Table Of ContentMAGNETIC PROPERTIES OF LA YERED TRANSITION METAL COMPOUNDS
Physics and Chemistry of Materials
with Low-Dimensional Structures
VOLUME 9
Editor-in-Chief
F. LEVY, Institut de Physique Appliquee, EPFL,
Departement de Physique, PHB-Ecublens, CH-IOI5 Lausanne, Switzerland
Honorary Editor
E. MOOSER, EPFL, Lausanne, Switzerland
International Advisory Board
J. V. ACRIVOS, San Jose State University, San Jose, Calif, U.S.A.
S. BARISIC, University ofZ agreb, Department ofP hysics, Zagreb, Yugoslavia
J. G. BEDNORZ, IBM Forschungslaboratorium, Riischlikon. Switzerland
C. F. van BRUGGEN, University ofGroningen. Groningen, The Netherlands
R. GIRLANDA, Universita di Messina, Messina, Italy
D. HAARER, University of Bayreuth, F.R.G.
A. J. HEBGER, University of California. Santa Barbara. Calif, U.S.A.
H. KAMIMURA, Dept. of Physics, University of Tokyo, Japan
W. Y. LIANG, Cavendish Laboratory, Cambridge, U.K.
P. MONCEAU, CNRS, Grenoble, France
J. ROUXEL, CNRS, Nantes, France
M. SCHLiiTER, AT & T. Murray Hill. N.J., U.S.A.
I. ZSCHOKKE, Universitiit Basel, Basel, Switzerland
The titles published in this series are listed at the end of this volume.
MAGNETIC PROPERTIES OF
LAYERED TRANSITION
METAL COMPOUNDS
Edited by
L. J. DE JONGH
Kamerlingh Onnes Laboratory,
Leiden University,
The Netherlands
KLUWER ACADEMIC PUBLISHERS
DORDRECHT / BOSTON / LONDON
Library of Congress Cataloging in I'ubliclilion Data
Mlgnel'c prep.rTles 01 livered TrlnSlllon .~t.1 cDlpounDs I edltld by
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-d,"ans,onll structure •. Serl •• A. LayereO SIrUClures)
Includes ,ndu .
ISBN·I:): 978·94-010_7329·$ ._ISBN_13: 978-94-009·1860-3
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TABLE OF CONTENTS
PREFACE Xl
L. J. DE JONGH / Introduction to Low-Dimensional Magnetic Systems I
I. Experimental realizations of 2-d magnetic systems 1
2. Magnetic model Hamiltonians 7
3. Survey of the predicted magnetic behaviour 10
4. Lattice- and spin-dimensionality crossovers in quasi 2-d magnetic
systems 19
5. Magnetic and nonmagnetic impurity doping in quasi 2-d magnets 33
References 47
V. L. POKROVSKY and G. V. UIMIN / Theory of Two-Dimensional
Magnets 53
1. Introduction 53
2. Ising magnets 53
2.1. Ising model. Excitations and phase transitions 53
2.2. Onsager solution 57
2.3. Critical exponents and scaling 60
2.4. Dual transformation. Order and disorder 60
3. Planar magnets 61
3.1. XY model 61
3.2. Excitations 62
3.3. Scaling and correlations 64
3.4. Phase transition 66
3.5. Magnetic vortices as a Coulomb gas 66
3.6. Relationships with other models 70
3.7. Planarantiferromagnets 71
4. Heisenberg magnets 75
4.1. Heisenberg model and real magnets 75
4.2. Renormailzation of the temperature 77
4.3. Heisenberg ferro magnets in an external magnetic field 79
4.4. Excitations of the 2-d Heisenberg model 80
4.5. Dipolar interactions 81
5. Experimental layered magnets 83
5.1. Ising layered magnets. ANNNI model: application to CeSb and
CeBi 83
5.2. Layered planar magnets 88
5.3. Layered Heisenberg magnets 89
6. Dynamics of 2-d magnets 90
v
vi TABLE OF CONTENTS
6.1. Equations of motion 90
6.2. Spin-wave dynamics 91
6.3. Spin-diffusion dynamics 94
6.4. Dynamics of localized excitations 95
6.5. Resonant paramagnetic excitation of vortex pairs 98
6.6. Summary 101
Acknowledgement 102
References 102
R. N A V A RRO / Application of High- and Low-Temperature Series
Expansions to Two-Dimensional Magnetic Systems 105
1. Introduction 105
1. 1. Series expansions 105
1.2. Methods applied in series analysis 109
1.2.1. Ratio methods 109
1.2.2. Pade approximant methods 111
1.2.3. Other methods of series analysis 113
2. Series expansions and predictions for the 2-d Ising model 114
2.1. Spin 1/2 model with nearest neighbours only (simple 2-d
lattices) 116
2.1.1. High-temperature series 116
