Solid State Physics: Essential Concepts

Contents
Preface
1 Electron Bands
1.1 Where Do Bands Come From? Why Solid State Physics Requires a New Way of Thinking
1.1.1 Energy Splitting Due to Wave Function Overlap
1.1.2 The LCAO Approximation
1.1.3 General Remarks on Bands
1.2 The Kronig–Penney Model
1.3 Bloch’s Theorem
1.4 Bravais Lattices and Reciprocal Space
1.5 X-ray Scattering
1.6 General Properties of Bloch Functions
1.7 Boundary Conditions in a Finite Crystal
1.8 Density of States
1.8.1 Density of States at Critical Points
1.8.2 Disorder and Density of States
1.9 Electron Band Calculations in Three Dimensions
1.9.1 How to Read a Band Diagram
1.9.2 The Tight-Binding Approximation and Wannier Functions
1.9.3 The Nearly Free Electron Approximation
1.9.4 k · p Theory
1.9.5 Other Methods of Calculating Band Structure
1.10 Angle-Resolved Photoemission Spectroscopy
1.11 Why Are Bands Often Completely Full or Empty? Bands and Molecular Bonds
1.11.1 Molecular Bonds
1.11.2 Classes of Electronic Structure
1.11.3 sp3 Bonding
1.11.4 Dangling Bonds and Defect States
1.12 Surface States
1.13 Spin in Electron Bands
1.13.1 Split-off Bands
1.13.2 Spin–Orbit Effects on the k-Dependence of Bands
References
2 Electronic Quasiparticles
2.1 Quasiparticles
2.2 Effective Mass
2.3 Excitons
2.4 Metals and the Fermi Gas
2.4.1 Isotropic Fermi Gas at T =0
2.4.2 Fermi Gas at Finite Temperature
2.5 Basic Behavior of Semiconductors
2.5.1 Equilibrium Populations of Electrons and Holes
2.5.2 Semiconductor Doping
2.5.3 Equilibrium Populations in Doped Semiconductors
2.5.4 The Mott Transition
2.6 Band Bending at Interfaces
2.6.1 Metal-to-Metal Interfaces
2.6.2 Doped Semiconductor Junctions
2.6.3 Metal–Semiconductor Junctions
2.6.4 Junctions of Undoped Semiconductors
2.7 Transistors
2.7.1 Bipolar Transistors
2.7.2 Field Effect Transistors
2.8 Quantum Confinement
2.8.1 Density of States in Quantum-Confined Systems
2.8.2 Superlattices and Bloch Oscillations
2.8.3 The Two-Dimensional Electron Gas
2.8.4 One-Dimensional Electron Transport
2.8.5 Quantum Dots and Coulomb Blockade
2.9 Landau Levels and Quasiparticles in Magnetic Field
2.9.1 Quantum Mechanical Calculation of Landau Levels
2.9.2 De Haas–Van Alphen and Shubnikov–De Haas Oscillations
2.9.3 The Integer Quantum Hall Effect
2.9.4 The Fractional Quantum Hall Effect and Higher-Order Quasiparticles
References
3 Classical Waves in Anisotropic Media
3.1 The Coupled Harmonic Oscillator Model
3.1.1 Harmonic Approximation of the Interatomic Potential
3.1.2 Linear-Chain Model
3.1.3 Vibrational Modes in Higher Dimensions
3.2 Neutron Scattering
3.3 Phase Velocity and Group Velocity in Anisotropic Media
3.4 Acoustic Waves in Anisotropic Crystals
3.4.1 Stress and Strain Definitions: Elastic Constants
3.4.2 The Christoffel Wave Equation
3.4.3 Acoustic Wave Focusing
3.5 Electromagnetic Waves in Anisotropic Crystals
3.5.1 Maxwell’s Equations in an Anisotropic Crystal
3.5.2 Uniaxial Crystals
3.5.3 The Index Ellipsoid
3.6 Electro-optics
3.7 Piezoelectric Materials
3.8 Reflection and Transmission at Interfaces
3.8.1 Optical Fresnel Equations
3.8.2 Acoustic Fresnel Equations
3.8.3 Surface Acoustic Waves
3.9 Photonic Crystals and Periodic Structures
References
4 Quantized Waves
4.1 The Quantized Harmonic Oscillator
4.2 Phonons
4.3 Photons
4.4 Coherent States
4.5 Spatial Field Operators
4.6 Electron Fermi Field Operators
4.7 First-Order Time-Dependent Perturbation Theory: Fermi’s Golden Rule
4.8 The Quantum Boltzmann Equation
4.8.1 Equilibrium Distributions of Quantum Particles
4.8.2 The H-Theorem and the Second Law
4.9 Energy Density of Solids
4.9.1 Density of States of Phonons and Photons
4.9.2 Planck Energy Density
4.9.3 Heat Capacity of Phonons
4.9.4 Electron Heat Capacity: Sommerfeld Expansion
4.10 Thermal Motion of Atoms
References
5 Interactions of Quasiparticles
5.1 Electron–Phonon Interactions
5.1.1 Deformation Potential Scattering
5.1.2 Piezoelectric Scattering
5.1.3 Fröhlich Scattering
5.1.4 Average Electron–Phonon Scattering Time
5.2 Electron–Photon Interactions
5.2.1 Optical Transitions Between Semiconductor Bands
5.2.2 Multipole Expansion
5.3 Interactions with Defects: Rayleigh Scattering
5.4 Phonon–Phonon Interactions
5.4.1 Thermal Expansion
5.4.2 Crystal Phase Transitions
