Progress in Solid State Chemistry 39 (2011) 1e50
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Progress in Solid State Chemistry
journal homepage: www.elsevier.com/locate/pssc
Simple rules for the understanding of Heusler compounds
Tanja Graf a, b, Claudia Felser a, *, Stuart S.P. Parkin c
a
Institute for Analytical and Inorganic Chemistry, Johannes Gutenberg-Universtity, 55099 Mainz, Germany
Graduate School Material Science in Mainz, 55099 Mainz, Germany
c
IBM Almaden Research Center, San Jose, CA 95120, USA
b
a b s t r a c t
Heusler compounds are a remarkable class of intermetallic materials with 1:1:1 (often called HalfHeusler) or 2:1:1 composition comprising more than 1500 members. Today, more than a century after
their discovery by Fritz Heusler, they are still a field of active research. New properties and potential
fields of applications emerge constantly; the prediction of topological insulators is the most recent
example. Surprisingly, the properties of many Heusler compounds can easily be predicted by the valence
electron count. Their extremely flexible electronic structure offers a toolbox which allows the realization
of demanded but apparently contradictory functionalities within one ternary compound. Devices based
on multifunctional properties, i.e. the combination of two or more functions such as superconductivity
and topological edge states will revolutionize technological applications. The subgroup of more than 250
semiconductors is of high relevance for the development of novel materials for energy technologies.
Their band gaps can readily be tuned from zero to z4 eV by changing the chemical composition. Thus,
great interest has been attracted in the fields of thermoelectrics and solar cell research. The wide range of
their multifunctional properties is also reflected in extraordinary magneto-optical, magnetoelectronic,
and magnetocaloric properties. The most prominent example is the combination of magnetism and
exceptional transport properties in spintronic devices. To take advantage of the extremely high potential
of Heusler compounds simple rules for the understanding of the structure, the electronic structure and
the relation to the properties are reviewed.
Ó 2011 Elsevier Ltd. All rights reserved.
Contents
1.
2.
3.
4.
5.
6.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Nomenclature of Heusler compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.
Half-Heusler compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2.
Heusler compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.1.
NowotnyeJuza phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.2.
Half-Heusler compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.3.
Heusler compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Structural properties and orderedisorder phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5.1.
Half-Heusler compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5.1.1.
Structure determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5.2.
Heusler compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.2.1.
Structure determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Magnetism and Heusler compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
6.1.
Half-metallic ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
6.1.1.
The SlaterePauling rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
* Corresponding author.
E-mail address: felser@uni-mainz.de (C. Felser).
0079-6786/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.progsolidstchem.2011.02.001
2
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
6.2.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
Properties of half-metallic ferromagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6.2.1.
Half-Heusler compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6.2.2.
Heusler compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.2.3.
Relation of disorder and spin polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.3.
Compensated ferrimagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.3.1.
Half-Heusler compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.3.2.
Heusler compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Magneto-optical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Heusler compounds in devices for spintronic applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
8.1.
The tunneling magnetoresitance effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
8.2.
Current-perpendicular-to-plane giant magnetoresitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
8.3.
Perpendicular magnetic anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
8.4.
Spin injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Shape-memory materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Thermoelectric materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Kondo systems and heavy-Fermion behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Topological insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Heusler goes nano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Heusler compounds in industrial applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
16.1.
Heusler compounds for spintronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
16.2.
Heusler compounds for thermoelectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Heusler compounds and related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
17.1.
Hexagonal analogues of Heusler compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
17.2.
REME phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
17.3.
Tetragonally distorted Heusler compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
17.4.
Related layered structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
17.5.
Relationship between Heusler compounds and perovskites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
1. Introduction
The history of one of the most exciting material classes can be
traced back to the year 1903 when Fritz Heusler discovered that an
alloy with the composition Cu2MnAl behaves like a ferromagnet,
although non of its constituent elements is magnetic by itself [1,2].
This remarkable material and its relatives, which by now comprise
a vast collection of more than 1000 compounds, are now known as
Heusler compounds. They are ternary semiconducting or metallic
materials with a 1:1:1 (also known as “Half-Heusler”) or a 2:1:1
stoichiometry. Fig. 1 shows an overview of possible combinations of
elements forming these materials.
Surprisingly, the properties of many Heusler compounds can
be predicted by simply counting the number of valence electrons
[3]. For example, non-magnetic Heusler compounds with
approximately 27 valence electrons are superconducting. Semiconductors display another major sub-class with more than 250
representatives and are considered to be novel materials for
energy technologies. Their band gaps can easily be tuned from 0 to
z4 eV by changing their chemical composition. Thus, they
attracted remarkable attention as potential candidates for both,
solar cell and thermoelectric applications. In fact, excellent thermoelectric properties have recently been demonstrated for
TiNiSn-based materials [4]. On the basis of their calculated electronic band structures a new class of Heusler compounds was
predicted only lately: multifunctional topological insulators, i.e.
a new state of matter, in which spin-polarized edge and surface
states are topologically protected against impurity scattering
[5,6]. The introduction of multifunctionality, i.e. the combination
of two or more functionalities, such as superconductivity and
topological edge states in one material, is easily possible in ternary
compounds [5]. The large class of magnetic X2YZ compounds
shows all kinds of magnetic behavior and multifunctional
magnetic properties, such as magneto-optical [7], magnetocaloric
[8] and magneto-structural characteristics [9]. The large family of
magneto-electrical Heusler compounds, the half-metallic ferromagnets are semiconducting for electrons of one spin orientation,
whereas they are metallic for electrons with the opposite spin
orientation. Such compounds exhibit nearly fully spin polarized
conduction electrons, making them suitable materials for spintronic applications. Half-metallic Co2-based Heusler compounds
continuously attract interest due to their high Curie temperatures
[10] and, in fact, are being used today in magnetic tunnel junctions
3
[11]. In this review article, we anticipate to give a detailed
description of all rules of thumb known about Heusler compounds
to provide an insight into this exceptional class of materials. Both,
the structure-to-property relations as well as the outstanding
properties of Heusler compounds are reviewed in context of
various possible applications.
This review article is organized as follows. First of all, we present
general statements for Heusler compounds and explain both, the
nomenclature and the crystal structure (Sections 2 and 3). After this
basic introductory part, we address their physical properties
starting with semiconductors (Section 4). In the following, their
structural properties with focus on disorder phenomena (Section
5), magnetic materials, their magneto-optical properties and their
applications in spintronics (Sections 6e8), shape memory alloys
(Section 9), superconductors and thermoelectrics (Sections 10 and
11), as well as Kondo systems and materials with heavy-Fermion
behavior (Section 12) are discussed. Finally, the recently discovered
topological insulators are reviewed (Section 13). After that, Section
14 gives an overview of some basic points concerning the synthesis
of Heusler materials. Additionally, the properties of Heusler nanoparticles are described in Section 15. Section 16 is dedicated to
research done by industrial companies to incorporate these new
materials into their products. Moreover, we discuss the relationship
of Heusler compounds with other material classes (Section 17), in
particular the hexagonal analogues, but also the tetragonally distorted variants, layered structures derived from the Heusler structure, and point out analogies between perovskites, REME phases
and Heusler compounds.
Additionally, short rules are formulated that simplify the
understanding of the properties of the Heusler compounds. Design
criteria to adjust the desired properties are given in “shopping lists”
at the beginning of each section.
2. Nomenclature of Heusler compounds
2.1. Half-Heusler compounds
In general, Half-Heusler materials XYZ can be understood as
compounds consisting of a covalent and an ionic part. The X and Y
atoms have a distinct cationic character, whereas Z can be seen as
the anionic counterpart. The nomenclature in literature varies
a lot, ranging from sorting the elements alphabetically, according to
their electronegativity or randomly, and thus, all three possible
Fig. 1. Periodic table of the elements. The huge number of Heusler materials can be formed by combination of the different elements according to the color scheme.
4
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
permutations can be found. In this article, we will stick to an order
reflecting the electronegativity. The most electropositive element is
placed at the beginning of the formula. It can be a main group
element, a transition metal or a rare earth element. The most
electronegative element, at the end, is a main group element from
the second half of the periodic table, e.g. Li AlSi, Zr NiSn, Lu AuSn
[12e14]. It has to be noted, that the lattice occupancy cannot be
directly derived from this nomenclature and care has to be taken to
assign the atomic parameters correctly (see Section 3 for details).
Often wrong lattice positions are used in theoretical models leading
to wrong results and predictions.
2.2. Heusler compounds
In the past, Heusler compounds were often understood as
intermetallic alloys, although the description as an intermetallic
compound is more appropriate due to their characteristic atomic
order. Ternary Heusler compounds have the general formula X2YZ,
where X and Y are transition metals and Z is a main group element.
However, in some cases Y is replaced be a rare earth element or an
alkaline earth metal. Traditionally, the metal, which exists twice, is
put at the beginning of the formula, whereas the main group
element is placed at the end, e.g. Co2MnSi, Fe2VAl [15,16]. Exceptions are those compounds, in which one element can definitively
be defined to be most electropositive, for instance LiCu2Sb and
YPd2Sb [17]. Here, the electropositive element is put at the beginning in agreement with the IUPAC nomenclature.
3. Crystal structure
There are two distinct families of Heusler compounds: one with
the composition 1:1:1 and the other one with 2:1:1 stoichiometry.
The compounds of the first family have the general formula
XYZ and crystallize in a non-centrosymmetric cubic structure
(space group no. 216, F43m, C1b) which is a ternary ordered
variant of the CaF2 structure and can be derived from the tetrahedral ZnS-type structure by filling the octahedral lattice sites
(Fig. 2). A characteristic feature of this Half-Heusler structure type
are three interpenetrating fcc sublattices, each of which are
occupied by the X, Y and Z atoms [18]. The corresponding occupied Wyckoff positions are 4a (0, 0, 0), 4b (1/2, 1/2, 1/2), and 4c
(1/4, 1/4, 1/4). In principle, three inequivalent atomic arrangements are possible within this structure type as summarized in
Table 1.
Generally, the Half-Heusler structure can be viewed as a ZnSsublattice (Wyckoff positions 4a and 4c) in which the octahedral
sites are occupied (4b). This description emphasizes the covalent
bonding interaction between two of the contained elements which
plays a major role for the electronic properties of the material. In
contrast, it is worth to mention that the atoms on position 4a and 4b
built a NaCl-type sublattice, i.e. their interaction has a strong ionic
character. The specific ordering of the atoms depends very much on
the chemical nature of the elements. Generally, atomic ordering
according two type I and II (see Table 1) is frequently observed. In
MgAgAs Ag and anionic As form the covalent ZnS-sublattice, while
the Mg and Ag built the NaCl-type lattice [19]. Consequently, As is
eightfold coordinated by monovalent and divalent cations. Even
though MgAgAs is the assigned prototype of all Half-Heusler
compounds, it has to be clarified that this material actually crystallizes with a different atomic order than most other Half-Heusler
compounds [20]. In this case a peculiar situation is present: The
assigned prototype itself is an exception! MgCuSb is an example
which represents the atomic arrangement in most Half-Heusler
materials correctly [19,21]; here, the Cu and the anionic Sb form the
ZnS-sublattice, and the electropositive Mg and the electronegative
Sb occupy the ionic NaCl-type sublattice. Thus, Cu is coordinated by
four Mg and four Sb atoms in form of an ideal cube.
Which of these two described atomic arrangements is preferred
depends on the one hand on the size difference between the
involved atoms, and on the other hand on the kind of interatomic
interaction. If the size difference of the involved cations is rather
small (Mg, Ag), the anion has eight cations in this coordination
sphere and every cation is surrounded by four anions. From metalorganic chemistry it is well known that some metals exhibit
Fig. 2. (a) Rock salt structure, (b) zinc blende structure and their relations to the Half-Heusler structure (c), and to the Heusler structure (d).
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
Table 1
Inequivalent site occupancies within the C1b-type structure. Atoms on Wyckoff
positions 4a and 4c form a ZnS-type sublattice; the atoms on 4b occupy the octahedral holes.
I
II
III
4a
4b
4c
X
Z
Y
Y
X
Z
Z
Y
X
a strong tendency to form covalent bonds, for instance Mg, Ag or Li.
This property supports the formation of the covalent ZnS-type
lattice if such elements are contained in the compounds. Examples
are LiAlSi, LiMgSb and the above discussed MgAgAs. However, if the
cations show distinct differences in size and metal-metal interaction is dominant, as it is the case in MgCuSb, the anion (Sb) is
coordinated by four cations (Cu), Cu for his part by four anions and
four cations, and Mg by four cations. Further examples for this kind
of order are all Half-Heusler compounds containing two transition
metals and the RE YZ materials. In literature, both variants are
labeled with the same prototype, LiAlSi-type [20]. Most HalfHeusler compounds containing two transition metals, however, are
designated with MgAgAs-type structure [20], which is actually
wrong. We would like to emphasize that the correct assignment of
the lattice positions is essential to understand the structure-toproperty relations of these materials and special care has to be
taken when performing theoretical studies to obtain correct results.
The Heusler compounds X2YZ crystallize in the cubic space
group Fm3m (space group no. 225) with Cu2MnAl (L21) as prototype [1,2,22,23]. The X atoms occupy the Wyckoff position 8c (1/4,
1/4, 1/4), the Y and the Z atoms are located at 4a (0, 0, 0) and 4b (1/2,
1/2, 1/2), respectively. Similar to Half-Heusler materials, this
structure consists of four interpenetrating fcc sublattices, two of
which are equally occupied by X. A rock salt-type lattice is formed
by the least and most electropositive element (Y and Z). Due to the
ionic character of their interaction, these elements are coordinated
octahedrally. On the other hand, all tetrahedral holes are filled by X.
This structure can also be understood as a zinc blende-type sublattice, build up by one X and Z, the second X occupies the
remaining tetrahedral holes, whereas Y is located in the octahedral
holes. These relations are illustrated in Fig. 2. In the literature,
Heusler compounds are often described by a CsCl-like superstructure. This is reasonable under the assumption of disorder on the Y
and Z sites and if the unit cell edges of the Heusler cell is shifted by
(1/4, 1/4, 1/4) with respect to the Fm3m cell. The combination of
both X-site fcc lattices leads to a simple cubic lattice. The Y and the Z
atoms occupy the centers of the simple cubic lattice, which results
in the CsCl-like superstructure. This kind of disorder between the Y
and Z site is often observed in half-metallic Heusler systems but
fortunately does not affect the properties significantly. The shifted
Heusler cell, as well as the CsCl-structure, is displayed in Fig. 3. This
description provides an intuitive understanding for one design
5
rule: The combination of two binary alloys crystallizing in the CsCltype structure leads to the formation of Heusler compounds [24].
In addition to the structure described above, an inverse Heusler
structure is observed, if the atomic number of Y is higher than the
one of X from the same period (Z(Y)>Z(X)), however, it may also
appear in compounds with transition metals from different periods
[25]. In all cases, the element X is more electropositive than Y.
Consequently, X and Z form a rock salt lattice to achieve an octahedral coordination for X. The remaining X atoms and Y atoms fill
the tetrahedral holes with fourfold symmetry. The structure is still
described by four interpenetrating fcc sublattices, however the X
atoms do not form a simple cubic lattice. Instead, they are placed on
the Wyckoff positions 4a (0, 0, 0) and 4d (3/4, 3/4, 3/4), while the Y
and the Z atoms are located at 4b (1/2, 1/2, 1/2) and 4c (1/4, 1/4, 1/4),
respectively. The prototype of this structure is CuHg2Ti with space
group F43m (Space group no. 216). It is also possible to emphasize
the difference to normal Heusler compounds by expressing the
formula as (XY) X0 Z. This inverse Heusler structure is frequently
observed for Mn2-based materials with Z(Y)>Z(Mn) as illustrated in
Fig. 5. A well-studied example is the compound Mn2CoSn or
(MnCo)MnSn [26,27]. In case of quaternary Heusler compounds
there are two different elements X and X0 . They are located at the 4a
and 4d positions, respectively, Y is placed on 4b and Z on 4c. This
structure has the prototype LiMgPdSn. An illustration of the inverse
Heusler structure and the quaternary variant is given in Fig. 4.
4. Semiconductors
Ternary semiconductors with 1:1:1 stoichiometry are closely
related to silicon and binary semiconductors such as GaAs. Starting
from the binary lattice, the ternary materials can be derived by
addition of atoms into vacant lattice sites. Therefore, these
compounds are termed “filled tetrahedral structures”. Within this
class of materials, several subgroups have to be differentiated: The
NowotnyeJuza phases AIBIICV with AI ¼ Li, Cu, Ag, BII ¼ Be, Mg, Zn, Cd,
and CV ¼ N, P, As Sb, Bi are well known wide band gap semiconductors
[28e31]. They were first reported by Juza and Hund in the 1940s
[28,29] and subjects of extensive theoretical examination in the 1980s
[32e35]. The nameless AIBIIICIV (for instance LiAlSi [36] and LiGaSi
[37]) and the AIIBIICIV phases (e.g. Mg2Si [38]) also belong to the group
of filled tetrahedral structures. If transition metals are contained in
the materials, they are referred to as Half-Heusler compounds.
The comparison of different filled tetrahedral structures yields
distinct differences in the charge density distribution as demonstrated in Fig. 6. The parent material Si is a covalent material with
directed bonds along the connection lines. In LiAlSi the major part
of the charge density is located at the silicon, however, a directed
bonding within the [AlSi] sublattice is still observed. This covalent
interaction gets weaker when going to LiMgN where the charge
density is accumulated at the nitrogen atoms. TiCoSb displays an
intermediate case, where a combination of undirected (ionic
interaction) and directed bonds (covalent interaction) plays a role.
These considerations show that the difference in electronegativity
is an important factor determining the bonding nature of materials
which, at first sight, seem to be very similar.
In the following sections we will discuss bonding models for
Nowonty-Juza, Half-Heusler, and Heulser phases in more detail and
review the exceptional properties of these semiconducting
materials.
4.1. NowotnyeJuza phases
Fig. 3. (a) CsCl structure and (b) the Heusler structure which is shifted by (1/4, 1/4, 1/4)
with respect to the standard cell to make the CsCl superstructure visible.
The properties of the Nowonty-Juza phases are strongly determined by their crystalline order and the resulting electronic
structure.
6
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
Fig. 4. (a) The inverse Heusler structure CuHg2Ti and (b) the quaternary version LiMgPdSn.
The chemical bonding in these compounds is illustrated in Fig. 7
using LiAlSi as an example. As described in Section 3, Al and Si form
a zinc blende-type sublattice. The ionic NaCl-type sublattice is
formed by Li and Al in analogy to MgAgAs. Due to the covalent
nature of the tetrahedral sublattice, the chemical bonding between
the corresponding atomic orbitals can be described by a molecule
orbital approach. The sp3 hybridized atomic orbitals of Al and Si
form a set of four degenerate bonding and antibonding orbitals,
which are separated by a distinct energy gap. On the other hand,
Lithium transfers its 2s electron to these orbitals, and as a result, the
empty 2s orbital of Li is located above the antibonding states of
[AlSi]. Based on this scheme, a simple electron counting rule can
be derived for these compounds: NowotnyeJuza phases with eight
valence electrons are semiconductors.
In 1985, Wood et al. predicted LiZnP to be a direct band gap
semiconductor, although its binary analog GaP (isoelectronic to
[ZnP]) is a strongly indirect-gap material [33]. The change from an
indirect to a direct-gap material is associated with a distortion of the
electronic structure caused by the insertion of Liþ at the interstitial
lattice sites. Theoretical studies by Carlsson et al. revealed that this
prediction can be expended to a general “interstition insertion rule”
for filled tetrahedral structures stating that the “degree of directness” of the band gap increases by placing closed shell ions at the
tetrahedral interstitial site [32]. Experimental examples are band
gaps of 1.25 eV for LiZnAs, 1.3 eV for LiCdP, and 2.1 eV and 2.43 eV for
LiZnP and LiMgP, respectively [39,40]. Electronic structure calculations for LiMgN, however, show an indirect band gap of 2.46 eV,
indicating that the situation is sometimes more complex [41].
The size of the band gap can be related to the difference in the
Pauling electronegativity between the Y and the Z atom for a given
element X. This relation was formulated by Van Vechten for binary
compounds [42] and is applicable to the ternary compounds discussed here, since they contain the binary [YZ]n sublattice, which is
partially filled with electropositive cations. The size of the band gap
increases with the electropositive character of X [43]. This result again
agrees with the picture of zinc blende-like sublattice where the
covalent bond of Y and Z is stabilized by the electron donated by X.
From an application point of view, the NowotnyeJuza phases are
promising candidates for opto-electronics, ranging from blue lasers
Fig. 5. Mn2-based Heusler compounds form both, the inverse and the regular structure, depending on the element on the Y position.
to Cd-free solar cell materials (substituting CdS, CdSe, and CdTe) and
buffer layer materials for chalcopyrite-based thin film solar cell
devices [43,44]. Particularly, the electronic structures of LiMgN and
LiZnN were proposed to fill the green gap left open by existing
InGaN-based emission devices [45]. The small lattice mismatch
between these materials combined with band gaps spanning the
visible range makes them good candidates. Additionally, the “lower”
band gap materials were investigated with regard to an application
in thermoelectrics having an intrinsicly p-type character. The
introduction of n-type carriers in LiZnSb by exchanging small
amounts of Zn or Sb for Ga or Te, respectively, should lead to an
optimized carrier concentration and a high thermoelectric figure of
merit ZT [46]. LiAlSi and LiAlGe are also promising, since they exhibit
a high Seebeck effect and a low thermal conductivity which is
attributed to mass fluctuation scattering or rattling of the Li ions [47].
4.2. Half-Heusler compounds
Besides the ternary relatives of classical binary semiconductors,
Heusler compounds incorporate also an impressive group of
unconventional semiconductors being comprised of metals and
Fig. 6. Charge density distribution of Si, and related filled tetrahedral structures.
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
7
Fig. 7. Schematic illustration of the hybridization of semiconducting NowotnyeJuza phases using the example of LiAlSi. The hybrid orbitals are formed by the covalent [AlSi]
sublattice, the empty anti-bonding Li s state is located above these hybrid orbitals.
containing at least one transition metal. Up to now, the properties
of these exceptional materials are nearly unexplored.
Among Half-Heusler compounds, TiNiSn and TiCoSb belong to the
group of non-magnetic and semiconducting materials, MnNiSb,
however, is a half-metallic ferromagnet [48]. Investigations of the
electronic structure of Half-Heusler compounds were carried out, to
gain an understanding of their physical properties. In fact, Pierre et al.
were among the first to recognize the importance of the valence
electron count in these compounds [49]. Jung et al. applied the
extended Hückel tight-binding method to study the non-spinpolarized electronic structure and described the bonding interaction
based on ionic arguments [50]. Since the X element is the most
electropositive element in XYZ, the authors formulate a model in
which X transfers its valence electrons to the more electronegative
elements Y and Z. In this simplified model they become stable closed
shell ions, i.e. a d10 configuration for Y and a s2p6 configuration for Z.
