acad)2 . As synthesized FePt nanoparticles possess a disordered face centered cubic (fcc) structure with
superparamagnetic behaviour but thermal annealing induces internal changes resulting in
a tetragonal structure (fct) with high magnetocrystalline anisotropy for the composition
Fe55 Pt45 . The same authors have shown that these nanoparticles can be self-assembled
into three-dimensional superlattices, which are chemically and mechanically robust having potential applications in ultrahigh-density magnetic recording. Here it is worth noting
the work of Jeyadevan et al. (2003) who were able to obtain directly from the synthesis,
FePt nanoparticles with a certain degree of tetragonal order by using tetraethylene glycol as
solvent and Fe(acac)3 as precursor. The solvent allows carrying out the reaction at temperatures as high as 300◦ C. The key to directly produce the tetragonal phase was to slow down
the reaction rate. Another point interesting to remark is that the use of Fe(acac)3 instead
of Fe(CO)5 is preferred because the latter is volatile, thermally unstable and a well-known
toxic material.
2.1.2. Organic reactors
2.1.2.1. Organized surfactant assemblies Surfactants are molecules with a polar hydrophilic head and a hydrophobic hydrocarbon chain (Pileni, 2003). If a surfactant is added
to water, the hydrocarbon chains tend to self-associate to minimize contact with the water
molecules, resulting in the formation of various aggregates (fig. 2.7). Organized surfactant
assemblies such as microemulsions, micelles, hexagonal phases, cubic phases, monolayers,
and vesicles are unique reaction media. Indeed, they can solubilize, concentrate, localize,
and even organize the reactants. Many naturally occurring biological reactions such as
photosynthesis (Fendler, 1983) take place in organized media, and the self-organization
obtained with surfactants in solution has opened a systematic exploration of their use to
control chemical reactions (Fendler, 1975).
Among all of these methods, water-in-oil microemulsions have been recently shown
to be an adequate and simple method to prepare nanosized particles (Lisiecki and Pileni,
1993; Pillai et al., 1995; Zhang et al., 1999; Tartaj and De Jonghe, 2000; Zarur and Ying,
2000; Tartaj and Tartaj, 2002b; Tartaj, 2005). A microemulsion may be defined as a thermodynamically stable isotropic dispersion of two immiscible liquids consisting of nanosized
domains of one or both liquids in the other stabilized by an interfacial film of surface-active
molecules (Pillai et al., 1995; Lopez-Quintela, 2003). Microemulsions may be classified as
420
P. TARTAJ et al.
Fig. 2.7. Surfactant shapes and various self-assemblies in colloidal solutions. a, Cone-shaped surfactant resulting
in b, normal micelles. c, Champagne cork shaped surfactant resulting in d, reverse micelles with control of their
size by the water content. e, Interconnected cylinders. f, Planar lamellar phase. g, Onion-like lamellar phase.
Adapted with permission from Pileni (2003).
© 2003 Nature Publishing Group.
water-in-oil (w/o) or oil-in-water (o/w) depending on the dispersed and continuous phases.
The surfactant-stabilized microcavities (typically in the range of 10 nm) provide a confinement effect that limits particle nucleation, growth, and agglomeration (Pileni, 1993;
Shchukin and Sukhorukov, 2004).
In w/o microemulsions, the aqueous droplets continuously collide, coalesce and break
apart resulting in a continuous exchange of solute content (Pillai et al., 1995). The collision process depend upon the diffusion of the aqueous droplets in the continuous media
(the oil), while the exchange process depends on the attractive interaction between the surfactant tails and the rigidity of the interface, as the aqueous droplets approach close to
each other. The basics of nanoparticles’ formation can be summarized as follows: if we
take two identical W/O microemulsions and dissolve two reactants in each of the respective aqueous phases, upon mixing, due to collision and coalescence of the droplets, the
reactants come in contact with each other and form a precipitate. This precipitate is confined to the interior of the microemulsion droplets and the size and shape of the particle
reflects the interior of the droplet. One of the advantages of the use of microemulsions for
the preparation of magnetic nanoparticles is that by changing the nature and/or amount
of the surfactant (ionic or non-ionic, different flexibility etc.), co-surfactant (presence or
absence), oil phase, water content and reacting conditions it is possible to access a variety
of materials and microstructures (Lopez-Quintela et al., 2004).
Examples of the preparation of magnetic iron oxide and metallic nanoparticles are numerous. For example, Fe and Ni nanoparticles have been prepared by this method using
SYNTHESIS, PROPERTIES, APPLICATIONS OF MAGNETIC NANOPARTICLES
421
a different series of reducing agent (Capek, 2004). Core-shell magnetic metallic particles
can also be prepared by using microemulsions. Lopez-Quintela and Rivas (1993) have prepared magnetite nanoparticles of around 4 nm in diameter by the controlled hydrolysis with
ammonium hydroxide of FeCl2 and FeCl3 aqueous solutions within the reverse micelle
nanocavities generated by using the anionic surfactant AOT (sodium bis(2-ethylhexyl) sulfosuccinate) and heptane as continuous oil phase. By changing the nature of the surfactant
(Brij-97) the same group has been able to tune the particle size of the magnetic nanoparticles from 2 to 8 nm (Lopez-Perez et al., 1997). Superparamagnetic maghemite nanoparticles have been also prepared by the controlled hydrolysis of FeSO4 with tetraethylamine
(TEA) within the aqueous pools of the microemulsions (Pillai et al., 1995).
Lee et al. (2005) have also recently reported the large-scale synthesis of uniform and
crystalline magnetite nanoparticles using reverse micelles as nanoreactors under reflux
conditions. Pileni and co-workers prepared nanosized magnetic particles with average sizes
from 5 to 15 nm and standard deviation ranging from 0.2 to 0.3 by using organized surfactant assemblies (e.g. Pileni, 2001a). Divalent dodecylsulfate, X(DS)2 (X = Fe, Co, Zn), is
dissolved in aqueous solution and forms mixed oil-in-water micelles. A base is added to
the micellar solution and the powder thus obtained consists of ferrite nanocrystals of different nature (Fe3 O4 , γ -Fe2 O3 , Cox Fey O4 , Cox Zny Fez O4 ). A similar approach has been
used for the preparation of Co nanocrystals though in this case they start from cobalt bis(2-ethyl hexyl) sulfosuccinate as precursor to prepare the micelles and sodium borohydride
is used to reduce the Co(II) to metallic Co. At the end of the process the Co nanocrystals
are extracted from reverse micelles, for example, by covalent attachment of oleic acid (e.g.
Pileni, 2001a).
2.1.2.2. Other organic and biological reactors Nature has developed a variety of protein components that function as carriers or storage devices for metal components. Of
these systems, the iron-storage protein ferritin is probably the most intensively studied
and best understood (Mann, 2001; Zhao et al., 2003). Ferritin consists of a central core
of hydrated iron(III) oxide encapsulated with a multisubunit protein shell. Ferritin can
be demineralized (so called apoferritin) leaving an approximately 8 nm internal cavity that can be used as a nanoreactor (fig. 2.8a). As a result of the inner diameter of
the nanoreactors, Mann and co-workers have been able to prepare magnetite and magnetite/maghemite nanoparticles of about 6–7 nm in diameter (Meldrum et al., 1992;
Wong et al., 1998). The magnetite/maghemite particles were generated by oxidation of apoferritin with trimethylamino-N-oxide, which was loaded with various amounts of iron(II)
ions. The biological template of ferritin has been also used to prepare Co, Ni and CoPt
nanometallic alloys (Allen et al., 2003; Hoinville et al., 2003; Okuda et al., 2003).
Recently, metallic nanomagnets (Fe and FeCo alloys) of different size randomly dispersed in spherical colloids of silica and alumina have been prepared by aerosol evaporation of an aqueous sol containing mineralized ferritin and a colloidal matrix precursor
(Tartaj et al., 2004a). Essential to the success of the method is the template effect provided by ferritin. Its internal cavity offers the ideal environment for the encapsulation of
multicomponent metallic precursors with simultaneous control in size. Moreover, the high
stability of the ferritin cage allows compatibility with different matrix precursor sols (i.e.
nature of colloidal matrix can be changed).
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P. TARTAJ et al.
Fig. 2.8. (A) Inner cavity of apoferritin. Adapted with permission from Ueno et al. (2004). © 2003 Wiley-VCH
Verlag. (B) Typical architecture of a third generation dendrimers. Adapted with permission from Aulenta et al.
(2003). © 2003 Elsevier Ltd. (C) Schematic representations of the most commonly found cyclodextrins. Adapted
with permission from Engeldinger et al. (2003). © 2003 American Chemical Society.
SYNTHESIS, PROPERTIES, APPLICATIONS OF MAGNETIC NANOPARTICLES
423
The term “dendrimer” is derived from the Greek ‘dendra’ = tree and ‘meros’ = part and
describes graphically the structure of this new class of macromolecules which have highly
branched, three-dimensional features that resemble the architecture of a tree (Tomalia
et al., 1990; Aulenta et al., 2003). A typical dendrimer consists of three main structural
components: a multifunctional central core, branched units and surface groups (fig. 2.8b).
