Condensed matter physics I

471 Pages·2002·1.445 MB·English

by  Riseborough P.S.

Condensed Matter Physics I
Peter S. Riseborough
October 21, 2002
Contents
1 Introduction 9
1.1 The Born-Oppenheimer Approximation . . . . . . . . . . . . . . 9
2 Crystallography 13
3 Structures 13
3.1 Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Crystalline Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 The Direct Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3.1 Primitive Unit Cells . . . . . . . . . . . . . . . . . . . . . 19
3.3.2 The Wigner-Seitz Unit Cell . . . . . . . . . . . . . . . . . 19
3.4 Symmetry of Crystals . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4.1 Symmetry Groups . . . . . . . . . . . . . . . . . . . . . . 21
3.4.2 Group Multiplication Tables . . . . . . . . . . . . . . . . 22
3.4.3 Point Group Operations . . . . . . . . . . . . . . . . . . . 23
3.4.4 Limitations Imposed by Translational Symmetry . . . . . 24
3.4.5 Point Group Nomenclature . . . . . . . . . . . . . . . . . 24
3.5 Bravais Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.5.1 Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.5.2 Cubic Bravais Lattices. . . . . . . . . . . . . . . . . . . . 28
3.5.3 Tetragonal Bravais Lattices. . . . . . . . . . . . . . . . . . 30
3.5.4 Orthorhombic Bravais Lattices. . . . . . . . . . . . . . . . 31
3.5.5 Monoclinic Bravais Lattice. . . . . . . . . . . . . . . . . . 32
3.5.6 Triclinic Bravais Lattice. . . . . . . . . . . . . . . . . . . . 32
3.5.7 Trigonal Bravais Lattice.. . . . . . . . . . . . . . . . . . . 33
3.5.8 Hexagonal Bravais Lattice. . . . . . . . . . . . . . . . . . 33
3.5.9 Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.6 Point Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.7 Space Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.8 Crystal Structures with Bases. . . . . . . . . . . . . . . . . . . . 39
3.8.1 Diamond Structure . . . . . . . . . . . . . . . . . . . . . . 39
3.8.2 Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.8.3 Graphite Structure . . . . . . . . . . . . . . . . . . . . . . 40
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3.8.4 Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.8.5 Hexagonal Close-Packed Structure . . . . . . . . . . . . . 41
3.8.6 Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.8.7 Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.8.8 Other Close Packed Structures . . . . . . . . . . . . . . . 43
3.8.9 Sodium Chloride Structure . . . . . . . . . . . . . . . . . 45
3.8.10 Cesium Chloride Structure . . . . . . . . . . . . . . . . . 45
3.8.11 Fluorite Structure . . . . . . . . . . . . . . . . . . . . . . 47
3.8.12 The Copper Three Gold Structure . . . . . . . . . . . . . 47
3.8.13 Rutile Structure . . . . . . . . . . . . . . . . . . . . . . . 48
3.8.14 Zinc Blende Structure . . . . . . . . . . . . . . . . . . . . 48
3.8.15 Zincite Structure . . . . . . . . . . . . . . . . . . . . . . . 49
3.8.16 The Perovskite Structure . . . . . . . . . . . . . . . . . . 50
3.8.17 Exercise 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.9 Lattice Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.9.1 Exercise 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.9.2 Exercise 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.10 Quasi-Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 Structure Determination 56
4.1 X Ray Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.1.1 The Bragg conditions . . . . . . . . . . . . . . . . . . . . 56
4.1.2 The Laue conditions . . . . . . . . . . . . . . . . . . . . . 57
4.1.3 Equivalence of the Bragg and Laue conditions . . . . . . . 59