This procedure requires 18 valence electrons and empties formally
the valence atomic orbitals of X. Consequently, the filled levels of the
d10 and s2p6 ions are stabilized by the empty levels of X in terms of
two-electron two-orbital stabilizing interactions. Due to the closed
shell configuration, Half-Heusler compounds with 18 valence electrons are particularly stable. Changing the valence electron number to
a different value mostly causes the compounds to become magnetic
and crystallize in a different crystal structure [51]. One exception is
displayed by the 22 electron system MnNiSb which shows the HalfHeusler structure and in which the ferromagnetism is attributed to
the strong tendency of the d electrons of Mn 3þ (d4) to localize [50]
(for further details on the magnetic properties see Section 6).
The description of the chemical bonding in Half-Heusler
materials corresponds to a covalent zinc blende sublattice [YZ]n,
filled with positive ions Xnþ. The importance of covalency, which is
one precondition for the existence of the rather open Half-Heusler
üt et al.,
structure, was also stressed by Tobola and Pierre [52]. Ög
however, emphasized that both, the ionic interaction in the XZ
rock salt-like substructure, as well as the symmetry breaking by
filling half of the tetrahedral holes with Y, are crucial factors for
the formation of the band gap [53]. Fig. 8 displays an illustration of
the chemical bonding in the semiconductor TiCoSb using the
molecule orbital approach. The covalent interaction of the zinc
blende sublattice [CoSb]4 is shown in Fig. 8(a). Here, the s and p
states of antimony are fully occupied and hybridize with the
unoccupied 5s and 5p states of cobalt, forming a set of low-energy,
bonding a1 and triple-degenerated t2 orbitals, as well as a set of
high-energy, anti-bonding, and unoccupied a1* and triple-degenerated t2* orbitals. The 3d orbitals of Co exhibit the octahedral
splitting, but they do not form hybrid orbitals with Sb. The
chemical bonding between this [CoSb]4 substructure and the Ti4þ
ion, which has a distinct ionic contribution, is sketched in Fig. 8(b).
The fully occupied 3d orbitals of Co form, together with the empty
Ti 3d orbitals, two sets of double degenerate e and triple degenerate t orbitals, one with a bonding and one with an anti-bonding
character, resulting in a weak covalent interaction. The hybrid
orbitals are well separated by an energy gap. The highest occupied
states have mainly Co character, while the lowest unoccupied
states have a strong Ti contribution. The calculated charge density
distribution supports this picture: a strong covalent bonding
interaction between cobalt and antimony (density between Co and
Sb) and stronger ionic character of the bonding between Ti and the
CoeSb-three-dimensional network (compare Fig. 6). The ionic
interaction between the zinc blende sublattice and the X atom
becomes stronger with increasing electropositive character of X.
Since in compounds with 18 valence electrons (here TiCoSb) only
bonding states are occupied, they are particularly stable, whereas
for 17 or 19 electron compounds the bonding states are not
completely occupied or anti-bonding states need to be populated.
This leads to a weakening of the bonding interaction, and thus, it is
not surprising that only few examples are known which show
a deviation from the 18-valence electron rule. This relationship
between stability and the compliance of valency rules known for
ionic or covalent compounds was first mentioned by Jeitschko in
1970 [54].
The size of the band gap is to a large extent related to the
energy difference between bonding and anti-bonding d states
[52]. This in turn depends on the differences in electronegativity
between the metals, which also determine the bonding strength
between them. Indeed, this relative intuitive scheme has been
verified using LMTO calculations [55]. Investigations on the
chemical bonding in TiCoSb-based on the crystal orbital Hamiltonian population (COHP) reveal that the CoeSb bonding interactions are strongest, however, the TieCo interactions, also play
a role. In this case, the electronic stability is supported by the
complete absence of anti-bonding states below the upper edge of
the valence band.
Considering the physical properties, Half-Heusler compounds
with VE ¼ 18, such as TiCoSb or TiNiSn (see above), are in the ideal
case diamagnetic semiconductors. By changing the valence electron number to VE ¼ 17 or 19, the system turns into a paramagnetic
or ferromagnetic metal, e.g. VCoSb and TiCoSn [56,57]. Consequently, various types of semiconductoremetal transitions
accompanied by a crossover from diamagnetism to paramagnetism
or ferromagnetism are observed [49]. These facts explain why most
Half-Heusler materials are 18 or 18þ localized n 4f valence electron
compounds. Apart from that, only very few compounds are known
for VE ¼ 16 or 20, which can be ascribed to instabilities in the
8
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
Fig. 8. Schematic illustration of the hybridization of semiconducting TiCoSb. (a) The covalent zinc blende sublattice [CoSb]4 is formed from the atomic Co and Sb states, and (b) the
[CoSb]4 hybrid orbitals interact with Ti4þ.
electronic structure as described above. Finally, compounds with
VE ¼ 22 exist only for X ¼ Mn and are stabilized by the strong
localization of the Mn d electrons.
Interestingly, the introduction of rare earth metals into a semiconducting Half-Heusler material does not change the electronic
structure and properties significantly. The reason for that is the fact
that the f states of the rare earth materials are strongly localized and
do not contribute to the density of states at eF. Formally, a rare earth
element adds only three electrons to the total electron number. An
example for semiconducting compounds with rare earth metals is
the family of RE PdBi materials [58], in which Pd contributes ten
valence electrons, the RE metal three and Bi five, respectively, which
again sums up to 18 valence electrons.
The great tunability of semiconducting Half-Heusler and NowotnyeJuza compounds is illustrated in Fig. 9 which displays the size
of the band gap as function of the average spineorbit coupling
expressed by the average nuclear charge over the atoms in the unit
cell [5]. This seems to be a suitable order parameter, which sorts the
materials almost along a straight line.
The flexibility of Half-Heusler semiconductors was already
demonstrated by fully-relativistic first-principle calculations [5]
showing the monotonous scaling of the direct band gap width
(distance between G8 and G6 eigenvalues) with a mean spineorbit
coupling strength (expressed through the average atomic nuclear
charge Z). Fig. 9 which combines the results from Ref. [5] and
presently calculated light Li-based members (using the same
approach [59]), demonstrates that Z is indeed rather suitable
parameter which sorts cubic semiconductors almost along the
straight line. Thus the spineorbit coupling is an important ingredient in their bang gap formation.
In general, Half-Heusler compounds are only stable if the
valence electron count sums up to 18, or if the addition of rare
earth elements leads to 18 þ 4fn valence electrons. Exceptions are
the valence electron number of 22 which is reached for 18 þ 3d4,
i.e. Mn in the formal oxidation state þ3, and very few materials
with different valance electron numbers, e.g. VCoSb [56]. Mn
seems to play an outstanding role in the Half-Heusler family (this
fact will be further discussed in Section 6).
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
9
Fig. 9. Band gaps as a function of their average nuclear charge for various Half-Heusler and NowotnyeJuza phases calculated using the optimized lattice parameters. The solar
energy spectrum is shown to emphasize the great potential of these materials for solar cell applications.
4.3. Heusler compounds
5. Structural properties and orderedisorder phenomena
Similar to the “18-electron-rule” for Heusler compounds with
C1b structure, a “24-electron-rule” was found for the family of
Heusler compounds containing more than one transition metal, i.e.
compounds with 24 valence electrons are semiconducting, e.g.
Fe2VAl [16,60,61]. As in the Half-Heusler alloys, the s and the p
states of the main group element are low in energy and are only
partially occupied [62]. Therefore, a partial uptake of transition
metal d electrons by these orbitals is possible which formally
reduces the number of electrons in the corresponding d states (five
d electrons for Z ¼ Al, Ga, and four for Z ¼ Si, Ge, Sn). The s-states are
separated from the p states by an energy gap whose size is
dependent on the main group element. It was shown that it is very
small for Al-containing compounds; for Sn-containing materials,
however, it is much larger [63,64].
In the following, a model based on a classical molecular orbital
approach is formulated to gain an insight into the electronic
structure from a chemist’s point of view: To explain the interaction of the d states, one first has to describe the hybridization of
the atoms occupying the zinc blende sublattice, as shown in
Fig. 10(a) using the example of Fe2VAl. The s and the p states of Fe
and Al hybridize, forming one set of bonding and one set of antibonding a1 and t2 orbitals resulting in an [FeAl]-substructure. The
Fe 3d states show a splitting between the dx2 y2 , dz2 and the dxy,
dxz, dyz orbitals which is typical for a tetrahedral surrounding.
These states form hybrid orbitals with the 3d states of the second
Fe atom resulting in two sets of e and t2g orbitals, i.e. the dx2 y2
and dz2 orbitals couple and a pair of degenerate bonding and antibonding e orbitals is created (compare Fig. 10 (b)). On the other
hand, the dxy, dxz, and dyz orbitals form sets of triple-degenerated
t2g orbitals. Finally, the atomic orbitals of V are inserted between
these Fe hybrid states and are located close to the Fermi level. The
small band gap between these V statesis responsible for the
semiconducting properties. Please note, that this gap in Heusler
compounds with 24 valence electrons is considerably smaller
than in Half-Heusler compounds. Already small amounts of
atomic disorder change the density of states in the vicinity of eF,
which in turn leads to a loss of the semiconducting properties and
the emergence of magnetism.
The properties of Heusler materials are strongly dependent on
the atomic arrangement of the atoms. Already a partial intermixture can alter the electronic structure distinctly. As described above,
Half-Heusler compounds are tetrahedrally filled structures, which
are closely related to binary semiconductors. Covalent bonding
interaction plays a significant role and their crystalline order is
retained up to the composition temperature [65]. Thus, structural
disorder leading to an occupation of the vacant lattice site occurs
only rarely in Half-Heusler compounds, whereas the X2YZ phases
often display considerable amounts of atomic disorder.
5.1. Half-Heusler compounds
Within the Half-Heusler structure different types of atomic disorder
are possible (compare Table 2). An overview of potential types of
disorder is displayed in Fig. 11, and a detailed description of all possible
atomic arrangements with the Heusler structure can be found in
Ref. [66]. A mixture of the atoms on Wyckoff positions 4a and 4b leads
to a CaF2-type structure (C1, space group Fm3m, no 225). In contrast to
this, the vacant sites can partially become occupied, while at the same
time, vacancies are introduced in the other sublattices. Thus, a partial
occupancy of the 4d sites accompanied by voids on the 4c sites yields
a Cu2MnAl-type structure (L21, space group Fm3m, no 225), and an
additional mixing of the atoms on positions 4a and 4b leads to a CsCltype of disorder (B2, Pm3m, no. 221). On the other hand, if the vacant
lattice site is partially occupied by atoms from the 4b site accompanied
by an intermixing the 4a and 4c positions, a NaTl-type structure is
obtained (B32a, Fd3m, no. 227). Finally, a completely random distribution of all three atoms on the four possible positions gives rise to
a tungsten-type disorder (W, Im3m, no. 229). Table 2 provides
a summary of the different structure types, and different notations
according to the Inorganic Crystal Structure Database (ICSD), the
Strukturberichte (SB), the Pearson database, as well as the space group.
5.1.1. Structure determination
As mentioned in the previous section, the electronic structure, and
therefore, the physical properties of Heusler compounds, are strongly
dependent on the order and distribution of the atoms within the
10
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
Fig. 10. Schematic illustration of the hybridization of semiconducting Fe2VAl. (a) The hybridization of one Fe atom and Al is shown. (b) The formed states interact with the orbitals
of V and second Fe atom.
crystal lattice. For this reason, a careful analysis of the crystal structure
is essential to understand, or even predict the properties of a material.
Band structure calculations show, that the size of the band gap
decreases with increasing amount of atomic disorder, and eventually
closes completely as shown for TiNiSn in Fig. 12.
The easiest experimental method to examine the crystal structure
of a compound is X-ray diffraction (XRD). Well-ordered Half-Heusler
compounds are characterized by the existence of the (111) and the
(200) reflection in the powder XRD pattern, as shown in Fig. 13.
However, for CaF2-type and Cu2MnAl-type disorder, the diffraction
patterns do not differ significantly from each other. The individual
reflection positions are identical, but there is a considerable difference
between the intensity ratios of the (111) and (200) reflections
(compare Table 3). On the other hand, for crystals with a CsCl-type
disorder the (111) reflection disappears completely. (Please note, that
the indices of the reflections changes due to the different space
group.). Finally, for a NaTl-type disorder, only the (111) reflection is
present, whereas in the tungsten-type of disorder only the cubic main
reflection remains.
5.2. Heusler compounds
Similar to the Half-Heusler materials, the properties of Heusler
compounds are strongly dependent on the atomic order. Band
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
11
Table 2
Site occupancy and general formula for differently ordered Half-Heusler compounds. The notations according to the Inorganic Crystal Structure Database (ICSD), the Strukturberichte (SB), the Pearson database, as well the space group are given. Wyckoff position 4d (3/4, 3/4, 3/4) denotes the second tetrahedral lattice site, which is void in ordered
materials.
Site occupancy
General formula
Structure type ICSD
SB
Pearson
Space group
4a, 4b, 4c
4a ¼ 4b, 4c
4a, 4b, 4c ¼ 4d,
4a ¼ 4b, 4c ¼ 4d
4a ¼ 4c, 4b ¼ 4d
4a ¼ 4b ¼ 4c ¼ 4d
XYZ
XZ2
X2YZ
XZ
YZ
X
LiAlSi (MgAgAs)a
CaF2
Cu2MnAl
CsCl
NaTl
W
C1b
C1
L21
B2
B32a
A2
cF16
cF12
cF16
cP2
cF16
cI2
F43m(No. 216)
Fm3m(No. 225)
Fm3m(No. 225)
Pm3m(No. 221)
Fd3m(No. 227)
Im3m(No. 229)
a
Please note that LiAlSi and MgAgAs represent only one variant of atomic arrangement on the three lattice position. In addition, a second variant is common which is
denoted also as MgAgAs. This is actually wrong as discussed in the text.
structure calculations show, that already small amounts of disorder
within the distribution of the atoms on the lattice sites cause
distinct changes in their electronic structure, and thus also in their
magnetic and transport properties [67e69]. Therefore, a careful
analysis of their crystal structure is essential to understand the
structure-to-property relation of Heusler compounds.
Fig. 14 shows the transition from the ordered to the most
prominent disordered Heusler structures, which will be explained
in the following [18,66,70e72]: If the Y and the Z atoms are evenly
distributed, the 4a and 4b positions become equivalent. This leads
to a CsCl-like structure, also known as B2-type disorder. As
a consequence, the symmetry is reduced and the resulting space
group is Pm3m. On the other hand, the random distribution of X and
Fig. 11. Overview of the most prominent types of disorder occurring in the HalfHeusler structure: (a) CaF2-type disorder, (b) NaTl-type disorder, (c) Cu2MnAl-type
disorder, (d) CsCl-type disorder, and (e) tungsten-type disorder. Please note, that only
for case (a) one vacant lattice site is retained, whereas in cases (c)e(e) all four lattices
are partially occupied.
Y or X and Z leades to a BiF3-type disorder (Space group no. 216:
Fm3m, DO3). Different from these types of disorder, the NaTl-type
structure is observed very rarely. In this structure type the X atoms,
which occupy one of the fcc sublattices, are mixed with the Y atoms,
whereas the X atoms on the second sublattice are mixed with the Z
atoms. This kind of disorder is also known as B32a disorder (Space
group no. 227, Fd3m). Here, the X atoms are placed at the Wyckoff
position 8a (0, 0, 0), while Y and Z are randomly distributed at
position 8b (1/2, 1/2, 1/2). In contrast to these partial disorder
phenomena all positions become equivalent in the tungsten-type
structure with a bcc lattice and reduced symmetry (Im3m(A2)).
Table 4 summarizes the different ordering variants of Heusler
compounds. The site occupancy is correlated with the corresponding general formula. Different notations of the crystal structures according to the Inorganic Crystal Structure Database (ICSD),
the Strukturberichte (SB), and the Pearson database, as well as the
space group are given.
Fig. 15 shows the groupesubgroup relationship between all
possible types of simple disorder in the Heusler structure. For
comparison, the Half-Heusler structure is also included in the
diagram. All disorder types, which can be found in the Heusler
structure, may also occur in Half-Heusler compounds, where the
vacancy is statistically distributed on all positions. Only in the CaF2type disorder, the vacant site is preserved. The numbers behind t
(“translationsgleich”) and k (“klassengleich”) specify the index of
the reduction in symmetry. One should note, that there are two
atoms per unit cell for the tungsten-type and the CsCl structure. For
all other structure types there are 16 atoms per unit cell due the
doubling of all cell axes.
Fig. 12. Density of states (DOS) for TiNiSn. The shaded area corresponds to the DOS of
the ordered compound. Already 5% of CaF2-type or Cu2MnAl-type disorder result in
a distinct decrease in gap size (straight lines). For 50% disorder, however, the band gap
is closed completely (dashed lines).
12
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
Fig. 13. Simulated X-ray diffraction patterns for TiNiSn assuming different structure
types.
In addition to the above-described types of disorder, a tetragonal distortion may also occur (compare also Section 17.3).
5.2.1. Structure determination
To understand the properties of Heusler compounds and to
correlate experimental and theoretical results, a detailed study of
the crystal structure of the compounds is essential. In the laboratory, powder XRD is the easiest experimental method to check both
structure and purity of a sample. In case of Heusler compounds this
is often not sufficient to exclude certain types of anti-side disorder:
Theoretical X-ray diffraction patterns of Co2MnSi, which were
simulated under the assumption of different crystal structures, are
displayed in Fig. 16 [72]. The ordered Cu2MnAl-type structure is
identified by the occurrence of the fcc-typical (111) and (200)
reflections, and their relation to the (220) reflection. However, the
intensity of these two fcc-typical reflections is very low in many of
the investigated compounds. This is, in particular, the case, if all
elements contained in the compound have a similar atomic
number. Unfortunately, the intensity of the (111) and the (200)
Table 3
Relative intensities of the reflections for TiNiSn in different structures.
Structure
(111)
(200)
(220)
(311)
(222)
(400)
C1b
CaF2
Cu2MnAl
CsCl
NaTl
Tungsten
11.14
15.44
5.70
e
11.22
0.40
43.23
15.65
43.23
43.23
e
0.42
100
100
100
100
100
100
5.75
8.20
3.11
e
5.67
0.22
6.42
4.74
6.42
6.42
e
0.06
17.09
17.09
17.09
17.09
17.09
17.00
reflections can then be below 1% of the scattered intensity of the
(220) reflection, which almost leads to the disappearence of the
(111) and the (200) reflections. For comparison, Table 5 provides
a survey of the relative intensities of the reflexes for the XRD
patterns displayed in Fig. 16. A difference in the intensities of the
(111) and (200) reflections is evident, when comparing the Cu2MnAl-type and the BiF3-type structure. In contrast, the Cu2MnAl and
the CuHg2Ti structure are hardly distinguishable by X-ray diffraction. Therefore, much care has to be taken in the structural analysis,
as both have general fcc-like symmetry. In the CsCl-type structure
only the (111) reflex vanishes, whereas in the NaTl-type structure
the (200) reflection disappears. In the tungsten-type structure,
however, both, the (111) and (200) reflections disappear. Of course,
most samples are not completely ordered or disordered, small
amounts of disorder in an ordered compound or a low degree of
order in a disordered structure may also occur. Unfortunately, this is
not always detectable by standard XRD methods. In such cases,
anomalous XRD investigations with synchrotron radiation lead to
a better structure determination. This allows for the direct observation of anti-site disorder, e.g. Co occupation on the Mn sublattice
could be quantified directly in the case of Co2MnGe thin films [73].
The quaternary alloy Co2FeAl(1x)Six is one example, in which an
increasing order is observed with the substitution of Si for Al [74].
Co2FeAl is CsCl-type disordered, i.e. Fe and Al atoms are randomly
distributed. With increasing Si concentration, the amount of
Cu2MnAl-type order increases, as indicated by the increasing
intensity of the (111) reflection (compare Fig. 17(a)). Differential
scanning calorimetry measurements revealed, that the phase
transition temperature decreases with increasing Si concentration
as shown in Fig. 17(b). Together with band structure calculations,
these results lead to the conclusion, that a composition of 50% Al
and 50% Si yields a stable and well-ordered half-metallic ferromagnet, and in fact, this material is used today in magnetic
tunneling junctions [75]. A similar situation was reported by
Umetsu et al. for the quarternary material Co2MnAl(1x)Six [76].
These findings indicate a general trend for Heusler compounds
with Z ¼ Al, which tend to show a considerable amount of CsCl-like
disorder.
A very useful method to further investigate the atomic disorder
is spin echo nuclear magnetic resonance (NMR). These measurements are able to probe the direct local environments of the active
atoms. NMR investigations provide a tool to obtain the local environment by measuring the resonance frequencies, and consequently, probing the local hyperfine magnetic fields. Thus, the
nature of the first neighboring shells of the active atoms is revealed
[77e79]. As an example, the quaternary substitution series Co2MnAl(1x)Six, which was theoretically predicted to show half-metallic
ferromagnetism, was studied using NMR techniques by Wurmehl
et al. [80,81]. In these reports the local environment of 55Mn nuclei
was investigated in detail, the obtained 55NMR spectrum of
Co2Mn0.5Fe0.5Si is displayed in Fig. 18. In agreement with the
expected random distribution of Mn and Fe on the 4b position,
several resonance lines are found. Each line can be correlated with
one particular Fe configuration in the third Mn coordination shell.
This important precondition for quaternary alloys was confirmed
by the NMR method, and thus, these compounds may show stable
half-metallicity, including a very high spin polarization. Additionally, in the case of Co2FeSi thin films XRD analysis revealed an
ordered Cu2MnAl-type structure, however 59Co NMR experiments
by Wojcik and coworkers yielded the presence of Fe anti-sites on
the Si sublattice corresponding to CsCl-type disorder [82].
The determination of the residual resistivity ratio RRR (r(300 K)/
r(5 K)) hints directly on the structural order. The introduction of
scattering centers by atomic displacements leads to a reduction of
RRR. Co2MnSi single crystals, for instance, exhibit a RRR of 6.5,
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
13
Fig. 14. Overview of the different types of disorder occurring in the Heusler structure: (a) CsCl-type disorder, (b) BiF3-type disorder, (c) NaTl-type disorder, and (d) tungsten-type
disorder.
while it is reduced to 2.7 for arc-melted polycrystalline samples
with CoeMn disorder [83]. Similarly, the quality of Co2FeSi single
crystals prepared by different synthetic methods can be distinguished by RRR [84].