The branched units are organized in layers called “generations”, and represent the repeating monomer unit of these macromolecules. Dendrimers are characterized by an ideally
branched structure and the presence of a high number of functional groups, which can
have a significant effect upon the physical properties both in the solid state and in solution. Furthermore, the macromolecular dimensions can be controlled, since the synthesis
involves a repetitive sequence of steps. The radial structure of the dendrimers was used
for the first time by Crooks and co-workers in 1998 to prepare dendrimers-encapsulated
metallic nanoparticles (Zhao et al., 1998). Briefly, these nanocomposite materials are synthesized by complexing metal ions within dendrimers and then reducing the composites
to yield zerovalent dendrimers-encapsulated metallic nanoparticles (Scott et al., 2005).
This group have prepared Au and Pd nanoparticles encapsulated in dendrimers. Pd and Au
nanoparticles apart from being of application in the catalysis field, have generated attention because as aforementioned these materials exhibit the onset of ferromagnetism at very
small particle sizes (Teranishi et al., 1997; Hori et al., 1999, 2004; Sampedro et al., 2003;
Shinohara et al., 2003; Crespo et al., 2004; Yamamoto et al., 2004).
Advantages of using dendrimers for the preparation of magnetic nanoparticles is that
the chemistry one can make when having dendrimers is very rich (number of surface
groups can be varied almost at will), which is specially useful in biomedical applications of magnetic nanoparticles. For example, iron oxide ferrite nanoparticles have been
recently prepared using dendrimers as templating hosts (Strable et al., 2001). Carboxylated
poly(amidoamine) PAMAM dendrimers were utilized for the synthesis and stabilization of
ferrimagnetic iron oxide nanoparticles. Oxidation of Fe(II) at slightly elevated pH and
temperature resulted in the formation of highly soluble nanocomposites of iron oxides and
dendrimers, which are stable under a wide range of temperatures and pH values. Thus, by
the judicious selection of the dendrimers it has been possible to prepare in a single-step biocompatible magnetic nanoparticles that could be used for in vivo biomedical applications
(Strable et al., 2001).
Cyclodextrins are a family of naturally occurring, water-soluble oligosaccharides forming a bucket-shaped macrocycle and made up of α-(+)-glucopyranose units, which adopt a
4 C chair conformation. The most common members which are produced on an industrial
1
scale contain 6, 7, or 8 units and are named α-, β-, and γ -cyclodextrins (Engeldinger et
al., 2003). They have a porous-shaped structure with a hydrophobic cavity and hydrophilic
rims made of hydroxyl groups; the former encapsulates smaller hydrophobic molecules,
and the latter makes the molecule water-soluble (fig. 2.8c). Cyclodextrin can be easily deprotonated and in this form is able to complex with different metal ions. Gateschi and
co-workers have found that the basic hydrolysis of ferrous salts in the presence of γ cyclodextrins produces small particles of iron oxide, presumably of maghemite (γ -Fe2 O3 ),
with very small average diameter (1.8 nm) and with a narrow size distribution entrapped
in pseudo-single crystals of γ -cyclodextrin (Bonacchi et al., 2004). Cyclodextrins, in this
case in the β-form have been used by Hou et al. (2005) to prepare spherical aggregates
(100 nm to 2000 nm) of magnetite nanoparticles.
424
P. TARTAJ et al.
Sukhoroukov et al. (1998a, 1998b) and Donath et al. (1998a, 1998b) introduced a
protocol to construct a novel type of microcontainer. This procedure consists in the templating of polyelectrolyte films through a layer-by-layer strategy (Decher et al., 1992;
Decher, 1997) on the surface of micrometer- and submicrometer-sized colloidal particles.
Core decomposition leads to the formation of hollow structures with a shell composed of
polyelectrolyte multilayers (Shchukin and Sukhorukov, 2004). The size and shape of the
hollow capsules are determined by the initial colloidal core size. On the other hand, the
shell is tunable in the nanometer range and is usually permeable for macromolecules and
nanoparticles at low pH (< 3) or high ionic strength, while it is close at high pH (> 9).
For example, iron oxide nanoparticles were synthesized inside 5 μm polyelectrolyte
capsules. The capsule wall, made of polyelectrolyte multilayers of poly(styrenesulfonate)
(PSS) and poly(allylamine hydrochloride) (PAH), is permeable for small ions but not for
polymers. To perform particle synthesis exclusively in the capsule interior, capsules were
loaded with PAH. Encapsulation of PAH was done by assembly of a double-shell structure on a decomposable 5 μm template. The outer shell is composed of stable PAH/PSS
multilayers, while the inner shell is a complex of PAH with multivalent anions. This complex can be dissolved, releasing small ions outside and keeping PAH inside the capsules.
The presence of polycations inside the capsules maintains a pH gradient across the capsule
wall according to Donnan equilibrium potential (basic inside). Fe(III) cations penetrating
capsule walls face a higher pH and precipitate in the interior, forming iron oxide particles
(Radtchenko et al., 2002).
Viruses also represent a typical example of biology-assisted syntheses of nanomaterials in general and magnetic nanoparticles in particular. For example, Mao et al. (2004)
have recently used a virus-based scaffold for the synthesis of single-crystal ZnS, CdS, and
freestanding chemically ordered CoPt and FePt nanowires, with the means of modifying
substrate specificity through standard biological methods. Peptides (selected through an
evolutionary screening process) that exhibit control of composition, size, and phase during
nanoparticle nucleation have been expressed on the highly ordered filamentous capsid of
the M13 bacteriophage. The incorporation of specific, nucleating peptides into the generic
scaffold of the M13 coat structure provides a viable template for the directed synthesis
of semiconducting and magnetic materials. Removal of the viral template by means of
annealing promoted oriented aggregation-based crystal growth, forming individual crystalline nanowires. The unique ability to interchange substrate-specific peptides into the
linear self-assembled filamentous construct of the M13 virus introduces a material tunability that has not been seen in previous synthetic routes.
In 1975, Blakemore discovered magnetotactic bacteria (Blakemore, 1975). Magnetotactic bacteria form a heterogeneous group of Gram-negative prokaryotes with morphological
and habitat diversity, which have an ability to synthesize fine (50–100 nm) intracellular
membrane-bound particles consisting of magnetite or greigite (Fe3 S4 ). Interesting, the
magnetic nanocrystals are covered with an intracellular phospholipid membrane vacuole,
forming structures called magnetosomes (Schuler and Frankel, 1999). Due to the presence
of the phospholipid layer the particles are biocompatible, their suspensions are very stable
and the particles can be easily modified. In fact, bacterial magnetite nanoparticles obtained
from magnetotactic bacteria after disruption of the cell wall and subsequent magnetic separation have been used for a variety of bioapplications (Safarik and Safarikova, 2002;
SYNTHESIS, PROPERTIES, APPLICATIONS OF MAGNETIC NANOPARTICLES
425
Matsunaga et al., 2004). Very recently bacterial magnetic nanocrystals have also been used
for the preparation of magnetic nanotubes (Banerjee et al., 2005).
2.1.3. Liquid aerosols
Aerosol-assisted routes represent a convenient procedure for obtaining finely dispersed
particles of predictable shape, size and variable composition. The resulting powders generally consist of spherical particles, the final diameter of which can be predetermined from
that of the original droplets. The method offers certain advantages over other more commonly used techniques (such as precipitation from homogenous solution) as it is simple,
rapid, and continuous. Recently, for example Lu et al. (1999) and Grosso et al. (2004) have
shown the production of mesoporous microspheres of silica and titania, while Xia et al.
(2001) have prepared phosphorescent nanoparticles of Eu-Y2 O3 .
Aerosol routes that can be classified as solution routes are processes in which a solid is
obtained by spraying a solution into a series of reactors where the aerosol droplets undergo
evaporation of the solvent and solute condensation within the droplet, followed by drying
(if this is the final step the technique is called aerosol evaporation) and thermolysis (if
this is the final step the technique is called aerosol pyrolysis or spray pyrolysis) of the
precipitated particle at higher temperature (Messing et al., 1993). This procedure gives rise
to microporous solids, which can be finally sintered to form dense particles.
Most of the pyrolysis based processes employed to produce maghemite nanoparticles
start with a Fe3+ salt and some organic compound that acts as the reducing agent. In alcoholic solutions, uniform γ -Fe2 O3 particles can be prepared with a wide variety of particle
morphologies and sizes, ranging from 5 to 60 nm, depending on the nature of the iron
precursor salt (Gonzalez-Carreño et al., 1993). Dense colloidal aggregates with spherical
shape composed of γ -Fe2 O3 subunits with a mean diameter of 6 and 60 nm have been
obtained using Fe(III) nitrate and Fe(III) chloride solutions, respectively. Here it is worthy
of noting that the degree of packing of these colloidal aggregates can be easily modified by
adding to the Fe(III) nitrate solutions different amounts of acetic acid (Tartaj et al., 2004b).
On the other hand, γ -Fe2 O3 obtained from acetylacetonate solutions resulted in monodispersed particles of about 5 nm in diameter while maghemite particles derived from Fe(II)
ammonium citrate appeared as hollow spheres with a mean diameter of 300 nm. The latter
consisted of small crystallites aggregated forming a shell, the size of which varied between
10 to 40 nm, depending on the heating temperature in the furnace. A typical microstructure of colloidal aggregates of magnetic nanoparticles produced by this method is shown
in fig. 2.9.
