Dynamic Spin-Fluctuation Theory of Metallic Magnetism (eBook)

Preface 6
Contents 7
List of Symbols 11
1 Introduction 17
References 20
2 Basics of Metallic Magnetism 22
2.1 Magnetic Susceptibility: Macroscopic Approach 22
2.1.1 Generalized Magnetic Susceptibility 22
2.1.2 Symmetry Relations 24
2.1.3 Dispersion Relations 25
2.2 Magnetic Susceptibility: Microscopic Approach 26
2.2.1 Magnetization and Spin 26
2.2.2 Linear Response Theory 28
2.2.3 Fluctuation-Dissipation Theorem 31
References 34
3 Many-Electron Problem 35
3.1 One-Electron States 35
3.2 Many-Electron States 37
3.3 Second Quantization 38
3.3.1 General Theory 38
3.3.2 Specific Operators 39
Charge Density 39
Spin Density: Wannier Representation 40
Spin Density: Bloch Representation 41
Single-Site Spin 42
Hamiltonian 42
3.4 Noninteracting Electrons 43
References 47
4 Mean-Field Theory 48
4.1 The Hubbard Model 48
4.2 Stoner Mean-Field Theory 50
4.2.1 Hartree-Fock Approximation 50
4.2.2 Magnetization: The T2 Law 52
4.2.3 Uniform Static Susceptibility 54
4.3 Band Calculations in Metals 55
References 56
5 Random-Phase Approximation 57
5.1 Magnetic Susceptibilities 57
5.1.1 Longitudinal Susceptibility 57
5.1.2 Transverse Susceptibility 60
5.2 Magnetic Excitations 62
5.2.1 Spin-Density Waves 62
5.2.2 Magnetization: The T3/2 Law 65
5.2.3 Stoner Spin-Flip Excitations 67
References 68
6 Green Functions at Finite Temperatures 70
6.1 Fermion-Type Green Functions 70
6.1.1 Real-Time Green Function 70
General Properties 70
Equation of Motion 71
Spectral Function 72
6.1.2 Temperature Green Function 74
Relation with Charge and Spin Density 74
Relation with the Real-Time Green Function 75
6.2 Boson-Type Green Functions 76
6.2.1 Dynamic Susceptibility 77
Longitudinal Susceptibility 77
Transverse Susceptibility 79
6.2.2 Thermodynamic Susceptibility 81
Relation with the Spin Correlator 81
Relation with the Dynamic Susceptibility 82
Noninteracting Electrons 83
Summation Rule 84
Measurement and Calculation Methods 85
References 86
7 Spin Fluctuation Theory in the Ising Model 87
7.1 Spins in the Fluctuating Field 87
7.2 Approximations of the Free Energy 89
7.2.1 Quadratic Part of the Free Energy 89
7.2.2 Higher-Order Terms of the Free Energy 90
7.3 Local Fluctuating Field 91
7.4 Magnetic Phase Diagrams 92
References 94
8 Functional Integral Method 95
8.1 Multiband Hubbard Hamiltonian 95
8.1.1 Intraatomic Interaction and Hund's Rule 95
8.1.2 Atomic Charge and Spin Density 97
8.2 Functional Integral over Fluctuating Fields 99
8.2.1 Thermodynamic ``Time'' Dependence 99
8.2.2 Electrons in the Fluctuating Field 100
8.2.3 Charge Fluctuations 102
8.3 Exact Relations 104
8.3.1 Field-Dependent Thermodynamic Potential 104
8.3.2 Mean Spin and Spin-Density Correlator 107
References 108
9 Gaussian Approximation 110
9.1 Motivation 110
9.2 Saddle-Point Approximation 111
9.3 Free Energy Minimum Principle 113
9.4 Optimal Gaussian Approximation 114
9.4.1 General Formulation 114
9.4.2 Ferromagnetic State 114
9.4.3 Self-Energy Equation 115
References 117
10 Single-Site Gaussian Approximation 118
10.1 Coherent Potential Equation 118
10.2 Single-Site Gaussian Fluctuating Field 120
10.3 Mean Single-Site Green Function 122
10.4 Basic Magnetic Characteristics 123
10.5 Application to Ferromagnetic Metals 125
10.5.1 Iron 125
10.5.2 Cobalt 127
10.5.3 Nickel 129
10.5.4 Comparison with Other Studies 129
References 130
11 High-Temperature Theory 131
11.1 Problem of Temperature Dependence 131
11.1.1 Discontinuous Jump of Magnetization 131
11.1.2 Instability Through Multiple Solutions 132
11.1.3 Temperature Hysteresis 133
11.2 Beyond the Gaussian Approximation 137
11.2.1 Renormalized Gaussian Approximation 137
11.2.2 Local and Uniform Fluctuations 139
11.2.3 Application to Fe and Fe–Ni Invar 140
Iron 140
Fe-Ni Invar 140
References 142
12 Low-Temperature Theory 144
12.1 Low-Temperature Region 144
12.1.1 Transverse Dynamic Susceptibility 144
12.1.2 Spin Waves and T3/2 Law 147
12.2 Beyond the Spin Waves 150
12.2.1 Low-Temperature DSFT 150
12.2.2 Application to Fe and Fe-Ni Invar 151
References 153
13 Temperature Dependence of Magnetic Characteristics 155
13.1 Temporal Correlation Function 155
13.2 Qualitative Analysis of Spin Correlations 157
13.3 Spin Correlations in the One-Electron Approximation 158
13.3.1 Unenhanced Susceptibilities 158
13.3.2 Computational Formulae 160
13.3.3 Band and Model Calculations 162
13.3.4 Interim Conclusions 164
13.4 Magnetic Properties in the DSFT 165
13.4.1 Local Magnetic Characteristics 165
13.4.2 Nuclear Spin Relaxation Rates 170
References 174
14 Neutron Scattering in Metals 176
14.1 Scattering Cross-Section 176
14.2 Scattering Potential Correlator 178
14.3 Neutron Scattering and Phonons 180
14.3.1 Lattice Vibrations 180
14.3.2 Debye-Waller Factor 182
References 184
15 Short-Range Order Above TC 185
15.1 Magnetic Neutron Scattering 185
15.1.1 Magnetic Interaction Potential 185
15.1.2 Nonpolarized Magnetic Scattering 186
15.1.3 Polarized Magnetic Scattering 187
15.2 Spin-Density Correlations 189
15.2.1 Spatial Spin Correlator 190
15.2.2 Spin Correlator in the DSFT 191
15.2.3 High-Temperature Approximation 192
15.3 Application to Iron 192
15.3.1 Comparison of DSFT and Experiment 192
15.3.2 Short-Range Order Analysis 195
References 197
16 Conclusion 199
Appendices 201
A Basic Mathematical Results 202
B Concepts Related to Functional Integral 227
C Fourier Transformations 237
D Dynamic Susceptibility in the RPA 248
E Proofs of Four Results in the DSFT 252
F Basic Approximations in Scattering Theory 259
G Lattice Vibrations in the Harmonic Approximation 264
H Numerical Integral Transformations 270
I DSFT Solution Methods and Software 277
Index 282