Additionally, Mössbauer spectroscopy is a useful characterization technique, which measures the hyperfine field at the core of
Mössbauer active atoms (commonly 57Fe). The nature of the
obtained spectra indicates the local environment of the probed
atomic species. As an example, the series Co(2x)Fe(1þx)Si exhibits
almost identical XRD patterns (see Fig. 19(a)), impeding a clear
differentiation between the CuHg2Ti and the Cu2MnAl-type structure. A definite assignment, however, is possible using Mössbauer
spectroscopy [85].
All Mössbauer spectra exhibit two sextets, which means that the
Fe atoms are present in two different magnetic environments.
Fig. 19(b) displays the Mössbauer spectrum of Co1.8Fe1.2Si as well as
the relative intensities of both sextets. It can be seen that the
intensity of the second sextet increases with increasing Fe content,
because the additional Fe atoms occupy the 4d lattice position. This
leads to the conclusion that Co2FeSi and CoFe2Si crystallize in the
Cu2MnAl-type (L21) and the CuHg2Ti-type structure, respectively.
Table 4
Site occupancy and general formula for different atomic order of Heusler
compounds. The notations according to the Inorganic Crystal Structure Database
(ICSD), the Strukturberichte (SB), the Pearson database, as well the space group are
given.
Site
occupancy
General
formula
Structure type
ICSD
SB
Pearson
Space
group
X, X0 , Y, Z
X ¼ X, Y, Z
X, X0 ¼ Y, Z
X ¼ X0 ¼ Y, Z
X ¼ X0 , Y ¼ Z
X ¼ Y, X0 ¼ Z
X ¼ X0 ¼ Y ¼ Z
XX0 YZ
X2YZ
XX20 Z
X3Z
X2Y2
X2X20
X4
LiMgPdSn
Cu2MnAl
CuHg2Ti
BiF3
CsCl
NaTl
W
Y
L21
X
DO3
B2
B32a
A2
cF16
cF16
cF16
cF16
cP2
cF16
cI2
F43m(No. 216)
Fm3m(No. 225)
F43m(No. 216)
Fm3m(No. 225)
Pm3m(No. 221)
Fd3m(No. 227)
Im3m(No. 229)
For the series Co2Mn(1x)Al, the existence of a short range order
was detected, although XRD studies indicated a complete disorder
on the Y and Z positions [86]. To get an insight into the structural
and magnetic properties of thin films, 57Fe conversion electron
Mössbauer spectroscopy (CEMS) is a very powerful method. CEMS
spectra of as-deposited expitaxial Co2 Cr0.6Fe0.4Al (CCFA) thin films
with an Fe buffer layer show distinguishable subspectra originating
from the CCFA and the Fe buffer layer (21(a)) [87]. Fig. 20(b)e(d)
presents the spectra of thin films without Fe buffer after annealing
at 450, 550, and 600 C, respectively. Spectra of samples with
a buffer layer have broader lines and look similar, except for the
presence of an Fe subspectrum, which is reduced by increasing
annealing temperature, and finally disappears after annealing
above 600 C. This result indicates a diffusion of Fe from the buffer
layer into the CCFA thin film, and apparently the related diffusion of
Cr atoms from the CCFA thin film into the Fe buffer layer occurs
simultaneously. The central peak, which is observed in the spectra
(b)e(d), results from Fe in a non-magnetic surrounding, i.e. from Fe
atoms substituting Co atoms in the B2-type structure. In summary,
CEMS studies demonstrate that annealing CCFA thin films not only
monotonously improves their crystallographic order in the
temperature range of 450e700 C, but also causes diffusion of Fe
atoms from the Fe buffer layer into the CCFA, thereby changing its
composition, and favors anti-site CoeFe disorder.
Extended x-ray absorption fine structure (EXAFS) analysis is
a sensitive method to determine short range chemical order around
atoms, which has also been successfully applied to investigate
Heusler materials. For instance a distinct amount of disorder
between the Co and the Mn sites has been revealed in Co2MnSi and
gave an explanation for the rather low measured spin polarization
of 55%, although band structure calculations predicted half-metallic
ferromagnetism for the ordered compound [83]. Another example,
that demonstrates the enormous potential of this characterization
technique for Heusler compounds, is displayed by the Co2FeZ
(Z ¼ Al, Si, Ga, Ge) system. For Z ¼ Al and Si the complete structure
could be revealed by XRD. However, for Z ¼ Ga or Ge XRD studies
showed only a cubic structure without the degree of ordering
14
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
Fig. 15. Bärnighausen tree for Heusler compounds illustrating the groupesubgroup
relations between different ordering variants. The indices of klassengleiche (k) and translationsgleiche (t) as well as the unit cell transformations and the origin shifts are given.
(compare Fig. 21(a)) [88]. EXAFS measurements were performed at
the Co K edges and the Fe K edges. The spectra could be fittet with
a CsCl-like disordered structure for Co2FeAl, but for the remaining
samples the fitting procedure yielded well-ordered structures as
shown in Fig. 21(b) and (c).
6. Magnetism and Heusler compounds
Heusler compounds first attracted interest among the scientific
community in 1903, when F. Heusler found, that the compound
Cu2MnAl becomes ferromagnetic, although non of its constituent
elements is ferromagnetic by itself [1,2]. However, it took three
decades until the crystal structure was determined to be ordered
with a face centered, cubic lattice [22,23]. Unfortunately, they faded
almost in oblivion in the following decades, and only few reports on
the synthesis of new Heusler compounds were published in the
1970s [89,90]. It was not until the prediction of half-metallic
ferromagnetism in MnNiSb by de Groot et al. [48] and in Co2MnSn
by Kübler et al. [91] in 1983, that scientific interest returned to
Heusler materials.
The XYZ materials exhibit one magnetic sublattice since only the
atoms on the octahedral sites can carry a magnetic moment, as
indicated in Fig. 22. In Section 4.2 it was already mentioned that
magnetic XYZ Half-Heusler materials exist only for X ¼ Mn, and RE.
This fact results from the localized nature of the four 3d electrons of
Mn3þ and the 4f electrons, respectively, which carry the magnetic
moment. Experimentally, a small induced magnetic moment is also
found on Nickel and the late transition metals. Indeed, this
circumstance can be neglected from the view point of simple rules.
Among the RE containing Heusler compounds known in literature
most compounds are semiconducting or semimetallic systems are
antiferromagnets with low Néel temperatures [58,92]. Since the
magnetic ions occupy the NaCl-sublattice, their distance is large
which hints at an magnetic interaction based on a super-exchange
Fig. 16. Theoretical XRD patterns for Co2MnSi under the assumption of different
crystal structures.
mechanism. Only very few ferromagnetic Half-Heusler compounds
are described in literature, for instance NdNiSb and VCoSb [56,93].
The Mn containing Half-Heusler compounds are half-metallic
ferromagnets with high Curie temperatures (See Section 6.2.1 for
detailed discussion.).
In the X2YZ Heusler compounds the situation is completely
different because of the two X atoms occupying the tetrahedral
sites which allows a magnetic interaction between the X atoms and
the formation of a second more delocalized magnetic sublattice
(compare Fig. 22). Due to the two different magnetic sublattices,
the X2YZ Heusler compounds can show all kinds of magnetic
phenomena, and in fact, today ferromagnetism, ferrimagnetism,
and half-metallic ferromagnetism are known.
6.1. Half-metallic ferromagnetism
In the eighties, unusual magneto-optical properties of several
Heusler compounds motivated the investigation of their electronic
structure, which lead to an unexpected result: Depending on the
Table 5
Relative intensities of the reflections for Co2MnSi in different structures.
Structure
(111)
(200)
(220)
(311)
(222)
(400)
Cu2MnAl
CsCl
BiF3
Tungsten
CuHg2Ti
NaTl
4.30
e
0.79
e
6.53
4.38
4.67
4.67
0.41
e
2.27
-
100
100
100
100
100
100
2.01
e
0.33
e
3.26
2.26
1.30
1.30
0.09
e
0.57
e
16.46
16.46
16.58
16.58
16.46
16.46
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
a
15
b
a
b
Fig. 17. (a) Powder diffraction of Co2FeAl(1x)Six. Shown are the powder patterns measured with Mo Ka at room temperature for selected compositions with x ¼ 0.3, 0.4, and 0.5. (b)
Phase transitions in Co2FeAl(1x)Six. Shown is the composition dependence of the phase transition temperature. The length of the vertical bars corresponds to the experimental
hysteresis. The insets (a) and (b) display typical DSC curves in low (0.1) and high (0.7) Si content compounds, respectively. Data taken from Ref. [74].
spin direction, certain Heusler materials show metallic as well as
insulating properties at the same time, a feature called half-metallic
ferromagnetism [48,91]. De Groot and coworkers developed
a classification scheme pointing out that three different types of
half-metallic ferromagnetism can be distinguished [94]. Fig. 23
displays a schematic illustration of the density of states (DOS) of
(a) a metal with a finite density of states at the Fermi energy, and
(b) the spin resolved representation of a metal: both spin channels
are identical and equally occupied. Fig. 23(c) shows the DOS of
a ferromagnet, in which the majority and minority states are shifted
against each other, leading to a measurable net magnetization of
the material. A half-metallic ferromagnet (HMF) behaves like
a metal for one spin direction and like an insulator for the other
spin direction (Fig. 23(d)). Formally, the complete spin polarization
of charge carriers in a HMF is only reached in the limiting case of
zero temperature and vanishing spineorbit interactions. Since
most of the Heusler compounds containing only 3d elements do not
show any spineorbit coupling, they are ideal candidates to exhibit
half-metallic ferromagnetism.
6.1.1. The SlaterePauling rule
Slater and Pauling discovered that the magnetic moment m of
the 3d elements and their binary alloys can be estimated on the
basis of the average valence electron number (NV) per atom [95,96].
The materials are divided into two areas depending on m(NV): The
first area of the SlaterePauling curve is the area of low valence
electron concentrations (NV8) and of localized magnetism. Here,
mostly bcc and bcc related structures are found. The second area is
the area of high valence electron concentrations (NV 8) and of
itinerant magnetism. In this area, systems with closed packed
structures (fcc and hcp) are found. Iron is located at the borderline
between localized and itinerant magnetism. Fig. 24(b) shows the
SlaterePauling curve for transition metals and some alloys. Heusler
compounds are situated in the localized part of this curve. Therefore, we focus on this area of the curve. The magnetic moment in
multiples of Bohr magnetons mB is given by
m ¼ NV 2nY
(1)
where 2nY denotes the number of electrons in the minority states.
The minimum in the minority density of states forces the number of
electrons in the d minority band to be approximately three.
Neglecting the s and p electrons, the magnetic moment in the
localized area of the SlaterePauling curve can be calculated
according to
mzNV 6
(2)
which means that magnetic moment per atom is just the average
number of valence electrons minus six. Half-metallic ferromagnets
exhibit per definition a band gap in the minority density of states at
the Fermi level. Due to this band gap, the number of occupied
minority states needs to be an integer, which is exactly fulfilled for
the case m ¼ NV 6 [97,98]. This rule may lead to non-integer
values, if the average valence electron concentration is not integer.
Thus, it is often more convenient to use the valence electron
number per formula unit NV.
For Half-Heusler compounds with three atoms per formula unit,
the SlaterePauling rule is given by
Fig. 18. Spin echo intensity as a function of frequency for 55Mn in Co2Mn0.5Fe0.5Si
(black dots) and the fit (red line) resulting from Gaussians (black lines). The distribution of Mn and Fe atoms in the third coordination shell of the 55Mn, as well as the
relative areas obtained from Gaussians are given for each line. Data taken from
Ref. [80].
mXYZ ¼ NV 18
(3)
In the case of X2YZ Heusler material, there are four atoms per
unit cell leading to the formula
16
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
a
b
Fig. 19. (a) Powder XRD of Co2xFe1þxSi. Shown are the powder pattern measured with Mo Ka at room temperature for selected compositions with x ¼ 0.2, 0.5, and 0.9. (b) 57Fe
Mössbauer spectra of Co2xFe1þxSi. Shown are the relative intensities of the two hyperfine magnetic fields related to the sextets I and II. The inset displays the 57Fe Mössbauer
spectrum of Co1.8Fe1.2Si. Data taken from Ref. [85].
mX2 YZ ¼ NV 24
(4)
The magnetic moment as function of the number valence
electrons per formula unit is shown in Fig. 24.
Fig. 20. 57Fe CEMS spectra of 100 nm CCFA thin films (light blue) deposited on MgO
substrates with a 10 nm Fe buffer layer (yellow) without annealing (a); spectra of
100 nm CCFA thin films (light blue) deposited on MgO substrates without a buffer layer
annealed at (b) 450 C, (c) 550 C, and (d) 600 C. The central peak dark blue corresponds to Fe atoms occupying Co sites. Data taken from Ref. [87]. (For interpretation of
the references to colour in this figure legend, the reader is referred to the web version
of this article).
These relations can be easily understood based on the molecular orbital diagrams for half-metallic Heusler compounds. First,
we discuss the example of MnNiSb (NV ¼ 22), which is illustrated
in Fig. 25 showing the hybridization scheme. The formation of the
[NiSb]3 substructure is very similar to the [CoSb]4 formation
described in Section 4. The coupling of these [NiSb]3 hybrid
orbitals with the Mn3þ atom leads to the formation of two sets of
bonding and anti-bonding orbitals. The bonding orbitals doubly
occupied and are filled with 18 valence electrons. The remaining
four valence electrons are located in the anti-bonding hybrid
orbitals, but now it is energetically favorable to single occupy
these orbitals, giving rise to a magnetic moment of 4 mB. This
model is also confirmed by theoretical calculations which show
that the valence band has Ni character for both spin directions, but
only majority Mn d states are observed, while the conduction band
contains also minority Mn d states. This justifies the single occupancy of the hybrid orbitals close to the Fermi level in Fig. 25.
We would like to emphasize that, besides very few exceptions,
magnetic Half-Heusler compounds are only stable for a valence
electron number of 22 with Mn or a rare earth element on the
octahedral lattice site which can be attributed to the high tendency
towards a localized magnetic moment of Mn as described by Kübler
et al. [91]. This Kübler rule plays an important role in all Heusler
compounds. Its oxidation state can be formally described to be
Mn3þ with a d4 configuration giving rise to a magnetic moment of
approximately 4 mB. The localized magnetic moment of Mn is also
represented in the calculated spin density distribution of MnNiSb
displayed in Fig. 26.
Changing the valence electron number to a different value
mostly results in the formation of a different crystal structure, e.g.
MnCrSb, FeMnSb and Mn2Sb do not crystallize in the Half-Heusler
structure [99,100]. A detailed list of magnetic moments located at
the Mn Y site is given in Ref. [91].
Now, we turn to the X2YZ Heusler compounds and as an
example discuss the case of Co2MnSi (NV ¼ 29) here in detail: The
hybridization scheme (Fig. 27) resembles the one for semiconducting Fe2VAl (NV ¼ 24) (compare Section 4). Co and Si, which
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
a
17
b
c
Fig. 21. (a) XRD of Co2FeZ with Z ¼ Al, Si, Ga, Ge. The XRD pattern has been excited by Mo Ka radiation. Note the different indexing of the reflections in the sc (Z ¼ Al) and fcc (Z ¼ Si)
crystal systems. The line below Z ¼ Ge is the difference between the measured data and the L21 Rietveld refinement for Co2FeGe. (b) EXAFS oscillations extracted from the X-ray
absorption measurements at the Co K edge. (c) Corresponding Fourier transforms (symbols) and best fitting results (gray line). The imaginary part of the Fourier transform is
displayed for the Co2FeGe compound (open circles). Data taken from Ref. [88].
are located on the zinc blende sublattice form two sets of bonding
and anti-bonding t2 and a1 orbitals. The atomic d orbitals of the
[CoSi] substructure and the second Co atom built two sets of t2g
and e hybrid orbitals. Mn, which is located at the octahedral lattice
site, inserts its d states between these hybrid states. These
molecular orbitals are filled up with 29 valence electrons
according to Hund’s rule and Pauli principle. Up to 24 valence
electrons, all orbitals are double occupied, resulting in an electron
configuration identical to semiconducting Fe2VAl. Please note, that
the energy difference between the orbitals above is so small that
a single electron occupancy with parallel spin orientation is
energetically favored, leading an half-metallic state and
a magnetic moment of 5 mB per formula unit. This similarity
between the Fe2VAl and Co2MnSi was recently confirmed by band
structure calculations, revealing that the minority band structure
hardly changes when going from a semiconducting Heusler
compound to a half-metallic ferromagnet [101].
Therefore, the magnetic moment of half-metallic Heusler materials scales linearly with the number of valence electrons according
to m ¼ VE 24 as shown in Fig. 24. Only a few of them with VE 24
are known, e.g. Mn2VAl which is a half-metallic ferromagnet with 22
valence electrons [102]. Substituting Co for half on the Mn atoms on
the X position, results in non-magnetic (Co0.5Mn0.5)2VAl with 24
electrons [103]. This example shows, that the SlaterePauling rule is
also valid for quaternary Heusler compounds.
Additionally, the Curie temperature (TC) of Co2-based Heusler
compounds shows a linear dependency on the magnetic moment
[10]. Due to the SlaterePauling behavior of the magnetic
moment, TC follows a linear trend, when viewed as a function of
valence electrons, as displayed in Fig. 28. The linear trend is
interrupted for materials with VE ¼ 27. Theoretical studies
revealed, that the magnetic moments on the Co and on the Y
sites increase simultaneously with VE which leads to a nonlinearity with m. This is compensated by changes in the Heisenberg exchange average resulting in a linear dependency on VE
[104,105]. It should be noted that the magnetization as a function
of temperature drops very slowly. For Co2MnSi a reduction of
less than 1% is observed when changing the temperature from
5 K to room temperature.
In fact, Co2FeSi is the Heusler compound with the highest
magnetic moment of 5.97 mB at 5 K and and the highest Curie
temperature of 1100 K [10,106].
Fig. 22. (a) XYZ Half-Heusler compounds exhibit only one magnetic sublattice since only the atoms on the octahedral sites carry a localized magnetic moment. (b) X2YZ Heusler
compounds, however, have two magnetic sublattices which can couple ferromagnetically or antiferromagnetically.
18
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
Fig. 23. Schematic illustration of the density of states of (a) a metal, (b) a metal (spin resolved), (c) a ferromagnet, (d) a half-metallic ferromagnet, and (e) a completely compensated
half-metallic ferrimagnet.
6.2. Properties of half-metallic ferromagnets
6.2.1. Half-Heusler compounds
At the beginning of the 1980s, the interest in fast and nonvolatile mass storage memory devices raised tremendous research
activity in the field of magneto-optics. Almost all existing magnetic
solids were studied with regard to the magneto-optic Kerr effect
(MOKE), leading to a maximum MOKE rotation of 1.27 of MnPtSb
[7]. This result motivated the investigation of the electronic structure of the isoelectronic Heusler compounds MnNiSb, MnPdSb, and
MnPtSb, which lead to the prediction of MnNiSb as the first
material being a half-metallic ferromagnet by de Groot and
coworkers in 1983 [48,108]. Indeed, many authors have verified this
prediction in the mean time [109e113]. Several explanations for
both the electronic structure and the band gap have been given, in
terms a NieMn interaction only, but these considerations did not
a
clarify why the octahedral coordination of manganese is essential
for the evolution of half-metallic ferromagnetism [114]. However,
Kübler summarized the chemical bonding, in relation to the band
gap, as a nickel-induced MneSb covalent interaction [97]. According to theoretical calculations, a minority band gap, located within
the larger gap of the NiSb-substructure is formed from the d states
[55]. The spin-polarized states at the Fermi energy strongly exhibit
Mn character.
A 100% spin polarization for bulk MnNiSb was confirmed by
polarized positron-annihilation experiments and inverse photoemission [115e117]. The surfaces, however, do not show halfmetallicity, which can partly be explained by the observation of
manganese segregations on the surface and its high affinity to
oxygen [118e121] but maybe also by the general tendency of
antimony to build surface segregations. But even for non-contaminated surfaces, a genuine half-matallic character was not observed
b
Fig. 24. (a) The magnetic moment per formula unit of Co2-based Heusler compounds is proportional to the number of valence electrons and follows the SlaterePauling curve which
is illustrated in (b). The values for 3d transition metals and their alloys are given for comparison. (Note: the A1xBx alloys are given as AB in the legend for shortness).
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
19
Fig. 25. Schematic illustration of the hybridization of the half-metallic ferromagnet MnNiSb. (a) The covalent zinc blende sublattice [NiSb]3 is formed from the atomic Ni and Sn
states, and (b) the [NiSb]3 hybrid orbitals interact with Mn3þ. Four unpaired electrons are located at eF which correspond to the observed magnetic moment of 4 mB.
[122,123]. This underlines again the sensitivity of half-metallic
properties to the crystal structure.
The transport properties of MnNiSb were studied thoroughly
and electrical resistivity data revealed a phase transition at
approximately 90 K [124e126]. One possible explanation for this
phase transition is the occurrence of thermal excitations, if the
Fermi energy is positioned close to a band edge. A crossing of
a magnon and a phonon branch, at an energy corresponding to
80 K, was also discussed in this context [127,128]. A final understanding of the phase transition, however, is still missing. The local
magnetic moments of Mn and Ni were examined using magnetic
circular dichroism, and revealed that the major portion of the
magnetic moment is located at the Mn site. A reduction of both, the
manganese and nickel moments around 80 K was observed indicating a loss of coupling between Mn and Ni [125]. Additionally, the
disappearance of the Ni moment at the transition temperature was
also found by computational studies [129]. Interestingly, none of
these anomalies was observed in the spontaneous magnetization of
bulk MnNiSb [124].
The compounds MnPtSb and MnPdSb are isoelectronic to
MnNiSb, and therefore, their electronic structures are similar. The
main difference is the higher nuclear charge of Pd and Pt with
respect to Ni. Therefore, relativistic effects leading to an energy shift
of the minority spin electrons have to be taken into account when
calculating the band structure of these compounds [130]. These
effects also provide an understanding for the differences in the
MOKE measurements for MnNiSb and MnPtSb [131]. In the case of
MnPdSb, the Fermi level intersects at the top of the valence band,
but further calculations are needed to clarify, if it is a half-metallic
ferromagnet or not. Maybe correlation effects have to be taken into
account since very small effects alter the spin polarization considerably. On the other side, MnPtSb is half-metallic, but in contrast to
Fig. 26. Charge and spin density distribution of MnNiSb.
20
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
Fig. 27. Molecular orbital diagram of Co2MnSi.
a
b
Fig. 28. (a) Temperature dependent magnetization measurements of selected Co2-based Heusler compounds [107] and (b) the Curie temperature of Co2-based Heusler compounds
is linearly dependent on the number of valence electrons.
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
Ni, Pt does not carry any magnetic moment and no magnetic
anomalies are expected. Angular-resolved photoemission measurements on MnPtSb single crystals yielded good agreement with the
calculated band structure, which is remarkable, since the ground
state in the experiment deviates from the occupations used for the
eigenvalue calculation in density functional theory [132].
