Introduction to Solid-State Theory

Introduction to Solid-State Theory is a textbook for graduate students of physics and materials science. It also provides the theoretical background needed by physicists doing research in pure solid-state physics and its applications to electrical engineering. The fundamentals of solid-state theory are based on a description by delocalized and localized states and - within the concept of delocalized states - by elementary excitations. The development of solid-state theory within the last ten years has shown that by a systematic introduction of these concepts, large parts of the theory can be described in a unified way. At the same time, this form of description gives a "pictorial" formulation of many elementary processes in solids, which facilitates their understanding.

Contenu

1. Fundamentals.- 1.1 Introduction.- 1.2 The Basic Hamiltonian.- 1.3 The Hartree-Fock Approximation.- 2. The One-Electron Approximation.- 2.1 The Electron Gas Without Interaction.- 2.1.1 Introduction.- 2.1.2 The Energy States.- 2.1.3 Excited States.- 2.1.4 The Fermi Distribution.- 2.1.5 Free Electrons in an Electric Field.- 2.1.6 Free Electrons in a Magnetic Field.- 2.1.7 Dia- and Paramagnetism of Free Electrons, the de Haasvan Alphen Effect.- 2.2 Electrons in a Periodic Potential.- 2.2.1 Introduction.- 2.2.2 The Symmetries of the Crystal Lattice.- 2.2.3 The Schrödinger Equation for Electrons in a Periodic Potential.- 2.2.4 The Reciprocal Lattice, Bragg Reflections.- 2.2.5 Consequences of Translational Invariance.- 2.2.6 Nearly Free Electron Approximation.- 2.2.7 Wannier Functions, LCAO Approximation.- 2.2.8 General Properties of the Function En(k).- 2.2.9 Dynamics of Crystal Electrons.- 2.2.10 The Density of States in the Band Model.- 2.2.11 The Band Structure of Metals, Fermi Surfaces.- 2.2.12 The Band Structure of Semiconductors and Insulators.- 2.2.13 Consequences of the Invariance of the Hamiltonian to Symmetry Operations of the Space Group.- 2.2.14 Irreducible Representations of Space Groups.- 2.2.15 Spin, Time Reversal.- 2.2.16 Pseudopotentials.- 3. Elementary Excitations.- 3.1 The Interacting Electron Gas: Quasi-Electrons and Plasmons.- 3.1.1 Introduction.- 3.1.2 The Coulomb Interaction.- 3.1.3 The Hartree-Fock Approximation for the Electron Gas.- 3.1.4 Screening, Plasmons.- 3.1.5 Quasi-Electrons.- 3.1.6 The Dielectric Constant of the Electron Gas.- 3.2 Electron-Hole Interaction in Semiconductors and Insulators: Excitons.- 3.2.1 Introduction.- 3.2.2 The Ground State of the Insulator in Bloch and Wannier Representation.- 3.2.3 Excited States, the Exciton Representation.- 3.2.4 Wannier Excitons.- 3.2.5 Frenkel Excitons.- 3.2.6 Excitons as Elementary Excitations.- 3.3 Ion-Ion Interaction: Phonons.- 3.3.1 Introduction.- 3.3.2 The Classical Equations of Motion.- 3.3.3 Normal Coordinates, Phonons.- 3.3.4 The Energy Content of the Lattice Vibrations, Specific Heat.- 3.3.5 Calculation of Phonon Dispersion Relations.- 3.3.6 The Density of States.- 3.3.7 The Long Wavelength Limit: Acoustic Branch.- 3.3.8 The Long Wavelength Limit: Optical Branch.- 3.4 Spin-Spin Interaction: Magnons.- 3.4.1 Introduction.- 3.4.2 Spin Waves in Ferromagnets: Magnons.- 3.4.3 Spin Waves in Lattices with a Basis, Ferri-, and Antiferromagnetism.- 3.4.4 Ferromagnetism Near the Curie Temperature.- 3.4.5 Ordered Magnetism of Valence and Conduction Electrons, the Collective Electron Model.- 4. Electron-Phonon Interaction: Transport Phenomena.- 4.1 The Interaction Processes.- 4.1.1 Introduction.- 4.1.2 Interaction of Electrons with Acoustic Phonons.- 4.1.3 Electron-Phonon Interaction in Polar Solids, Polarons.