introduction to solid state physics - sfu to solid state physics - sfu

Introduction to Solid State Physics

David M. Broun

Department of PhysicsSimon Fraser University

Quantum Materials ProgramCanadian Institute for Advanced Research

ii

Introduction to Solid State Physics D. M. Broun SFU/CIFAR QM

Contents

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

1 Introduction 1

1.1 The free Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Sommerfeld free electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 The classical to quantum crossover . . . . . . . . . . . . . . . . . . . . . . . 5

2 Thermodynamics and statistical mechanics 7

2.1 Review of thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Maxwell relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Review of statistical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Statistical mechanics of ideal quantum gases . . . . . . . . . . . . . . . . . . 11

2.4.1 Example: statistics of an impurity in a semiconductor . . . . . . . . . 13

2.4.2 Example: interacting electrons in a metal . . . . . . . . . . . . . . . . 15

3 Thermal properties of the Fermi gas 17

3.1 Specific heat of an electron gas . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 The Sommerfeld expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Specific heat of the ideal Fermi gas . . . . . . . . . . . . . . . . . . . . . . . 21

4 Review of single-particle quantum mechanics 25

4.1 Single-particle quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . 25

4.2 The harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Many-particle quantum systems 33

5.1 Quantum mechanics of many-particle systems . . . . . . . . . . . . . . . . . 33

5.2 Second quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.2.1 Occupation number representation . . . . . . . . . . . . . . . . . . . 35

5.2.2 Representation of states in second quantization . . . . . . . . . . . . 37

iiiIntroduction to Solid State Physics D. M. Broun SFU/CIFAR QM

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5.2.3 Representation of operators in second quantization . . . . . . . . . . 38

5.2.4 Field operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2.5 Examples of second quantized operators . . . . . . . . . . . . . . . . 40

6 Applications of second quantization 41

6.1 The tight-binding model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.2 The jellium model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

7 The Hartree–Fock approximation 47

7.1 A model two-electron system. . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7.2 The Hartree–Fock approximation . . . . . . . . . . . . . . . . . . . . . . . . 48

7.3 Hartree–Fock theory for jellium . . . . . . . . . . . . . . . . . . . . . . . . . 51

8 Screening and the random phase approximation 55

8.1 Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

8.2 Thomas–Fermi theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8.3 The density operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

