Neutron Scattering and Thermodynamic Studies
of Quantum Magnetism on the Kagome Lattice.,.,..
by
MASSACHUSETTS INSTITUTE
OF TECI'fl\~'J~(;3Y
-~"--
Robin Michael Daub Chisnell
NOV 10 2014
B.S. Physics
Washington University in St. Louis (2008)
LIBRi\RJES
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2014
© Massachusetts Institute of Technology 2014.
All rights reserved.
Signature redacted
Author ............... "........ ___ .................................... .
Department of Physics
July 24, 2014
Signature redacted
C~rtified
by .......... .
v V
Signature redacted
Young S. Lee
Professor
Thesis Supervisor
Accepted by ...... .
/ ~
Krishna Rajagopal
Professor, Associate Department Head" for Education
2
Neutron Scattering and Thermodynamic Studies of
Quantum Magnetism on the Kagom4 Lattice
by
Robin Michael Daub Chisnell
Submitted to the Department of Physics
on July 24, 2014, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
The geometry of the kagome lattice leads to exciting novel magnetic behavior in both
ferromagnetic and antiferromagnetic systems. The collective spin dynamics were investigated in a variety of magnetic materials featuring spin-1 and spin-1 moments on
kagome lattices using neutron scattering and thermodynamic probes. Both ferromagnetic and antiferromagnetic systems were studied.
Cu(1,3-bdc) is an organometallic material, where the Cu2 + ions form a ferromagnetic S = . kagom6 system. Synthesis techniques were developed to produce
-mg-sized deuterated single crystals, and ~2,000 crystals were partially coaligned
to create a sample for neutron scattering measurements. Elastic neutron scattering measurements show the existence of long range magnetic ordering below T =
1.77 K. Integrated Bragg peak intensities were analyzed to determine the structure
of ordered magnetic moments. Inelastic neutron scattering measurements show the
magnon dispersion spectrum, which consists of a flat high energy band and two dispersive, lower energy bands. The application of a magnetic field perpendicular to
the kagome plane opens gaps between these three bands and distorts the flatness of
the highest energy band. The system was modelled as a nearest-neighbor Heisenberg
ferromagnet with Dzyaloshinskii-Moriya(DM) interaction. The model dispersion and
scattering structure factor were calculated and fit to the data to precisely determine
the strengths of the nearest-neighbor coupling and DM interaction. The observed
manon band structure is a bosonic analog to the band structure of the topological
insulator systems.
Antiferromagnetic kagome systems can exhibit novel magnetic ground states such
as quantum spin liquids and spin nematics. Thermodynamic measurements were
performed on the antiferromagnetic kagome materials Mg.Cu 4s (OH)6 Cl 2 , featur-
ing S = 1 moments. These measurements reveal magnetic ordering at low values
of x that is suppressed with increasing x. At x = 0.75, this ordering is not fully
suppressed, but susceptibility and specific heat measurements reveal behavior similar to that of the quantum spin liquid candidate herbertsmithite. Thermodynamic
and neutron scattering measurements were performed on the kagome lattice mate3
rial BaNi 3 (OH) 2 (VO 4 )2 , which features S = 1 moments. These measurements reveal
competing interactions, which result in a spin glass ordering transition.
Thesis Supervisor: Young S. Lee
Title: Professor
4
Acknowledgments
I would not have been able to complete this thesis without the help and support of a
number of people. I am grateful to them for their contributions to my education and
to my life.
I would first like to thank my advisor Young Lee for his mentorship and guidance.
I am always impressed by the depth of his understanding of neutron scattering techniques, and his help with scattering experiments has been invaluable. During these
experiments he would rearrange his schedule to be availalbe at all times of the day
and night to help me make the most of my beam time. I would also like to thank
Young for all the effort he has put into helping me develop as an independent scientist. Young has allowed me the freedom to wrestle with understanding my data on my
own, but has always been available to help me think through an idea. He has taught
me to be more confident in my abilities as a scientist, particularly in presentations of
my results.
I would also like to thank Senthil Todadri and Nuh Gedik for serving on my thesis
committee. I am grateful for the time and thought they have put into my projects
and thank them for many helpful discussions.
I also thank the many graduate students and postdocs who have worked with
me during my time at MIT. In particular, I thank Joel Helton and Deepak Singh
for their help throughout my thesis work. They both helped to introduce me to
experimental physics techniques during my first year as a member of Young's group.
Joel was a patient teacher, always willing to take time to explain something to the new
grad student. Deepak introduced me to single crystal growth and characterization
techniques. I later collaborated with both of them at the NIST Center for Neutron
Research, where they taught me a great deal about neutron scattering. Deepak's
knowledge of the workings of the instrument I used was invaluable, and he would go
out of his way to be sure to introduce me to other scientists. Joel was always there to
think through problems with me or to run the experiment for a few hours while I got
some sleep. I am very grateful to them both. I thank Andrea Prodi and Harry Han
5
for teaching me new measurement and crystal growth techniques. Craig Bonnoit and
Dillon Gardner have been wonderful people to work with. They have been available
throughout my time at MIT as sounding boards for both good and bad ideas. I have
learned a lot through discussions with them. They have also been good friends and
have made my life more enjoyable, in and out of the lab. I also thank Drew Potter
and Evelyn Tang for entertaining my questions about theoretical physics.
My work would not have been'possible without my colaborators from Dan Nocera's
group in the MIT Department of Chemistry. I thank Tyrel McQueen and Danna
Freedman for synthesizing some of the samples I used for this research. I especially
want to thank Danna for all her help with my own crystal growth. I would not have
succeeded in growing the crystals I did without her input. I thank her for taking
time to teach me about chemistry, and for walking me through different synthesis
procedures.
From my time outside the lab I want to thank my friends for all they have done
to make my life more enjoyable. Thanks to my roommates from over the years, Matt
Edwards, Brent Dorr, and Dan Pilon, as well as to Tim Curran, Christina Ignarra,
and Simon Lee, and to the guys from MIT Ultimate. Most importantly, I want to
thank Lucia Marconi. Her friendship, love, and support have helped me get through
times when graduate school and thesis-writing seemed too daunting.
Finally, I want to thank my family. My parents, John Chisnell and Margo Daub,
have always encouraged me to be curious and to investigate answers to questions
on my own. When I was little, they took me on countless trips to museums, zoos,
aquariums, and botanical gardens, and were always enthusiastic about teaching me
new things. My mother has inspired me to pursue a career as a scientist and a
professor. My father has always insisted on tinkering with and fixing broken things
himself and on having me help. As a child I hated this but it is a skill that has
helped me succeed as an experimental scientist and I am grateful to him for helping
me develop it. Throughout my life they have believed in me and pushed me to do
as well as I could. My brother Peter has been a good friend and during my time in
graduate school has found ways to spend time with me even from across the country.
6
I would not be where I am without my family, and I thank them for all they have
done for me.
7
8
Contents
21
. . . . . . . . . . . . . . . . . .
. .
21
.
. .
Geometric Frustration
1.1.2
Zero-Energy Modes . . . . . . . . . . . . . . . . . . . .
. . .
25
1.1.3
The Quantum Spin Liquid Ground State . . . . . . . .
. . .
26
Ferromagnetism on the Kagome Lattice . . . . . . . . . . . . .
. . .
29
1.2.1
Flat Mode in the Kagome Ferromagnet . . . . . . . . .
. . .
29
1.2.2
Kagom6 Lattice Flat Band and the Fractional Quantum
Hall
.
.
.
.
1.1.1
. . . . . . . . . . . . . . . . . . . . . . . . . . .
34
Thesis O utline . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
Magnon Hall Effect and Topological Edge Modes
. .
.
1.2.3
39
Experimental Techniques
39
.
. . . . . . .
Properties of the Neutron
. . . . . .
. . . . . . .
40
2.1.2
The Neutron Scattering Cross Section
. . . . . . .
41
2.1.3
Bragg Scattering . . . . . . . . . . .
. . . . . . .
46
2.1.4
Inelastic Magnetic Scattering
. . . .
. . . . . . .
47
2.1.5
Neutron Spetrometers
. . . . . . . .
. . . . . . .
49
2.1.6
Instrumental Resolution . . . . . . .
. . . . . . .
55
Sample Preparation . . . . . . . . . . . . . .
. . . . . . .
59
2.2.1
Single Crystal Growth . . . . . . . .
. . . . . . .
60
2.2.2
Sample Holder and Assembly
. . . . . . .
61
.
.
.
.
.
.
2.1.1
.
2.2
Neutron Scattering . . . . . . . . . . . . . .
.
2.1
32
.
E ffect
1.3
. . . . . . . . . .
.
1.2
Antiferromagnetism on the Kagom6 Lattice
.
1.1
2
19
Introduction
9
. . . .
.
1
Crystal Structure . . . . . . . . . . . . .
. . . . . . . . . .
65
3.2
Thermodynamic Measurements
. . . . .
. . . . . . . . . .
67
3.2.1
Magnetic Measurements . . . . .
. . . . . . . . . .
67
3.2.2
Specific Heat Measurements . . .
. . . ... . . . .
71
. . . . . . . . . .
75
Elastic Scattering Measurements
. . . . . . . . . .
75
3.3.2
Magnetic Structure . . . . . . . .
. . . . . . . . . .
81
3.3.3
Inelastic Scattering Measurements
. . . . . . . . . .
87
Conclusion . . . . . . . . . . . . . . . . .
. . . . . . . . . .
91
.
.
3.3.1
93
Magnetic Excitations in Cu(1,3-bdc)
.
Inelastic Neutron Scattering Measurements . . . . . . . . . . . . . .
4.4
102
. . . . . . . . . . . . . . . . . . . . . . . . .
104
The Spin Hamiltonian
4.2.1
The Dzyaloshinskii-Moriya Interaction
. . . . . . . . . . . .
105
4.2.2
Spin Wave Dispersion Calculation . . . . . . . . . . . . . . .
109
Determining Hamiltonian Parameters . . . . . . . . . . . . . . . . .
113
4.3.1
Structure Factor Calculation . . . . . . . . . . . . . . . . . .
113
4.3.2
Instrumental Resolution . . . . . . . . . . . . . . . . . . . .
116
4.3.3
Fits to Data . . . . . . . . . . . . . . . . . . . . . . . . . . .
119
4.3.4
Specific Heat Calculation . . . . . . . . . . . . . . . . . . . .
124
C onclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
128
.
.
.
.
.
.
.
4.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
erations
4.2
93
Magnetic Form Factor and Spin Polarization Direction Consid-
.
4.1.1
.
4.1
133
Magnesium Paratacamite . . . . . . . . . .
. . . . . . . . . . . 133
5.1.1
M otivation . . . . . . . . . . . . . .
. . . . . . . . . . . 133
5.1.2
Magnetization Measurements
. . .
. . . . . . . . . . . 134
5.1.3
Specific Heat Measurements . . . .
. . . . . . . . . . . 141
5.1.4
Conclusion . . . . . . . . . . . . . .
. . . . . . . . . . . 143
.
.
5.1
.
Studies of New Kagom6 Antiferromagnets
.
5
.
.
.
.
Neutron Scattering Measurements . . . .
.
3.4
.
3.1
3.3
4
65
Magnetic Order in Cu(1,3-bdc)
.
3
10
5.2
Nickel Vesignieite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.2.1
M otivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.2.2
Thermodynamic Measurements of BaNi 3 (OH) 2 (VO 4 ) 2 . . . . .
149
5.2.3
Thermodynamic Measurements of BaNi 3 (OD) 2 (VO 4 ) 2 . . . . .
156
5.2.4
Inelastic Neutron Scattering Measurements . . . . . . . . . . .
161
5.2.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
A Spin Wave Calculation Details
171
A.1 Calculation of Spin Wave Dispersion
A.2
Calculation of S(Q,w)
. . . . . . . . . . . . . . . . . .
171
. . . . . . . . . . . . . . . . . . . . . . . . . .
176
B Cu(1,3-bdc) Calculated vs Measured Structure Factor
11
179
12
List of Figures
1-1
The kagome lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
1-2
Spin ordering configurations on square and triangular plaquettes . . .
22
1-3
Spin ordering configurations on the triangular and kagome lattices . .
24
1-4
Zero-energy modes on the kagom6 lattice . . . . . . . . . . . . . . . .
25
1-5
Localized exciations in the kagome ferromagnet
. . . . . . . . . . . .
29
1-6
Localized exciations in the kagome lattice hopping model . . . . . . .
31
1-7
Spin wave dispersion of the Heisenberg ferromagnet on a kagome lattice 32
1-8
Kagom4 lattice unit cell with fictitious fluxes . . . . . . . . . . . . . .
35
2-1
Schematic of a standard triple-axis neutron spectrometer . . . . . . .
51
2-2
Schematic of a direct geometry time-of-flight neutron spectrometer . .
53
2-3
Properties of the time-of-flight energy resolution function . . . . . . .
58
2-4
Neutron scattering sample assembly . . . . . . . . . . . . . . . . . . .
63
3-1
Crystal structure of Cu(1,3-bdc) . . . . . . . . . . . . . . . . . . . . .
66
3-2
Susceptibility of single crystal Cu(1,3-bdc) . . . . . . . . . . . . . . .
68
3-3
Magnetization of single crystal Cu(1,3-bdc) as a function of applied field 70
3-4
Specific heat of Cu(1,3-bdc) under applied field
. . . . . . . . . . . .
72
3-5
Zero-field specific heat and estimated magnetic entropy of Cu(1,3-bdc)
74
3-6
Elastic scans through (0 0 L) Bragg positions under zero magnetic field
76
3-7
Longitudinal scans through (0 0 L) Bragg positions under applied mag-
netic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
3-8
6 scans at (0 0 L) Bragg positions under applied magnetic field . . . .
78
3-9
Magnetic Bragg peak intensity as a function of applied magnetic field
79
13
3-10 Magnetic Bragg peak intensity as a function of temperature
. . . . .
80
3-11 Scaling of measured Bragg peak intensity due to vertical beam divergence 84
3-12 Integrated (0 0 L) Bragg peak intensities . . . . . . . . . . . . . . . .
86
3-13 Schematic of the ground state spin configuration in the magnetically
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
3-14 Temperature dependence of the inelastic spectrum . . . . . . . . . . .
89
ordered state.
4-1
Inelastic scattering spectrum measured using the LET spectrometer
with incident neutron energy 6.01 meV . . . . . . . . . . . . . . . . .
4-2
Inelastic scattering spectrum measured using the LET spectrometer
with incident neutron energies 3.53 meV and 12.4 meV . . . . . . . .
4-3
96
Energy scans through the flat mode with nonzero out-of-plane momentum transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-4
94
98
Energy scans through the flat mode with magnetic field applied parallel
to the kagom6 plane
. . . . . . . . . . . . . . . . . . . . . . . . . . .
99
. . . . . . . . . . .
99
. . . . . . . . . . . . . . . .
100
4-5
Decrease in flat mode intensity with applied field
4-6
Scans of the low-energy mode dispersion
4-7
Flat mode peak intensity as a function of total momentum transfer
4-8
Local environment of the magnetic ions in Cu(1,3-bdc)
4-9
Dzyaloshinskii-Moriya vectors on the Cu(1,3-bdc) kagome lattice.
101
. . . . . . . .
107
. .
108
4-10 Spin wave dispersion of the Heisenberg ferromagnet with DM interaction on the kagom6 lattice
. . . . . . . . . . . . . . . . . . . . . . . .
112
4-11 Spin wave structure factor of the Heisenberg ferromagnet with DM
. . . . . . . . . . . . . . . . . . . .
115
. . . . . . . . . . . . . . . . . . . . . . . . . .
116
. . . . . . . . . . . . . . . . . . . . . .
117
4-14 Calculated magnetic contribution to the scattered intensity . . . . . .
118
4-15 Calculated intensity fit to data . . . . . . . . . . . . . . . . . . . . . .
120
4-16 Scans of the low-energy mode dispersion including calculated intensity
123
interaction on the kagomd lattice
4-12 LET energy resolution
4-13 SPINS momentum resolution
14
4-17 Calculated and measured specific heat of Cu(1,3-bdc) as a function of
tem perature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4-18 Calculated and measured specific heat of Cu(1,3-bdc) as a function of
applied field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4-19 Kagom6 lattice unit cell with fictitious fluxes due to DM interaction . 131
5-1
Susceptibility of Mg.Cu 4 _.(OH) 6 Cl2
5-2
Inverse susceptibility of MgCu 4 -,(OH) 6 Cl2
5-3
Magnetization of Mgo. 7 5 Cu3 .2 5 (OH) 6 C
5-4
AC susceptibility of Mgo. 75 Cu 3 .25 (OH) 6 Cl 2 as a function of temperature
and applied field
. . . . . . . . . . . . . . . . . .
2
135
. . . . . . . . . . . . . .
136
as a function of applied field .
137
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5-5
Scaled AC susceptibility of Mgo.75Cu 3 .2 5 (OH)6 C2
. . . . . . . . . . .
140
5-6
Specific heat of Mg.Cu 4 _,(OH) 6 Cl 2 . . . . . . . . . . . . . . . . . . .
142
5-7
Specific heat and estimated magnetic entropy of MgCu4 _.(OH) 6 Cl2 .
144
5-8
Specific heat of Mg2Cu 4 _.(OH) 6 Cl 2 under applied field . . . . . . . .
145
5-9
Susceptibility of BaNi 3 (OH) 2 (VO 4 ) 2 . . . . . . . . . . . . . . . . . . .
148
5-10 Magnetization of BaNi 3 (OH) 2 (V04)2 as a function of applied field . . 150
5-11 AC susceptibility of BaNi 3 (OH) 2 (VO 4 ) 2 as a function of temperature
152
5-12 Magnetization and AC susceptibility of BaNi 3 (OH) 2 (VO 4 ) 2 as a func-
tion of temperature under applied magnetic field . . . . . . . . . . . . 153
5-13 Specific heat of BaNi3 (OH) 2 (VO 4 ) 2 . . . . . . . . . . . . . . . . . . .
155
5-14 Susceptibility of BaNi3 (OD) 2 (VO 4 ) 2 . . . . . . . . . . . . . . . . . . .
157
5-15 Magnetization of BaNi3 (OD) 2 (V0 4 ) 2 as a function of applied field . . 158
5-16 AC susceptibility of BaNi 3 (OD) 2 (VO4 ) 2 as a function of temperature
160
5-17 Inelastic scattering spectrum of BaNi 3 (OD) 2 (V04)2 measured with neutrons of initial energy Ej = 10 meV . . . . . . . . . . . . . . . . . . .
162
5-18 Elastic scattering for BaNi3 (OD) 2 (VO 4 ) 2 . . . . . . . . . . . . . . . .
163
5-19 Temperature dependence of the inelastic spectrum of BaNi 3 (OD) 2 (V0 4 )2 164
5-20 Inelastic scattering spectrum of BaNi 3 (OD) 2 (V0 4 )2 measured with neu-
trons of initial energy Ei = 3 meV . . . . . . . . . . . . . . . . . . . .
15
166
5-21 Temperature dependence of the inelastic spectrum of BaNia(OD)2(VO 4 ) 2
measured with neutrons of initial energy Ej = 3 meV . . . . . . . . .
167
B-1 LET Ej = 3.53 meV 1(7 T)-I(2 T) . . . . . . . . . . . . . . . . . . . 181
B-2 LET Ej = 3.53 meV 1(0 T)-I(7 T) . . . . . . . . . . . . . . . . . . . 182
B-3 LET Ej = 6.01 meV 1(7 T)-I(2 T) . . . . . . . . . . . . . . . . . . . 183
B-4 LET E = 6.01 meV 1(0 T)-I(7 T) . . . . . . . . . . . . . . . . . . . 184
B-5 LET E = 12.4 meV 1(7 T)-I(2 T) Low Q . . . . . . . . . . . . . . 185
B-6 LET E = 12.4 meV 1(7 T)-I(2 T) High Q . . . . . . . . . . . . . . 186
B-7 LET E = 12.4 meV 1(0 T)-I(7 T) Low Q . . . . . . . . . . . . . . 187
B-8 LET E = 12.4 meV 1(0 T)-I(7 T) High Q . . . . . . . . . . . . . . 188
B-9 SPINS 1(7 T)-I(0 T) ..........................
16
189
List of Tables
3.1
Crystallographic data for Cu(1,3-bdc) . . . . . . . . . . . . . . . . . .
67
5.1
Curie-Weiss temperature of MgCu 4 _,(OH) 6 Cl 2 as a function of x . .
135
17
18
Chapter 1
Introduction
In condensed matter research, there continues to be great interest in strongly correlated electron systems. In these materials, the collective behavior of interacting
electrons can lead to quasiparticle excitations whose behavior bears little resemblance
to that of the individual constituent parts of the material. These materials exhibit a
wide variety of exotic properties and are of interest as test models for fundamentally
new physics.
Of particular attention are systems where the collective behavior is strongly influenced by the geometry of the underlying crystal lattice. In this thesis, we focus
on magnetic materials that feature the kagome lattice. The kagom6 lattice, shown
in Figure 1-1, is a two-dimensional (2D) lattice, which consists of a three-atom basis
on a triangular lattice. The basis is arranged such that the lattice tiles the plane
with corner-sharing triangles. In real materials, the crystal lattice is of course threedimensional (3D). The materials we will examine consist of layers of kagom6 planes
that are separated such that interactions between layers are weak and the magnetic
behavior can be approximated as 2D.
The geometry of the kagom6 lattice leads to a wide range of interesting behaviors.
Kagome lattice antiferromagnets have long been of interest due to the high degree of
geometric frustration. Recently, however, the case of ferromagnetism on the kagome
lattice has been considered as well. In this thesis we will present measurements of
both ferromagnetic and antiferromagnetic kagome systems.
19
AmIk
Ank
AdIlk
x
x
x
mw
v
v
v
Figure 1-1: The kagome' lattice
20
1.1
1.1.1
Antiferromagnetism on the Kagome Lattice
Geometric Frustration
Frustrated magnetism refers to a broad class of magnetic materials where there exists
no spin configuration that will satisfy all pairwise spin interactions simultaneously[1,
2]. Frustration can be present due to competing interactions, such as a ferromagnetic
nearest-neighbor coupling combined with an antiferromagnetic next-nearest-neighbor
coupling. Frustration can also be a result of disorder, as is the case in many spin glass
systems[3, 4]. Geometric frustration refers to systems where the underlying geometry
of the crystal lattice is the source of the frustration. In other words, geometrically
frustrated materials are inherently frustrated, even in the absence of competing interactions or disorder.
To illustrate the concept of geometric frustration, let us consider the simple case of
antiferromagnetic nearest-neighbor coupling on small plaquettes, as shown in Figure
1-2. The energy of each bond can be expressed as JSj - Sj with J > 0. Thus the
lowest energy of each bond is -JS 2 and is achieved when nearest-neighbor spins are
oriented antiparallel to each other. On a square lattice this condition is easily satisfied
for every bond simultaneously by alternating up and down spins, and therefore the
square lattice is not geometrically frustrated. On the triangular lattice not all bonds
can have energy -JS 2 simultaneously. In the Ising case[5], where spins are restricted
to point only parallel or anitiparallel to a specified axis, it is only possible for two
bonds on a single triangle to be in the low energy state. The third bond will be in
the high energy state, with energy +JS2 . Flipping a spin to put the unsatisfied bond
in the low energy state results in raising the energy of another bond. The frustration
can be slightly relaxed in the Heisenberg case, where spins are treated as vectors. In
this case the ground state will consist of nearest-neighbor spins rotated 1200 from
each other. For this configuration no single bond is in its lowest energy state, but the
total energy of a triangle is lowered as compared to the Ising case.
It is now clear that lattices formed from triangles with antiferromagnetic nearestneighbor exchange will be geometrically frustrated. An obvious example is the trian21
Square
V-JS 2
_jS 2
-Js 2
-JS 2
Trianglular - Ising Spins
x
+Js
1Y/
V/
_Js 2
2
-JS 2
_jS 2
-JS 2
+JS
2
x
Trianglular - Heisenberg Spins
-
1/
2 Js
- 1/ 2 Js2
2
-
1/2JS
2
Figure 1-2: Spin ordering configurations on square and triangular plaquettes. The
square lattice is not frustrated and spins can be configured so that each bond has its
minimum energy of -JS 2 . The triangular lattice is frustrated. In the Ising case, the
spin in the bottom left corner cannot satisfy the bonds with both its nearest neighbors
simultaneously. In the Heisenberg case the ground state configuration consists of spins
rotated 1200 from their neighbors.
22
gular lattice, which consists of edge-sharing triangles. As was discussed above, the
ground state configuration for Ising spins on a single triangle will have two bonds in
the low energy state and one in the high energy state. This is true not just for an
isolated triangle but for the extended lattice as well, as shown by Wannier in 1950[6].
Wannier found that any spin configuration where each triangle possessed two low energy bonds and one high energy bond would be a ground state. Therefore the system
possesses not a single ground state but an extended manifold of degenerate ground
states. This degenerate manifold prevents the system from selecting any particular
ground state and thus from ordering. In fact Wannier calculated that for the triangular Ising model, the system is disordered at all finite temperatures and has finite
entropy, even at zero temperature. He calculated this entropy to be 0.323kB per spin.
By employing triangles as building blocks, we can assemble other geometrically
frustrated lattices. The kagome lattice is the other natural choice for 2D lattices.
There are also 3D analogs to the triangular and kagome lattices, built by tiling space
with tetrahedra. The face-centered-cubic lattice consists of edge-sharing tetrahedra
while the pyrochlore lattice consists of corner-sharing tetrahedra. Even among these
lattices, the kagom6 lattice geometry drives behavior that is unique. This unique
behavior is the motivation for the focus of this thesis on the kagome lattice.
One of the most significant features of the kagome geometry as compared to the
triangular geometry is its lowered connectivity. In the triangular lattice, each spin
has six nearest-neighbor spins, whereas in the kagome lattice each spin has only four
nearest neighbors. This lowered connectivity means that the kagom6 lattice is in some
sense 'more frustrated' than the triangular lattice. A calculation performed on the
kagom6 Ising model[7] found a zero-temperature entropy of 0.502kB per spin, larger
than that of the triangular Ising model.
The effects of the lowered connectivity of the kagome lattice become even more
pronounced when we consder the case of vector spins, as shown in Figure 1-3. In this
case a ground state spin configuration is one where the three spins of every triangle
are rotated by 1200 from each other. Seen another way, the vector sum of the three
spins around a triangle is zero, E
s Si = 0. On the triangular lattice, neighboring
23
Triangular
Kagome
Figure 1-3: Spin ordering configurations on the triangular and kagom6 lattices. In
the triangular lattice neighboring triangles share two spins, so fixing the spins on the
top triangle uniquely determines the spins on the neighboring triangles and thus on
the entire lattice. In the kagom6 lattice neighboring triangles share only one spin, so
fixing the top triangle does not determine the spin configuration on the neighboring
triangles.
triangles share two spins. Therefore, once the spin configuration on a single triangle is
fixed the spin configurations on every neighboring triangle, and subsequently on every
triangle in the plane, are uniquely determined. On the kagom6 lattice, neighboring
triangles share only one spin. Therefore, fixing the spin configuration on a single
triangle does not uniquely determine the spin configuration even on the neighboring
triangles. If the spins are confined to a plane (XY), then there are two choices for the
configuration of each neighboring triangle, as shown by the red and blue arrows in
Figure 1-3. If the spins can point in any direction, then there are an infinite number
of configurations available for the neighboring triangle. Any arbitrary rotation of
two spins that maintains their 120* separation will also be a ground state. Therefore
the kagom6 lattice has a much larger manifold of degenerate ground states than the
triangular lattice.
24
Figure 1-4: Zero-energy modes on the kagom6 lattice. The spin configuration in the
left figure supports zero-energy modes along an infinite line. The spin configuration
in the right figure supports zero-energy modes around closed hexagonal loops.
1.1.2
Zero-Energy Modes
One consequence of the kagome lattice's lowered connectivity and resulting increased
ground state degeneracy is that the kagom6 lattice Heisenberg antiferromagnet supports a large density of zero-energy modes[8, 9]. These modes connect the states of
the ground state manifold. As discussed above, the only constraint on the ground
state spin configuration is that for each triangle
S = 0.
$ 4 If we consider a copla-
nar spin arrangement, then every spin must point in one of three directions. We can
therefore describe the spin configuration in terms of three sublattices, one for each
spin orientation. If we keep the spins of one of these sublattices fixed and rotate the
spins of the other two sublattices simultaneously about the axis defined by the spin
of the first sublattice, we have not changed the energy of the configuration. These
rotations are thus known as zero-energy modes. In fact, we need not rotate the entire
sublattice.
Consider a closed loop or infinte open path formed by spins of our two
chosen sublattices. We can rotate the spins along this loop, completely independent
of the rest of the lattice. Because these modes are localized in real space they will be
dispersionless in momentum space, as confirmed by spin wave calculations[10, 11].
25
Figure 1-4 shows two coplanar ground state spin arrangements on the kagome
lattice and a zero-energy mode in each arrangement. The left figure shows the q-= 0
structure, in which all spins point either into or out of a triangle. In this spin arrangment, the zero-energy modes take the form of infinite lines. The right figure shows the
V-3 x
V3
structure. In this arrangement, the zero-energy modes are localized loops
surrounding a single hexagon. Each line or hexagon can be rotated independently.
Therefore the kagom6 lattice supports a large density of zero-energy modes. This
lies in stark contrast to the triangular lattice, which will not support any zero-energy
modes except for a trivial rotation of all spins at once.
Experimentally, a flat mode at zero energy cannot be observed. However, in the
antiferromagnetic kagome lattice material iron jarosite, KFe 3 (OH) 6 (SO 4 ) 2 , the zero-
energy mode is lifted to finite energy due to the presence of a Dzyaloshinskii-Moriya
interaction[12]. This mode has been detected using neutron scattering measurements[13,
14] and NMR measurements[15].
1.1.3
The Quantum Spin Liquid Ground State
In the previous section, we described the behavior of a kagom6 antiferromagnet with
large, classical spins. As the value of S decreases, quantum fluctuations become more
significant, and the behavior of the system changes. Kagome lattice antiferromagnets
with small spins are ideal systems in which to study quantum spin liquid physics.
To understand the quantum spin liquid state, let us first consider the more familiar
ground state for a conventional antiferromagnet.
As proposed by Louis Neel[16],
the ground state of an antiferromagnetic material will consist of spins frozen into
a periodic pattern made up of two or more sublattices. In this state the average
magnetization at each site is nonzero, and the spin-spin correlation length diverges
at the ordering temperature, called the N6el temperature TN.
This state breaks
spin rotation symmetry and can break lattice translational symmetry as well. The
Neel state is not an eigenstate of the Hamiltonian, and so was initially thought to be
impossible because it would be destroyed by quantum fluctuations. However, neutron
scattering experiments were able to demonstate that the Neel ordered state did in fact
26
exist[17]. Following these experiments, a more complete theory of antiferromagnetism
demonstrated that most 2D and 3D antiferromagnetic systems should exhibit Neel
order[18].
In constrast, the quantum spin liquid state is made up of spins that are dynamic
even at zero temperature due to quantum fluctuations.