If other than isoelectronic elements are substituted in MnNiSb,
the total valence electron number has to be kept constant (NV ¼ 22),
as discussed above. MnAuSn is also a half-metallic ferromagnet but
it is not as intensively investigated as MnNiSb [133e135]. Mn
cannot be replaced by other element in the formal oxidation state
3þ, since this leads to the formation of a crystal structure. In fact, all
Half-Heusler compounds, which are known with X ¼ rare earth
elements, are also known for X ¼ Mn3þ. However, in the rare earth
containing materials, correlation effects have to be considered to
describe their electronic structure correctly [136]. Therefore, the
following question needs to be solved in future: What is so special
about Mn3þ and are these materials strongly correlated?
Theoretical calculations show that MnCoSb is a half-metallic
ferromagnet with a reduced magnetic moment of 3 mB [109].
Experimental results, however, demonstrated that MnCoSb crystallizes in a cubic superstructure with doubled lattice parameter
and Co displacements [137]. This structure can be illustrated by
alternating MnSb and Co2MnSb cells as displayed in Fig. 29. The
magnetic moment is 3.8 mB and consequently, MnCoSb is not a halfmetallic ferromagnet. Unfortunately, pure MnFeSb does not exist,
and only a substitution of up to 10% of Fe for Ni retains the HalfHeusler structure [99]. The compounds MnFeSb, MnMnSb and
MnCrSb do not exist in the Half-Heusler structure, they form
antiferromagnetically ordered materials and are briefly discussed
in Section 6.3.
A different route to induce half-metallic ferromagnetism in HalfHeusler compounds is provided by electron doping of semiconducting TiCoSb resulting in a dilute magnetic semiconductor
[138]. The partial replacement of Ti by Cr or Fe (10%) converts the
semiconductor TiCoSb into a half-metallic ferromagnet. Both,
calculations and experiments indicate that only the atoms replacing Ti contribute to the total magnetic moment which is in good
agreement with the magnetic sublattice located at the octahedral
positions. For the Cr containing material, the experimental
magnetic moment is distinctly smaller than expected from calculations, which can be explained by partial antiferromagnetic
coupling of the Cr atoms. Since the Curie temperature of these
materials is well above room temperature (700 K for Ti0.9Fe0.1CoSb),
Fig. 29. Crystal structure of MnCoSb. Alternating units of MnSb and Co2MnSb form the
unit cell with a doubled lattice parameter compared to the regular Half-Heusler
structure.
21
they are interesting materials for future applications in magnetoelectronics and spintronics.
6.2.2. Heusler compounds
In the very same year as the discovery of half-metallicity in
MnNiSb by de Groot and coworkers, in 1983, ab inito calculations
performed by Kübler et al. revealed that the density of states of
ferromagnetic Co2MnSn and Co2MnAl nearly vanishes for one spin
direction at eF resulting in a high spin polarization. They concluded
that this leads to peculiar transport properties [91]. Indeed, these
results were verified by many authors and extended to a large
group of Co2-based Heusler compounds [69,98,139e143].
One design recipe for new half-metallic ferromagnets with L21
structure developed by Butler is fairly simple [24]: Alloys with B2type structure that are found in the localized part of the SlaterePauling curve can be combined to form a L21 ordered Heusler
compounds. For instance, the combination of binary FeTi and FeAl
results in the half-metallic ferromagnet Fe2TiAl. However, the story
of success of Heusler compounds in spintronics started with
Co2Cr0.6Fe0.4Al (CCFA). The idea behind this material was the
combination of a large band gap in the minority density of states
with a large density of states (van Hove singularity) in the majority
states. The appearance of a van Hove singularity at or close to the
Fermi energy is an important requirement for a stable half-metallic
ferromagnet insensitive to disorder [144]. The same fingerprint is
also observed in many colossal magnetoresistive compounds with
high spin polarization [144,145]. Band structure calculations
revealed that this is fulfilled for Co2-based Heusler compounds
with 27.8 or 28.5 valence electrons and that, in these cases, the
Fermi energy is located in the middle of the minority band gap
which makes half-metallic ferromagnetism stable against temperature fluctuations [140,146]. This non-integer number of valence
electrons can among others be implemented by quaternary alloys
of the type Co2Y1xYx0 Z or Co2YZ1xZx0 .
Due to the rather disappointing results with Heusler compounds
in the early days of GMR multilayers, there were only a few groups
working in this area. Interest in Heusler compounds grew enormously with the discovery of a high magnetoresistance effect in
CCFA together with its success in tunnel magnetoresistance
devices. Band structure calculations assuming ordered compounds
Co2CrAl and CCFA predicted a full spin polarization at the Fermi
energy [139]. In case of CCFA a peak in the density of states indicates a half-metallic ferromagnetic state with a Van Hove singularity in the majority channel near the Fermi energy and a band gap
in the minority channel. Due to this special band structure,
Co2Cr0.6Fe0.4Al is a promising candidate for a high magnetoresistance (MR) ratio. Indeed, in powder pellets a high (MR) ratio of 30%
was observed in a low external magnetic field of 0.1 T at room
temperature [140]. This was the starting point for extensive
investigations to gain an understanding of the extraordinary electronic structure of CCFA.
To confirm the initial assumption band structure calculations
proved the half-metallic ferromagnetism occurring in the Cr-rich
compounds [67,114,147,148]. For example, Zhang et al. found that
Co2CrAl is a half-metallic ferromagnet with 3 mB in agreement with
the Salter-Pauling rule [149]. However, the substitution of Cr for Fe
results in a loss for half-metallic ferromagnetism in Co2Cr(1x)FexAl
for x 0.625 [150]. The replacement of Cr by Fe can be understood
as an electron doping that leads to a shift of the Van Hove singularity relative to the Fermi energy. This circumstance opens the
possibility to tune the electronic properties according to different
demands. However, it has to be noted that such a simple rigid-band
model is often too trivial to describe the results of such a doping
procedure [147,148]. On the other side, band structure calculations
for the disordered compound verified a distinct reduction of the
22
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
magnetic moment and a loss of half-metallic ferromagnetism.
Applying the coherent potential approximation (CPA), Miura et al.
studied the consequences of disorder on the electronic and
magnetic properties of Co2Cr(1x)FexAl [67]. They found that the
complete spin polarization is retained up to 10% of CsCl-type
disorder, while a stronger reduction of the spin polarization occurs
for the other types of disorder. Using single point contacts a spin
polarization of 81% was measured [151]. Galanakis investigated, in
addition to the bulk properties, the influence of surface states on
the half-metallic properties of Heusler compounds and reported
that the CrAl-terminated surface of Co2CrAl behaves differently
compared to most other Heusler alloys, i.e. that the half-metallic
properties are preserved [152].
Experimental data of Co2CrAl reveal a magnetic moment of
1.56 mB per formula unit, as already observed by Buschow and Van
Engen [153]. It was assumed, that the magnetic moment is mainly
carried by the cobalt atoms, while the contribution of Al and Cr is
small. According to recent band structure calculations, all constituents of the compound should carry a magnetic moment: 0.77 mB
per Co atom, 1.63 mB per Cr atom and 0.10 mB per Al atom.
Consequently, a value of 3 mB is then expected for the total magnetic
moment in agreement with the SlaterePauling rule. The element
specific investigation of magnetic moments has turned out to be
a useful tool to discriminate different types of disorder [154].
However, the comparison of experimental data and calculations
reveals that disorder on lattice sites is the biggest drawback for the
production of half-metallic ferromagnets, especially for thin films.
Unfortunately, the preparation of well-ordered Co2Cr(1x)FexAl
films is extremely difficult. The reason for the reduced magnetic
moment and the problem of disorder, especially in CreAl-containing compounds is their sensitivity against oxygen which might
trigger disorder and phase separation effects.
Although the family of X2YZ Heusler compounds is very large,
there are only few compounds exhibiting half-metallic ferromagnetism that are not based on X ¼ Co. The first authors to report on
experiments with Co-based materials were Webster and Ziebeck
[89,155]. Since that time, especially the Co- and Mn-based compounds
evoked great interest due to their high Curie temperatures.
As already stated above, the electronic structure plays an
important role for the determination of magnetic properties and the
prediction of half-metallic ferromagnets. Therefore, band structure
calculations need to be carried out carefully and all approximations
need to be considered when the results are discussed.
Unfortunately, the first efforts to calculate the band structure of
Co2MnSn, Co2TiSn, and Co2TiAl did not yield half-metallic ferromagnetism [156]. Instead, the calculations showed a crossing of the
minority bands at eF although a minimum of the density of states at
the Fermi level was observed. At that time, the calculations were
based on a spherical potential, and the exchange correlation of the
local spin density approximation (LSDA) was used in a rather
simple form [157e159]. The first clear indications of half-metallic
ferromagnetism were found by Ishida et al. for Co2MnZ and Ru2MnZ
(Z ¼ Al, Si, Sn, Sb) [160,161]. Mohn et al., on the other hand, found
a magnetic ground state for Co2TiZ (Z ¼ Al, Sn) by a full-potential
method, however, no half-metallic ground state was revealed [162].
Galanakis et al. reported half-metallic ferromagnetism in different
Co2YZ compounds, but not for Co-based ones with Y ¼ Ti or Fe [114].
Their results are in good agreement with the results by Picozzi et al.
who used a generalized gradient approximation (GGA) correction
instead of pure LSDA [163]. The GGA by Perdew et al. does not only
consider the exchange correlation potential of the local density
approximation, like in pure LSDA, but additionally its gradient
[164e167]. It was not possible to verify the half-metallic ferromagnetism in Co2FeAl using a spherical potential and GGA [67,168].
GGA-calculations with a full potential, however, yield a half-
metallic ground state for the complete substitution series
Co2Cr(1x)FexAl [147]. This illustrates, that the correct electronic
ground state is only obtained if the full potential and the generalized gradient approximation are taken into account.
Based on these results, the properties of Co-based Heusler
compounds were calculated, leading to the result that most Cobased compounds follow the SlaterePauling rule (compare Fig. 24).
The question, however, in which way the electrons have to be
distributed for the formation of a half-metallic ferromagnet, remains
to be solved. The s and p electrons do not contribute to the magnetic
moment, they are fully delocalized. The d electrons, however, are
sufficiently localized to be attributed to specific atoms. For the
compounds Co2YZ (Y ¼ (Sc,.,Fe), Z ¼ Al, Si) approximately 7.5
d electrons are localized at Co, i.e. Co has approximately a d7.5
configuration [141]. On the other side, the number of d electrons at
Y increases linearly with the atomic number. The magnetic moment
at the Co position is z1 mB for Co2YSi with Y ¼ Ti,., Mn and a little
bit lower for Co2YAl with Y ¼ V,. Fe. In both cases, the Y elements Ti
and Sc do not contribute to the magnetic moment, independent of Z.
The total magnetic moment, however, follows the SlaterePauling
rule, which means that the Co moment is reduced for a lighter
transition metal Y. The moment at the cobalt site needs to increase
for Co2FeSi to achieve the total magnetic moment of 6 mB. These
relations are illustrated in Fig. 30. Regrettably, for this compound
a wrong result is obtained by LSDA-GGA-calculations, even with the
full potential. The behavior of the Co magnetic moments (z1 mB)
illustrate that the CoeCo interaction plays a crucial role for halfmetallic ferromagnetism. On the contrary, this interaction is absent
in case of Half-Heusler compounds.
To explain the properties of Co2FeSi, a partial localization and
correlation of d electrons needs to be considered. The relative
relevance of itinerant compared to localized magnetism of d electrons in intermetallic alloys was already discussed by Slater
[95,169], van Vleck [170,171], and Goldmann [172]. Particularly,
a localization of the d electrons in Heusler compounds is unquestioned as already mentioned by Pauling for Cu2MnAl [96]. However,
the following question needs to be answered: To which extent
do the Coulomb interactions between the d electrons persist,
despite the increasing screening effect by delocalized electrons, so
that as a result a conservation of important atomic properties, such
as Hund’s rule is achieved [173]?
Fig. 30. Element specific magnetic moments in Co2YZ (Z ¼ Al, Si). Shown is the evaluation of the local moments at the Co and Y ¼ Sc, Ti, V, Cr, Mn, and Fe sites as a function
of the valence electron number at the Y sites. Data taken from Ref. [141].
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
Fig. 31. Spin resolved band structure of Co2FeAl. Compared are the band structures
calculated in the LSDA-GGA (a, b) and the LDA þ U (c, d) approaches. Data taken from
Ref. [176].
To answer this question, calculations with the LDA þ U-method
which is the simplest and most popular method to incorporate
electron-electron correlations on transition metal positions were
performed [174,175]. The LDAþU method takes the orbit dependency of the Coulomb and exchange interaction into account,
which is, on the other hand, not considered in a pure LSDA calculation. The effective Coulomb exchange interaction, Ueff ¼ U J, was
used to correct for the double counted terms. U and J correspond to
the exchange and the Coulomb integral, respectively. Fig. 31
compares the spin resolved band structure of Co2FeAl calculated
in the LSDA-GGA and the LDAþU approach [176]. It can be seen, that
the inclusion of Ueff in the calculation does not cause pronounced
changes in the majority bands. Even the flat band at z4 eV below
the Fermi energy is only shifted by 200 meV to higher binding
energies. This is remarkable, since this band is mainly responsible
for the localized magnetic moment at the Fe atom. However, the
major impact of the Coulomb parameter is found in the minority
bands, and, in particular, on their unoccupied part. The gap is
clearly opened up and the flat, lowest conduction bands at the G
point are shifted by z1 eV to higher energies.
Additional calculations were performed for the series
Co2Mn(1x)FexSi (0x1) by Kandpal et al. [174]. Independent of
the Fe concentration x, the following values were choosen for Ueff:
UCo ¼ 1.9, UFe ¼ 1.795, and UMn ¼ 1.768 eV. Previous results showed,
that the experimental magnetic state of Co2MnSi and Co2FeSi is
well describe with exactly these semi-empirical values within the
LDA þ Umethod. The values correspond to those for the Coulomb
interaction Udd between d electrons in elemental 3d transition
23
metals, determined by Bandyopadhyay and Sarma already prior to
the introduction of the LDA þ U method [177].
The curve for the magnetic moment m as a function of the iron
concentration x, obtained by LDA þ U calculations, matches well
with the experimental data as shown in Fig. 32. A shift of the Fermi
energy within the minority band gap, from the upper edge of the
minority valence band to the bottom edge of the minority
conduction band, is observed. The main impact of dynamical
correlations is usually seen in the spectral (energy-dependent)
properties. In contrast, the electronic structure near the Fermi level
and the related quantities must be much less affected due to the
Fermi liquid character of dynamical self-energy. Indeed, as shown
in Fig. 32 the full account of correlation effects within the
LDA þ DMFT approach does not significantly change the values of
magnetic moments (as energy-integrated quantities) calculated
within an account of the static part of correlations only, i.e. with
LDA þ U [178]. Co2Mn(1x)FexSi stays a half-metallic ferromagnet
for the whole range of x within the LDA þ DMFT approach contrary
to calculations that ignore correlation effects confirming the
assumption, that the electron-electron correlation cannot be
neglected in Heusler compounds.
The electronic structure of the substitution series Co2Mn(1x)FexSi was also studied experimentally by high resolution, highenergy X-ray photoelectron spectroscopy (HXPES) [146,179]. The
high photon energy of 8 keV ensures a real bulk sensitivity for the
valence band spectrum due to the high escape depth of the emitted
electrons (115 Å). In Fig. 33 displays a comparison of the calculated
density of states to the HXPES valence band spectra. Most interesting in these results is the behavior of the calculated DOS and the
measured spectra close to eF as this might give an indication of the
gap in the minority states. The majority band structure contributes
only a few states to the density at eF emerging from strongly
dispersed bands. This region of low density is surrounded by a high
DOS arising from flat bands at the upper and lower limit of the
minority band gap. The onset of the minority valence band is clearly
seen in the total DOS as well as the low majority density at eF. In fact,
the same behavior is observed in the measured valence band
spectra. The Fermi energy level can be estimated to be approximately 0.5 eV above the minority valence band. This is a strong
evidence that all compounds of the Co2Mn(1x)FexSi series exhibit
half-metallic ferromagnetic behavior. However, the desired method
Fig. 32. Comparison of the total magnetic moments for Co2Mn(1x)FexSi calculated
within LDA (green open squares), LDA þ U (blue open diamonds) and LDA þ DMFT
(orange open triangles) with the results of the SQUID magnetic measurements (red
circles). Data taken from Ref. [178]. (For interpretation of the references to colour in
this figure legend, the reader is referred to the web version of this article).
24
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
a
b
c
d
e
f
Fig. 33. Valence density of Co2Mn(1x)FexSi, (x ¼ 0, 0.5, 1). (a)e(c) show the photoelectron spectra exited with hn ¼ 7.939 eV and (d)e(e) display the calculated total DOS convoluted
by a FermieDirac distribution unsing T ¼ 20 K. Data taken from Ref. [146].
to finally prove half-metallic ferromagnetism in Heusler compounds is spin resolved photoemission.
6.2.3. Relation of disorder and spin polarization
The effect of atomic disorder on the electronic structure in HalfHeusler alloys was for the first time studied by Ebert and Schütz for
MnPtSb [180]. The local density of states (LDOS) was calculated
using the spin-polarized relativistic KorringaeKohneRostocker
(KKR) method within an atomic sphere approximation (ASA). The
comparison of the minority spin band gap at eF for an ordered and
anti-site disordered compound revealed that impurity states fill the
band gap locally. This effect is most pronounced in case of Mn
atoms on Pt sites, in which a peak arises in the minority spin gap,
resulting in a distinct reduction of the spin polarization and the
magnetic moment.
A more quantitative investigation on the effect of atomic
disorder was carried out by Orgassa et al. for MnNiSb [181,182].
Here, the effect of random atom distributions on each lattice site
was investigated by the layer KKR-ASA method combined with the
coherent potential approximation (CPA). The authors considered
different types of atomic disorder: The partial interchange of Ni and
Mn, the partial occupation of the vacant lattice site by Mn and Ni or
Sb. In all cases, disorder-induced states appear in the minority spin
gap, resulting in a band gap narrowing and a reduction of the spin
polarization at the Fermi level. Although the spin polarization
remains 100% for disorder levels lower than a few percent,
a considerable decrease is initiated for higher amounts ( 5%). For
instance, the spin polarization is reduced to 52% for 5% MneNi
interchange. When Mn and Sb occupy the vacant lattice site,
a reduction to values as low as 24% takes place. Thus, a supression of
anti-site disorder and very careful structure analysis is necessary to
obtain high spin-polariztions in Half-Heusler materials. Intrinstic
defects in MnNiSb and their consequences for the spin polarization
were also investigated theoretically by Attema et al. [183]. The
authors showed that most types of defects which are likely to occur
do not influence the spin polarization distinctly.
The tunneling magnetoresistance (TMR) effect was observed for
the first time at room temperature in magnetic tunnel junctions
with CsCl-type (B2) disordered CCFA electrodes [184]. To clarify the
relationship between atomic disorder and the spin polarization,
Miura et al. theoretically investigated the electronic structure of
disordered Co2CrxFe(1x)Al, based on first-principle density functional calculations with the KKR-CPA [67,185]. This study revealed
that in the parent phase Co2CrAl the spin polarization remains high
(more than 90%), even for a complete interchange of Cr and Al.
A detailed analysis of the density of states (DOS) and the atom
orbital projected local density of states (LDOS) of Co 3d revealed
that the energy gap of the minority DOS near the Fermi energy level
is mainly constructed of Co 3d states. Thus, the disorder between Cr
and Al does not significantly affect the electronic structure near the
Fermi level, and therefore, the semiconducting character of the
minority bands is still kept even in the disordered B2 structure.
The spin polarization of Co2CrxFe(1x)Al is reduced with increasing
Fe concentration for both ordered L21 and disordered B2 structures.
For CCFA the spin polarization is evaluated as 90% and 77% for the
L21 and B2-type structures, respectively [185]. Contrary to the
CreAl interchange, the Co-Cr interchange gives rise to a considerable reduction of the spin polarization, due to the appearance of
anti-side Co 3d states in the minority spin band gap. The total
magnetic moment is reduced linearly from 3 mB per formula unit
with increasing amount of Co-Cr disorder. Wurmehl et al. showed,
that this reduction of the total magnetic moment can be attributed
to ferrimagnetic order with an antiparallel alignment of the antiside Cr moments to the other magnetic moments on the ordinary
Cr and Co sides [148].
6.3. Compensated ferrimagnets
Half-metallic ferrimagnetic materials are desirable candidates
for the application in magnetoelectronic devices. The advantage of
a ferrimagnetic interaction is the resulting reduction of the
magnetic moment due to the compensation of the moments
carried by the different sublattices. These materials offer distinct
advantages over their ferromagnetic counterparts which are mostly
due to their small magnetic moment. The ideal compensated
ferrimagnet would exhibit a total magnetic moment of zero. For
such compensated ferrimagnetis which were initially named
“compensated antiferromagnets” single spin superconductivity
was observed by Pickett [186]. Further interesting applications can
be envisioned, since they do not give rise to strong stray fields and
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
are less affected by external magnetic fields. An ideal case for
application would be a half-metallic compensated ferrimagnet
since it would be a perfectly stable spin-polarized electrode in
a junction device, especially for current-induced magnetic
switching, which uses the spin-transfer effect. Spin-transfer torque
(STT) which provides an ultra-low-power switching (writing)
solution and makes a down-scaling of the individual bit cell below
10 nm possible, is predicted to be the next key step towards the
development of practical spintronic devices. For radio frequency
devices a new type of an integrated spin-transfer torque nano
oscillator (STTNO) has been proposed for telecommunication. In
such a device the STT causes a magnetization precession of the free
magnetic layers, leading to the generation of microwaves at GHz
frequencies. Furthermore, the use of such a material as a tip in
a spin-polarized scanning tunneling microscope (STM), would not
give rise to stray flux, and thus, would not distort the domain
structure of a soft magnetic material.
6.3.1. Half-Heusler compounds
Half-Heusler compounds possess only one magnetic sublattice
since only the atoms on the octahedral sites can carry a magnetic
moment as shown in Fig. 22. In the literature, there are many
examples for ferrimagnetic or antiferromagntic compounds, which
are easily assumed to be Half-Heusler materials. In fact, most of
these materials, e.g. CrMnSb, FeMnSb, crystallize in structure types
different from the Half-Heusler structure. Only for rare earth containing materials antiferromagnets with low Néel temperatures are
known [92].
6.3.2. Heusler compounds
In Heusler alloys, two magnetic sublattices allow the antiferromagentic coupling of the atomic magnetic moments, leading
to ferrimagnetic or even completely compensated ferrimagnetic
materials (see Fig. 22).
