Table Of ContentSTRUCTURAL PHASE TRANSITIONS IN LAYERED TRANSITION
METAL COMPOUNDS
PHYSICS AND CHEMISTRY OF MATERIALS
WITH LOW-DIMENSIONAL STRUCTURES
Series A: Layered Structures
Managing Editor
F. LE VY, Institut de Physique Appliquee, EPFL,
Departement de Physique, PHB-Ecublens, CH-I015 Lausanne
Advisory Editorial Board
J. V. ACRIVOS, San Jose State University, San Jose, California
H. AREND, Laboratorium jUr Festkorperphysik ETH, Ziirich
H. W. MYRON, Katholieke Universiteit, Nijmegen
A. D. YOFFE, Cavendish Laboratory, University of Cambridge
GENERAL EDITOR: E. MOOSER
STRUCTURAL PHASE
TRANSITIONS IN
LAYERED TRANSITION
METAL COMPOUNDS
Edited by
KAZUKO MOTIZUKI
Department ofM aterial Physics,
Faculty ofE ngineering Science,
Osaka University, Japan
D. REIDEL PUBLISHING COMPANY
A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP
DORDRECHT/BOSTON/LANCASTER/TOKYO
Ubrary or Congress Cataloging in Publication Data
SOfle-on'r reprinl oflhe hanko ....· r I5t edition 1986
Structural phase transitions in layered uansition metal compounds.
(Physics and chemistry of materials with low--dimensional
structures. Series A, Layered structures)
Bibliography: p.
Includes index.
I. Transition metal compounds. 2. Layer structure (Solids)
3. Phase transformations (Statistical physics) I. Motizuki,
Kazuko,l928- II. Series.
QOln.T6S77 1986 530.4'1 86-22017
ISIJN-13:978-94-O I0 -8533-5 dSIlN·13:978-94-009-4H6-{)
001:10.1007978-94-009-4S76-{)
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TABLE OF CONTENTS
PREFACE ~
K. MOTIZUKI and N. SUZUKI / Microscopic Theory of Structural Phase
Transitions in Layered Transition-metal Compounds 1
1. Introduction 1
2. General theory of electron-,Iattice interaction and lattice dynamics
based on the none>rthogonal tight-binding approximation 2
2.1. Electron-lattice interaction 3
2.2. Generalized electronic susceptibility 9
2.3. Interatomic force, phonon dispersion, and lattice instability 12
2.4. Electronic structure of the CDW state 19
2.5. Otber methods oflattice dynamical calculation 23
3. 1T -type transition-metal dicha1cogenides 29
3.1. TiSez 30
3.1.1. Formation of superlattice 30
3.1.2. Electronic structure 31
3.1.3. Electron-lattice interaction and generalized elec-
tronic susceptibility 33
3.1.4. Lattice dynamics 41
3.1.5. The CDW state 46
3.2. TiSz 55
3.3. M~ed compounds 55
3.4. VSez and CrSez 61
3.4.1. Summary of experimental evidence regarding struc-
tural transformation 61
3.4.2. Electronic structure 62
3.4.3. Electron-lattice interaction and lattice instability 64
3.5. TaSz and TaSez 67
3.5.1. Summary of successive phase transitions 67
3.5.2. Electronic structure 68
3.5.3. Lattice instability 73
4. 2H-type transition-metal dichalcogenides 76
4.1. Formation of superlattice 76
4.2. Electronic structure 77
4.3. Electron-lattice interaction and generalized electronic
susceptibility 80
4.4. Lattice dynamics and phonon anomaly 87
4.5. Discussion 91
5. Transition-metal trichlorides MCl3 (M = Ti, V, Cr) 92
5.1. Summary of experimental results 92
5.2. Electronic structure 92
5.2.1. Tight-binding calculation 92
v
vi TABLE OF CONTENTS·
5.2.2. The Wannier function 94
5.2.3. Estimation of band parameters 96
5.3. Electron-lattice interaction and lattice instability in TiCl 97
3
.5.3.1. Generalized electronic susceptibility 97
5.3.2. Effective d-electron-Iattice interaction and lattice
instability 100
5.3.3. The transition temperature 103
5.3.4. Some remarks regarding VCl and CrCl 103
3 3
5.4. Phase transition 104
5.4.1. Calculation of free energy 104
5.4.2. Magnetic susceptibility 105
Appendix A. Perturbation theory in nonorthogonal representation 107
Appendix B. Electronic free energy expansion in the adiabatic
approximation and derivation of generalized electronic
susceptibility 113
Appendix C. Frohlich model and Bloch model 116