The average spin moment
at each site is zero, and the state does not break spin rotation or lattice translation
symmetry[19]. Spin-spin correlations are short ranged, but spins separated by a long
distance can be quantum mechanically entangled.
Consider a 1D chain of antiferromagnetically coupled quantum mechanical spins.
The energy of each nearest-neighbor bond is JSj - Sj with J > 0. Because of the
lowered dimensionality of this system we expect Noel order to be destabilized by
quantum fluctuations. This becomes clear when we examine the energies of different
states.
In the Neel ordered state, where neighboring spins alternate up and down,
the energy per spin is -JS 2 , which is -0.25J for spin-i moments.
This energy is
lowered significantly if neighboring spins instead form singlet pairs with wave function
(Lt) - I4t)), where the energy per spin is -0.375J for spin-! moments. Therefore
the Neel ordered state is unstable. The true ground state of the spin-j 1D antiferromagnetic chain can be solved exactly[2] using a variational method. This ground
state is made up of a linear superposition of singlet dimers, including pairings of spins
further apart than nearest-neighbors, and has per-spin energy -0.443J.
The quantum spin liquid state supports charge-neutral spin-1 quasi-particle excitations, which are known as spinons[20].
Spinons are a remarkable example of
emergent collective behavior in magnetic systems. 1D antiferromagnetic chains with
integer spins will exhibit gapped spinon excitations, while chains with half-integer
spin will exhibit gapless spinons[21, 22]. Spinons are created by the breaking of a
single dimer singlet. Spinons are therefore always created two at a time, and their
excitation spectrum is described by a continuum. Spinons are also deconfined. The
two spinons created by the breaking of a dimer do not interact with the remaining
dimers, and are thus free to move through the lattice independently of one another.
The quantum spin liquid state can also exist in two-dimensional systems. Though
27
Neel order is stable for most 2D antiferromagnets, this order is suppressed in geometrically frustrated lattices. Anderson proposed that the ground state of antiferromagnetically coupled spin-j moments on the geometrically frustrated triangular lattice
could be described in terms of valence bond singlet pairs[23]. Due to strong quantum
fluctuations, the pattern of dimers covering the lattice could easily fluctuate. Therefore the true ground state wavefunction is a superposition of all possible patterns of
dimers covering the lattice. Anderson named this ground state the resonating valence
bond (RVB) state. This state does not break lattice translational symmetry or spin
rotation symmetry and supports deconfined spinon excitations. Therefore, the RVB
state is a 2D analog of the ID quantum spin liquid state described above.
The RVB state can be constructed on lattices other than the triangular lattice.
In fact, the RVB state on a square lattice was proposed to explain the phenonmenon of high T, superconductivity in doped cuprate materials[24]. However, both
the square lattice and triangular lattice antiferromagnets have been shown to have
ordered ground states[25, 26, 27]. The spin-j kagom6 lattice antiferromagnet stands
out as a promising candidate to display spin liquid behavior because of its geometric frustration and lowered connectivity as compared to the triangular lattice.
Though the exact ground state of the spin-1 kagom6 lattice antiferromagnet is still
a matter of debate, the theoretical consensus is that the ground state should be
disordered[28, 29, 30, 31, 32]. The details of this ground state are still an open question of great interest. Therefore, real materials featuring antiferromagnetic coupling
on a kagom6 lattice are desireable in order to experimentally investigate this question.
Many different spin-1 kagome lattice antiferromagnets have been investigated
in recent years.
Among them are the copper-based minerals vesignieite[33] and
herbertsmithite[34]. In this thesis we present studies of isostructural analogs of both
materials. Herbertsmithite has been demonstrated to be the best experimental candidate to display spin liquid physics. Herbertsmithite displays no magnetic ordering
down to the lowest measured temperatures, and there is no evidence of a spin gap
down to very small energy scales of J/170[35]. Furthermore, the low-energy magnetic
excitation spectrum is dominated by a continuum of spinon excitations[36].
28
VJ 4
X
Figure 1-5: Localized excitations in the kagome ferromagnet. Left: the kagom6
ferromagnet has a simple colinear ground state. Right: A localized excitation from
the ground state. The spin flip is shared among the six spins around the hexagon
(red). Neighboring spins (black) cannot lower the total energy by rotating, making
this excitation independent of the rest of the lattice.
1.2
Ferromagnetism on the Kagome Lattice
Research on kagom6 lattice systems has historically been focused mostly on the antiferromagnetic case due to the effects of geometric frustration, as described in the
previous section. Recently there has been increasing interest in the ferromagnetic
case as well. Though the ferromagnet does not have a frustrated ground state, the
kagom6 geometry drives novel behaviors, some similar to the antiferromagnetic case
and some distinct. Kagome lattice ferromagnets are rare, and much of this thesis is
focused on the study of a new ferromagnetic kagome material.
1.2.1
Flat Mode in the Kagom6 Ferromagnet
The effects of the kagome geometry on the behavior of a ferromagnetic system are not
immediately obvious. Consider Heisenberg spins on a kagom6 lattice. As shown in
Figure 1-5, the ground state is a simple colinear arrangement of all the spins. Rather
than having a large manifold of degenerate ground states, the kagome ferromagnet
has a single ground state, up to a trivial global spin rotation.
29
Though the ground state is simple, the effects of the kagome geometry become
apparent when considering the excited states. Just like in the antiferrmagnetic case,
the kagom6 ferromagnet supports localized modes. Unlike the antiferromagnetic case,
these localized modes are excitations at finite energy. Consider an excited state that
consists of a single spin flip divided among the six spins around a hexagon, as shown in
Figure 1-5. Each of the six spins is rotated by 300 so that the total change in angular
momentum is equivalent to that of reversing one spin, with nearest-neighbor spins
around the hexagon rotated in opposite directions. This exciation clearly costs a finite
energy, as it has raised the energy of the six bonds connecting these rotated spins
as well as the twelve bonds connecting the rotated spins to their unrotated nearestneighbor spins. Consider these unrotated nearest-neighbor spins, colored black in
Figure 1-5.
Each of these spins has both a left-rotated spin and a right-rotated
spin among its nearest neighbors. Because these two spins are rotated by the same
amount from the unrotated spin but in opposite directions, the overall energy of the
system cannot be lowered by rotating the unrotated spins. Therefore this excitation
is completely isolated from the rest of the lattice.
Furthermore, the six spins can be rotated simultaneously without changing the
energy of the excitation, as shown by the dashed lines in Figure 1-5. This is reminiscent of the zero-energy mode that exists in the vF3 x v/5 antiferromagnetic arrangement (Figure 1-4). Therefore, the kagome ferromagnet also supports a large
density of localized modes, though they are excited states at finite energy. Like the
antiferromagnetic case, the existence of this localized exciation is a consequence of the
geometry of the kagom6 lattice. Every spin that neighbors this excitation necessarily
neighbors two spins, with one spin rotated in each direction. This is not the case
even for the triangular lattice, in which it would be possible for a spin to neighbor
only one of the excited spins.
Above we have discussed the localized excitation in terms of classical spins, but
this behavior also survives a more quantum mechanical treatment.
The Heisen-
berg ferromagnet can be directly related to a bosonic hopping model by use of the
Holstein-Primakoff transformation[37, 38]. Hopping models on the kagome lattice
30
Figure 1-6: Localized excitations in the kagom6 lattice hopping model. Excitations
are superpositions of creation operators around closed loops or along infinite lines
with creation operators on neighboring sites being of opposite phase. The modes are
localized by destructive interference.
show that localized states can be constructed consisting of linear combinations of
creation operators[39].
The creation operators are arranged around a closed loop
with operators on nearest-neighbor sites being of opposite phase, as shown in Figure
1-6.
Thus the localization of the mode can be understood in terms of destructive
interference. The amplitude to hop to an external site is exactly canceled by the two
neighboring sites. In fact these states can be constructed not only on closed loops
but on inifinite lines in the lattice, just like the zero-energy mode that exists in the
antiferromagnetic q = 0 ordering (Figure 1-4).
We can use this hopping model representation to calculate the dispersion of the
kagome ferromagnet excitation spectrum. The dispersion is shown in Figure 1-7. As
we expect from the localization of the excited states there exists a dispersionless mode
at finite energy.
An interesting result from the treatment of the hopping model in
Reference [39] is that the dispersive mode must touch the flat mode at q = 0. This
touch point is protected and survives many perturbations to the Hamiltonian. Any
perturbation to the Hamiltonian which does break this degeneracy and open a gap
must necessarily destroy the flatness of the band.
31
3
2
1
100
0.5
0.5
t-k 2k 0) (r.Lu.)
0.6.0
1.0
{h 0 0) (r.lu.)
Figure 1-7: Spin wave dispersion of the Heisenberg ferromagnet on a kagom6 lattice.
The highest energy mode is completely flat and touches the dispersive mode at q'= 0
1.2.2
Kagom4 Lattice Flat Band and the Fractional Quantum Hall Effect
Systems that contain flat bands are of great interest because the interactions between
particles in these bands become completely nonperturbative. As all particles possess
the same kinetic energy, their interactions will dominate the behavior of the collective
system. Thus these systems provide an ideal environment to search for interesting
many-body behavior. One example of the exciting many-body behavior that can be
found in flat band systems is the fractional quantum Hall effect.
The quantum Hall effect[40] is the quantization of electrical transport in 2D electron gasses placed in a magnetic field. At specific magnetic field strengths the longitudinal resistance vanishes while the transverse (Hall) resistance is quantized to a
rational multiple of h/e 2 . The integer quantum Hall effect, where the Hall resistance
is quantized to h/(ie2 ) with i an integer, can be described in terms of single particles filling Landau levels, which are just the result of placing a 2D electron gas in a
magnetic field[38]. These Landau levels are highly degenerate and dispersionless, and
in real space can be pictured as localized electron orbits. The fractional quantum
Hall effect[41] is a many-body effect, resulting from interactions between electrons in
fractionally filled Landau levels. Because these Landau levels are dispersionless, the
32
J
interactions dominate and are able to drive a number of novel collective behaviors,
among them particles with only a fraction of the electron charge[42].
The quantum Hall effect is not only a property of 2D electron gasses in a magnetic
field. It was shown in 1982 that the integer quantum Hall effect could exist even in
a strong periodic potential provided the chemical potential falls in a gap[43]. In this
case the Hall conductance could be mapped to a topological invariant associated with
filled bands, called the Chern number. Furthermore, it was shown by Haldane in
1988 that the Hall effect could exist in a lattice model without any net magnetic
field, provided time reversal symmetry is broken[44]. Haldane's model is built on a
honeycomb lattice with a periodic magnetic field such that the total magnetic flux
through the unit cell is zero. A particle which hops around a closed loop will encircle
magnetic flux and thus gain a phase due to the Aharonov-Bohm effect[45]. In this
model there exist inequivalent closed loops within the unit cell such that a particle
can gain a different phase depending on its path. This inequivalence is what allows
the existence of the Hall effect.
Much more recently, it has been shown that the fractional quantum Hall effect can
also exist in a strong periodic potential and zero net magnetic field. Called fractional
Chern insulators, these models show that particles in fractionally filled bands with
nonzero Chern number can produce fractional excitations and exhibit phenomena
such as the fractional quantum Hall effect in the absence of Landau levels[46]. A
number of different models have been proposed[47, 48, 49], all of which include nearly
flat bands with nonzero Chern number. The bands are tuned to be flat in order for
the electron interaction to dominate the behavior.
The model considered in Reference [47] is of particular interest in our case because
it is built on a kagome lattice. Similar to Haldane's model on the honeycomb lattice,
the kagom6 lattice unit cell contains inequivalent loops, which leads to a Hall effect
when spin-orbit coupling is included.
In this system again, the geometry of the
kagome lattice contributes to the emergence of novel behavior.
33
1.2.3
Magnon Hall Effect and Topological Edge Modes
The Hall effect in conducting systems is driven by the Lorentz force. Magnons are
charge neutral and thus do not feel the Lorentz force. However, the geometry of the
kagom6 lattice can lead to a Hall effect even for charge neutral particles, such as
magnons[50, 51, 52] and phonons[53, 54].
Magnons on a kagom6 lattice can experience a Hall effect due to the presence
of either a ring exchange interaction caused by an external magnetic field[50] or a
Dzyaloshinskii-Moriya (DM) interaction, which arises from spin-orbit coupling[51, 52].
If we consider the kagome ferromagnet in terms of a hopping model via the HolsteinPrimakoff transformation, both of these interactions result in a complex hopping
parameter. In other words, as a magnon particle hops from site to site it picks up
a complex phase. In traversing a closed loop the particle can pick up a nontrivial
phase. This can be thought of as analogous to the phase picked up by a charged
particle encircling a magnetic flux due to the Aharonov-Bohm effect. In this sense,
we can think of a closed loop in the kagom6 lattice as enclosing a finite 'fictitious
flux,' as shown in Figure 1-8. Due to the geometry of the kagom6 lattice, there exist
inequivalent loops within the unit cell, such that a magnon can pick up a different
phase depending on the path it traverses.
This fictitious flux will affect a magnon in exactly the same way that a true
magnetic flux would affect a charged particle. Thus the magnon hopping model with
ring exchange or DM interaction is analogous to Haldane's model, which produces the
Hall effect for electrons on the honeycomb lattice due to inequivalent loops within the
unit cell. We can then think of the DM interaction as producing an effective Lorentz
force which acts on charge neutral magnons.
The magnon Hall effect has been observed experimentally through measurements
of the transverse thermal conductivity[51, 52]. Measurements of thermal conductivity
serve as an effective probe for magnon behavior as they are sensitive to any excitations
that carry energy and not only to those that carry charge. Pyrochlore ferromagnets
Lu 2V 2 O 7 , Ho 2 V 2 0 7 , and In 2 Mn 2 O7 all show a nonzero thermal Hall responses. The
34
-2$
Figure 1-8: Kagom6 lattice unit cell (highlighted in yellow). 4 and -2# are the
fictitious fluxes enclosed by the triangle and hexagon, respectively, when traversing
the loop in the direction of the arrows.
35
pyrochlore can be thought of as a 3D analog to the kagome lattice, as it is made up of
corner-sharing tetrahedra. Additionally, the pyrochlore can be visualized as a lattice
consisting of alternating kagome and triangular planes along the crystallographic
[1 1 1] direction.
Another consequence of the DM interaction in kagome ferromagnets is the presence of topologically protected magnon modes that are confined to the edge of the
crystal[55].
The presence of a DM interaction leads to a nonzero Chern number
for the highest and lowest energy bands. Thus the kagome ferromagnet with DM
interaction has a topological band structure analogous to that of the topological insulator systems, except that the excitations are bosons rather than fermions. Just as
with electronic systems, edge modes are predicted to exist in the gaps between bulk
magnon modes. Because adjacent bands have different Chern numbers, these modes
are topologically protected. Furthermore, these modes are unidirectional, such that
magnons will propagate in only one direction along an edge, preventing backscattering. Bosonic topological edge modes have been observed in a photonic crystal[56],
but have yet to be observed experimentally in a magnon system.
The kagome lattice geometry drives novel behavior in magnetic systems. Changing
the sign of the magnetic coupling from antiferromagnetic to ferromagnetic results in
a drastic change in the nature of the magnetic behavior. Kagome antiferromagnets
feature a highly degenerate frustrated ground state, which for small spins can be
disordered even at zero temperature.
The kagome ferromagnet features a simple
ground state, but displays novel magnetic behavior in its excited states. The exciation
spectrum includes a flat mode, and the addition of a Dzyaloshinskii-Moriya interaction
results in a band structure analogous to that of the topological insulator systems.
However, real materials featuring ferromagnetic coupling on a kagom6 lattice are
rare. Many of the proposals to investigate the behavior of the kagome ferromagnet
involve the use of thin films of a pyrochlore system to approximate a kagom6 lattice.
In this thesis we present studies of a new material that is an ideal model system in
which to investigate the behavior of the kagome ferromagnet.
36
1.3
Thesis Outline
In this thesis we report the results of thermodynamic and neutron scattering experiments on three low-spin kagome compounds. The majority of this thesis focuses on the
S = } kagom6 ferromagnet Cu(1,3-bdc). However, we also present the results of studies of two different kagome antiferromagnets: the S = I2 series MgCu 4 _,(OH)6 Cl 2
.
and the S = 1 compound BaNi 3 (OH) 2 (VO4 ) 2
In Chapter 2, we provide a description of the experimental techniques. General
properties of the neutron and its scattering cross section are described. Properties
of the scattering structure factor are reviewed with a focus on magnetic scattering.
The principles of operation of the triple-axis and time-of-flight neutron spectrometers
are described. Aspects of the two instruments' resolution effects and approximations
that will be used for analysis of our data are also discussed. We then describe the
process of growth of single crystal samples, as well as the partial alignment of many
single crystals to form a sample large enough for neutron scattering measurements.
In Chapter 3, we describe the crystal structure of the kagome ferromagnet Cu(1,3bdc) and show that it is an ideal model kagom6 system. We present magnetization and
specific heat measurements that suggest a ferromagnetic transition. We also present
the results of neutron scattering studies of the magnetic structure. Magnetic Bragg
peaks show that while each kagom6 plane orders ferromagnetically, neighboring planes
order antiferromagnetically. An analysis of the integrated intensities of the Bragg
peaks, taking into account the effects of the vertical beam divergence, shows that
spins are oriented parallel to the kagom6 plane at zero applied field. We also present
the temperature dependence of the inelastic scattering spectrum which demonstrates
that 2D correlations persist well above the 3D transition temperature.
In Chapter 4, we present the results of inelastic neutron scattering measurements
perfomed on Cu(1,3-bdc), which reveal the magnon excitation spectrum. These measurements were performed with magnetic fields applied both parallel to and perpendicular to the kagom6 plane. An out-of-plane field opens gaps in the magnon spectrum
while an in-plane field does not. We introduce the Dzyaloshinskii-Moriya (DM) in37
teraction and discuss how the symmetries of the Cu(1,3-bdc) crystal lattice restrict
the possible orientations of the DM vectors. We model our system as a Heisenberg
ferromagnet with DM interaction. By fitting this model to our data we precisely
determine the parameters of the spin Hamiltonian.
In Chapter 5, we present the results of thermodynamic and neutron scattering
studies on two antiferromagnetic kagome systems. The series MgCu4 _ (OH)6 C
2
is
isostructural to the paratacamite family, Zn.Cu 4 _2(OH) 6 Cl 2 . The Mg series displays
much of the same behavior as the paratacamite series, but has the advantage that
magnesium and copper are easily distinguised by conventional neutron and X-ray
scattering techniques. The second material, BaNi 3 (OH) 2 (V0 4 )2 , is a structural analog
to the mineral vesignieite, BaCu3 (OH) 2 (VO 4 ) 2 . The Ni2 + ion carries a spin-1 moment
as compared to the spin-A moment of the Cu 2+ ion. Ni-vesignieite material shows
evidence of competing antiferromagnetic and ferromagnetic couplings, which lead to
a spin glass transition.
38
Chapter 2
Experimental Techniques
2.1
Neutron Scattering
For over 100 years, scattering techniques have been used to probe the microscopic
properties of condensed matter systems. In 1912, Max von Laue developed the technique of X-ray diffraction, a breakthrough that led to advances in crystallography
and other fields. X-ray scattering remains a powerful tool for studying materials even
today, but does have its limitations.
Neutron scattering has proved an invaluable
tool in the study of magnetic structures and low energy excitations, two areas where
X-ray scattering can be difficult to apply.
In the early 1930's it was found that bombardment of light elements such as
beryllium and boron with alpha particles produced radiation that was more penetrating than any known -- rays[57, 58]. In 1932 James Chadwick performed a series
of experiments showing that this radiation was not a gamma ray at all, but rather a
chargeless particle with mass similar to that of the proton[59]. Chadwick is credited
with the discovery of the neutron, and was awarded the Nobel Prize in Physics in
1935. Simple experiments perfomed in 1936 showed that scattering neutrons off of
crystals produced diffraction patterns similar to those produced by X-rays[60]. In the
1940's, newly built nuclear reactors provided neutron sources with far greater flux,
allowing for the development of more sophisticated scattering techniques by Clifford
Shull and Bertram Brockhouse. Neutron scattering has been so vital to the field of
39
condensed matter physics that Shull and Brockhouse were awarded the Nobel Prize
in Physics in 1994.
In the following sections an overview of the principles of neutron scattering is
provided. A more detailed treatment of neutron scattering theory can be found in
the textbooks by Lovesey[61] and Squires[62].
2.1.1
Properties of the Neutron
In their award to Shull and Brockhouse, the Nobel Prize committee described neutron
scattering as a technique that can show us "where atoms are" and "what atoms do."
Neutrons also allow us to measure the local magnetic moments of unpaired spins.
The neutron possesses an amazing combination of fundamental properties that make
it an ideal probe with which to study condensed matter systems.
Firstly, the neutron has a mass comparable to that of the proton, m" = 1.675x 10-27
kg. This large mass makes it relatively easy to control the energy of neutrons by allowing them to collide with a collection of atoms of similar mass, such as hydrogen,
called a moderator. This results in a distribution of neutron energies that is approximately a Maxwell-Boltzmann distribution characterized by the temperature of the
moderator. The neutron mass also dictates that thermal neutrons, which are produced by a room temperature (T ~ 300 K) moderator, have de Broglie wavelength
1 - 4 Aand energy 5 - 100 meV. These wavelengths are similar to the interatomic
distances in most condensed matter systems, which means that scattering of neutrons
from a condensed matter system will exhibit interference effects that can be used to
gain information about the system. In addition, the neutron energies are comparable
to the energy scales of many structural and magnetic exciations in condensed matter
systems. Thus, changes in the neutron energy due to a scattering process that creates
or annihilates one of these excitations will be easily detectable, as the change will be
a large fraction of the neutron's initial energy. This leads to high energy resolution
and is markedly different from inelastic X-ray scattering, where
-
100 meV repre-
sents only a small change in the X-ray energy of order 1 - 50 keV, and is much more
difficult to detect. Even larger ranges of neutron energies are accessible by using 'hot'
40
and 'cold' moderators.
Secondly, the neutron has zero net charge.
The absence of charge makes the
neutron interact weakly with matter, which provides several advantages. It allows
the neutron to penetrate deeply into the sample, so that the scattered signal is a
function of the bulk properties and not due to surface effects. The absence of Coulomb
repulsion also allows the neutron to pass close enough to nulcei to interact via the
short-range strong force interaction. The weakness of the interaction with matter
is also beneficial in that the neutron may be treated as a small perturbation to the
system, allowing for a relatively straightforward interpretation of the scattered signal
with few theoretical assumptions.
Finally, the neutron has a magnetic moment A2
nuclear gyromagnetic ratio, pN
=
'
=
-- YIN& where y = 1.913 is the
is the nuclear magneton, and 8 is the Pauli
spin operator for the neutron. Therefore the neutron will interact with magnetic
moments in the sample arising from unpaired electron and nuclear spins and from
orbital electron moments via the dipole-dipole interaction. The effective magnetic
scattering length for interaction with a single electron (moment 1 [B) is y
10-12
-
0-2695 x
cm where ro is the classical electron radius. This scattering length is comparable
to the scattering length of most typical elements. Thus the nuclear and magnetic
scattering cross sections will be similar, and both scattering processes will contribute
significantly to the measured signal. This fact sets neutron scattering apart from
other scattering probes and makes it extremely valuable in the study of magnetic
systems.
2.1.2
The Neutron Scattering Cross Section
In any neutron scattering experiment, a neutron with incident momentum ki and
energy E is scattered by a sample, leaving with final momentum kf and energy Ef.
The parameters of interest are:
Q =i
W
2
- kf
1
2m,
(ki - k2)
41
(2.1)
(2.2)
(-.2
Which are the momentum and energy lost by the neutron during the scattering process. By conservation of momentum and energy, Q and w are the momentum and
energy, respectively, gained by the sample. Here we have used units where h = 1.
The quantity directly measured by neutron scattering spectrometers is the partial
differential cross section,
d
2
. This quantity is the rate at which neutrons with
final energy between Ef and Ef + dEf are scattered into a solid angle d2, normalized
to the incident neutron flux. It is straightforward to calculate this cross section using Fermi's golden rule because the weakness of the neutron interaction with matter
makes it unlikely that any neutron will be scattered more than once while traveling
through the sample. The partial differential cross section for a scattering process in
which the neutron energy, wavevector, and spin polarization are changed from Ej, kj,
ao to Ef, kf, af and the sample state is changed from Ai to Af is
d
2kd
d~dEf xf
v
.
- f
27rh2)
(kfaf Af
kiauA)
6(hw - Ej + Ef)
(2.3)
Where V is the the interaction operator between the neutron and the sample. In
a scattering experiment, we do not measure the cross section for a specific transition. Rather, the measured signal is obtained by averaging over the available initial
sample states and summing over all possible final states. Thus the measured partial
differential cross section is
d2
d~dEf
-
kf
kf
)
27rh2
A~ ir
p
ia
aKAf V kriuA) 6 (hw -E+Ef)
If
(
(2.4)
Where p\ is the probability that the sample is initially in state Ai and p, is the
probability that the neutron is initially in spin state ai.
The calculation is further simplified by the weakness of the neutron-sample interaction because the neutron wavefunction is not significantly perturbed by the scattering process. This allows the use of the first Born approximation, which treats both
the incident and final neutron states as plane waves. Using this approximation, the
42
interaction matrix element becomes
(ifAf Vf~ kiAj)
V (Q) (Af IZeQaAi)
=
(2.5)
Where 'y are the coordinates of the scattering centers and V(Q) is the Fourier transform of the interaction potential.
V(Q)
=
JdiV(-) e
(2.6)
Nuclear Scattering
The scattering of neutrons from nuclei is due to the strong force interaction. While
there is no complete theory of this interaction, it is known to have a very short range
(~
10-3 cm), which is about 10 times smaller than the typical thermal neutron wave-
length. Therefore the interaction can be treated as pointlike, making it spherically
symmetric[62] and characterized by a single parameter called the scattering length,
b. b can be complex, with the imaginary component corresponding to the absorption
cross section. We model the strong force interaction using a Fermi pseudo-potential,
which gives isotropic scattering when applying the first Born approximation.
The
interaction potential between a neutron and a lattice of nuclei is then given by
2wr
b 6(r- f r)
Z
Mn
(2.7)
.
V(i) =
Where f is the location of the jth nucleus and bj is the scattering length of that
nucleus.
Using this potential along with (2.4), it is possible to calculate the partial differential cross section in terms of correlation functions, as developed by Van Hove in
1954[63]. The cross section for neutrons scattering from a monatomic lattice becomes
d~dEf
= N
nuclear
1-[cS (Q, w) + -iSi(Q, w)
ki 47r
(2.8)
Where N is the number of nuclei and a, and a- are the coherent and incoherent parts,
43
respectively, of the total scattering cross section for the nucleus. The incoherent
scattering arises because not all nuclear scatterers are identical due to variations of
isotope or nuclear spin orientation. These variations cause differences in scattering
length at different atomic locations. o, is due to coherent scattering and is proportional to the square of the average scattering length
mean-square deviation of the scattering length
(
-
1b1
I
2
. oa is proportional to the
2). S(Q, w) is the dynamic
nuclear structure factor and is given by
1
S(Q,w) = 27rN
Where
] dr e
fr' r (.9
dt
-wt)f d ' (,(i'- ", 0)p(f', t))
(2.9)
(r', t) gives the number density of particles at position f and time t. The dy-
namic nuclear structure factor is the time and space Fourier transform of of the timedependent pair correlation function ((F'
- F, 0)1 (i',t)). The incoherent scattering
function Si (Q, w) is the Fourier transform of the self time-dependent pair correlation
function,
(3(i',
O),(f', t)), which measures correlations between the same particle at
different times.
Magnetic Scattering
Magnetic scattering arises from the interaction of the neutron with the extended
dipole magnetic field produced by unpaired electrons on magnetic ions. Specifically,
the interaction potential of a neutron in the magnetic field H, produced by a single
electron is
ma()
= -An
(2.10)
'He
where[64]
He = V x (
X
r3
+ (e
C
r3
(2.11)
and r' is the neutron-electron separation, Ae is the electron magnetic moment, and
Je
is the electron velocity. The first term is due to the spin of the electron and the second
due to its orbital motion. For most magnetic transition metal ions, the presence of
a crystal field lifts the ground state degeneracy of the d-orbitals.
As a result, the
orbital angular momentum is 'quenched' and only the spin moment will interact with
44
the neutron.
As with nuclear scattering, the partial differential scattering cross section for magnetic scattering can be expressed in terms of a dynamic structure factor. For unpolarized neutrons, the cross section for scattering by localized spin moments is given
by
f( r) 2 [gf(Q)e-W
E 8~
d2 d
pin = N ki(2a"3
2
(sa - Q
where a, 3 refer to the xy,z vector components.
)Sa
'(Q,W)
(2.12)
This expression differs slightly
from the expression for the nuclear scattering cross section. Electron orbitals extend
over much larger length scales than nuclei, so the interaction between the neutron
and electron spins cannot be treated as pointlike. This results in the prefactor
f(Q)
called the magnetic form factor, which is the Fourier transform of the normalized
unpaired spin density p, (i) on a single magnetic ion:
f(Q)
=
Jd-- p.(r)e"'
(2.13)
The Debye-Waller factor e-w is an attenuation that occurs due to thermal fluctuations of atoms about their mean positions.
For measurements presented in this
thesis, temperature and Q are low enough that this factor may be neglected. The
factor (J,, - QaQg) arises because of the dipole-dipole interaction, and expresses
the fact that only the components of electron spin perpendicular to the momentum
transfer Q will contribute to the scattered signal.
For magnetic scattering, the dynamic structure factor is
Saa(Qw) =
3
dt e-ut Z eQ4r(Sa(o)S (i",t))
(2.14)
which is the time and space Fourier transform of the spin pair correlation function
(S"(0, 0)Sfl(?, t)). Just as with nuclear scattering, spin correlations lead to both
incoherent and coherent magnetic scattering. Correlations between the same spin
45
at different times (Sa(O, O)SO(O, t)) lead to incoherent scattering. We will consider
the coherent magnetic scattering by separating it into elastic and inelastic scattering
contributions. The elastic contributions arise due to time-independent correlations
between two different spins, (Sa(O,O)SO(y,0)). Inelastic contributions are due to
dynamic correlations, (Sa(0, 0)S(?, t)) - (Sa(0, 0)S,(8, 0)).
2.1.3
Bragg Scattering
Bragg scattering arises from nuclear and magnetic correlations that are static, i.e.
that persist in the long-time (t -+ oo) limit. These correlations result in scattering
characterized by 3-functions at w = 0 and Q = G where G is a reciprocal lattice
vector. To obtain expressions for the elastic scattering differential cross sections, we
consider the time average of the density function ,(f) and spin operator S(r) and
integrate the partial differential cross section over final neutron energies Ef. For
nuclear scattering, the coherent elastic differential cross section is
do.