2.1.4. Other solution routes
In sonochemistry, the acoustic cavitation, that is, the formation, growth, and implosive collapse of a bubble in an irradiated liquid, generates a transient localized hot
spot, with an effective temperature of 5000 K and a nanosecond lifetime (Suslick,
1990). The cavitation is a quenching process, and hence the composition of the particles formed is identical to the composition of the vapor in the bubbles, without phase
separation. The sonochemical decomposition of metal carbonyls in an alkene solvent has
been used to prepare iron metallic nanoparticles (Suslick et al., 1991, 1996). Equiaxial
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Fig. 2.9. Typical TEM microstructure of colloidal aggregates of magnetic nanoparticles produced by aerosol
pyrolysis. The symbols (D) from dense and (H) from hollow represent different degrees of hollowness. Reprinted
with permission from Tartaj et al. (2004a).
© 2004 Wiley-VCH Verlag.
ferrite nanoparticles and Ni nanoparticles encapsulated in carbon have also been produced by this method (Cao et al., 1997; Shafi et al., 2001, 2002; Lai et al., 2004;
Oliete et al., 2004). Special mention deserves the work of the Gedanken group on the
use of the sonochemical-assisted synthesis for the production of magnetic nanoparticles.
This group has obtained by this technique highly stable iron metallic nanoparticles due to
the presence of an iron carbide shell (Nikitenko et al., 2001), and even they report the formation of highly uniform magnetite nanorods (50 × 15 nm) when carrying the ultrasound
irradiation of iron(II) acetate in the presence of cyclodextrins (Vijaya-Kumar et al., 2001).
The microwave processing of nanoparticles results in rapid heating of the reaction mixtures, particularly those containing water. As a consequence, the precipitation of particles
from such solutions tends to be rapid and nearly simultaneous. This leads to very small
SYNTHESIS, PROPERTIES, APPLICATIONS OF MAGNETIC NANOPARTICLES
427
particle sizes and narrow size distributions within the products. The method offers the additional benefit of requiring very short reaction times (Cushing et al., 2004). Microwave
methods have been used for the synthesis of Ni nanoparticles stabilized with polyvinyl
pyrrolidone (Palchik et al., 2001; Tsuji et al., 2002).
Electrochemical methods have also been used for the production of maghemite nanoparticles in an organic medium (Pascal et al., 1999). The size was directly controlled by the
imposed current density, and the resulting particles were stabilized as a colloidal suspension by the use of cationic surfactants. The size distributions of the particles were narrow,
with the average sizes varying from 3 to 8 nm. Strontium ferrite nanoparticles have been
also prepared by electrolyzing acidic aqueous solutions of Sr2+ and Fe(II) chlorides and
nitrates in a 40–80◦ C temperature range (Asenjo et al., 2001).
Electrochemical methods are also used for the production of magnetic nanowires (e.g.
Kovtyukhova and Mallouk, 2002). The method involves metal deposition on or inside
preformed nanoscale templates. The electrochemical replication of porous membranes, introduced by Moskovitz (Al-Mawlawi et al., 1994) and Martin (Nishizava et al., 1995),
involves the evaporation of a metal film on the backside of an alumina or polycarbonate
membrane, which is then used as the cathode in an electrochemical cell. The cylindrical
pores determine the diameter of the wires, and the charge passed determines the wire length
with almost atomic precision (Garcia et al., 1999, 2000). Membranes with pore diameters
ranging from about 15 to 350 nm are commercially available, and can be synthesized in the
laboratory by anodic treatment of aluminum metal. Freestanding wires are obtained after
dissolving the backside metal film and the membrane itself.
2.2. Gas-phase routes
Gas-phase synthesis is a well-known chemical manufacturing technique for an extensive
variety of nanoscale particles (Gutsch et al., 2005). In gas-phase synthesis a supersaturated
vapor of condensable gaseous species is formed either as the result of chemical reactions
that create new species or as the result of physical processes such as cooling that reduce
the vapor pressure of condensable species (Kodas and Hampden-Smith, 1999). A supersaturated vapor consists of atoms or molecules of given species that are present at a partial
pressure higher than the vapor pressure of that species. At sufficiently high supersaturation,
new particles form by homogenous nucleation.
Metallic magnetic nanoparticles are routinely prepared using gas-phase routes (Del
Bianco et al., 2002; Theil-Jun et al., 2002; Hendy et al., 2003; Wang et al., 2003;
Chen et al., 2004; Li et al., 2004a, 2004b; Poddar et al., 2004; Sakiyama et al., 2004;
Huber, 2005). Gas-phase routes are also used to produce metallic nanomagnets encapsulated in carbon matrices (encapsulation in carbon matrix protects metallic nanomagnets
against corrosion). For example, nanocrystallites of iron embedded in an amorphous carbon matrix were prepared by in situ pyrolysis of maleic anhydride (C4 H2 O3 ) and ferrocene
(C5 H5 )2 Fe. Thermally assisted chemical vapor is used to decompose and chemically react
the introduced precursors. This method provides relative size control of the individual particles by varying the carbon/Fe concentration in precursors and the pyrolysis temperature
during the co-decomposition process (Sajitha et al., 2004). Pol et al. (2005) have prepared
Ni nanospheres encapsulated in a fullerene-like carbon by using the thermal decomposition of nickel acetylacetonate, Ni(C5 H7 O2 )2 . Moreover, gas-phase methods are adequate
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for the production of metallic magnetic nanowires either isolated or encapsulated in carbon
nanotubes (Rao et al., 2001; Guan et al., 2004).
Among the different gas-phase methods the Laser pyrolysis of carbonyl precursors has
been recently shown to be an excellent technique for the direct and continuous production
of γ -Fe2 O3 , Fe-based metallic, Co and Ni nanoparticles (Veintemillas-Verdaguer et al.,
1998, 2001; Martelli et al., 2002; Bomati-Miguel et al., 2005a, 2005b; He et al., 2005; Zhao
et al., 2005). The method firstly reported by Cannon et al. (1982) involves heating a flowing
mixture of gases with a continuous wave carbon dioxide laser, which initiates and sustains
a chemical reaction. Above a certain pressure and laser power, a critical concentration of
nuclei is reached in the reaction zone, which leads to homogeneous nucleation of particles
that are further transported to a filter by an inert gas. Three characteristics of this method
must be emphasized: (a) the small particle size, (b) the narrow particle size distribution and
(c) the nearly absence of aggregation (Cauchetier et al., 1994).
For example, to obtain γ -Fe2 O3 or Fe-based metallic nanoparticles by Laser-Pyrolysis,
Fe(CO)5 is used as precursor. Due to fact that this precursor does not absorb the radiation
at the laser wavelength (10.60 ± 0.05 μm), ethylene is used as absorbent as well as the
carrier to transport the carbonyl vapor to the reaction zone. Ethylene does not decompose
at the energy density used (652 W cm−2 ) but simply absorbs the laser radiation heating the
Fe(CO)5 , which is decomposed into iron and carbon monoxide. Either uniform γ -Fe2 O3
or Fe-based metallic nanoparticles are obtained by controlling the oxidation procedure.
Thus, when the airflow is introduced in the system together with the reactant gases, a violent oxidation of the initial iron nuclei takes place immediately and γ -Fe2 O3 nanoparticles
are obtained. However, when the airflow is introduced after the formation of the iron nuclei, the oxidation is milder, producing an iron oxide passivation layer surrounding the
iron metallic core. Increasing the laser power, the pressure into the reactor cell and/or the
temperature of the iron pentacarbonyl solution produces particles with bigger sizes and
better crystallinity. A typical microstructure of the materials obtained by this technique is
displayed in fig. 2.10.
2.3. Solid routes
Solid routes mainly mechanical milling and mechanochemical synthesis are currently used
for the production of magnetic nanoparticles. In mechanical milling the raw powder particles with a size of several microns experience severe plastic deformation, i.e. undergo
a repetitive cold welding and fracturing mechanism (Koch, 1993). Vibratory mill, planetary mill, uniball mill and attritor mill are commonly used to perform mechanical alloying.
A variety of magnetic nanoparticles from metallic to oxides have been prepared by this
technique (Eckert et al., 1992; Ponpandian et al., 2005). The disadvantage of ball-milling
for making nanocrystalline powders is the contamination of products from the milling media (balls and vial) and atmosphere.
Mechanochemical synthesis involves the mechanical activation of solid-state displacement reactions in a ball mill. Thus, mechanical energy is used to induce chemical reactions.
The chemical precursors typically consist of mixtures of oxides, chlorides and/or metals
that react either during milling or during subsequent heat treatment to form a composite powder consisting of the dispersion of ultrafine particles within a soluble salt matrix
(Tjong and Chen, 2004). The ultrafine particle is then recovered by selective removal of
SYNTHESIS, PROPERTIES, APPLICATIONS OF MAGNETIC NANOPARTICLES
429
Fig. 2.10. Typical TEM microstructure of magnetic nanoparticles produced by laser pyrolysis.
the matrix phase through washing with an appropriate solvent. Mechanochemical synthesis
is generally based on the following displacement reaction:
Ax C + yB → xA + By C,
where Ax C and B are precursors, A is the desired product and By C is a by-product of the
reaction. For example, pure Ni nanoparticles of 10–20 nm can be synthesized by milling
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NiCl2 with Na in a mill for various times followed by washing with distilled water, and
rinsing with methanol (Ding et al., 1996). This route has also been used for the production
of ferrite nanoparticles of different composition (Mahmoud et al., 2000; Manova et al.,
2004).
