| |

| |

| |

Metals: the Drude and Sommerfeld models | |

| |

| |

| |

Introduction | |

| |

| |

| |

What do we know about metals? | |

| |

| |

| |

The Drude model | |

| |

| |

| |

Assumptions | |

| |

| |

| |

The relaxation-time approximation | |

| |

| |

| |

The failure of the Drude model | |

| |

| |

| |

Electronic heat capacity | |

| |

| |

| |

Thermal conductivity and the Wiedemann-Franz ratio | |

| |

| |

| |

Hall effect | |

| |

| |

| |

Summary | |

| |

| |

| |

The Sommerfeld model | |

| |

| |

| |

The introduction of quantum mechanics | |

| |

| |

| |

The Fermi-Dirac distribution function | |

| |

| |

| |

The electronic density of states | |

| |

| |

| |

The electronic density of states at E [approximate] E[subscript F] | |

| |

| |

| |

The electronic heat capacity | |

| |

| |

| |

Successes and failures of the Sommerfeld model | |

| |

| |

| |

The quantum mechanics of particles in a periodic potential: Bloch's theorem | |

| |

| |

| |

Introduction and health warning | |

| |

| |

| |

Introducing the periodic potential | |

| |

| |

| |

Born-von Karman boundary conditions | |

| |

| |

| |

The Schrodinger equation in a periodic potential | |

| |

| |

| |

Bloch's theorem | |

| |

| |

| |

Electronic bandstructure | |

| |

| |

| |

The nearly-free electron model | |

| |

| |

| |

Introduction | |

| |

| |

| |

Vanishing potential | |

| |

| |

| |

Single electron energy state | |

| |

| |

| |

Several degenerate energy levels | |

| |

| |

| |

Two degenerate free-electron levels | |

| |

| |

| |

Consequences of the nearly-free-electron model | |

| |

| |

| |

The alkali metals | |

| |

| |

| |

Elements with even numbers of valence electrons | |

| |

| |

| |

More complex Fermi surface shapes | |

| |

| |

| |

The tight-binding model | |

| |

| |

| |

Introduction | |

| |

| |

| |

Band arising from a single electronic level | |

| |

| |

| |

Electronic wavefunctions | |

| |

| |

| |

Simple crystal structure | |

| |

| |

| |

The potential and Hamiltonian | |

| |

| |

| |

General points about the formation of tight-binding bands | |

| |

| |

| |

The group IA and IIA metals; the tight-binding model viewpoint | |

| |

| |

| |

The Group IV elements | |

| |

| |

| |

The transition metals | |

| |

| |

| |

Some general points about bandstructure | |

| |

| |

| |

Comparison of tight-binding and nearly-free-electron bandstructure | |

| |

| |

| |

The importance of k | |

| |

| |

| |

hk is not the momentum | |

| |

| |

| |

Group velocity | |

| |

| |

| |

The effective mass | |

| |

| |

| |

The effective mass and the density of states | |

| |

| |

| |

Summary of the properties of k | |

| |

| |

| |

Scattering in the Bloch approach | |

| |

| |

| |

Holes | |

| |

| |

| |

Postscript | |

| |

| |

| |

Semiconductors and Insulators | |

| |

| |

| |

Introduction | |

| |

| |

| |

Bandstructure of Si and Ge | |

| |

| |

| |

General points | |

| |

| |

| |

Heavy and light holes | |

| |

| |

| |

Optical absorption | |

| |

| |

| |

Constant energy surfaces in the conduction bands of Si and Ge | |

| |

| |

| |

Bandstructure of the direct-gap III-V and II-VI semiconductors | |

| |

| |

| |

Introduction | |

| |

| |

| |

General points | |

| |

| |

| |

Optical absorption and excitons | |

| |

| |

| |

Excitons | |

| |

| |

| |

Constant energy surfaces in direct-gap III-V semiconductors | |

| |

| |

| |

Thermal population of bands in semiconductors | |

| |

| |

| |

The law of mass action | |

| |

| |

| |

The motion of the chemical potential | |