8.4 The random phase approximation . . . . . . . . . . . . . . . . . . . . . . . . 59

8.5 Collective excitations of the electron gas . . . . . . . . . . . . . . . . . . . . 61

9 Scattering and periodic structures 63

9.1 The Born approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

9.2 Periodic structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

9.2.1 Convolution as a means of replication . . . . . . . . . . . . . . . . . . 65

9.2.2 The convolution theorem . . . . . . . . . . . . . . . . . . . . . . . . . 66

9.3 The reciprocal lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

10 The nearly free electron model 69

10.1 Bloch’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

10.2 Free electron in 1–D as a Bloch wave . . . . . . . . . . . . . . . . . . . . . . 71

10.3 Nearly free electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

10.3.1 Properties of UG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

10.4 The Schrodinger equation in momentum space . . . . . . . . . . . . . . . . . 74

10.5 The periodic potential as a weak perturbation . . . . . . . . . . . . . . . . . 75

10.5.1 The nondegenerate case . . . . . . . . . . . . . . . . . . . . . . . . . 75

11 The nearly free electron model and tight binding 77

11.1 Degenerate electrons in a periodic potential . . . . . . . . . . . . . . . . . . 77

11.2 The tight binding method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

11.2.1 Introduction to tight binding . . . . . . . . . . . . . . . . . . . . . . 82


CONTENTS v

12 The tight-binding approximation 87

12.1 Degenerate tight-binding theory . . . . . . . . . . . . . . . . . . . . . . . . . 88

12.1.1 Example 1: s-like band in a face-centred cubic crystal . . . . . . . . . 91

12.1.2 Example 2: p-like bands in a face-centred cubic crystal . . . . . . . . 92

13 Energy bands and electronic structure 97

13.1 Review of the nearly free electron model . . . . . . . . . . . . . . . . . . . . 97

13.2 Free electron Fermi surfaces in 2D . . . . . . . . . . . . . . . . . . . . . . . . 100

13.3 Electrons and holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

13.4 Metals, semimetals, insulators and semiconductors . . . . . . . . . . . . . . . 102

14 Electronic structure calculations 105

14.1 Orthogonalized plane waves and pseudopotentials . . . . . . . . . . . . . . . 105

14.1.1 Example: the need for pseudopotentials . . . . . . . . . . . . . . . . . 105

14.2 Orthogonalized plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

14.3 Relativistic effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

15 Density functional theory 111

15.1 The Hohenberg–Kohn theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 111

15.1.1 Proof of the Hohenberg–Kohn theorem . . . . . . . . . . . . . . . . . 112

15.1.2 Proof of the second Hohenberg–Kohn theorem . . . . . . . . . . . . . 113

15.2 Application of the Hohenberg–Kohn theorem . . . . . . . . . . . . . . . . . . 113

15.3 The Kohn–Sham equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

15.4 The local density approximation . . . . . . . . . . . . . . . . . . . . . . . . . 115

15.5 Thomas–Fermi as a density functional theory . . . . . . . . . . . . . . . . . 115

15.6 What can be calculated with density functional theory? . . . . . . . . . . . . 116

16 The dynamics of Bloch electrons I 119

16.1 Energy bands and group velocity . . . . . . . . . . . . . . . . . . . . . . . . 119

16.2 Rules of the semiclassical model . . . . . . . . . . . . . . . . . . . . . . . . . 121

16.3 The k·P method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

16.3.1 First order terms: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

16.3.2 Second order terms: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

16.4 Consequences of the semiclassical model . . . . . . . . . . . . . . . . . . . . 125

16.4.1 Electrical current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

16.4.2 Thermal current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

16.4.3 Filled bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

16.4.4 Electrons and holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

16.5 Semiclassical motion in a uniform dc electric field . . . . . . . . . . . . . . . 126


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17 The dynamics of Bloch electrons II 129