A combination of the above explained SlaterePauling rule and
the Kübler rule allows the prediction of half-metallic completely
compensated ferrimagnetism in Heusler compounds with 24
valence electrons [187]. In these compounds, the two atoms on the
X site have to compensate the magnetic moment of the atom at
the Y site (mostly Mn). The precondition for Mn to be located on the
Y position is that it is the more electropositive transition metal in
the compound (compare Section 3). The only possible elements to
occupy the X position are, therefore, Fe, Co, Ni, Cu, and Zn, as well as
Mn itself. The total valence electron number of 24 restricts the
possible combinations to the binary Mn2MnZ compounds, with
Z being an element from the third main group of the periodic table.
Wurmehl et al. first designed the material Mn2MnGa (or simply
Mn3Ga) indeed leading to totally compensated half-metallic ferrimagnet with the L21 Heusler structure with Mn3þ on the octahedral
position (Kübler rule) [187]. The two magnetic moments of the
manganese atoms on the tetrahedral positions cancel the moment
of the Mn3þ leading to zero net magnetization.
The synthesis of Mn3Ga, however, revealed, that this compound
does not crystallize in the cubic and ordered Heusler structure, but
in a tetragonal distorted structure with prototype Al3Ti (for details
see Section 17.3) [188,189]. Ab initio calculations for the distorted
crystal structure show that Mn3Ga is ferrimagnetically ordered
with an total magnetic moment of z1.7 mB. A higher DOS of the
minority electrons compared to the majority channel at eF indicates
a distinct difference in the conductivity between the two spin
directions. Furthermore, a pronounced magnetic anomaly was
observed, which suggests that Mn3Ga is a magnetically frustrated
ferrimagnet. Due to the tetragonal distortion, this compound is
a promising candidate as an electrode material with perpendicular
magnetic anisotropy for spin torque devices (see also Section 8).
25
Ferrimagnetic behavior is also observed for Mn2YZ compounds
with valence electron numbers unequal 24, depending on the
crystallographic order. If the inverse Heusler structure is formed
and one Mn atom is located at an octahedrally coordinated lattice
site, the local magnetic moment of Mn is strongly localized
(Kübler’s rule) and can be partially compensated by the atoms on
the X positions resulting in a ferrimagnetic order. The inverse
Heusler structure is formed if the nuclear charge of Y is higher than
the one of Mn, i.e. only for Z(Y) Mn (see Fig. 5 for comparison).
However, an incommensurate magnetic spiral structure with
zero magnetization was reported for the compound Mn3Si
[190,191]. This was explained by a direct exchange between the Mn
atoms on the Y positions stimulating the formation of a collinear
antiferromagnetic configuration of the moments of this sublattice
[192] and an exchange interaction with the Mn atoms on the X2
positions, which stimulate a parallel orientation of the other Mn
moments. Thus, a competition between effective ferromagnetic
and antiferromagnetic interactions leads to the formation of the
collinear spiral configuration.
7. Magneto-optical properties
A very important aspect of Heusler compounds is their
magneto-optical (MO) behavior. Magneto-optical effects comprise
various changes in the polarization state if light upon interaction
with materials possessing a net magnetic moment, including
rotation of the plane of linearly polarized light (Faraday, Kerr
rotation), and the complementary differential absorption of left and
right circularly polarized light (circular dichroism). The discovery of
an extremely large Kerr rotation for the Half-Heusler material
MnPtSb (1.27 at RT and 5 at 80 K) showed the technological
relevance of such compounds in the context of magneto-optical
reading and recording [7,193]. This value, which was totally unexpected for a 3d-based material, was for many years the record Kerr
rotation observed in such a system, and therefore called “giant”.
Almost simultaneously with the experimental discovery, the
theoretical finding of the half-metallic nature of MnPtSb was
reported [48]. The MO Kerr rotation of the isoelectronic MnNiSb,
however, was much smaller, although experimental evidence for
half-metallic ferromagnetism was found for both compounds
[116,194].
On the theoretical side, several model explanations of MO
spectra for these compounds were proposed: One was based on
a possible loss of the half-metallic character due to spineorbit
coupling [195], while another explanation was based on the
differences of the semi-relativistic effects in MnNiSb and MnPtSb
[130], or the enhancement of the MO Kerr spectra near the plasma
resonance [196]. Although all these proposed models contained
interesting physical mechanisms, one of the remaining puzzling
issues was the explanation of the measured differences in the MO
spectra.
Only owing to the development of ab initio calculations of the
MO spectra the detailed quantitative comparison between experiment and first-principles spectra became feasible [112,168,197,198].
The calculated MO spectra, however, spread rather widely due to
the nature of the calculations and the contained approximations.
Antonov et al. explained the Kerr spectra based on their electronic
structure [199,200]: They found that the anomalies in the Kerr
spectra of MnNiSb and MnPtSb arise from an interplay of
compound related differences in the spineorbit interaction, in the
half-metallic character, and also in the relative positions of energy
bands.
Not only Half-Heusler compounds were investigated but there
has also been tremendous interest in the class of Heusler materials
for magneto-optics. Bushow and van Engen studied the MO
26
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
properties of X2YAl and X2YGa experimentally [153,201], and
showed that the polar Kerr rotation angle qK for Co2FeAl and
Co2FeGa has a strong minimum near 1.5 eV, the value of the jqK j
corresponding to this minimum being almost as large as in pure
CoeFe compounds. Additionally, studies on the effect of structural
defects and disorders on the properties of these compounds were
reported, in which the effect of point defects and anti-sides on the
magnetic and magneto-optical properties were investigated [168].
Kumar et al. found that the optical transitions in Co2FeX Kerr
spectra are governed by the absorptive parts of the optical
conductivity [198]. In this study, the main peak was explained by
the behavior of the frequency-dependent absorptive parts of the
optical conductivity tensor element. The observation of peaks in the
MOKE spectra at high energies is very promising for possible
technological applications for high density MO recording.
8. Heusler compounds in devices for spintronic applications
The discovery of the giant magnetoresistance (GMR) effect in
magnetic multilayers and sandwiches in 1986 by P. Grünberg [202]
and A. Fert [203] revolutionized the field of information technology.
For this outstanding discovery they were honored with the Nobel
prize in physics in 2007. Today, we are in contact with spintronics in
our everyday life, in form of spin valves based on the GMR effect,
which are used in magnetic hard disk drives. In such a spin valve,
two magnetic layers sandwich a very thin non-magnetic metallic
spacer. If the magnetization of both ferromagnetic layers is aligned
in parallel direction, the resistance of the device is low, whereas
a high resistance state is present, if the ferromagnetic layers are
aligned antiparallel. At the top of Fig. 34 an example of such
a multilayer FeeCreFe system is shown [203]. By applying
a magnetic field, the resistivity of these multilayers can be suppressed by orders of magnitude. Depending on the thickness of the
Cr spacing layers, the interlayer exchange coupling between the Fe
layers changes from a ferromagnetic (parallel) to an antiferromagnetic (antiparallel) state. Measurements of the electrical
resistivity show, that an antiferromagnetic exchange leads to a high
resistance, which can be altered by applying large external
magnetic fields. The resistivity decreases, when the configuration
of the magnetization in neighboring Fe layers is changed from
antiparallel to parallel. The spin oriented electrons of the Fe layers
are accelerated by an applied electrical field, until they encounter
a scattering center. Provided that the interlayer thickness is less
than the coherence length, the electron arrives at the interface of
the neighboring ferromagnetic layer, still carrying its initial spin
orientation. In the case of ferromagnetically coupled Fe layers, the
arriving electron has a high probability of entering the adjacent
layer, due to the matching spin orientation. If the exchange
coupling is antiferromagnetic, the electrons are scattered strongly
at the interface, resulting in a high resistance. The magnetoresistance ratio of a FeeCreFe multilayer reaches 79% at 4 K and small
fields, and is still 20% at room temperature for systems with a 9 Å
thick Cr layer [203].
A read head or a magnetic sensor device, on the other hand,
consists of an artificial multilayer thin film material with alternating ferromagnetic and non-magnetic metals. In fact, IBM
introduced these devices in 1997 and the market for them is now
one billion dollars per year [204]. In a GMR device the current can
either flow perpendicular to the interfaces (CPP, current-perpendicular-to-plane), or parallel to the interfaces (CIP, current-inplane). The GMR was originally discovered in a CIP configuration;
however the CPP configuration shows even larger effects. A spin
valve consists of two ferromagnetic layers sandwiching a thin nonmagnetic metal layer. One of the magnetic layers is “pinned” by
an antiferromagnetic material, and is, therefore, insensitive to
moderate magnetic fields; the second layer is “free”, i.e. its
magnetization can be rotated by the application of small magnetic
fields.
GMR spin valves led to a dramatic increase in area storage
density, but as emerging technologies are developed with incredibly high speed, the era of GMR is superseded by spin-dependent
tunneling devices. The replacement of the metallic spacer by an
insulating material lead to a rise in magnetoresistance by a factor of
10 compared to GMR spin valves. Since the effect is based on the
tunneling of electrons through the insulating barrier, these new
devices are known as magnetic tunnel junctions (MTJs) or as
tunneling magnetoresistance (TMR) devices (compare schematic
illustration in Fig. 34 and see review [205] for further details).
Interestingly, the ultimate goal of spintronics, i.e. a tunneling device
with a magnetoresistance effect of several thousand percent, can be
reached by two different courses: One way is to engineer the
insulation barrier, and the other way is to develop new electrode
materials with 100% spin polarization. Potential candidates include
half-metallic ferromagnetic oxides as well as half-metallic ferromagnetic metals, such as Heusler compounds.
Particularly, Co2-based Heusler materials were intensely studied
due to their high potential as new electrode materials in spintronic
devices, such as magnetic tunnel junctions (MTJs) [184,206e211],
giant magnetoresistance (GMR) devices [212e216], and for spin
injection from ferromagnetic electrodes into semiconductors [217]
as discussed in the following sections.
8.1. The tunneling magnetoresitance effect
Fig. 34. Illustration of the basic spintronic devices. At the top, GMR multilayers are
shown; the magnetic coupling can be adjusted by varying the thickness of the nonmagnetic spacer layer. At the bottom, a TMR device is illustrated. The tunneling
current, that follows perpendicular to the film surface, experiences a high resistivity in
case of antiparallel magnetization of the ferromagnetic electrodes, while the resistivity
is low for a parallel magnetization direction.
Early pioneering investigations on the problem of spin-dependent tunneling were performed in the 1970s by P. M. Tedrow and
R. Meservey [218], by M. Jullière [219], and by S. Maekawa and
U. Gäfvert [220]. Twenty years later, however, the first large
magnetoresistance in magnetic tunnel junctions was observed at
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
27
room temperature by J. S. Moodera [221] and T. Miyazaki [222].
Following the Jullière model [219], the TMR ratio of a junction is
related to the spin polarization P of the electrodes according to:
DR
2P1 P2
¼
1 þ P1P2
RTMR
(5)
where P1 is the polarization of one electrode and P2 is the polarization of the second electrode. Moreover, the spin polarization P is
defined by
P ¼
N[ NY
N[ þ NY
(6)
N[ and NY are the densities of the majority and the minority
electrons at the Fermi energy level. The Julliére model is a simple
approximation for the tunnel effect. However, the model is commonly
used to estimate the spin polarization of the electrodes since a high
spin polarization is required for high TMR ratios. The largest ratio of
1800% was measured by Fert’s group in a tunnel junction with
a manganite electrode. This corresponds to an electrode spin polarisation of at least 95%, but unfortunately only at 4 K [223].
The first theoretical prediction of half-metallicity in MnNiSb
stimulated tremendous research interest, aiming at the utilization
of Heusler compounds in MTJs. In fact, for an MnNiSb bulk single
crystal a spin polarization of almost 100% at eF was observed by
means of spin-polarized positron-annihilation (SSPA) [115,116]. The
preparation of thin films of this material, however, turned out to be
not without difficulties. Therefore, different growth methods,
comprising co-sputtering and molecular beam epitaxy (MBE), had
to be employed to prepare epitaxial films. Finally, the crystal
structure was confirmed by XRD, and the presence of a magnetocrystalline anisotropy. Furthermore, a maximum spin polarization
of z60% at 1.6 K was observed by point-contact Andreev reflection
(PCAR) measurements [230e232]. The first integration of an
epitaxial MnNiSb thin film into a MTJ yielded a low TMR effect of 9%
at room temperature and 18% at 4.2 K, which corresponds to a spin
polarization of only 25% [231]. Since another MnNiSb thin film that
was grown in a similar manner, showed a spin polarization of 60%,
a considerable contribution of atomic disorder at the empty lattice
sites in the vicinity of the tunnel barrier is assumed [181]. This is in
good agreement with the reported vanishing of the energy gap for
the minority spins at eF with more than 7% atomic disorder and the
fragility of the surface state due to reduced symmetry and surface
reconstruction [233,234].
Similar arguments can also be applied for other Half-Heusler
thin films. For example, sputtering MnPtSb on Al2O3 (001)
substrates leads to the formation of spin valves showing a MR ratio
of only 0.47% at RT [235,236].
Scientific interest in this field was further stimulated by investigation carried out by Block et al., who discovered a large negative
MR at RT in the quaternary Heusler compound Co2Cr0.6Fe0.4Al
(CCFA) [140,237,238], which demonstrated the tunability of the
spin density of states at the Fermi level by substituting constituent
elements. Consequently, in pressed powder compacts that act as
a series of MTJs, a MR of 30% was reported in a small external field
of 0.1 T. This discovery triggered enormous research efforts
focusing on the implementation of this material into spintronic
devices [151,184,239,240]. Shortly after that, the first TMR using
a B2 sputtered CCFA electrode was reported to be 26.5% at 5 K (16%
at RT) by Inomata et al. [184]. The incorporation of Heusler
compounds into TMR devices led to a dramatic increase in the TMR
ratio in the following years as shown in Fig. 35. One breakthrough
was the discovery of a large magnetoresistance effect of nearly
600% in Co2MnSi with an AlOx tunnel barrier at low temperature
[225]. However, the temperature dependence was strong and the
Fig. 35. Development of the TMR ratio for MTJs with Heusler electrodes. Open symbols
denote the TMR value at 5 K, while filled symbols display the value at room temperature. Data taken from Ref. [75,184,206,224e229].
TMR value decreased to only 70% at RT. The discovery of Co2FeSi,
the half-metallic Heusler compound with the highest magnetic
moment of 5.97 mB and the highest Curie temperature of 1100 K
[106], and the adjustment of the Fermi level to the middle of
the gap using Co2FeAle0.5Si0.5 or Co2Fe0.5Mn0.5Si improved the
temperature dependence considerably [176,241]. Nevertheless, the
list of promising candidates is long, and many different materials
have been tested, e.g. Co2FeSi, Co2MnSi, Co2MnGe, Co2Fe0.5Mn0.5Si,
Co2FeAl0.5Si0.5 [227,242e245]. Subsequently an improvement of
the film quality led to a distinct improvement of the MTJs based on
Heusler compounds, as displayed in Fig. 35. For tungsten-type
disordered Co2FeAl, a TMR at room temperature of 47% was
obtained, while Co2FeAl electrodes in the CsCl-type structure
yielded only a TMR of 27% [224]. These results are in good agreement with calculations that predicted a spin polarization (P) of 62%
for the tungsten-type and 30% for the CsCl-type structure [67]. It
turned out, that not only a sufficient crystallinity of the thin films
plays a major role in MTJs, but that also the surface roughness and
the interface morphology between the Heusler electrode and the
barrier has a great influence on the TMR value [246]. As a result,
epitaxially grown Heusler alloys with flat surfaces lead to distinctly
enhanced TMR ratios. Apart from that, a spin polarization of 0.49
was measured for an ordered Co2FeSi thin film by the point
Andreev reflection technique (PCAR). TMR values obtained from
the MTJs with such Co2FeSi electrodes and an alumina barrier were
67.5% at 5 K and 43.5% at 298 K, respectively [242]. The P value
estimated from the TMR, using Jullière’s model, matches the spin
polarization measured by PCAR very well, indicating that the TMR
value from the MTJ is governed by the intrinsic value of P of the
electrode material for incoherent tunneling. But in fact, PCAR is
a technique in which the spin polarization of a material depends on
fitting parameters. Therefore, the spin polarization of tunnel junctions often differs from the results obtained by PCAR. On the other
hand, a TMR ratio 159% at 2 K was observed for an alumina barrier
with an epitaxially grown L21-ordered Co2MnSi electrode and
a Co75Fe25 top electrode [246]. According to Jullière’s formula, the
spin polarization of the Co2MnSi bottom electrode in these MTJs
was estimated to be P ¼ 0.89. Unfortunately, the observed TMR
value shows a large temperature dependency. However, the
replacement of the top Co75Fe25 electrode by Co2MnSi led to
comparable TMR values at room temperature, but a dramatical
increase is observed with decreasing temperature to 570% at
28
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
2 K [225]. If the spin polarization of the lower Co2MnSi electrode is
assumed to be 0.89%, the spin polarization for the upper Co2MnSi
electrode can be estimated to be 0.83%. This strong temperature
dependency is commonly attributed to spin-flip tunneling caused
by magnetic impurities at the Co2MnSi/AlOx interface or pinholes in
the barrier. Especially, for MTJs containing Co2MnSi, the creation of
magnetic impurities is a critical problem, as Mn and Si both have
a high affinity to oxygen compared to other 3d transition metals.
Furthermore, the location of eF in the half-metallic energy gap is an
important factor in the temperature dependency of the TMR ratio.
Therefore, the large decrease in the TMR ratio is also attributed to
the small energy separation between the Fermi level and the
bottom of the conduction band, since the thermal fluctuations at
room temperature are twice as big as this energy separation
(Fig. 36).
A different approach for obtaining much larger TMR values was
established by the using single-crystalline MgO as barrier material
[247,248]. Butler et al. predicted a much larger TMR ratio, as high as
6000% for the Fe/MgO/Fe MTJ using a first-principles layer KKR
approach. The authors found reported that the main contribution to
the large magnetoresistance was caused by the coherent tunneling
of highly spin-polarized Fe s-like states (D1 states), propagating
along the direction perpendicular to the plane due to their
symmetry matching with complex D1 MgO bands within the
energy gap and the slow decay in MgO. Indeed, a larger TMR
(z410%) ratio was obtained at room temperature for the MTJ of Fe/
MgO/Fe(001) [249,250]. Theoretical investigations show that the
principle of coherent tunneling is also transferable to Co2MnSi/
MgO junctions [251]. Here, the introduction of MgO barriers into
MTJs with Heusler electrodes led to a dramatic improvement in
their performance. In addiation to that, the relatively small lattice
mismatch between Co2YZ and MgO for a 45 in-plane rotation (e.g.
z3.7% for Co2Cr0.6Fe0.4Al, and 5.1% for Co2MnSi) makes the
fabrication of fully epitaxial MTJ trilayers possible, featuring
smooth and abrupt interfaces [208,252e254]. As a result, relatively
high TMR ratios of 109% at room temperature (317% at 4.2 K) were
Fig. 36. Design criteria for half-metallic ferromagnets and their application in
spintronic devices.
demonstrated for Co2Cr0.6Fe0.4Al/MgO/Co50Fe50 MTJs [253], and
90% at ambient temperature (192% at 4.2 K) for Co2MnSi/MgO/
Co50Fe50 MTJs [208]. A further increase of the TMR ratio was achieved by the fabrication of MTJs with Co2MnSi electrodes as both,
the lower and upper electrode, leading to TMR ratios of 179% at
room temperature and 638% at 4.2 K [255]. An even higher value of
753% at 2 K was obtained for a Co2MnSi/MgO/Co50Fe50 junction
[227]. Although these results are promising, a large decrease with
increasing temperature to 217% at room temperature was still
obeserved, which was explained with the occurrence of inelastic
tunneling events, caused by magnon excitations due to the low
Curie temperature at the Co2MnSi surface, and by magnetic
impurity scatterings, resulting from the presence of Mn and Si
oxide impurities. However, the highest reproted TMR ratio so far of
340% at room temperature was observed in a CsCl-type disordered
Co2FeAl based MTJ [11]. Since this Heusler material does not exhibit
a complete spin polarization, the high TMR value is a strong indication for pronounced coherent tunneling. Additionally, a TMR
oscillation as a function of the MgO layer thickness was observed.
First-principles electronic band structure calculations confirm, that
the CsCl-type disordered Co2FeAl behaves like a half-metal in terms
of the D1 symmetry in the (001) direction. It should be pointed out,
that this remarkable result was achieved with a Heusler alloy displaying considerable disorder. This result strongly indicates that
Heusler compounds are promising materials for a giant TMR due to
coherent tunneling and their tunable electronic and magnetic
properties.
8.2. Current-perpendicular-to-plane giant magnetoresitance
In addition to the fabrication of TMR devices, current-perpendicular-to-plane (cpp) GMR devices with Heusler electrodes
recently emerged in the field of spintronics. Compared to TMR, cppGMR is expected to be insensitive to the electronic state at the
interfaces, where half-metallicity is often destroyed. In fact, the first
cpp-GMR devices consisted of two Co2MnSi electrodes, sandwiching a 3 nm Cr spacer [212]. For this trilayer system a maximum MR
ratio of 36.4% was obtained at room temperature (67.2% at 110 K)
[256]. It should be noted that the choice of the spacer layer is an
important issue, since an epitaxial growth of the Heusler thin film
on the spacer material is required to form fully epitaxial Heusler/
spacer/Heusler trilayers. A large spin-diffusion length and low
resistivity are also necessary for the spacer layer to obtain large
cpp-GMR values. These considerations, combined with a small
lattice mismatch, led to the selection of silver as an ideal spacer
layer. Consequently, a cpp-GMR ratio of 6.9% at room temperature
(14% at 6 K) was realized for a Co2FeAl0.5Si0.5/Ag/Co2FeAl0.5Si0.5
structure [213]. Additionally, an enhanced cpp-GMR ratio of 34%
at 290 K (80% at 14 K) for the same system was just reported
recently [257].
A different approach is given by the using copper as spacer
material. Despite the large lattice mismatch between Cu and
Heusler compounds, cpp-GMR values of 8.6% at room temperature
(30.7% at 6 K) were obtained for the system Co2MnSi/Cu/Co2MnSi
[215]. To minimize the lattice mismatch and to optimize the
interface scattering properties, “all-Heusler” cpp-GMR devices with
the trilayer Co2MnSi/Ru2CuSn/Co2MnSi were proposed. These
devices yielded a MR ratio of 6.7% at room temperature [216].
Narrow cpp-GMR read heads, incorporating Heusler materials as
reference layers, were successfully tested using a conventional
spin-stand system. Thus, the capability of the cpp-GMR technology
for ultra-high density magnetic recording was demonstrated,
further development of the cpp-GMR stag materials, however, is
necessary, to make the heads superior to TMR heads [258]. From an
applications point of view, a stable cpp-GMR effect 30% at room
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
29
temperature is perfectly suitable to manufacture high performance
devices.
