1 Metals: the Drude and Sommerfeld models 1.1 Introduction 1.2 What do we know about metals? 1.3 The Drude model 1.3.1 Assumptions 1.3.2 The relaxation-time approximation 1.4 The failure of the Drude model 1.4.1 Electronic heat capacity 1.4.2 Thermal conductivity and the Wiedemann-Franz ratio 1.4.3 Hall effect 1.4.4 Summary 1.5 The Sommerfeld model 1.5.1 The introduction of quantum mechanics 1.5.2 The Fermi-Dirac distribution function 1.5.3 The electronic density of states 1.5.4 The electronic density Of states at E≈EF 1.5.5 The electronic heat capacity 1.6 Successes and failures of the Sommerfleld model 2 The quantum mechanics of particles in a periodic potential: Bloch’s theorem 2.1 Introduction and health warning 2.2 Introducing the periodic potential 2.3 Born-von Karman boundary conditions 2.4 The Schrödinger equation in a periodic potential 2.5 Bloch's theorem 2.6 Electronic bandstructure 3 The nearly-free electron model 3.1 Introduction 3.2 Vanishing potential 3.2.1 Single electron energy state 3.2.2 Several degenerate energy levels 3.2.3 Two degenerate free-electron levels 3.3 Consequences of the nearly-free-electron model 3.3.1 The alkali metals 3.3.2 Elements with even numbers of valence electrons 3.3.3 More complex Fermi surface shapes 4 The tight-binding model 4.1 Introduction 4.2 Band arising from a single electronic level 4.2.1 Electronic wavefunctions 4.2.2 Simple crystal structure. 4.2.3 The potential and Hamiltonian 4.3 General points about the formation of tight-binding bands 4.3.1 The group ⅠA and ⅡA metals; the tight-binding model viewpoint 4.3.2 The Group Ⅳ elements 4.3.3 The transition metals 5 Some general points about bandstructure 5.1 Comparison of tight-binding and nearly-free-electron bandstructure 5.2 The importance of k 5.2.1 ħk is not the momentum 5.2.2 Group velocity 5.2.3 The effective mass 5.2.4 The effective mass and the density of states 5.2.5 Summary of the properties of k 5.2.6 Scattering in the Bloch approach 5.3 Holes 5.4 Postscript 6 Semiconductors and Insulators 6.1 Introduction 6.2 Bandstructure of Si and Ge 6.2.1 General points 6.2.2 Heavy and light holes 6.2.3 Optical absorption 6.2.4 Constant energy surfaces in the Conduction bands of Si and Ge 6.3 Bandstructure of the direct-gap Ⅲ-Ⅴ and Ⅱ-Ⅵ semiconductors 6.3.1 Introduction 6.3.2 General points 6.3.3 Optical absorption and excitons 6.3.4 Excitons 6.3.5 Constant energy surfaces in direct-gap Ⅲ-Ⅴ semiconductors 6.4 Thermal population of bands in semiconductors 6.4.1 The law of mass action 6.4.2 The motion of the chemical potential 6.4.3 Intrinsic carrier density 6.4.4 Impurities and extrinsic carriers 6.4.5 Extrinsic carrier density 6.4.6 Degenerate semiconductors 6.4.7 Impurity bands 6.4.8 Is it a semiconductor or an insulator? 6.4.9 A note on photoconductivity 7 Bandstructure engineering 7.1 Introduction 7.2 Semiconductor alloys 7.3 Artificial structures 7.3.1 Growth of semiconductor multilayers 7.3.2 Substrate and buffer layer 7.3.3 Quantum wells 7.3.4 Optical properties of quantum wells 7.3.5 Use of quantum wells in opto-electronics 7.3.6 Superlattices 7.3.7 Type Ⅰand type Ⅱ superlattices 7.3.8 Heterojunctions and modulation doping 7.3.9 The envelope-function approximation 7.4 Band engineering using organic molecules 7.4.1 Introduction 7.4.2 Molecular building blocks 7.4.3 Typical Fermi surfaces 7.4.4 A note on the effective dimensionality of Fermi-surfacesections 7.5 Layered conducting oxides 7.6 The Peierls transition 8 Measurement of bandstructure 8.1 Introduction 8.2 Lorentz force and orbits 8.2.1 General considerations 8.2.2 The cyclotron frequency 8.2.3 Orbits on a Fermi surface 8.3 The introduction of quantum mechanics 8.3.1 Landau levels 8.3.2 Application of Bohr's correspondence principle to arbitrarily-shaped Fermi surfaces in a magnetic field 8.3.3 Quantisation of the orbit area 8.3.4 The electronic density of states in a magnetic field 8.4 Quantum oscillatory phenomena 8.4.1 Types of quantum oscillation 8.4.2 The de Haas-van Alphen effect 8.4.3 Other parameters which can be deduced from quantum oscillations 8.4.4 Magnetic breakdown 8.5 Cyclotron resonance 8.5.1 Cyclotron resonance in metals 8.5.2 Cyclotron resonance in semiconductors 8.6 Interband magneto-optics in semiconductors 8.7 Other techniques 8.7.1 Angle-resolved photoelectron spectroscopy (ARPES) 8.7.2 Electroreflectance spectroscopy 8.8 Some case studies 8.8.1 Copper 8.8.2 Recent Controversy: Sr2RuO4 8.8.3 Studies of the Fermi surface of an organic molecular metal 8.9 Quasiparticles: interactions between electrons 9 Transport of heat and electricity in metals and semiconductors 9.1 A brief digression; life without scattering would be difficult! 