4.1.4 The Ewald Construction . . . . . . . . . . . . . . . . . . . 60
4.1.5 X-ray Techniques . . . . . . . . . . . . . . . . . . . . . . . 61
4.1.6 Exercise 10 . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.1.7 The Structure and Form Factors . . . . . . . . . . . . . . 62
4.1.8 Exercise 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1.9 Exercise 12 . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1.10 Exercise 13 . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.1.11 Exercise 14 . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.1.12 Exercise 15 . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.1.13 Exercise 16 . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.1.14 Exercise 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Neutron Diffraction. . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3 Theory of the Differential Scattering Cross-section . . . . . . . . 75
4.3.1 Time Dependent Perturbation Theory . . . . . . . . . . . 76
4.3.2 The Fermi-Golden Rule . . . . . . . . . . . . . . . . . . . 78
4.3.3 The Elastic Scattering Cross-Section . . . . . . . . . . . . 79
4.3.4 The Condition for Coherent Scattering . . . . . . . . . . . 82
4.3.5 Exercise 18 . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3.6 Exercise 19 . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3.7 Exercise 20 . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4 Elastic Scattering from Quasi-Crystals . . . . . . . . . . . . . . . 85
4.5 Elastic Scattering from a Fluid . . . . . . . . . . . . . . . . . . . 87
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5 The Reciprocal Lattice 89
5.0.1 Exercise 21 . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.1 The Reciprocal Lattice as a Dual Lattice. . . . . . . . . . . . . . 90
5.1.1 Exercise 22 . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Examples of Reciprocal Lattices. . . . . . . . . . . . . . . . . . . 93
5.2.1 The Simple Cubic Reciprocal Lattice . . . . . . . . . . . . 93
5.2.2 The Body Centered Cubic Reciprocal Lattice . . . . . . . 94
5.2.3 The Face Centered Cubic Reciprocal Lattice . . . . . . . 94
5.2.4 The Hexagonal Reciprocal Lattice . . . . . . . . . . . . . 95
5.2.5 Exercise 23 . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3 The Brillouin Zones . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3.1 The Simple Cubic Brillouin Zone . . . . . . . . . . . . . . 96
5.3.2 The Body Centered Cubic Brillouin Zone . . . . . . . . . 97
5.3.3 The Face Centered Cubic Brillouin Zone . . . . . . . . . . 97
5.3.4 The Hexagonal Brillouin Zone. . . . . . . . . . . . . . . . 97
6 Electrons 99
7 Electronic States 99
7.1 Many Electron Wave Functions . . . . . . . . . . . . . . . . . . . 100
7.1.1 Exercise 24 . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.2 Bloch’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.4 Plane Wave Expansion of Bloch Functions . . . . . . . . . . . . . 112
7.5 The Bloch Wave Vector . . . . . . . . . . . . . . . . . . . . . . . 114
7.6 The Density of States . . . . . . . . . . . . . . . . . . . . . . . . 116
7.6.1 Exercise 25 . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.7 The Fermi-Surface . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8 Approximate Models 121
8.1 The Nearly Free Electron Model . . . . . . . . . . . . . . . . . . 121
8.1.1 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . 121
8.1.2 Non-Degenerate Perturbation Theory . . . . . . . . . . . 122
8.1.3 Degenerate Perturbation Theory . . . . . . . . . . . . . . 125
8.1.4 Empty Lattice Approximation Band Structure . . . . . . 128