2.1.2. Low-temperature series 119
2.1.3. Properties in nonzero parallel field 123
2.1.4. Properties in nonzero perpendicular field 127
2.2. Ising model with general S 129
2.3. Other series for 1(112) 131
2.3.1. Restricted dimensionality systems 131
2.3.2. Further-neighbour interactions 134
2.3.3. Crossover from 2-d to 3-d behaviour 136
3. Series expansions and predictions for the Heisenberg model 140
3.1. Series for S = 1/2, arbitrary Sand S = 00 141
3.1.1. Properties at nonzero field 145
3.2. Other series for the Heisenberg model 147
3.2.1. Restricted dimensionality 147
3.2.2. Further-neighbour interactions 149
3.2.3. Crossover from 2-d to 3-d behaviour 149
4. Series expansion in the XY and Ising-Heisenberg models 150
4.1. Series for the 2-d XY model 151
4.2. Series for the 2-d Ising-Heisenberg model 153
5. Applications to magnetic systems 155
5.1. Ising model 155
5.2. Heisenberg model 158
5.2.1. Spin 1/2 159
5.2.2. Spin 1 162
5.2.3. Spin 3/2 and spin 2 164
T ABLE OF CONTENTS VII
5.2.4. Spin 5/2 167
5.2.5. Restricted dimensionality 169
5.3. XY and Ising-Heisenberg models 170
Acknowledgements 173
References 184
A. F. M. ARTS and H. W. DE WIJN / Spin Waves in Two-Dimensional
Magnetic Systems: Theory and Applications 191
1. Introduction 191
2. Magnetic structures and spin Hamiltonians 1 93
3. Spin wave theory of model systems 197
4. Dispersion relation 204
5. Thermodynamic properties 213
6. Impurities in antiferromagnets 222
References 22 7
K. HIRAKAWA and H. IKEDA / Neutron Scattering Experiments on
Two-Dimensional Heisenberg and Ising Magnets 231
1. Introduction 231
2. 2-d systems with Ising and Heisenberg interactions 232
2.1. KzCoF 4: a 2-d Ising system 232
2.2. KzFeF4: a 2-d planar antiferromagnet 235
2.3. K2MnF4 and KzNiF4: weakly anisotropic Heisenberg magnets 237
2.4. RbzCrC14: a planar Heisenberg ferromagnet with small anisot-
ropy 238
2.5. K CuF4: a planar Heisenberg ferromagnet 238
2
3. 2-d random magnetic systems 243
3.1. Phase transitions and critical phenomena 243
3.2. Excitations 248
3.3. Random field effects 250
3.4. Relaxation from 2-d to 3-d order 254
3.5. Competing anisotropies and interactions 256
4. Triangular latticc antifcrromagnct (TALAF) 259
4.1. Fluctuations 259
4.2. An additional degree of freedom 260
4.3. Perturbation 261
4.4. Quantum effect
RbFeCI] and CsFeCl]
VX2 (X = Cl, Br, I)
AMX2 (A = Li, Na, K; M = 3d metal ion; X = 0, S, Se) 261
References 267
L. P. REGNAULT and J. ROSSAT-MIGNOD / Phase Transitions in Quasi
Two-Dimensional Planar Magnets 271
1. Introduction 271
2. Phase transition and excitations in the 2-d XY model 272
viii TABLE OF CONTENTS
3. Crystallographic properties of BaM2(X)4)2 compounds 277
4. Magnetic properties of BaNiz(P04)2 278
4.1. Static properties 278
4.2. Dynamic properties 282
4.3. Critical properties 285
5. Magnetic properties of BaCo2(As04)2 288
5.1. Static properties 288
5.2. Magnetic phase diagrams 294
5.3. Dynamic properties 296
6. Magnetic properties of BaNi (As0 )z 304
2 4
6.1. Static properties 304
6.2. Dynamic properties 306
7. Magnetic properties of BaCo2(P04)2 310
8. Other experimental realizations of the 2-d planar model 312
8.1. K CuF 312
2 4
8.2. NiCl and CoCL graphite intercalated compounds 313
2 2
NiCI -GIC
2
CoCI -GIC
2
9. Concluding remarks 315
Acknowledgement 318
References 318
H. BENNER and J. P. BOUCHER / Spin Dynamics in the Paramagnetic
Regime: NMR and EPR in Two-Dimensional Magnets 323
1. Introduction 323
1.1. Dynamics of the 2-spin correlation functions 324
1.2. Nuclear magnetic resonance (NMR) 325
1.3. Electron paramagnetic resonance (EPR) 326
2. General formalism 328
2.1. Diffusion and dimensionality 328
2.2. Cut-off and EPR linewidth 330
3. EPR spectrum 331
3.1. Diffusion of 4-spin correlation functions 334
3.2. Secular contribution @o 336
3.3. Nonsecular contributions 338