(21r)3
elastic
nuclear
-
-
(>
N
-
(Q -G)
-
2
(2.15)
FN(Q)
G
where FN is the static nuclear structure factor
e(2.16)
e
FN(Q)
N is the number of unit cells, V is the volume of one unit cell, and v sums over the
atoms in the unit cell. dv, b, and e-WV are the position, nuclear scattering length,
and Debye-Waller factor, respectively, for atom v.
For magnetic scattering, the coherent elastic differential cross section is
= NM
dQ
magnetic
6(Q -GM)
(7
M
GM
46
Fm(Q)
2
(2.17)
where FM is the static magnetic structure factor
Fm(Q)
=
f
2
e4de-wv
(2.18)
V
=
x (S, x Q) is the f,(Q)
componentisofthe
the form factor of the vth atom and
Q
spin perpendicular to the scattering vector Q. The subscript m denotes the magnetic
unit cell and reciprocal lattice vectors. It is possible for magnetic and structural unit
cells to be different, and this is often the case for antiferromagnets. For Cu(1,3-bdc),
we will see that the structural and magnetic unit cells are the same and so for further
calculations will often drop the subscripts.
2.1.4
Inelastic Magnetic Scattering
We will restrict our discussion of inelastic neutron scattering to the scattering due to
the magnetic interaction. For a discussion of inelastic scattering due to the nuclear
interaction, see [61] or [62].
The inelastic contribution to the scattering cross section is obtained by subtracting
the elastic contribution from the total magnetic cross section. From (2.12):
d2. inelastic
d~~r=NK
dQdEf
spin
k
yro
k
22
2
[
W
gf (Q)e[g6
QSOJSO
2
(a
0W]
ng
S(Q,)
"(Q,
(2.19)
It is often beneficial to represent the inelastic part of the scattering function in
terms of the generalized dynamic susceptibility, which makes use of the language of
linear response theory[65]. Linear response theory is concerned with the change in a
given variable in response to a time-dependent perturbation. When both the perturbation and the response are small, only terms to linear order need be included. Since
the magnetic interaction between neutron and sample is weak, we can consider the linear response of the spin system to the perturbation provided by the magnetic moment
of the neutron. The neutron provides a time- and wavelength-dependent magnetic
field
H(Q,
t) = H(Q)e't and we define the generalized susceptibility function x(Q, w)
47
=)
in terms of the change in magnetization[66]:
(M (-, t)) - W (,
dt' HO(Q, t)XO3(Q, t - t')
t))j=o
(2.20)
where X(Q, w) is the time Fourier transform of x(Q, t). The imagniary part of the
generalized susceptibility, X"(Q, w), describes energy dissipation and can be related
to the inelastic dynamic structure factor by the fluctuation-dissipation theorem[67]
ceiatcQ
w)
=
-
e
IWkB
(2.21)
(n(w) + 1)X""Q(Q)
where n(w) is the Bose occupation factor.
We now consider the specific case of spin wave excitations in an ordered ferromagnet. By noting that spin waves involve small displacements of spins from an average
spin direction, we can simplify (2.19) with the substitution[61]
(50 - QaQ)S elastic(QW) =
-(1 + Q)Ssw(Qw)
(2.22)
where Q2 is the component of the Q unit vector that points parallel to the ordered
spin direction and Ssw(Q, w) is the inelastic scattering function for spin waves. For
a ferromagnet with a 1-atom basis[68],
Ssw(Q, w) = S E [(n(wq.) + 1)J(Q - q- GM)6(w - wo)
+nWT6Q+q-'_ G)6(w +wry)]
(2.23)
where q' represents a wavevector in the first Brillouin zone and WT is the magnon
dispersion relation. The first term in (2.23) corresponds to the creation of a magnon,
while the second term corresponds to the annihilation of a magnon. This expression
changes slightly when considering a lattice with a multi-atom basis. We will discuss
this expression in detail for the specific case of the kagom6 lattice in Chapter 4.
The delta function in energy in the expression for Ssw is a result of only including
48
spin terms up to quadratic order in our spin wave theory. In this picture, magnons
do not interact with each other and therfore have infinite lifetimes. In real systems,
interactions between magnons will lead to finite lifetimes and the scattering function
will not be infintely sharp in energy. To account for this effect, we can model magnons
as damped harmonic oscillators. For a full discussion of this model, see the text by
Chaikin and Lubensky[69]. The effect of damping can be accommodated by replacing
the delta functions in (2.23) with Lorentzians and simultaneously renormalizing the
magnon energies:
JWW
--WI
IF
w.
(2.24)
j
+ r72
where
W
2.1.5
=
q -
r2
(2.25)
Neutron Spetrometers
Neutron scattering measurements require high flux beams of neutrons, which are produced in one of two types of neutron sources.
Research reactors, such as the one
located at the NIST Center for Neutron Research, produce neutrons as a byproduct of
2
1
5
U nuclear fission. Spallation sources, such as those located at Oak Ridge
National Laboratory (ORNL) and at Rutherford Appleton Laboratory (RAL) produce neutrons by bombarding a heavy target, such as tungsten or mercury, with high
energy protons produced in an accelerator ring.
Many different types of neutron spectectrometers have been developed, making
neutron scattering a versatile tool relevant to many fields of study. Neutron scattering
data presented in this thesis were obtained using three different neutron sources:
the NIST Center for Neutron Research (NCNR), the ISIS facility at RAL, and the
Spallation Neutron Source (SNS) at ORNL. The instruments used can be classified
either as triple-axis or time-of-flight spectrometers. Following is an overview of the
49
principles of operation of these two basic types of spectrometer.
Triple-Axis Spectrometers
The triple-axis spectrometer is one of the most versatile types of neutron spectrometers, allowing the experimenter to probe scattering at carefully controlled values
of momentum and energy transfer. The concept was developed by Brockhouse in 1961
at Chalk River Nuclear Laboratory in Canada. A detailed description of the tripleaxis spectometer and its use in experiments can be found in the text by Shirane,
Shapiro, and Tranquada[68].
A schematic of a standard triple-axis spectrometer is shown in Figure 2-1. The
three axes are those of the monochromater, sample, and analyzer, as shown in the figure. At each axis the instrument allows a rotation of the scattering element (mochromater, sample, or analyzer) about its axis. This rotation angle is referred to as 0. The
arm of the instrument following each element also rotates, defining an angle referred
to as 20. The monochromater and analyzer are used to select the initial and final
neutron energies, respectively. The sample 29 angle determines the relative directions
of ki and k 1 and thus the magnitude of the momentum transfer Q. The sample 0
angle selects the direction of Q with respect to the sample's crystallographic axes.
The monochromater and analyzer select neutrons of a particular wavelength by
Bragg diffraction from a given set of lattice planes.
The SPINS instrument at the
NCNR used for experiments presented in this thesis uses pyrolytic graphite (PG)
for both monochromater and analyzer.
Pyrolytic graphite is composed of many
stacked hexagonal graphite layers with good c axis alignment but random in-plane
alignment[68].
The randomness reduces multiple Bragg scattering and incoherent
scattering, which helps to keep the background signal low.
PG crystals also have
high reflectivity, which helps to keep useable neutron flux high. For our experiments,
the (0 0 2) reflection was used. The SPINS instrument employs a vertically focused
monochromater, which further increases flux at the expense of the resolution of incident neutron momentum in the direction perpendicular to the scattering plane. The
analyzer can be used as a single crystal or can be focused in the scattering plane.
The focused mode improves intensity but significantly broadens the Q resolution.
50
Reactor
Collimator
Filter /
Monochromater
Monitor
k,
Sample
kf
Analyzer
Detector
Figure 2-1: Schematic of a standard triple-axis neutron spectrometer
51
The use of Bragg scattering to select neutron wavelength results in a beam that
contains the desried wavelengh as well as higher harmonics. Specifically, if a neutron
of wavelength A satisfies the Bragg condition for the PG (0 0 2) reflection at scattering
angle 20, then a neutron of wavelength A/2 will satisfy the Bragg condition at the
same scattering angle for the (0 0 4) reflction. Therefore, after passing through the
monochromater the beam contains neutrons of wavelength A but also wavelengths A/2,
A/3, etc. To remove these higher harmonics, a filter is placed after the monochromater. For thermal neutrons, PG is often used as a filter because transmission of
neutrons passing through PG parallel to the aligned c axis is highly dependent on
neutron wavelength. For cold neutrons, beryllium (Be) and beryllium oxide (BeO)
are commonly used. These materials take advantage of the fact that Bragg scattering
can only occur for wavelengths less than twice the largest d-spacing in the crystal,
known as the Bragg cutoff wavelength. The crystal will be nearly transparent to
wavelengths longer than this cutoff. These materials make effective low-pass filters
that only transmit neutrons with energies below 3.7 meV (BeO) or 5.2 meV (Be).
Be and BeO filters are often cooled with liquid nitrogen to increase transmission by
reducing scattering from thermally excited phonons.
Two other important elements used in the triple-axis spectrometer are the collimator and monitor. Collimators are used to reduce angular divergence of the neutron
beam. This defines the neutron flight path and also reduces background signal due to
randomly directed neutrons. Collimators consist of parallel thin plates of steel coated
with cadmium, which is a good neutron absorber. Most collimators have a horizontal
divergence of 10' - 80'. Tight collimation improves instrumental Q resolution but
will cut down on flux significantly. Typcially collimators are used only to restrict
the horizontal divergence, while vertical divergence is kept much wider to increase
neutron flux. A collimator is placed after each scattering element. The monitor is a
low-efficiency neutron detector placed in the beam after the monochromater. Scattered intensities are normalized to the monitor count in order to account for changes
in the incident neutron flux.
52
I
Choppers
Detectors
k,
Monitor
Sample
Figure 2-2: Schematic of a direct geometry time-of-flight neutron spectrometer
Time-of-Flight Spectrometers
The second class of neutron spectrometer used to collect data presented in this
thesis is the time-of-flight spectrometer. In 1935 it was shown that mechanical choppers could be used to select neutrons of a particular velocity[70].
A chopper is a
rotating disc that will block neutrons unless a small hole in the chopper overlaps
with the neutron beam path. Time-of-flight spectrometers make use of short pulses
of neutrons and thus are well suited to pulsed neutron sources. Choppers can be used
to create pulses of neutrons from the continuous beam produced by a reactor source,
but this has the disadvantage of removing a significant fraction of the neutron flux.
Two classes of time-of-flight instruments are used, called direct geometry and
inverse geometry spectrometers. Both types of spectrometers were used for measurements presented in this thesis. The LET instrument at ISIS and the CNCS instrument
at SNS are both direct geometry spectrometers, while the Iris instrument at ISIS is
an inverse geometry instrument.
Figure 2-2 shows a schematic of a direct geometry spectrometer.
In a direct
geometry spectrometer, a series of choppers and filters is used to create a pulsed
monochromatic neutron beam. This beam is scattered from the sample, just as in
a triple-axis spectrometer.
However, instead of a single detector, a time-of-flight
spectrometer has an array of detectors, and each of these detectors is time-resolved.
53
Because the neutron beam is pulsed, the flight time of the neutron from source to
detector can easily be calculated. The neutrons all have the same velocity to begin
with, so differences in flight time are due to different neutron velocities following
scattering from the sample. The time can therefore be used to calculate the final
neutron energy and, by conservation of energy, the energy transfered to the sample.
The neutron flight path can be broken into two legs: source to sample and sample
to detector. The total flight time of the neutron is then
(2.26)
t = ti + t1
where
tif
= --
(2.27)
m L?,f
'-
2Ejf
Vif
Where Mn is the neutron mass, Lif are the distances from source to sample and
from sample to detector, and vi, 1 and Ejf are the initial and final neutron speeds and
energies, respectively. The initital neutron velocity is set by the choppers, so tj is
known and is only a function of vi (or Es). tf is calculated from the measured value
of the flight time t, which gives the final energy. The energy transfer is then
hw
=
E-
Ef =
M
-
- (
t)]
(2.28)
An inverse geometry spectrometer operates on a similar principle. However, the
initial neutron pulse is a 'white' beam containing neutrons of many different energies.
In these instruments the choppers are used to create a short neutron pulse and to
define a range of initial neutron energies, but not to monochromate the beam. After
the beam scatters from the sample, it is scattered off an array of analyzer crystals
before arriving at the detectors. As in a triple-axis instrument, the analyzer scatters
neutrons of a selected wavelength onto the detectors. In the case of inverse geometry
spectrometers, differences in arrival time of the neutron allow for the calcuation of
initial neutron energy.
Time-of-flight instruments are very powerful tools in that they allow for simulta54
neous measurements over large areas of Q and w space. The time resolution provides
the ability to measure multiple energy transfers while the large array of detectors provides measurements at many different values of 20 and thus of Q. The main drawback
of these instruments is that their neutron flux tends to be lower than that availalbe
in triple-axis instruments. Time-of-flight instruments at reactor sources must cut out
much of the neutron flux in order to create a pulsed beam. At pulsed sources, these
instruments can take advantage of more of the available flux, but pulsed neutron
sources generally produce weaker beams than reactor sources. Therefore time-offlight instruments are advantageous when one is interested in measuring over a wide
range of Q and w but over a limited range of other variables, such as temperature,
applied magnetic field, or sample orientation. A triple-axis instrument is generally
preferable for an experiment targeting specific values of Q and w over a larger range
of temperature or magnetic field.
2.1.6
Instrumental Resolution
Because of the small cross sections for neutron scattering and the limited neutron
fluxes available, measurements are usually performed using neutron beams with finite angular divergence in order to increase neutron flux. On a triple-axis instrument,
relaxed collimations are used, and monochromaters and analyzers have finite mozaic
spreads. On a time-of-flight instrument, the holes in the choppers have finite size and
will overlap the beam for a finite amount of time allowing for an angular beam divergence and a pulse that is not infinitely narrow in time. This means that with both
types of instruments the neutron beam will have non-zero angular divergence and ki
and kf will not be perfectly well-defined. Spectrometers will accept a distribution of
neutrons with momenta near ki from the neutron source, and will detect a distribution of neutrons with momenta near If. Therefore the measured intensity will be a
convolution of the scattering function with the instrumental resolution function[68].
I(wo, Qo)
=
J
dw dQ -- R(w - wo, Q- Q0 )S(Q, w)
55
(2.29)
where
(2.30)
Vi = dki Pi(kz - k0)
and P(ki - 2) is the probability distribution of the incident neutrons with wavevector
ki and mean wavevector k2. Here, for simplicity, we have assumed prefactors such as
the nuclear scattering cross section are included in S(Q, W).
The resolution function R(w - wo, Q - Qo) is peaked at (wo, Qo) and decreases
as Q and w deviate from the mean values. The shape of the resolution function will
depend on the type of instrument and on instrumental parameters, and is in general
a complicated function of Q and w. Here we discuss briefly the resolution functions
for the triple-axis and time-of-flight instruments.
Triple-Axis Resolution Function
On a triple-axis spectrometer, if one assumes the transmission functions of collimators and the mozaics of monochromater and analyzer crystals are well described
by Gaussian distributions, then the resolution function can be expressed as a fourdimentional Gaussian distribution function. This distribution was first derived by
Cooper and Nathans in 1967[71], and is given by
-
#
1
-
#
R(w - wo, Q - Qo) = Roexp(--AQMAQ)
2
(2.31)
where
AQ =
"hQ
(W - wo), Q11 - Q0, QL, QZ)
(2.32)
and M is a 4x4 matrix. Qii - Qo, Qj-, and Qz are the deviations of Q from Qo along
the direction of Qo, perpendicular to Q0 but within the scattering plane, and perpendicular to both Q0 and the scattering plane, respectively. For a detailed explanation
of the triple-axis resolution function, the reader should see the text by Shirane, et
al.[68].
If we make the further assumption that the beam divergence is small, the resolution in the vertical (Q,) direction becomes decoupled from the other three coordinates
and M becomes a 3x3 matrix for w, Qii, and Qj and a 1x1 matrix for Q,. When
56
analyzing the intensities of Bragg peaks, it is important to take into account the
effects of the vertical component of the resolution function. This will be discussed in
more detail in Chapter 3.
The inelastic measurements presented in this thesis were taken on a partially
aligned crystal sample of a two dimensional material. Because of the partial alignment
and the two-dimensional nature of the material, we will be interested in the intensity
as a function of the magnitude of the in-plane wavevector, and not the full wavevector
Q. Additionally, we are measuring with broad collimations, a large sample mozaic,
and a focusing analyzer. For simplicity, we will make a further approximation
R(w - wo, Q - Qo) ~ R0(w - wo)RQ(IQI -
I$oj)
(2.33)
Where R, and RQ are described by Gaussians. This is a rough approximation but
will suffice for our analysis.
Time-of-Flight Resolution Function
The instrumental resolution function for a time-of-flight instrument is significantly
more complicated. The energy resolution function is not well modeled by simple Gaussian functions, as it is asymmetric in energy transfer and has a width that is strongly
energy dependent[72]. For a direct geometry instrument, the energy resolution will
tighten at larger energy transfers. This can be understood by noting that the spectrometer determines energy transfer by measuring arrival time of scattered neutrons
and calculating the speed of the detected neutron. Higher energy transfers correspond
to slower neutrons, which are further separated in time when they are detected. If
the detectors have a fixed time resolution, then the energy resolution will tighten at
higher energy transfers. Additionally, small deviations in the measured flight time
due to the finite pulse time width are less significant for the slower neutrons, which
further tightens resolution at higer energy transfers.
In Chapter 4 we present measurements taken using the direct geometry time-offlight instrument LET. In our analysis of these measurements, we will again make
the approximation (2.33).
However, we will approximate the instrumental energy
57
1.0
0.8
1.0
arVE Gaussiam
T
a)
b)
Time Gaussiau
0.l
OA
0.4'
O0A
0.2
0.2
0.0
0.4
-0.2
0.0
0.2
0.0
OA
-4
-2
w (MeV)
0
2
4
t (WeV)
1.0-
C)
=025
0.8-
0.5
-=0.75
0.5
0.4
0.2
0.0
0.2
0A
Aoi/Eq
0.6
0A
1.0
Figure 2-3: Properties of the time-of-flight energy resolution function. (a),(b) Time
Gaussians plotted with energy Gaussians of the same FWHM, with Ot (a) equal to the
fit value for our measurements on LET and (b) enlarged to better show the asymmetry
of the time Gaussian function. (c) The time Gaussian at different values of wO. The
width shrinks at larger wo and vanishes as hwo -+ Ej
58
resolution function (R,) as a Gaussian in time rather than a Gaussian in energy.
This function was chosen because it reproduces both the asymmetric lineshape of the
true resolution function and the tightening of peak width in energy at higher energy
transfers. Specifically, the LET energy resolution function is modeled as
2
(t(w)-t(wO))
Rw(w - wo) = Ae
(2.34)
2'cr
Where A is a normalization constant that depends on wo and o-t is a Gaussian width.
t(w) depends on both Ej and w and is found from (2.28):
L2
t(w) = ti +
ti
(2.35)
2rw
m
Figure 2-3 illustrates the key features of the energy resolution function, captured
by the time Gaussian. The time Gaussian is compared to an energy Gaussian of the
same full width at half maximum (FWHM) to show that it is asymmetric about wo.
This function also has a width that diminishes with increasing wo and vanishes as
hwo -+ Ej.
2.2
Sample Preparation
When working with powder samples, a scattering experiment measures an average of
scattering signals from different local crystal momenta due to the random orientation
of powder grains. This averaging obscures information about the local interactions in
the sample. Measurements of single crystals provide the ability to observe scattering
signal along specific crystallographic directions, providing much more clarity as to
the nature of observed crystal and magnetic structures as well as the microscopic
interactions and their associated dynamics. Large samples are required for neutron
scattering measurements because of the small neutron scattering cross section. However, the small cross section also makes the neutron a bulk probe that is not sensitive
to sample surfaces. Therefore, multiple single crystals can be aligned such that their
59
local crystal axes are oriented in the same directions and neutrons will scatter from
this collection of crystals as if it were a single large crystal. An important part of the
work presented in this thesis was the growth and alignment of many single crystals to
create a partially aligned sample. Most of the neutron scattering measurements presented in Chapters 3 and 4 were performed on this sample. Following is a description
of the process of single crystal growth and assembly of the sample used for neutron
scattering measurements.
2.2.1
Single Crystal Growth
Single crystals of Cu(1,3-bdc) were initially grown by Nytko, et al.[73]. Cu(1,3-bdc)
contains hydrogen, which is problemtic for neutron scattering measurements. The
incoherent scattering and absorption cross sections for hydrogen are very large[74], so
measurements of samples containing hydrogen have a reduced signal from absorption
and increased background from incoherrent scattering. In order to work around this
problem, deuterium is often substituted for hydrogen. Deuterium has significantly
smaller incoherent scattering and absorption cross sections than hydrogen, and its
substitution does not strongly alter the crystal structure of most materials.
Deuterated crystals of Cu(1,3-bdc) were grown using a similar procedure to that
reported by Nytko, et al. Initial treatment of Cu(OH) 2 with deuterated isophthalic-d4
acid (1,3-bdcH 2 ) under hydrothermal conditions led to formation of small crystals of
Cu(1,3-bdc), following the condensation reaction:
Cu(OH) 2 + 1, 3-bdcH 2 -+ Cu(1, 3-bdc) + 2H 2 0
(2.36)
To increase crystal size the reaction rate was slowed by using dilute nitric acid in
place of water. This can be understood by noting that the condensation takes place
once the isophthalic acid is deprotonated in solution. Increasing the concentration of
H+ ions already in solution by adding nitric acid reduces the concentration of 1,3bdc2- ions available to react with the Cu 2+ ions and thus slows the reaction. This
reaction was run multiple times with varying nitric acid concentrations to maximize
60
the crystal size. Crystals increased in size with acid concentration up to - 2% w/w
HNO 3 . Stronger concentrations prevented crystals from forming at all. The 2% HNO 3
was made by diluting 2.65 g 70% HNO 3 in 90 g deionized water.
To make enough crystals for neutron scattering measurements, many reactions
were carried out in 23 mL and 125 mL Teflon-lined pressure vessels. 23 (125) mL liners
were charged with 145 (435) mg Cu(OH) 2 , 250 (750) mg deuterated isophthalic acid,
and 8.5 (25.5) g 2% nitric acid in water. Liners were placed into steel hydrothermal
bombs. Bombs were heated to 1500 C and maintained at this temperature for 21 days,
removed from the furnace at temperature and cooled in air, resulting in clusters of
crystals of Cu(1,3-bdc) forming at the bottoms of the liners. Cooling was performed
in air because slow cooling of the bombs resulted in the dissolution of the Cu(1,3bdc) crystals and the formation of an impurity phase. Crystal clusters were washed in
deionized water and dried in air. Single crystal pieces were were manually separated
from clusters under a microscope, resulting in crystals with a mass typically 0.1 - 1
mg but up to 3 mg. This process was repeated to produce a collection of crystals
with total mass 1 g.
2.2.2
Sample Holder and Assembly
Single crystal pieces of Cu(1,3-bdc) formed as flat flakes with the c axis perpendicular
to the plane of the flat face. Crystals were arranged on flat aluminum plates to coalign
the ' axes. The orientation of the kagom6 plane of each crystal was not aligned, and
is assumed to be random. As discussed in the following chapter, Cu(1,3-bdc) consists
of 2D layers of Cu 2 + ions stacked along the ' axis with very little magnetic coupling
between neighboring planes.
A well-defined ' axis allows for measurements that
distinguish interactions within each Cu2+ plane from interactions between neighboring
planes.
Figure 2-4 shows the collection of single crystals on six aluminum plates. Crystals
were attached to both sides of the plates using Fomblin Y oil. Fomblin Y was chosen
because it is viscous enough to hold the crystals to the plates and because it contains
no hydrogen. Plates were then wrapped in aluminum foil to secure the crystals. The
61
six plates contain ~ 2,000 individual crystals with total mass 1 g. The plates were
held parallel to each other and mounted to an aluminum holder, as shown in Figure
2-4(b).
62
(a)
1
2
I
I
3
4
5
6
2cm
(b)
Figure 2-4: Sample assembly. (a) Six aluminum plates covered front and back with
single crystals of deuterated Cu(1,3-bdc). The ' axis of all crystals is aligned by
placing them flat on the plates. The orientation of the a and b axes is random.
(b) The assembled sample holder, with the six plates held parallel.
63
64
Chapter 3
Magnetic Order in Cu(1,3-bdc)
Cu(1,3-benzenedicarboxylate(bdc)) is a hybrid organometallic compound featuring
S=
j Cu2+ ions on a kagome lattice. Prior magnetic and specific heat measurements
on powder samples show evidence of a ferromagnetic ordering transition near T
=
1.8 K despite a negative Curie-Weiss temperature that suggests antiferromagnetic
nearest-neighbor interactions[73]. In contrast, muon spin resonance(/pSR) measurements have suggested that below the transition temperature the fluctuation rate of
the spins was slowed, but that there was no long range ordering of the moments[75].
In this chapter, we present thermodynamic measurements performed on single crystal
samples of Cu(1,3-bdc) as well as neutron scattering measurements on a powder sample and on the partially aligned single crystal sample described in Chapter 2, which
show the existence of a long range magnetic ordering transition. We use an analysis
of the measured Bragg peak intensities to determine the orientations of the ordered
magnetic moments.
3.1
Crystal Structure
The crystal structure of Cu(1,3-bdc) makes the material in many ways an ideal model
system in which to investigate the behavior of the 2D kagome lattice. The crystal
structure of Cu(1,3-bdc) is shown in Figure 3-1. Cu 2+ ions form 2D kagome planes,
with adjacent copper ions bridged by carboxylate groups. The kagome lattice is per65
a)
b)
C
Cu
0
0
0
C
H
--
---
MW
a
Figure 3-1: Crystal structure of Cu(1,3)-bdc.
(a) The hexagonal unit cell.
The
positions of copper (blue), oxygen (red), carbon (cyan) and hydrogen (white) atoms
are shown along with the unit cell basis vectors a, b, and c (b) An expanded view
of the crystal structure showing that copper ions form kagome planes separated by
benzenedicarboxylate (bdc) molecules. (1,3-bdc) molecules not conncected to the
center hexagons and all hydrogen atoms have been removed for clarity.
66
Empirical formula
C8 H 4 CuO 4
Formula weight
Crystal system
227.65 g/mol
Hexagonal
Space group
P63 /m
a
b
c
9.1081(2) A
9.1081(2) A
15.9432(5) A
a
90
#
-y
Unit cell Volume
1200
900
1145.41(5) A3
Table 3.1: Crystallographic data for Cu(1,3-bdc), determined by single-crystal X-ray
diffraction at 100 K. (From Ref. [73])
fect in the sense that it is composed of equilateral triangles with equivalent coupling
pathways between each nearest-neighbor copper ion pair. Kagome planes are stacked
in an AA fashion, with neighboring kagome planes well separated by large organic
(1,3-bdc) molecules. The long interplane coupling pathway suggests that the magnetic interactions between different kagome planes will be weak and thus the magnetic
behavior should be quasi two-dimensional. As a further benefit, the absence of metal
ions between planes suggests that the magnetic behavior of the system should be
dominated by that of the kagome planes. Crystallographic data on Cu(1,3-bdc) are
summarized in Table 3-1.
3.2
3.2.1
Thermodynamic Measurements
Magnetic Measurements
Magnetization measurements were performed on single crystal samples of Cu(1,3-bdc)
using a Quantum Design Magnetic Property Measurement System(MPMS). Measurements were done on both deuterated and protonated crystals. Magnetization was
measured as a function of temperature over the range 1.8 K - 350 K at a number
67
25
a
Hin plane Deuterated
-A-H in plane Protonated
e H out of plane Deuterated
v- H out of plane Protonated
20-
15-
E
10-
5-V-
0
6
5
4
3
2
1
0
Temperature (K)
800-
b)
6001
E
*X
-
400
E
X-X
200-
I-u--
0
50
100
200
150
250
300
350
400
Temperature (K)
Figure 3-2: Susceptibility of single crystal Cu(1,3-bdc), approximated as M/H.
(a) Susceptibility of deuterated and protonated samples measured as a function of
temperature with field H = 100 Oe applied both parallel to and perpendicular to the
copper kagom6 planes. (b) Inverse susceptibility of a deuterated sample measured
with field H = 5,000 Oe applied parallel to the kagom6 plane. Lines are Curie-Weiss
fits over the range 150 K - 350 K to the measured susceptibility (blue) and the
susceptibility corrected for the molecular diamagnetic contribution (orange).
68
of applied fields ranging from 20 Oe to 50,000 Oe. Measurements were performed
with the field applied parallel to and perpendicular to the kagome plane. Low temperature measurements were performed under both field-cooled and zero-field-cooled
conditions.
No difference was observed between these two measurements down to
temperatures of 1.8 K and fields as low as 20 Oe for either applied field direction.
Figure 3-2(a) shows the susceptibility X of Cu(1,3-bdc) approximated as the mag-
netization M divided by the applied field H, measured at a field of H = 100 Oe. No
difference in susceptibility is observed between protonated and deuterated samples.
This figure shows an upturn at low T, which is in qualitative agreement with previous
measurements performed on a powder sample at the same applied field[73]. However,
our single crystal measurements allow us to observe a dependence on the direction of
the applied field. The susceptibility shows an upturn at low T for both applied field
directions, but below ~ 4 K the susceptibility is greater for the field applied parallel
to the kagome plane.
Figure 3-2(b) shows the inverse susceptibility of a deuterated crystal measured at
H = 5,000 Oe. The field was applied parallel to the kagome plane. This plot shows
both the raw data (X) and the data corrected for the molecular diamagnetism of the
sample (X-Xo) by use of Pascal's constants[76].
the susceptibility due to the copper spins.
This correction is done to isolate
Both sets of data were fit to a Curie-
Weiss function over the temperature range 150 K < T < 350 K. Surprisingly, the
raw inverse susceptibility data is better fit by a linear model than the data corrected
for the sample's diamagnetism. The data also deviate less from the fit line at lower
temperatures. The fit to the raw data gives a Curie-Weiss temperature of Ecw = 3 t
1 K, while the fit to the corrected data gives
Ecw
= -10
1 K. A positive Curie-Weiss
temperature suggests a ferromagnetic nearest-neighbor exchange coupling, while a
negative temperature suggests antiferromagnetic coupling.
This sign descrepancy
makes it difficult to determine the coupling from the inverse susceptibility data, but
suggests that the coupling is small,
IJI ~ 0.1 - 1 meV.
The magnetization of Cu(1,3-bdc) was also measured as a function of field, as
shown in Figure 3-3. Measurements were taken at T = 1.8 K, 5 K, and 30 K with
69
a)
1.0-
0.5-
co 0.0-
-- a- H in plane Deuterated
H in plane Protonated
-e- H out of plane Deuterated
v H out of plane Protonated
-0.5-
-
A
-1.0
I
-2000
-1000
I
*
0
I
*
~
3000
2000
1000
Field (Oe)
b)0-
1.00-
H in kagome plane
* T=1.8K
*T=5K
0.75-
A T=30K
H out of kagome plane
v T=1.8K
4 T=5K
4 T=30K
: 0.50-
0.254
4-
0.100
0
20000
40000
60000
80000
Field (0e)
Figure 3-3: Magnetization of single crystal Cu(1,3-bdc) as a function of field applied
both parallel to and perpendicular to the copper kagom6 planes. (a) Magnetization of
deuterated and protonated single crystals measured at T = 1.8 K. (b) Magnetization
of a deuterated single crystal at T = 1.8 K, 5 K, and 30 K.