17.1 Semiclassical motion in a uniform magnetic field . . . . . . . . . . . . . . . . 129

17.1.1 The cyclotron frequency . . . . . . . . . . . . . . . . . . . . . . . . . 131

17.2 Limits of validity of the semiclassical model . . . . . . . . . . . . . . . . . . 133

17.3 Magnetic breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

18 Quantum oscillatory phenomena 139

18.1 Quantum mechanics of the orbital motion . . . . . . . . . . . . . . . . . . . 139

18.2 Degeneracy of the Landau levels . . . . . . . . . . . . . . . . . . . . . . . . . 140

18.3 Landau levels in a periodic potential . . . . . . . . . . . . . . . . . . . . . . 141

18.4 Visualizing Landau quantization . . . . . . . . . . . . . . . . . . . . . . . . . 143

18.5 Quantum oscillations as a Fermi surface probe . . . . . . . . . . . . . . . . . 144

19 Electronic structure of selected metals 149

19.1 Construction of free-electron Fermi surfaces . . . . . . . . . . . . . . . . . . 149

19.1.1 Free-electron Fermi surfaces in two dimensions . . . . . . . . . . . . . 149

19.1.2 Free-electron Fermi surfaces in three dimensions . . . . . . . . . . . . 150

19.2 The alkali metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

19.3 The noble metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

19.4 Divalent metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

19.5 The trivalent and tetravalent metals . . . . . . . . . . . . . . . . . . . . . . . 160

19.6 Transition metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

20 Strongly correlated systems & semiconductors 163

20.1 Interactions and the Hubbard U . . . . . . . . . . . . . . . . . . . . . . . . . 163

20.2 Measuring the Hubbard U . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

20.3 3d transition metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

20.4 Semiconductor structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

20.5 Semiconductor chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

20.6 Bonding, nonbonding and antibonding states . . . . . . . . . . . . . . . . . . 169

20.7 Electronic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

20.7.1 Spin–orbit coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

20.8 Other semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172


CONTENTS vii

21 Semiconductors 175

21.1 Homogeneous semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . 175

21.1.1 Carrier density in thermal equilibrium . . . . . . . . . . . . . . . . . 175

21.1.2 The nondegenerate case . . . . . . . . . . . . . . . . . . . . . . . . . 176

21.1.3 The intrinsic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

21.1.4 The extrinsic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

21.1.5 Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

21.1.6 Population of impurity levels in thermal equilibrium . . . . . . . . . . 179

21.1.7 Thermal equilibrium carrier density . . . . . . . . . . . . . . . . . . . 180

21.1.8 Transport in nondegenerate semiconductors . . . . . . . . . . . . . . 180

21.2 Inhomogeneous semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . 181

21.2.1 The p-n junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

21.3 The p-n junction as a rectifier . . . . . . . . . . . . . . . . . . . . . . . . . . 184

22 The Boltzmann transport equation 187

22.1 The distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

22.2 The continuity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

22.3 The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

22.4 The linearized Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . 190

22.5 The relaxation time approximation . . . . . . . . . . . . . . . . . . . . . . . 191

22.6 Transport properties of metals . . . . . . . . . . . . . . . . . . . . . . . . . . 192

22.6.1 Electrical conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . 192

22.6.2 Thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

22.7 The Wiedemann–Franz law . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

23 Time-dependent perturbation theory 195

23.1 Time-dependent perturbation theory . . . . . . . . . . . . . . . . . . . . . . 195

23.2 Sudden perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

23.3 Adiabatic perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

23.4 Periodic perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

23.5 Fermi’s golden rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

23.6 The Kubo formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

24 Density response of the electron gas 203

24.1 Time and space dependent perturbations . . . . . . . . . . . . . . . . . . . . 203

24.2 Density response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

24.3 Energy loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

24.4 Screening and the dielectric function . . . . . . . . . . . . . . . . . . . . . . 207

24.5 Properties of the RPA dielectric function . . . . . . . . . . . . . . . . . . . . 208


viii CONTENTS

25 Electrons in one dimension 211

25.1 One dimensional conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

25.2 The Peierls instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

25.2.1 Static lattice distortions . . . . . . . . . . . . . . . . . . . . . . . . . 212

25.2.2 The energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

25.3 Kohn anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

25.4 Nesting of the Fermi surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

25.5 Spin–charge separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

25.6 The Luttinger–Tomonaga model . . . . . . . . . . . . . . . . . . . . . . . . . 217

26 Collective modes and response functions 221

26.1 Response functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

26.2 Collective modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

26.3 Inelastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

26.4 Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

26.5 Optical conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

26.6 Oscillator strength sum rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

27 The electron spectral function 229

27.1 The Schrodinger representation . . . . . . . . . . . . . . . . . . . . . . . . . 229

27.2 The Heisenberg representation . . . . . . . . . . . . . . . . . . . . . . . . . . 229

27.3 Particles and quasiparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

27.4 A single free particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

27.5 The spectral function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

27.6 Interacting systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

27.7 Angle resolved photoemission spectroscopy . . . . . . . . . . . . . . . . . . . 235

28 Landau’s Fermi Liquid theory 239

28.1 The noninteracting Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . 239

28.2 Excited states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

28.2.1 Heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

28.2.2 Pauli spin susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . 242

28.3 Landau quasiparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

28.4 Quasiparticle decay rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

28.5 Landau’s Fermi liquid theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

28.5.1 Interactions between quasiparticles . . . . . . . . . . . . . . . . . . . 246

28.6 Experimental consequences of Fermi liquid theory . . . . . . . . . . . . . . . 247

28.6.1 Heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

28.6.2 Spin susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248


CONTENTS ix

29 Superconductivity I — phenomenology 251

29.1 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

29.2 Perfect conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

29.3 The Meissner–Ochsenfeld effect . . . . . . . . . . . . . . . . . . . . . . . . . 252

29.4 The London theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

29.5 Flux trapping and quantization . . . . . . . . . . . . . . . . . . . . . . . . . 255

29.6 The Josephson effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

30 Superconductivity II — pairing theory 259

30.1 The Cooper problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

30.2 The origin of the attractive interaction . . . . . . . . . . . . . . . . . . . . . 261

30.3 Bardeen–Cooper–Schrieffer theory . . . . . . . . . . . . . . . . . . . . . . . . 263

30.4 The Bogoliubov transformation . . . . . . . . . . . . . . . . . . . . . . . . . 264

31 Superconductivity III — exotic pairing 267

31.1 Conventional superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . 267

31.2 Pairing glue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

31.3 Anderson’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

31.4 Unconventional pairing in specific materials . . . . . . . . . . . . . . . . . . 270

31.4.1 Cuprate superconductivity . . . . . . . . . . . . . . . . . . . . . . . . 270

31.4.2 Heavy fermion superconductivity . . . . . . . . . . . . . . . . . . . . 274


x CONTENTS


List of Figures

1.1 Allowed momentum states in two dimensions . . . . . . . . . . . . . . . . . . 3

1.2 The Fermi sea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Evaporative cooling of trapped atoms . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Gibbs’ energy surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Dopant energy levels in a semiconductor . . . . . . . . . . . . . . . . . . . . 14

2.3 Statistics of the dopant states with and without interactions . . . . . . . . . 15

2.4 Quasiparticle distribution functions . . . . . . . . . . . . . . . . . . . . . . . 16

3.1 Density of states of the electron gas . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 The Fermi function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.1 Energy eigenstates of the harmonic oscillator. . . . . . . . . . . . . . . . . . 28

6.1 Tight binding on a square lattice . . . . . . . . . . . . . . . . . . . . . . . . 41

7.1 Hartree–Fock self consistency loop . . . . . . . . . . . . . . . . . . . . . . . . 50

7.2 Hartree–Fock function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

7.3 Energy dispersion in the Hartree–Fock model . . . . . . . . . . . . . . . . . . 53

8.1 The response of the electron gas to an external charge. . . . . . . . . . . . . 57

8.2 Excitations of the electron gas . . . . . . . . . . . . . . . . . . . . . . . . . . 61

10.1 Bragg scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

10.2 The extended zone scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

10.3 The reduced zone scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

11.1 Degenerate energy bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

11.2 Band gaps in the energy dispersion . . . . . . . . . . . . . . . . . . . . . . . 78

11.3 States at the zone boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

11.4 Construction of energy bands in one dimension . . . . . . . . . . . . . . . . . 80

11.5 Free electron bands of the FCC structure . . . . . . . . . . . . . . . . . . . . 81

xiIntroduction to Solid State Physics D. M. Broun SFU/CIFAR QM

xii LIST OF FIGURES

11.6 Fermi surface construction in two dimensions . . . . . . . . . . . . . . . . . . 82

11.7 A model of a one-dimensional metal. . . . . . . . . . . . . . . . . . . . . . . 83

11.8 Orthogonalization of atomic orbitals . . . . . . . . . . . . . . . . . . . . . . 84

12.1 s-like band in a FCC crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

12.2 p-like bands in a FCC crystal . . . . . . . . . . . . . . . . . . . . . . . . . . 93

