Models of Itinerant Ordering in Crystals: An Introduction
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Table of contents
- Copyright Pageiv
- Table of Contentsvii
- Prefacexi
- Part One. Introduction to Theory of Solids1
- Chapter 1. Periodic Structures3
- 1.1 Fundamental Types of Lattices3
- 1.2 Diffraction of Waves by a Crystal and the Reciprocal Lattice4
- 1.2.1 Reciprocal lattice vectors7
- 1.3 Brillouin Zones8
- References10
- Chapter 2. Various Statistics11
- Appendix 2A: Fermi–Dirac And Bose–Einstein Distribution Functions14
- References17
- Chapter 3. Paramagnetism and Weiss Ferromagnetism19
- 3.1 Paramagnetism19
- 3.2 Weiss Ferromagnetism25
- References26
- Chapter 4. Electron States27
- 4.1 The Nearly Free Electron Model27
- 4.1.1 General result for ρ(ε)31
- 4.1.2 Use of DOS for evaluating lattice sums in momentum space32
- 4.1.3 Heat capacity of the free electron gas: an introduction34
- 4.2 The Tight-Binding Method35
- 4.2.1 Cohesion energy40
- 4.3 Bloch Theorem43
- Appendix 4A: Nearly Free Electrons, Two-Plane Waves Model44
- References48
- Part Two. Models of Itinerant Ordering in Crystals49
- Chapter 5. The Hubbard Model51
- 5.1 Simple Hubbard Model51
- 5.2 Extended Hubbard Model54
- References58
- Chapter 6. Different Approximations for Hubbard Model59
- 6.1 Chain Equation for Green Functions60
- 6.2 Hartree–Fock Approximation63
- 6.3 Hubbard I Approximation64
- 6.3.1 Atomic limit64
- 6.3.2 Finite bandwidth limit66
- 6.4 Extended Hubbard III Approximation69
- 6.5 Coherent Potential Approximation72
- 6.5.1 Relation between CPA, Hubbard III and extended Hubbard III approximations80
- 6.5.2 Different applications of the CPA81
- 6.6 Spectral Density Approach81
- 6.7 Modified Alloy Analogy86
- 6.8 Dynamical Mean-Field Theory88
- 6.9 Hubbard Model Extended by Inter-site Interactions90
- 6.9.1 Modified Hartree–Fock approximation90
- 6.9.2 Coherent potential approximation for the extended Hubbard model94
- Appendix 6A: Equation of Motion for the Green Functions95
- Appendix 6B: Hubbard Solution for the Scattering and Resonance Broadening Effects97
- 6B.1 The scattering effect97
- 6B.2 The resonance broadening effect100
- Appendix 6C: Modified Hartree–Fock Approximation for the Inter-site Interactions106
- References113
- Chapter 7. Itinerant Ferromagnetism115
- 7.1 Periodic Table – Ferromagnetic Elements115
- 7.1.1 Ferromagnetic elements118
- 7.2 Introduction to Stoner Model125
- 7.2.1 Static magnetic susceptibility129
- 7.3 Stoner Model for Ferromagnetism131
- 7.4 Stoner Model for Rectangular and Parabolic Band134
- 7.4.1 Rectangular band134
- 7.4.2 Parabolic nearly free electron band136
- 7.5 Modified Stoner Model138
- 7.5.1 Modified Stoner Model for a semi-elliptic band141
- 7.6 Beyond Hartree–Fock Model144
- 7.6.1 General formalism144
- 7.6.2 Enhancement of magnetic susceptibility148
- 7.6.3 Critical values of interactions149
- 7.6.4 Numerical results150
- 7.7 The Critical Point Exponents154
- 7.8 Spin Waves in Ferromagnetism157
- 7.8.1 Energy of spin-wave excitations161
- 7.8.2 Dynamic susceptibility of ferromagnets161
- 7.8.3 Curie temperature164
- References165
- Chapter 8. Itinerant Antiferromagnetism167
- 8.1 Phenomenological Introduction167
- 8.2 Simple Model of Itinerant Antiferromagnetism168
- 8.3 Free Energy and the Magnetic Susceptibility174
- 8.4 Antiferromagnetism Induced by On-site and Inter-site Correlations175
- 8.5 Free Energy and the Magnetic Susceptibility Including Correlation Effects180
- 8.5.1 Longitudinal and transversal susceptibility182
- 8.6 Onset of Antiferromagnetism186
- 8.6.1 The case of zero Coulomb correlation: U = 0187
- 8.6.2 The case of the strong correlation: U >> D189
- 8.7 Numerical Results for Magnetization and Néel’s Temperature191
- 8.8 Spin-Density Waves193
- Appendix 8A: Antiferromagnetism in the Presence of On-site and Inter-site Coulomb Correlation199
- References202
- Chapter 9. Alloys, Disordered Systems203
- 9.1 Introduction203
- 9.2 Order–Disorder Transformation and Bragg–Williams Approximation205