2.4. Size selection methods
Independently of the synthetic route used to obtain magnetic nanoparticles, the relative
amount of the particles with the desired size can be increased by the fractionation of
magnetic fluids (Rheinländer et al., 2000a). Common methods currently used for the fractionation of magnetic fluids are centrifugation (Sjögren et al., 1997) and size-exclusion
chromatography (Nunes and Yu, 1987). All these methods separate the particles via nonmagnetic properties like density or size.
Lefebure et al. (1998) have proposed a size sorting procedure based on the thermodynamic properties of aqueous dispersions of nanoparticles prepared by the Massart method.
The positive charge of the maghemite surface allows its dispersion in aqueous acidic solutions and the production of dispersions stabilized through electrostatic repulsions. By
increasing the acid concentration (in the range 0.1–0.5 mol l−1 ), interparticle repulsions
are screened and phase transitions are induced. Using this principle, these authors describe
a two-step size sorting process, in order to obtain significant amounts of monodisperse
particles with diameters typically between 6 and 13 nm. As the surface of the latter is not
modified by the size sorting process, usual procedures are used to disperse them in several
aqueous or oil-based media. Another size-selection process involves the addition of a poor
solvent to precipitate the larger particles. Fried et al. (2001) have been able to prepare iron
oxide ferrites of mean particle diameter of 6 nm with σ ∼ 10% using the Massart method
through four size-selective precipitations.
Preference should be given, however, to partitions based on the properties of interest,
in this case the magnetic properties. So far, magnetic methods have been used only for
the separation of magnetic fluids, for example, to remove aggregates by magnetic filtration (O’Grady et al., 1986). Field–flow fractionation is a family of analytical separation
techniques in which the separation is carried out in a flow with a parabolic profile running
through a thin channel (Giddings, 1993). An external field is applied at a right angle to
force the particles toward the so-called accumulation wall (Rheinländer et al., 2000a). Recently, the fractionation of magnetic nanoparticles by field–flow fractionation was reported
(Williams et al., 1999).
3. Colloidal properties of magnetic nanoparticles
Many of the applications of magnetic nanoparticles require the preparation of stable dispersions in liquid media. For example, applications in the biotechnology field require
magnetic nanoparticles to be stable in water at neutral pH and physiological salinity.
The Derjaguin–Landau–Verwey–Overbeek (DLVO) theory gives a classical description
of the stability of charged colloidal particles based on the combination of two forces
(Quesada-Perez et al., 2002). The dispersion forces are due to the difference in polarizability between particles and solvent, and are always attractive for identical particles. There also exists an electrostatic contribution to the interaction potential, which
SYNTHESIS, PROPERTIES, APPLICATIONS OF MAGNETIC NANOPARTICLES
431
Fig. 3.1. pH stability range of dispersions of maghemite nanoparticles. , the surface charge density of nanoparticles is a function of the pH of the medium. Through acid-base equilibria, can be progressively and reversibly
modified: if || is large enough, nanoparticles are dispersable, leading to a SOL. Around the Point of Zero Charge
(PZC) where = 0, interparticle electrostatic repulsion is too small and nanoparticles flocculate (FLOC). The
PZC value is related to the chemical nature of superficial charges. For example, when adding citrate ligands (L)
the nanoparticles are stable above pH = 4. Reprinted with permission from Bacri et al. (1990).
© 1990 Elsevier Science Publishers.
is repulsive in nature for identical particles. Colloidal stability of dipersions has to
be evaluated by considering all particle-particle interactions, that is, the DLVO theory
must be extended to include magnetic dipole attractions (Charles and Popplewell, 1986;
Philipse et al., 1994), non-DLVO forces (Ninham, 1999) and steric repulsion terms derived from the presence of surfactant or polymeric layers (Charles and Popplewell, 1986;
Russel et al., 1989; Hunter, 1990; Scholten, 1992; Galindo-Gonzalez et al., 2005).
The colloidal stability of dispersions, thus, depends on factors such as the nature of
the dispersion liquid, the dimensions of the particles, surface characteristics and the presence of stabilizing agents (Philipse et al., 1994; Butter et al., 2003; Lalatonne et al., 2004;
Richardi et al., 2004; Hsu et al., 2005; Klokkenburg et al., 2004). As an example of the
influence of the coating we illustrate in fig. 3.1, a study carried out by Bacri et al. (1990) related to the pH stability range of maghemite dispersions in aqueous media. We can observe
that the coating of magnetic nanoparticles with citrate ligands allows us for example stable
dispersions at neutral pH. The nature of the coating could be diverse and its selection firstly
depends on the nature of the solvent in which the magnetic nanoparticles are dispersed to
form the dispersions. For those applications involving apolar solvents, hydrophobic coatings of organic nature (organic coatings also contribute to an additional stabilization term
derived from steric repulsion) are used. On the other hand, for applications involving polar
solvents (normally aqueous media) hydrophilic coatings either organic or inorganic in nature are used. A detailed description of the different types of coatings will be given in the
section devoted to the surface modification of magnetic nanoparticles. We can also modify
the colloidal stability of dispersions by changing the magnitude of the magnetic dipole–
dipole interactions. Because these interactions scale as r 6 /d 3 (where r is the particle
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radius and d is the distance between the magnetic dipoles) their magnitude is substantially lowered when reducing the particle size (decreasing r) and/or through coating with
non-magnetic shells (increasing d).
From technological applications, ferrofluids are the dispersions of magnetic nanoparticles most employed. Ferrofluids are fluid materials with properties tailored on the nanometer level (Rosensweig, 1985; Vorobiev et al., 2004). They have the characteristic of being
stable as fluid even at very high magnetic fields. Ferrofluids have a content in magnetic
nanoparticles typically between 10 and 25 vol%. The most commonly studied ferrofluid
consists of an apolar solvent (typically kerosene or cyclohexane), in which magnetite particles with a typical diameter of 10 nm are dispersed. A monolayer of oleic acid grafted on
the surface of these particles prevents massive coagulation of the nanoparticles. The viscosity in ferrofluids depends upon external magnetic fields giving rise to their well-known
non-Newtonian hydrodynamic behavior (Odenbach, 2000; Melle et al., 2001, 2003). The
magnetic control of ferrofluids opens the possibility to position the fluid inside a technical
device allowing their use in a broad range of technological applications (Horng et al., 2001;
Odenbach, 2003), some of which are of importance in everyday life. For example, in high
power loudspeakers or speakers with a reduced size (e.g. car HiFi systems) the ohmic
heat produced in the voice coil leads to a critical limitation of the maximum power of the
speaker. Placing a ferrofluid in the magnetic field around the voice coil increases thermal
conductivity in this region and enables thus an increased heat transfer to the speakers’
structure.
A key issue in research on dispersions of magnetic nanoparticles is the effect of magnetic dipolar interactions on their structure and phase behavior (Teixeira et al., 2000;
Tlusty and Safran, 2000), which is not only important for practical applications but gives
fundamental insight into dipolar fluids in general. Nanomagnets dispersed in solution are
subjected to two major types of attractive forces: Van der Waals and magnetic dipolar
interactions. In the absence of the dipolar interactions, large isotropic Van der Waals interactions induce formation of spherical aggregates. For weak Van der Waals interactions, the
strength of the dipolar moments controls the formation of chain-like structures (Lalatonne
et al., 2004). In 1970, De Gennes and Pincus (1970) predicted a Van der Waals-like phase
diagram and the presence of linear chains of particles in dispersions in zero magnetic
field. Although simulations (Chantrell et al., 1982; Tavares et al., 1999, 2002; Weis, 1998;
Satoh et al., 2000) clearly demonstrated the existence of chain-like structures, no direct
evidence of their presence has been reported in the absence of magnetic field until very
recently (Butter et al., 2003; Klokkenburg et al., 2004). Philipse and co-workers have used
cryogenic transmission electron microscopy to the direct observation of dipolar chains in
dispersions of iron and magnetite nanoparticles coated with a surfactant layer that weakens
the Van der Waals interaction (Butter et al., 2003; Klokkenburg et al., 2004). In particular, these authors observe the onset of dipolar chains for iron particle sizes above 7 nm
(fig. 3.2) while for magnetite nanoparticles dipolar chains are observed for particle sizes
above 20 nm (fig. 3.3). The observation of the onset of dipolar chains in iron at smaller
particle sizes than those of magnetite is consistent with its higher magnetic moment.
There are two approaches for fabrication of systems at nanometric or mesoscopic scales
(Whitesides and Grzybowski, 2002; Richardi et al., 2004). The so-called top-down approach is based on lithography techniques and is widely used to fabricate microelectronic devices. Due to the continuous trend in minimization, top-down techniques become
SYNTHESIS, PROPERTIES, APPLICATIONS OF MAGNETIC NANOPARTICLES
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Fig. 3.2. Typical in situ cryo-TEM images (zero field) of iron dispersions. (A) The 2.1 nm particles. (B) The
6.0 nm particles. (C) The 6.6 nm particles. (D) The 6.9 nm particles. (E) The 8.2 nm particles. D and E, that is,
6.9 and 8.2 nm particles form dipolar chains. Scale bars: 100 nm. Reprinted with permission from Butter et al.
(2003).
© 2003 Nature Publishing Group.
extremely expensive. A promising alternative approach, called bottom-down, uses selforganization, which is defined as an autonomous assembly of components into structures.