13.1 Bragg planes and reciprocal lattice vectors . . . . . . . . . . . . . . . . . . . 98

13.2 Energy dispersion in one and two dimensions . . . . . . . . . . . . . . . . . . 98

13.3 Brillouin zones of the square lattice . . . . . . . . . . . . . . . . . . . . . . . 99

13.4 Fermi surface of a tetravalent square lattice . . . . . . . . . . . . . . . . . . 100

13.5 Second and third zone Fermi surface pockets of the tetravalent square lattice 101

13.6 The Harrison construction in three dimensions. . . . . . . . . . . . . . . . . 102

13.7 Schematic views of the density of states . . . . . . . . . . . . . . . . . . . . . 103

13.8 Broadening of atomic levels into energy bands . . . . . . . . . . . . . . . . . 104

14.1 Pseudopotentials and pseudowavefunctions . . . . . . . . . . . . . . . . . . . 107

14.2 Pseudopotentials and pseudowavefunctions for silver . . . . . . . . . . . . . . 108

16.1 The hierarchy of length scales in a Bloch wavepacket. . . . . . . . . . . . . . 120

16.2 Displacement of the Fermi surface in k-space . . . . . . . . . . . . . . . . . . 127

16.3 Bloch oscillations of the electron velocity . . . . . . . . . . . . . . . . . . . . 128

17.1 Cyclotron orbits in a magnetic field . . . . . . . . . . . . . . . . . . . . . . . 130

17.2 The relation between real and k-space orbits . . . . . . . . . . . . . . . . . . 131

17.3 Construction for calculating the cyclotron period . . . . . . . . . . . . . . . 132

17.4 The energy dispersion near a Bragg plane . . . . . . . . . . . . . . . . . . . . 135

17.5 The conditions for magnetic breakdown . . . . . . . . . . . . . . . . . . . . . 136

17.6 A Fermi surface undergoing magnetic breakdown . . . . . . . . . . . . . . . 137

18.1 Quantum states in a strong magnetic field . . . . . . . . . . . . . . . . . . . 143

18.2 Oscillations of the thermodynamic potential . . . . . . . . . . . . . . . . . . 146

18.3 Periodicity of quantum oscillations in inverse field . . . . . . . . . . . . . . . 146

18.4 De Haas–van Alphen oscillations in SrRu2O4 . . . . . . . . . . . . . . . . . . 147

18.5 The Fermi surface of SrRu2O4 . . . . . . . . . . . . . . . . . . . . . . . . . . 147

19.1 Bragg planes and Brillouin zones in two dimensions . . . . . . . . . . . . . . 150

19.2 Constructing the two dimensional Fermi surface . . . . . . . . . . . . . . . . 151

19.3 Free-electron Fermi spheres for the FCC structure . . . . . . . . . . . . . . . 152

19.4 Free-electron Fermi surfaces for FCC structures of different valence . . . . . 152

19.5 Free-electron Fermi surfaces for BCC structure . . . . . . . . . . . . . . . . . 153


LIST OF FIGURES xiii

19.6 Free-electron Fermi surfaces for HCP structure . . . . . . . . . . . . . . . . . 153

19.7 The periodic table showing crystal structures at room temperature I . . . . . 154

19.8 The periodic table showing crystal structures at room temperature II . . . . 155

19.9 Band structure of sodium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

19.10First Brillouin zone of the BCC structure . . . . . . . . . . . . . . . . . . . . 157

19.11The Fermi surface of copper . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

19.12Band structure of copper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

19.13The Fermi surface of beryllium . . . . . . . . . . . . . . . . . . . . . . . . . 159

19.14Band structure of calcium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

19.15The Fermi surface of aluminium . . . . . . . . . . . . . . . . . . . . . . . . . 160

19.16Free electron Fermi surfaces for the FCC structure . . . . . . . . . . . . . . . 161

20.1 Density of states in a Mott insulator . . . . . . . . . . . . . . . . . . . . . . 164

20.2 Ionization processes and the Hubbard U . . . . . . . . . . . . . . . . . . . . 164

20.3 The diamond structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

20.4 sp3 hybridization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

20.5 Bonding and antibonding orbitals . . . . . . . . . . . . . . . . . . . . . . . . 169

20.6 The band structure of silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

20.7 Constant energy surfaces in germanium and silicon . . . . . . . . . . . . . . 171

20.8 Spin–orbit interaction and semiconductor band structure . . . . . . . . . . . 172

20.9 The band structure of germanium . . . . . . . . . . . . . . . . . . . . . . . . 173

20.10The band structure of GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