| |

| |

| |

Intrinsic carrier density | |

| |

| |

| |

Impurities and extrinsic carriers | |

| |

| |

| |

Extrinsic carrier density | |

| |

| |

| |

Degenerate semiconductors | |

| |

| |

| |

Impurity bands | |

| |

| |

| |

Is it a semiconductor or an insulator? | |

| |

| |

| |

A note on photoconductivity | |

| |

| |

| |

Bandstructure engineering | |

| |

| |

| |

Introduction | |

| |

| |

| |

Semiconductor alloys | |

| |

| |

| |

Artificial structures | |

| |

| |

| |

Growth of semiconductor multilayers | |

| |

| |

| |

Substrate and buffer layer | |

| |

| |

| |

Quantum wells | |

| |

| |

| |

Optical properties of quantum wells | |

| |

| |

| |

Use of quantum wells in opto-electronics | |

| |

| |

| |

Superlattices | |

| |

| |

| |

Type I and type II superlattices | |

| |

| |

| |

Heterojunctions and modulation doping | |

| |

| |

| |

The envelope-function approximation | |

| |

| |

| |

Band engineering using organic molecules | |

| |

| |

| |

Introduction | |

| |

| |

| |

Molecular building blocks | |

| |

| |

| |

Typical Fermi surfaces | |

| |

| |

| |

A note on the effective dimensionality of Fermi-surface sections | |

| |

| |

| |

Layered conducting oxides | |

| |

| |

| |

The Peierls transition | |

| |

| |

| |

Measurement of bandstructure | |

| |

| |

| |

Introduction | |

| |

| |

| |

Lorentz force and orbits | |

| |

| |

| |

General considerations | |

| |

| |

| |

The cyclotron frequency | |

| |

| |

| |

Orbits on a Fermi surface | |

| |

| |

| |

The introduction of quantum mechanics | |

| |

| |

| |

Landau levels | |

| |

| |

| |

Application of Bohr's correspondence principle to arbitrarily-shaped Fermi surfaces in a magnetic field | |

| |

| |

| |

Quantisation of the orbit area | |

| |

| |

| |

The electronic density of states in a magnetic field | |

| |

| |

| |

Quantum oscillatory phenomena | |

| |

| |

| |

Types of quantum oscillation | |

| |

| |

| |

The de Haas-van Alphen effect | |

| |

| |

| |

Other parameters which can be deduced from quantum oscillations | |

| |

| |

| |

Magnetic breakdown | |

| |

| |

| |

Cyclotron resonance | |

| |

| |

| |

Cyclotron resonance in metals | |

| |

| |

| |

Cyclotron resonance in semiconductors | |

| |

| |

| |

Interband magneto-optics in semiconductors | |

| |

| |

| |

Other techniques | |

| |

| |

| |

Angle-resolved photoelectron spectroscopy (ARPES) | |

| |

| |

| |

Electroreflectance spectroscopy | |

| |

| |

| |

Some case studies | |

| |

| |

| |

Copper | |

| |

| |

| |

Recent controversy: Sr[subscript 2]RuO[subscript 4] | |

| |

| |

| |

Studies of the Fermi surface of an organic molecular metal | |

| |

| |

| |

Quasiparticles: interactions between electrons | |

| |

| |

| |

Transport of heat and electricity in metals and semiconductors | |

| |

| |

| |

A brief digression; life without scattering would be difficult! | |

| |

| |

| |

Thermal and electrical conductivity of metals | |

| |

| |

| |

Metals: the 'Kinetic theory' of electron transport | |

| |

| |

| |

What do [tau subscript [sigma] and [tau subscript [kappa] represent? | |

| |

| |

| |

Matthiessen's rule | |

| |

| |

| |

Emission and absorption of phonons | |

| |

| |

| |

What is the characteristic energy of the phonons involved? | |

| |

| |

| |

Electron-phonon scattering at room temperature | |

| |

| |

| |

Electron-phonon scattering at T [double less-than sign] [theta subscript D] | |

| |

| |

| |

Departures from the low temperature [sigma] [proportional to] T[superscript -5] dependence | |

