Models of Itinerant Ordering in Crystals (eBook)

Front Cover 1
Models of ITINERANT ORDERING IN CRYSTALS 4
Copyright Page 5
Table of Contents 8
Preface 12
Part One. Introduction to Theory of Solids 14
Chapter 1. Periodic Structures 16
1.1 Fundamental Types of Lattices 16
1.2 Diffraction of Waves by a Crystal and the Reciprocal Lattice 17
1.2.1 Reciprocal lattice vectors 20
1.3 Brillouin Zones 21
References 23
Chapter 2. Various Statistics 24
Appendix 2A: Fermi–Dirac And Bose–Einstein Distribution Functions 27
References 30
Chapter 3. Paramagnetism and Weiss Ferromagnetism 32
3.1 Paramagnetism 32
3.2 Weiss Ferromagnetism 38
References 39
Chapter 4. Electron States 40
4.1 The Nearly Free Electron Model 40
4.1.1 General result for ?(?) 44
4.1.2 Use of DOS for evaluating lattice sums in momentum space 45
4.1.3 Heat capacity of the free electron gas: an introduction 47
4.2 The Tight-Binding Method 48
4.2.1 Cohesion energy 53
4.3 Bloch Theorem 56
Appendix 4A: Nearly Free Electrons, Two-Plane Waves Model 57
References 61
Part Two. Models of Itinerant Ordering in Crystals 62
Chapter 5. The Hubbard Model 64
5.1 Simple Hubbard Model 64
5.2 Extended Hubbard Model 67
References 71
Chapter 6. Different Approximations for Hubbard Model 72
6.1 Chain Equation for Green Functions 73
6.2 Hartree–Fock Approximation 76
6.3 Hubbard I Approximation 77
6.3.1 Atomic limit 77
6.3.2 Finite bandwidth limit 79
6.4 Extended Hubbard III Approximation 82
6.5 Coherent Potential Approximation 85
6.5.1 Relation between CPA, Hubbard III and extended Hubbard III approximations 93
6.5.2 Different applications of the CPA 94
6.6 Spectral Density Approach 94
6.7 Modified Alloy Analogy 99
6.8 Dynamical Mean-Field Theory 101
6.9 Hubbard Model Extended by Inter-site Interactions 103
6.9.1 Modified Hartree–Fock approximation 103
6.9.2 Coherent potential approximation for the extended Hubbard model 107
Appendix 6A: Equation of Motion for the Green Functions 108
Appendix 6B: Hubbard Solution for the Scattering and Resonance Broadening Effects 110
6B.1 The scattering effect 110
6B.2 The resonance broadening effect 113
Appendix 6C: Modified Hartree–Fock Approximation for the Inter-site Interactions 119
References 126
Chapter 7. Itinerant Ferromagnetism 128
7.1 Periodic Table – Ferromagnetic Elements 128
7.1.1 Ferromagnetic elements 131
7.2 Introduction to Stoner Model 138
7.2.1 Static magnetic susceptibility 142
7.3 Stoner Model for Ferromagnetism 144
7.4 Stoner Model for Rectangular and Parabolic Band 147
7.4.1 Rectangular band 147
7.4.2 Parabolic nearly free electron band 149
7.5 Modified Stoner Model 151
7.5.1 Modified Stoner Model for a semi-elliptic band 154
7.6 Beyond Hartree–Fock Model 157
7.6.1 General formalism 157
7.6.2 Enhancement of magnetic susceptibility 161
7.6.3 Critical values of interactions 162
7.6.4 Numerical results 163
7.7 The Critical Point Exponents 167
7.8 Spin Waves in Ferromagnetism 170
7.8.1 Energy of spin-wave excitations 174
7.8.2 Dynamic susceptibility of ferromagnets 174
7.8.3 Curie temperature 177
References 178
Chapter 8. Itinerant Antiferromagnetism 180
8.1 Phenomenological Introduction 180
8.2 Simple Model of Itinerant Antiferromagnetism 181
8.3 Free Energy and the Magnetic Susceptibility 187
8.4 Antiferromagnetism Induced by On-site and Inter-site Correlations 188
