A Compendium of Solid State Theory

Description

Designed to sit alongside more conventional established condensed matter physics textbooks, this compact volume offers a concise presentation of the principles of solid state theory, ideal for advanced students and researchers requiring an overview or a quick refresher on a specific topic. The book starts from the one-electron theory of solid state physics, moving through electron-electron interaction and many-body approximation schemes, to lattice oscillations and their interactions with electrons. Subsequent chapters discuss transport theory and optical properties, phase transitions and some properties of low-dimensional semiconductors. This extensively expanded second edition includes new material on adiabatic perturbation theory, kinetic coefficients, the Nyquist theorem, Bose condensation, and the field-theoretical approach to non-relativistic quantum electrodynamics.   Throughout the text, mathematical proofs are often only sketched, and the final chapter of the book reviews some of the key concepts and formulae used in theoretical physics. Aimed primarily at graduate and advanced undergraduate students taking courses on condensed matter theory, the book serves as a study guide to reinforce concepts learned through conventional solid state texts.  Researchers and lecturers will also find it a useful resource as a concise set of notes on fundamental topics. Preface to the Second Edition Preface to the First Edition Contents 1 Introduction 2 Non-Interacting Electrons 2.1 Free Electrons 2.2 Electron in Electric and Magnetic Fields 2.2.1 Homogeneous, Constant Electric Field 2.2.2 Homogeneous, Constant Magnetic Field 2.2.2.1 Magnetization 2.2.3 Motion in a One-Dimensional Potential Well 2.3 Electrons in a Periodical Potential 2.3.1 Crystal Lattice 2.3.2 Bloch Functions 2.3.3 Periodical Boundary Conditions 2.3.4 The Approximation of Quasi-Free Electrons 2.3.5 The Kronig–Penney Model 2.3.6 Band Extrema, kp: Perturbation Theory and Effective Mass 2.3.7 Wannier Functions and Tight-Binding Approximation 2.3.8 Bloch Electron in a Homogeneous Electric Field 2.4 Electronic Occupation of States in a Crystal 2.4.1 Ground State Occupation of Bands: Conductors and Insulators 2.4.2 Spin–Orbit Coupling and Valence Band Splitting 2.5 Electron States Due to Deviations from Periodicity 2.5.1 Effective Mass Approximation 2.5.2 Intrinsic Semiconductors at Finite Temperatures 2.5.3 Ionic Impurities 2.5.4 Extrinsic Semiconductors at Finite Temperatures: Acceptors and Donors 2.6 Semiconductor Contacts 2.6.1 Electric Field Penetration into a Semiconductor 2.6.2 p–n Contact 3 Electron–Electron Interaction 3.1 The Exciton 3.1.1 Wannier Exciton 3.1.2 Exciton Beyond the Effective Mass Approximation 3.2 Many-Body Approach to the Solid State 3.2.1 Self-Consistent Approximations 3.2.2 Electron Gas with Coulomb Interactions 3.2.3 The Electron–Hole Plasma 3.2.4 Many-Body Perturbation Theory of Solid State 3.2.5 Adiabatic Perturbation Theory 4 Phonons 4.1 Lattice Oscillations 4.2 Classical Continuum Phonon-Model 4.2.1 Optical Phonons in Polar Semiconductors 4.2.2 Optical Eigenmodes 4.2.3 The Electron–Phonon Interaction 4.2.3.1 The Franck–Condon Effect 4.2.3.2 The Quantized Interaction of Electrons with Phonons 5 Transport Theory 5.1 Non-Equilibrium Phenomena 5.2 Classical Solvable Model of an Electron in a d.c. Electric Field Interacting with Phonons 5.3 The Boltzmann Equation 5.3.1 Classical Conductivity 5.4 Kinetic Coefficients 5.5 Master and Rate Equations 5.5.1 Master Equations 5.5.2 Rate Equations 5.6 Hopping Transport 5.6.1 Hopping Diffusion on a Periodic Cubic Lattice 5.6.2 Transverse Magneto-Resistance in Ultra-Strong Magnetic Field 5.6.3 Seebeck Coefficient for Hopping Conduction on Random Localized States 6 Optical Properties 6.1 Linear Response to a Time-Dependent External Perturbation 6.2 Equilibrium Linear Response 6.3 Dielectric Response of a Coulomb Interacting Electron System 6.4 The Full Nyquist Theorem 6.5 Dielectric Function of an Electron Plasma in the Hartree Approximation 6.6 The Transverse, Inter-Band Dielectric Response of an Electron–Hole Plasma 6.7 Ultra-Short-Time Spectroscopy of Semiconductors 6.8 Third Order Non-Linear Response 6.9 Differential Transmission 6.10 Four Wave Mixing 7 Phase Transitions 7.1 The Heisenberg Model of Ferro-Magnetism 7.2 Bose Condensation 7.2.1 Bose Condensation in Real Time 7.3 Bogoliubov's Self-Consistent Model of Repulsive Bosons at T=0 7.4 Time-Dependent Bogoliubov and Gross–Pitaevskii Equations 7.5 Superconductivity 7.5.1 The Phenomenological Theory of London 7.6 Superconducting Phase Transition in a Simple Model of Electron–Electron Interaction 7.6.1 Meissner Effect Within Equilibrium Linear Response 7.6.2 The Case of a Contact Potential: The Bogoliubov–de Gennes Equation 8 Low Dimensional Semiconductors 8.1 Exciton in 2D 8.2 Motion of a 2D Electron in a Strong Magnetic Field 8.3 Coulomb Interaction in 2D in a Strong Magnetic Field 8.3.1 Classical Motion 8.3.2 Quantum Mechanical States 9 Extension of the Solid-State Hamiltonian: Current–Current Interaction Terms of Order 1/c2 9.1 Classical Approach 9.2 The Second Quantized Version 10 Field-Theoretical Approach to the Non-Relativistic Quantum Electrodynamics 10.1 Field Theory 10.2 Classical Maxwell Equations Coupled to a Quantum Mechanical Electron 10.3 Classical Lagrange Density for the Maxwell Equations Coupled to a Quantum Mechanical Electron 10.4 The Classical Hamiltonian in the Coulomb Gauge 10.5 Quantization of the Hamiltonian 10.6 Derivation of the 1/c2 Hamiltonian 11 Shortcut of Theoretical Physics 11.1 Classical Mechanics 11.2 One-Particle Quantum Mechanics 11.2.1 Dirac's ``bra/ket'' Formalism 11.3 Perturbation Theory 11.3.1 Stationary Perturbation 11.3.2 Time-Dependent Adiabatic Perturbation 11.4 Many-Body Quantum Mechanics 11.4.1 Configuration Space 11.4.2 Fock Space (Second Quantization) 11.4.2.1 Fermions 11.4.2.2 Bosons 11.5 Density Matrix (Statistical Operator) 11.6 Classical Point-Like Charged Particles and Electromagnetic Fields 12 Homework 12.1 The Kubo Formula 12.2 Ideal Relaxation 12.3 Rate Equation for Bosons 12.4 Bose Condensation in a Finite Potential Well