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DUAON MA YAD RAKHIYA GA
CONTENTS
Preface to the second edition xi
Preface to the first edition xiii
1 Electrons in One-Dimensional Periodic Potentials 1
1.1 The Bloch Theorem for One-Dimensional Periodicity 2
1.2 Energy Levels of a Single Quantum Well and of a Periodic Array
of Quantum Wells 5
1.3 Transfer Matrix, Resonant Tunneling, and Energy Bands 12
1.4 The Tight-Binding Model 25
1.5 Plane Waves and Nearly Free-Electron Model 34
1.6 Some Dynamical Aspects of Electrons in Band Theory 38
Appendix A. Solved Problems and Complements 49
Further Reading 64
2 Geometrical Description of Crystals: Direct and Reciprocal Lattices 67
2.1 Simple Lattices and Composite Lattices 67
2.2 Geometrical Description of Some Crystal Structures 72
2.3 Wigner-Seitz Primitive Cells 83
2.4 Reciprocal Lattices 84
2.5 Brillouin Zones 88
2.6 Translational Symmetry and Quantum Mechanical Aspects 91
2.7 Density-of-States and Critical Points 99
Further Reading 104
3 The Sommerfeld Free-Electron Theory of Metals 107
3.1 Quantum Theory of the Free-Electron Gas 107
3.2 Fermi-Dirac Distribution Function and Chemical Potential 112
3.3 Electronic Specific Heat in Metals and Thermodynamic Functions 116
3.4 Thermionic Emission from Metals 118
Appendix A. Outline of Statistical Physics and Thermodynamic Relations 120
Appendix B. Fermi-Dirac and Bose-Einstein Statistics for Independent Particles 125
Appendix C. Modified Fermi-Dirac Statistics in a Model of Correlation Effects 131
Further Reading 133
4 The One-Electron Approximation and Beyond 135
4.1 Introductory Remarks on the Many-Electron Problem 136
4.2 The Hartree Equations 137
4.3 Identical Particles and Determinantal Wavefunctions 139
4.4 Matrix Elements Between Determinantal States 140
4.5 The Hartree-Fock Equations 144
Contents
Preface to the first edition xiii
1 Electrons in One-Dimensional Periodic Potentials 1
1.1 The Bloch Theorem for One-Dimensional Periodicity 2
1.2 Energy Levels of a Single Quantum Well and of a Periodic Array
of Quantum Wells 5
1.3 Transfer Matrix, Resonant Tunneling, and Energy Bands 12
1.4 The Tight-Binding Model 25
1.5 Plane Waves and Nearly Free-Electron Model 34
1.6 Some Dynamical Aspects of Electrons in Band Theory 38
Appendix A. Solved Problems and Complements 49
Further Reading 64
2 Geometrical Description of Crystals: Direct and Reciprocal Lattices 67
2.1 Simple Lattices and Composite Lattices 67
2.2 Geometrical Description of Some Crystal Structures 72
2.3 Wigner-Seitz Primitive Cells 83
2.4 Reciprocal Lattices 84
2.5 Brillouin Zones 88
2.6 Translational Symmetry and Quantum Mechanical Aspects 91
2.7 Density-of-States and Critical Points 99
Further Reading 104
3 The Sommerfeld Free-Electron Theory of Metals 107
3.1 Quantum Theory of the Free-Electron Gas 107
3.2 Fermi-Dirac Distribution Function and Chemical Potential 112
3.3 Electronic Specific Heat in Metals and Thermodynamic Functions 116
3.4 Thermionic Emission from Metals 118
Appendix A. Outline of Statistical Physics and Thermodynamic Relations 120
Appendix B. Fermi-Dirac and Bose-Einstein Statistics for Independent Particles 125
Appendix C. Modified Fermi-Dirac Statistics in a Model of Correlation Effects 131
Further Reading 133
4 The One-Electron Approximation and Beyond 135
4.1 Introductory Remarks on the Many-Electron Problem 136
4.2 The Hartree Equations 137
4.3 Identical Particles and Determinantal Wavefunctions 139
4.4 Matrix Elements Between Determinantal States 140
4.5 The Hartree-Fock Equations 144
Contents
4.6 Overview of Approaches Beyond the One-Electron Approximation 154
4.7 Electronic Properties and Phase Diagram of the Homogeneous Electron Gas 155
4.8 The Density Functional Theory and the Kohn-Sham Equations 163
Appendix A. Bielectronic Integrals Among Spin Orbitals 171
Appendix B. Outline of Second Quantization Formalism for Identical Fermions 172