- 4.2 The Boltzmann Equation.- 4.2.1 Introduction.- 4.2 Boltzmann Equations for the Electron and Phonon Systems.- 4.2.3 The Relaxation Time Approximation.- 4.2.4 The Variational Method.- 4.3 Formal Transport Theory.- 4.3.1 The Transport Equations.- 4.3.2 Transport Coefficients Without a Magnetic Field.- 4.3.3 Transport Coefficients with a Magnetic Field.- 4.4 Transport in Metals and Semiconductors.- 4.4.1 The Electrical Conductivity.- 4.4.2 Transport Coefficients in the Relaxation Time Approximation.- 4.4.3 Limits of Validity and Possible Extensions of the Approximations Used.- 5. Electron-Electron Interaction by Exchange of Virtual Phonons: Superconductivity.- 5.1 Introduction.- 5.2 Cooper Pairs.- 5.3 The Ground State of the Superconducting Electron Gas.- 5.4 Excited States.- 5.5 Comparison with Experiment.- 5.6 Thc Meissner-Ochsenfeld Effect.- 5.7 Further Theoretical Concepts.- 6. Interaction with Photons: Optics.- 6.1 Fundamentals.- 6.1.1 Introduction.- 6.1.2 Photons.- 6.1.3 Polaritons.- 6.1.4 The Complex Dielectric Constant.- 6.2 Electron-Photon Interaction.- 6.2.1 Introduction.- 6.2.2 Direct Transitions.- 6.2.3 Indirect Transitions.- 6.2.4 Two-Photon Absorption.- 6.2.5 Exciton Absorption.- 6.2.6 Comparison with Experimental Absorption and Reflection Spectra.- 6.2.7 Absorption by Free Charge Carriers.- 6.2.8 Absorption and Reflection in a Magnetic Field.- 6.2.9 Magneto-Optics of Free Charge Carriers.- 6.3 Phonon-Photon Interaction.- 6.3.1 Introduction.- 6.3.2 One-Phonon Absorption.- 6.3.3 Multi-Phonon Absorption.- 6.3.4 Raman and Brillouin Scattering.- 7. Phonon-Phonon Interaction: Thermal Properties.- 7.1 Introduction.- 7.2 Frequency Shift and Lifetime of Phonons.- 7.3 The Anharmonic Contributions to the Free Energy, Thermal Expansion.- 7.4 The Thermal Conductivity of the Lattice.- 8. Local Description of Solid-State Properties.- 8.1 Localized and Extended States.- 8.2 The Chemical Bond.- 8.2.1 Introduction.- 8.2.2 The Localized Single Bond.- 8.2.3 Localized and Delocalized Bonds.- 8.2.4 Solids with Localized Bonds: Insulators and Semiconductors.- 8.2.5 The Dielectric Theory of the Covalent Bond.- 8.2.6 Solids with Delocalized Bonds: Metals.- 8.3 Local Versus Nonlocal Description in Unperturbed Lattices.- 8.3.1 Introduction.- 8.3.2 Correlations, the Hubbard Model.- 8.3.3 Metal-Insulator Transitions.- 8.3.4 Limits of the Boltzmann Equation, the Kubo and Kubo-Greenwood Formulae.- 8.3.5 The Small Polaron.- 8.3.6 Hopping Conductivity in Polar Solids.- 9. Localized States.- 9.1 Point Imperfections.- 9.1.1 Introduction.- 9.1.2 Description Within the Framework of the Band Model.- 9.1.3 Crystal Field Theory.- 9.1.4 Localized Lattice Vibrations.- 9.1.5 Defect Statistics, Reaction Kinetics.- 9.1.6 Disorder Equilibria.- 9.1.7 Diffusion and Ionic Conduction.- 9.1.8 Recombination Processes at Imperfections.- 9.1.9 Optical Transitions at Imperfections, Configuration Coordinates.- 9.1.10 Electron-Phonon Interaction at Imperfections.- 9.1.11 Bound Excitons.- 9.1.12 Imperfections as Scattering Centres, the Kondo Effect.- 9.2 Localized States and Elementary Excitations at Surfaces.- 9.2.1 Introduction.- 9.2.2 Electronic Surface States.- 9.2.3 Surface-Phonons, -Polaritons, and -Plasmons.- 10. Disorder.- 10.1 Localized States in Disordered Lattices.- 10.1.1 Introduction.- 10.1.2 Localized States.- 10.1.3 Density of States.- 10.2 Transport in Disordered Lattices.- 10.2.1 Transport in Extended States.- 10.2 The Hopping Probability.- 10.2.3 Fixed Range and Variable Range Hopping.- 10.2.4 Conductivity in Impurity Bands and in Amorphous Semiconductors.- Appendix: The Occupation Number Representation.- Problems to Chapters 1-9.