It is of large interest for fundamental and applied research to understand the basic rules,
which govern organization at a nanometer and mesoscopic scale. In the particular case of
magnetic nanoparticles, self-organization is strongly dependent on their colloidal properties.
In the past few years, several groups have concentrated their efforts on producing selforganized magnetic nanocrystals in a 2D superlattice (Lisiecki et al., 2004). When magnetic nanocrystals are deposited on a surface, collective properties due to long-range dipolar interactions are observed (Petit et al., 1998; Black et al., 2000; Ngo and Pileni, 2000;
Legrand et al., 2001; Murray et al., 2001a, 2001b; Puntes et al., 2004; Vazquez et al.,
2004). In 3D mesoscopic structures built up with maghemite particles, which are not or-
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Fig. 3.3. Typical in situ cryo-TEM images (zero field) of vitrified magnetite dispersions. (A) The 16 nm particles
(surface fraction including organic layer: 0.64) appear to repel each other. (B and C) The 21 nm particles (surface
fraction including organic layer: 0.17) form dipolar chains. Scale bars: 100 nm. Reprinted with permission from
Klokkenburg et al. (2004).
© 2003 American Chemical Society.
ganized in well-defined superlattices, the influence of the film structure on the collective
properties has been clearly demonstrated (Ngo and Pileni, 2000). In all these cases, the
influence of the colloidal properties of the nanocrystals has been studied. Recently, Pileni
and co-workers have shown that cobalt nanocrystals self-organize into long-range scale in
fcc supra-crystals (Lisiecki et al., 2003). The same group have also recently shown that the
adjustment of the experimental conditions of the deposition process leads to either disordered or highly ordered 3D assemblies (Lisiecki et al., 2004). These are built of the same
nanocrystal population (i.e., the same size and the same size distribution).
SYNTHESIS, PROPERTIES, APPLICATIONS OF MAGNETIC NANOPARTICLES
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Isotropic Van der Waals and anisotropic magnetic dipolar forces are also important
terms to consider when explaining the self-organization of nanomagnets during evaporation of dispersions under a magnetic field. For example, Pileni and co-workers have
recently shown abrupt transitions of maghemite nanocrystal organizations from chain-like
to random structures when nanoparticle solutions are evaporated under a magnetic field
(Lalatonne et al., 2004). The resulting organizations were explained by Brownian dynamics simulations in terms of a variation of the strength of Van der Waals interactions with
the particle contact distance, which is tuned by the length of the molecules coating the
particles. The weak dipole–dipole interactions between the maghemite particles are usually not sufficient to result in the chain formation observed here. However, due to the Van
der Waals interactions, when the nanocrystal contact distance is short enough, clusters of
nanocrystals are formed during the evaporation process. These clusters exhibit large dipole
moments compared with a single particle explaining the formation of chain-like structures.
Conversely, when the nanocrystal contact distance is too long, no nanocrystal aggregation
occurs, and a random distribution of maghemite nanocrystals is obtained.
4. Surface modification or encapsulation of nanoparticles
This section is intended to describe some of the protocols reported in the bibliography for
the coating or encapsulation of magnetic nanoparticles. As aforementioned these processing routes are of importance to obtain stable dispersions of magnetic nanoparticles either
in non-polar or polar solvents. Because of the growing interest in the use of iron oxide
ferrite nanoparticles in biomedicine, most of this description is centered in the protocols
followed to render these nanoparticles useful for these applications (hydrophilic coatings)
though some words will be also given at the end of this section in the topic of hydrophobic
coatings. For biomedical applications the benefits of encapsulation of magnetic nanoparticles either in liposomes or artificial capsules have prompted us to devote a single section
to this topic. We also briefly describe recent reports that mention protocols to transform
hydrophobic monodispersed magnetic nanoparticles prepared by precipitation from organic solutions (FePt and ferrites) into hydrophilic ones. We should keep in mind that
as aforementioned the method of precipitation from organic solutions is the most adequate
to obtain samples with good size control, narrow size distribution and good crystallinity
(Hyeon, 2003) and making these particles hydrophilic could open the possibility of their
use in biomedical applications, for example.
4.1. Organic hydrophilic coatings
Stability in water at neutral pH and physiological salinity of magnetic nanoparticles is
usually achieved by coating with organic substances (Mornet et al., 2004). Two main approaches have been described in the bibliography to produce biostable dispersions of magnetic nanoparticles coated with organic substances. One of these approaches aims to produce, in a single-step, coated nanoparticles as in the case of iron oxide superparamagnetic
nanoparticles encapsulated in biopolymers of the dextran family (Josephson et al., 1988;
Poulquen et al., 1991; Palmacci and Josephson, 1993; Chouly et al., 1996). Starch and synthetic polymers such as polyvinyl alcohol (PVA) or poliacrylic acid (PAA) have also been
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used to produce biocompatible magnetic compounds in a single-step (Olsson et al., 1986;
Bergemann, 1998; Kellar et al., 1999; Nestorova et al., 2000; Pardoe et al., 2001;
Kim et al., 2003a).
Alternatively the coating can be made onto the previously obtained magnetic nanoparticles. For example, by using small organic molecules with phosphate groups that coordinate
well with the surface of the iron oxide ferromagnetic nanoparticles (Portet et al., 2001) or
by using compounds having carboxylate moieties. In this case compounds such as sodium
oleate (Kim et al., 2003b) or 6-mercaptohexadecanoic acids have been used (Liu and Xu,
1995). In this latter example, the compound binds through the carboxylate group to the iron
oxide ferromagnetic nanoparticles leaving the reactive thiol or disulphide groups pointing
outward. Synthetic polymers such as polyethylene (Chaterjee et al., 2002), poly(ethylene
glycol) (PEG) or its derivatives (Kim et al., 2002; Zhang et al., 2002), poly(D,L-lactideco-glycolide) (Lee et al., 2004a, 2004b) or poly(ethyl-2-cyanoacrylate) (Arias et al., 2001)
have been also used. Among the synthetic polymers applied to coat magnetic nanoparticles,
polyethylene glycol (PEG) is the most used. Surfaces covered with PEG are biocompatible, nonimmunogenic, nonantigenic and protein resistant. An additional advantage is that
particles coated with PEG can cross cell membranes because PEG can dissolve in both
polar and non-polar solvents and it has high solubility in cell membranes (Zhang et al.,
2002).
Sometimes, in order to give an extra functionality to the nanoparticles secondary surface
modifiers are used (Weissleder et al., 1991; Kresse et al., 2003). Finally pharmaceutical
drugs for example: antiallergic, chemotherapeutic agents, hormones, cytostatic agents can
be added to the magnetic nanoparticles as optional components or can be coupled to the
targeting polymers (the conjugate of polymer and medical substance is then used as the
targeting polymer).
4.1.1. Polysaccharides
Some applications based on magnetic separation have little requirements on the type
of polymer shell, however others such as those that require intravenous administration
must meet strict criteria of nontoxicity, nonimmunogenic, nonantigenic and biodegradability (with non toxic byproducts). Polysaccharides fulfill these criteria (Arshady et al., 2002).
The most common natural additive of this family is dextran that can be added either unsubstituted, monosubstituted as carboxydextran or polysubstituted as polycarboxymethyldextran (Molday and Mackenze, 1982; Josephson et al., 1988; Poulquen et al., 1991; Noguchi
et al., 1992; Palmacci and Josephson, 1993; Chouly et al., 1996; Hasegawa et al., 1998;
Chen et al., 1999). Dextran is a polymer of glucose made exclusively by 1:6 glycosidic
linkages (fig. 4.1).
Usually the way dextran is added to the magnetic nanoparticles is by coprecipitation
from aqueous solutions containing the iron oxide precursors and the polysaccharides. The
presence of these polysaccharides not only dictates the surface properties of the resulting composite but also affects their particle size (Widder et al., 1987). One of the first
examples of such a synthesis was given by Molday and Mackenze (1982). Particularly,
magnetic iron-dextran particles were routinely prepared by mixing aqueous solution of
dextran (molecular weight 40 000 g mol−1 ) with deaerated aqueous solution containing
Fe(III) and Fe(II) chlorides in adequate proportions. The mixture was titrated to pH 10–11
SYNTHESIS, PROPERTIES, APPLICATIONS OF MAGNETIC NANOPARTICLES
437
Fig. 4.1. Dextran (glucose monosaccharides linked through 1:6 glycosidic bonds).
by the dropwise addition of ammonium hydroxide and heated to 60–65◦ C in a water bath
for 15 min. Aggregates were then removed by three cycles of centrifugation (the colloidal
composites obtained in such a way are presently called ferumoxides).
Modifications of the method of Molday and Mackenze (1982) include the cold gelation
process in which the precipitation is carried out at 0◦ C and heated after at 60 or 100◦ C
under N2 atmosphere (Palmacci and Josephson, 1993). The effect of this modification is to
produce more densely coated nanoparticles. The colloids so obtained are stable over a pH
range 3–10, have a superparamagnetic iron core (maghemite-magnetite) of 10–20 nm and
the dispersion consists in colloidal aggregates of particles of 50–150 nm (Jung and Jacobs,
1995). These colloidal dispersions are known as SuperParamagnetic Iron Oxides (SPIO)
and also Ferumoxtran.