8.3. Perpendicular magnetic anisotropy
The magnetoresistance phenomena discussed in the previous
section (GMR or TMR) allows to control an electron flow through
a magnetic nanostructure by its magnetic state. The reciprocal
phenomenon also exists. A spin-polarized current flowing through
a magnetic nanostructure can influence its magnetic state. This socalled spin-transfer torque is one of the most promising technologies today to satisfy the increasing demand for faster, smaller and
non-volatile electronics. Convoluting this development towards
smaller device sizes is the fact that power-consumption requirements are increasing as transistor sizes shrink to the sub-100 nm
regime (Fig. 37).
Switching the spin with a current is possible due to the
exchange interaction between the spin of the incoming conduction
electrons and the spin of the electrons responsible for the local
magnetization, as schematically sketched in Fig. 38. A magnet
usually responds to an electric current because of the magnetic
field generated by the current. But if the magnet is small (typically
less than 100 nm), a new force emerges [259e261]. When the
electrons constituting the current pass through a magnetic
conductor, their spins will become preferentially aligned to its
magnetic direction, i.e. they are spin polarized. These spins may be
repolarized into a new direction when they encounter another
magnet (Fig. 38). In repolarizing the current, the nanomagnet
experiences a torque (or turning force) associated with the charge
in angular momentum that occurs due to the rotation of the electron spins. This spin-transfer torque can pump enough energy into
the nanomagnet to cause a precession of its magnetic moment, i.e.
it moves at microwave frequencies around the symmetry axis with
ever-increasing amplitude until it reverses its orientation, accomplishing a magnetic switch.
From an applications point of view, the thermal stability of ultrahigh density magnetic memory storage devices is a crucial point. To
overcome the superparamagnetic limit when decreasing the device
size, thin films with perpendicular magnetic anisotropy (PMA), i.e.
with the easy magnetization axis pointing perpendicular to the film
surface are advantageous. Suitable materials need to exhibit a high
spin polarization, and simultaneously, a low saturation magnetization. These prerequisites make Mn3xGa a promising material,
due to the predicted high PMA property based on studies of the
Fig. 37. As conduction electrons pass a magnet, their spins preferentially align in the
magnet’s direction. As the electrons encounter a nanomagnet, sandwiched between
layers of non-magnetic material close to the fixed orientation magnet, the direction of
their spins is repolarized to match that of the nanomagnet. As a result, the nanomagnet’s magnetic moment begins to percess, turning like a spinning-top about its
axis.
Fig. 38. Design criteria for materials with potential application in spin-torque devices.
MeH curves of the polycrystalline alloy [188,189]. The ferrimagnetic
coupling of the Mn atoms results in a low saturation magnetization,
while the Curie temperature is higher than 770 K. The theoretical
calculated spin polarization of 88% is sufficient for the desired
application. As an example, tetragonally distorted Mn2.5Ga films
were grown on Cr-buffered MgO substrates with the tetragonal
c-axis pointing along the normal direction, resulting in a giant
PMA with an effective magnetic anisotropy energy of
Kueff ¼ 1:2 107 erg=cm3 [262]. The search for new materials with
suitably designed properties is an active field ongoing research.
Especially tetragonally distorted Heusler materials are in focus as
new magnetic layers in spin-torque devices. A detailed description
of their crystal structure and properties is given in Section 17.3.
8.4. Spin injection
In the field of spintronics, spin injection into degenerate semiconductors such as GaAs is also an area of great scientific interest
[263]. In fact, the technological applications of spin injection are
myriad and include the manipulation of classical information
carried by spin, initialization, and readout of a spin qubits [264] and
coherent manipulation of spin in the proposed spin field effect
transistor [265]. Single element ferromagnetic transition metals
such as Fe are attractive spin injectors as they possess a high Curie
temperature and exhibit well understood thin film magnetism.
However, the sed hybridized nature of the band structures of these
metals automatically leads to a limitation of the Fermi level
polarization to a range of 30%e40%.
Pioneering work on the growth of Heusler alloys/semiconductor
hybrid structures was performed by Ambrose et al. [266]. The
authors successfully demonstrated the epitaxial growth of
Co2MnGe on GaAs(001) with minor enhancement of the lattice
constant as compared to the bulk value. However, calculations
suggested for this particular system, that a strong reduction of the
magnetic moment may occur near the Co2MnGe/GaAs interface
due to the Co-As and Co-Ga bonding [267]. A different example for
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T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
the incorporation of Heusler compounds into spin injection devices
was given by Hirohata vet al., who prepared Co2Cr(1x)FexAl thin
films on GaAs(001) substrates by MBE [268]. In this system, the
initial formation of the A2-type structure has been observed upto
a thickness of z2 monolayers, followed by a B2-type structure
above 3.5 monolayers and the stable L21 phase above 13e14
monolayers [269]. Even though the half-metallicity should be
preserved at the Co/As interface on the (001) surface [270], the
initial growth of the A2 structure may introduce unfavorable
interfaces, such as CrAl/As, and therefore, reduce the half-metallicity in the vicinity of the interface.
Spin injection experiments with off-stoichiometric Co1.6Mn2.4Ga
revealed a injected electron spin polarization into InGaAs of 13% at
5 K [271]. An injected spin polarization of 27% at 2 K was reported
for the system Co2MnGe/GaAs [217], contrast to the value of 40%
reached with a Fe injector.
The growth of the Half-Heusler material MnNiSb on semiconducting substrates offers another opportunity to build spininjection devices [78,272]. In this context, the epitaxial growth on
InP was demonstrated which is favored compared to GaAs due to
a smaller lattice mismatch [273,274].
Apart from that, the injected polarization of Heusler compounds is
significantly below the value of 100% that would be expected for
a half-metal. Possible explanations for this phenomenon comprise
a local atomic disorder and small band gaps for the minority spins, e.g.
z200 meV in Co2MnGe [163]. Consequently, Heusler compounds
with larger minority spin gaps, such as Co2MnSi [275,276], may be
more efficient injectors. Since spin injection experiments probe the
spin polarization at the interface, a realistic theory does not only need
to consider the electronic structure of the interface, but also the
presence of atomic disorder as well as the effects of non-zero
temperature. Indeed, these factors play an essential role in interpreting spin injection measurements on new materials.
9. Shape-memory materials
Today, the Ni2MnGa system is one of the most intensively
investigated materials owing to its shape memory behavior and its
potential application in actuator devices, in which strains are
controlled by the application of an external magnetic field. In this
system, the cubic phase undergoes a ferromagnetic transition at
TC ¼ 376 K [277]. Additionally, stoichiometric Ni2MnGa undergoes
a structural phase transition at TM ¼ 202 K from the high-temperature cubic L21 structure to a low-temperature martensite phase
[277]. Due to the reversibility of this structural transition, a shape
memory effect is observed in this system. Moreover, the crystal
structure of the martensite phase in NieMneGa-system can be
modulated or unmodulated to be orthorhombic, tetragonal, or
monoclinic [278e281]. In case of a tetragonal martensite phase, the
cubic unit cell is contracted along one (001) axis and extended
anlong the other two. Since this transformation is diffusionless,
large stresses have to be stored and accommodated in the
martensite microstructure. As a consequence, the minimization of
the strain energy leads to the formation of a number of crystallographic domains, known as variants. In this case, the original cubic
cell allows the formation of three different variants, depending on
which axis is contracted. A typical martensite microstructure
consists of mixtures of the three variants, in which two adjacent
variants meet at one of the two possible well-defined interfaces,
called twin planes (compare Fig. 39). While each of these variants
has a unique orientation, defined by its c-axis, the martensite phase
is essentially in a polycrystalline state, composed of variable
volume fractions of these three variants. In ferromagnetic shape
memory alloys, such as Ni2MnGa, a magnetic field can move these
twin planes. Variants, in which the easy-axis of magnetization is
Fig. 39. Schematic views of a sample in the martensite phase are shown at the top. The
direction of magnetization is indicated by the arrows. At the bottom, the orientation of
the martensitic unit cell is sketched on the left hand side, and the twinned martensitic
cell is displayed on the right.
aligned in parallel to the external field, grow preferentially at
expense of variants with different orientation, resulting in macroscopic strains of up to 10% [282].
To provide the necessary driving force for twin boundary motion
in moderate magnetic fields (z1 to 2 T), the presence of strong
magnetoelastic coupling on the mesoscopic length-scale is essential.
This condition is fulfilled in NieMneGa shape memory alloys, in
which the magnetocrystalline anisotropy energy and the magnetization are sufficiently high in the martensite phase [283,284]. On
a microscopic length-scale, however, strong magnetic coupling
leads to large strains. In contrast to the NieMneGa system, where
the saturation magnetization in the martensite phase is higher than
in the L21 austenite phase, the magnetization in NieMneZ, Z ¼ In,
Sn, or Sb is lower in the martensite phase than in the L21 austenite
phase [285e288]. Therefore, a magnetic field applied to the
martensite phase can shift the transition to sufficiently low
temperatures and stabilize the austenite phase giving rise to a fieldinduced reverse martensite transformation (FIRMT) [9,289]. In fact,
such transitions were observed in the Heusler-based materials
Ni50Mn36Sn14 and Ni50Mn36In14 by neutron and x-ray diffraction
measurements in magnetic fields [290,291], and are also classified as
metamagnetic transitions [292].
On the other hand, in NieMneGa-based shape-memory alloys
the volume does not change when the structural phase transition
takes place, and the rate of change in the martensite start temperature range with applied field is comparably small (jdTM =dHjz1 to
2 KT1) [293,294]. Therefore, field-induced strains superimpose on
the larger strain, which is caused by twin boundary motion, and
thus, strains associated with any field-induced transformation
become negligible compared to those originating from twinboundary motion. Contrary to this, the structural phase transition is
distinctly affected by an applied magnetic field in NieMneZ (Z ¼ In,
Sn, or Sb) based materials and a unit cell volume change between
the austenite and and the martensite transition of z0.4% [291,295].
Such volume changes can lead to large strains and barocaloric
effects at the FIRMT [295]. The rate of change in the transtion
temperature with the magnetic field can be as high as z10 KT1 in
some of these materials, which can result in magnetic superelasticity (large reversible magnetic field induced strains) [290].
Another interesting feature is the tunability of both critical
temperatures by alloying in the NieMneGa-based system:
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
Changing the relation between TM and TC results in different
properties, which makes these alloys promising for technological
applications. For example, in Ni2þxMn1xGa with 0.18 x 0.20,
a coupling of the magnetic and the structural transition takes place,
because the transition temperatures are close to each other [296].
Consequently, it becomes possible not only to achieve a shape
memory effect by applying an external magnetic field, but also to
induce attractive properties such as the giant magnetocaloric effect,
magnetostriction, and magnetoresistance, which are important
for magnetic refrigeration or magnetostrictive transducers
[282,297e300]. However, for x 0.3 TM is higher than TC, and
therefore, the martensite transition occurs in the paramagnetic
region. Since TM increases dramatically with increasing x, alloys
with a high Ni excess can be used as high temperature shape
memory alloys. Here again, the total electron count is an easy, but
reliable way to qualitatively understand the relation between
composition and transition temperatures. For example, it was
shown, that an increasing number of valence electrons lowers TM
[301]. However, a profound understanding of the physical properties is needed, to design new materials with predictable properties.
For this purpose, first-principles calculations can give an insight
into the complex relation between concentration dependent
properties and the transition temperatures [302].
The ternary phase diagram of the NieMneGa system was
mapped to search for new shape memory alloys and for a systematic relation between TC and TM in a wide range of compositions
[303]. Generally, the transition temperature increases as the
molecular percentage of Ga is decreased, which in turn results in
a non-Heusler composition. A typical composition is Ni43Mn47Ga10
whose martensite transition starts at 400 K making this system
particularly interesting for technological applications. Based on
these findings it can be concluded that a Ga-induced structural
instability in the ferromagnetic/antiferromagnetic transition region of Ni1xMnx is the origin of the martensite transition in the
NieMneGa system.
For a technical application of magnetic shape memory materials,
such as actuators with a long stroke and high precision, NieMneGabased materials are extremely well suitable due to their very high
magnetic field induced strain (up to 10%) and their full shape recovery
over 108 mechanical cycles [304]. For a long time these very large
effects could only be achieved for single crystals. Compared to
monocrystalline NiMnGa, fine-grained NiMnGa is much easier to
process but shows near-zero strains because twin boundary motion is
inhibited by constraints imposed by grain boundaries [305e307].
A new approach to maintaining the ease of processing and reduce the
constrains imposed by grain boundaries by introducing porosity in
NieMneGa [308]. This leads to magnetic field induced strains of
2.0e8.7% being stable for more than 200,000 cycles and which are
larger than those of any polycrystalline, active material.
In addition to the above mentioned giant magnetocaloric effect
in NieMneGa materials, where the structural and magnetic transition temperatures are close to each other, samples close to the
Ni2MnGa stoichiometry show an inverse magnetocaloric effect
[298], in which the adiabatical application of a magnetic field leads
to a cooling of the sample. In the case of Ni2MnGa, the effect,
however, vanishes as the magnetic field increases, and the standard
magnetocaloric effect is observed at high fields. On the other hand,
for the Ni0.50Mn0.50xSnx system with compositions lying in the
narrow range of 0.13 x 0.15 an inverse magnetocaloric effect
was oberseved that is three times larger than in NieMneGa based
compounds [8]. The potential technological application of inverse
magnetocaloric effects are manifold since they open uo the possibility of increasing room-temperature refrigeration efficiency by
using materials showing this effect in combination with a conventional magnetocaloric material.
31
Recently, the observation of a giant barocaloric effect in the
NieMn-In system was reported [309]. This effect is based on the
isothermal entropy change or adiabatic temperature change by
application or withdrawal of external pressure. Today, this effect is
used in most present cooling technologies that are based on the
compression and expansion cycles of gases. The value for the barocaloric effect of 24.4 J kg1 K1 under a hydrostatic pressure of
2.6 kbar at ambient temperatures is 20 times larger than the value
resulting from elastic heating [309] and is comparable to the best
values reported in giant magnetocaloric materials [310,311]. Interestingly, the physical origin of the barocaloric effect found in the
NieMneIn system is the same as that reported for the inverse
magnetocaloric effect, i.e. the large entropy change that accompanies the structural (martensite) transition, which can be induced
either by magnetic field or by hydrostatic pressure, or even
a combination of both (multicaloric effect).
Shape memory materials were expanded to a large number
of compounds, for instance Ni2MnAl, Co2NbSn and Fe2MnGa
[312e314] and also quaternary compounds were investigated in
this context [315,316], e.g. a magnetic-field induced shape recovery
was reported for compressively deformed NiCoMnIn [9]. Stresses of
more than 100 MPa can be generated in this material by application
of a magnetic field. The observed deformation of z3% is fully
recovered to the original shape of the material which is attributed
to a reverse transformation from the antiferromagnetic (or paramagnetic) martensite state to the ferromagnetic parent phase in
Ni45Co5Mn3.67In13.3 single crystals.
10. Superconductors
The family of Heusler compounds includes not only metallic and
semiconducting materials, but also superconducting compounds
(Fig. 40). The first superconducting Heusler materials Pd2RESn and
Pd2REPb (RE ¼ rare earth) were reported by Ishikawa et al. in 1982
[318]. Up to now several new superconductors within the Heusler
family have been reported, their critical temperature, however, being
Fig. 40. Design criteria for superconducting Heusler compounds.
32
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
Fig. 41. Electronic structure of ZrNi2Ga. (a) Displays the band structure and (b) the density of states. The inset in (b) shows the dispersion of the bands that cause the van Hove
singularity at the L point on an enlarged scale. Data taken from Ref. [317].
too low from an applications point of view. Generally, superconductivity is often found in Heusler compounds with 27 valence
electrons. Band structure calculations reveal a common feature in
their electronic structure, i.e. a saddle point at the L point in the
energy dispersion curve at or close to eF, resulting in a high density of
states (DOS). These saddle points are often referred to as van Hove
singularities [319]. According to the BCS theory for superconductivity, an exponential increase of the transition temperature with
increasing DOS is expected, assuming that the Debye frequency and
the Cooper-pairing interaction are independent of the DOS [320]. It
should be mentioned, that this van Hove scenario is also used to
explain the unusually high transition temperatures of the intermetallic A15 superconductors [321]. As an example for all superconducting Heusler compounds with 27 valence electrons, the band
structure of ZrNi2Ga, which exhibits a van Hove singularity at the L
point just above eF, is displayed in Fig. 41 [317].
In fact, among superconducting Heusler materials based on Pd2,
Au2 or Ni2, Pd2YSn is up to now the one with the highest transition
temperature of 4.9 K [322]. On the other side, for RE containing
compounds, the coexistence of superconductivity and a magnetically ordered state has been reported, e.g. Pd2YbSn shows a superconducting transition at TC ¼ 2.46 K and a magnetic transition at
TM ¼ 0.23 K that does not destroy the superconducting state [323].
Similar results were observed for Pd2ErSn with TC ¼ 1.17 K and
TM ¼ 1.00 K [324]. As shown for the system Pd2Er(1x)Yx Sn, the
substitution of one RE metal for a different RE element leads to
a linear variation of TC between the transition temperatures of the
ternary compounds [324]. In contrast, a maximum in TC was found
for the series Au(2x)PdxYIn for x ¼ 0.7 [322]. Additionally, the
successful prediction of new superconductors according to the
above explained van Hove scenario was reported, both experimentally and theoretically by Winterlik et al. [317,325]. In these
studies Pd2YZ with Y ¼ Zr, Hf and Z ¼ Al, In exhibit critical
temperatures in the range of 2.4e3.8 K. Doping experiments were
carried out in order to study the influence of TC on the energy of of
the van Hove singularities. Assuming a rigid-band model and
a fixed lattice parameter, either electron or hole doping should shift
the van Hove singularity onto the Fermi energy level, and thus lead
to a rise in TC due to an enhanced DOS at eF. Unfortunately, doping
led to a distinct amount of disorder resulting in spacial fluctuations
of the superconducting gap and a suppression of the superconducting state. Additionally, superconductivity was also found in
systems based on Ni2, with Ni2NbSn showing the highest TC of 3.4 K
[322], although nickel-containing Heusler compounds with a high
nickel concentration are intuitively expected to show magnetic
order rather than superconductivity. Contrary to the above
mentioned Pd2-based systems, the Ni2-based superconductors do
not show any indications for magnetic order.
Within the class of Half-Heusler compounds, no superconductor
is known, since they are non-centrosymmetric materials. The only
exception is LaPtBi with a critical transition temperature of 0.9 K
[326]. For such a semimetal with a very low carrier density,
superconductivity was not expected and is currently discussed in
the context of topological insulators. However, a clear understanding of the origin of superconductivity, magnetism, and their
coexistence in Heusler compounds is still missing.
11. Thermoelectric materials
In recent years, Heusler compounds have attracted great scientific
interest due to their possible application in the field of thermoelectrics (Fig. 42). As explained in Section 6.2.1, Half-Heusler materials
with 18 valence electrons exhibit semiconducting properties. Band
structure calculations revealed narrow bands, leading to a high
effective mass and a large thermopower [327]. A great advantage of
Heusler compounds is the possibility to dope each of the three
occupied fcc sublattices individually in order to optimize the thermoelectric properties. For example, it is possible to alter the number
of charge carriers by doping on the Z position, and simultaneously
introduce disorder by doping on the X and Y position, resulting in
mass fluctuations, which can decrease the thermal conductivity k.
The most attractive properties of Half-Heusler materials for thermoelectrics are their high Seebeck coefficient S up to z300 mV K1 at
room temperature and their high electrical conductivity (z1000 to
10000 Scm1) [327e331]. The only drawback is the relatively high
thermal conductivity, which can be as high as 10 Wm1 K1.
Many different Half-Heusler compounds were investigated in
the past with regard to improve their thermoelectric properties
[327,329,332e335]. Fig. 43 provides an overview on the most
promising materials: n-type TiNiSn-based compounds have been
most intensively investigated, but recently great progress was also
made for p-type TiCoSb-based materials that posses a very high
potential for a significant increase in ZT due to their still high k.
Table 6 provides, additionally to the ZT value, the corresponding k,
which is inversely proportional to ZT. It shows that the p-type
materials exhibit 2e3 times larger a thermal conductivity
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
Fig. 42. Design criteria for thermoelectric materials.
compared to n-type materials. A reduction of k in ZrCoSb0.9Sn0.1 by
a factor of three without changing the other properties would
result in a maximum ZT of 1.2. Therefore, various synthetic methods
were applied to reduce the thermal conductivity of this material,
Fig. 43. State of the art in thermoelectric efficiency of Half-Heusler compounds.
33
however, the introduction of nanostructures into the material being
one of the most promising methods. Indeed, this task can be
addressed by a melt spinning process or by ball milling followed by
spark plasma sintering. An increasing ZT value was recently
reported for TiNiSn which was treated by high-energy ball milling
followed by spark plasma sintering and annealing at elevated
temperatures by Gelbstein et al. [336]. The authors attributed this
enhancement of the thermoelectric performance to the grain
structure leading to a reduced thermal conductivity and the
reduction of secondary phases resulting in improved electronic
properties.
Furthermore, it is possible to introduce electrically non-active
nanocomposites, for instance TiO2 into the material to create
additional phonon scattering centers [337]. Recently, a different
approach to optimize the thermoelectric properties was applied to
the Ti(1x)MnxCoSb system [338,339]. In the present case, the
system undergoes a phase separation during the solidification
process from the melt into a TiCoSn and a MnCoSb phase. Thus,
a distinct microstructure leads to significant reduction of the
thermal conductivity due to boundary scattering mechanisms as
shown in Fig. 44. However, these phase separation mechanisms
require further investigation in future by physical metallurgy
methods since the controlled introduction of phase boundaries into
a material will be of great benefit for the design of new thermoelectric compounds with enhanced properties.
Apart from the processes described above, the preparation of
thin films and multilayers represents a further approach to optimize
the thermoelectric properties of Half-Heusler compounds. Jäger
et al. reported the epitaxial growth of TiNiSn and Hf0.5Zr0.5NiSn thin
films [340]. The authors found that a distinct dependence of the
thermoelectric properties on the epitaxial quality of the films. The
successful preparation of multilayers containing both, TiNiSn and
Hf0.5Zr0.5NiSn is a promising step towards thin film thermoelectric
devices since a considerable reduction of the cross-plane thermal
conductivity is expected due to interface effects.