3.4. Satellite line 349
4. Experiments on quasi 2-d Heisenberg magnets 343
4.1. NMR experiments 344
4.2. EPR experiments 346
4.2.1. Angular dependence of linewidth 346
4.2.2. Frequency dependence of magic angle linewidth 348
4.2.3. Dynamic shift 349
4.2.4. Lineshape of the main line 349
4.2.5. Satellite lines at half resonance field 351
5. Critical dynamcis 353
T ABLE OF CONTENTS IX
5.1. Critical behaviour of the NMR line 355
5.1.1. Isotropic regime 355
5.1.2. Anisotropic regime 359
5.1.3. Experimenb 360
5.2. Critical behaviour of the EPR linewidth 361
5.2.1. Ferromagnets 362
5.2.2. Antiferromagnets 366
5.3. AC susceptihility 370
6. Conclusions 373
References 376
H. J. M. DE GROOT and L. J. DE JONGH / Field-Induced Phenomena in
Two-Dimensional Weakly Anisotropic Heisenherg Antiferromagnets 379
1. Introduction 379
2. Effective, field-dependent anisotropies 379
3. The phase diagram 383
4. Random fields and domain walls (solitons) 489
5. The spin flop transition 394
6. The bicritical point 400
7. Concluding remarks 403
Acknowledgements 403
References 403
INDEX OF NAMES 405
INDEX OF CHEMICAL COMPOUNDS 415
INDEX OF SUBJECTS 417
PREFACE
In the last two decades low-dimensional (low-d) physics has matured into a major
branch of science. Quite generally we may define a system with restricted
dimensionality d as an object that is infinite only in one or two spatial directions
(d = 1 and 2). Such a definition comprises isolated single chains or layers, but also
fibres and thin layers (films) of varying but finite thickness. Clearly, a multitude of
physical phenomena, notably in solid state physics, fall into these categories. As
examples, we may mention:
• Magnetic chains or layers (thin-film technology).
• Metallic films (homogeneous or heterogeneous, crystalline, amorphous or
microcristalline, etc.).
• I-d or 2-d conductors and superconductors.
• Intercalated systems.
• 2-d electron gases (electrons on helium, semiconductor interfaces).
• Surface layer problems (2-d melting of monolayers of noble gases on a
substrate, surface problems in general).
• Superfluid films of ~He or 'He.
• Polymer physics.
• Organic and inorganic chain conductors, superionic conductors.
• I-d or 2-d molecular crystals and liquid crystals.
• I-d or 2-d ferro-and antiferro electrics.
The study of these systems has substantially advanced our fundamental knowledge
of physics, apart from the technological applications (materials science) to which it
may have led. No matter how widely different the above physical problems may
appear at first sight, there are many parellellisms and characteristics that they all
share, which is inherent in fact to the common intrinsic property of restricted
dimensionality. It is for this very reason that we may justifiably consider low-d
physics as an entity, as a recognizable field of science.
In this larger field then, the sub-branch of low-d magnetism has played a quite
important role right from the outset. Theoretical work on magnetic chains and
layers started as early as the twenties with the Ising chain, and was followed by
studies of 2-d Ising magnets in the thirties and forties. About twenty-five years ago
systematic experimental studies on quasi low-d magnetic systems were started in a
few research groups. The maximum effort was attained in the seventies, the level
having remained about constant since then. In most cases the systems studied are
magnetic compounds consisting of weakly coupled chains or layers. Often the
predicted behaviour for the isolated chain or layer has been found to be
approximated to a surprisingly high degree. Ultrathin magnetic films of a few
monolayers have also been studied. In general these systems should be dis tin-
xi
L. J. De Jongh (Ed.), Magnetic Properties of Layered Transition Metal Compounds xi-xiii.
© 1990 Kluwer Academic Publishers.