- 9.2.1 Bragg–Williams approximation206
- 9.3 Relation with the Band Model208
- 9.4 Transition Metal Alloys213
- 9.5 Different Types of Disorder in Bragg–Williams Approximation219
- References224
- Chapter 10. Itinerant Superconductivity227
- 10.1 Phenomenological Introduction and Historical Background228
- 10.2 Physical Properties of the High-Temperature Superconductors230
- 10.2.1 General properties230
- 10.2.2 Crystal structure of the HTS231
- 10.2.3 Symmetry of the energy gap232
- 10.2.4 Dependence of the critical temperature on concentration235
- 10.2.5 Phase diagrams of the ordering236
- 10.3 Classic (BCS) Model for Superconductivity237
- 10.4 Electron–Electron Interaction as a Source of Superconductivity240
- 10.4.1 Introduction240
- 10.4.2 Single-band model243
- 10.4.2.1 Model Hamiltonian243
- 10.4.2.2 Moments method for the superconductivity equation245
- 10.4.2.3 Analysis of the solution: critical temperature dependence on concentration248
- 10.4.2.4 Effect of internal pressure on superconductivity252
- 10.4.2.5 Symmetry of the energy gap257
- 10.4.3 Three-band model259
- 10.4.3.1 Introduction259
- 10.4.3.2 The Model Hamiltonian259
- 10.4.3.3 Hamiltonian diagonalization and the pairing interaction263
- 10.4.3.4 Results266
- Appendix 10A: Transformation of Superconductivity Hamiltonian to Momentum Space269
- Appendix 10B: Green Function Equations for Singlet Superconductivity274
- Appendix 10C: Effective Pairing Potential in the Single-Band Model277
- Appendix 10D: Bogoliubov Transformation280
- References282
- Chapter 11. The Coexistence Between Magnetic Ordering and Itinerant Electron Superconductivity287
- 11.1 Experimental Evidence of Coexistence Between Magnetic Ordering and Superconductivity288
- 11.2 Coexistence of Ferromagnetism and High-Temperature Superconductivity297
- 11.2.1 Model Hamiltonian297
- 11.2.2 Green function solutions300
- 11.2.2.1 General equations300
- 11.2.2.2 Ferromagnetism coexisting with singlet superconductivity301
- 11.2.2.3 Ferromagnetism coexisting with triplet opposite spins pairing superconductivity304
- 11.2.2.4 Ferromagnetism coexisting with triplet parallel (equal) spins pairing superconductivity304
- 11.2.3 Comparison with experimental results (for UGe2, ZrZn2, URhGe)306
- 11.3 Coexistence of Antiferromagnetism and High-Temperature Superconductivity309
- 11.3.1 Model Hamiltonian310
- 11.3.2 Formalism of the model312
- 11.3.3 Numerical examples313
- 11.4 Superconducting Gap in Stripe States315
- Appendix 11A: Coexistence of Ferromagnetism and Singlet Superconductivity318
- Appendix 11B: Coexistence of Antiferromagnetism and Singlet Superconductivity321
- References323
- Subject Index325
This book is devoted to the mathematical description of interesting phenomena which occur in solids, such as ferromagnetism, antiferromagnetism and superconductivity. Superconductivity and its interaction with ferro and antiferromagnetism is of special importance since over the last 15 years the temperature of superconductivity existence has been raised from 15-20 K to 100 K, which will allow in the near future numerous practical applications of this phenomenon. Although the book is written in a rather rigorous mathematical language it is made easy to read by detailed derivation for those having only an undergraduate background in physics.
Key Features:
- new field of research
- common formalism for superconductivity and magnetism
- easy and simple models
- easy reading which includes all derivations
- good for graduate students and young researchers
* A new field of research
* Common formalism for superconductivity and magnetism
* Easy reading and simple models, which includes all derivations
Key Features:
- new field of research
- common formalism for superconductivity and magnetism
- easy and simple models
- easy reading which includes all derivations
- good for graduate students and young researchers
* A new field of research
* Common formalism for superconductivity and magnetism
* Easy reading and simple models, which includes all derivations