70
applied fields up to 7 T. As with the susceptibility measurement, the behavior observed here is simlar to the behavior measured on a powder sample, but we observed
a dependence on the direction of the applied field at low temperatures, as shown in
Figure 3-3(a). The spins are more easily polarized with the field applied parallel to
the kagome plane. In either direction, the spins are easily polarized, with the magnetization saturated by fields smaller than 20,000 Oe at T = 1.8 K. Very little hysteresis
is observed, with a coercive field less than 20 Oe at T = 1.8 K. As shown in Figure
3-3(b), at higher temperatures the difference between different applied field directions
goes away, with little difference seen at 30 K.
As shown in Figure 3-3(a), both protonated and deuterated samples were measured with very little difference observed between the two. At low field the magnetization is identical, and the two samples show the same dependence on field. The
only difference observed was a small variation in the saturation magnetization.
3.2.2
Specific Heat Measurements
The specific heat of a protonated single crystal sample of Cu(1,3-bdc) was measured
using a Quantum Design Physical Property Measurement System(PPMS). Measurements were performed in applied fields up to 14 T with field applied parallel to and
perpendicular to the kagom6 plane. Figure 3-4(a) displays the data measured with
the field perpendicular to the kagome plane. A peak is observed near T = 1.8 K in
zero applied field, which is similar to that observed in previous measurements and
consistent with the onset of magnetic order at that temperature. With increasing
applied field, the peak broadens and shifts to higher temperatures. Figure 3-4(b)-(d)
compare measurements with the field applied in the two different directions. No difference is observed at temperatures far away from the peak in the specific heat. At
temperatures near the peak, the specific heat is enhanced when the field is applied
parallel to the kagom6 plane. This effect is suppressed at high fields, and is no longer
detectable for fields greater than 5 T.
To investigate the nature of the magnetic transition, we examine the zero-field
magnetic specific heat as shown in Figure 3-5. The specific heat was measured as a
71
U
6
-
7-
-T
-
'0.7T
(a)
-*-0.8T
5-
A-0.OT -0-0.9T
0.2T-- IT
v-0.3T+1.5T
-0 O.4T -x- 5T
-
E
- -0.6T -
4-
-14T
E
-
-
4-
1OT
Ie
- -0.5T
-
6-
(b)
-
%H
3-
3-
2-22-
f
1
2
3
4
5
6
7
8
9
10
0
1
2
3
Temperature (K)
4
5
6
7
8
9
C
Temperature (K)
(d)
-
(c)
5-
5-
4-
4-
0~ 3-
0~ 3-
E
E
2-2-
2-20
0
1
2
3
4
5
6
7
8
9
10
Temperature(K)
0
1
2
3
4
5
6
7
8
9
10
Temperature (K)
Figure 3-4: Specific heat of Cu(1,3-bde). (a) Specific heat under applied fields up to
14 T with field perpendicular to the kagome plane. (b)-(d) Comparison of specific
heat with field applied perpendicular to the kagome plane (black) and parallel to the
kagome plane (red) at applied fields (b) 0.1 T, (c) 0.5 T, and (d) 5 T.
72
function of temperature with small spacing (AT = 0.02 K) between measured points
through the transition to map the shape of the zero-field peak. An applied field of 14
T shifts the magnetic feature in the specific heat to higher temperatures (- 10 K) and
suppresses the specific heat at low temperatures. The specific heat measured at 14 T
is thus assumed to be dominated by the phonon contribution in the temperature range
T ; 5 K. A 5th order polynomial was fit to the 14 T data and this fit function was
subtracted from the zero-field data to obtain a measure of the magnetic contribution
to the zero-field specific heat, shown in Figure 3-5(c).
Although there is a clear peak in the zero-field specific heat, it does not sharply
diverge.
Specific heat data near the transition temperature are not well fit by a
power law divergence. Below the ordering temperature Tc, the data are well fit by a
logarithmic divergence, as shown in Figure 3-5(c). The 2D Ising model results in a
logarithmic divergence of the specific heat near the transition temperature, as shown
by Onsager[77, 78]:
C(T) oc - In 11 -
) - (1 + -)(3.1)
- I+ In( k
4
2c
Tc
Where e is the coupling between nearest-neighbor Ising spins. Fitting this formula
to the specific heat of Cu(1,3-bdc) only below the transition gives a value of fe~
0.01 meV. The fit is far from perfect, but may suggest that the sytem could display
an Ising symmetry. We will see later that this value of e is at least an order of
magnitude smaller than the energies that determine the spin wave behavior. However,
it is possible the Dzyaloshinskii-Moriya vector in Cu(1,3-bdc) possesses an in-plane
component, which could be responsible for Ising behavior.
Above the transition temperature, the specific heat does not follow a logarithmic
divergence even allowing for a different set of fit parameters than those used below
Tc. There is a significant magnetic contribution to the specific heat at temperatures
well above the transition temperature, which decreases only approximately linearly
with temperature. Using the estimated zero-field specific heat, the magnetic entropy
was calculated and is plotted in Figure 3-5(d).
73
The magnetic entropy released in
Ali
o poH=OT
(b)
0
=14T
3
S30
S3
20
m
C
2
r.
t0
14
S=14T
0
10
30
20
40
2
1
0
50
5
4
3
Temperature (K)
Temperature (K)
1
(C)
(d)
0.8
.3
mem
+ 0.6
2
.20.4
0.2
0
0.0
-
Lt.
1
2
3
4
5
0
1
2
3
4
Temperature(K)
Temperature (K)
Figure 3-5: Zero-field peak in specific heat of Cu(1,3-bdc). (a),(b) Specific heat
measured at zero field and 14 T field applied perpendicular to the kagom6 plane. The
14 T data are assumed to be d ominated by the phonon contribution with negligible
magnetic contribution in the temperature range T < 5 K. (c) Estimate of zero-field
magnetic contribution to the specific heat obtained by subtracting 14 T data from
zero-field data. Purple line is a fit of the low temperature data to a logarithmic
divergence as described in the text. (d) Magnetic entropy starting at T = 0.4 K
calculated from the data in (c).
74
zero field below the transition temperautre T = 1.77 K down to the lowest measured
temperature of T = 0.4 K was found to be only 24% of kBln(2) per spin, while about
40% is released between 5 K and the transition temperature. At higher temperatures,
the approximation of the 14 T data as nonmagnetic becomes invalid, but the zero-field
magnetic contribution to the specific heat is nonzero at least up to T = 6.5 K, where
the zero-field and 14 T data intersect. Some entropy may also be lost at temperatures
lower than could be measured, but it is clear that a large fraction of the magnetic
entropy is released at temperatures well above the transition temperature.
3.3
3.3.1
Neutron Scattering Measurements
Elastic Scattering Measurements
Elastic neutron scattering measurements were performed on the deuterated ' axisaligned sample described in Chapter 2. Measurements were taken on the triple-axis
spectrometer SPINS at the NIST Center for Neutron Research(NCNR). The sample
was placed in a He-4 cryostat and oriented so that the coaligned ' axis was in the
scattering plane. Measurements were taken using a flat analyzer with configuration
80'-sample-80'-analyzer-open. To reduce contamination from high energy neutrons, a
Be filter was placed before the sample and a BeO filter after the sample. Neutrons of
energy 3 meV were used.
Longitudinal scans and 6 scans were performed through the (0 0 L) Bragg peak
positions for L = 1 - 5 at the base temperature of T = 1.6 K and at T = 24 K well
above the transition temperature, as shown in Figure 3-6. The (0 0 L) structural
peaks are forbidden for odd values of L because there are two identical kagome layers
per unit cell. At base temperature, Bragg peaks were observed at all five measured
positions, but at T = 24 K peaks at odd-L Bragg positions were seen to disappear.
To further examine the nature of these Bragg peaks, neutron diffraction measurements were performed on the same ' axis-aligned sample under an applied field.
Measurements were again taken on the SPINS spectrometer at the NCNR. The sam75
7n n(M)
I
211 IMMI
* T= 1.6 K
15000
150001
* T=24 K
U
2
U
10000
10000
0
:
us.
..e .s. .
U
5000
5000
-
0.8
0.9
1.0
1.1
80
1.2
100
90
L (r.Lu.)
0000
250000
M
(b)
2 50000
E 2C 0000
2
m 15 0 000
C
C
0
1. 50 000
U
10000
50000
1.8
1.9
2.1
2.0
90
2.2
110
100
L (r.Lu.)
0
- -- -
4000-
. . . . .
.
4000 -.
(c)
3000
134
120
0(0)
-
I
C
T= 1.6 K
*T=24K
I 00000
0 000
2
a
(g)
*
2 00000
0
120
300000
-
30
110
0(0)
(h)
3000
I
2000
.T=1.6K
*T =24 K
*
2000
10
2.8
3.0
2.9
3.1
100
3.2
140
130
120
110
L (r.Lu.)
0(0)
7000
6000
C
0
5000
4000
4000
3000
C
C
-
S
T=1.6 K
60000)
5000
3000
2000
2000
1000
1000 -
.a mr
*33535
3.9
3.8
4.1
4.0
T =24 K
Ua
mOODO
120
4.2
L (r.Lu.)
0(0)
2000--C
?nM0
*
(e)
1500
1500
(j)
1000
13
500
ag".
a
u T=1.6K
[11]
T=24K
2
~1000
-500
150
140
130
Em
4.8
4.9
*
..
5.0
:
u
U
.
wsa
C
C
0
5.1
130
5.2
L (r.Lu.)
*".
150
140
s.
160
0(0)
Figure 3-6: Elastic scans through (0 0 L) Bragg positions above and below the transition temperature under zero magnetic field. (a)-(e) Longitudinal scans and (f)-(j)
0 scans. Each 0 scan was taken at the peak L position of the neighboring longitudinal
scan. Peaks at odd values of L disappear above T,
76
3500Cr.
000
(0
3025000
(
30
25 000
.
01)
poH =0
I
(002)
= 0.025T
150000
=0.ST
20000
m
100 000
.
015000
10000000
*
:5000
0.9
0.8
aO
55000
1.0
1.9
1.8
1.2
1.1
2.0
2.1
2.2
L (rM.u.)
L (r.l.u.)
0
an5000
10000
7000
(0 03)
6000
'"eOH =0
C
-
d
S=0.025T
+
(0
8000'
04)
=0.5T
e-_ 6000-
4000
S3000
300
400040000
10;e+
S2000
a
oH=0
2.9
2.8
(006)
(005)
4.2
6.1
6.2
33
8000
= 0.025T
*
=0.5T
2500
MenC
2000
4.1
L (r.l.u.)
L (r..u.)
3000
4.0
3.9
3.8
3.2
3.1
3.0
Owosso
aa
2000
1000
20000
1500
1000
-
1 000
10000
.
00
4.8
4.9
5.0
L (r.I.u.)
5.1
5.2
5.8
5.9
6.0
L (r.I.u.)
Figure 3-7: Longitudinal scans through (0 0 L) Bragg positions at T = 70 mK.
Application of a magnetic field quickly suppresses the Bragg peaks at the odd-L
positions while increasing intensity at the (0 0 4) position.
ple was placed in a dilution insert in a 10 T magnet and oriented so that the coaligned
re etete,
netrns f ay nery
tha scattering
was applied parallel to the
fieldwllusull
the magnetic ignl
and athughthedeectd
plane
c axis was in the
kagome plane. Measurements were taken in two-axis mode with fixed initial neutron
energy 5.0 meV and collimator configuration 80'-sample-80'. A Be filter was placed
before the detector to reduce contamination from high energy neutrons.
Diffraction measurements were used in order to gain a precise measure of the
integrated Bragg peak intensity. In a diffraction measurement, no analyzer crystal
is used and the detector is placed directly in the path of the scattered neutrons, so
77
200 000
35000
30000
E poH= 0
(001)
(002)
=0.025T
S
25000
1-
.
0
150000
20000
.
U
15000
U
+U
ca
05
B
B
100000
+e
10000
50000
UR
5000
90
80
100
90
110
100
120
110
0(0)
0(0)
MAiW.1
6000
(003)
UP
S
U
025000
1
*
0
8000
= 0.025T
=0.5T
+
(004)
-"
6000
4000
03000
4000
0-U
*
Us
"
"m
,S
-
2000
2000
1000
100
110
120
110
00
130
120
140
130
0(0)
0(0)
3500
(005)
3000
*
C
2500
poH=0
= 0.025T
0
=0.ST
+
B
(006)
30000
2000
=
0
0
20000B
1500
10
~10000
1000
500.
gum
110
120
130
120
140
130
~.
U..
140
150
160
0(0)
0(0)
Figure 3-8: 0 scans at (0 0 L) Bragg positions at T = 70 mK. Application of a magnetic field quickly suppresses the Bragg peaks at the odd-L positions while increasing
intensity at the (0 0 4) position.
78
3x10 4
-1 - H decreasing
H~L ir
ire Inf
C~)
CrO
2x0
x
0
*~1X104
--
0~
-0.10
-0.05
0.00
0.05
0.10
POH (T)
Figure 3-9: (0 0 1) peak intensity plotted as a function of applied magnetic field at
T = 70 mK. Line is a guide to the eye.
be dominated by the elastic scattering. As will be discussed in Section 3.3.2, the
integrated intensity of a Bragg peak in a diffraction experiment has a very simple
dependence on the scattering angle 26. Including an analyzer crystal complicates this
dependence, especially for crystals with a large mozaic[68].
Longitudinal scans and 0 scans were performed through the (0 0 L) Bragg peaks
for L = 1 - 6 at the base temperature of 70 mK at zero field and applied fields up to
10 T. The results are shown in Figures 3-7 and 3-8. Bragg peaks were observed at all
six measured positions, consistent with our previous measurements. Peaks at (0 0 L)
for odd values of L were quickly suppressed by application of a magnetic field. These
peaks also disappeared above T = 1.8 K. Application of a magnetic field enhanced
the intensity of the peak at (0 0 4). The structural peaks at (0 0 2) and (0 0 6) are
very intense, making it difficult to resolve magnetic field dependence.
The field dependence of the (0 0 1) peak intensity is shown in Figure 3-9. The
peak is fully suppressed by poH ~ 0.05 T. The falloff in intensity is symmetric about
H = 0 and shows no sign of hysteresis. The peak at (0 0 4) reaches its peak intensity
79
-
20000-
15000- -
C)
0
)A
10000--
5000--
W 0
-
c
* Dilution Fridge T Stable
* Dilution Fridge T Sweeping
He-4 Cryostat T Stable
C2000 --=
1500 --
8
b)
1000_500--
-
0
0.0
0.5
1.0
1.5
2.0
2.5
Temperature (K)
Figure 3-10: Magnetic Bragg peak intensities plotted as a function of temperature.
(a) (0 0 1) peak intensity with zero applied field. Line is a fit to the data of the
function (1- f)2,3 x (normalization factor) with parameter values T, = 1.77(2) K and
/
=
0.246(3). (b) (0 0 4) peak intensity with applied field poH = 0.05 T. The (0 0 4)
peak intensity measured at T = 0.07 K and H = 0 was subtracted to remove the
structural peak and background intensities.
at the same field and is not enhanced further up to puOH = 10 T.
To examine the critical behavior of the magnetic transition, a temperature scan
of the zero-field (0 0 1) peak intensity was performed. Temperature control of the
magnet cryostat became unstable above 1.3 K. In order to investigate the behavior
near the transition temperature of -1 .8 K, we combined three different measurements
of the (0 0 1) peak intensity. First, below 1.3 K measurements of the peak intensity
were taken with the temperature stable at a set point. Second, measurements were
perfomed continuously in 30 s increments while cooling the sample from above 2
K to base temperature.
For these points a temperature error bar was included.
This error bar was estimated by comparing the temperatures of the previous and
80
following points. The third measurement was a temperature scan of the (0 0 1)
peak intensity performed during the initial elastic scattering measurements with the
sample in the He-4 cryostat. This allowed for measurement of the peak intensity
at stable temperatures near 1.8 K. To compare the intensitites of the diffraction
measurements taken in the 10 T magnet and the elastic scattering measurements
taken in the He-4 cryostat, background signal was estimated by fitting a constant to
data points measured at temperatures above 3 K. After subtracting the background
from each data set, the overlapping data point at T = 1.63 K was scaled to be the
same in both data sets. By combining these three measurements, as shown in Figure
3-10(a) we were able to examine the behavior of the (0 0 1) peak as a function of
temperature through the transition. This combined temperature scan was fit to the
function
(1-_L)2,8
x (normalization factor), resulting in
#
=
0.246(3) and Tc
=
1.77(2)
K. This value of T is consistent with the transitions observed in our specific heat
measurements and in pSR[75] measurements.
We also examined the temperature dependence of the magnetic field-induced peak
at (0 0 4). A field of magnitude poH = 0.05 T was applied and the (0 0 4) peak
intensity was measured while cooling from 2.4 K to 70 mK. The field strength of 0.05
T was chosen because it was strong enough to produce the maximum intensity of the
(0 0 4) peak at T = 70 mK. To isolate the magnetic component of the scattering, the
(0 0 4) peak intensity measured at T = 70 mK and zero applied field was subtracted
to remove the contributions from the structural peak and from background. The field
broadens the transition and shifts it to higher temperatures.
3.3.2
Magnetic Structure
In contrast with the pSR result[75], our observation of the emergence of magnetic
Bragg peaks below T = 1.77 K is a clear sign of a transition to a state with long range
magnetic order. The magnetic peaks at (0 0 L) for odd values of L, where structural
peaks are forbidden, suggest antiferromagnetic ordering between neighboring kagome
planes. The suppression of these peaks with magnetic field along with the growth of
the (0 0 4) peak shows the material is easily pushed into a fully ferromagnetic state.
81
This is consistent with the easily saturated magnetization (Figure 3-3) and suggests
ferromagnetic in-plane ordering. In-plane ferromagnetic ordering is confirmed by our
observation of magnons consistent with a ferromagnetic nearest-neighbor coupling, as
presented in Chaper 4.
At T = 70 mK a magnetic field of only 0.05 T is able to fully polarize the spins,
suggesting the interplane antiferromagnetic interaction is of the order ~ 1 pIeV. As
will be shown in Chapter 4, the in-plane nearest neighbor coupling J = 0.6 meV is
much stronger than this energy scale. Furthermore, 1 peV is within a factor of order
unity of the dipole-dipole interaction expected for the S = 1 moments. Therefore
the antiferromagnetic ordering could be caused simply by the magnetic fields of the
dipoles themselves, suggesting that the interplane coupling may be even weaker than
1 peV.
To investigate the zero-field ordered magnetic state, we examined the integrated
intensities of the measured Bragg peaks. The scattered intensity from a magnetically
ordered crystal is proportional to the component of the spin that is perpendicular
to the momentum transfer
Q[68].
If the spins point parallel to the C axis, there
will be no magnetic scattering at (0 0 L) positions. Therefore, in the ground state
configuration, spins must have some component parallel to the kagome plane for the
(0 0 L) magnetic peaks to be observed.
The integrated intensity of a 0 scan through a Bragg reflection in a typical neutron
diffraction experiment is proportional to
I Oc IF(hkl) 12
sin 29
(3.2)
where 20 is the angle between the incident and diffracted beam and F is the static
nuclear or magnetic structure factor as defined in Chapter 2 (Equations (2.16) and
(2.18)). The proportionality constant depends on neutron wavelength, incident flux,
sample volume, and counting time[68]. For our measurements the first three factors
are held constant and the intensity is normalized to monitor count to correct for the
counting time, so this proportionality constant is just an overall scaling factor.
82
A 0 scan rotates the sample through the Bragg reflection. Integrating over this
scan accounts for the horizontal divergence of the beam and for the mozaic width of
the sample in the scattering plane. The SPINS spectrometer also includes a verticallyfocusing monochromater, so the incident beam includes neutrons with some component of their momentum perpendicular to the scattering plane. This beam divergence
combined with the sample mozaic results in a decrease in measured intensity at higher
values of
IQI, due
to the finite detector size.
To account for the effect of the vertical beam divergence on measured Bragg peak
intensity, we calculated the fraction of the scattered beam that would be incident on
the detector as a function of |Q|:
Jf
Pmono(kg)
x
Psampie(ZkZ,
QI)
x
Pdet(k') dkidk'
(3.3)
Where Pmono is the distribution of neutrons with initial vertical component of momentum (kf) leaving the monochromater, Psample is the probability a scattered neutron
has its vertical component of momentum changed by AkZ = k' - kf, and Pdet selects
the neutrons with the correct k' to be incident upon the detector.
Figure 3-11(a) shows different models that were assumed for Pmono. The SPINS
monochromater consists of nine pyrolytic graphite blades, each with a mozaic width
of 30'. We modeled the incoming vertical distribution as nine sources, each with a
Gaussian distribution about a different mean value of kf. We tried both equal weighting and a triangular weighting of these nine Gaussians. We also tried models that
ignored the details of the monochromater and just treated it as a source with finite
size. With these models we also tried both a square and a triangular distribution.
To estimate the strength of the effect of the vertical focusing, we also considered the
case of perfect vertical collimation, where Pmono is modeled as a 6-function and all
neutrons are assumed to have no initial vertical component of momentum, kF = 0.
Figure 3-11(b) shows a 0 scan at the (0 0 1) Bragg position at zero applied field.
The scan taken at poH = 0.5 T was used as background and subtracted from the
zero-field scan. This scan provides a measurement of the mozaic width of the sample
83
15
C,)
C)
a)
2.0-
b)
1.5-
10
5
0
0.2
-0.1
0.1
0.0
.0
1.0-
C)
0_
0.50.0
0.2
80
Monochromater kz (A-')
90
100 110 120
8 (degrees)
1.00
--
~--Triangle.
C)
0.75
---Delta
Square
071
0.50
0.25
0.00
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
IQI (A')
Figure 3-11: (a) Models of probability distribution of z-component of neutron initial
momentum due to the focusing monochromater. (b) Background-subtracted 0 scan
at the (0 0 1) Bragg peak position showing the mozaic width of the ' axis-aligned
crystal sample in the scattering plane. The 0 scan taken at applied field of pa0H = 0.5
T was used as background. Line is a fit to a Gaussian with width 4.60 (c) Calculated
scaling of measured intensity due to the vertical beam divergence as a function of IQI.
84
in the scattering plane. The mozaic is approximately Gaussian with width 4.60. The C
axis-aligned crystals that make up the measured sample have no preferred orientation
in the plane perpendicular to c Therfore the mozaic perpendicular to the scattering
plane should be similar to the mozaic in the scattering plane. We model the vertical
mozaic as a Gaussian with width 4.60. Psampie depends both on the mozaic width
and on the momentum of the Bragg reflection. A neutron scattering from a crystal
inclined by an angle
# from the scattering
momentum changed by Ak" = -Qz
=
plane will have its vertical component of
-IQ
sin(#). Using this relation we convert
the mozaic distribution to a IQ-dependent distribution of Akz.
Pdet was assumed to be a square distribution to reflect the finite size of the detector.
It depends only on the size of the detector, the sample to detector distance, and the
energy of the diffracted neutrons. Figure 3-11(c) shows the calculated intensity as a
function of
IQI
for our different models. There was very little difference between the
square distribution of monochromater blades and the overall square distribution, and
very little difference between the two triangular distributions as well. At low values
of
IQI,
the calculated intensity is slightly dependent on the chosen model of Pmono.
At higher
IQI, the calculation
becomes independent of the model of Pmono, but more
dependent on the Gaussian width used in the model of Psample. To account for these
differences, we included an error bar of 10% of the calculated value in calculations
using this predicted scaling.
We calculated the expected intensity for each of the measured Bragg peaks:
2
IF(OOL)1
sin 20 x V(OOL)
(3.4)
Where V(OOL) is the calculated fraction of total intensity that will be measured due
to the vertical divergence of the beam.
For odd values of L, F(OOL) was calcu-
lated assuming spins were ferromagnetically ordered within each kagom6 plane and
antiferromagnetically ordered between planes, and that they were confined to the
kagome plane. We also calculated the intensity for the structural peaks and for the
field-induced ferromagnetic peak at (0 0 4). For this we assumed full ferromagnetic
85
2.0 2.0
I
I.,.I.I.I.
E Structural Peak
* H=0 Magnetic Peak
A pOH=0.5T Magnetic Peak
1.0
-
-- ---
-- ---
- --
---
E
0.5
1
0.0
0
1
2
3
4
5
6
L (r.I.u.)
Figure 3-12: Integrated intensities of 0 scans through (0 0 L) Bragg peaks as a fraction
of calculated intensity, normalized to the (0 0 2) Bragg peak intensity. The dashed
line at 1.0 is a guide to the eye, and indicates the value where measured and calculated
intensities are equal.
ordering again with the spins confined to the kagome plane. The magnetic form factor was assumed to be the free Cu 2 + ion form factor, and the g-value of gxy = 1.9
from spin wave fits to the inelastic spectrum (presented in Chapter 4) was used. All
calculated intensities were normalized to the (0 0 2) calculated intensity to account
for the unknown constant of proportionality.
0 scans of the (0 0 L) Bragg peaks were background-subtracted and integrated as
a measure of the total peak intensity. For odd values of L, the scans taken at puOH
= 0.5 T were used as a measure of the background. For the structural peaks at even
values of L, a linear fit to the four points furthest from the peak was used to estimate
background.
To get a measure of the field-induced magnetic intensity at (0 0 4),
the zero-field scan was subtracted from the 0.5 T scan to remove both background
and the structural peak signal. Integrated values were normalized to the measured
intensity of the (0 0 2) peak and then divided by the calculated intensity.
Figure 3-12 shows the ratio of measured peak intensity to calculated peak intensity.
86
The value of 1.0 indicates that the measured and calculated intensities agree. The
structural peaks were included as a check of our calculation, and the field-induced
peak at (0 0 4) was included to check that the use of the free Cu2 + ion form factor
was a reasonable approximation of the true form factor. The antiferromagnetic peak
intensities (red points) provide a measured value for the ordered moment. Since the
calculated value of the intensity assumed spins were aligned parallel to the kagome
plane, the intensities shown in Figure 3-12 can be interpreted as the ratio of the
square of the measured moment to the square of the full moment:
'measured
_
(gxySmeasured)
(gxyS)
Xcalculated
2
2
(3.5)
From our measurements we can determine a value of the ordered moment of gxySmeasured
= (0.95 t 0.2) PB, which suggests that the spins point entirely within the kagome
plane.
Figure 3-13 shows a schematic of the ground state spin configuration, which summarizes the results of our diffraction measurements. At zero field, spins within each
kagom6 plane are ferromagnetically orderered and point parallel to the kagom6 plane,
while neighboring kagome planes are antiferromagnetically ordered. Due to the existence of two copper layers per unit cell, the magnetic unit cell is equivalent to the
structural unit cell. A small magnetic field (poH > 0.05 T at T = 70 mK) reorients
the spins so that neighboring planes are ferromagnetically ordered.
3.3.3
Inelastic Scattering Measurements
Inelastic neutron scattering measurements were performed on a deuterated powder
sample of Cu(1,3-bdc) on the Iris time-of-flight spectrometer at the ISIS facility at
Rutherford Appleton Laboratory. An aluminum can was filled with 3.9 g of powder
and placed in a dilution refrigerator. A final neutron energy of 7.38 meV was selected,
giving an energy resolution of ~ 70 peV FWHM. Measurements were taken over a
temperature range of 0.1 K to 40 K.
At low temperatures a non-dispersive excitation around energy transfer 1.8 meV
87
(a)
(b)
Q
Figure 3-13: Schematic of the ground state spin configuration in the magnetically
ordered state at (a) zero applied field and (b) applied field pOH > 0.05 T parallel to
the kagome plane. The Cu 2 + ions are ferromagnetically ordered within each kagome
plane and constrained to point in the kagome plane. Neighboring planes are antiferromagnetically ordered at zero field but are easily aligned by a small magnetic
field.
was observed. The field dependence and dispersion of the inelastic spectrum will be
presented in Chapter 4. Here we focus on the temperature dependence of the highest
energy flat mode.
Figure 3-14(a) shows the inelastic neutron scattering data integrated over a range
of momentum transfers 1.0
A
Qj
K 3.4
A-
at 100 mK and 40 K. Intensities
at different temperatures were normalized by integrating over the measured energy
range -3.7
meV < hw < 6.1 meV. We assume the total scattering intensity to be
constant over this interval. The peak at 1.8 meV appears at low temperatures and
is due to the non-dispersive magnetic excitation. The smaller peak seen in the 40K
data around 1.9 meV is background signal most likely due to scattering from the
cryostat. As discussed in Chapter 2, the inelastic scattering signal is proportional
to the dynamic structure factor S(Q, w) =
[n(w)+1]X"(Q, w), where n(w) is the Bose
occupation factor and x"(Q, w) is the imaginary part of the dynamic susceptibility.
To isolate the inelastic signal due to scattering from the sample we applied the following procedure. For negative energy transfers at low temperatures, the scattered
88
II
I
151
Co
*6-9
C
a)
*
I
*
S
CU
C,,
C
Mi-
S
M
-o 10-
I
aOM
5-
Q)
=e
4-a
C
0
a
I
a
. T=lOOmK
* T=40K
6
b)
4
C,)
C
4--
2
0
---------------- --------------2
1
0
-1
-2
o (meV)
30
U)
20
c)
.c
10
--- -- -- - -
--
-
I...
0
0
2
6
4
8
10
Temperature (K)
Figure 3-14: (a) Inelastic neutron scattering data measured on a powder sample inte-
I
grated over momentum transfers 1.0 V
3.4
. (b) X"(w) extracted from
the data as described in the text. (c) X"(w), integrated over the energy range 1.5 meV
<w < 2.1 meV spanning the low temperature peak, as a function of temperature.
The dashed line indicates the value at T = 40 K.
89
intensity is only background because the scattering from the sample is suppressed by
the Bose factor. Therefore, x"(w, T = 40 K) can be calculated by subtracting the
intensity measured at 100 mK from that measured at 40 K and dividing by the Bose
factor. Then X"(w, T = 40 K) is known for positive energy transfers because X"(w)
is an odd function of w. The positive energy transfer background can be calculated
by subtracting the calculated signal at 40 K from the measured intensity at 40 K.
Assuming this background is temperature-independent in the range 100 mK to 40
K, this background can be subtracted from the intensities measured at other temperatures to arrive at S(w,T). X"(w,T) is then calculated by dividing by the Bose
factor.
Figure 3-14(b) shows X"(w, T = 40 K). Points at small energy transfer, Iwi <
0.5 meV were removed because resolution effects lead to some of the elastic or positive
energy transfer signal at 100 mK being measured at small negative energy transfers.