8.1.5 Exercise 26 . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.1.6 Degeneracies of the Bloch States . . . . . . . . . . . . . . 133
8.1.7 Exercise 27 . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.1.8 Brillouin Zone Boundaries . . . . . . . . . . . . . . . . . . 142
8.1.9 The Geometric Structure Factor . . . . . . . . . . . . . . 144
8.1.10 Exercise 28 . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8.1.11 Exercise 29 . . . . . . . . . . . . . . . . . . . . . . . . . . 148
8.1.12 Exercise 30 . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.1.13 Exercise 31 . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.1.14 Exercise 32 . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.2 The Pseudo-Potential Method . . . . . . . . . . . . . . . . . . . . 151
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8.2.1 The Scattering Approach . . . . . . . . . . . . . . . . . . 155
8.2.2 The Ziman-Lloyd Pseudo-potential . . . . . . . . . . . . . 156
8.2.3 Exercise 33 . . . . . . . . . . . . . . . . . . . . . . . . . . 157
8.3 The Tight-Binding Model . . . . . . . . . . . . . . . . . . . . . . 158
8.3.1 Tight-Binding s Band Metal. . . . . . . . . . . . . . . . . 163
8.3.2 Exercise 34 . . . . . . . . . . . . . . . . . . . . . . . . . . 166
8.3.3 Exercise 35 . . . . . . . . . . . . . . . . . . . . . . . . . . 166
8.3.4 Exercise 36 . . . . . . . . . . . . . . . . . . . . . . . . . . 167
8.3.5 Exercise 37 . . . . . . . . . . . . . . . . . . . . . . . . . . 168
8.3.6 Exercise 38 . . . . . . . . . . . . . . . . . . . . . . . . . . 168
8.3.7 Exercise 39 . . . . . . . . . . . . . . . . . . . . . . . . . . 169
8.3.8 Wannier Functions . . . . . . . . . . . . . . . . . . . . . . 169
8.3.9 Exercise 40 . . . . . . . . . . . . . . . . . . . . . . . . . . 171
9 Electron-Electron Interactions 172
9.1 The Landau Fermi Liquid . . . . . . . . . . . . . . . . . . . . . . 172
9.1.1 The Scattering Rate . . . . . . . . . . . . . . . . . . . . . 173
9.1.2 The Quasi-Particle Energy . . . . . . . . . . . . . . . . . 173
9.1.3 Exercise 41 . . . . . . . . . . . . . . . . . . . . . . . . . . 176
9.2 The Hartree-Fock Approximation . . . . . . . . . . . . . . . . . . 176
9.2.1 The Free Electron Gas. . . . . . . . . . . . . . . . . . . . 180
9.2.2 Exercise 42 . . . . . . . . . . . . . . . . . . . . . . . . . . 188
9.3 The Density Functional Method. . . . . . . . . . . . . . . . . . . 188
9.3.1 Hohenberg-Kohn Theorem. . . . . . . . . . . . . . . . . . 189
9.3.2 The Variational Principle . . . . . . . . . . . . . . . . . . 190
9.3.3 The Electrostatic Terms . . . . . . . . . . . . . . . . . . . 192
9.3.4 The Kohn-Sham Equations . . . . . . . . . . . . . . . . . 193
9.3.5 The Local Density Approximation . . . . . . . . . . . . . 195
9.4 Static Screening. . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
9.4.1 The Thomas-Fermi Approximation . . . . . . . . . . . . . 199
9.4.2 Linear Response Theory . . . . . . . . . . . . . . . . . . . 201
9.4.3 Density Functional Response Function . . . . . . . . . . . 204
9.4.4 Exercise 43 . . . . . . . . . . . . . . . . . . . . . . . . . . 205
9.4.5 Exercise 44 . . . . . . . . . . . . . . . . . . . . . . . . . . 206
10 Stability of Structures 207
10.1 Momentum Space Representation . . . . . . . . . . . . . . . . . . 207
10.2 Real Space Representation . . . . . . . . . . . . . . . . . . . . . . 211
11 Metals 215
11.1 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
11.1.1 The Sommerfeld Expansion . . . . . . . . . . . . . . . . . 216
11.1.2 The Specific Heat Capacity . . . . . . . . . . . . . . . . . 218
11.1.3 Exercise 45 . . . . . . . . . . . . . . . . . . . . . . . . . . 221