Appendix C. An Integral on the Fermi Sphere 175
Further Reading 176
5 Band Theory of Crystals 179
5.1 Basic Assumptions of the Band Theory 180
5.2 The Tight-Binding Method (LCAO Method) 182
5.3 The Orthogonalized Plane Wave (OPW) Method 189
5.4 The Pseudopotential Method 197
5.5 The Cellular Method 204
5.6 The Augmented Plane Wave (APW) Method 207
5.7 The Green’s Function Method (KKR Method) 211
5.8 Iterative Methods in Electronic Structure Calculations 217
Appendix A. Matrix Elements of the Augmented Plane Wave Method 228
Appendix B. Solved Problems and Complements 232
Appendix C. Evaluation of the Structure Coefficients of the KKR
Method with the Ewald Procedure 235
Further Reading 240
6 Electronic Properties of Selected Crystals 243
6.1 Band Structure and Cohesive Energy of Rare-Gas Solids 244
6.2 Electronic Properties of Ionic Crystals 251
6.3 Covalent Crystals with Diamond Structure 263
6.4 Band Structures and Fermi Surfaces of Some Metals 266
6.5 Carbon-Based Materials and Electronic Structure of Graphene 272
Appendix A. Solved Problems and Complements 277
Further Reading 284
7 Excitons, Plasmons, and Dielectric Screening in Crystals 287
7.1 Exciton States in Crystals 288
7.2 Plasmon Excitations in Crystals 296
7.3 Static Dielectric Screening in Metals within the Thomas-Fermi Model 298
7.4 The Longitudinal Dielectric Function within the Linear Response Theory 301
7.5 Dielectric Screening within the Lindhard Model 304
7.6 Quantum Expression of the Longitudinal Dielectric Function in Crystals 312
7.7 Surface Plasmons and Surface Polaritons 314
Appendix A. Friedel Sum Rule and Fumi Theorem 318
Appendix B. Quantum Expression of the Longitudinal Dielectric
Function in Materials with the Linear Response Theory 320
Appendix C. Lindhard Dielectric Function for the Free-Electron Gas 325
Appendix D. Quantum Expression of the Transverse Dielectric Function
in Materials with the Linear Response Theory 328
Further Reading 331
4.7 Electronic Properties and Phase Diagram of the Homogeneous Electron Gas 155
4.8 The Density Functional Theory and the Kohn-Sham Equations 163
Appendix A. Bielectronic Integrals Among Spin Orbitals 171
Appendix B. Outline of Second Quantization Formalism for Identical Fermions 172
Appendix C. An Integral on the Fermi Sphere 175
Further Reading 176
5 Band Theory of Crystals 179
5.1 Basic Assumptions of the Band Theory 180
5.2 The Tight-Binding Method (LCAO Method) 182
5.3 The Orthogonalized Plane Wave (OPW) Method 189
5.4 The Pseudopotential Method 197
5.5 The Cellular Method 204
5.6 The Augmented Plane Wave (APW) Method 207
5.7 The Green’s Function Method (KKR Method) 211
5.8 Iterative Methods in Electronic Structure Calculations 217
Appendix A. Matrix Elements of the Augmented Plane Wave Method 228
Appendix B. Solved Problems and Complements 232
Appendix C. Evaluation of the Structure Coefficients of the KKR
Method with the Ewald Procedure 235
Further Reading 240
6 Electronic Properties of Selected Crystals 243
6.1 Band Structure and Cohesive Energy of Rare-Gas Solids 244
6.2 Electronic Properties of Ionic Crystals 251
6.3 Covalent Crystals with Diamond Structure 263
6.4 Band Structures and Fermi Surfaces of Some Metals 266
6.5 Carbon-Based Materials and Electronic Structure of Graphene 272
Appendix A. Solved Problems and Complements 277
Further Reading 284
7 Excitons, Plasmons, and Dielectric Screening in Crystals 287
7.1 Exciton States in Crystals 288
7.2 Plasmon Excitations in Crystals 296
7.3 Static Dielectric Screening in Metals within the Thomas-Fermi Model 298
7.4 The Longitudinal Dielectric Function within the Linear Response Theory 301
7.5 Dielectric Screening within the Lindhard Model 304
7.6 Quantum Expression of the Longitudinal Dielectric Function in Crystals 312
7.7 Surface Plasmons and Surface Polaritons 314
Appendix A. Friedel Sum Rule and Fumi Theorem 318
Appendix B. Quantum Expression of the Longitudinal Dielectric