Therefore the measured intensity at 100 mK is not only background at small negative
energy transfers. The peak in the scattered intensity at w
-
-1.4 meV (see Figure
3-14(a)) is background and its effects are not fully canceled out by subtracting the two
data sets. Points in the range 1.25 meV < IwI < 1.55 meV were removed and replaced
by fitting a smooth function to the remaining data points. X"(w, T = 40 K) was then
used to calculate X"(w) for the other measured temperatures. X"(w, T = 100 mK) is
also shown in Figure 3-14(b).
X"(w) was integrated over the range 1.5 meV < Iwj < 2.1 meV for each temperature to get a measure of the flat mode intensity. The results are shown in Figure
3-14(c). Significant spectral weight remains in this energy transfer range well above
the 3D transition temperature of 1.77 K seen in the temperature scan of the magnetic
Bragg peaks (Figure 3-10(a)). This suggests the existence of 2D correlations within
the kagom6 planes at temperatures above the ordering transition temperature.
90
3.4
Conclusion
In conclusion, Cu(1,3-bdc) provides an ideal model system in which to investigate
the behavior of ferromagnetism on the kagom6 lattice. Its crystal structure provides
a structurally perfect kagom6 lattice of S = 1 Cu2 + ions. Adjacent kagome layers are
connected by large organic molecules, which provide two benefits. The long coupling
pathway means that there is little magnetic coupling between adjacent planes, and
the absence of interplane metal ion sites means that the magnetic behavior of the
system will be dominated by the kagome layers.
Our magnetization measurements reveal behavior consistent with a ferromagnet
above its ordering temperature. A small anisotropy was observed, with spins more
easily polarized parallel to the kagom6 plane than perpendicular to the kagom6 plane.
Both deuterated and protonated samples were measured, with no significant difference observed.
This suggests the substitution of deuterium for hydrogen has not
affected the magnetic behavior, and the two materials can be used interchangeably
to investigate this system.
Specific heat measurements provide further evidence for a magnetic transition and
also show a small anisotropy between field applied in-plane and out-of-plane. This
anisotropy is only seen in temperatures near the peak in the specific heat. The zerofield magnetic specific heat displays a sharp peak at T, = 1.77 K but this peak does
not diverge and cannot be fit to a power law. The data are well fit by a logarithmic
divergence below T, but not above T,.
At least 40% of the magnetic entropy is
released above Tc, demonstrating that significant spin correlations exist well above
the transition.
Elastic neutron scattering measurements show clear evidence for a long range
mangetic ordering transition. Magnetic Bragg peaks show that while each kagome
plane orders ferromagnetically, neighboring planes order antiferromagnetically. Our
analysis of the integrated intensity of these Bragg peaks shows that in the ordered
ground state the spins lie parallel to the kagome plane. This is consistent with the
small anisotropy observed in our magnetic measurements.
91
A very small magnetic field of ~ 0.05 T fully suppresses the antiferromagnetic
Bragg peaks, demonstrating that the spins are fully polarized.
This suggests an
interplane coupling strength of ~ 1 peV or less. This is a factor of 600 smaller than
the in-plane nearest-neighbor coupling. Therefore this system will be well described
by a 2D model that neglects interactions between kagome planes.
The temperature dependence of the antiferromagnetic Bragg peaks shows a sharp
3D magnetic ordering transition. The intensity, which is proportional to the square of
the magnetization, is well fit by a power law with exponent 3 = 0.246(3). The power
law fits the data well over a surprisingly large temperature range 0.07 K < T < 1.77
K.
Inealstic neutron scattering measurements on a powder sample were used to investigate the temperature dependence of a dispersionless magnetic excitation. This
excitation persists well above the 3D magnetic transition temperature, which demonstrates that 2D correlations exists in Cu(1,3-bdc) well above the transition. This is
consistent with our observation from specific heat measurements that a large fraction
of the magnetic entropy is released above the transition temperature.
92
Chapter 4
Magnetic Excitations in
Cu(1,3-bdc)
Inelastic neutron scattering provides a direct measurement of the magnetic excitation
spectrum of a material. The dispersion of these excitations reveals details about the
microscopic interactions. In this chapter we present inelastic scattering measurements
performed on the ' axis-aligned sample of Cu(1,3-bde) described in Chapter 2. We
present a simple model microscopic Hamiltonian to describe the magnetic interactions.
By isolating the magnetic signal from the data and fitting to the spin wave dispersion,
we are able to precisely determine the parameters of this spin Hamiltonian.
4.1
Inelastic Neutron Scattering Measurements
Inelastic neutron scattering measurements were performed on the deuterated, ' axisaligned sample of Cu(1,3-bdc) described in Chapter 2. Measurements were taken
using the cold neutron time-of-flight spectrometer LET at the ISIS facility at Rutherford Appleton Laboratory. The sample was placed in a dilution refrigerator insert in
a 7 T magnet and oriented so that the powder-like d/b plane was in the scattering
plane. Measurements were taken below the magnetic transition temperature at T =
1 K with the magnetic field applied perpendicular to the kagom6 plane. Measurements were also taken above the transition temperature at T = 4 K with no applied
93
Intensity (arb. units)
200
100
3.0
3.0
2.5
2.5
2.0
2.0
E.
1.5
1.0
1.0
0.5
0.5
0.0
0
0.0.
0.0
0.Z
U.4
3.0
3.0
2.5
2.5
2.0
2.0
Z 1.5
S1.5
1.0
1.0
0.5
0.5
0.0
02
0.4
0.6
08
U.O
1.0
1.4
JQJ (A~')
IQI (A')
0.0
U.O
1.0
1.2
1.4
IQI (A')
0.0
0.0
02
0.4
0.6
0.8
QI (A~')
Figure 4-1: Inelastic scattering spectrum measured using the LET spectrometer with
incident neutron energy 6.01 meV at (a)-(c) T = 1 K and (d) T = 4 K. (a) and (d)
were taken at zero field and (b) and (c) were taken with applied fields poH = (b) 2
T and (c) 7 T with the field applied perpendicular to the kagom6 plane. IQI denotes
the magnitude of the momentum transfer in the kagome plane. Black stripes are due
to missing detectors.
94
field. We used neutrons primarily of incident energy 6.01 meV. The LET instrument
can collect scattering data for multiple incident energies simultaneously, so data from
neutrons of incident energy 3.53 meV, 12.4 meV, and 39.3 meV were also collected,
though the neutron flux at these energies is less than the flux at 6.01 meV.
Figure 4-1 shows the inelastic excitation spectrum at various applied magnetic
fields. The plots show scattered intensity as a function of momentum transfer 1Q1
and energy transfer hw. The sample is oriented so the powder-like plane is in the
scattering plane, so JQJ represents the magnitude of the momentum transfer in the
kagom6 plane. At zero applied field we observe a dispersive excitation extending from
zero energy and momentum transfer up to
IQI
~ 0.8
A,
hw ~ 1.8 meV where it
connects with a non-dispersive excitation. The non-dispersive peak is significantly
broader than the intstrumental energy resolution. With an applied field the entire
spectrum sharpens and is shifted to higher energy by approximately the Zeeman
energy gPBH. Gaps open in the excitation spectrum forming three distinct bands.
The highest energy band extends to higher and lower values of
nearly flat, broadenening slightly as
IQI
IQI
and remains
approaches 0.8 A'. At T = 4 K the sharp
excitations have disappeared, but there is still increased intensity in the region of the
flat excitation.
Figure 4-2 shows the measurements with initial energies 3.53 meV and 12.4 meV.
These measurements reveal the same excitations seen in the 6.01 meV measurement.
The 3.53 meV measurement provides a sharper resolution but a limited range of momentum and energy transfers. The high intensity above ~2.6 meV is due to the
arrival of the next neutron pulse. The 12.4 meV measurement has a much broader
resolution but probes a large range of momentum and energy space. In the measurement performed with initial energy 39.3 meV, the energy resolution is so broad
that all three excitations are obscured by the elastic peak. However, it does allow
us to probe up to energy scales much larger than the energy scale of the observed
excitations. No significant scattering is observed above the dispersionless excitation
up to the highest measured energies.
95
Intensity (arb. units)
0
100
2(
~~AW
I~ A.
1
' &4
*.1
1.5
-
1.5
.
0,
101(A7)
NN(A,)
101 (A')
3.0
3.0
3.0
2.5
2.5
2.5
2.0
2.0
2.0
1.5
1,5
1.0
I.0
0.5
0.5
1Q(I~')
1.0
1Q1 (A~,)
1Q1 (A-,)
Figure 4-2: Inelastic scattering spectrum measured using the LET spectrometer with
incident neutron energy (a)-(c) 3.53 meV and (d)-(f) 12.4 meV at T = 1 K. Magnetic
fields voH = (a)/(d) 0 T, (b)/(e) 2 T, and (c)/(f) 7 T were applied perpendicular
to the kagom6 plane. IQJ denotes the magnitude of the momentum transfer in the
kagom6 plane. Black stripes are due to missing detectors.
96
Neutron scattering measurements were also performed on the same sample using
the triple-axis spectrometer SPINS at the NIST Center for Neutron Research. The
sample was placed in a He-4 cryostat but for this measurement oriented so that
the scattering plane was defined by the ' axis and an arbitrary vector within the
powder-like plane. The sample was cooled to a base temperature of -1.6 K. Inelastic
measurements were taken in a focused analyzer geometry with fixed final neutron
energy 3 meV and collimator configuration 80'-sample-radial-open. A Be filter was
placed after the sample to reduce contamination from high energy neutrons.
With the sample oriented so that the single-crystalline ' axis was in the scattering plane, measurements were taken at nontrivial out-of-plane momentum transfer.
Constant-Q energy scans were measured through the flat mode at a number of different values of in-plane and out-of-plane momentum transfers. For a fixed in-plane
momentum (INkagI) scans were taken at different values of out-of-plane Q (L, where
1 r.l.u. = 0.39 A-'). Figure 4-3 shows scans taken at IQkagI = 0.8
A-',
where the
flat mode is most intense. The intensity of the mode decreases with increasing L,
which can be attributed to the falloff in the copper ion form factor at higher total
However, no change in the location of the peak is observed. This is expected
because of the weak interplane coupling.
QI.
Further neutron scattering measurements were performed on the SPINS spectrometer with the same sample orientation but with improved sample environment
capabilities. For these measurements, the sample was placed in a dilution insert in
a 10 T magnet with the magnetic field applied parallel to the kagome plane. The
sample was cooled to a base temperature of -0.07 K. Inelastic measurements were
taken in a focused analyzer geometry with fixed final neutron energy 2.9 meV and
collimator configuration 80'-sample-radial-open, giving an energy resolution of -0.15
meV FWHM and a momentum resolution of ~0.15 A-' FWHM. A Be filter was
placed after the sample to reduce contamination from high energy neutrons.
Constant-Q energy scans were measured through the flat mode at a number of applied magnetic fields. Figure 4-4 shows scans at zero out-of-plane momentum transfer
and in-plane transfers IQkag = 0.6 A', 0.7 A-', and 0.8 A'. For comparison, Fig97
w L=0
200
-A-
L
0.5
L= 1
--v- L =1.5
-L=2
-4-L=2.5
-b0-- L =4
-- L 5
150-
100
T
500
1.2
1.4
1.6
1.8
2.0
2.2
2.4
hw (meV)
Figure 4-3: Energy scans through the flat mode measured at T = 1.6 K. All scans
were performed at the same in-plane momentum transfer IQkagI = 0.8 A- but at
different values of out-of-plane momentum transfer. No dispersion is observed as a
function of L.
ure 4-4(d) shows cuts through the LET data at the same magnetic fields integrated
over a range of 0.72 A- 1
IQI
0.88
A-
as to approximate the IQ-resolution on
SPINS with the focusing analyzer. In contrast to the measurment taken with the
field applied perpendicular to the kagome plane, when the field is applied parallel
to the kagom6 plane no gap between the flat mode and the lower energy dispersive
mode opens as the field is applied. The peaks are all slightly broader than the instrumental resolution and show no sharpening with field, but do decrease in intensity
with increasing field up to poH = 0.05 T, as shown in Figure 4-5. For pOH > 0.05 T
the peak intensity is approximately constant. This is the same field required to fully
polarize the spins at this temperature, as shown in Chapter 3.
In order to investigate the dispersion of the lowest energy excitation, constant-Q
energy scans were measured at low IQkagI at an applied field of 7 T. These measurements were performed at an out-of-plane momentum transfer L = 1.75 r.l.u. By
including a non-zero out-of-plane component of momentum transfer, we were able to
measure all the way down to IQkagI = 0. The value of L was chosen so that even at
98
7 AC
400
* poH=OT
200
(a)
0
02
= 2T
(b)
0 300.
a..
150
0
4
200
02
100
=
0
50
2.5
2.0
1.5
1.0
0.5
100-
0.5
3.0
1.0
2.5
2.0
1.5
3.0
11w (meV)
1w (meV)
40
500
u poH=OT
60
=2T
(
4
300-
4
0
41 30
4
200
~(d)+
Z 50
0
40
+
400
20
100
0.5
10
1.5
1.0
2.0
2.5
0.5
3.0
1.0
2.0
1.5
2.5
3.0
Aw (meV)
hw (meV)
Figure 4-4: (a)-(c) Energy scans taken on SPINS with magnetic field applied parallel
1
to the kagome plane at in-plane momentum transfers IQkag = (a) 0.6 A- , (b) 0.7
A-1, and (c) 0.8 A 1 and zero out-of-plane momentum transfer. (d) Inelastic neutron
1
scattering data taken on LET and integrated over momentum transfers 0.72 A< 0.88 A- 1 with magnetic field applied perpendicular to the kagom6 plane.
IQI
500
400
m pOH=OT
3001
*
=0.015T
200
V
=0.05T
0.025T
~4
-. 5T
=T
I4
100
-
A1
1.4
1.6
1.8
2.0
2.2
hw (meV)
Figure 4-5: Energy scans taken on SPINS with magnetic field applied parallel to the
1
kagom6 plane at in-plane momentum transfer IQkagl = 0.8 A- . The intensity of the
excitation decreases with applied field up to a field of 0.05 T and remains constant
at higher fields. Lines are guides to the eye.
99
400
*
a)
p
400
HoH=OT
+
300
300
200
200
100
100
0.4
0.6
0.8
(b)
=7T
1.0
0.6
0.4
1.2
4010
(c)
oH=0
1.0
1.2
1.4
400
T
1.6
1.8
(d)
+=7T
'U
=
0.8
hw (meV)
1w (meV)
300
30
6
10.
'U
2
10
200
0
100
'U
'U
0.4
0.6
0.8
1.0
1.2
U.
1.4
1.0
0.8
1.2
tuo (meV)
1.4
kw (meV)
300
400
=
=
*poH=OT
(e)
+
'U 200
'U
300-
6
'U
I-
'U
200-
'U
'U
'U 100 F
'U
1.0
1.2
1.4
1.6
1.8
0 lQkg1= 0
(e)
250
=7T
2.0
a:=0.2 A-
150.
100,
+
0.3A-
50 [-
0.5
1.0
1.5
2.0
1w (meV)
tua (meV)
,
Figure 4-6: Scans of the low-energy mode dispersion. (a)-(e) Energy scans taken on
SPINS with magnetic field applied parallel to the kagome plane at in-plane momentum
transfers Qkagl = (a) 0 A-1 , (b) 0.1 A- 1 (c) 0.2 A- 1, (d) 0.3 A-1 , and (e) 0.4 A 1
and out-of-plane momentum transfer L = 1.75 r.l.u. (e) Intensity measured at 7 T
minus intensity measured at 0 T, isolating the magnetic signal at 7 T.
100
400
u OH =0 T
Z
300
,
7T
200
100
1.5
1.0
0.5
2.5
2.0
|Q| (A-')
Figure 4-7: Flat mode peak intensity as a function of total momentum transfer measured at constant IQkagI = 0.8 A-' while scanning out-of-plane momentum tranfer.
Red line shows the squared free Cu2 + ion form factor. Black and blue lines show
the squared free Cu2 + ion form factor multiplied by scaling factors due to the vertical
beam divergence and to the spin polarization direction at different fields, as described
in the text.
IQkagl = 0, 20 would be large enough to keep the detector arm out of the main beam,
which reduces the background signal. A non-integer value was used to avoid contamination from Bragg peaks due to the instrument's energy resolution. The 7 T field
was used to increase the energy of the dispersive mode by the Zeeman energy (-0.8
meV) so that the mode could be measured down to its lowest energies. Scans were
performed at IQkagi = 0
A-',
0.1
A-',
0.2 A-1, 0.3 A-, and 0.4 A-', and are shown
in Figure 4-6. Scans were repeated at zero applied field, where no magnetic signal
is observed in the measured energy range. Using the zero-field scans as background,
we get a measure of the low energy dispersive mode all the way down to its lowest
energies at zero in-plane momentum transfer, as shown in Figure 4-6(e).
Finally, the intensity of the flat mode was measured as a function of the magnitude
of the total momentum transfer IQI. Energy transfer was kept constant at the location
of the flat mode peak intensity and IQkagI kept constant at 0.8
101
A-',
where the mode
is most intense. Intensity was measured while changing L. As there is no dispersion
along the out-of-plane direction, this measures the same excitation while changing
the total magnitude of the momentum transfer. This scan was performed at both
zero field and 7 T. The results are shown in Figure 4-7.
4.1.1
Magnetic Form Factor and Spin Polarization Direction
Considerations
As shown in Figure 4-5, the flat mode peak decreases in intensity with increasing field
up to pOH = 0.05 T. For pOH > 0.05 T the intensity is approximately constant. This
is the same field required to fully polarize the spins at this temperature. To account
for this fall off in intensity we note that neutron scattering is only sensitive to the
components of spin fluctuation perpendicular to Q. As discussed in Chapter 2, for
spin waves in a ferromagnet:
(4.1)
IOc (1+Q2)
where I is the scattered intensity and where
QZ
is the component of the Q unit
vector that points parallel to the ordered spin direction. The kagom6 planes of the
component crystals of the sample are randomly oriented, so at zero field we expect
Qz will be randomly oriented with respect to the scattering plane and thus (Q) =
At high fields the spins are aligned perpendicular to the scattering plane so 0Z = 0.
Therefore we expect
I(poH = 0)
I(poH > 0.05 T)
3
2
(4.2)
which is consistent with the decrease in intensity we observe.
From our measurements of the magnetic Bragg peak intensities presented in Chap+
ter 3, we expect the magnetic form factor to be well approximated by the free Cu 2
ion form factor. However, Figure 4-7 shows a peak intensity that dies off much faster
than the square of the free ion form factor. To account for this, we calculated the
fraction of the scattered beam that would be incident on the detector as a function
of
IQI.
See Chapter 3 for details, as this calculation was very similar to the one
102
performed to account for the effects of the vertical beam divergence on measured
Bragg peak intensities. For this calculation we only add one further assumption. In
the measurements of the Bragg peaks no analyzer crystal was used, so the detector
was directly in the path of the scattered beam. For our inelastic measurements an
analyzer was used, which introduces another scattering object that could potentially
change the vertical component of the neutron's momentum. However, the analyzer
crystal mozaic of 30' is much smaller than the sample mozaic. We therefore neglect
the effect of the analyzer in the vertical divergence calculation and simply treat the
detector as if it were further away from the sample. Taking into account the scaling due to the vertical beam divergence, we see that the form factor is indeed well
approximated by the free ion form factor.
We can also account for the difference in measured intensity at the two applied
fields in terms of the difference in spin polarization direction as described by (4.1).
When Q is parallel to the kagome plane the difference in intensities is largest, and
follows (4.2). At larger values of L, the difference decreases because the component
of Q perpendicular to the kagome plane is always perpendicular to the polarization
direction regradless of the applied field. The difference in intensity is then due only
to the component of Q parallel to the kagome plane. At zero applied field
2
1Q1 2
while at paoH = 7 T, Q, = 0. Therefore the difference between intensities at the two
applied fields dies off as 1/1Q12.
103
4.2
The Spin Hamiltonian
It was shown by Heisenberg in the 1920's that the electrostatic Coulomb interaction
between electrons combined with the Pauli exclusion principle can lead to a strong
coupling between the spins of nearby atoms[79]. Called direct exchange, this coupling
is a consequence of significant overlap of the electronic orbitals of neighboring metal
ions. This exchange can be understood by noting that electrons are fermions, and thus
the total wavefunction describing multiple electrons must be antisymmetric under
exchange. The Coulomb interaction leads to a lower energy when electron orbitals
have less overlap, which selects an antisymmetric spatial wavefunction. For the total
wavefunction to be antisymmetric the spin wavefunction must be symmetric, leading
to a ferromagnetic coupling between spins.
However, in many magnetic systems magnetic ions are separated by large distances and thus have little orbital overlap. Metal ions are also often separated by
nonmagnetic ions. This case was first addressed by Hendrik Kramers to explain
magnetic interactions between Mn ions in oxides[80]. The theory was later refined by
Philip Anderson[81, 82]. Known as (Kramers-Anderson) superexchange, the magnetic
interaction between magnetic ions is indirect and mediated by the intermediate nonmagnetic ion. The exchange can be either ferromagnetic or antiferromagnetic, with
-
the sign depending on the angle formed by the two bonds defining the magnetic ion
nonmagnetic ion - magnetic ion pathway, as described by the Goodenough-Kanamori
rules[83, 84, 85].
In Cu(1,3-bdc) the nearest-neighbor copper ions are well separated by a distance
of 4.55
A.
The coupling pathway between coppers is more complicated than in most
magnetic oxides, as the nearest-neighbor coppers are bridged not by a single nonmagnetic ion but by a carboxylate molecule. Superexchange between Cu2 ions mediated
by carboxylate groups has been observed previously[86, 87], but in these cases the
coupling is antiferromagnetic, whereas our system is clearly ferromagnetic.
104
We model the magnetic behavior of Cu(1,3-bdc) as a nearest-neighbor Heisenberg
ferromagnet:
=-
7iFM
ZSij
(4.4)
(i,j)
Where the summation (i, j) is over pairs of nearest-neighbor spins and J is the (positive) nearest-neighbor exchange coupling.
The long coupling pathways combined
with the fact that pathways connecting further than nearest-neighbor copper ions
will pass through nearest-neighbor sites suggests that we can neglect interactions
between further than nearest-neighbor pairs.
This simple Hamiltonian does not capture all of the magnetic behavior we observe
in Cu(1,3-bdc).
In particular, the gap opened in the spin wave spectrum and the
strong dependence of the gap on the applied field direction require an explanation.
To address this, we consider a perturbation to our model Hamiltonian in the form of
the Dzyaloshinskii-Moriya interaction.
4.2.1
The Dzyaloshinskii-Moriya Interaction
In the 1950's it was observed that a number of materials, such as a-Fe2O 3 , developed
a weak spontaneous ferromagnetic moment at low temperatures despite antiferromagnetic order[88]. Attempts to explain this behavior in terms of impurities or domain
wall effects proved to be unsatisfactory. It was demonstrated by Dzyaloshinskii in
1958 that crystal symmetry would allow for the existence of an anisotropic superexchange interaction between two ions driven by the spin-orbit coupling, provided there
was not an inversion center between the two ions[89]. This interaction could lead
to a canted antiferromagnetic state with a weak ferromagnetic moment. Making use
of Anderson's description of superexchange[82], Moriya was able to develop a microscopic description of this interaction[90, 91], now called the Dzyaloshinskii-Moriya
(DM) interaction. We will include a DM term in our model Hamiltonian:
7-DM= Dij.(St XSj)
0i1j)
105
(4.5)
Where Dij is the DM vector between the magnetic ions at sites i and j.
Moriya's calculations provide several rules that determine constraints placed on
the DM vector by the symmetries of the crystal structure. These rules do not guarantee the existence of the DM vector or determine its strength. Consider two magnetic
ions located at sites A and B and the point C, which bisects the line segment AB
connecting A and B. The following rules apply to the DM vector D between the ions
at A and B:
1. If C is a center of inversion, then D = 0.
2. If the plane perpendicular to AB and passing through C is a mirror plane, then
D is perpendicular to AB.
3. If there exists a mirror plane containing both A and B, then D is perpendicular
to this mirror plane.
4. If there exists a two-fold rotation axis perpendicular to AB that passes through
C, then D is perpendicular to this two-fold rotation axis.
5. If AB is an n-fold rotation axis for n > 1, then D is along AB.
Let us consider the application of these rules to the crystal lattice of Cu(1,3bdc). (1) For two nearest-neighbor sites on the kagom6 lattice, the center of the bond
connecting them is not an inversion center, so a DM vector is allowed. (3) For a
single kagome plane or a lattice consisting only of stacked kagom6 layers, the kagome
plane would be a mirror plane and thus the DM vector would be constrained to
point perpendicular to the kagome plane. However, it is important to remember that
the magnetic behavior in Cu(1,3-bdc) is not entirely determined by the metal ions.
The carboxylate groups that mediate the magnetic exchange must also be taken into
account. As shown in Figure 4-8, the carboxylate groups reside either above or below
the kagom6 plane. This breaks the mirror symmetry of the kagome plane and allows
D to have a component parallel to the kagom6 plane. (4) A two-fold rotation about
the axis perpendicular to AB and passing through C rotates above-plane carboxylate
106
Figure 4-8: The local environment of the magnetic ions in Cu(1,3-bdc). Carboxylate
groups consisting of oxygen (red) and carbon (cyan) mediate the magnetic coupling
between the copper (blue) ions and break the mirror symmetry of the kagom6 plane.
groups to below the plane and therefore is not a symmetry. (5) AB is not an n-fold
rotation axis for n > 1. (2) In the case of a number of kagom6 lattice magnets, such
as iron jarosite and herbertsmithite, the perpendicular bisecting plane of nearest
neighbor bonds is a mirror plane[14, 92]. This constrains the in-plane component
of the DM vector to point either into or out of each triangle, perpendicular to the
bond.
However, as can be seen in Figure 4-8, this plane is not a mirror plane in
the case of Cu(1,3-bdc). This is because the locations of the two oxygen atoms in
the Cu - 0 - C - 0 - Cu coupling pathway are not in symmetrically positioned on
either side of the carbon atom.
To summarize, Moriya's rules tell us that a DM vector is allowed by the symmetries
of the lattice in Cu(1,3-bdc) but do not place any constraints on the direction of the
vector. In general, microscopic calculations of the exact strength and direction of
the DM vector are quite difficult.
However, we can glean some information from
the symmetries of the crystal lattice. Before we do, because the DM interaction is
antisymmetric under the exchange of two spins, Dij needs to be defined relative to
the order in which we consider the spins. We define Dij such that for the triangle
with vertices labeled 1, 2, and 3 in Figure 4-9 the contribution to the Hamiltonian
is of the form D 12 - (S1 x
S2) + D 23 -
(S2 x
107
S3) + D31 - (S3
x $).
In this manner
3
OD
0
0
1
2
Figure 4-9: Dzyaloshinskii-Moriya vectors on the Cu(1,3-bdc) kagom6 lattice. DM
vectors are defined for spins considered in a counter-clockwise order around a triangle
(1 -+ 2 -+ 3 -+ 4 -+ 5 -* 3). The DM vectors can be parameterized in terms of outof-plane component D2, in-plane component Dp, and the in-plane orientation angle
OD. Neighboring kagome planes have the same sign of D. but the opposite sign of
DP.
we consider the spins in a counterclockwise order around a triangle. This can also be
thought of as considering the spins in clockwise order around a hexagon.
Now let us consider what we can learn from the symmetries of the Cu(1,3-bde)
crystal lattice and its P6 3/m space group. Consider a single bond, for example the top
bond of the bottom down-pointing triangle in the right figure of Figure 4-9, and define
an arbitrary D vector for this bond. The three fold rotation axis about the center
of the triangles then gives us the D vector for the other two bonds of that triangle.
Translation symmetries then give us the D vectors for every down-pointing triangle.
Next we recognize that the P6 3/m space group includes a screw axis, such that a
six-fold rotation plus a translation of
is a symmetry. j is the vector that translates
between neighboring kagom6 planes, so the screw axis symmetry then defines the
D vectors for every up-pointing triangle on the adjacent kagom6 planes. To fill in
the final half of the triangles, we recognize that the plane parallel to the kagome
108
planes but halfway in between neighboring planes is a mirror plane. A reflection
through this plane reverses whether the carboxylate groups are above or below the
kagome plane, but does not change their in-plane shape. Therefore this reflection
will flip the component of D parallel to the kagom6 plane but keep the component
perpendicular to the plane constant. So, by defining the DM vector on a single bond,
the crystal symmetries determine the DM vector for every other bond. We can thus
parameterize the D vector for Cu(1,3-bdc) in terms of only three quantities: The
out-of-plane magnitude (D,), the in-plane magnitude (Dp), and the angle by which
the in-plane vector deviates from parallel to the bond (OD). A schematic is shown in
Figure 4-9.
4.2.2
Spin Wave Dispersion Calculation
The total spin Hamiltonian we will consider is then:
4FM
=
+
DM
+
7t
(4-6)
Zeeman
[- J S - Sj + Dii - (Si
x S
)] - gPB
'
i
-
=
(i,j)i
where Si is the Cu2 + spin moment at site i, Di is the DM vector between sites i and
j, and (ij) indicates summation over pairs of nearest-neighbor Cu atoms. The final
term is the Zeeman energy from the coupling of each spin to the external magnetic
field H.
To calculate the spin wave dispersion, we use the Holstein-Primakoff transformation [37,
38]. This transformation recognizes that the spin raising and lowering operators Sf
and S-, in changing the angular momentum at site i by AS = +1, can be treated as
operators that anihilate and create, respectively, a spin-1 particle. Thus, Sf, S,- and
109
Sf can be represented in terms of bosonic operators:
S =
(2S-atai)ai
S7 = ac
(4.8)
(2S - atai)
Sf = S - atai
Where at (ai) are the operators that create (anihilate) a boson at site i. At low
temperatures, the the number of excitations is small, and the thermal average (alai)
is expected to be of the order 0(1/N)
<
Sj e,_
2S. In this limit we make the approximation
2Sa
(4.9)
S;- ~ v/25- at
Si= S - ajai
Next we recognize that for a fully polarized ferromagnetic state, components of
the DM vector perpendicular to the spin polarization direction (S) do not affect the
Hamiltonian to quadratic order in deviation of S[51].
To understand this, let us
represent each spin as an average value plus a small deviation:
(4.10)
Si = (S) + JS$
where 6JS _L (S). Then 6JS x 6JS
(S) and for terms that are quadratic in deviation
of S:
Di-
where
x
3S6) = (Dij - i) 6 SijSS
(4.11)
is the direction of the spin polarization. As was shown in Chapter 3, small
fields can fully polarize the spins in Cu(1,3-bdc) in any direction at low temperatures.
Therefore we can select the spin polarization direction 2
=
H/H, and retain only the
component of the DM vector (Di- 2)2.