| |

| |

| |

Very low temperatures and/or very dirty metals | |

| |

| |

| |

Summary | |

| |

| |

| |

Electron-electron scattering | |

| |

| |

| |

Electrical conductivity of semiconductors | |

| |

| |

| |

Temperature dependence of the carrier densities | |

| |

| |

| |

The temperature dependence of the mobility | |

| |

| |

| |

Disordered systems and hopping conduction | |

| |

| |

| |

Thermally-activated hopping | |

| |

| |

| |

Variable range hopping | |

| |

| |

| |

Magnetoresistance in three-dimensional systems | |

| |

| |

| |

Introduction | |

| |

| |

| |

Hall effect with more than one type of carrier | |

| |

| |

| |

General considerations | |

| |

| |

| |

Hall effect in the presence of electrons and holes | |

| |

| |

| |

A clue about the origins of magnetoresistance | |

| |

| |

| |

Magnetoresistance in metals | |

| |

| |

| |

The absence of magnetoresistance in the Sommerfeld model of metals | |

| |

| |

| |

The presence of magnetoresistance in real metals | |

| |

| |

| |

The use of magnetoresistance in finding the Fermi-surface shape | |

| |

| |

| |

The magnetophonon effect | |

| |

| |

| |

Magnetoresistance in two-dimensional systems and the quantum Hall effect | |

| |

| |

| |

Introduction: two-dimensional systems | |

| |

| |

| |

Two-dimensional Landau-level density of states | |

| |

| |

| |

Resistivity and conductivity tensors for a two-dimensional system | |

| |

| |

| |

Quantisation of the Hall resistivity | |

| |

| |

| |

Localised and extended states | |

| |

| |

| |

A further refinement- spin splitting | |

| |

| |

| |

Summary | |

| |

| |

| |

The fractional quantum Hall effect | |

| |

| |

| |

More than one subband populated | |

| |

| |

| |

Inhomogeneous and hot carrier distributions in semiconductors | |

| |

| |

| |

Introduction: inhomogeneous carrier distributions | |

| |

| |

| |

The excitation of minority carriers | |

| |

| |

| |

Recombination | |

| |

| |

| |

Diffusion and recombination | |

| |

| |

| |

Drift, diffusion and the Einstein equations | |

| |

| |

| |

Characterisation of minority carriers; the Shockley-Haynes experiment | |

| |

| |

| |

Hot carrier effects and ballistic transport | |

| |

| |

| |

Drift velocity saturation and the Gunn effect | |

| |

| |

| |

Avalanching | |

| |

| |

| |

A simple resonant tunnelling structure | |

| |

| |

| |

Ballistic transport and the quantum point contact | |

| |

| |

| |

Useful terminology in condensed matter physics | |

| |

| |

| |

Introduction | |

| |

| |

| |

Crystal | |

| |

| |

| |

Lattice | |

| |

| |

| |

Basis | |

| |

| |

| |

Physical properties of crystals | |

| |

| |

| |

Unit cell | |

| |

| |

| |

Wigner-Seitz cell | |

| |

| |

| |

Designation of directions | |

| |

| |

| |

Designation of planes; Miller indices | |

| |

| |

| |

Conventional or primitive? | |

| |

| |

| |

The 14 Bravais lattices | |

| |

| |

| |

Derivation of density of states in k-space | |

| |

| |

| |

Introduction | |

| |

| |

| |

Density of states | |

| |

| |

| |

Reading | |

| |

| |

| |

Derivation of distribution functions | |

| |

| |

| |

Introduction | |

| |

| |

| |

Bosons | |

| |

| |

| |

Fermions | |

| |

| |

| |

The Maxwell-Boltzmann distribution function | |

| |

| |

| |

Mean energy and heat capacity of the classical gas | |

| |

| |

| |

Phonons | |

| |

| |

| |

Introduction | |

| |

| |

| |

A simple model | |

| |

| |

| |

Extension to three dimensions | |

| |

| |

| |

The Debye model | |

| |

| |

| |

Phonon number | |

| |

| |

| |

Summary; the Debye temperature as a useful energy scale in solids | |

| |

| |

| |

A note on the effect of dimensionality | |

| |

| |

| |

The Bohr model of hydrogen | |

| |

| |

| |

Introduction | |

| |

| |

| |

Hydrogenic impurities | |

| |

| |

| |

Excitons | |

| |

| |

| |

Experimental considerations in measuring resistivity and Hall effect | |

| |

| |

| |

Introduction | |

| |

| |

| |

The four-wire method | |

| |

| |

| |

Sample geometries | |

| |

| |

| |

The van der Pauw method | |

| |

| |

| |

Mobility spectrum analysis | |

| |

| |

| |

The resistivity of layered samples | |

| |

| |

| |

Canonical momentum | |

| |

| |

| |

Superconductivity | |

| |

| |

| |

Introduction | |

| |

| |

| |

Pairing | |

| |

| |

| |

Pairing and the Meissner effect | |

| |

| |

| |

List of selected symbols | |

| |

| |

| |

Solutions and additional hints for selected exercises | |

| |

| |

Index | |