8.5 Free Energy and the Magnetic Susceptibility Including Correlation Effects 193
8.5.1 Longitudinal and transversal susceptibility 195
8.6 Onset of Antiferromagnetism 199
8.6.1 The case of zero Coulomb correlation: U = 0 200
8.6.2 The case of the strong correlation: U > >
8.7 Numerical Results for Magnetization and Néel’s Temperature 204
8.8 Spin-Density Waves 206
Appendix 8A: Antiferromagnetism in the Presence of On-site and Inter-site Coulomb Correlation 212
References 215
Chapter 9. Alloys, Disordered Systems 216
9.1 Introduction 216
9.2 Order–Disorder Transformation and Bragg–Williams Approximation 218
9.2.1 Bragg–Williams approximation 219
9.3 Relation with the Band Model 221
9.4 Transition Metal Alloys 226
9.5 Different Types of Disorder in Bragg–Williams Approximation 232
References 237
Chapter 10. Itinerant Superconductivity 240
10.1 Phenomenological Introduction and Historical Background 241
10.2 Physical Properties of the High-Temperature Superconductors 243
10.2.1 General properties 243
10.2.2 Crystal structure of the HTS 244
10.2.3 Symmetry of the energy gap 245
10.2.4 Dependence of the critical temperature on concentration 248
10.2.5 Phase diagrams of the ordering 249
10.3 Classic (BCS) Model for Superconductivity 250
10.4 Electron–Electron Interaction as a Source of Superconductivity 253
10.4.1 Introduction 253
10.4.2 Single-band model 256
10.4.2.1 Model Hamiltonian 256
10.4.2.2 Moments method for the superconductivity equation 258
10.4.2.3 Analysis of the solution: critical temperature dependence on concentration 261
10.4.2.4 Effect of internal pressure on superconductivity 265
10.4.2.5 Symmetry of the energy gap 270
10.4.3 Three-band model 272
10.4.3.1 Introduction 272
10.4.3.2 The Model Hamiltonian 272
10.4.3.3 Hamiltonian diagonalization and the pairing interaction 276
10.4.3.4 Results 279
Appendix 10A: Transformation of Superconductivity Hamiltonian to Momentum Space 282
Appendix 10B: Green Function Equations for Singlet Superconductivity 287
Appendix 10C: Effective Pairing Potential in the Single-Band Model 290
Appendix 10D: Bogoliubov Transformation 293
References 295
Chapter 11. The Coexistence Between Magnetic Ordering and Itinerant Electron Superconductivity 300
11.1 Experimental Evidence of Coexistence Between Magnetic Ordering and Superconductivity 301
11.2 Coexistence of Ferromagnetism and High-Temperature Superconductivity 310
11.2.1 Model Hamiltonian 310
11.2.2 Green function solutions 313
11.2.2.1 General equations 313
11.2.2.2 Ferromagnetism coexisting with singlet superconductivity 314
11.2.2.3 Ferromagnetism coexisting with triplet opposite spins pairing superconductivity 317
11.2.2.4 Ferromagnetism coexisting with triplet parallel (equal) spins pairing superconductivity 317
11.2.3 Comparison with experimental results (for UGe2, ZrZn2, URhGe) 319
11.3 Coexistence of Antiferromagnetism and High-Temperature Superconductivity 322
11.3.1 Model Hamiltonian 323
11.3.2 Formalism of the model 325
11.3.3 Numerical examples 326
11.4 Superconducting Gap in Stripe States 328
Appendix 11A: Coexistence of Ferromagnetism and Singlet Superconductivity 331
Appendix 11B: Coexistence of Antiferromagnetism and Singlet Superconductivity 334
References 336
Subject Index 338