In order to obtain aggregates with smaller hydrodynamic size and to increase the blood
circulation time of the preparations, the addition of a higher proportion of dextran combined with ultrafiltration with Amicon YM300 membrane was carried out. Ultrafiltration
assures the presence of smaller aggregates in the final dispersions while the addition of
an extra amount of dextran assures invisibility against the plasma proteins, that is, longer
blood circulation times (Weissleder et al., 1994). Using this approach Weissleder (1991)
obtained preparations known as MION-37 and Shen et al. (1993) obtained preparations
known as MION-46. The resulting colloidal composites are also known as Ultrasmall SuperParamagnetic Iron Oxides (USPIO) due to their extremely small hydrodynamic size
(20 nm). The difference between MION-37 and MION-46 is the proportion of iron, the
first one has 47% in iron content while the latter one has only 27%, that is, MION46 has higher amount of dextran than MION-37. Accordingly to the content in dextran, MION-37 and MION-46 have half-live times in blood of 90 and 180 min, respectively.
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Fig. 4.2. Schematic representation of the adsorption mechanism of dextran on superparamagnetic iron oxide
nanoparticles.
An important factor in the choice of dextran appears to be the favorable size of dextran
chains, which enables optimum polar interactions (mainly chelation and H-bonding) with
iron oxide surfaces. Although single H-bond are relatively weak, total bonding energy of
H-bonds over the length of polysaccharide molecule can be very high because of the large
number of hydroxy groups per molecule. Jung (1995) has considered a model for polymer adsorption in which the interactions take place at different segments of the polymer
(fig. 4.2). It is worthy noting that dextran molecules can be desorbed from the iron oxide
surface by heating at 120◦ C (Palmacci and Josephson, 1993) or by dilution (Bautista et al.,
2005). The usually way to avoid desorption of dextran is to use epichlorhydrine, that is,
agents with crosslinking capacities (Palmacci and Josephson, 1993; Fournier et al., 1995).
Carboxydextran coating of iron oxide nanoparticles can be carried out under similar
conditions to those aforementioned for dextran (Goetze et al., 2002). The resulting colloidal composites have smaller particle size than those obtained using dextran (Chen et al.,
1999). Ordinary dextran can be replaced by poly-substituted dextrans to improve stability.
For example, the use of carboxymethyldextran, in which carboxymethyl group replaces the
hydroxyl groups present in the glucose unit, allows a multi-side attachment through their
negative charges (Hasegawa et al., 1998; Goetze et al., 2002). On the other hand, the use
of diethylaminoethyldextran changes surface charge values from the ordinary negative into
positive at physiological pH (Chouly et al., 1996).
Starch has also been employed for coating the iron oxide nanoparticles (Fahlvik et al.,
1990; Kellar et al., 1999; Kim et al., 2003a, 2003b). Starch is a complex natural polysaccharide consisting of two components: amylose and amylopectin. Like dextran both are
polymers of the monosaccharide glucose. Amylose is a linear molecule containing alpha
SYNTHESIS, PROPERTIES, APPLICATIONS OF MAGNETIC NANOPARTICLES
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glycosidic linkages from carbon number one (C-1) to carbon number four (C-4) of the adjacent glucose units. Amylopectin is a branched molecule; it consists of several hundred
of short amylose chains cross-linked to each other by glycosidic linkages from C-1 to C-6
of the adjacent glucose units. Similar to dextran the high number of hydroxyls in starch
favors the hydrogen bonding between the hydroxyl groups and the surface of nanoparticles
(Khosla et al., 1984). When searching for a special affinity of nanoparticles against a group
of cells with highly specific receptors, a specific polysaccharide is used instead of dextran
or starch. For example to attach nanoparticles to the hepatocyte cells of the liver, we can
use galactose that contains polysaccharides as arabinogalactan because of their affinity to
the asialoglycoprotein present in hepatocyte cells (Josephson et al., 1990).
As aforementioned dextran can be adsorbed in a second step onto the surface of the iron
oxide superparamagnetic nanoparticles (Sjögren et al., 1997). In some cases the methodology employed for the synthesis of the magnetic nanoparticles prevent the addition of
the polysaccharides during the formation of the magnetic nanoparticles. Thus, their surface modification must be performed after the magnetic nanoparticles have been prepared.
This is the case of ferrite nanoparticles prepare by sonochemical synthesis (Kim et al.,
2005a) and both ferrite and iron metallic nanoparticles prepared by Laser pyrolysis of
iron pentacarbonyl vapors (Veintemillas-Verdaguer et al., 2004; Bomati-Miguel et al.,
2005a, 2005b). In the latter case, for example, stable aggregates (hydrodynamic size of 46
nm) of maghemite nanoparticles coated with dextran were obtained when the nanoparticles
were dispersed in a strong alkaline solution in the presence of dextran (VeintemillasVerdaguer et al., 2004).
4.1.2. Synthetic polymers
A variety of magnetic polymer spheres have been produced from synthetic polymers
with the purpose of incorporating functional groups on their surfaces to mainly perform
magnetic separation. Coating or encapsulation with synthetic polymers offers better chemical stability with respect to natural ones (polysaccharides for example) but the resulting
composites are less biocompatible. This is the reason why these composites have commonly been used in in vitro applications. The strategies employed in the preparation of
these materials are: (1) synthesis of the nanoparticles in the presence of the polymer;
(2) polymerization in dispersions containing magnetic nanoparticles (so-called dispersion
polymerization) and (3) grafting the polymer on the magnetic particles once synthesized
(polymeric surfactants).
Precipitation of superparamagnetic iron oxide nanoparticles into the porous structure of
synthetic polystyrene latexes is the commercial method for the production of Dynabeads® ,
the magnetic composites most frequently used in magnetic separation. These micron sized
particles combine low densities (and consequently dispersability) with good magnetic
properties, which make easy their separation using small permanent magnets. The polymer particles used as starting material are prepared by emulsion polymerization resulting
in particles in the range up to about 0.6 microns in diameter. Bigger sizes can be attained
using a seeding process (Ugelstad et al., 1983). Water insoluble substances are added to
the system to develop porosity in the polymer particles. Ordinary monomers are styrene,
maleic anhydride, methyl methacrylates and vinyl acetate. Initiators are potassium persulphate or benzoyl peroxide. Normally sulphonic or carboxylic acid groups are incorporated
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to the polymer to increase the adsorption capacity of the polymer for the iron precursors
(Ugelstad et al., 1983). Modifications of this process relay on the production of polymer
spheres by miniemulsions (Ugelstad et al., 1992). Coprecipitation of magnetite/maghemite
nanoparticles can also be performed in the presence of water-soluble polymers as polyvinyl
alcohol (Pardoe et al., 2001). The particles form aggregates in a polymer matrix in form of
strands or necklaces suggesting iron oxide precipitation along unfolded polymer strands.
Recent advances in the synthesis of magnetic nanoparticles in the presence of polymers
are based on the use of polymer gels (Zhang et al., 2004). The advantages of using polymer gels are: (1) they have a network structure of pores that can be used for nanoparticle
nucleation and growth, (2) they undergo volume transitions in response to external stimuli
(e.g. temperature, pH, ionic strength or the action of a electric field) and (3) functional
species other than nanoparticles, for example, small proteins or drug molecules, can be
incorporated and released from the interior of microgels.
Pluronics® are a family of poly(ethylene oxide)-b-poly(propylene oxide)-b-poly(ethylene
oxide) triblock copolymers used as nonionic surfactants in biotechnological and pharmaceutical industries for their unique surfactant abilities, low toxicity and minimal immune
response (Nace, 1996). These block copolymers have been used for the thermal synthesis of iron based magnetic nanoparticles (Lai et al., 2004). Here it is worthy noting that
poly(ethylene oxide) (PEO) and poly(ethylene glycol) (PEG) are differenced by their range
of molecular weight covered, commonly this polyether is named PEO when the molecular
weight is higher than 50 000 and PEG otherwise.
The second approach (dispersion polymerization) consists of polymerizing the monomer
in the presence of a magnetic material dispersed in a liquid. The coating takes place if
the monomer is solubilized in the dispersion liquid but the polymer does not. This technique was mainly developed to overcome problems of hydrophobicity in the magnetic
nanoparticles encapsulated inside styrene divinyl benzene polymers that were produced by
the aforementioned Ugelstad method. These composites exhibit unspecific protein adsorption, which produces an undesirable background signaling in immunoassays. Using this
approach, Zaitsev et al. (1999) have reported the preparation of biocompatible magnetic
poly(2-hydroxyethyl methacrylate)-based microspheres with low nonspecific adsorption.
Horák et al. (2003, 2005) have optimized the process to improve the functionability of the
particles by introducing reactive oxirane groups. The miniemulsion process is used to obtain magnetic polymers of smaller particle sizes. This process is based on the formation of
a miniemulsion of the reactant dispersion (formed in this case by hydrophobized iron oxide
and styrene) in water to then carry out the polymerization process in each of the new formed
nanodroplets (30–100 nm) in parallel (Ugelstad et al., 1983; Landfester and Ramirez, 2003;
Ramirez and Landfester, 2003).
If polymerization initiators are chemically attached to the surface of the magnetic
nanoparticles, the graft polymerization resultant forms a dense layer of polymer chains
with low polydispersity of molecular weights. Vestal and Zhang (2002) reported the formation of magnetic MnFe2 O4 polystyrene nanoparticles using this technique that the authors called atom transfer radical polymerization. The encapsulated material was MnFe2 O4
nanoparticles with a mean size of 9 nm prepared by the microemulsion technique. A similar
procedure was reported by Matsuno et al. (2003) to coat magnetite nanoparticles.