21.1 The density of states of a doped semiconductor. . . . . . . . . . . . . . . . . 176

21.2 Two views of the spatially varying electronic structure in a p-n junction. . . 182

21.3 Carrier density, charge density and potential in a p-n junction . . . . . . . . 183

21.4 The current–voltage characteristic of a p-n junction, showing diode action. . 185

22.1 Evolution of electron states in the semiclassical phase space . . . . . . . . . . 188

22.2 Electron distributions for electrical and thermal currents . . . . . . . . . . . 191

22.3 The Weidemann–Franz law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

24.1 Static susceptibility in one, two and three dimensions . . . . . . . . . . . . . 209

24.2 Friedel oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

25.1 Free electrons in one dimension . . . . . . . . . . . . . . . . . . . . . . . . . 212

25.2 Dimerization in a one-dimensional solid . . . . . . . . . . . . . . . . . . . . . 214

25.3 Phonon softening and the Kohn anomaly . . . . . . . . . . . . . . . . . . . . 215

25.4 Fermi surface nesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216


xiv LIST OF FIGURES

26.1 Inelastic scattering as a probe of collective excitations . . . . . . . . . . . . . 223

26.2 Contours for evaluating the Fourier transform. . . . . . . . . . . . . . . . . . 224

26.3 Contours used for evaluating the Cauchy integral. . . . . . . . . . . . . . . . 225

27.1 Electron spectral function of a noninteracting system . . . . . . . . . . . . . 233

27.2 Electron spectral function of an interacting system . . . . . . . . . . . . . . 234

27.3 The energetics of photoemission . . . . . . . . . . . . . . . . . . . . . . . . . 235

27.4 Experimental geometry for angle-resolved photoemission . . . . . . . . . . . 236

27.5 Synchrotron-based photoemission experiment . . . . . . . . . . . . . . . . . . 237

27.6 ARPES spectra for Sr2RuO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

28.1 The ground state distribution function. . . . . . . . . . . . . . . . . . . . . . 240

28.2 Particle and hole excitations near the Fermi surface . . . . . . . . . . . . . . 241

28.3 Partial spin polarization of a Fermi gas by the application of a magnetic field. 243

28.4 The adiabatic principle for the noninteracting case . . . . . . . . . . . . . . . 244

28.5 Kinematics of quasiparticle decay . . . . . . . . . . . . . . . . . . . . . . . . 245

28.6 Polarizations of the distribution function . . . . . . . . . . . . . . . . . . . . 248

29.1 The superconducting transition of mercury . . . . . . . . . . . . . . . . . . . 252

29.2 The Meissner–Ochsenfeld effect. . . . . . . . . . . . . . . . . . . . . . . . . . 253

29.3 Superconducting ring containing a Josephson junction . . . . . . . . . . . . . 256

29.4 Multiple branches of the Cooper pair kinetic energy . . . . . . . . . . . . . . 257

29.5 Ground state supercurrent in superconducting ring . . . . . . . . . . . . . . 257

30.1 Effective interaction in an elastic solid . . . . . . . . . . . . . . . . . . . . . 262

30.2 Dynamic overscreening as a source of electron–electron attraction . . . . . . 262

30.3 Coefficients of the BCS variational wavefunction . . . . . . . . . . . . . . . . 265

30.4 Excitation energies of the normal and superconducting states. . . . . . . . . 266

30.5 Density of states in a BCS superconductor . . . . . . . . . . . . . . . . . . . 266

31.1 Magnetic fluctuations as a pairing glue . . . . . . . . . . . . . . . . . . . . . 268

31.2 Doping–temperature phase diagrams of the cuprates . . . . . . . . . . . . . . 270

31.3 Cuprate structure and antiferromagnetism . . . . . . . . . . . . . . . . . . . 271

31.4 d-wave pairing by antiferromagnetic fluctuations . . . . . . . . . . . . . . . . 272

31.5 d-wave density of states and superfluid density . . . . . . . . . . . . . . . . . 273

31.6 Quantum critical superconductivity in CePd2Si2 . . . . . . . . . . . . . . . . 274


List of Tables

12.1 Some two-centre integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

12.2 Complete table of two-centre integrals . . . . . . . . . . . . . . . . . . . . . 96

xvIntroduction to Solid State Physics D. M. Broun SFU/CIFAR QM

xvi LIST OF TABLES


Lecture 1

Introduction

While research in other areas of physics focuses on extremes of size and energy, condensedmatter physics explores the middle ground. At first sight, this might seem like a strangeplace to make new discoveries, as the fundamental particles (electrons and nuclei) have beenknown for over 100 years and the basic equation that governs their behaviour can be writtendown directly:

Helec = −∑i

~2

2m∇2i +

∑I

P 2I

2MI

−∑i,I

KZIe2

|ri −RI |+

1

2

∑i6=j

Ke2

|ri − rj|+

1

2

∑I 6=J

KZIZJe2

|RI −RJ |, (1.1)

where K = 1/4πε0, ri and RI label the coordinates of the electrons and nuclei respectively,and ZI and MI are the nuclear charge and mass. This equation controls so much of whatgoes on in the universe that some people call it ‘The Theory of Nearly Everything’. If wewere taking a reductionist approach, in which we were trying to find the simplest unifieddescription of a basic phenomenon, our task would essentially be done. However, it is onething to have identified the fundamental particles (electrons and nuclei) and the hamilto-nian responsible for their evolution, and quite another to solve the equation, even for asmall number of particles, with sufficient accuracy to predict and understand the behaviourthat emerges. It is these emergent properties of condensed matter that make the subject sointeresting. Phenomena such as superconductivity and the fractional quantum Hall effect(FQHE) were not predicted by solving Eq. 1.1, but were understood after the experimentaldiscoveries using approximate models and cleverly chosen wavefunctions. In both supercon-ductivity and the FQHE, the characteristic physical properties are highly nonintuitive and,in the latter case, the fundamental particles are strikingly different from those of Eq. 1.1 —they are excitations with one third the charge of the electron.

There are two basic approaches to understanding electrons in solids, each with its own merits,both of which we will look at during this course. The first is more traditional, and involvesmaking approximations to the full Schrodinger equation of the solid and then developingschemes for carrying out calculations. This is the approach we will begin with. The secondmethod is more concerned with phenomena than calculations, and is therefore less prejudicedabout possible outcomes. This approach goes by the name of ‘Collective Phenomena’ andseeks to understand the new excitations that emerge in many body systems, excitations thatoften have all the properties of real particles — i.e. they are weakly interacting, with well

1Introduction to Solid State Physics D. M. Broun SFU/CIFAR QM

2 LECTURE 1. INTRODUCTION

defined charge and statistics. This will be the topic of later parts of the course.