For now we consider the case of the field applied perpendicular to the kagom6
plane. We keep only the D_ components and neglect the Dp components, so that
110
Dij -+ +D, . We insert (4.9) into (4.7) and neglect any terms that contain the product of four ai operators. Finally, we make use of the Bloch theorem[38] to represent
?H in terms of crystal momentum k. Then, (4.7) becomes:
(4Js + () +a,(,)7 4(k)a(kk)] -fl =
A
Eo +
k
(4.12)
+
Wi,,
where y, v sum over the basis sites of the kagom4 lattice, at (k) are the magnon
creation operators, and:
0
=
-2JS
cos(i -)
0
c )s (k D
co (k~
cos (k -)
cos (k -+b)
0
0
Cos (k -
-cos (k-
(4.13)
-cos (2-)
0
-cos (' - 6
)
+z2DzS
cos ( -)
cos(k.-2)
)
R (k)
cos ( - )
0
Where a and b are the crystal lattice vectors. The energies of the magnon modes wi(k)
are simply the eigenvalues of the matrix 7R() plus the constant term (4JS+ giBH).
The details of this calculation are discussed in Appendix A.
Figure 4-10 shows the effects of the DM interaction on the ferromagnetic spin
wave dispersion. The degeneracies between neighboring bands are broken, and the
flatness of the high energy mode is distorted near the points where the bands touch
if Dz = 0.
111
3
2
hw
I
J
0
1.0
0.5
(-k 2k 01 (rll.
0.5
0.0.0
1.0
1.5
2.0
{h 0 0} (r.Lu.)
3
2
J
0
1.0
0.5
(-k 2k 0) (r.l.u.)
05
0.0. 0
1.0
1.5
2.0
{h 0 0) (r.l.u.)
Figure 4-10: Spin wave dispersion of the Heisenberg ferromagnet with DM interaction
on the kagome lattice with spins polarized perpendicular to the kagom6 plane with
(a) D, = 0 and (b) DZ/J = 0.15.
112
4.3
4.3.1
Determining Hamiltonian Parameters
Structure Factor Calculation
To compare with our neutron scattering data, we calculated the dynamic spin wave
structure factor Ssw (Q, w), which is proportional to the inelastic scattering intensity.
As discussed in Chapter 2 (Equation (2.22)), for spin waves in an ordered magnet:
( 60
-
-. 1
QaQ3)Selastic(Q,W) = -(1 + QO)Ssw(Qw)
(4.14)
a43
where
, is the component of the Q unit vector that points parallel to the ordered
spin direction and Ssw(Q, w) is the inelastic scattering function for spin waves. For a
magnetic crystal with a single spin per unit cell, Ssw is just a 6-function in momentum
and energy weighted by a Bose occupation factor. When the unit cell contains more
than one spin, the expression becomes slightly more complicated. Specifically,
Ssw(Q, W)
=
S
(z
u'
) [n(wi,) + 1) 6(w - wij) + n(w )3(w + wig)
(4.15)
where i sums over the multiple magnon bands, wig is the energy of the ill band at
momentum Q and i4(Q) is the corresponding eigenvector of R() (Equation (4.13))
at k = Q. This differs from the single spin per unit cell case only by the factor in
parentheses, which depends on the eigenvectors of 7(k), and by the sum due to the
existence of multiple bands. The factor in parentheses arises because 7R(k) is not
diagonal in the basis described by separate magnon excitations being restricted to
separate unit cell basis spins. Since the Hamiltonian is not diagonal, the excitations
will involve mixings of all three basis spins, which can lead to destructive interference
of the scattered neutron wave. The calculation of Ssw(Q, w) is presented in Appendix
A.
as a function of Q for each
Figure 4-11 shows plots of the factor
magnon band. This figure contains no information about the energy of the mode, but
113
instead displays the relative scattered intensity of the mode at each value of Q. While
all three modes have a well defined energy at every value of Q, it is clear that neutron
scattering will only detect this mode at specific locations in Q space. Ssw has a very
sharp structure for the pure Heisenberg model, and this structrue is broadened by
the addition of a DM term for the two higher energy bands. Ssw of the lowest energy
band is not noticeably affected by the addition of the DM interaction.
To model the data taken on the C axis-aligned sample, this calculated Ssw was
averaged over Q directions in the kagom6 plane to get a calculated scattered intensity
as a function of energy and magnitude of in-plane momentum transfer, Scalc(IQ|, w).
114
Relative Intensity
1.5
0
3
-0
-1.02
1
-2
-
1.0
2
2
-2
-1
0
(H00)
(- 0)
1.0
0.
0.
C
C
0.
-0.
-1.0
1
-1 U
-2
2
.
-0.
(BO00)
(H0 0)
1-1
1-1
CD
-0.
-0.5
-1.0 2
(BOO)
-1
0
1
2
(1010)
Figure 4-11: Spin wave structure factor of the Heisenberg ferromagnet with DM
interaction and spins polarized perpendicular to the kagome plane. Plots show relative scattered intensity for each magnon band as a function of in-plane momentum
transfer.
(a)-(c) D, = 0 and (d)-(f) DZ/J = 0.15.
(a)/(d) Highest energy (flat)
band, (b)/(e) middle band, and (c)/(f) lowest energy band. White lines indicate the
boundaries of the first two Brillouin zones.
115
10000
* Ei = 3.53 meV
= 6.01 meV
*
8000
S=
12.4 meV
6000
4000
&
2000
-0.4
0.2
0.0
-0.2
0.4
/oi (meV)
Figure 4-12: LET energy resolution. Neutron scattering data taken on LET and
integrated over momentum transfers 0.32 A- < |Q < 0.48 A-' with 7 T field
applied perpendicular to the kagome plane. Lines are fits to the time Gaussian
function described in Chapter 2. The fits return similar values of the time width
at across all three E; and are consistent with fits done at other values of IQ.
4.3.2
Instrumental Resolution
To compare Scaic(PQ, w) with the data, we convolve the function with each instrumental resolution function.
LET Resolution Function
To obtain an estimate of the LET energy resolution function, energy cuts were
taken through the data measured at MOH = 7 T. The 7 T field was used to move any
inelastic magnetic scattering to higher energies away from the elastic peak. Data were
binned over
IQI windows of width AjQ
= 0.16
A'
centered at a number of different
IQ but avoiding the structural Bragg peak at IQ ~ 0.8 A-'. The data near the
incoherent elastic peak were fit to the time Gaussian function described in Chapter
2. Figure 4-12 shows the fits to the cuts centered at
IQI
= 0.4 A' measured at three
different values of Ej. The fits yeild a Gaussian width a- that is fairly consistent
across different values of Ej. The fit values of at are slightly higher for Ej = 3.53
meV and slightly lower for Ej = 12.4 meV. However, all fits give a Gaussian width
in the range a- = 1.3 t 0.2 x 10-5 s, so it appears that the time Gaussian function
will provide a good approximation of the energy resolution function.
116
'uu
600
Cu
400
200
0.7
0.8
IQI
0.9
1.0
(A- 1
)
0.6
Figure 4-13: SPINS momentum resolution. Longitudinal scan performed through the
(1 0 0) Bragg peak position. Blue line is a fit to a Gaussian function and red line is
a square wave. These two functions produce similar results in our analysis.
The momentum resolution was estimated by fitting a Gaussian to a QI-cut through
the (1 0 0) Bragg peak yeilding a FWHM of 0.029
much narrower than the
IQI
-' for E = 6.01 meV. This is
ranges over which we will bin the data for our fitting, so
it will have a negligible effect on our analysis.
SPINS Resolution Function
The pyrolytic graphite analyzer used on the SPINS spectrometer has a wellcharacterized behavior. At the measured final energy of Ef = 2.9 meV it provides
an energy resolution of ~0.15 meV FWHM. We will approximate the SPINS energy
resolution as a Gaussian with the same FWHM, or a Gaussian width of UE
=
0.064
meV.
The inelastic scattering measurements performed on the SPINS spectrometer
made use of a focusing analyzer, which gives up Q resolution to gain more flux.
We estimate the IQI resolution by performing a longitudinal scan through the (1 0 0)
structural Bragg peak position. The focusing analyzer combined with the sample's
relatively large mozaic leads to a very broad peak, as shown in Figure 4-13. A Gaussian fit to this peak yeilds a width of o-Q = 0.063 A-, (FWHM of 0.15
1- ).
We
also model this resolution function as a square wave with width 0.16 A- 1 . The two
functions result in little difference to our analysis.
117
Intensity (arb. units)
0
0.0
2.51
100
200
(a)
2.0
1.5
3
1.0
0.5
0.0
M.4
1.4
IQI (P-)
Figure 4-14: Calculated magnetic contribution to the scattered intensity. (a) Inelastic
neutron scattering data measured with incident neutron energy 6.01 meV and PoH
= 7 T with the field applied perpendicular to the kagome plane. Black stripes are
due to missing detectors. (b) Calculated inelastic structure factor for the Heisenberg
ferromagnet with DM interaction model convolved with the instrumental resolution
function with parameters J = 0.6 meV, g, = 2.2, D, = 0.09 meV, and pOH = 7 T with
the spins aligned perpendicular to the kagome plane. The calculation is displayed over
the measureable range for the LET instrument.
118
4.3.3
Fits to Data
Now that we have a powder-averaged calculated structure factor and models of the
instrumental resolution functions, we can begin to compare our calculation to the
data. Figure 4-14(b) shows the powder-averaged Scaic(IQ, w) for MoH = 7 T calculated using appropriate values of the Hamiltonian parameters and convolved with
the LET resolution function. It matches qualitatively the observed intensity at 7 T
(Figure 4-14(a)).
To compare our model quantitavely with the data, we take cuts of intensity vs
energy transfer through the data, integrated over a range 0.72
A-1
; Q1 < 0.88
-i.
To isolate the magnetic signal, we examine the difference in intensities measured at
the two applied fields, 1(7 T)-I(2 T), as the background and any nonmagnetic signal
should be field-independent. We use the data measured at non-zero applied fields to
examine the behavior of the system when the spins are polarized perpendicular to the
kagom6 plane. Recall that the spins are aligned parallel to the kagome plane at zero
applied field but are easily polarized perpendicular to the kagome plane with a small
field, as shown in Chapter 3. We calculated the powder-averaged S,(IQ 1, w) for our
model at poH = 2 T and paOH = 7 T, convolve the calculations with the LET resolution
function, integrate over the same range of
IQI
and subtract to get a calculated value
of 1(7 T)-I(2 T). This was done for a range of values of the parameters g, J, and
DZ. An overall scaling factor was applied to the calculated intensity to minimize the
2
X2 value of the model, and the minimum value of x was found for the parameter
values g_ = 2.2 0.04, J = 0.6 0.01 meV, and D, = 0.09 0.01 meV. These are
the parameter values used in the color plot of the calculation in Figure 4-14(b).
Figure 4-15 shows the calculated intensity plotted against the data for the cut
used for fitting as well as cuts at two other values of
IQ .
The high-field data are
well fit by this model. At zero field, the spins are aligned in the kagom6 plane, so the
out-of-plane component of the DM vector, D, does not contribute to the dispersion.
To examine the behavior of the system with the spins aligned in the kagome
plane, we apply a similar procedure to the data taken on SPINS with the magnetic
119
100
(a)
50-
0-1
-50--
)(b)
-50--
-6c-50-'
C
10-
-5
0-
100-
0
2
1
12
3
ho (meV)
Figure 4-15: Points: (a)-(c) Intensity at pi0H
=
7 T minus intensity at p0 H
=
2T
measured on LET with field perpendicular to the kagom6 plane integrated over IQI i
(a) 0.2 A-', (b) 0.6 A- 1 , (c) 0.8 A-' (d) Intensity at potH = 7 T
0.08 A-' for |I=
minus intensity at puoH = 0 T measured on SPINS with field parallel to the kagome
plane at Qdl = 0.8 A- Lines: Calculated structure factor for Heisenberg ferromagnet
= 0, integrated over
with DM interaction (a)-(c) 15 - = + 0.09 meV and (d) D
I0Qt 0.08 A- 1 and convolved with the respective intstrumental resolution function,
assuming 6-function spin waves (blue) and damped spin waves with damping I' =
0.03 meV (red)
120
field applied parallel to the kagome plane. We compare the measurements taken
at 1 aOH = 0 T and at poH = 7 T, as shown in Figure 4-15(d). We take intensity
vs energy transfer cuts at constant
IQI
through our model convolved with Gaussian
energy and momentum resolution functions. We first consider the case where Dp =
0. As described in Section 4.1.1, due to the difference in spin polarization directions,
we expect 1(0 T)/I(7 T) = 3/2. We therefore scale the calculations to reflect this.
Applying an overall scaling factor to the calculated difference, we see that the data
are well fit by our model with the same value of J = 0.6 meV but with no DM term.
A slightly differnt value of gxy = 1.9
0.05 provides the best fit.
In both measurements, peaks are slightly broader than the respective instrumental
resolution functions. This broadening can be accounted for by including the effects
of magnon damping, as discussed in Chapter 2. To summarize, the effect of damping
can be accommodated by replacing delta functions in Ssw(Q, w) with Lorentzians
and renormalizing magnon energies:
i4
[w
IW
2
(4.16)
+]
F2
where
W12
ir2(4.17)
A Lorentzian half width at half maximum of F = 0.03
0.01 meV improves the
fit of the calculated Ssw(Q, w) to the high-field data from both measurements and to
the zero-field data measured on SPINS, as shown in Figure 4-15. Peaks measured at
zero field on LET are significantly broader, and require a damping term of F
-
0.08
meV. This width is too broad to be consistent with the width of the zero-field peak
observed on SPINS, and the difference in broadening is likely due to the difference
in sample temperature during the two measurements. The LET measurement was
performed at 1 K while the SPINS measurement was performed at 70 mK.
121
Here we have only presented plots of selected cuts through the data. Our simple
model well reproduces the behavior overall. Using the same Hamiltonian parameters
and the same time Gaussian width, the measurements taken on LET at Ej = 3.53
meV and Ej = 12.4 meV are also well reproduced. The only difference between
measurements taken at different values of Ej is a slight difference in the dependence
of each cut's overall scaling factor as a function of
IQI.
This suggests a possible
dependence of scattering intensity on scattering angle 20. However, this difference is
small and does not affect how well the model's dispersion recreates the data. Scans
performed on SPINS at other momentum transfers are also well reproduced. A full
compliment of plots can be found in Appendix B.
So far we have neglected the in-plane component of the DM vector, assuming Dp
=
0. As discussed previously, a nonzero Dp is also allowed by the crystal symmetries
of Cu(1,3-bdc). Dp would affect the dispersion measured at zero field and with the
field applied parallel to the kagom6 plane. We considered the effect of a Dp term on
the magnon dispersion and found that with the spins aligned in plane the Dp term
would open gaps in the dispersion. However, this gap would be very small even for
large values of Dp (Eg,,p
0.06 meV if D, = J), which would not be measureable with
the SPINS energy resolution of -0.15 meV FWHM. This gap varied slightly with the
direction of the in-plane component 9 D, but was always small. There is no significant
change in the resolution-convolved calculated structure factor for jbpj < J/3, so we
are unable to rule out the presence of a significant in-plane component of the DM
vector.
Finally, we compare our calculated Scac (IQ, w) to the measurements of the low
energy dispersive mode as measured on SPINS. The raw data were shown previously in
Figure 4-6, and the background-subtracted data are plotted alongside our calculation
in Figure 4-16. The calculation recreates the overall shape of the observed peaks, but
consistently misses the measured peak by ~0.06 meV. The cause of this difference
is unclear. These measurements were taken at a Q with a nontrivial out-of-plane
component of momentum transfer, but the difference is much larger than the upper
bound on the interplane coupling of ~1 peV. It is therefore unlikely that this difference
122
2001
150
(a)
(b)
150
100
3..
100.
50
50,
=
0
1.5
1.0
0.5
2.
0
-501
0
1.0
0.5
1.5
2.0
1.5
2.0
Aw (meV)
hkw (meV)
onnE
200
(d)
150[
cW
150 100.
a-
100
'U
50
-
50
'U
0.5
1.0
+
T
0
'U
1.5
-Nil
2. 0
-
-
0.5
-
-
-
-50
0
1.0
1w (meV)
1kw (meV)
2001
150
(e)
100
50
0
0.5
1.0
1.5
2.0
1kW (meV)
Figure 4-16: Scans of the low-energy mode dispersion. Energy scans taken on SPINS
with magnetic field applied parallel to the kagome plane at in-plane momentum transfers IQkagl = (a) 0 A 1 , (b) 0.1 A 1 , (c) 0.2 A- 1 , (d) 0.3 A 1 , and (e) 0.4 A-' and
out-of-plane momentum transfer L = 1.75 r.l.u. Lines are calculated structure factor
for Heisenberg ferromagnet with DM interaction but with spins polarized parallel to
the kagom6 plane, D - = 0. Calculated structure factor is integrated over IQ t
0.08 A 1 and convolved with the SPINS intstrumental resolution function, assuming
6-function spin waves (blue) and damped spin waves with damping F = 0.03 meV
(red)
123
is due to dispersion along c5*. A gap in the low energy dispersion appears inconsistent
with the data measured on LET. Including a nonzero Dp in the calculation actually
pushes the manon energy down, which would make the discrepancy worse. This
fact at least suggests that Dp is small in our system. An easy-plane anisotropy is
consistent with our magnetization measurements and with the fact that the ground
state configuration has the spins aligned parallel to the kagome plane. However, this
would result in a lowering of the energies of the magnons when the spins are pushed
out of the kagome plane by a field with respect to the energies of the magnons with
the spins aligned parallel to the plane. This is inconsistent with our measurements
of the high energy dispersionless mode with the two different field orientations. It is
also possible that the anisotropy observed in magnetic measurements is due to the
shape of the sample and not due to anisotropic exchange.
Specific Heat Calculation
4.3.4
We can also use our model Hamiltonian to calculate the magnetic contribution to the
specific heat. The total energy of the magnetic system can be expressed as
Jd2k n [wi(k)]w(k)
U = A2
(4.18)
where A is the sample area, i sums over the three magnon bands, and n[w] is the
Bose thermal occupation factor. From this we can calculate the specific heat
Ck
#
=
T 412
=47r
j
1
kT
2
F
(j-2
e 3Win
(4.19)
1wi)(k)
d)k
(e/i
)
2
A
(ew)_1)2
Ow
kBT
.
where
dU
dT
Figure 4-17 shows plots of the specific heat of Cu(1,3-bdc) along with the calculated magnetic contribution to the specific heat as a function of temperature at a
variety of applied fields. At low temperatures the calculated specific heat matches the
measurement. At higher temperatures the calculated value starts to deviate from the
measured value. This calculation includes no free parameters. It uses only the unit
124
4
(b)
(a)
2M
3
3
Sol*
3
U
0
2
*EEEEU.
2
*
2-
U
M
U
M
2
1
0
4
3
0 1
5
3
2
4
5
8
10
Temperature (K)
Temperature (K)
ff
MEMO
(C)
(d)
U
3-
3
0
0
2-
2A
1
.
-,
......... .
-
-
0
-
a
... . .
2
3
45
2
4
6
Temperature(K)
Temperature (K)
Figure 4-17: Specific heat of Cu(1,3-bdc) as a function of temperature under applied
fields (a) 0 T, (b) 0.1 T, (c) 0.5 T, and (d) 14 T, with field applied perpendicular
to the kagom6 plane. Black points are measured specific heat and red lines are the
calculated magnetic contribution to the specific heat.
C)
125
17
0.31
i
(a)
0.8
S
2
(b)
1.0
0.2-
0.6
0.4
0.1
0.2t
0.0
1.0
05
1.5
2.
0
0.0
0.5
poH (T)
1
4
1.0
1.5
2.0
poH (T)
I
4
()
3-
(d)
(~3
S
*
2
2
1
0
2
4
6
8
0
10
po1H (T)
2
4
6
8
10
12
14
poH (T)
Figure 4-18: Specific heat of Cu(1,3-bdc) as a function of field applied perpendicular
to the kagome plane at T = (a) 0.4 K, (b) 1 K, (c) 1.8 K, and (d) 4 K. Black points
are measured specific heat binned over a temperature range (a)-(c) T t 0.05 K and
(d) T 0.1 K. Red lines are the calculated magnetic contribution to the specific heat.
126
cell dimensions from X-ray measurements[73] and the spin Hamiltonian parameters
determined from our fits to inelastic neutron scattering data. The specific heat was
calculated for two cases: all spins aligned in-plane so the D, term does not affect the
dispersion, and all spins aligned out-of-plane so the Dz term does affect the dispersion.
The lowest energy magnon band dominates the specific heat at low temperatures, but
this band is not strongly affected by the addition of the D, term. Therefore the difference between the two calculations is negligible, less than the width of the lines in
Figure 4-17.
We can also examine the specific heat as a function of applied field by binning
the data presented in Chapter 3 over small temperature ranges. Figure 4-18 shows
plots of the measured specific heat along with the calculated magnetic specific heat
as a function of applied field. At the lowest measured temperature, the calculation
reproduces the data at all applied fields. As the temperature is raised, the calculation
deviates from the measurement at low fields but reproduces the data at higher fields.
This is true even at temperatures above Tc.
From these specific heat calculations we can see that our model Hamiltonian describes the magnetic behavior of Cu(1,3-bdc) very well at low temperatures. As the
temperature is raised, more magnons are thermally excited and our assumption that
the occupation number is small (Equation (4.9)) becomes invalid. Thus the calculation begins to deviate from the measured value. Applying a magnetic field raises
the energy of all magnons due to the Zeeman energy and restores the validity of our
assumption. Therefore our model also describes the behavior at higher temperatures
under large applied field. Even above the transition temperature, a strong magnetic
field can fully polarize the spins. Once the spins are polarized the specific heat is
reproduced by our magnon model.
127
4.4
Conclusion
We have presented inelastic neutron scattering measurements performed on our Jaxisaligned sample of Cu(1,3-bdc). These measurements reveal the magnon excitation
spectrum. This spectrum consists of two dispersive modes and a higher energy flat
mode. At zero applied field these modes touch each other. Application of a magnetic
field shifts the entire spectrum to higher energy due to the Zeeman energy, and
decreases the scattered intensity due to a realignment of the spins. A field aligned
perpendicular to the kagom6 plane opens gaps between the modes, while application
of a field parallel to the kagom6 plane does not open an observable gap.
The flat mode was measured at nontrivial values of out-of-plane momentum transfer L, and no dispersion was observed along this direction. The fall off in intensity
of the flat mode as a function of L is well described by the Cu2 + free ion form factor
combined with the effects of the vertical divergence of the neutron beam. The dispersion of the lowest energy band was also investigated down to zero in-plane momentum
transfer by measuring at nontrivial L.
We have modeled our system as a nearest-neighbor Heisenberg ferromagnet with
Dzyaloshinskii-Moriya (DM) interaction. We have introduced the DM interaction and
discussed the restrictions placed on the orientation of the DM vector by Moriya's rules
and by the symmetries of the Cu(1,3-bdc) crystal lattice. The Holstein-Primakoff
transformation was used to represent this model Hamiltonian in terms of a magnon
hopping model. From this model the magnon dispersion was calculated. This model
accounts for the dependence of the dispersion on applied field direction, because the
dispersion is only sensitive to the component of the DM vector parallel to the ordered
spin moment.
To compare our model to the data, we calculated the spin wave structure factor
and averaged it over in-plane Q directions because of our sample's ' axis alignment.
We convolved this model with the appropriate instrumental resolution function and
took energy cuts through the model. These energy cuts were compared to the data
while varying the Hamiltonian parameters.
128
Our model reproduces the observed magnon spectrum well for Hamiltonian parameter values J
9Z = 2.2 and g,
=
=
0.6 meV, D2 = 0.09 meV, and a slightly anisotropic g factor
1.9. Fits are further improved by including a magnon damping
term of F = 0.03 meV for measurements performed at T = 70 mK and for measurements performed at T = 1 K at high fields poH > 2 T. The measurement performed
at T = 1 K at zero field requires a slightly larger damping F = 0.08 meV.
We also considered the effects of an in-plane component of the DM vector, Dp.
Our measurements of the flat mode with the magnetic field applied parallel to the
kagom6 plane are consistent with no in-plane component, Dp = 0. However, because
the gap opened in the magnon spectrum by the Dp term is much smaller than the
gap opened by the Dz term, we cannot completely rule out a nonzero value. Our
measurements of the dispersion of the lowest energy magnon mode differ slightly
from the calculation at low jQj. The source of this difference is unclear, but including
a nonzero D, makes the discrepancy worse, suggesting that Dp is small in our system.
We have calculated the magnetic contribution to the specific heat due to our model
spin Hamiltonian using the parameters from the fits to the neutron scattering data.
When the spins are well polarized due to low temperatures or high magnetic fields,
the calculation matches the measured data. The calculation deviates from the data
at higher temperatures and low magnetic fields, when many magnons are thermally
excited and the spins are not fully polarized. At zero applied magnetic field, the
calculation begins to deviate from the measured specific heat at -1 K, which may
help to explain the larger magnon width observed in the inelastic neutron scattering
measurements at zero field and T = 1 K.
We have shown that the magnetic behavior of Cu(1,3-bdc) is well described by
a simple model spin Hamiltonian. This Hamiltonian is the Heisenberg ferromagnet
with out-of-plane DM interaction on the 2D kagome lattice. The strength of the DM
interaction is DZ/J = 0.15. As discussed in Chapter 1, this model predicts a number
of interesting novel behaviors.
Firstly, our magnon hopping model is equivalent to the nearest-neighbor fermionic
hopping model considered in Reference [47] except with bosonic excitations. The top
129
and bottom bands have non-zero Chern number for D_
$
0, i/ 2 J. In the fermionic
model, the non-zero Chern numbers lead to topologically protected edge modes in
the gaps between bands, as well as an integer quantum Hall effect when the chemical
potential lies in the gaps. If the bottom band is tuned to be flat then the fractional
quantum Hall effect can exist for partial fillings of the band.
Returning to the magnetic system, our spin Hamiltonian leads to a magnon Hall
effect driven by the DM interaction.
The magnon Hall effect has been observed
experimentally in the ferromagnetic pyrochlore materials Lu2 V2 0 7 , Ho 2V 2 O7 , and
In 2 Mn 2O7 51, 52]. The pyrochlore lattice can be viewed as an alternating stacking of
kagom6 and triangular lattices along the [1 1 1] direction. When the magnetic field is
applied along the i = [1 1 1] direction, the projection of the DM vector D
is zero for
bonds between sites of neighboring layers and nonzero and equal for bonds between
sites of the same layer. As the perpendicular component of D does not contribute to
the spin wave Hamiltonian up to quadratic order, this system may be considered as
consisting of ferromagnetic kagome layers with a DM interaction separated by layers
that have no DM interaction. In the limit that interlayer coupling goes to zero, this
is precisely the spin Hamiltonian that describes our system. We therefore expect our
system to display the magnon Hall effect as well in the field-induced state where the
spins are polarized out of plane.
Just as with the fermionic case, the non-zero Chern numbers of the top and bottom bands should lead to topologically protected edge modes in the gaps between
bands. This has been shown numerically for our spin Hamiltonian with DM interaction strengths DZ/J = 0.1 and DZ/J = 0.4[55]. Our system lies in between these two
values, with Dz/J = 0.15. Therefore, we expect our system to include these modes
as well. These modes propagate only in a single direction and so are protected from
backscattering, which allows the modes to persist even in the presence of defects.
Also, if a thermal gradient is applied across the material, magnons will propagate
from cold to hot along one of the edges. The edge modes exist only in a region up
to ~5 nm from the edge of the crystal. These modes have yet to be observed experimentally in any system. Our sample is made up of crystals with typical area ~1
130
-2$
Figure 4-19: Kagom6 lattice unit cell (highlighted in yellow). Due to the DM interaction, when spins are polarized out-of-plane, a hopping magnon aquires a phase 0/3.
Closed loops can be thought of as enclosing 'fictitious fluxes'. # and -2# are the
fictitious fluxes enclosed by the triangle and hexagon, respectively, when traversing
the loop in the direction of the arrows.
mm2 , so the egde modes will exist in an area only -0.002% of the area in which the
bulk modes exist. Therefore we do not expect to be able to observe these modes in
our neutron scattering measurements.
To understand how the DM interaction gives rise to these behaviors, let us examine
the effect of adding an out-of-plane DM vector term to the Heisenberg ferromagnet
model. When the spins are polarized out-of-plane, Dj -2 =
D,. The effect of this on
our magnon hopping model is to add a complex exchange paramter. In other words,
when adding the DM interaction, J
-+
J
zDz. We can also express this in terms of
a complex exponential:
J
where tan
zD, =
(4.20)
cos (#/3)
=
-
In this way we can see that in the presence of the DM interaction, a hopping
131
particle picks up a complex phase &0/3. In traversing a closed loop the particle can
pick up a nontrivial phase. This can be thought of as analagous to the phase picked
up by a charged particle encircling a magnetic flux due to the Aharonov-Bohm effect.
In this sense, we can think of a closed loop in the kagome lattice as enclosing a finite
'fictitious flux,' as shown in Figure 4-19, and we can think of the DM interaction as
producing an effective Lorentz force that acts on charge neutral magnons. Due to the
geometry of the kagom6 lattice, there exist inequivalent loops within the unit cell,
such that a magnon can pick up a different phase depending on the path it traverses.
This fictitious flux will affect a magnon in exactly the same way that a true
magnetic flux would affect a charged particle. Thus our model of the Heisenberg
ferromagnet with out-of-plane DM interaction is analagous to Haldane's model, which
produces the Hall effect for electrons on the honeycomb lattice due to inequivalent
loops within the unit cell.
The kagome lattice geometry drives novel magnetic behaviors in ferromagnetic
systems. Cu(1,3-bdc) represents an ideal system in which to investigate these behaviors. It has negligible out-of-plane coupling, making it an ideal model system in which
to study two-dimensional physics. It is well described by a simple magnetic Hamiltonian with only nearest-neighbor interactions. Our measurements have allowed us to
precisely quantify this simple Hamiltonian. From our model Hamiltonian, we predict
that Cu(1,3-bdc) will display the magnon Hall effect, and will feature topologically
protected edge modes.
132
Chapter 5
Studies of New Kagom4
Antiferromagnets
Low spin kagome antiferromagnets are predicted to display novel magnetic behavior.
The spin-j system is widely believed to have a disordered spin liquid ground state,
though the details of this ground state are still a matter of debate. New materials
featuring antiferromagnetically coupled small spins on a kagome lattice are desireable
in order to investigate this behavior experimentally.
In this chapter, we present
the results of thermodynamic and neutron scattering measurements on two newly
synthesized antiferromagnetic kagome lattice systems.
5.1
5.1.1
Magnesium Paratacamite
Motivation
The mineral family paratacamite, ZnCu4 _,(OH) 6 Cl 2 , has been studied extensively.
Of particular attention is the mineral herbertsmithite, at the x = 1 end of the
series[34, 35, 36, 92, 93, 94]. For x > -, paratacamite features spin-1 Cu 2 + ions
arranged on a structurally perfect kagome lattice. Kagom planes are separated by a
mixture of spin-1 Cu 2+ ions and nonmagnetic Zn 2 + ions.
Herbertsmithite is currently one of the best experimental candidates to display
133
spin liquid physics. However, due to the chemical similarity of Cu2+ and Zn2+, it was
thought that site-mixing of the two ions could occur[94] and explain the behavior of
herbertsmithite without the presence of an interesting quantum spin liquid state[95].