11.1.4 Exercise 46 . . . . . . . . . . . . . . . . . . . . . . . . . . 221
11.1.5 Pauli Paramagnetism . . . . . . . . . . . . . . . . . . . . 221
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11.1.6 Exercise 47 . . . . . . . . . . . . . . . . . . . . . . . . . . 224
11.1.7 Exercise 48 . . . . . . . . . . . . . . . . . . . . . . . . . . 224
11.1.8 Landau Diamagnetism . . . . . . . . . . . . . . . . . . . . 224
11.1.9 Quantum Considerations . . . . . . . . . . . . . . . . . . 226
11.1.10The Diamagnetic Susceptibility . . . . . . . . . . . . . . . 227
11.2 Transport Properties . . . . . . . . . . . . . . . . . . . . . . . . . 229
11.2.1 Electrical Conductivity . . . . . . . . . . . . . . . . . . . 229
11.2.2 Scattering by Static Defects . . . . . . . . . . . . . . . . . 229
11.2.3 Exercise 49 . . . . . . . . . . . . . . . . . . . . . . . . . . 236
11.2.4 The Hall Effect and Magneto-resistance. . . . . . . . . . . 237
11.2.5 Multi-band Models . . . . . . . . . . . . . . . . . . . . . . 245
11.2.6 The Quantum Hall Effect . . . . . . . . . . . . . . . . . . 247
11.2.7 Exercise 50 . . . . . . . . . . . . . . . . . . . . . . . . . . 254
11.2.8 The Fractional Quantum Hall Effect . . . . . . . . . . . . 254
11.2.9 Quasi-Particle Excitations . . . . . . . . . . . . . . . . . . 256
11.2.10Skyrmions . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
11.2.11Composite Fermions . . . . . . . . . . . . . . . . . . . . . 267
11.3 Electromagnetic Properties of Metals . . . . . . . . . . . . . . . . 269
11.3.1 The Longitudinal Response . . . . . . . . . . . . . . . . . 271
11.3.2 Electron Scattering Experiments . . . . . . . . . . . . . . 277
11.3.3 Transverse Response . . . . . . . . . . . . . . . . . . . . . 280
11.3.4 Optical Experiments . . . . . . . . . . . . . . . . . . . . . 283
11.3.5 Kramers-Kronig Relation . . . . . . . . . . . . . . . . . . 284
11.3.6 Inter-Band Transitions . . . . . . . . . . . . . . . . . . . . 286
11.3.7 Exercise 51 . . . . . . . . . . . . . . . . . . . . . . . . . . 287
11.3.8 Exercise 52 . . . . . . . . . . . . . . . . . . . . . . . . . . 287
11.4 Measuring the Fermi-Surface . . . . . . . . . . . . . . . . . . . . 288
11.4.1 Semi-Classical Orbits . . . . . . . . . . . . . . . . . . . . 288
11.4.2 de Haas van Alphen Oscillations . . . . . . . . . . . . . . 292
11.4.3 Exercise 53 . . . . . . . . . . . . . . . . . . . . . . . . . . 294
11.4.4 The Lifshitz-Kosevich Formulae . . . . . . . . . . . . . . . 295
11.4.5 Other Fermi-Surface Probes . . . . . . . . . . . . . . . . . 300
11.4.6 Cyclotron Resonances . . . . . . . . . . . . . . . . . . . . 301
12 Insulators 306
12.1 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
12.1.1 Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
12.1.2 Intrinsic Semiconductors . . . . . . . . . . . . . . . . . . . 312
12.1.3 Extrinsic Semiconductors . . . . . . . . . . . . . . . . . . 314
12.1.4 Exercise 54 . . . . . . . . . . . . . . . . . . . . . . . . . . 317
12.2 Transport Properties . . . . . . . . . . . . . . . . . . . . . . . . . 318
12.3 Optical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 318
13 Phonons 319
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14 Harmonic Phonons 319
14.1 Lattice with a Basis . . . . . . . . . . . . . . . . . . . . . . . . . 325
14.2 A Sum Rule for the Dispersion Relations. . . . . . . . . . . . . . 325
14.2.1 Exercise 55 . . . . . . . . . . . . . . . . . . . . . . . . . . 327
14.3 The Nature of the Phonon Modes . . . . . . . . . . . . . . . . . . 327
14.3.1 Exercise 56 . . . . . . . . . . . . . . . . . . . . . . . . . . 329
14.3.2 Exercise 57 . . . . . . . . . . . . . . . . . . . . . . . . . . 329