1.1 The free Fermi gas

We can make several simplifying although not particularly well justified approximations toEq. 1.1. We can assume the ion cores are stationary, and throw away their dynamics. Thisis called the Born–Oppenheimer approximation. It is often quite valid, although in makingit we will always miss out on some important things, such as the physics of lattice vibrationsand the phonon-mediated interactions responsible for conventional superconductivity.

The next approximation we make is to assume the electrons move independently of eachother, by removing the Coulomb interaction between them. With this approximation, Eq. 1.1separates into a sum of identical Hamiltonians, one for each electron:

Helec → Hindep =∑i

Hi, (1.2)

Hi = − ~2

2m∇2i −

∑I

KZIe2

|ri −RI |+ const. (1.3)

This equation is the basis for the one-electron band theory of solids, although it is quite a badapproximation, and without interactions between electrons we miss out on many interestingproperties, such as magnetism.

As a final level of approximation, we can hammer the ion cores out flat and assume theelectrons move in a constant potential that confines them to the interior of the sample. Thisis the basis of the Sommerfeld theory of metals, in which the only parts of the originalproblem we keep are the electron kinetic energy and Fermi statistics.

Hindep → Hfree =∑i

Hi, (1.4)

Hi = − ~2

2m∇2i + const. (1.5)

1.2 Sommerfeld free electrons

We begin by calculating the ground state properties of N free electrons confined to a volumeV . Because the electrons do not interact with one another, we can find the N -electron groundstate by calculating the single particle energy levels and then filling these levels subject tothe constraints of the Pauli exclusion principle. The Schrodinger equation for a single freeelectron is

− ~2

2m∇2ψ(r) = εψ(r), (1.6)

where ψ(r) is the single electron wavefunction. By imposing suitable boundary conditions wecan represent the confinement of the electron to a volume V . One possibility is to insist thatthe wavefunction ψ(r) vanish at the boundary of the solid. This is not the most convenient


1.2. SOMMERFELD FREE ELECTRONS 3

kx

ky

2π/L

Figure 1.1: The allowed states in momentum space of a two-dimensional solid of areaA = L2. Periodic boundary conditions cause the states to be spaced by ∆k = 2π/L. Thek-space area per state is (2π)2/L2, shown as the shaded region.

choice, as it leads to solutions that are standing waves, and we would eventually like todescribe transport properties in terms of the single electron states. A better choice is to useperiodic boundary conditions. Without any loss of generality, we choose our solid to be acube of side L = V 1/3. The periodic boundary condition is then

ψ(r + Lx) = ψ(r + Ly) = ψ(r + Lz) = ψ(r), (1.7)

where x, y and z are unit vectors along the axes of the cube. We now solve the single particleSchrodinger equation, subject to this boundary condition, and find

ψk(r) =1√V

eik·r × spin function, (1.8)

with energy

εk =~2k2

2m. (1.9)

(The spin function represents the two possibilities — spin up and spin down.)

ψk(r) is also an eigenfunction of the momentum operator, with momentum p = ~k, andcorresponds to an electron moving with velocity v = ~

mk.

Substitution of ψk(r) into the boundary condition gives

eikxL = eikyL = eikzL = 1. (1.10)

This will only be satisfied by certain discrete values of k, given by:

kxL = 2πnx, kyL = 2πny, kzL = 2πnz, (1.11)



Figure 1.2: (After A. J. Schofield [1]) For temperatures small compared to the Fermitemperatuer TF the electron gas is a filled Fermi sea in momentum space. The low energyexcitations of the system are electron–hole pairs, formed by exciting an electron across theFermi surface. The energy available for creating these excitations, kBT , is much smaller thanthe Fermi energy EF = kBTF.

where nx, ny and nz are integers.

Figure 1.1 shows the allowed k values plotted in two dimensions. They are separated by2π/L. The area per point is therefore (2π/L)2. In three dimensions, the volume per point is(2π/L)3 = 8π3/V . This will be important when counting up the contribution of each stateto the bulk physical properties.

Because in this model the electrons are noninteracting, we can build up the N -electronground state by placing the electrons in the one-electron levels, subject to the Pauli exclusionprinciple. This limits the occupation of each level (each k-state) to one electron of each spin.The lowest energy configuration is a sphere of filled states centred on k = 0, as shownin Fig. 1.2. The radius of the sphere in k-space is called the Fermi wave vector kF. Themagnitude of kF is determined by the condition that the sphere contain N electrons.

2∑|k|<kF

1 = N, (1.12)

where the factor of 2 accounts for the two possible spin states. k-space summations will oftenappear in our calculations. These are easiest to evaluate by converting them to integrals overthe continuous variable k. In order that an integral over a volume of k-space containing asingle k-point (i.e. k-space volume = (2π)3/V ) return the correct value (i.e. 1), we multiplythe integral by a prefactor V/(2π)3. In problems where there is spin degeneracy, we alsoinclude a factor of 2.