Despite the high power factor of up to 70 mWcm1 K2 at 650 K
for Sb-doped TiNiSn, a ZT value of only 0.45 at 650 K was reached
due to the relatively high thermal conductivity [334]. According to
the theory of Callaway et al., isoelectronic alloying enhances the
phonon scattering by point defects due to mass differences (mass
fluctuations) and size differences (strain flied impurities) between
the impurity atoms and the host atoms, without introducing charge
disorder [345]. Thus, alloying on the Ti lattice site resulted in
a reduction of the thermal conductivity to 3.6e4.9 Wm1 K1 at
room temperature for the composition XX0 NiSn (X, X0 ¼ Ti, Zr, Hf)
[13,346]. Sb turned out to be an efficient dopant on the Sn sites
resulting in an electrical resistivity and thermal conductivity for
Zr0.5Hf0.5NiSn0.9Sb0.01 at room temperature of 0.8 mUcm and
6.6 Wm1 K1, respectively [327]. In fact, an increasing Sb content
shifts the maximum of the Seebeck coefficient to higher temperatures, and therefore enhances the power factor. A further
improvement was made by partial substitution of Ni for Pd, resulting
in the n-type material Zr0.5Hf0.5Ni0.8Pd0.2Sn0.99Sb0.01 with a ZT value
of z0.7 at 800 K [347]. Apart from that, the maximum ZT value of 1.4
at 700 K was reported for the compound (Zr0.5Hf0.5)0.5Ti0.5NiSn0.998Sb0.002 [4]. A further important issue to enhance the thermoelectric performance of Half-Heusler compounds is the
preparation of single crystals. Indeed, Zr0.5Hf0.5NiSn single crystals
show a high intrinsic ZT value of 0.4 at 350 K [348].
In addition to TiNiSn-based materials, systems based on TiCoSb
are promising candidates, due to large S values and relatively large
theoretical band gaps of 0.95 eV, which is larger than those of most
other Half-Heusler compounds [349]. This is an advantage for the
optimization of the power factor by electron or hole doping.
Generally, an increase in the electrical conductivity simultaneously
34
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
Table 6
Overview of state of the art half-Heusler compounds for thermoelectric applications. Given are the thermal conductivity k and the maximum figure of merit ZT.
M NiSn (n-type)
M CoSb (p-type)
Composition
k [Wm1 K1]
ZT
Hf0.5Zr0.5NiSn
Ti0.5Zr0.25Hf0.25NiSn0.998Sb0.002 3
Zr0.25Hf0.75NiSn0.975Sb0.025 6-7
Ti0.3Zr0.35Hf0.35NiSn
3
1.4 (700 K) [4]
0.8 (1073 K) [342]
e
0.9 (960 K) [341]
1.2 (696 K) [4]
ZrCoSb0.9Sn0.1
Zr0.5Hf0.5CoSb0.8Sn0.2
TiFe0.15Co0.85Sb
7-10
3.6e4.1
5.7
0.45 (958 K) [343]
0.51 (1000 K) [344]
0.45 (800 K) [347]
causes a decrease in S, but for TiCoSb the relative large band gap
creates space for an improvement of the power factor with
a retained high S value. In addition to that, a maximum ZT of 0.51
was reached for Ni-doped Ti0.5Zr0.25Hf0.25CoSb, which is mainly
attributed to the low thermal conductivity of 2.29 W/mK at 900 K
[329]. In fact, the highest ZT value for an n-type TiCoSb-based
material, reported so far, is 0.7 at 900 K for Ti0.6Hf0.4Co0.87Ni0.13Sb
with a power factor of 23.4 mWcm1 K2 [335]. Recently, a transition
from a n-type to a p-type material was observed for Ti(1x)MxNiSn
(M ¼ Sc, V) which opens the door to produce thermocouples based
on the same material which is very favorable, since, for example,
differences in thermal expansion can be neglected [350].
Among half-metallic Heusler compounds, the family of Co2TiZ
(Z ¼ Al, Si, Ge, Sn) shows unusual transport properties [63,351]. For
instance, the Seebeck coefficient remains constant over a wide
temperature range above the respective Curie temperature in
these materials making them promising candidates for an application in thermocouples due to the linear dependency of the
thermovoltage on temperature. Additionally, the working range of
these materials can be tuned by changing the valence electron
number [352]. The Co2TiZ system exhibits high Seebeck coefficients in a metallic system and thus, is regarded as a potential
material for the combination of half-metallic ferromagnetism and
thermoelectric effect in the new research field of spincalorics
[107]. Besides these half-metallic ferromagnets, the semiconductor
Fe2VAl has been thoroughly studied [16,60,61]. In this case, the
presence of a pseudogap due to hybridization effects was, in fact,
predicted by several theoretical studies [353e356]. The pseudogap
formation arises from an indirect band overlap at the Fermi energy
eF, and thus it can be viewed as a semimetal. An experimental
verification of this assumption was achieved by optical conductivity measurements [357], however, NMR investigations revealed
a small Fermi-level DOS within the pseudogap, which is consistent
with the material being a semimetal [358]. In comparison with
band structure calculations, low-temperature specific heat
measurements yielded a huge enhancement of the effective mass
[359]. Generally, semimetals with heavy band mass are expected
to show large Seebeck coefficients. Indeed, stoichiometric Fe2VAl
has a positive Seebeck coefficient of z25 mV K1 at room
temperature indicating a hole-type carrier dominated heat transport. On the other hand, for non-stoichiometric samples
Fe(2x)V(1þx)Al, the an enhancement in S accompanied by a sign
change is observed [360]. Such an effect can be attributed to
a rigid-bandlike shift of eF from the central region of the pseudogap and a modification of electron and hole pockets near the
band edges. Similarly, a substitution of Ge for Al causes significant
changes in the low-temperature resistivity and an enhancement in
S, reaching 130 mV K1 for 5% Ge [361]. Recently, the fabrication
of Fe2VAl thin films with high Seebeck values and low thermal
conductivity, caused by the grain structure of the films, was
reported allowing for the application in thin film thermoelectric
devices [362].
12. Kondo systems and heavy-Fermion behavior
The field of quantum criticality is of extensive current interest in
condensed matter physics, because a rich variety of phenomena is
observed including the coexistence of unconventional superconductivity and magnetism, hidden order, and non-Fermi-liquid
behavior. In particular, the discovery of the heavy-Fermion
Fig. 44. Ti(1x)MnxCoSb undergoes a phase separation into TiCoSb and MnCoSb. (a) Mircostructure of Ti0.5Mn0.5TiSb detected by EDX; dark areas consist of TiCoSb, while bright areas
comprise MnCoSb. (b) The thermal conductivity of the phase-separated material is reduced by factor three compared to single phase TiCoSb. Data taken from Ref [339].
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
Fig. 45. Design criteria for topological insulators.
compound YbPtBi in 1991 by Canfield et al. [363,364] with a huge
specific heat coefficient g has motivated the search for new enormous heavy-Fermion systems and anomalous metallic materials
[365,366]. One central question in these Yb-based compounds is
whether the large g can be associated with the existence of very
heavy renormalized quasi-particles. In fact, there are different
mechanisms that may lead to a large specific heat coefficient g. One
aspect, which may significantly influence the value of g is the
stability of different electronic configurations, i.e. Yb3þ or Yb2þ.
Indeed, the exceptionally large specific heat coefficient of
g ¼ 8 Jmol1 K2 in YbPtBi attracted considerable attention among
the scientific community. This value is one order of magnitude
larger compared to typical heavy-Fermion compounds [367], and
Fig. 46. Schematic illustration of the band structure for a trivial semiconductor (CdTe)
and a topological insulator (HgTe).
35
three orders of magnitude larger than that of conventional metals,
corresponding to a small Kondo temperature TK of z1 K. Interestingly, YbPtBi undergoes a magnetic order transition at TC ¼ 0.4 K
with a very small ordered moment of 0.1 mB/Yb ion, which was
detected by muon-spin-rotation measurements. Furthermore,
YbPtBi exhibits an anisotropic resistivity below TC. This indicates,
that a spin density wave transition occurs at TC, and this transition
partially gaps the Fermi surface [364]. Additionally, YbPtBi seems to
be a low carrier concentration metal. The massive electronic state in
YbPtBi can be associated with these factors. Generally, the Yb ions
tend to be in the magnetic Yb3þ state, but in fact, the possibility of
the Yb ion to fluctuate between non-magnetic Yb2þ (J ¼ 0) and
magnetic Yb3þ (J ¼ 7/2) states is a key issue in these Yb-based
systems. Since the magnetic volume of the magnetic Yb3þ state is
smaller than that of non-magnetic Yb2þ, the substitution effect of
Sb for Bi on the 4f electronic state is expected to be substantial.
Recently, ultrasonic measurements provided evidence for strong
renormalization of quasi-particles in YbPtSb, probably due to nonFermi liquid characteristics formed close to the quantum critical
point [368].
Fe2VAl and related materials have attracted considerable
attention, since it was claimed, that these 3d-electron systems with
small carrier concentration exhibit a significant carrier mass
enhancement and non-Fermi liquid behavior [16]. Stoichiometric
Fe2VAl is non-magnetic and related alloys, such as Fe2þxV1xAl and
Fe2VAl1d exhibit ferromagnetic transitions [369e371]. Consequently, the samples with x z 1 and d z 0 are located at the brink
of ferromagnetic order, close to the ferromagnetic quantum critical
point [16,359]. Stimulated by these results, theoretical investigations claimed Fe2VAl to be a non-magnetic semimetal with
a pseudogap at eF [354]. Indeed, experimental evidence was found
in electrical resistivity data of stoichiometric Fe2VAl, which exhibits
a negative temperature coefficient up to 1300 K and a constant
value of the Hall coefficient at low temperatures [16,369,372]. In
this case, the considerable mass enhancement of conduction
carriers was attributed to excitonic correlations or spin fluctuations
[353,355]. However, subsequent specific heat, C (T) measurements
in applied magnetic fields showed that the upturn in C/T at low
temperatures is caused by an Schottky contribution of magnetic
clusters in Fe2VAl [359]. Another interesting aspect of Fe2VAl is its
potential for an application in thermoelectric devices, as discussed
in more detail in Section 11. Additionally, Kondo-lattice behavior
was reported for Fe2TiSn [373,374].
Another example for the features explained above is provided
by UPd2Sb, antiferromagnet with TN ¼ 55 K. This material shows
some characteristic features for strongly correlated electron
systems, such as a reduced effective magnetic moment, a large
negative paramagnetic Curie temperature, a negative magnetoresistance, logarithmic decrease in the electrical resistivity, and an
enhanced low-temperature electronic specific heat [375]. The
Seebeck coefficient and the field dependency of the magnetoresistance show a variation, which is typical for systems with strong
hybridization of 5f electrons with the conduction band. These
properties are consistent with a Kondo behavior and therefore,
UPd2Sb can be classified as a low effective-mass heavy-Fermion
metal. Moreover, heavy-Fermion behavior was also observed for
UPd2Sn and UPd2Sb [376,377].
Investigations of the Ag(2x)CuxCeIn series revealed that the
hybridization between local states and the conduction band
continuously increases from Ag2CeIn with an antiferromagnetic
Kondo lattice to Cu2CeIn, which is a heavy-Fermion system [378].
As a function of the Cu concentration, the paramagnetic Curie
temperature turns more and more negative reflecting the increase
in the Kondo temperature TK when going from Ag2CeIn (TK ¼ 2 K) to
Cu2CeIn (TK ¼ 6 K). In fact, among all Ce-based heavy-Fermion
36
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
Fig. 47. Band structures of CdTe and HgTe compared with ScPtSb and ScPtBi. Red color marks the bands with G8 symmetry, blue with G6. Comparison reveals obvious similarity
between binary systems and their ternary equivalents: both CdTe and ScPtSb are trivial semiconductors with G6 situated above G8, which is located at the Fermi energy (set to zero).
Both HgTe and ScPtBi are topological with inverted band order; the band with G6 symmetry is situated below G8. Data taken from Ref. [5]. (For interpretation of the references to
colour in this figure legend, the reader is referred to the web version of this article).
systems, Cu2CeIn is one of the heaviest and no magnetic order was
observed down to 0.5 K.
13. Topological insulators
In 2006, Zhang and coworkers theoretically predicted [379] and
later Molenkamp and coworkers experimentally verified [380]
a quantum spin Hall state in quantum wells of HgTe/CdTe. This
new state of matter in “topological insulators” has started immense
research activities in fundamental condensed matter physics and
material science (Fig. 45). To design a topological insulator a direct
band gap at the center of the Brillouin zone, the G point is favorable.
It is worth to mention that the name topological insulators is
slightly misleading, the systems are, in fact, low band gap
Fig. 48. (a) Photograph of an arcmelter equipped with a water-cooled copper crucible plate and a tungsten electrode, (b) polycrystalline Heusler ingot as it is obtained after
arcmelting, and (c) Co2FeSi single crystal prepared by the floating zone method.
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
37
experimental realization of many new quantum phenomena, such
as the quantized anomalous Hall effect or topological superconductivity. They also open new research directions towards multifunctional topological devices for spintronics and fault-tolerant
quantum computing (Fig. 48).
14. Synthesis
Fig. 49. TEM image of Co2FeGa nanoparticles; (a) displays a particle with a size of
18 nm, (b) shows a part of the nanoparticle (see square in (a)) on an enlarged scale, and
(c) is the Fourier transform of the image. Data taken from Ref. [381].
semiconductors such as Bi2Te3 (300 meV) or even zero band gap
semimetals such as HgTe. HeTe is a zero band gap semiconductor
due to partially degenerated p states at eF. This degeneracy can be
lifted by strain application. Recently, it was demonstrated that
around 50 Heusler compounds show a band inversion, similar to
that reported for HgTe (compare Fig. 46) [5,6]. The topological state
in these zero-gap semiconductors can be created by applying strain
or by designing an appropriate quantum well structure. Many of
these ternary zero-gap semiconductors contain a rare earth
element with strongly correlated f electrons, yielding additional
properties ranging from superconductivity (e.g. LaPtBi [326]) to
magnetism (e.g. GdPtBi [363]) and heavy-Fermion behavior (e.g.
YbPtBi [364]).
In particular, the compounds YPtSb, YPdBi, and ScAuPb are close
to the border between being trivial or topological insulators are
exceptionally interesting because a quantum phase transition can be
induced by changing the lattice constant or the composition slightly.
Fig. 47 compares the band structures of the topological system
HgTe to the Half-Heusler compound ScPtBi. These band structures
reveal clear fingerprints: the band with G6 symmetry (blue) is situated below G8 (red) that is located at the Fermi level. This means
that the parity changes compared to a trivial semiconductor, such
as ScPtSb, which is the necessary condition for the topological state
(for more details see Refs. [5,6]).
These results show that Half-Heusler compounds are a highly
tunable and flexible class of materials that may allow the
The most common method to synthesize bulk Heusler
compounds is represented by arcmelting stoichiometric amounts of
high purity elements. The obtained ingots need to be turned over
and be remelted several times to ensure a homogeneous element
distribution throughout the sample. Special care has to be taken to
avoid oxygen contamination, in particular in case of materials
containing elements with a high oxygen affinity, such as manganese. Hence, a vacuum level of at least 104 mbar in combination
with high purity argon (5.0) is required, whereby the use of oxygen
getter material, e.g. Ta or Ti, which is melted prior to the actual
samples can further improve the sample quality. Apart from that,
the weight loss during the melting process has to be monitored,
since some elements, such as Sb, Mn, and Bi tend to evaporated
during the melting process, which results in non-stoichiometric
samples. In these cases, it is either possible to adjust the starting
composition appropriately or to chose a different synthetic method,
for instance the preparation in a closed crucible in an induction
furnace. The phase purity as well as the crystal structure can be
improved by subsequent annealing of the as-cast sample in a sealed
silica tube. To choose the corresponding annealing temperature,
differential scanning calorimetry measurements need to be carry
out. In general, high annealing temperatures are preferred since the
diffusion velocity is increased and impurities vanish more easily.
However, some Heusler compounds undergo structural phase
transitions at elevated temperatures which need to be taken into
account before starting the annealing process. Depending on the
individual material, quenching the sample into ice water may lead
to the desired structure. Finally, the samples can be crushed in
a mortar, cut into disks or sticks, and can be polished determined by
the various needs for the desired measurements.
The preparation of high quality Heusler thin films has constantly
improved during the last years. For this purpose, one central issue is
the use of an ultra-high vacuum chamber to avoid oxygen
contamination of the films. Up to now, mainly Co-based Heusler
thin films were grown for the application in spintronic devices. In
this case, MgO (100) single crystal substrates turned out to be best
applicable due to a small lattice mismatch leading to an epitaxial
growth of the Heusler thin film in (100) direction, rotated by 45
with respect to the MgO unit cell. To generate a clean surface, it is
possible to deposit a thin layer of fresh MgO on the substrate.
Moreover, the application of Cr-buffer layers led to outstanding
results. Similar to bulk materials, the crystal structure of the thin
films can be distinctly improved by annealing. However, in case of
a multilayer device the interlayer diffusion needs to be minimized.
Therefore, subsequent annealing steps after depositing each layer
are performed, whereby the highest temperature is applied to the
Heusler bottom layer. Another crucial point is the engineering of
the interfaces, since here the spin polarization is reduced in many
cases due to defects or termination effects.
To investigate the physical properties of Heusler compounds
thoroughly, it is essential to prepare high quality single crystals,
which is possible by the Czochralski-technique or the floating-zone
method. In general, the latter involves a number of distinct
advantages: (i) the exact temperature control, (ii) the preparation
of crystals from incongruently melting starting materials, and (iii)
the better tightness of the apparatus leading to a reduced oxygen
contamination compared to the Czochralski-technique. It should be
38
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
Fig. 50. The family tree of cubic Heusler compounds, starting from the diamond structure in comparison with its hexagonal analogues.
mentioned that the use of high purity elements is a prerequisite for
the successful preparation of Heusler single crystals, impurities,
however, are often found at grain boundaries.
15. Heusler goes nano
There is no doubt today that the evolution of nanotechnology
has had an enormous impact on many different scientific areas. The
Fig. 51. Design criteria for Heusler compounds with a tetragonal distortion.
reason for this circumstance is the fact that nanocrystals materials
exhibit physical properties that are quite different from their bulk
counterparts. Finite size effects, which originate from the quantum
confinement inside nano-sized crystallites, lead to the evolution of
novel magnetic phenomena that can be exploited in a vast variety
of different applications. Especially, magnetic nanoparticles have
gained enormous interest for applications in various fields such as
data storage devices, catalysis, drug delivery, and biomedical
imaging [382e388]. As an example, nanocrystals of ferromagnetic
compounds, such as Fe3O4, g-Fe2O3, or FePt have been studied
intensively during the past decades since they behave like paramagnets above the magnetic blocking temperature TB. In general,
the magnetization of the nanoparticulate probe decreases, if the
particle size is reduced, owing to a reduction of the corresponding
domain size and, therefore, the number of magnetic spins. On the
other hand, nanoparticles of antiferromagnetic materials, including
MnO, NiO and FeO, show an increased magnetic moment, if the
particle size is decreased [389e392]. This behavior is often
explained with the presence of uncompensated magnetic spins on
the particle surface, leading to a measurable magnetization of the
nanoparticles in an external magnetic field. Since a reduced particle
size corresponds to a larger surface-to-volume ratio, more
uncompensated spins exist on smaller particles, therefore resulting
in enhanced magnetic moments.
A common method to fabricate nanoparticles from a bulk
material is the ball-milling technique [393e398]. For instance,
ferromagnetic Ni2MnGa nanoparticles were prepared using the
ball-milling method combined with a post-annealing process
[395]. These investigations revealed that a phase transition from
the tetragonal to a cubic disordered phase occurs during the ball
milling procedure. An intermediate phase, which controls the
transformation kinetics, was detected and distinct differences
between the behavior of coarse-grained and nano-sized material
were observed.
Only recently, ternary Heusler nanoparticles were successfully
synthesized from precursors and their magnetic and structural
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
39
Fig. 52. Crystal field splitting for a d4 ion in an octahedral coordination sphere: (a) non distorted octahedron, (b) elongated octahedron, (c) compressed octahedron. The distortion in
(b) and (c) is also known as JahneTeller distortion.
properties were investigated [381,399]. Co2FeGa nanoparticles
were found to exhibit the ordered L21 structure with a slightly
reduced lattice parameter compared to the bulk material. The
crystal structure was further studied by high-resolution transmission electron microscopy (HR-TEM), a typical HR-TEM image is
shown in Fig. 49. The complete Co2FeGa nanoparticle as shown in
(a) is nearly spherical and has a diameter of d¼18 nm. A part of the
particle on an enlarged scale is shown in (b) to visualize the lattice
planes. The lattice distance d220 of the 220-planes of about 0.2 nm
in agreement with the expected value. Additionally, Fig. 49(c)
displays the Fourier transformed image, revealing a sixfold
symmetry. It corresponds to six (110) planes perpendicular to the
(111)-like direction of the cubic lattice. Magnetic measurements
revealed that the particles are soft magnetic with a Curie
Fig. 53. Relation between the cubic Heusler cell and a tetragonally distorted cell for Mn2YZ. (a) Transition of the cubic Heusler cell to a tetragonal distorted cell with elongation
along the one axis, (b) the unit cell edges of the tetragonal unit cell are marked within the cubic cell, (c) top view of the 45 rotation between the cubic and the tetragonal unit cell,
(d) tetragonal unit cell with space group I4/mmm.
40
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
Fig. 54. Theoretical XRD patterns for Mn2FeGa under the assumption of different
degrees of distortion. Please note, that the c/a ratio of 1.41 corresponds to a cubic
structure. Therefore, the indices of the reflections are different in this case.
temperature far above room temperature. The saturation magnetization at low temperatures is similar to the bulk value, which
indicates, that the half-metallic properties are preserved in the
nanostructured material.
16. Heusler compounds in industrial applications
16.1. Heusler compounds for spintronics
Spintronics is one of the emerging disciplines that continue to
revolutionize the thriving field of information technology. Its
commercial impact to date has been mainly in the area of spin
valves used in hard drive disks. Although spintronic devices today
are small and the data storage density is quite high, the ever-
increasing demand for ultrafast, high density data storage and
processing devices requires the development of new concepts to
commercialize nanoscale devices. In the past 50 years, the only way
to switch or excite magnetic moments was the use of a magnetic
field, but magnetic fields are extremely detrimental from a device
perspective. The problem is that as devices shrink in size, larger and
larger magnetic anisotropies are necessary to prevent them from
being disturbed by thermal fluctuations as the superparamagnetic
limit is approached, which means larger magnetic fields are also
necessary to write and switch them. A novel method of switching
magnetic tunnel junctions is the exploration of spin-transfer torque
effects, which enables the scaling of magnetic random access
memory (MRAM). These phenomena will likely have a technological impact in near future. However, developments in the field of
spintronics continue to be strongly dependent on the exploration
and discovery of novel material systems. In recent years, many
business companies took notice of the outstanding research results
and the vast tunability of Heusler materials. Therefore, more and
more companies jump into the field of Heusler compounds and
develop new products. The growing number of patents issued on
Heusler-based discoveries reflects the impact of Heusler
compounds for industrial research and product development.