14.3.3 Exercise 58 . . . . . . . . . . . . . . . . . . . . . . . . . . 329
14.3.4 Exercise 59 . . . . . . . . . . . . . . . . . . . . . . . . . . 330
14.3.5 Exercise 60 . . . . . . . . . . . . . . . . . . . . . . . . . . 330
14.3.6 Exercise 61 . . . . . . . . . . . . . . . . . . . . . . . . . . 331
14.3.7 Exercise 62 . . . . . . . . . . . . . . . . . . . . . . . . . . 332
14.4 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
14.4.1 The Specific Heat . . . . . . . . . . . . . . . . . . . . . . 335
14.4.2 The Einstein Model of a Solid . . . . . . . . . . . . . . . . 336
14.4.3 The Debye Model of a Solid . . . . . . . . . . . . . . . . . 336
14.4.4 Exercise 63 . . . . . . . . . . . . . . . . . . . . . . . . . . 338
14.4.5 Exercise 64 . . . . . . . . . . . . . . . . . . . . . . . . . . 338
14.4.6 Exercise 65 . . . . . . . . . . . . . . . . . . . . . . . . . . 339
14.4.7 Exercise 66 . . . . . . . . . . . . . . . . . . . . . . . . . . 339
14.4.8 Lindemann Theory of Melting. . . . . . . . . . . . . . . . 339
14.4.9 Thermal Expansion . . . . . . . . . . . . . . . . . . . . . 342
14.4.10Thermal Expansion of Metals . . . . . . . . . . . . . . . . 343
14.5 Anharmonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
14.5.1 Exercise 67 . . . . . . . . . . . . . . . . . . . . . . . . . . 344
15 Phonon Measurements 346
15.1 Inelastic Neutron Scattering . . . . . . . . . . . . . . . . . . . . . 346
15.1.1 The Scattering Cross-Section . . . . . . . . . . . . . . . . 347
15.2 The Debye-Waller Factor . . . . . . . . . . . . . . . . . . . . . . 352
15.3 Single Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . 354
15.4 Multi-Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . 355
15.4.1 Exercise 68 . . . . . . . . . . . . . . . . . . . . . . . . . . 356
15.4.2 Exercise 69 . . . . . . . . . . . . . . . . . . . . . . . . . . 356
15.4.3 Exercise 70 . . . . . . . . . . . . . . . . . . . . . . . . . . 356
15.5 Raman and Brillouin Scattering of Light . . . . . . . . . . . . . . 356
16 Phonons in Metals 359
16.1 Screened Ionic Plasmons . . . . . . . . . . . . . . . . . . . . . . . 360
16.1.1 Kohn Anomalies . . . . . . . . . . . . . . . . . . . . . . . 361
16.2 Dielectric Constant of a Metal. . . . . . . . . . . . . . . . . . . . 361
16.3 The Retarded Electron-Electron Interaction . . . . . . . . . . . . 364
16.4 Phonon Renormalization of Quasi-Particles . . . . . . . . . . . . 365
16.5 Electron-Phonon Interactions . . . . . . . . . . . . . . . . . . . . 367
16.6 Electrical Resistivity due to Phonon Scattering . . . . . . . . . . 368
16.6.1 Umklapp Scattering . . . . . . . . . . . . . . . . . . . . . 373
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16.6.2 Phonon Drag . . . . . . . . . . . . . . . . . . . . . . . . . 374
17 Phonons in Semiconductors 375
17.1 Resistivity due to Phonon Scattering . . . . . . . . . . . . . . . . 375
17.2 Polarons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
17.3 Indirect Transitions. . . . . . . . . . . . . . . . . . . . . . . . . . 376
18 Impurities and Disorder 378
18.1 Scattering By Impurities . . . . . . . . . . . . . . . . . . . . . . . 381
18.2 Virtual Bound States . . . . . . . . . . . . . . . . . . . . . . . . . 383
18.3 Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
18.4 Coherent Potential Approximation . . . . . . . . . . . . . . . . . 387
18.5 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
18.5.1 Anderson Model of Localization. . . . . . . . . . . . . . . 389
18.5.2 Scaling Theories of Localization. . . . . . . . . . . . . . . 390
19 Magnetic Impurities 393
19.1 Localized Magnetic Impurities in Metals . . . . . . . . . . . . . . 393