Function in Materials with the Linear Response Theory 320
Appendix C. Lindhard Dielectric Function for the Free-Electron Gas 325
Appendix D. Quantum Expression of the Transverse Dielectric Function
in Materials with the Linear Response Theory 328
Further Reading 331
8 Interacting Electronic-Nuclear Systems and the Adiabatic Principle 333
8.1 Interacting Electronic-Nuclear Systems and Adiabatic
Potential-Energy Surfaces 334
8.2 Non-Degenerate Adiabatic Surface and Nuclear Dynamics 337
8.3 Degenerate Adiabatic Surfaces and Jahn-Teller Systems 342
8.4 The Hellmann-Feynman Theorem and Electronic-Nuclear Systems 356
8.5 Parametric Hamiltonians and Berry Phase 359
8.6 The Berry Phase Theory of the Macroscopic Electric Polarization in Crystals 364
Appendix A. Simplified Evaluation of Typical Jahn-Teller and Renner-Teller Matrices 371
Appendix B. Solved Problems and Complements 377
Further Reading 389
9 Lattice Dynamics of Crystals 391
9.1 Dynamics of Monoatomic One-Dimensional Lattices 391
9.2 Dynamics of Diatomic One-Dimensional Lattices 396
9.3 Dynamics of General Three-Dimensional Crystals 400
9.4 Quantum Theory of the Harmonic Crystal 407
9.5 Lattice Heat Capacity. Einstein and Debye Models 410
9.6 Considerations on Anharmonic Effects and Melting of Solids 412
9.7 Optical Phonons and Polaritons in Polar Crystals 415
Appendix A. Quantum Theory of the Linear Harmonic Oscillator 430
Further Reading 436
10 Scattering of Particles by Crystals 437
10.1 General Considerations 437
10.2 Elastic Scattering of X-rays from Crystals and the
Thomson Approximation 440
10.3 Compton Scattering and Electron Momentum Density 455
10.4 Inelastic Scattering of Particles and Phonons Spectra of Crystals 459
10.5 Quantum Theory of Elastic and Inelastic Scattering of Neutrons 463
10.6 Dynamical Structure Factor for Harmonic Displacements and
Debye-Waller Factor 467
10.7 Mössbauer Effect 474
Appendix A. Solved Problems and Complements 476
Further Reading 481
11 Optical and Transport Properties of Metals 483
11.1 Macroscopic Theory of Optical Constants in Homogeneous
Materials 484
11.2 The Drude Theory of the Optical Properties of Free Carriers 490
11.3 Transport Properties and Boltzmann Equation 499
11.4 Static and Dynamic Conductivity in Metals 502
11.5 Boltzmann Treatment and Quantum Treatment of Intraband Transitions 508
11.6 The Boltzmann Equation in Electric Fields and Temperature Gradients 509
Appendix A. Solved Problems and Complements 523
Further Reading 527
12 Optical Properties of Semiconductors and Insulators 529
12.1 Transverse Dielectric Function and Optical Constants in
Homogeneous Media 530
12.2 Quantum Theory of Band-to-Band Optical Transitions and Critical Points 534
12.3 Indirect Phonon-Assisted Transitions 539
12.4 Two-Photon Absorption 544
12.5 Exciton Effects on the Optical Properties 547
12.6 Fano Resonances and Absorption Lineshapes 553
12.7 Optical Properties of Vibronic Systems 559
Appendix A. Transitions Rates at First and Higher Orders of Perturbation Theory 569
Appendix B. Optical Constants, Green’s Function and Kubo-Greenwood Relation 574
Further Reading 575
13 Transport in Intrinsic and Homogeneously Doped Semiconductors 577
13.1 Fermi Level and Carrier Density in Intrinsic Semiconductors 577
13.2 Impurity Levels in Semiconductors 582
13.3 Fermi Level and Carrier Density in Doped Semiconductors 590
13.4 Non-Equilibrium Carrier Distributions 594
13.5 Generation and Recombination of Electron-Hole Pairs in
Doped Semiconductors 599
Appendix A. Solutions of Typical Transport Equations in Uniformly
Doped Semiconductors 601
Further Reading 608
14 Transport in Inhomogeneous Semiconductors 609
14.1 Properties of the p-n Junction at Equilibrium 609
14.2 Current-Voltage Characteristics of the p-n Junction 615
14.3 The Bipolar Junction Transistor 621
14.4 Semiconductor Heterojunctions 624
14.5 Metal-Semiconductor Contacts 627