2∑k

→ 2V

(2π)3

∫d3k. (1.13)


1.3. THE CLASSICAL TO QUANTUM CROSSOVER 5

1970

doppler cooling Chu, Hollberg

Bjorkhom, Cable,& Ashkin

Magneto-optical Trap Raab, Prentiss, Cable, Chu, & Pritchard

J = 2

+1

0

−1

mJ = −2

+2

ν0

νrfmagnetic field gradient

Brief history of laser-cooling:three experiments that changed everything


J = 2

+1

0

−1

mJ = −2

+2

ν0

νrf

Brief history of laser-cooling:thre erything

Kap& Diraelectron

fro

1933


J = 2

+1

0

−1

mJ = −2

+2

ν0

• an idea borrowed from the community working to achieve BEC with atomic hydrogen

Figure 1.3: (After K. Madison) The evaporative cooling process used to form Bose–Einsteincondensates in the nanokelvin temperature range. mJ denotes the magnetic sublevels of aJ = 2 atomic species. A spatially varying magnetic field gives rise to spatially varying Zee-man energies, which are used to confine the mJ = +2 species in a magnetic trap. Radiowavesof frequency ν are resonant, in certain positions, with the energy-level spacing of the mag-netic trap. The most energetic atoms are kicked out of the trap first, lowering the averageenergy. The radiofrequency is gradually reduced, kicking out more atoms and cooling thoseremaining. Eventually the system becomes degenerate and a macroscopic fraction of theatoms enter the ground-state of the trap.

For the free Fermi gas, this leads straight away to the condition on kF.

N = 2V

(2π)3

4π

3k3

F (1.14)

⇒ kF =

(3π2N

V

) 13

(1.15)

Points to note about the free Fermi gas:

• the Fermi velocity vF = ~kF/m is much greater than the classical thermal velocity ofthe electron v = (3kBT/m)1/2.

• the Fermi energy EF = ~2k2F/2m corresponds to a Fermi temperature TF = EF/kB that

is tens of thousands of Kelvin in most metals. At room temperature we are thereforenot very far from the ground-state configuration.

1.3 The classical to quantum crossover

For a system of identical particles it is convenient to think of the crossover from classical toquantum statistics in terms of the thermal de Broglie wavelength. The de Broglie wavelengthis given by Planck’s constant over the average momentum, h/p. For a particle with a typicalthermal energy, the thermal de Broglie wavelength is λdB = h/

√3mkBT . λdB grows as



the particles are cooled, and is much larger for electrons than atoms due to the differencein mass. The crossover from classical to quantum behaviour occurs when λdB exceeds theaverage spacing between particles, d ≡ n−1/3. Why does this occur? Well, for a system tocease behaving classically, the particles must be able to explore their quantum statistics.They do this by coherently exchanging places with a neighbouring particle, hence the needfor λdB > d.1 When two fermions exchange places, the many-particle wavefunction changessign. For bosons, the sign is unchanged.

• Electrons are so light that at densities found in a typical solid they form a quantum fluidup to temperatures of the order of 104 K. Solids melt before the quantum-to-classicalcrossover is reached.

• Atoms are much heavier, so there is a chance of observing the classical to quantumcrossover. For noble gas atoms such as neon and argon, the liquids freeze beforeλdB can exceed the average atomic spacing. Solidification fixes the atoms in placeand effectively prevents the particle exchanges required for the atoms to explore theirquantum nature.

• Helium on the other hand is much lighter, which lengthens λdB and lowers the freezingtemperature. In fact, at ambient pressure, helium does not freeze and remains a liquiddown to T = 0. It therefore becomes a quantum fluid. Helium comes in two isotopes,3He, a fermion, and 4He, a composite boson. What is quite remarkable is the differencein properties that arises solely from the quantum statistics. At T = 2.15 K, 4He becomesa Bose superfluid. 3He, on the other hand, is a Fermi liquid with TF ≈ 100 mK and,below T ≈ 2 mK, a p-wave superfluid with Cooper pairs of 3He atoms. This is thestrongest isotope effect of any element.

• At even lower temperatures we enter the realm of trapped atom optics, shown inFig. 1.3. Using laser and radio-frequency cooling it is now possible to cool below1 nanokelvin. The goal in this field is to make the atoms enter the quantum limit, byincreasing λdB beyond the interparticle spacing. However, this must be done carefully.Cooling by simply removing the hot atoms is not sufficient — the particle densitydecreases faster than λdB can increase. Collisions between atoms, although rare, arerequired to keep the system in thermal equilibrium in order to achieve more efficientcooling. Atoms with even numbers of protons, neutrons and electrons Bose condenseon reaching the quantum limit. Fermion condensation is more difficult, and has onlyjust been achieved — in addition to temperatures well into the quantum regime, itrequires an attractive interaction between the atoms in order to form Cooper pairs.Using trapped atoms, experimentalists have been able to resolve a long standing issue— the time taken to form a Bose condensate. Interestingly, it is the time taken for apair of particles to interchange.

1Incoherence occurs when the wavefunction of the quantum particle gets out of phase. The distance overwhich this occurs is inversely proportional to the spread in wavevector, and the spread in wavevector is dueto thermal smearing.


Lecture 2

Thermodynamics and statisticalmechanics

There are many good books on this subject. One of the classic textbooks is ‘Fundamentals of Statistical and

Thermal Physics’, by Frederick Reif [2]. This short review is based on Chapter 2 of ‘Quantum Theory of

Many Particle Systems’, by Fetter and Walecka [3].