19.2 Mean Field Approximation . . . . . . . . . . . . . . . . . . . . . 393
19.2.1 The Atomic Limit . . . . . . . . . . . . . . . . . . . . . . 396
19.3 The Schrieffer-Wolf Transformation . . . . . . . . . . . . . . . . . 396
19.3.1 The Kondo Hamiltonian . . . . . . . . . . . . . . . . . . . 399
19.4 The Resistance Minimum . . . . . . . . . . . . . . . . . . . . . . 400
20 Collective Phenomenon 405
21 Itinerant Magnetism 405
21.1 Stoner Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
21.1.1 Exercise 71 . . . . . . . . . . . . . . . . . . . . . . . . . . 407
21.1.2 Exercise 72 . . . . . . . . . . . . . . . . . . . . . . . . . . 407
21.2 Linear Response Theory . . . . . . . . . . . . . . . . . . . . . . . 407
21.3 Magnetic Instabilities . . . . . . . . . . . . . . . . . . . . . . . . 409
21.4 Spin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
21.5 The Heisenberg Model . . . . . . . . . . . . . . . . . . . . . . . . 414
22 Localized Magnetism 415
22.1 Holstein - Primakoff Transformation . . . . . . . . . . . . . . . . 416
22.2 Spin Rotational Invariance. . . . . . . . . . . . . . . . . . . . . . 419
22.2.1 Exercise 73 . . . . . . . . . . . . . . . . . . . . . . . . . . 422
22.3 Anti-ferromagnetic Spinwaves . . . . . . . . . . . . . . . . . . . . 423
22.3.1 Exercise 74 . . . . . . . . . . . . . . . . . . . . . . . . . . 425
23 Spin Glasses 425
23.1 Mean Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 427
23.2 The Sherrington-Kirkpatrick Solution. . . . . . . . . . . . . . . . 428
7
24 Magnetic Neutron Scattering 432
24.1 The Inelastic Scattering Cross-Section . . . . . . . . . . . . . . . 432
24.1.1 The Dipole-Dipole Interaction. . . . . . . . . . . . . . . . 432
24.1.2 The Inelastic Scattering Cross-Section . . . . . . . . . . . 432
24.2 Time Dependent Spin Correlation Functions . . . . . . . . . . . . 436
24.3 The Fluctuation Dissipation Theorem . . . . . . . . . . . . . . . 438
24.4 Magnetic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 440
24.4.1 Neutron Diffraction . . . . . . . . . . . . . . . . . . . . . 440
24.4.2 Exercise 75 . . . . . . . . . . . . . . . . . . . . . . . . . . 441
24.4.3 Exercise 76 . . . . . . . . . . . . . . . . . . . . . . . . . . 442
24.4.4 Spin Wave Scattering . . . . . . . . . . . . . . . . . . . . 442
24.4.5 Exercise 77 . . . . . . . . . . . . . . . . . . . . . . . . . . 443
24.4.6 Critical Scattering . . . . . . . . . . . . . . . . . . . . . . 443
25 Superconductivity 445
25.1 Experimental Manifestation . . . . . . . . . . . . . . . . . . . . . 445
25.1.1 The London Equations. . . . . . . . . . . . . . . . . . . . 446
25.1.2 Thermodynamics of the Superconducting State . . . . . . 448
25.2 The Cooper Problem . . . . . . . . . . . . . . . . . . . . . . . . . 450
25.3 Pairing Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
25.3.1 The Pairing Interaction . . . . . . . . . . . . . . . . . . . 454
25.3.2 The B.C.S. Variational State . . . . . . . . . . . . . . . . 456
25.3.3 The Gap Equation . . . . . . . . . . . . . . . . . . . . . . 458
25.3.4 The Ground State Energy . . . . . . . . . . . . . . . . . . 459
25.4 Quasi-Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
25.4.1 Exercise 78 . . . . . . . . . . . . . . . . . . . . . . . . . . 464
25.5 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
25.6 Perfect Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 466
25.7 The Meissner Effect . . . . . . . . . . . . . . . . . . . . . . . . . 468
25.8 Domain Wall Energy . . . . . . . . . . . . . . . . . . . . . . . . . 469
8
1 Introduction
CondensedMatterPhysicsisthestudyofmaterialsinSolidandLiquidPhases.