14.6 Metal-Oxide-Semiconductor Structure 632
14.7 Metal-Oxide-Semiconductor Field-Effect Transistor (MOSFET) 637
Further Reading 640
15 Electron Gas in Magnetic Fields 643
15.1 Magnetization and Magnetic Susceptibility 644
15.2 Energy Levels and Density-of-States of a Free Electron Gas in
Magnetic Fields 646
15.3 Landau Diamagnetism and de Haas-van Alphen Effect 655
15.4 Spin Paramagnetism of a Free-Electron Gas 661
15.5 Magnetoresistivity and Classical Hall Effect 662
15.6 Quantum Hall Effects 668
Appendix A. Solved Problems and Complements 686
Further Reading 694
16 Magnetic Properties of Localized Systems and Kondo Impurities 697
16.1 Quantum Mechanical Treatment of Magnetic Susceptibility 698
16.2 Permanent Magnetic Dipoles in Atoms or Ions with Partially Filled Shells 701
16.3 Paramagnetism of Localized Magnetic Moments 704
16.4 Localized Magnetic States in Normal Metals 709
16.5 Dilute Magnetic Alloys and the Resistance Minimum Phenomenon 714
16.6 Magnetic Impurity in Normal Metals at Very Low Temperatures 724
Further Reading 729
17 Magnetic Ordering in Crystals 731
17.1 Ferromagnetism and the Weiss Molecular Field 732
17.2 Microscopic Origin of the Coupling Between Localized
Magnetic Moments 741
17.3 Antiferromagnetism in the Mean Field Approximation 748
17.4 Spin Waves and Magnons in Ferromagnetic Crystals 750
17.5 The Ising Model with the Transfer Matrix Method 756
17.6 The Ising Model with the Renormalization Group Theory 761
17.7 Itinerant Magnetism 769
Appendix A. Solved Problems and Complements 775
Further Reading 787
18 Superconductivity 789
18.1 Some Phenomenological Aspects of Superconductors 790
18.2 The Cooper Pair Idea 799
18.3 Ground State for a Superconductor in the BCS Theory at Zero Temperature 805
18.4 Excited States of Superconductors at Zero Temperature 813
18.5 Treatment of Superconductors at Finite Temperature and Heat Capacity 820
18.6 The Phenomenological London Model for Superconductors 824
18.7 Macroscopic Quantum Phenomena 828
18.8 Tunneling Effects 837
Appendix A. The Phonon-Induced Electron-Electron Interaction 845
Further Reading 848
Index 851
8.1 Interacting Electronic-Nuclear Systems and Adiabatic
Potential-Energy Surfaces 334
8.2 Non-Degenerate Adiabatic Surface and Nuclear Dynamics 337
8.3 Degenerate Adiabatic Surfaces and Jahn-Teller Systems 342
8.4 The Hellmann-Feynman Theorem and Electronic-Nuclear Systems 356
8.5 Parametric Hamiltonians and Berry Phase 359
8.6 The Berry Phase Theory of the Macroscopic Electric Polarization in Crystals 364
Appendix A. Simplified Evaluation of Typical Jahn-Teller and Renner-Teller Matrices 371
Appendix B. Solved Problems and Complements 377
Further Reading 389
9 Lattice Dynamics of Crystals 391
9.1 Dynamics of Monoatomic One-Dimensional Lattices 391
9.2 Dynamics of Diatomic One-Dimensional Lattices 396
9.3 Dynamics of General Three-Dimensional Crystals 400
9.4 Quantum Theory of the Harmonic Crystal 407
9.5 Lattice Heat Capacity. Einstein and Debye Models 410
9.6 Considerations on Anharmonic Effects and Melting of Solids 412
9.7 Optical Phonons and Polaritons in Polar Crystals 415
Appendix A. Quantum Theory of the Linear Harmonic Oscillator 430
Further Reading 436
10 Scattering of Particles by Crystals 437
10.1 General Considerations 437
10.2 Elastic Scattering of X-rays from Crystals and the
Thomson Approximation 440
10.3 Compton Scattering and Electron Momentum Density 455
10.4 Inelastic Scattering of Particles and Phonons Spectra of Crystals 459
10.5 Quantum Theory of Elastic and Inelastic Scattering of Neutrons 463
10.6 Dynamical Structure Factor for Harmonic Displacements and
Debye-Waller Factor 467
10.7 Mössbauer Effect 474
Appendix A. Solved Problems and Complements 476
Further Reading 481
11 Optical and Transport Properties of Metals 483
11.1 Macroscopic Theory of Optical Constants in Homogeneous
Materials 484
11.2 The Drude Theory of the Optical Properties of Free Carriers 490