SYNTHESIS, PROPERTIES, APPLICATIONS OF MAGNETIC NANOPARTICLES
441
Fig. 4.3. Cryo-TEM pictures of maghemite-loaded liposomes. Reprinted with permission from Lesieur et al.
(2003).
© 2003 American Chemical Society.
4.1.3. Encapsulation in liposomes and artificial capsules
Encapsulation of magnetic nanoparticles in stealth liposomes or artificial polyelectrolyte
capsules seems a promising approach to improve the applicability of magnetic nanoparticles in biomedical applications. Advantages of the use of these devices are that through
conjugation with biological ligands different biomedical functions can be implemented
and also the possibility to increase the circulation time of magnetic nanoparticles in the
blood (Mahato, 2005).
Lipids usually form lamellar bilayer structures over the majority of their phase diagram,
and in dilute solutions they form vesicles (from the Latin vesicle, small bubble) (Antonietti
and Förster, 2003). Liposomes can be considered as a special class of vesicles, in which
the building blocks units are lipids. Magnetoliposomes consist of nanometer-sized iron
oxide particles encapsulated in liposomes (fig. 4.3). Several methods have been described
for the preparation of magnetoliposomes: dialysis of single unilamellar vesicles (SUVs)
in the presence of magnetite nanoparticles (De Cuyper and Valtonen, 2001); lauric acidstabilized magnetite cores (Bulte et al., 1999); or extrusion of a mixture of USPIOs with
phospholipid–cholesterol large unilamellar vesicles (LUVs) (Bulte et al., 1993).
Polyelectrolyte multilayer (PEM) micro- and nanocapsules were introduced (Keller et
al., 1995; Donath et al., 1998a, 1998b; Sukhorukov et al., 1998a, 1998b) as an extension of PEM on flat substrate surfaces (Decher and Hong, 1991; Decher et al., 1992;
Decher, 1997). When decomposable colloids are used as templates they can be removed
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after completion of the PEM coating and hollow capsules or shells are obtained. Surprisingly, they are very stable in different aqueous and organic solvents. One of the advantages
of PEM capsules is their versatility of physical and chemical properties by variation of
composition and fabrication as well as application conditions.
Several dozen of natural and synthetic polyelectrolytes or other charged substances
are suitable as layer components, e.g. chitosan, poly(diallyldimethylammonium chloride),
chitosansulfate, poly(L-lysine), poly(acrylic acid), dextransulfate, and proteins (Prokop
et al., 1998). Many of them can be functionalized to provide special surface structures of biological or technical relevance. In addition, hollow capsules can be applied
as containers for inorganic and organic matter, e.g. drugs, macromolecules or luminescent nanoparticles (Caruso et al., 2000; Radtchenko et al., 2000; Duan et al., 2005;
Zebli et al., 2005). Recently, Voigt et al. (2001) have integrated superparamagnetic magnetite nanoparticles mounted to the multilayer shell itself or as a component of the capsule
interior. The PEM is formed at different (decomposable) colloidal templates, e.g. melamine
formaldehyde resin, glutaraldehyde fixed red blood cells, emulsion oil droplets.
4.2. Inorganic hydrophilic coatings
Inorganic coatings present a series of advantages with respect to the organic ones. For
example, there are no swelling or porosity changes with a change in pH, and they are
not vulnerable to microbial attack (Weetal, 1970). They also effectively protect doped
molecules (enzymes, drugs, etc.) against denaturation induced by extreme pH and temperature (Jain et al., 1998; Roy et al., 2003). Moreover, the use of inorganic matrices, in
particular of silica, as dispersion media of magnetic nanocrystals is an effective way to
modulate the magnetic properties by a simple heating process (Del Monte et al., 1997;
Ennas et al., 1998; Corrias et al., 2004). An appropriate tuning of the magnetic properties
is essential for the potential use in biomedicine of magnetic composites as reported by
Tartaj and Serna (2002). Another advantage of having a surface enriched in silica is the
presence of surface silanol groups that can easily react with alcohols and silane coupling
agents to produce dispersions that are not only stable in non-aqueous solvents but also
provide the ideal anchorage for covalent bonding of specific ligands (Ulman, 1996). The
strong binding makes desorption of these ligands a difficult task. In addition, as noticed
by Mulvaney et al. (2000), the silica surface confers high stability to suspensions of the
particles at high volume fractions, changes in pH or electrolyte concentration.
Many protocols exist in the bibliography to obtain magnetic nanoparticles coated with
silica. For example, Philipse et al. (1994) have developed a two-step process that consist
first in the precipitation of sodium silicate followed by the basic hydrolysis of tetraethyl
orthosilicate to prepare silica coated magnetic particles. Correa-Duarte et al. (1998) have
assembled SiO2 -coated Fe3 O4 nanoparticles prepared by the Philipse method on silicon
and glass substrates following the principle of the layer-by-layer self-assembly. Modifications of the Philipse method are in the direction of using polyelectrolytes (Tartaj, 2003) or
citrate anions (Sun et al., 2005; Wang et al., 2005a, 2005b) to improve the efficiency of
the coating. Shao et al. (2005) have recently reported the preparation of magnetic hollow
silica nanospheres. First, silica coated magnetite nanoparticles were obtained by the basic
hydrolysis of tetraethyl orthosilicate in methanol. Then, the composites were deposited in
spherical calcium carbonate particles between 20–100 nm under alkaline conditions. The
SYNTHESIS, PROPERTIES, APPLICATIONS OF MAGNETIC NANOPARTICLES
443
as-synthesized nanospheres were immersed in an acidic solution to remove nanosized calcium carbonate and to generate the silica coated magnetite hollow nanospheres. Kobayashi
et al. (2003) have also developed a method to obtain silica coated Co nanoparticles. Silica
coating was performed by addition of an ethanolic solution containing 3-aminopropyltrimethoxysilane and tetraethyl orthosilicate to an aqueous solution containing Co nanoparticles, which were obtained by reduction of CoCl2 with NaBH4 . In the water/ethanol
mixture, attachment of 3-aminopropyl-trimethoxysilane onto the surface of cobalt particles
takes place, followed by hydrolysis/condensation of both 3-aminopropyl-trimethoxysilane
and tetraethyl orthosilicate which results in the formation of the silica shells. For crystallization of cobalt, the silica-coated cobalt particles were separated from dispersion by
centrifugation and drying, and the resulting powder was annealed in air at temperatures
between 200 and 700◦ C for 2 h.
Yi et al. (2005) have reported the synthesis of a novel soluble hybrid material consisting of quantum dots (QD) and magnetic nanoparticles (MP) encapsulated in a silica
shell. The SiO2 /MP-QD nanocomposite particles preserved the unique magnetic property of maghemite and optical properties of CdSe QDs. Microemulsions have been used
for the preparation of silica coated magnetic nanoparticles (Tartaj and Serna, 2002;
Gao et al., 2003; Tartaj and Serna, 2003). Yang et al. (2004) have used the microemulsion
method for the preparation of monodisperse silica-coated iron oxide superparamagnetic
nanoparticles and the further entrapment of biological macromolecules in the porous of
the nanoparticles. Recently, Son et al. (2005) synthesized silica nanotubes, prepared in the
pores of a template film with a layer of magnetite nanoparticles on the inner surface of the
nanotube using porous alumina film as template (60 and 200 nm pore diameters). To form
the layer of magnetite nanoparticles, the silica nanotube alumina film was dip-coated in a
solution containing Fe(II) and Fe(III) chlorides and then immersed in ammonium hydroxide.
Aerosol routes have been used to prepare submicronic silica coated maghemite hollow
spheres and submicronic silica coated maghemite/iron dense spheres both with a high loading of magnetic material (Tartaj et al., 2001, 2002b, 2004c). Silica coated γ -Fe2 O3 hollow
spherical particles with an average size of 150 nm (fig. 4.4(a) and (b)) were prepared by the
aerosol pyrolysis of methanol solutions containing iron ammonium citrate and tetraethyl
orthosilicate at a total salt concentration of 0.25 M. On the other hand, the aerosol pyrolysis of iron nitrate and tetraethyl orthosilicate at a total salt concentration of 1 M produced
silica coated γ -Fe2 O3 dense spherical particles with an average size of 250 nm.
An illustration of the possible formation mechanism of the silica coated magnetic hollow
spheres is shown in fig. 4.4c. During the first stage the rapid evaporation of the methanol
solvent favours the surface precipitation (i.e. formation of hollow spheres) of components.
The low solubility of the iron ammonium citrate in methanol when compared with that of
tetraethyl orthosilicate promotes the initial precipitation of the iron salt solid shell. During
the second stage the probable continuous shrinkage of this iron salt solid shell facilitates
the enrichment at the surface of the silicon oxide precursor (tetraethyl orthosilicate). In the
third stage, the thermal decomposition of precursors produces the silica coated γ -Fe2 O3
hollow spheres. The formation of the γ -Fe2 O3 is associated with the presence of carbonaceous (reducing) species coming from the decomposition of the methanol solvent and from
the iron ammonium citrate and tetraethyl orthosilicate. The increase in salt concentration to
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Fig. 4.4. (a) TEM picture of hollow silica/iron oxide composites prepared by aerosol pyrolysis of a mixture of
iron ammonium citrate and tetraethoxysilane. (b) Details of a hollow spherical particle showing an outer particle
layer mainly constituted by SiO2 . (c) Illustration of the formation mechanism of the silica coated γ -Fe2 O3 hollow
particles. Reprinted with permission from Tartaj et al. (2001).