Today, Heusler compounds are mainly investigated in the
context of magnetic recording [216,258,400e402]. Read heads
utilizing the current-in-plane (cip) geometry have recently been
replaced by heads based on the current-perpendicular-to-plane
(cpp) tunneling MR effect [403]. The higher signal to noise ratio
Fig. 55. Comparison between the regular and the inverse Heusler structures and the corresponding tetragonally distorted unit cells.
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
41
Fig. 56. (a) By combination of two unit cells with CuAu-type structure (L10), the tetragonally distorted Heusler structure is obtained. (b) Similarly, the disordered variant of the cubic
Heusler structure can be formed by combining eight CsCl-type (B2) unit cells.
provided by TMR sensors was a main reason for this transition. As
the recording density continues to increase and sensor dimensions
shrink accordingly, cpp-GMR emerges as a candidate for next
generation read heads since the lower resistivity of these metallic
devices enables higher data rates. However, for cpp-GMR sensors
based on conventional 3d metals, the low amplitude remains
a major drawback for practical applications. High bias currents are
necessary to achieve sufficient output which raises reliability
concerns due to the onset of current-driven instabilities and electromigration. Therefore, it is critical to increase the MR ratio to
achieve higher output at moderate current densities. Toshiba has
designed an all Heusler cpp-GMR device with Co2MnGe electrodes
and a non-magnetic Ru2CuSn spacer layer which yielded a MR ratio
of 6.7% for a bottom spin valve configuration [216]. A new design
scheme for cpp-GMR junction was invented recently; half-metallic
ferromagnets can be combined with non-magnetic, semiconducting quaternary Heusler materials that can be derived from
the half-metal by exchanging only one element, for instance
Co2MnAl and CoMnVAl [404]. Thus it becomes possible to engineer
the interfaces and create non-destructive interfaces which preserve
the half-metallicity. Hitachi has developed cpp-GMR devices based
on Heusler alloys which exploit their high spin polarization but
produce minimal current-induced noise without loss of magnetoresistance of sensor resolution [401]. Furthermore, they applied
band structure calculations to study the influence of impurities and
distortions on Co2MnGe, yielding that the spin polarization is
retained even under significant strains and distortions, whereas
impurity concentrations as low as 3% affect the spin polarization
destinctly [405]. Seagate technology invented a memory cell based
on spin tranfer torque effects (ST-RAM) which incorporates
magnetic Heusler layers [406]. TDK designed a multilayer device
with perpendicular magnetic anisotropy incorporating Heusler
materials with high spin ploarization and low magnetic damping
[407]. Spin-stand testing of narrow-track recording heads
confirmed compatibility of these materials with the hard disk drive
reader technology [258].
Apart from that, Heusler compounds were intensively studied by
Toshiba to develop new metal to semiconductor spin injection
devices [271,408,409]. For a stoichiometry optimized Co2MnGa thin
film grown on GaAs (100), a spin polarization transfer of z6.4% at
5 K was observed in the current of a GaAs peien diode even with
compositinal disorder at the interface [409]. Furthermore, an
anisotropic magnetoresistance (AMR) effect of 1% at 300 K and its
weak magnetic anisotropy make this material combination promising candidates for convential magnetic sensors working at RT
[410].
16.2. Heusler compounds for thermoelectrics
Apart from spintronics, companies are very interested in the
incorporation of Half-Heusler material in thermoelectric elements.
Especially materials based on M NiSn are among the most promising candidates for n-type thermoelectric applications, since they
consist of non-toxic elements, are easy to produce and process
42
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
Fig. 57. Starting from the Cu2MnAl-type structure, the layered structures of PbFCl (LiFeAs) and MnP (FeSe) can be derived by removing atoms systematically. Adopted from
Ref. [433].
(compare Section 11 for further details). Based on first-principles
calculations, Toyota successfully designed a new Y-Sb co-doped
Ti0.5Zr0.25Hf0.25NiSn material, which reached a ZT value of 0.96 at
773 K [411]. Recently, they patented a TiNiSn-based material, with
various possible substitutions on all three lattice positions, as
well as a manufacturing method [412]. Additionally, it was
observed, that ZrNiSn can be converted from n-type to a p-type
material by addition of Co and Ir which simultaneously reduces
the thermal conductivity by the solid solution effect due to
vacancy site occupation of Co and Ir [413]. This conversion is
particularly important for the fabrication of thermoelectric
converters, which combine n-and p-type materials, since for
instance thermal expansion effects can be minimized by the
utilization of the same parent material for both components. The
influence of different synthetic methods, i.e. arcmelting,
mechanical alloying with subsequent spark plasma sintering was
investigated and revealed that the latter procedure leads to
a bulk material with a distinct grain structure which improves
the thermal properties of the Half-Heusler thermoelectric materials [414,415].
tree of Heusler compounds starting from the diamond lattice.
Carbon in the diamond modification crystallizes in an fcc lattice, in
which half of the tetrahedral holes are periodically populated. The
structure can be converted into the binary zinc blende structure:
The S-anions (green) form the fcc lattice, while the smaller Zncations (red spheres) are located on the tetrahedral interstitials.
Based on zinc blende, plenty of different crystal structures can be
derived, depending on the location of additional atoms. The
17. Heusler compounds and related structures
The Heusler compounds are closely related to many different
crystal structures. Fig. 50 displays, on the left hand side, the family
Fig. 58. The crystal structure of Heusler compounds and perovskites are compared.
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
43
Table 7
Comparison of characteristic properties of classical semiconductors, Heusler compounds and perovskites.
Semiconductors
Heusler compounds
Perovskites
Covalent
No
No
Quantum well
MBE
Covalent & ionic
Yes
Yes (Mn, RE)
PbOCl (Cu2Sb)
Sputtering
Ionic
Yes
Yes
Ruddlesden Popper
PLD
Bonding
Phase transitions
Correlations
From 2D to 3D
Deposition technique
population of the remaining void tetrahedral sites leads to the antifluorite structure, assuming the yellow and red spheres denote
identical atoms. This is favored, if the difference in electronegativity
between the elements is big and the bonding interaction has
a strong ionic character, for instance in Mg2Si. Filling the octahedral
sites only, results in filled tetrahedral structure structure, which is
also known as the “Half-Heusler” structure (compare Section 3). In
a ternary compound, the most electropositive atoms usually occupy
the octahedral position, while the most electronegative atom
determines the fcc lattice. Thus, the elements with the highest
difference in electronegativity form an energetically favored ionic
rock salt sublattice. Filling both, the remaining empty tetrahedral
and the octahedral positions of zinc blende, leads to the Heuslercommon Cu2MnAl-type lattice.
17.1. Hexagonal analogues of Heusler compounds
On the right hand side of Fig. 50, the hexagonal analogues of the
above-described cubic structures are displayed. Starting from the
hexagonal modification of diamond, the wurtzite structure can be
deduced by placing the S-anions on the hcp lattice and filling one
half of the tetrahedral holes periodically with the Zn-cations. Thus,
wurtzite represents the hexagonal analogue of zinc bende. In the
hexagonal lattice, however, no structure with fully occupied
tetrahedral sites is known. Additional atoms may be placed on the
octahedral holes, resulting in the LiGaGe-type structure. This
structure type is frequently observed, if rare earth elements are
contained in Half-Heusler compounds, as discussed in detail in
Section 17.1. Please note, that slight variations of the z coordinate of
Fig. 59. Overview on the different aspects of Heusler compounds discussed in this review article.
44
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
Fig. 60. Bonding interaction in Heusler materials: depending on the composition and
stoichiometry, their characteristics are dominated by covalent and ionic interaction,
covalent and metallic or they are even found to incorporate a mixture of all three
bonding natures.
the Y and Z atom may occur which introduces an additional degree
of freedom into these hexagonal systems, i.e. the degree of puckering of the YZ-layers. However, filling all interstitial lattice sites
completely, as in case of the Cu2MnAl-type structure, is not possible
in the hexagonal crystal system.
Hexagonal XYZ compounds, with X being a rare earth metal (RE),
evoked considerable interest in the last 20 years due to their
exceptional physical properties. EuPtP, for instance, shows fluctuations of the mean europium valency which can be correlated to
two first-order phase transitions [416]. Similarly, a mixed-valent
state is reported for EuNiP that undergoes a Verwey-type charge
delocalization [417]. In these compounds, two inequivalent Eu
atoms reside in planes that are well separated by layers of Pt or Ni
and P. On the other side, CeAuGe undergoes a transition from
a paramagnetic state to long range ferromagnetic order below
a Curie temperature of 10.0(2) K [418]. Investigations of the
magnetic properties of the RE PdSb system (RE ¼ La to TM) show,
that CePdSn orders ferromagnetically with on ordering temperature of 17 K, whereas the compounds with RE ¼ Nd, Sm, Eu, and Ga
order antiferromagnetically with Néel temperatures between 11
and 17 K, while all remaining compounds are paramagnetic [419].
Apart from that, the properties of RE rhodium stannides RE RhSn
are characterized by the hybridization between the rare earth 4f
electron states and the conduction electron states. Especially,
compounds with RE ¼ Ce or Yb are known to be strongly correlated
electron systems [420,421]. In YbRhSn the competition between
the
intrasite
Kondo
effect
and
intersite
RudermaneKitteleKasuyaeYoshida interactions leads to the formation of
a magnetically ordered heavy-Fermion ground state [421]. In this
case, the magnetic properties show strong single-ion anisotropy
induced by crystalline-electric field effects, which compete with
frustrated magnetic interactions due to the topology of the
underlying structure. This competition leads to double magnetic
transitions and suggests the presence of complex magnetic structures. CePtSi is another system displaying both, the properties of
a heavy-Fermion and a coherent dense Kondo lattice, but, in
contrast to YbPhSn, crystallizes in a tetragonal LaPtSi-type structure [422]. The so-called Kondo semiconductor CePhAs undergoes
successive structural phase transitions from the hexagonal LiGaGetype to the orthorhombic e-TiNiSi-type structure [423]. Studies of
the energy gap under pressure reveal, that the formation of the gap
cannot be explained by the hybridization of the 4f electrons and the
conduction band, and a sort of charge density wave transition is
proposed for the origin of the gap formation.
Another important feature that needs to be addressed is that the
exact arrangement of the atoms within the hexagonal unit cell has
a strong influence on the electronic properties. For the LaCuSn
system, detailed theoretical studies, assuming different z coordinates for Cu and Sn, reveal, that the degree of puckering of the
CuSn-layers is a crucial factor for the system to behave like
a semiconductor or a metal [424]. In general, XYZ with the LiGaGetype structure are non-magnetic materials without rare earth
metals. However, the introduction of f-electrons leads to enhanced
magnetic behavior at low temperatures. On the other hand, the
LiGaGe-type structures are mainly metallic, but become semiconducting, depending on the degree of puckering of the YZ
hexagonal layers [136].
17.2. REME phases
Fig. 61. Vision: design of an all Heusler multifunctional device which combines all
advantages of half-metallic ferromagnetism, non-magnetic semiconductors, shape
memory alloys and topological insulators, allowing the manipulation of one material
by another.
A big class of ternary compounds with 1:1:1 stoichiometry are
the so-called REME phases, where RE ¼ rare earth, alkali, alkaline
earth, or from groups 3 and 4, M ¼ late transition metal, from
groups 8e12, E ¼ main group element, from groups 13e15. Sine
more than 60% of these compounds contain RE ¼ lanthanide, RE is
used for the classification of these materials. The RE sublattice
interacts with the rest of the framework in primarily an ionic way.
The remaining anionic counterpart is formally discirbed as [ME]n,
where n denoted the oxidation state of RE. These remant [ME]n
asseblies build 2-D slabs, which are then arranged in the third
dimension (physically connected by bonds) in various patterns.
This gives rise to cubic, hexagonal, tetragonal, orthorhombic,
trigonal and monoclinic crystal structures.
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
In general, three different kinds of 2-D slabs are observed. (a)
Planar, graphitic layers (hexagonal), where atoms are three-connected, and there are only ME bonds. These sheets are arranged
eclipsed with respect to each other. (b) Slightly puckered layers, in
which three adjacent atoms of a hexagon go “up” with respect to
a median plane, and are able to form three bonds with a layer
above, while three other atoms go “down”, potentially generating
three bonds with the layer beneath. The [ME]n sublattices are, in
principle, four-connected nets with ME intralayer bonds, and ME or
MM and EE interlayer contacts. They are in general orthorhombic,
and are stacked in an eclipsed way. (c) Puckered layers with still
greater distortions (close to classical “chair” hexagons) which,
when stacked, form diamond-type structures by linking three
alternate atoms of the six-membered ring upward, and the other
three downward. The choices are either simple diamond (forgetting
for the moment the M, E difference), with staggered chair-conformation hexagons along the stacking direction, or hexagonal diamond, forming eclipsed boat-hexagons along the same axis. This is
the series that includes the cubic REME s, which also contain the
XYZ Half-Heusler compounds with X ¼ RE. In both of these cases,
there are exclusively ME bonds (when M and E are distinct).
Nearly all the REME compounds have unusual magnetic and
electric properties. The cation for most of these phases is a rare
earth metal, sometimes magnetic, and in the anionic net there are
transition metals, magnetic as well. There is much research concerned with the determination of these magnetic and electrical
properties of REME phases, or even investigating the superconducting properties of certain phases. For further details there
are several reviews which analyze these properties in depth, along
with theoretical considerations on a number of such systems
[425e431].
17.3. Tetragonally distorted Heusler compounds
In addition to the well-known cubic structures of Heusler
compounds, tetragonally distorted Heusler compounds have
recently attracted great scientific interest in the field of spintronics,
especially for spin-torque applications [262,432] (Fig. 51). A
tetragonal distortion is observed for Mn2YZ compounds crystallizing in the inverse Heusler structure. In this structure, the Mn
atoms occupy two different lattice sites, one with tetragonal and
one with octahedral coordination. Theoretical investigations by
Kübler showed, that the Mn atom on the octahedral site formally
possesses an oxidation state of þ3 (Mn3þ, d4) [91]. The electronic
configuration for a single d4 high spin ion in an octahedral environment, according to crystal field theory, is displayed in Fig. 52(a).
The triple-degenerated t2g orbitals and one of the double degenerated eg orbitals are each occupied by a single electron. In fact, this
electron configuration is energetically not favored, and energy can
be gained by a distortion of the octahedron. Both, an elongation and
a compression are possible, as shown in Fig. 52(b) and (c). These
distortions lead to a lowering of the occupied orbitals resulting in
an energy gain a phenomenon often referred to as JahneTeller
distortion. Alternatively, a double degenerate van Hove singularity,
i.e. a saddle point in the band structure, can lead to a tetragonal
distortion since this singularity maximizes the band energy,
leading to an unfavorable condition, which is avoided by a tetragonal lattice distortion. In the case of Mn2YZ compounds, the cubic
unit cell undergoes an elongation along the c axis, as shown in
Fig. 53(a). Consequently, the symmetry of the crystal changes from
the cubic space group F43m to the tetragonal spacegoup I4/mmm
(space group no. 139). Fig. 53(b) and (c) illustrate the relation
between the tetragonal and the cubic unit cell. The tetragonal unit
cell can be derived form the cubic cell, by rotation of the cell edges
by 45 . The resulting tetragonal structure is displayed in Fig. 53(d).
45
Similar to the Heusler structure, a regular and an inverse variant of
the tetragonal cell are known (see Fig. 55). The tetragonal cell
derived from the Cu2MnAl-type structure, is displayed at the
bottom. The X atoms occupy the Wyckoff position 4d (0, 1/2, 1/4),
the Y are placed at 2b (0, 0, 1/2) and the Z atoms are located at
2a (0, 0, 0). The prototype of this structure is Ni2MnSn. As
mentioned above, the inverse structure is frequently observed in
case of Mn2YZ materials. Therefore, an inverse variant of the
tetragonal unit cell is also possible, as shown at the top of Fig. 55.
Here, the first Mn atom is located at the Wyckoff position 2b, while
the second Mn atom and the Y atom are placed at the Wyckoff
position 4d. Finally, the Z atom occupies the 2a position.
Experimentally, these features can be revealed by XRD where
the tetragonal distortion becomes evident by a splitting of the cubic
(220) reflection into the (112) and (200) reflections in the corresponding diffraction pattern. Depending on the direction of the
distortion (elongation or compression), the reflections move to
smaller or bigger scattering angles compared to the cubic (220)
reflection. Therefore, the distance between the (112) and the (200)
reflection is a measure for the degree of distortion. Fig. 54 provides
an overview over different degrees of distortion and the correpffiffiffi
sponding XRD patterns. Please note, that the c/a ratio of 1.41 ðz 2Þ
corresponds to a cubic structure.
Up to now, only few tetragonal distorted Heusler materials have
been studied thoroughly, Mn3Ga being the most prominent
example [188,189]. These materials are particularly interesting due
to the perpendicular magnetic anisotropy which can be achieved in
thin films [262] opening the door to spin-torque devices. Therefore,
it is essential to design new materials that fulfill the corresponding
criteria, i.e. low saturation magnetization, high spin polarization as
well as low magnetic damping (compare Fig. 51). A very intuitive
route towards new tetragonal Heusler materials is sketched in
Fig. 56. The tetragonal unit cell is closely related to the cubic fcc
CuAu-type cell (L10), since doubling the cubic unit cell in one
direction yields a disordered variant of the tetragonal cell. A similar
relationship can be deduced for the conventional cubic Heusler
materials, which can be divided into eight bcc CsCl-like subcells.
These relationships make it easy to design new materials, since the
combination of two materials with CuAu-type structure leads to
new compounds with a tetragonally distorted unit cell. The
combination of the binaries MnGa and NiMn, for instance, results in
the well-known shape memory alloy Mn2NiGa. This design scheme
leads to a huge variety of new materials which can be explored in
future research.
17.4. Related layered structures
In addition to the hexagonal structures described above,
a number of layered crystal structures are also closely related to the
Heusler structure. Fig. 57 displays that the removal of every second
layer of atoms on the tetrahedral sites of the Cu2MnAl-type structure leads to the tetragonal LiFeAs structure, which is characterized
by FeAs layers interlaced with a Li charge reservoir. The binary
prototype of this structure is Cu2Sb, whereas the ternary version is
represented by the compound PbFCl. A further removal of all atoms
from the octahedral positions results in the tetragonal MnP-type
structure, the parent structure the superconductor FeSe. All crystal
structures shown in Fig. 57 are dominated by the same structural
motif, i.e. infinte [XZ] layers with a considerable covalent bonding
interaction as reported in detail by Fässler and coworkers [433].
The Heusler structure can thus be regarded as a “filled” variant of
the MnP-type structure, while the Cu2Sb or PbFCl-type structure
displays a “partially filled” variant.
In the Cu2Sb-type structure, which does not exhibit an inversion
center, the discovery of the “111” superconductor LiFeAs attracted
46
T. Graf et al. / Progress in Solid State Chemistry 39 (2011) 1e50
considerable interest due to its high critical temperature of 16e18 K
[434]. Since the structure of this compound is rather simple, it can
be used to build multilayered superconductors or to study the
underlying superconducting mechanism in iron arsenides. Another
example for a superconductor that, in contrast to LiFeAs, contains
only main group metals is NaAlSi with a critical temperature of 7 K
[435]. Moreover, several semiconductors can be found in this
particular structure type, for instance NaCuS, or NaAuTe [436].
The compound FeSe is directly related to the new superconductors based on FeAs, which all share the same structural motif,
i.e. the FeAs layers. FeSe has the same iron pnictide layered structure, but without the separating layers. Therefore, this compound is
regarded as a simple model system for the pnictide superconductors. Investigations of the superconducting properties under pressure revealed, that the critical temperature increases from 8.5 K up
to 36.7 K at an applied pressure 8.9 GPa [437]. The volume of the
unit cell changes with applied pressure due to a collapse of the
separation of the Fe2Se2 layers. The compound undergoes a structural phase transition into a hexagonal NiAs-type structure, which
is completed at 38 GPa. The exploration of the electronic phase
diagram revealed that there is no region showing a spin density
wave or static magnetism. Thus the increase of TC under pressure is
not associated with the suppression of a magnetically ordered
phase, but is attributed to the considerable decrease in volume, and
the following collapse of the space between the Fe2Se2 planes.
performances, e.g. the application of a current to the half-metallic
ferromagnet leads to a spin injection into the semiconducting
Heusler compound, or an externally applied magnetic field causes
the shape memory alloy to deform which can influence the behavior
of the topological insulator on top. Such a device could be designed
according to the specific needs of the corresponding application and
the new, unknown multifunctional properties could be developed,
all within the one material class, the Heusler compounds.
Acknowledgments
The authors thank B. Balke, A. Beleanu, C. G. F. Blum, F. Casper, S.
Chadov, G. H. Fecher, T. Gruhn, V. Jung, J. Kübler, V. Ksenofontov, S.
Ouardi, T. D. Schladt, M. Schwall, J. Winterlik, and S. Wurmehl for
providing data and for many fruitful discussions.
Financial support by the Deutsche Forschungsgemeinschaft
(Project TP 2.3-A in research unit FOR 1464 “ASPIMATT”), and the
Graduate School of Excellence “Material Science in Mainz” is
gratefully acknowledged.
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17.5. Relationship between Heusler compounds and perovskites
The families of Heusler compounds and perovskites comprise
plenty of compounds with multiple properties since both families are
outstanding due to their wide range of tunability. The structural
similarities of Heusler compounds and perovskites are illustrated in
Fig. 58. They both contain compounds composed of main group
elements or transition metals. Furthermore, metal to insulator transitions are observed in both cases, and a large variety of magnetic
properties can be found due to two different magnetic sublattices.
Additionally, low dimensional variants are known for both classes,
i.e. the RuddlesdenePopper phases for the perovskites and the
PbOCl-type materials for the Heusler compounds. Despite all these
analogies, characteristic differences appear in the analysis of the
chemical bonding. While Heusler compounds combine covalent and
ionic interaction (compare Section 3), perovskites are purely ionic
materials. They are dominated by the octahedral coordinating of the
metals by oxygen, whereas in Heusler compounds the octahedral and
tetrahedral coordination spheres play a major role for their generic
properties. Correlation effects have to be considered in perovskites to
describe their electronic properties in a suitable way. In the case of
Heusler compounds, correlations have only to be considered in Mn
and RE containing materials. Table 7 provides a concluding overview
on the central points concerning the comparison of classical
semiconductors, Heusler compounds and perovskites.
18. Summary and outlook
This review article gives a broad overview of an outstanding
class of materials, the Heusler compounds. Fig. 59 summarizes all
important aspects concerning these exceptional materials, ranging
from semicondutors, over metals and magnets to topological
insulators with plenty of technological applications in spintronics,
thermoelectrics, opto-electronics and much more (Fig. 60).
Many fascinating research projects will certainly emerge in
future which take advantage of their countless functionalities. One
possible multifunctional device is illustrated in Fig. 61. Depositing
several Heusler materials on top of each other leads to a device, in
which various externally applied forces result in different
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