11.3 Transport Properties and Boltzmann Equation 499
11.4 Static and Dynamic Conductivity in Metals 502
11.5 Boltzmann Treatment and Quantum Treatment of Intraband Transitions 508
11.6 The Boltzmann Equation in Electric Fields and Temperature Gradients 509
Appendix A. Solved Problems and Complements 523
Further Reading 527
12 Optical Properties of Semiconductors and Insulators 529
12.1 Transverse Dielectric Function and Optical Constants in
Homogeneous Media 530
12.2 Quantum Theory of Band-to-Band Optical Transitions and Critical Points 534
12.3 Indirect Phonon-Assisted Transitions 539
12.4 Two-Photon Absorption 544
12.5 Exciton Effects on the Optical Properties 547
12.6 Fano Resonances and Absorption Lineshapes 553
12.7 Optical Properties of Vibronic Systems 559
Appendix A. Transitions Rates at First and Higher Orders of Perturbation Theory 569
Appendix B. Optical Constants, Green’s Function and Kubo-Greenwood Relation 574
Further Reading 575
13 Transport in Intrinsic and Homogeneously Doped Semiconductors 577
13.1 Fermi Level and Carrier Density in Intrinsic Semiconductors 577
13.2 Impurity Levels in Semiconductors 582
13.3 Fermi Level and Carrier Density in Doped Semiconductors 590
13.4 Non-Equilibrium Carrier Distributions 594
13.5 Generation and Recombination of Electron-Hole Pairs in
Doped Semiconductors 599
Appendix A. Solutions of Typical Transport Equations in Uniformly
Doped Semiconductors 601
Further Reading 608
14 Transport in Inhomogeneous Semiconductors 609
14.1 Properties of the p-n Junction at Equilibrium 609
14.2 Current-Voltage Characteristics of the p-n Junction 615
14.3 The Bipolar Junction Transistor 621
14.4 Semiconductor Heterojunctions 624
14.5 Metal-Semiconductor Contacts 627
14.6 Metal-Oxide-Semiconductor Structure 632
14.7 Metal-Oxide-Semiconductor Field-Effect Transistor (MOSFET) 637
Further Reading 640
15 Electron Gas in Magnetic Fields 643
15.1 Magnetization and Magnetic Susceptibility 644
15.2 Energy Levels and Density-of-States of a Free Electron Gas in
Magnetic Fields 646
15.3 Landau Diamagnetism and de Haas-van Alphen Effect 655
15.4 Spin Paramagnetism of a Free-Electron Gas 661
15.5 Magnetoresistivity and Classical Hall Effect 662
15.6 Quantum Hall Effects 668
Appendix A. Solved Problems and Complements 686
Further Reading 694
16 Magnetic Properties of Localized Systems and Kondo Impurities 697
16.1 Quantum Mechanical Treatment of Magnetic Susceptibility 698
16.2 Permanent Magnetic Dipoles in Atoms or Ions with Partially Filled Shells 701
16.3 Paramagnetism of Localized Magnetic Moments 704
16.4 Localized Magnetic States in Normal Metals 709
16.5 Dilute Magnetic Alloys and the Resistance Minimum Phenomenon 714
16.6 Magnetic Impurity in Normal Metals at Very Low Temperatures 724
Further Reading 729
17 Magnetic Ordering in Crystals 731
17.1 Ferromagnetism and the Weiss Molecular Field 732
17.2 Microscopic Origin of the Coupling Between Localized
Magnetic Moments 741
17.3 Antiferromagnetism in the Mean Field Approximation 748
17.4 Spin Waves and Magnons in Ferromagnetic Crystals 750
17.5 The Ising Model with the Transfer Matrix Method 756
17.6 The Ising Model with the Renormalization Group Theory 761
17.7 Itinerant Magnetism 769
Appendix A. Solved Problems and Complements 775
Further Reading 787
18 Superconductivity 789
18.1 Some Phenomenological Aspects of Superconductors 790
18.2 The Cooper Pair Idea 799
18.3 Ground State for a Superconductor in the BCS Theory at Zero Temperature 805
18.4 Excited States of Superconductors at Zero Temperature 813
18.5 Treatment of Superconductors at Finite Temperature and Heat Capacity 820
18.6 The Phenomenological London Model for Superconductors 824
18.7 Macroscopic Quantum Phenomena 828
18.8 Tunneling Effects 837
Appendix A. The Phonon-Induced Electron-Electron Interaction 845
Further Reading 848
Index 851
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