9.2 Thermal and electrical conductivity of metals 9.2.1 Metals: the‘Kinetic theory’of electron transport 9.2.2 What do τσ and τκ represent? 9.2.3 Matthiessen’s rule 9.2.4 Emission and absorption of phonons 9.2.5 What is the characteristic energy of the phonons involved? 9.2.6 Electron-phonon scattering at room temperature 9.2.7 Electron-phonon scattering at T«θD 9.2.8 Departures from the low temperature σ∝T-5 dependence 9.2.9 Very low temperatures and／or very dirty metals 9.2.10 Summary 9.2.11 Electron-electron scattering 9.3 Electrical Conductivity of semiconductors 9.3.1 Temperature dependence of the carrier densities 9.3.2 The temperature dependence of the mobility 9.4 Disordered systems and hopping conduction 9.4.1 Thermally-activated hopping 9.4.2 Variable range hopping 10 Magnetoresistance in three-dimensional systems 10.1 Introduction 10.2 Hall effect with more than one type of carrier 10.2.1 General considerations 10.2.2 Hall effect in the presence of electrons and holes 10.2.3 A clue about the origins of magnetoresistance 10.3 Magnetoresistance in metals 10.3.1 The absence of magnetoresistance in the Sommerfeld model of metals 10.3.2 The presence of magnetoresistance in real metals 10.3.3 The use of magnetoresistance in finding the Fermi-surface shape 10.4 The magnetophonon effect 11 Magnetoresistance in two-dimensional systems and the quantum Hall effect 11.1 Introduction: two-dimensional systems 11.2 Two-dimensional Landau-level density of states 11.2.1 Resistivity and conductivity tensors for a two-dimensional system 11.3 Quantisation of the Hall resistivity 11.3.1 Localised and extended states 11.3.2 A further refinement-spin splitting 11.4 Summary 11.5 The fractional quantum Hall effect 11.6 More than one subband populated 12 Inhomogeneous and hot carrier distributions in semiconductors 12.1 Introduction: inhomogeneous carrier distributions 12.1.1 The excitation of minority carriers 12.1.2 Recombination 12.1.3 Diffusion and recombination 12.2 Drift, diffusion and the Einstein equations 12.2.1 Characterisation of minority carriers; the Shockley-Haynes experiment 12.3 Hot carrier effects and ballistic transport 12.3.1 Drift velocity saturation and the Gunn effect 12.3.2 Avalanching 12.3.3 A simple resonant tunnelling structure 12.3.4 Ballistic transport and the quantum point contact A Useful terminology in condensed matter physics A.1 Introduction A.2 Crystal A.3 Lattice A.4 Basis A.5 Physical properties of crystals A.6 Unit cell A.7 Wigner-Seitz cell A.8 Designation of directions A.9 Designation of planes; Miller indices A.10 Conventional or primitive ? A.11 The 14 Bravais lattices B Derivation of density of states in k-space B.1 Introduction B.1.1 Density of states B.1.2 Reading C Derivation of distribution functions C.1 Introduction C.1.1 Bosons C.1.2 Fermions C.1.3 The Maxwell-Boltzmann distribution function C.1.4 Mean energy and heat capacity of the classical gas D Phonons D.1 Introduction D.2 A simple model D.2.1 Extension to three dimensions D.3 The Debye model D.3.1 Phonon number D.3.2 Summary; the Debye temperature as a useful energy scale in Solids D.3.3 A note on the effect of dimensionality E The Bohr model of hydrogen E.1 Introduction E.2 Hydrogenic impurities E.3 Excitons F Experimental considerations in measuring resistivity and Hall effect F.1 Introduction F.2 The four-wire method F.3 Sample geometries F.4 The van der Pauw method. F.5 Mobility spectrum analysis F.6 The resistivity of layered samples G Canonical momentum H Superconductivity H.1 Introduction H.2 Pairing H.3 Pairing and the Meissner effect I List of selected symbols J Solutions and additional hints for selected exercises Index
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