Appendix D. Expressions for the overlap and transfer integrals and
their derivatives in terms of Slater-Koster integrals 121
N. SUZUKI and K. MOTIZUKI / Microscopic Theory of Effects of Lattice
Fluctuation on Structural Phase Transitions 135
1. Introduction 135
2. Formulation 136
2.1. Hamiltonian and Green's functions for electron and phonon
systems 136
2.1.1. Hamiltonian 136
2.1.2. Thermal Green's functions for electrons 137
2.1.3. Thermal Green's functions for phonons 140
2.1.4. Self-energy ~ and polarization function .7C 142
2.2. Random phase approximation 143
2.3. Mode-mode coupling 146
2.3.1. Lattice fluctuation 147
2.3.2. Transition temperature 149
2.3.3. Coherence length 150
2.3.4. Electron self-energy 151
2.4. Spin susceptibility and electrical resistivity 152
3. Effects oflattice fluctuation on CDW transition 153
3.1. Model 153
3.2. Calculated results for the 1D system 155
3.3. Calculated results for the 3D system 156
4. Effects of lattice fluctuations on the electronic density of states,
spin susceptibility, and electrical resistivity 161
4.1. The 1D system 161
4.2. The 3D system 164
TABLE OF CONTENTS Vll
5. Supplementary remarks 169
Appendix A. Feynman rules for ~ and:rr 170
Appendix B. Evaluation of frequency sum with the use of contour
integral in the complex plane 172
H. SHIBA and K. NAKANISHI / Phenomenological Landau Theory of
Charge Density Wave Phase Transitions in Layered Compounds 175
1. Introduction 175
2. Construction of Landau free energy 177
2.1. 2H-TaSe 177
2
2.2. IT-TaS and TaSe 181
2 2
3. A simple example: single-q CDW 185
3.1. Successive phase transitions and discommensurate state 185
3.2. Fluctuation modes 191
4. 2H-TaSe 194
2
4.1. Basic features of phase transitions 194
4.2. Single-layer properties 199
4.3. Commensurate phases with various symmetries 208
4.4. Discommensuration structures oftwo-Iayer stacking 212
4.5. Reentrant lock-in transition caused by pressure 218
4.6. Discommensuration diagram and dislocations 224
4.7. Fluctuation modes 237
4.8. 2H-NbSe 242
2
5. IT-TaS and TaSe 243
2 2
5.1. Brief summary of observed phase transitions 243
5.1.1. IT-TaS 243
2
5.1.2. IT-TaSe 244
2
5.2. Single-layer properties 246
5.2.1. Commensurate state 246
5.2.2. Incommensurate states and discommensuration
structures 247
5.3. Three-dimensional orderings of charge density waves 253
5.3.1. Commensurate states 253
5.3.2. Incommensurate and discommensurate states 258
5.4. New phase of IT-TaS 260
2
6. Concluding remarks 261
F. C. BROWN / Charge Density Waves in the Transition-metal Dichal-
cogenides: Recent Experimental Advances 267
1. Introduction 267
2. Charge density wave transformations observed in the Group Vb
compounds 268
2.1. 2H structures 268
2.2. 1T structures 271
V111 TABLE OF CONTENTS
3. The 2ao superlattice in the Group Nb compound 1T - TiSe 273
2
3.1. General features 273
3.2. Infrared reflectivity-free carrier and phonon effects 277
3.3. Phonon dispersion at low temperature 280
3.4. Angle-resolved photoemission and the electronic structure
of TiSe 281
2
3.5. The transformed band structure of TiSe 285
2
4. Recent developments 289
INDEX OF NAMES 293
INDEX OF SUBJECTS 295
PREFACE
The structural phase transition is one of the most fundamental problems in solid
state physics. Layered transition-metal dichalcogenides provide us with a most
exciting area for the study of structural phase transitions that are associated with
the charge density wave (CDW). A large variety of structural phase transitions,
such as commensurate and incommensurate transitions, and the physical proper
ties related to the formation of a CDW, have been an object of intense study made
for many years by methods employing modem microscopic techniques. Rather
recently, efforts have been devoted to the theoretical understanding of these
experimental results.