Due to their similar atomic weight and number, Cu and Zn are difficult to distinguish
by means of conventional X-ray and neutron scattering techniques. Though recent
anamalous X-ray scattering measurements have demonstrated that no Zn2+ ions occupy the kagom6 sites[96], a similar material where the magnetic and nonmagnetic
ions are more easily distinguishable is desireable.
A number of samples in the series MgCu4 -_(OH)6 Cl 2 have been synthesized and
structurally characterized as described in Reference [97]. These materials are isostructural with materials from the paratacamite series. Spin-! Cu 2+ ions form structurally
+
perfect kagome layers with neighboring layers separated by a mixture of spin-j Cu 2
ions and nonmagnetic Mg 2 + ions. Like Zn, Mg is non-Jahn-Teller active and thus
prefers the interplane sites over the Jahn-Teller distorted kagome layer sites, so the
amount of intersite mixing is expected to be small. Unlike Zn, Mg is easily distinguished from Cu by conventional X-ray diffraction techniques. Refinements of X-ray
diffraction measurements performed on samples of Mg.Cu4 _.(OH)6 Cl2 demonstrated
that < 3% of kagom6 sites are occupied by Mn2+ ions[97].
5.1.2
Magnetization Measurements
Magnetization measurements were performed on three polycrystalline samples of
MgxCu 4 -x(OH) 6 Cl2 with x = 0.39 (1), 0.54 (2), and 0.75 (3). Measurements were
performed on a Quantum Design Magnetic Property Measurement System (MPMS).
Magnetization was measured as a function of temperature at an applied field of
500 Oe under both field-cooled (FC) and zero-field-cooled (ZFC) conditions. The
data were corrected for the molecular diamagnetism of the sample by use of Pascal's
constants[76].
Figure 5-1 shows the magnetic susceptibility of the measured samples, approximated as M/H. The susceptibility of a powder sample of herbertsmithite measured
at the same field of 500 Oe (from Ref. [92]) is shown for comparison. Below ~5-6
134
0.20
0. 15-
0. 10
-
-
=0.39
---- - x
x=0.54
-A-x = 0.75
+ Herbertsmithite
20-
0.020-
A
0
'AA
A
09
\
0.015
A
A
0.010
0 0
%
E
I
-m- x = 0.39
x = 0.54
x = 0.75
0.005- ---
-
0.05
0.000.
00 j
2
4
6
+
Herbertsmithite
2
10
8
-
4
-
0
-.-
0.025
0.25-
6
8
10
Temperature (K)
Temperature (K)
Figure 5-1: Susceptibility of MgCu4 _,(OH) 6 Cl 2 , approximated as M/H, measured at
H = 500 Oe. Both zero-field-cooled (open symbols) and field-cooled (filled symbols)
data are shown. Susceptibility of herbertsmithite measured at the same field H =
500 Oe is shown for comparison (from Ref. [92]). (a) and (b) show the same data
but with different scales on the M/H axis.
X
0.39
0.54
0.75
eCW (K)
-190
-203
-297
Table 5.1: Curie-Weiss temperature of MgxCu4 -x(OH)6 Cl 2 as a function of x
K, the susceptibility increases sharply and FC and ZFC measurements split. With
increasing x, the magnitude of the increase in susceptibility decreases, as does the
splitting between FC and ZFC data. However, the temperature of the transition is
approximately composition-independent.
Figure 5-2 shows the inverse susceptibility of MgxCu4 _x(OH) 6 C 2 . Figure 5-2(a)
shows data for all three samples alongside the inverse susceptibility of herbertsmithite.
Figure 5-2(b)-(d) show Curie-Weiss fits to the data.
All fits were done over the
temperature range T > 150 K. The resulting values of the Curie-Weiss temperature
(1cw) are listed in Table 5-1. A strong dependence of Ecw on x is observed. In
all samples, Ecw is large and negative, indicating strong antiferromagnetic coupling.
Larger values of x lead to even stronger antiferromagnetic coupling.
Although the transition observed in the magnetic susceptibility is not fully sup-
135
E
1200-
1000-
1000-
m
800-
e
e
A
v
gi
g
600
-
()
1200-
E
x = 0.39
b)
800-
x =0.54
x =0.75
0)
600-
Herbertsmithite
U
400-
200
200.
-
-
400
U
0.
50
100
150
200
250
300
350
50
I
100
150
200
250
300
350
300
350
Temperature (K)
Temperature (K)
1200
1000-
100-
-
-
1200-
800
-
800E
0)
0-
600-
0
S-
E
E
-
0
400-
I
600-
400200-
200.
0
0
50
100
150
200
250
300
350
4
I
0
50
100
150
200
250
Temperature (K)
Temperature (K)
Figure 5-2: (a) Inverse susceptibility of MgCu4 _,(OH) 6 Cl 2 , approximated as H/M,
measured at H = 500 Oe. Inverse susceptibility of herbertsmithite measured at the
same field H = 500 Oe is shown for comparison (from Ref. [92]). (b)-(d): CurieWeiss fits over the range T > 150 K for samples with Mg concentration (b) x = 0.39,
(c) x = 0.54, and (d) x = 0.75.
136
0.10 -
T=2K
4K
A 6K
v 8K
+ 10K
*
V V Qo
.
A
20K
''0.06-
.
50K
+
0.08
V
44
V
4t
, .4
-
0
-
-
0.00
100000
50000
150000
H (Oe)
Figure 5-3: Magnetization of Mgo.75Cu 3 .2 5 (OH) 6 C
several temperatures.
2
as a function of applied field at
pressed even in the x = 0.75 sample (3), the feature is significantly weaker than that
observed in the samples with lower values of x. Above ~5-6 K, the susceptibility
of sample 3 is very similar to that of herbertsmithite.
In order to explore the ex-
tent of the similarities between these two materials, we performed further magnetic
measurements on sample 3.
Magnetization and AC susceptibility of 3 were measured using a Quantum Design
Physical Property Measurement System (PPMS). Parameters of these measurements
were chosen to match measurements taken on powder samples of herbertsmithite presented in Reference [92]. Magnetization was measured as a function of applied field
up to 14 T at a number of temperatures ranging from 2 K to 50 K. AC susceptibility
was measured as a function of temperature under applied fields up to 14 T. Measurements were performed with an oscillating field of amplitude 17 Oe and frequency
100 Hz. Further AC measurements were performed as a function of field up to 14 T
at a number of temperatures ranging from 2 K to 50 K using an oscillating field of
amplitude 0.5 Oe and frequency 654 Hz.
Results of the magnetization measurements are shown in Figure 5-3. The mag-
137
* a
I.
0.015-
A 1T
v 3T
S5 T
47T
10 T
* 14T
mus
Ul.
I
0
-
*I'oH=OT
* 0.5 T
0.010E
X
'E'
I t.,....
-
0.005
-
0.020-
0.000
0
5
15
10
310
25
20
0.004-
a T=2K
o 4K
6K
8K
b)
y10K
-
0.006
0
E'
4
E
-
Temperature (K)
'
50K
20K
0.002-
0.000- .1.
0
4
50000
100000
-
C)
150000
H (0e)
Figure 5-4: AC susceptibility of Mgo. 7 5Cu3 .2 5(OH) 6 Cl 2.
(a) AC susceptibility vs.
temperature, measured using an oscillating field of amplitude 17 Oe and frequency
100 Hz. (b) AC susceptibility vs. applied field, measured using an oscillating field of
amplitude 0.5 Oe and frequency 654 Hz.
138
netic moment is only -10% of the saturated magnetic moment even at the lowest
measured temperature of 2 K and highest measured field of 14 T. This demonstrates
the strong antiferromagnetic interactions that exist in this material. At high temperatures the magnetization is approximately linear with field, while at low temperatures
it resembles a Brillouin function. Herbertsmithite shows similar behavior, reaching a
magnetic moment that is ~13% of saturation at T = 1.8 K and pOH = 14 T.
Results of AC susceptibility measurements are shown in Figure 5-4. The data are
qualitatively similar to the AC susceptibility of herbertsmithite. At low fields, the
susceptibility diverges at low temperatures. At fields above 3 T, the susceptibility
has a local maximum at finite temperature. The temperature of this local maximum
is fairly consistent with the temperature maximum expected for a free spin-1 moment
in a magnetic field.
The AC susceptibility of herbertsmithite has been shown to obey a scaling relationship [98]. The quantity X'T' can be expressed as a universal function of H/T.
This type of scaling is of interest because it is associated with proximity to a quantum critical point in quantum antiferromagnets[99] and heavy-fermion metals[100].
In herbertsmithite, for a = 0.66 the data collapse onto a single curve over a broad
range of pBH/kBT. The scaling is valid up to temperatures of roughly T = 35 K at
small fields (p-oH ~ 0.5 T) and as high as T = 55 K at larger fields (OH > 5 T). Figure 5-5 shows the AC susceptibility of sample 3 (Mgo.75Cu 3 . 2 5 (OH)6 Cl2 ), plotted as
X'T' vs.
[LB H/kBT
with a = 0.66. Most of the data collapse onto a single function of
IIBH/kBT, which is qualitatively similar to the behavior observed in herbertsmithite.
Figure 5-5(a) shows the data measured over the temperature range 2 K - 50 K. At
low fields the lowest temperature data points deviate significantly from the rest of the
data, demonstrating that the scaling fails at these temperatures and fields. However,
as seen in the measurements of the DC susceptibility (Figure 5-1), 3 undergoes a
magnetic transition below T = 6 K. Above the transition temperature, the magnetic
behavior of 3 is very similar to that of herbertsmithite. If we remove the data points
measured below T = 6 K, the scaling improves, as shown in Figure 5-5(b). As in
herbertsmithite, the scaling begins to fail at high temperatures.
139
a)
P4
0.01
CIS
0.5T
*
X0
A
1T
v
3T
5T
7T
1 OT
14T
4
*
1 E-3
0.01
0.1
1
PBH/kBT
b)
4
0.01
e
'0
E-4
X
0.5T
A
1T
v
3T
5T
4
7T
*
10T
14T
1 E-3
0.01
0.1
1
PBH/kBT
Figure 5-5: Scaled AC susceptibility of Mgo.7mCu 3 .2 5 (OH) 6 Cl 2 plotted as X'T1 vs.
PBH/kBT with a = 0.66. (a) Includes AC susceptibility measured over the tem-
perature range 2 K - 50 K. (b) Excludes data points below 6 K, including only
measurements taken above the temperature of the transition observed in the DC
susceptibility, over the temperature range 6 K - 50 K.
140
5.1.3
Specific Heat Measurements
The specific heat of the powder samples 1-3, and of a single crystal sample with
x = 0.31 (4) were measured using a Quantum Design Physical Property Measurement
System (PPMS). Measurements were performed under zero applied field as a function
of temperature from T = 0.4 K up to T = 25 K for 2 and 4 and up to T = 40 K for
1 and 3.
Figure 5-6 shows the results of these measurements. A sharp peak is observed in
sample 4 (x = 0.31). This peak has a similar shape to the peak observed in the x = 0
material clinoatacamite[92], but occurs at a lower temperature. With increasing x
this peak is suppressed and moved to slightly higher temperatures. This is in contrast
with the transition observed in DC susceptibility measurements, which occurs at a
relatively composition-independent temperature. A broad peak at lower temperatures
is also observable in the higher-x samples, most noticeably in the x = 0.54 sample.
The x = 0.75 sample provides an interesting case. Although the FC and ZFC
measurements of DC susceptibility show a splitting below -5-6 K, there is no observable feature in the specific heat data at this temperature. In fact, the specific
heat of 3 is remarkably similar to that of herbertsmithite, as shown in Figure 5-6(b).
Below T = 2 K the data lie on top of each other.
It is difficult to isolate the magnetic contribution to the specific heat, because there
is no known isostructural nonmagnetic compound to use as a direct measure of the
lattice contribution. We can estimate the lattice contribution by using a polynomial
fit to the high temperature specific heat, where we expect the lattice contribution
to dominate. Figure 5-7(a) shows the specific heat of Mg.Cu 4 ,(OH) 6 Cl 2 up to a
temperature of 40 K. In the temperature range 30 K - 40 K, the specific heat of 1
and 3 are the same. We therefore restrict our polynomial fit to this range, and assume
the lattice contribution will be approximately composition-independent. Because of
the large magnetic energy scales present in these materials, we expect some magnetic
contribution to be present in this energy range. Thus we are likely overestimating the
phonon specific heat and underestimating the magnetic contribution. This estimate
141
-
9
x = 0.31
87-
0
a)
x = 0.39
A x = 0.54
v x = 0.75
6
-
LL5
E 4%
2, 3-
20-
1
0
2
.. 0
AAAIA
3
4
5
6
7
8
9
10
Temperature (K)
-
2.0
1.6 -
1.4
y x =0.75
*
Herbertsmithite
-
1.8-
b)
D1.2-
E
1.0-
0.80.60.40.2-
-
---
0 .0 0
*
2
4
6
8
10
Temperature (K)
Figure 5-6: Specific heat of MgCu 4s(OH)6 C 2 as a function of temperature under
zero applied field (a) Comparison of samples with different Mg concentration x.
(b) Specific heat of the sample with highest Mg concentration x = 0.75 plotted
alongside specific heat of herbertsmithite (from Ref. [92])
142
still offers some information about the low temperature magnetic specific heat. Using
the estimated value of magnetic specific heat, we can calculate the magnetic entropy
released.
This is plotted in Figure 5-7(b).
In all samples, the magnetic entropy
released below the transition temperature of ~5-6 K is < 15% of kBln(2) per spin.
Thus, we expect a significant fraction of the magnetic entropy is released at higher
temperatures.
Further specific heat measurements were performed on 1 and 3 under applied
fields up to 14 T. The results are shown in Figure 5-8. For sample 1, the sharp peak
in the specific heat is broadened and shifted to higher temperatures with applied
field. Sample 3 again shows similar behavior to that of herbertsmithite. The broad
low-temperature feature is pushed to higher temperatures, becoming a broad peak at
higher fields.
5.1.4
Conclusion
We have presented the results of thermodynamic studies of the kagome lattice materials Mg.Cu 4 _
1(OH)6Cl 2 . These materials are isostructural to the paratacamite family
of minerals, ZnxCu 4 _x(OH)6 Cl 2 . They feature spin-. Cu2 + ions on a kagome lattice.
Unlike Zn, Mg is easily distinguishable from Cu by conventional X-ray scattering
techniques. It can therfore be easily verified that there is little Mg occupancy of the
kagome sites.
The materials Mg2Cu 4 _-(OH)6 Cl 2 display strong antiferromagnetic coupling as
seen from the Curie-Weiss temperature, which increases with increasing x. At low
temperatures, DC susceptibility measurements reveal a magnetic transition as evidenced by an upturn in x and a splitting between the FC and ZFC data. This
transition is suppressed with increasing x, with the magnitudes of both the upturn
in susceptibility and splitting between FC and ZFC data decreasing. However, the
temperature of this transition appears relatively composition-independent.
The specific heat of these materials was also measured. In all samples except
the sample with highest Mg concentration (x = 0.75), a peak is observable near the
temperature of the magnetic transition observed in the susceptibility measurements.
143
40m x = 0.31
* x=0.39
x=0.54
A
30-
V x = 0.75
a)
6 20-
E
10-
040
30
20
10
0
Temperature (K)
)-
x = 0.31
-
0.20
b)
0.15-
-
x = 0.39
-
x = 0.54
-x=
0.75
C,)
C
0.10-
0.05-
0.00-
0
5
10
15
20
25
30
35
40
45
Temperature (K)
Figure 5-7: Specific heat and estimated magnetic entropy of MgCu 4 _,(OH)6 C2(a) Specific heat as a function of temperature under zero applied field. Orange line is
an odd-order polynomial fit to the data in the temperature range 30 K - 40 K, used
as an estimate of the lattice contribution to the specific heat. (b) Magnetic entropy
released calculated from the data in (a).
144
5
*RoH
0T
S2T
4
v
a)
14 T
3L
-
II.
E
2C)
SI-
-
I
0
0
4
2
10
8
6
12
14
Temperature (K)
2.0
i
1.8
*IoH = OT
1.6-
4 1T
* 2T
6T
0 .5T
L
1.4-
1.2-
y 14T
LL
1.0E
b)
0.8-
AP00
0.60.40.2-
0.0-
=
i
0
1
2
3
4
5
Tempearture (K)
Figure 5-8: Specific heat of MgCu 4 _ (OH)6 Cl 2 under applied field. (a) x = 0.39 and
(b) x = 0.75.
145
This peak is sharp at x = 0.31 but is suppressed and broadened with increasing x.
No peak is observed in the x = 0.75 sample.
The x = 0.75 sample displays behavior very similar to that of herbertsmithite.
Above the -5-6 K transition, the DC susceptibilities are similar. The magnetization
and AC susceptibility of Mgo. 75 Cu3 .25 (OH) 6 C
2
are also qualitatively similar to those
of herbertsmithite, particularly at high magnetic fields. The AC susceptibility obeys
a similar scaling relationship to herbertsmithite, though the scaling fails below the
magnetic transition temperature. The zero-field specific heat of Mgo.75Cu 3 .2 5 (OH)6 C
2
is identical to that of herbertsmithite below -2 K and is qualitatively similar at higher
temperatures and fields.
The material Mgo.7Cu 3.2 5 (OH)6 Cl 2 may provide another experimental system in
which to investigate quantum spin liquid physics. Although the DC magnetic susceptibility suggests the existence of a magnetic ordering transition, other thermodynamic
probes reveal behavior strikingly similar to that of the quantum spin liquid candidate
herbertsmithite. This material has the added benefit that the Cu and Mg ions are
more easily distinguishable than the Cu and Zn ions in herbertsmithite. Samples of
Mg.Cu 4 .(OH) 6 Cl 2 with higher values of x are desireable, but so far attempts to
increase x beyond 0.75 have been unsuccessful.
146
5.2
5.2.1
Nickel Vesignieite
Motivation
Kagome lattice antiferromagnets with half-integer spins have been studied extensively.
Integer spin states have been less extensively studied, but are expected to display
distinct behavior in some cases[101].
Spin-1 systems provide an interesting case.
S = 1 is the smallest possible integer spin, and thus will have the strongest quantum
fluctuations of integer-spin systems. These fluctuations combined with the geometric
frustration of the lattice can lead to novel magnetic behavior. For example, the spin1 kagome lattice antiferromagnet with single-ion anisotropy is expected to display a
spin-nematic ordering[102].
Spin-1 kagome lattice compounds are rare. The materials KV 3 (OH) 6 (SO 4 ) 2 and
YCa 3 (VO) 3 (BO 3 ) 4 both display ferromagnetic coupling between the nearest-neighbor
kagome spins [103, 104]. The organic kagom6 compound [C6 N 2 H8 ] [NH 4 ] 2 [Ni 3F 6 (SO 4 ) 2]
features antiferromagnetic coupling but undergoes a long range magnetic ordering
transition at 10 K[105, 106]. The spin-1 antiferromagnet Ni3 V 2 0 8 displays a rich
phase diagram due to geometric frustration[107, 108], but features a kagome staircase
lattice rather than a true kagom6 lattice.
Here we present the results of thermodynamic and inelastic neutron scattering
measurements on the newly synthesized compound BaNi 3 (OH) 2 (VO4 )2 [109]. This
compound is similar to the copper mineral vesignieite, BaCu3 (OH) 2 (VO 4 ) 2 [33, 110].
Vesignieite is a spin-! antiferromagnet with a nearly perfect kagome lattice. Vesignieite has no long range magnetic order down to T = 1.5 K and has been investigated
as a candidate material to display quantum spin liquid behavior. The synthesis of a
similar material with nickel in place of copper results in a spin-1 kagome system. We
performed measurements on both protonated and deuterated samples of this material,
.
BaNi3 (OH) 2 (V0 4 ) 2
147
I
a)
I
*
H=200e
500e
A 100 e
V 50000e
*
m
I
6-
0..
z
E 4-
-
8-
-
.
*A*
C)
E
AU
2-
A*.
--
0
0
5
10
A,&
20
15
25
30
35
40
350
400
Temperature (K)
I'
I*
300-
I*E*
I-m-
b)
250E 200-
z
0
15010050-
0
50
100
150
200
250
300
Temperature (K)
Figure 5-9: (a) Susceptibility of BaNi 3 (OH) 2 (VO 4 ) 2 , approximated as M/H. Both
zero-field-cooled (open symbols) and field-cooled (filled symbols) data are shown.
(b) Inverse susceptibility, measured at H = 5000 Oe. Red line is a Curie-Weiss fit
over the range 50 K - 350 K.
148
5.2.2
Thermodynamic Measurements of BaNi3 (OH) 2 (VO 4 ) 2
,
Magnetic measurements were performed on a powder sample of BaNi3 (OH) 2 (VO4 )
2
synthesized as described in Reference [109]. Magnetization was measured under applied magnetic fields ranging from 20 Oe to 5000 Oe using a Quantum Design Magnetic Property Measurement System (MPMS) SQUID magnetometer. Measurements
were performed under field-cooled (FC) and zero-field-cooled (ZFC) conditions. The
data were corrected for the molecular diamagnetism of the sample by use of Pascal's
constants[76].
Figure 5-9(a) shows the magnetic susceptibility, approximated as M/H, as a function of temperature. Below T = 25 K there is a sharp upturn in the susceptibility,
and a splitting of the FC and ZFC data. The magnitudes of the upturn and splitting
decrease with increasing field. By H = 5000 Oe there is no observable difference
between FC and ZFC data down to 2 K. Figure 5-9(b) shows inverse susceptibility
data. A Curie-Weiss fit over the range 50 K - 350 K yeilds a Curie-Weiss temperature
of Ecw = +12 K. While this value of Ecw is indicative of a small ferromagnetic coupling, it can also be explained by competing ferromagnetic and antiferromagnetic interactions. In vesignieite, there are multiple oxide superexchange coupling pathways
between nearest-neighbor metal ions[109]. The sign and strength of the couplings
will depend on the angles of these bonds, as described by the Goodenough-Kanamori
rules[83, 84, 85]. The bond angles suggest that both strong ferromagnetic and strong
antiferromagnetic couplings will exist between nearest-neighbor nickel ions. In the
copper mineral vesignieite, the antiferromagnetic coupling pathway dominates due
to a Jahn-Teller distortion of the Cu octehedra. Therefore vesignieite has a larger,
negative Curie-Weiss temperature, 9 cw = -77 K[33]. In contrast, the Ni analog
should have increased electron density in the ferromagnetic coupling pathway, which
effectively lowers ECW but does not weaken the antiferromagnetic exchange.
Further magnetization measurements were performed using an MPMS Vibrating Sample Magnetometer (VSM) and using a Quantum Design Physical Property
Measurement System (PPMS). VSM data were normalized by comparing to mea149
0.4-a
0.3-
0.2 -0.1
-0.1
-0~~
2A,
T = 1.8
-5K
K
-0.3 -,-.-
afi
-0.4
-10000
-5000
0
K
5000
10000
1.5 -
)
H (Oe)
b)
4 44
-
1.0 - Z4
e4
0.5 -m
1
A
T =1.8 K
5K
5 K
0.0
0
50000
100000
150000
H (0e)
Figure 5-10: Magnetization of BaNi 3 (OH) 2 (VO4)2 as a function of applied field.
(a) Low-field magnetization data measured on an MPMS using a VSM, showing a
hysteresis loop. (b) High-field magnetization data. Black and red data points are
the same as in (a). Blue data points were measured using a PPMS in order to reach
higher applied fields.
150
surements performed on the PPMS at the same temperature. Figure 5-10(a) shows
hysteresis loops measured using the VSM at T = 1.8 K and T = 5 K. At 1.8 K a
coercive field of -3,000 Oe is observed. This coercive field shrinks to ~1,000 Oe at
5 K. At higher fields, there is no difference between the magnetizations at the two
measured temperatures, as shown in Figure 5-10(b). Magnetization was measured up
to fields of 14 T at T = 5 K. Even under a 14 T field, the magnetization is less than
75% of the expected saturation magnetization for spin-1 moments. This suggests the
existence of strong antiferromagnetic coupling, and supports the idea of competing
strong couplings rather than weak ferromagnetic coupling.
AC susceptibility was measured using a PPMS. Measurements were performed
at zero field in the temperature range of the magnetic transition seen in the magnetization data, using an oscillating field of amplitude 10 Oe. Measurements were
performed using oscillating fields of 20 different frequencies ranging from 10 Hz to
10,000 Hz in logarithmic intervals. The results are shown in Figure 5-11. The real
component of the susceptibility (Fig. 5-11(a)) features a peak around T = 20 K,
which is slightly below the temperature of the upturn in DC susceptibility. There
is also a shoulder at lower temperatures (-15 K). The signal is strongly frequencydependent in the region of the peak. Above the peak temperature, data measured at
different frequencies quickly collapse onto a single function. At lower temperatures,
the frequency dependence also decreases, although much more slowly than at high
temperatures. The imaginary component of the susceptibility (Fig. 5-11(b)) clearly
displays two peaks, at the temperatures of the peak and shoulder, respectively, seen
in the real component. The imaginary component goes quickly to zero above the transition temperature. The imagniary component is only slightly frequency-dependent.
The frequency dependence and asymmetry observed in the AC susceptibility indicate
a spin glass ordering transition. This behavior is a result of a continuum of relaxation
times, which exists in a spin glass due to the different domain sizes created by a glassy
ordering[4].
Additional magnetization and AC susceptibility measurements were performed
under applied fields up to 14 T. Application of a 0.5 T field removed the frequency
151
2.
a)
10Hz
1.5-
Z
E
10
1.0-
10,000 Hz
E
0.5-
1
,
I
0
5
.
,
10
15
,
25
20
30
35
41
)
0.0
I,I,.,'i
Temperature (K)
0.15
E
E
0.10
0.05
-
-I
*
z
b)
0.00
0
5
10
15
20
25
30
35
40
Temperature (K)
Figure 5-11: AC susceptibility of BaNi 3 (OH) 2 (VO4 ) 2 as a function of temperature
showing the (a) real and (b) imaginary components of the response function. Measurements were taken with zero DC applied field and an oscillating field of amplitude
10 Oe. Measurements were performed using 20 different oscillating field frequencies
ranging from 10 Hz to 10,000 Hz in logarithmic increments.
152
0.4
-
L*oH 0.5 T
A
1T
v 2T
a)
0.3-
3T
4T
=z
-5
SST
S10 T
* 14T
j
0.2-
0.1
V
9
4
70
80
V.0
0
20
10
30
40
50
60
Temperature (K)
0.15
b)
z
*
*
T
=0
mstoH
I
*
0.10
75
4 4T
E
5T
S10 T
* 14 T
M.
0.05
0.00
10
,
-I
0
20
30
40
50
60
70
80
Temperature (K)
Figure 5-12: Magnetization and AC susceptibility of BaNi 3 (OH) 2 (VO 4 ) 2 as a function
of temperature under applied magnetic fields. (a) M/H (b) Real part of the AC
susceptibility measured using an oscillating field of amplitude 10 Oe and frequency
10,000 Hz.
153
dependence of the AC susceptibility and suppressed the imaginary component of
the susceptibility. Figure 5-12 shows the magnetization divided by the applied field
(M/H) and the real part of the AC susceptibility as a function of temperature at
a variety of applied fields. The M/H data show an upturn at low temperatures,
and the magnitude of the upturn decreases with increasing field, continuing the trend
observed at lower fields (Fig. 5-9). The magnitude of the AC susceptibility is strongly
suppressed by the application of a relatively small field. The peak height at paOH =
0.5 T is only 1/10 the peak height at zero field. AC susceptibility decreases further
with increasing field, but the rate of decrease is smaller. The peak observed in zero
field is still apparent up to MOH ~ 3 T, but is shifted to higher temperatures with
increasing field. Another peak is also observed at lower temperatures T
-
5 K. At
higher fields (10 T and 14 T) no peaks are observed. Rather, an upturn is observed
at low temperatures, with the temperature of the upturn decreasing with increasing
field.
Specific heat of a pressed powder pellet of BaNi 3 (OH) 2 (V0 4 )2 was also measured
using a PPMS. Measurements were performed as a function of temperature at a
number of applied fields up to 14 T. The results are shown in Figure 5-13. A slight
peak is observed in the zero-field data at T = 26 K, which is consistent with the
temperature of the upturn in the DC susceptibility. This feature is enhanced and
broadened by a 2 T applied field and suppressed by larger fields [LOH > 5 T. Below T
= 10 K a broad hump is observed in the zero-field data. This feature is suppressed by
application of a field, suggesting it is also due to magnetic behavior, but is not fully
suppressed even by a 14 T field. No distinct feature is observed at this temperature
in the magnetization or susceptibility measurements.
154
14-
a)
12-
10-
z
EE68
6 --
.jiGH1 0 T
4
*
2
2T
5T
10T
.14T
0
5
10
15
20
25
30
35
Temperature (K)
-b)
-
100
E.
80-
z
75
Em
60
E
40
-
2
o
U
2000
50
100
150
200
Temperature (K)
Figure 5-13: Specific heat of BaNi 3 (OH) 2 (VO 4 ) 2 . (a) Low-temperature specific heat
at a number of applied fields. (b) Zero-field specific heat up to 200 K.
155
5.2.3
Thermodynamic Measurements of BaNi 3 (OD) 2 (VO 4 ) 2
In preparation for neutron scattering measurements, a deuterated powder sample
of nickel vesignieite (BaNi 3 (OD) 2 (VO 4 ) 2 ) was synthesized by the method described
in [109], except substituting Ni(OD) 2 for Ni(OH) 2 . Deuterium is advantageous for
neutron scattering measurements because it has much lower incoherent scattering and
absorption cross sections than hydrogen[74]. Substitution of deuterium usually has
little effect on the structural and magnetic properties of materials. In the case of the
nickel vesignieite analog, we found the deuterated compound showed behavior that
was qualitatively similar to that of the protonated compound, but with significant
quantitative differences.
.
Magnetic measurements were performed on a powder sample of BaNi 3 (OD) 2 (VO4 ) 2
Magnetization was measured under applied magnetic fields of 50 Oe and 5000 Oe
using an MPMS SQUID magnetometer. Measurements were performed under fieldcooled (FC) and zero-field-cooled (ZFC) conditions. The data were corrected for the
molecular diamagnetism of the sample by use of Pascal's constants[76].
Figure 5-14(a) shows the magnetic susceptibility, approximated as M/H, as a
function of temperature. As in the protonated compound, at low temperatures the
susceptibility increases and FC and ZFC data split. This splitting is suppressed by
a 5,000 Oe magnetic field. However, this upturn occurs at a significantly lower temperature of T ~ 15 K, as compared to a temperature of T ~ 25 K in the protonated
compound.
A Curie-Weiss fit over the range 50 K - 350 K yeilds a Curie-Weiss
temperature Ocw = -20
K, which now suggests antiferromagnetic coupling.
The
change in Curie-Weiss temperature from positive to negative suggests a relative increase either of the strength of the antiferromagnetic coupling or of the occupation of
the antiferromagnetic coupling pathway as compared to the protonated compound.
The decrease in the magnetic transition temperature is consistent with this. As the
antiferromagnetic coupling becomes more significant, the geometric frustration of the
kagome lattice will suppress magnetic ordering.