Itencompassesthestudyoforderedcrystallinephasesofsolids,aswellasdisor-
dered phases such as the amorphous and glassy phases of solids. Furthermore,
it also includes materials with short-ranged order such as conventional liquids,
andliquidcrystalswhichshowunconventionalorderintermediatebetweenthose
of a crystalline solid and a liquid. Condensed matter has the quite remarkable
property that, due to the large number of particles involved, the behavior of
the materials may be qualitatively distinct from those of the individual con-
stituents. The behavior of the incredibly large number of particles is governed
by (quantum) statistics which, through the chaotically complicated motion of
theparticles,producesnewtypesoforder. Theseemergentphenomenaarebest
exemplified in phenomenon such as magnetism or superconductivity where the
collective behavior results in transitions to new phases.
In surveying the properties of materials it is convenient to separate the
properties according to two (usually) disparate time scales. One time scale
is a slow time scale which governs the structural dynamics, and a faster time
scale that governs the electronic motion. The large difference between the time
scales is due to the large ratio of the nuclear masses to the electronic mass,
M /m ∼ 103. The long-ranged electromagnetic force binds these two con-
n e
stituents of different mass into electrically neutral material. The slow moving
nuclear masses can be considered to be quasi-static, and are responsible for
defining the structure of matter. In this approximation, the fast moving elec-
trons equilibrate in the quasi-static potential produced by the nuclei.
1.1 The Born-Oppenheimer Approximation
Thedifferenceintherelevanttimescalesforelectronicandnuclearmotionallows
onetomaketheBorn-OppenheimerApproximation. Inthisapproximation,the
electronicstatesaretreatedasifthenucleiwereatrestatfixedpositions. How-
ever, when treating the slow motions of the nuclei, the electrons are considered
asadaptinginstantaneouslytothepotentialofthechargednuclei,therebymin-
imizing the electronic energies. Thus, the nuclei charges are dressed by a cloud
of electrons forming ionic or atomic-like aggregates.
A qualitative estimate of the relative energies of nuclear versus electronic
motion can be obtained by considering metallic hydrogen. The electronic en-
ergies are calculated using only the Bohr model of the hydrogen atom. The
equation of motion for an electron of mass m has the form
e
Z e2 m v2
− = − e (1)
a2 a
whereZ isthenuclearchargeandaistheradiusoftheatomicorbital. Thestan-
9
dardsemi-classicalquantizationconditionduetoBohrandSommerfeldrestricts
the angular momentum to integral values of ¯h
m v a = n ¯h (2)
e
These equations can be combined to find the quantized total electronic energy
of the hydrogen atom
Z e2 m v2
E = − + e
e a 2
Z e2
= −
2 a
m Z2 e4
= − e (3)
2 n2 ¯h2
which is a standard result from atomic physics. Note that the kinetic energy
term and the electrostatic potential term have similar magnitudes.
Nowconsiderthemotionofthenuclei. TheforcesconsistofCoulombforces
betweenthenucleiandelectrons,andthequantummechanicalPauliforces. The
electrostaticrepulsionsandattractionshavesimilarmagnitudes,sincetheinter-
nuclear separations are of the same order as the Bohr radius. In equilibrium,
the sum of the forces vanish identically. Furthermore, if an atom is displaced
from the equilibrium position by a small distance equal to r, the restoring force
is approximately given by the dipole force
Z e2
− α r (4)
a3
where α is a dimensionless constant. Hence, the equation of motion for the
displacement of a nuclei of mass M is
n
Z e2 d2 r
− α r = M (5)
a3 n dt2
which shows that the nuclei undergo harmonic oscillations with frequency
Z e2
ω2 = α (6)
M a3
n
The semi-classical quantization condition
I
dr M v = n ¯h (7)
n
yields the energy for nuclear motion as
E = n ¯h ω
N
= n me Z2 e4 α12 (cid:18) me (cid:19)12 (8)
¯h2 Mn
10