Thus, McMillan, for example, has developed an elegant phenomenological
theory on the basis of the Landau free energy expansion. An extension of
McMillan's theory has provided a successful understanding of the successive phase
transitions observed in the IT- and 2H-compounds. In addition, a microscopic
theory of lattice instability, lattice dynamics, and lattice distortion in the CDW
state of the transition-metal dichalcogenides has been developed based on their
electronic structures. As a result, the driving force of the CDW formation in the
IT- and 2H-compounds has become clear. Furthermore, the effect of lattice
fluctuations on the CDW transition and on the anomalous behavior of various
physical properties has been made clear microscopically.
This volume reviews these theoretical investigations and discusses some
interesting topics arising from recent experimental advances. The first two articles,
contributed by Motizuki and Suzuki, are devoted to the microscopic theory. The
first article discusses the general theory of electron-lattice interaction and lattice
dynamics on the basis of the tight-binding approximation. In addition, a study of
lattice instability, lattice dynamics, and the CDW state is presented for the IT- and
2H-type transition-metal dichalcogenides and for transition-metal trichlorides. The
second article is devoted to a discussion of the microscopic theory of lattice
fluctuations. The effect of mode-mode coupling due to electronically induced
lattice anharmonicity on the CDW transition and on the temperature dependences
of various physical quantities are studied in detail. The third article, contributed by
Shiba and Nakanishi, deals with the phenomenological Landau theory. The general
expressions for the Landau free energy for different CDW states in layered
transition-metal dichalcogenides are derived, and theoretical investigations of
successive phase transitions in 2H-TaSe IT- TaSe and IT- TaS are reviewed.
2, 2, 2
The final article, contributed by Brown, reviews recent experimental advances. In
particular, IT- TiSe is discussed in detail with special emphasis on the agreement
2
between microscopic theory and experiment.
It might be added that the general theory that constitutes the first three articles
in this volume should be widely applicable to other materials enabling an under
standing of their structural phase transitions.
K. MOTIZUKI Osaka, October 1985
ix
K. Motizuki (ed.), Structural Phase Transitions in Layered Transition-metal Compounds, ix.
© 1986, by D. Reidel Publishing Company.
MICROSCOPIC THEORY OF STRUCTURAL
PHASE TRANSITIONS IN LAYERED
TRANSITIONAL-METAL COMPOUNDS
K. MOTIZUKI AND N. SUZUKI
Dept. ofM aterial Physics,
Faculty ofE ngineering Science,
Osaka University,
Japan
1. Introduction
Layered transition-metal compounds such as transition-metal dichalcogenides
provide a most fascinating area of the study of the structural phase transition
associated with a charge density wave (CDW). Transition-metal dichalcogenides
are compounds of a transition metal in groups IV (Ti, Zr, Hf), V (V, Nb, Ta), and
VI (Cr, Mo, W) of the Periodic Table and one of the chalcogens S, Se, and Te. The
basic structure of these compounds consists of hexagonal layers of transition-metal
ions, with each layer sandwiched between two hexagonal layers of the chalcogen.
Each metal ion has six nearest-neighbor chalcogen ions arranged either in the form
of a nearly regular octahedron (T-type) or in the form of a trigonal prism (H-type).
Different ways of stacking sandwiches lead to the formation of various polytypes
such as IT, 2H, etc.
The transition-metal dichalcogenides form a structurally and chemically well
defined family. Electrically, however, they cover a wide range of properties from
insulators like HfS2, through semiconductors like MoS2 and semimetals like TiSe2,
to true metals like NbS2• All of the Nb and Ta compounds are superconducting at
very low temperatures.
The structural phase transitions of the transition-metal dichalcogenides have
been intensively studied by various kinds of experiments, such as neutron
scattering, X-ray and electron diffraction, and photoemission. The Group V metal
compounds which are metallic reveal successive phase transitions from the normal
phase which exists at high temperatures to distorted incommensurate and then
commensurate phases as the temperature is decreased. For the Group IV metal
compounds, with the exception of TiSe2, however, no structural transformations
have been reported. The semimetallic TiSe forms a supedattice below 202 K. In
2
the Mo and W compounds of Group VI, the normal structure is unstable even
when the temperature is higher than that of room temperature. These facts indicate
that the origin of the structural phase transitions of the transition-metal dichal
cogenides has to be found in their electronic structures.
The electronic band structures of these compounds have been extensively
studied, both theoretically and experimentally. A large number of different
1
K. Motizuki (ed.), Structural Phase Transitions in Layered Transition-metal Compounds, 1-133.
© 1986, by D. Reidel Publishing Company.