5.5 Electron–Electron Interactions
5.5.1 Semiclassical Estimation of Screening Length
5.5.2 Average Electron–Electron Scattering Time
5.6 The Relaxation-Time Approximation and the Diffusion Equation
5.7 Thermal Conductivity
5.8 Electrical Conductivity
5.9 Thermoelectricity: Drift and Diffusion of a Fermi Gas
5.10 Magnetoresistance
5.11 The Boltzmann Transport Equation
5.12 Drift of Defects and Dislocations: Plasticity
References
6 Group Theory
6.1 Definition of a Group
6.2 Representations
6.3 Character Tables
6.4 Equating Physical States with the Basis States of Representations
6.5 Reducing Representations
6.6 Multiplication Rules for Outer Products
6.7 Review of Types of Operators
6.8 Effects of Lowering Symmetry
6.9 Spin and Time Reversal Symmetry
6.10 Allowed and Forbidden Transitions
6.10.1 Second-Order Transitions
6.10.2 Quadrupole Transitions
6.11 Perturbation Methods
6.11.1 Group Theory in k · p Theory
6.11.2 Method of Invariants
References
7 The Complex Susceptibility
7.1 A Microscopic View of the Dielectric Constant
7.1.1 Fresnel Equations for the Complex Dielectric Function
7.1.2 Fano Resonances
7.2 Kramers–Kronig Relations
7.3 Negative Index of Refraction: Metamaterials
7.4 The Quantum Dipole Oscillator
7.5 Polaritons
7.5.1 Phonon-Polaritons
7.5.2 Exciton-Polaritons
7.5.3 Quantum Mechanical Formulation of Polaritons
7.6 Nonlinear Optics and Photon–Photon Interactions
7.6.1 Second-Harmonic Generation and Three-Wave Mixing
7.6.2 Higher-Order Effects
7.7 Acousto-Optics and Photon–Phonon Interactions
7.8 Raman Scattering
References
8 Many-Body Perturbation Theory
8.1 Higher-Order Time-Dependent Perturbation Theory
8.2 Polarons
8.3 Shift of Bands with Temperature
8.4 Line Broadening
8.5 Diagram Rules for Rayleigh–Schrödinger Perturbation Theory
8.6 Feynman Perturbation Theory
8.7 Diagram Rules for Feynman Perturbation Theory
8.8 Self-Energy
8.9 Physical Meaning of the Green’s Functions
8.10 Finite Temperature Diagrams
8.11 Screening and Plasmons
8.11.1 Plasmons
8.11.2 The Conductor–Insulator Transition and Screening
8.12 Ground State Energy of the Fermi Sea: Density Functional Theory
8.13 The Imaginary-Time Method for Finite Temperature
8.14 Symmetrized Green’s Functions
8.15 Matsubara Calculations for the Electron Gas
References
9 Coherence and Correlation
9.1 Density Matrix Formalism
9.2 Magnetic Resonance: The Bloch Equations
9.3 Optical Bloch Equations
9.4 Quantum Coherent Effects
9.5 Correlation Functions and Noise
9.6 Correlations in Quantum Mechanics
9.7 Particle–Particle Correlation
9.8 The Fluctuation–Dissipation Theorem
9.9 Current Fluctuations and the Nyquist Formula
9.10 The Kubo Formula and Many-Body Theory of Conductivity
9.11 Mesoscopic Effects
References
10 Spin and Magnetic Systems
10.1 Overview of Magnetic Properties
10.2 Landé g-factor in Solids
10.3 The Ising Model
10.3.1 Spontaneous Symmetry Breaking
10.3.2 External Magnetic Field: Hysteresis
10.4 Critical Exponents and Fluctuations
10.5 Renormalization Group Methods
10.6 Spin Waves and Goldstone Bosons
10.7 Domains and Domain Walls
10.8 Spin–Spin Interaction
10.8.1 Ferromagnetic Instability
10.8.2 Localized States and RKKY Exchange Interaction
10.8.3 Electron–Hole Exchange
10.9 Spin Flip and Spin Dephasing
References
11 Spontaneous Coherence in Matter
11.1 Theory of the Ideal Bose Gas
11.2 The Bogoliubov Model
11.3 The Stability of the Condensate: Analogy with Ferromagnets
11.4 Bose Liquid Hydrodynamics
11.5 Superfluids versus Condensates
11.6 Constructing Bosons from Fermions
11.7 Cooper Pairing
11.8 BCS Wave Function
11.9 Excitation Spectrum of a Superconductor
11.9.1 Density of States and Tunneling Spectroscopy
11.9.2 Temperature Dependence of the Gap
11.10 Magnetic Effects of Superconductors
11.10.1 Critical Field
11.10.2 Flux Quantization
11.10.3 Type I and Type II Superconductors
11.11 Josephson Junctions
11.12 Spontaneous Optical Coherence: Lasing as a Phase Transition
11.13 Excitonic Condensation
11.13.1 Microcavity Polaritons
11.13.2 Other Quasiparticle Condensates
References
Appendix A Review of Bra-Ket Notation
Appendix B Review of Fourier Series and Fourier Transforms
Appendix C Delta-Function Identities
Appendix E Second-Order Perturbation Theory
Appendix F Relativistic Derivation of Spin Physics
Index