Magnetization was also measured as a function of applied field, as shown in Figure
156
0.6
0.61
II
O.
0.5-
a)
H= 500e
9
y 5000 Oe
*
0.4-
e.
E5
0.3C.)
E
S-
0.2
-
E
0.1
0.0
*1
(D
-**
0
5
10
15
25
20
Temperature (K)
.
.
400
350-
b)
300250-
z
200-
75 150100500.
a
0
50
100
150
200
250
300
350
400
Temperature (K)
Figure 5-14: (a) Susceptibility of BaNi 3 (OD) 2 (V0 4 ) 2 , approximated as M/H. Both
zero-field-cooled (open symbols) and field-cooled (filled symbols) data are shown.
(b) Inverse susceptibility, measured at H = 5,000 Oe. Red line is a Curie-Weiss fit
over the range 50 K - 350 K.
157
0.08-
a)
0.04-
--0.00
T = 1.8 K
5K
-0.04-
-0.08
-7500
-5000
-2500
0
2500
5000
7500
H (0e)
0.6
b)
0.5
0.4
a
Zw
U
0.3 -
T=1.8K
5K
0.20.10.0
0
20000
40000
60000
80000
H (0e)
Figure 5-15: Magnetization of BaNi 3 (OD) 2 (VO 4 ) 2 as a function of applied field.
(a) Low-field magnetization data showing a hysteresis loop. (b) High-field magnetization data.
158
5-15. As with the protonated sample, a hysteresis is observed at T = 1.8 K. However,
the coercive field is much smaller, and the hysteresis is nearly gone by T = 5 K. Aso
like the protonated compound, the high-field behavior is the same at T = 1.8 K and
T = 5 K. However, the magnetization increases much more slowly with field. At paOH
= 7 T, the protonated sample reaches a magnetic moment of -1 PB/Ni. At this field
the deuterated sample only reaches a magnetization of ~0.5 suB/Ni. This is consistent
with an increase in antiferromagnetic coupling suggested by the negative Curie-Weiss
temperature.
Finally, AC susceptibility was measured under zero applied field using a PPMS.
Measurements were performed in the temperature range of the magnetic transition
seen in the magnetization data, using an oscillating field of amplitude 10 Oe. Measurements were performed using oscillating fields of 10 different frequencies ranging
from 10 Hz to 10,000 Hz in logarithmic intervals. The results are shown in Figure
5-16. As with the protonated sample, the real part of the AC susceptibility shows an
upturn at the same temperature as the upturn in the DC susceptibility. Slightly below
this temperature at T
-
12 K the AC susceptibility is peaked and develops a strong
dependence on the frequency of the oscillating field. Unlike the protonated sample,
the signal does not appear to be dominated by two peaks. The highest-temperature
peak is well defined, but at lower temperatures the signal decreases slowly and the
frequency dependence remains down to the lowest measured temperature of T = 2
K. A feature that resembles two broad peaks can be seen in the temperature range
~ 4 K - 8 K. The imagniary component also does not die off at low temperatures.
This behavior still suggests a spin glass transition.
159
0.30-
.a)
10
0.25-
Hz
0.20-
Z
-
~
0.15-
E
0.10
0.05
0.000
2
4
6
8
10
12
14
16
18
20
14
16
18
20
Temperature (K)
0.035-
.b)
0.030 -b
Z
0.025-
E
E(
0.020-
0.0150.0100.0050.0000
2
4
6
8
10
12
Temperature (K)
Figure 5-16: AC susceptibility of BaNi3 (OD) 2 (VO 4 ) 2 as a function of temperature
showing the (a) real and (b) imaginary components of the response function. Measurements were taken with zero DC applied field and an oscillating field of amplitude
10 Oe. Measurements were performed using 10 different oscillating field frequencies
ranging from 10 Hz to 10,000 Hz in logarithmic increments.
160
5.2.4
Inelastic Neutron Scattering Measurements
Inelastic neutron scattering measurements were performed on the deuterated powder
sample described in the previous section on the Cold Neutron Chopper Spectrometer
(CNCS) at the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory.
An aluminum can was filled with 4.7 g of powder, sealed in a helium atmosphere, and
placed in a He-4 cryostat. Measurements were performed in High Flux mode with
chopper speeds of 300 Hz.
Measurements were performed with initial neutron energy 10 meV over a temperature range of 1.9 K to 58 K. Figure 5-17 shows the inelastic scattering spectrum
as a function of energy and momentum transfer. A broad feature can be seen in
the low-energy region. This feature extends out from the elastic line and disappears
above ~3 meV. In the low temperature measurements there is a peak near -3 meV,
which is less pronounced at higher temperatures. There is no obvious dispersion or
strong jM-dependence. A second feature can be seen at higher energies (~
5 - 8
meV). This broad peak is likely due to phonon scattering, as its intensity increases
with increasing
IQI
and with increasing temperature.
An estimate of the elastic scattering signal was obtained by integrating the inelastic data over small energy transfers, -0.5 meV < hw < 0.5 meV. Figure 5-18
shows the JQI-dependence of the elastic scattering. We see no evidence of long range
magnetic ordering, as no new magnetic peaks appear at low temperatures.
To further investigate the inelastic scattering spectrum, we integrate the data
over a large region of momentum transfer 1.0 A-'
QI
5 3.0 A-' and plot as a
function of energy transfer, as shown in Figure 5-19(a). As discussed in Chapter 2,
the inelastic scattering signal is proportional to the dynamic structure factor S(Q,w)
= [n(w)+1]X"(Q,w), where n(w) is the Bose occupation factor and x"(4,w) is the
imaginary part of the dynamic susceptibility. To isolate the inelastic signal due to
scattering from the sample we applied the following procedure. For large enough negative energy transfers at low temperatures, the scattered intensity is only background
because the scattering from the sample is suppressed by the Bose factor. Therefore,
161
Intensity (arb. units)
4
0
E
8
8r
6
6
E
4
4
3
2
0
2
1
2
3
0
4
1
|QI (A-')
E
8
6
6
E
4
3
4
3
4
4
1
2
3
0
4
1
2
|Q| (KP)
IQ| (A-)
8
8
6
6
4
3
4
2
2
0
4
2
2
3
3
IQ| (A-')
8
0
2
1
2
3
4
0
1
2
IQ| (A
)
|QI (A-)
Figure 5-17: Inelastic scattering spectrum of BaNi 3 (OD) 2 (VO4 )2 measured with neutrons of initial energy Ej = 10 meV at temperatures (a) 1.9 K, (b) 5.5 K, (c) 7.7 K,
(d) 11 K, (e) 19 K, and (f) 58 K.
162
5000
Z
*58K
4000
1
1
3000
~urip
I5~
2000
f
1000
0
2
1
3
4
IQI(A-')
Figure 5-18: Elastic scattering for BaNi 3 (OD) 2 (VO 4 ) 2 , obtained by integrating inelastic neutron scattering data measured with neutrons of initial energy Ej = 10 meV
over energy transfers -0.5 meV < hw < 0.5 meV, plotted as a function of IQI.
X"(w, T = 58 K) can be calculated by subtracting the intensity measured at 1.9 K
from that measured at 58 K and dividing by the Bose factor. Then x"(w, T = 58 K)
is known for positive energy transfers because X"(w) is an odd function of w. The
positive energy transfer background can be calculated by subtracting the calculated
signal at 58 K from the measured intensity at 58 K. Assuming this background is
temperature-independent in the range 1.9 K to 58 K, this background can be subtracted from the intensities measured at other temperatures to arrive at S(w, T).
X"(w, T) is then calculated by dividing by the Bose factor. X" can also be calculated
by using the 19 K data or 38 K data in place of the 58 K data. These calculations
all yield similar results.
Figure 5-19(b) shows x"(w, T) as a function of energy transfer at the different
measured temperatures. Points at small energy transfers (IhwI < 1 meV) were removed because resolution effects lead to some of the elastic or positive energy transfer
signal at 1.9 K being measured at small negative energy transfers. Therefore the measured intensity at 1.9 K is not only background at small negative energy transfers.
The broad peak centered around hw ~ 3 meV is suppressed with increasing temperature. There is still some signal even at temperatures above the transition temperature
163
( T= 1.9 K
.5.5
20
K
*7.7 K
rI~
=
11K
15
1519 K
I-
38 K
=
10
,58
K
5
J
01
-5
-10
0
5
10
hIo (meV)
4-
4(b)
2
0
=
8
6
2
0
2
4
6
0
8
H
hw (meV)
10
(c)
8
C
6
6
a
--
--
4
2
0
10
20
30
50
40
60
Temperature(K)
Figure
5-19:
Temperature
BaNi3 (OD) 2 (VO 4 ) 2.
dependence
of
the
inelastic
spectrum
of
(a) Inelastic neutron scattering data measured with ini-
tial neutron energy Ej = 10 meV, integrated over momentum transfers 1.0
A- < IQI K 3.0 A-'. (b) x"(w) extracted from the data as described in the text.
(c) X"(w), integrated over the energy range 1 meV < hw < 5 meV spanning the low
energy peak, as a function of temperature. The dashed line indicates the value at
T = 58 K.
164
of -15 K seen from thermodynamic measurements. X"(w) was integrated over the
range 1 meV < hw < 5 meV for each temperature to get a measure of the integrated
intensity of the low-energy excitation. The results are shown in Figure 5-19(c).
At higher energy transfers, X" collapses onto a single function for T < 19 K. The
38 K and 58 K data differ slightly, but show qualitatively the same behavior. This
supports the idea that the higher-energy peak is due mostly to phonon scattering, as
the differences in scattered intensity at different temperatures should be due only to
the Bose factor. The difference between the low and high temperature measurements
at high energy transfers may indicate the presence of some magnetic scattering in this
energy window as well.
To investigate the inelastic spectrum at smaller energy transfers, further measurements were performed with an initial neutron energy of Ej = 3 meV. The lower value
of Ej improves the energy resolution and so narrows the incoherent elastic peak. This
allows us to resolve inelastic features at lower energies. Measurements were performed
at temperatures of 1.9 K, 7.7 K, and 19 K. Figure 5-20 shows the inelastic scattering
spectrum as a function of energy and momentum transfer. No distinct features are
seen at these lower energy transfers.
Figure 5-21(a) shows the inelastic spectrum measured at E = 3 meV integrated
over momentum transfer range 0.7 A' < IQI
1.5 A-'. The signal is dominated
by quasielastic scattering, which becomes broader at higher temperatures. X" can be
calculated as described above using the 1.9 K and 19 K data sets. The results are
shown in Figure 5-21(b). At these energy transfers, X" is mostly flat, although the
19 K value is slightly lower at the lowest energies. Above 2 meV the data become
quite noisy, but are consistent with the value of X" calculated from the Ej = 10 meV
data. At low energy transfers (hw ~ 1 - 2 meV) the calculations from different Ej
measurements differ slightly.
165
Intensity (arb. units)
1
0
2
4
3
2
U) 2
2
3
3
1
0
0.5
1
1.5
1
0
2
0.5
1
1.5
2
IQI (A')
IQI (A-')
2
E
3
1
0
0.5
1
1.5
2
|QI (K')
Figure 5-20: Inelastic scattering spectrum of BaNi 3 (OD) 2 (VO4 )2 measured with neutrons of initial energy Ej = 3 meV at temperatures (a) 1.9 K, (b) 7.7 K, and (c) 19
K.
166
. . . . .
.
30
*T=1.9K.
T= .K
(a)
+7.7
25
K
19 K
20
4
N
1510.
-
5
-2
-3
-1
0
1
2
3
2.0
2.5
3.0
hw (meV)
(b)
15
10
IHI
I
0.0
0.5
1.0
1.5
Aw (meV)
Figure
QJ : 1.5
5-21:
Temperature
dependence
of
the
inelastic
spectrum
of
BaNi 3 (OD) 2 (VO 4 ) 2. (a) Inelastic neutron scattering data measured with initial neu<
tron energy Ej = 3 meV, integrated over momentum transfers 0.7 A-'
A-'.
(b) X"(w) extracted from the data as described in the text.
167
5.2.5
Conclusion
We have presented the results of thermodynamic and neutron scattering studies on the
compound BaNi 3 (OH) 2 (VO4 ) 2 . The compound is a structural analog of the kagom6
antiferromagnet vesignieite, but the kagom6 lattice is formed from spin-1 Ni2 + ions
in place of spin-} Cu 2+ ions.
The Ni - 0 - Ni bond angles suggest that nearest-neighbor Ni ions will be cou-
pled by both strong ferromagnetic and strong antiferromagnetic coupling pathways,
and this is supported by our magnetization and AC susceptibility measurements. A
small positive (ferromagnetic) Curie-Weiss temperature is observed, but the material
is not fully polarized by a relatively strong magnetic field even at low temperatures,
suggesting the presence of a strong antiferromagnetic interaction. AC susceptibility
displays the characteristic behavior of a spin glass transition, which is also consistent
with competing interactions. Specific heat measurements display a magnetic feature near the spin glass transition temperature, but also show a significant magnetic
contribution at much lower temperatures.
The deuterated compound BaNi 3 (OD) 2 (VO4 ) 2 displays qualitatively similar be-
havior to BaNi 3 (OH) 2 (V0 4 ) 2 . However, the relative strength of the antiferromagnetic
coupling is increased with respect to the protonated compound. The Curie-Weiss temperature has a negative (antiferromagnetic) value, and the magnetization increases
more slowly with field than the magnetization of the protonated compound. Additionally, the ordering transition is shifted to a lower temperature due to the geometric
frustration of the kagom6 lattice.
Neutron scattering measurements reveal no signs of long range magnetic ordering.
The inelastic spectrum contains a broad, IQj-independent peak that is suppressed as
the temperature is raised above the spin glass transition temperature. A peak that
is broad in energy and flat in
IQI
is consistent with a spin glass, which is made up of
small magnetic domains with varying sizes and relaxation times.
In summary, frustrated spin-1 systems are of great interest but experimental realizations are rare. The nickel analog of vesignieite is a new frustrated magnetic system
168
featuring spin-1 moments. The magnetism is frustrated by the kagome geometry as
well as by competing ferromagnetic and antiferromagnetic interactions.
The com-
peting interactions lead to a spin glass transition. However, the relative strengths
of these competing interactions appear readily tunable in this system, as they are
affected strongly even by the substitution of deuterium for hydrogen. This makes
BaNi 3 (OH) 2 (VO 4 ) 2 an exciting system in which to study the behavior of spin-1 mo-
ments on the kagome lattice, as well as the effects of competing interactions combined
with geometric frustration.
169
170
Appendix A
Spin Wave Calculation Details
A.1
Calculation of Spin Wave Dispersion
We begin with our spin Hamiltonian
=
FM
+
7
iDM
+
(A.1)
fHZeeman
(-iS)i -S + D-
St
-gpB
(A.2)
'Si
(idj)
To calculate the spin wave dispersion, we first want to represent our Hamiltonian
in terms of the spin raising and lowering operators. We make the substitutions
(A.3)
st - S-
(A.4)
SiY =
0-,
*
ST =si+ +2 Sl-,
We also recall that to quadratic order in deviation of S, we only need to include the
component of D parallel to the spin polarization direction (2) and therefore
D*j - (i x S.) -÷ (5 - 2)[($
x
S) -2]
= D
x
(A.5)
It is important not to confuse Df. and D_. Df- is the component of the vector Dij along
171
the spin polarization direction, which can be controlled by changing the direction of
the applied field. D, is the component of the DM vector perpendicular to the kagom6
plane, which is intrinsic to the material. These two are the same when the spins are
polarized perpendicular to the kagome plane.
Substituting (A.3-5) into the components of (A.1):
71,FM
(A.6)
S;-St + 2Sj-S)
ZSt S3
=-~
(i,j)
ZD(S%-.St
Df-
=D
-
(A.7)
St S)
(i,3)
IiZeeman = -glB
H
S
(A.8)
Next we use the Holstein-Primakoff transformation[37, 38] and make the assumption that at low temperatures (a~ai)
2S.
(2S - alai) ai ~, v/25 ai
SV=
S
<
= a-
(A.9)
(2S - alai) ~ v25 a T
Siz = S - atai
Substituting (A.9) into (A.6-8) and neglecting any terms containing the product
of four a operators:
4M=
EfM - JS E(aiaj + ata3 ) + 8JS
atai
(A.10)
(iJ
?DM =
liZeeman =
Where EM
=
-ZS E Df (aa - ajaj)
-9g
BH(NS - E
aaja)
(A.11)
(A.12)
-4NJS2 is the ground state energy of the ferromagnet with 4 nearest
neighbors.
To see the spin wave dispersion, we want to examine this Hamiltonian as a function
172
of momentum rather than position. To do this we perform the standard canonical
transformation suggested by the Bloch theorem[38]
ai = an,=
at= at
1
1
e,( fi+W,)av(q
1
(A. 13)
e-TRfn+d,,)at(qJ
where n specifies the unit cell at location Rn and v specifies the basis atom within
the unit cell at location d. E {0, , }. Additionally,
a,[at.
= JT,4'
(A.14)
By substituting (A.13) into (A.10-12) we can arrive at an expression for 7
as a
function of momentum. Here we will go through this process explicitly for the middle
term of (A. 10). The calculations for the other terms follow in a similar fashion. First
we represent the operator locations i in terms of their unit cell vector and basis vector:
S(aiaj + aa,.) =1
(ij)
an,a , + h.c.
(A.15)
n,v m,pE{n,v}nn
Where m and /t are selected such that Rm + d,, is a nearest neighbor to Rn + d,, and
h.c. designates the Hermitian conjugate. The factor of i2 is added because this sum
will count each bond twice. Substituting (A. 13) into (A. 15) gives
1+
n,v mIE{n,v}n.
a(qJa ("') + h.c.
lid'
(A.16)
2N
e
+ h.c.
~
(A.17)
We can bring out the sum over n because the term [(Rm + dl) - (Rn + d,)] defines
the vector to the nearest-neighbor site. The set of these vectors will depend on v but
173
not on n. From here we replace [(Rm + dt) - (Rn +
I which is in the set of
Ld)] with
vectors that define the nearest neighbors of the site v.
Summing over n:
Se*(q')Rn
(A.18)
= N6g~g,
which we can substitute into (A.17) and then sum over q' to get
1
sq
2
-t(
E
)
~ I~a,0t
(q3) + h.c.
(A.19)
We will now perform the sum over v and 6:
+e-'
a~)
/
e-'T A I(q)
a,(q)
e
2a
1
a2(q)
+
2a3(q-) +
e-*
" e-Tc bf3l(q)
(q)
e-q2 a'(q)
e-
+ e-
+ +-eaf a(0-~
2a
S 2
2
(A.20)
+ h.c.
a(q)
a(q
a (q-)
/
a3(q)
+ e-
f lq)
e-6
Since there are no terms of the form avat we can switch the order of all creation
and anihilation operators. Also, the Hamiltonian is real so the Hermitian conjugate
is identical to the term we have just worked out. Therefore we just multiply by two
and arrive at:
(A.21)
(aia- + ata3 ) =
(ij)
0
2
(a'(q)
a'(qJ
a'(q)
cos(q - 6)
cos(q-
0
cos(q'
cos(q. 6+6)
-
IT
cos(q - )
cos(q .
0
(q
)
a2(q)
Ja(q
The other terms of ?- can be worked out in a similar fashion, with the result that
174
the Hamiltonian (A.1) can be expressed as
'
=
[(4JS + gpBH)Jyna1 k)a,( #)+a1(k )'1y,(k)a,(k)]
Eo+Z
(A.22)
k~ "t'L
where
cos(k
0
0
cos (k -()
cos(k-)
0
cos (k -6)2
2)
D 2 cos (k -)
0
+z2S
-D
2
)
Dj COS (k -
-Dz cos (k-
0
cos (k.)
-Diz, COS (k -
(A.23)
)
cos (k
(k-)
Dz2
COS
(
J
0
)
'h() = -2JS
)cos
In the case where the spins are polarized perpendicular to the kagome plane and
D2 = DZ = Dz = Dz, the second term reduces to
z2DZS
-cos (k-)
cos (k-)
-COS (k
COS (k -)
- cos
(A.24)
cos (k -i
0
)
0
(k.-)
0
In this case (A.23) can also be represented in the simpler form
0
-.
-
'c(k) =
2JS
COS
cos (' _ )e-k./3
cos (k - )2/
cos (k - 2)e-%0/3
0
3
cos (k- )e-%0/ cos (k
cos (k - )e/
e,0/3
(A.25)
0
where tan! = P. An analytic solution for the eigenvalues and eigenvectors of (A.25)
is presented in the Supplementary Information for Reference [50].
175
A.2
Calculation of S(Q,w)
We follow the treatment by Lovesey[61] but modified to include the multi-atom basis
of the kagome lattice. Scattered intensity is proportional to
y(Jpa -
(A.26)
M3 ) SO (Q,
a,/3
where
dt e-tEe(4+&)
S~a (c,6w) =
KSa(O)S3 +a (t))
(A.27)
Where n sums over unit cells and v sums over the basis vectors within each unit
cell. Because S2 commutes with '-, the inelastic spectrum depends only on the terms
(S,*Sf
(t)). Then, (A.26) reduces to
1+
)
dt e-t (KSS3(t) + S_-
+ (t))
(A.28)
We perform the inverse of the canonical transformations (A. 13) on the spin operators
S-
e Fnd+)S
_
_
(A.29)
and on the boson creation operators
at
-
1
E e-(fn+Z)at
(A.30)
so that under the Holstein-Primakoff transformation
S = v\ 2
Q
(A.31)
a
Qii
176
And eigenstates of the hamiltonian can be represented as
(A.32)
t , 10)
Q, i) = 2
Where 10) represents the fully aligned ground state.
We calculate the thermal average by summing over the eigenstates weighted by
the Bose occupation factor n(w)
,
Zi a
,) (Q'
V
i,4'
= 2S 1
n(w ,,)
a,
(01
_
ezt/hagu
a ,
0) e-(hwi+EL)1h
/L'A'
i4'.L
n(wjo)
iE
\'
US?)U4)
9
e- to
(A.33)
/
= 2S
Q'
e/,,e-"/h
/
KS- (0)St(t)) = 2S 1 n(w
Which is the result from [61] multiplied by the factor in parentheses and summed
over the different bands due to the multi-atom basis of the kagome lattice. Then,
following [61]:
(St(0)S- (t)
no,)+
=2
2S n~j'c)
14)
1)
uU?. J- (M
Z'V
e44
(A.34)
and (A.28) becomes
2
(A.35)
(1 + Z) Ssw(Q, w)
with
Ssw(Q, w) =
I
i
(
,
)
n(wi) + i) 6(w
-
+ n(wgg)6(w +
wi)1
(Av
(A.36)
177
178
Appendix B
Cu(1,3-bdc) Calculated vs
Measured Structure Factor
In Chapter 4 we presented only selected cuts of our data compared with our calculation. Here we present a full complement of these plots. LET measurements were
taken at a temperature of T = 1 K. SPINS measurements were taken at T = 70 mK.
All plots contain the following:
1. Black Points:
(a) LET Plots: Intensity integrated over
IQI
0.08
A-1
(b) SPINS Plot: Constant-c scans
Intensities measured at two different applied fields are subtracted to isolate the
magnetic signal.
2. Blue Lines:
Calculated structure factor for Heisenberg ferromagnet with DM interaction D,
=
0.09 meV and D, = 0. For fits to LET data at H 5 0, D -
=
tD, and
for fits to all SPINS data and LET data at H = 0, D - i = 0. Calculation is
integrated over
IQI
t 0.08 A- and convolved with the LET or SPINS intstru-
mental resolution function, assuming 6-function (infinite lifetime) spin waves.
Calculated structure factors for fits to zero-field data are increased by a factor
179
of 3/2 to account for the difference in intensity due to the difference in spin
polarization direction as described in Chapter 4. Calculated structure factors
for two different fields are subtracted and an overall scaling factor is applied to
each cut. This overall scaling factor changes with
but is consistent across
IQ
.
fields for data measured at the same
IQI
3. Red Lines:
Same as blue lines except including a magnon damping term F = 0.03 meV for
fits to all SPINS data and LET data at H f 0 and F = 0.08 meV for fits to
LET data at H = 0.
180
0
8
------..
xx::::::::::::---..::::::--...:::::::::::.xx
xx
xxx.xx
xxxxxxx
- xxx
xx.
-
e:..ee
~~..
nee
------........
-..-- -.......-----.
.-.-.
xxxxxxxx
..
00
.ep
Ho
cDn
H
(D
i
-
I
4
K
I'
K
Intensity (arb. units)
4
Intensity (arb. units)
i
-
i
I
e
____
Intensity (arb. units)
e
Intensity (arb. units)
i
-
I
Intensity (arb. units)
g.
Intensity (arb. units)
00
tH-
(D
oq
i
-
---- -
I
Intensity (arb. units)
0
Intensity (arb. units)
-
I
..............
-
7
Intensity (arb. units)
Intensity (arb. units)
i
p
..
.. ....
II
Intensity (arh. units)
*0,
Intensity (arb. units)
00
L-i
i
----------
1IP
->
.........................
....
i
Intensity (arb. units)
Intensity (arb. units)
-
I
2
0-
~-u~--
-
Intensity (arb. units)
I
Intensity (arb. units)
8
i
9
0
0
Intensity (arb. units)
ig
Intensity (arb. units)
8
a
i
I
a
a
Intensity (arb. units)
-
Intensity (arb. units)
I
I
0
#
0
#
Intensity (arb. units)
8
Intensity (arb. units)
00
H
C
I-
I
-
o
'A
-8
0
S
.. ..
....
.. ..
..
..
.....
I
Intensity (arb. units)
Cr
cz 00
I
Intensity (ark. units)
i
I
-
II
I
a
'0
II
Intensity (arb. units)
Intensity (arb. units)
8
C
-
RC
I
Intensity (arb. units)
.4
-----
Intensity (arb. units)
-8
i
-
I
Intensity (arb. units)
5.
Intensity (arb. units)
-
o CR
oC
I
C
CR
0p
Intensity (arb. units)
Intensity (arb. units)
-0
C;'
00
....
.....
...
...
.....
tH
(D
O-q
p
I
II
I
Intensity (arb. units)
71C
*
CR
I
Intensity (arb. units)
-
pa
I-
-
PA
0
I
P
Intensity (arb. units)
--L---- - - -
Ii
Intensity (arb. units)
i
IA
p.
*
*0c
I
II
p
T1C
Intensity (arb. units)
CR
Intensity (arb. units)
5
!1A
CR
C
P
CC
A
*
I
T,
Intensity (arb. units)
I
Intensity (arb. units)
30
.
20
10
2
[IIQ=1.A
3
10
20
30
10
IU
0
0.0
0.5
1.0
1.5
2.0
0
-10
U
71-INV,
V
nTQ=1.1A
2
U
U
Tilt
-20
U
-30
2.5
0.
3.0
0.5
1.0
. I
0- 1 . I
3
2
0
1.5
2.0
2.5
3.0
2.5
3.0
2.5
3.0
2.5
3.0
hw (meV)
Aw (meV)
oI|LQI= 1.A-'
0
20
U
.U|i
10
10
-U
0
-U
-10
-20
-30
)
0.
I
-1
0
-2 0
-3
1.0
0.5
-U
1.5
2.0
2.5
0
0.0
3.0
0.5
1.0
IQI=1.4A
Ii,-
10
20
U
U
10*
a-
U
-U
0
-1
-2
-3
0
0
U
TT
U
0
0.0
2.0
30
30
20
1.5
hw (meV)
Aw (meV)
I
f
I
a
-10
-20
-30
0.5
1.0
1.5
2.0
2.5
0.
3.0
0.5
1.0
hw (meV)
1.5
2.0
1k (meV)
30
20
20
I-1|=1.6A~
10
10
0 di -I
S0
-a-10
-10
I
-20
p
S-20
-3
-30
0.0
0.5
1.0
1.5
2.0
2.5
01
0.0
3.0
kw (meV)
0.5
1.0
1.5
2.0
hw (meV)
Figure B-6: LET Ej = 12.4 meV 1(7 T)-I(2 T). Note Intensity scale change from
Fig. B-5
186
. . ........
.................
............
. ....
.....
....
0o
-]
(D
Hrj
(D
oq
ft
a
P
P
Ki
a5
Intensity (arb. units)
K-
Intensity (arb. units)
i
i
-
(A
th
(A
aS
~ F
;p
A
(
a
CD
r
P
-4
2
-
;5
Intensity (arb. units)
0
Intensity (arb. units)
P
th
Ile
I
T
a
c
(A
Intensity (arb. units)
-
Intensity (arh units)
5
1,
V
0
j
I(
I
I
;5
Intensity (arb. units)
>7
Intensity (arb. units)
i
.... .....
............
30
30
IQ|=I.A~
20
10
10
.-
1~
0)
0)
0)
IQI=1.1A-,
20
Y
-10-1
]
-20
-30j
0.0
0.5
1.0
1.5
2.0
2.5
-Y
-)
-10
-20
-30,
3.0
0.0
0.5
1.0
30
1
20
0)
=
IQI=1.2A-'
I
1.5
2.0
2.5
3.0
Aw (meV)
hw (meV)
-
0
2
0
10
I.
0)
0)
0)
01
J
-10
-3 0
-2
-20
IQI=1.5A~
0
-30.
0.0
0.5
1.0
1.5
2.0
2.5
00.0
3.0
4 0.5
1.0
hw (meV)
2.5
3.0
0
3
2 0m
0)
0)
0
0
aO
4i-
-2
-3 0
0.0
IQI=1.5A~'
20
10
I0)
#,
0
0)
0)
-10
-20
-30
0.5
1.0
1.5
hw
2.5
0.
3.0
0.5
1.0
1.5
hw
(meV)
.5
0.0 0. 1.0
2.0
2.
=2.A
2.0
2.5
3.0
(meV)
3.
0.
3
31
IQ=1.6A-1
20
0)
0)
1.5
2.0
Ae (meV)
0)
=
IQI=1.7A'
2
0
14
a-
I0)
0)
0
0)
0)
-1
-2
0)
4.'
0)
0 T
-3
-20
-30
0.0
0.5
1.0
1.5
hw
2.0
2.5
0.0
3.0
0.5
1.0
1.5
2.0
2.5
3.0
hw (meV)
(meV)
Figure B-8: LET Ej = 12.4 meV 1(0 T)-I(7 T). Note Intensity scale change from
Fig. B-7
188
IQI=0.6A- 1
200
100.
1.
1
0
-100
-200
-300
400
0.5
-- ----
1.0
-
1.5
-
2.0
2.5
3.0
3.5
3.0
3.5
hco (meV)
200
IQI=0.78-'
100
6
0
-100
-200
-4001
0.5
-
-3001
1.0
2.0
1.5
2.5
hw (meV)
C7
200
- - - - . - - - - . - - - - .I - - -. - - - -. - - -
-
300
igS=0. A-(
100
0
-100
-200
-300
0.5
---1.0
1.5
--- --- * -2.5
2.0
hto (meV)
Figure B-9: SPINS I(7 T)-I(O T)
189
- -3.0
-
- 400
3.5
190
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