© 2001 Wiley-VCH Verlag.
a value of 1 M favours the formation of dense spherical particles. To obtain α-Fe nanoparticles encapsulated in silica the dense spheres containing maghemite are thermally treated
in a reducing atmosphere (fig. 4.5).
The encapsulation of magnetic nanoparticles into silica have been efficiently achieved in
a single and continuous process by laser-induced pyrolysis of ferrocene and TEOS aerosols
as shown by Bomati-Miguel et al. (2005b). In this system TEOS acts as radiation absorber.
The process resulted in homogeneous iron/magnetite particles smaller than 10 nm encapsulated by a silica coating of about 20 nm.
Gold is another inorganic coating highly adequate to implement functionality to the
magnetic nanoparticles as well that to improve their stability in aqueous dispersions. In
fact, the functionalization of gold surfaces is widely reported in the bibliography (Katz
and Willner, 2004). Some protocols exist in the literature to obtain magnetic nanoparticles coated with gold. Water-soluble Au-coated magnetite nanoparticles with diameters of
about 60 nm were synthesized by the reduction of Au(III) onto the surface via iterative
hydroxylamine seeding (Lyon et al., 2004). The Au-coated particles exhibit a surface plasmon resonance peak that blue-shifts from 570 to 525 nm with increasing Au deposition.
Carpenter’s group has prepared metallic iron particles coated by a thin layer of gold by
microemulsions (Carpenter, 2001; Carpenter et al., 2003). The reverse micelle reaction is
SYNTHESIS, PROPERTIES, APPLICATIONS OF MAGNETIC NANOPARTICLES
445
Fig. 4.5. α-Fe nanomagnets encapsulated in silica colloids and prepared by spray pyrolysis. Reprinted with permission from Tartaj et al. (2004b).
© 2004 American Physical Society.
carried out using cetyltrimethylammonium bromide (CTAB) as the surfactant, octane as
the oil phase, n-butanol as co-surfactant and aqueous reactants as the water phase. The
metal particles are formed inside the reverse micelle by the reduction of a metal salt using
sodium borohydride. The sequential synthesis offered by reverse micelles is utilized to first
prepare an iron core by the reduction of ferrous sulfate by sodium borohydride. After the
reaction has been allowed to go to completion, the micelles within the reaction mixture are
expanded to accommodate the shell using a larger micelle containing additional sodium
borohydride.
Activated carbon is another inorganic material that is routinely used to encapsulate
magnetic nanoparticles (normally metallic iron). The applicability of these composites in
biotechnological applications relies mainly on the high porosity of the resulting composite, which allows the adsorption of a variety of biocompounds. Moreover, the resulting
composites have low toxicity (Kuznetsov et al., 1999). For example, Rudge et al. (2000)
studied the fabrication of magnetic drug delivery materials made up of active carbon-iron
composites (250 m2 g−1 ), which had been obtained by means of a high-energy milling
technique.
Apart from biotechnology, the entrapment of magnetic nanoparticles into solid matrices is nowadays attracting much scientific and technological interest in different fields.
For instance, transparent magnets are among other magnetic composites, a very interesting
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challenge because of their novel potential applications in magneto-optical switches, modulators, optical circulators, laser isolators, magnetic field and electric current sensors based
on the Faraday effect. A narrow size distribution of the magnetic particles and its homogeneous distribution over the matrix assure the achievement of composites with magnetic
and optical homogeneity.
The sol-gel method has shown to be very useful for the preparation of these magnetic
nanocomposites. By this method, γ -Fe2 O3 nanoparticles in the range of sizes between 4
and 15 nm, can be embedded in an inert, inorganic, transparent and temperature resistant
silica matrix. First, a gel is formed by heating an ethanol solution containing tetraethylortosilicate (TEOS) and iron nitrate at 45◦ C for 7 days. Then, γ -Fe2 O3 nanoparticles are
obtained after a direct heat treatment of the gels at a temperature of 400◦ C. The porous
nature of the matrix formed by sol-gel provides the sites for nucleation of the iron oxide particles, minimises their aggregation and imposes an upper limit on their size. The
presence of organic species inside the gel pore has been determined as being essential to
obtain the magnetic iron oxide phase which mechanism of formation takes place through
a reduction–oxidation process. On the other hand, particle size depends on the nature and
concentration of the iron salt and the thermal treatment of the gel (Del Monte et al., 1997;
Guerrero et al., 1997; Rosa et al., 2005).
Transparent and magnetic iron oxide/Vycor-glass composites have also been prepared
by impregnation of slices of porous Vycor-glass rods with an iron nitrate solution followed
by a thermal treatment and a reduction process (Zayat et al., 2003). The magnetic behaviour
of the sample accounts for the presence of maghemite nanoparticles of 10 nm with a narrow
particle size distribution dispersed in the glass matrix. One of the main advantages of using
Vycor-glass is its rigid porous structure, which limits the size of the particles avoiding their
aggregation during the thermal treatment. In addition, the small pores, their narrow size
distribution, and their homogeneous dispersion over the glass matrix leads to very uniform
composites at the nanoscopic and macroscopic levels.
4.3. Hybrid hydrophilic coatings
Siloxanes are routinely used to coat magnetic nanoparticles. In most of the cases these
coatings are intended to improve the functionality of the magnetic nanoparticles. Many
protocols exist in the bibliography to obtain magnetic nanoparticles coated with siloxanes
(Whitehead et al., 1985; Bourgeat-Lami, 2002). The most frequently used takes place in
two steps. In the first step, the selected siloxane precursors are condensed to form polymers by addition to a solvent such as methanol plus water in acid conditions. The siloxane
precursors used for polymerization have general formula R–Si(OX)3 , wherein (OX)3 represents a trialkoxy group typically trimethoxy or triethoxy and R represents an aryl, alkyl
or aralkyl group terminating in aminophenyl, amino hydroxyl, sulphydryl or other organic
group suitable for covalent coupling to a bioaffinity adsorbent.
The second step of the coating consists in the union of the condensed species with the
iron oxide magnetic nanoparticles, probably through a covalent bond with surface OH
groups. This association is accomplished by heating the Si-based polymer and the magnetic
iron oxide nanoparticles in presence of a wetting agent miscible in both the organic solvent
and water (glycerol at 160◦ C). During this process all rest of organics and water evaporate
and the presence of glycerol prevents the aggregation (Whitehead et al., 1985). The col-
SYNTHESIS, PROPERTIES, APPLICATIONS OF MAGNETIC NANOPARTICLES
447
loidal dispersions prepared in such a way using aminoethyl aminopropyl trimethoxysilane
are called Ferumoxsil® (Jung and Jacobs, 1995).
4.4. Hydrophobic coatings
This topic is more developed than the one related to hydrophilic coatings because the first
technological applications of ferrofluids involved the use of nonpolar solvents. For example, a detailed review of the protocols employed to disperse magnetic nanoparticles in
nonpolar liquids was already given by Charles and Popplewell in 1980 (Charles and Popplewell, 1980).
The stability of suspensions of solid particles in a liquid medium is ensured by repulsion
between the particles. The interparticle repulsion can be either electrostatic if the polarity
of the solvent is high enough, or steric if a suitable surfactant or polymer is used as a
steric barrier (Russel et al., 1989; Hunter, 1990). For a nonpolar solvent, steric repulsion
is the only way to stabilize the dispersions. The surfactant employed to make hydrophobic
nanoparticles must contain one or more polar groups (functional group) to adhere to the
surface of the particle and another group, for example a long enough hydrocarbon chain
to render the surfactant-coated particle compatible with the nonpolar carrier. The chain or
tail of the surfactant provides a permanent distance between the particles and compatibility
with the liquid carrier. The functional group can be cationic, anionic or nonionic in nature
and is attached to the outer layer of the magnetic particle by either chemical bonding or
physical force or a combination of both. Most of the magnetic fluids employed today have
one to three types of surfactants arranged in one, two or three layers around the particles.
Ferrofluids in which one of the dispersants is a fatty acid such as oleic acid are susceptible
to oxidative degradation resulting in gelation and therefore in an increased viscosity with
time. Surface modifiers such as isobutyltrimethoxysilane has been found to be particularly
useful to allow the use of various surfactants different to fatty acids to keep the particles
dispersed in silicone oil-based, hydrocarbon oil-based or ester oil-based carried liquids
(Borduz et al., 2001).
Stabilization in nonpolar solvents can be attained using three different routes. In the first
route, surfactants or polymers can be introduced during the synthesis, which can be performed directly in the organic solvent (Sun et al., 1999; Yee et al., 1999a). In the second
route, particles can be synthesized in water and then make them hydrophobic by coating
(Lala et al., 2001; Pileni, 2001b; Yao et al., 2001; Zhao et al., 2002). In the third route, hydrophobic nanoparticles are simply obtained by grinding the bulky material in the presence
of surfactant or the particles previously prepared in water (Berkovsky, 1996). Grinding,
a method that is used industrially, is not convenient since the final size distribution of the
nanoparticles cannot be forecast and controlled. On the other hand coating also raises numerous problems. This methodology often involves precipitation of the dispersions in order
to remove the excess surfactant; thus the quantities both of adsorbed and of free surfactant
molecules are not fully