2.1 Review of thermodynamics

The change in internal energy E of a system can be related to small, independent changesin the entropy S, volume V and particle number N , by the fundamental thermodynamicrelation

dE = TdS − pdV + µdN. (2.1)

This shows that the internal energy is a thermodynamic function of these three variables,E = E(S, V,N). The temperature T , pressure p and chemical potential µ are given bypartial derivatives of E:

T =

(∂E

∂S

)V N

p = −(∂E

∂V

)SN

µ =

(∂E

∂N

)SV

. (2.2)

In order to prevent confusion the variables that are kept constant must be explicitly specifiedwhen differentiating.

The internal energy is useful for studying adiabatic (isentropic) processes. However, weusually perform experiments at fixed temperature, so it is often better to make a Legendretransformation to the variables (T, V,N) or (T, p,N). The resulting functions are calledthe Helmholtz free energy F (T, V,N) = E − TS and the Gibbs free energy G(T, p,N) =E − TS + pV . The differentials of these functions are

dF = −SdT − pdV + µdN, and (2.3)

dG = −SdT + V dp+ µdN, (2.4)


8 LECTURE 2. THERMODYNAMICS AND STATISTICAL MECHANICS

E

S

V

Figure 2.1: The internal energy E is a surface is S–V –N space. (Particle number N issuppressed in this diagram.) Temperature and pressure are derivatives of the internal energywith respect to S and V . Because E is a well-behaved function of S and V , it does not matterwhich order second partial derivatives are taken in. This leads to the Maxwell relations.

showing that F and G are indeed thermodynamic functions of the specified variables. Thechemical potential is given by:

µ =

(∂F

∂N

)TV

=

(∂G

∂N

)Tp

. (2.5)

Gibbs had the clever idea of representing thermodynamic functions graphically. In Fig. 2.1,the particle number coordinate N is suppressed and the internal energy E sketched as afunction of S and V . Temperature and pressure are given by the gradient of the surface, andLegendre transforms change the shape of the surface and map it onto different coordinates.

Another important set of independent variables is (T, V, µ), which is appropriate for variableN . This is useful when our experimental system is able to exchange particles with a reservoir,as is the case when we make electrical measurements. A further Legendre transformationleads to the thermodynamic potential or grand potential

Ω(T, V, µ) = F − µN = E − TS − µN, (2.6)

which has differentialdΩ = −SdT − pdV −Ndµ. (2.7)

In terms of Ω, the entropy, pressure and particle number are given by:

S = −(∂Ω

∂T

)V µ

p = −(∂Ω

∂V

)Tµ

N = −(∂Ω

∂µ

)TV

. (2.8)


2.2. MAXWELL RELATIONS 9

Although E,F,G and Ω provide equivalent ways of describing the same system, their inde-pendent variables differ in one important way. The set (S, V,N) consists entirely of extensivevariables, which scale with the size of the system, while (T, p, µ) are all intensive variables.This allows us to obtain an important result. If we scale the size of the system by λ, we have

λE = E(λS, λV, λN). (2.9)

Differentiating with respect to λ we obtain

E = S

(∂E

∂S

)V N

+ V

(∂E

∂V

)SN

+N

(∂E

∂N

)SV

= TS − pV + µN. (2.10)

From this, the remaining thermodynamic functions are found to be

F = −pV + µN G = µN Ω = −pV (2.11)

In particular, we see that the chemical potential µ is the Gibbs free energy per particle,and that the thermodynamic potential per unit volume is the negative of the pressure. TheGibbs free energy is particularly important for studying phase transitions.

2.2 Maxwell relations

Maxwell relations provide a very useful application of the above ideas, stemming from abasic property of differentiable functions of several variables. To demonstrate this, considerchanges in the internal energy with respect to entropy and volume.

dE = TdS − pdV (2.12)

On purely mathematical grounds we can write:

dE =

(∂E

∂S

)V

dS +

(∂E

∂V

)S

dV (2.13)

from which we saw above that

T =

(∂E

∂S

)V

and p = −(∂E

∂V

)S

. (2.14)

We can differentiate again to get:(∂T

∂V

)S

=

(∂

∂V

(∂E

∂S

)V

)S

(2.15)

−(∂p

∂S

)V

=

(∂

∂S

(∂E

∂V

)S

)V

(2.16)

We are differentiating a well behaved surface, so it does not matter what order we take thederivatives in. As a result, we have the Maxwell relation:(

∂T

∂V

)S

= −(∂p

∂S

)V

. (2.17)



This illustrates the basic principle. There are many other Maxwell relations, some moreuseful that others. Particularly important are the ones involving entropy, which is notdirectly accessible in experiments. For example:(

∂S

∂V

)T

=

(∂p

∂T

)V

(2.18)

−(∂S

∂p

)T

=

(∂V

∂T

)p

(2.19)

Relations like the last one are particularly powerful. It relates the volume coefficient ofthermal expansion to a derivative of the entropy. From the third law of thermodynamics weknow that the entropy vanishes as T → 0. This must occur at all pressures, so ∂S/∂p mustalso vanish. As a result we see that the coefficient of thermal expansion goes to zero in thislimit for very fundamental reasons.

2.3 Review of statistical mechanics

Macroscopic thermodynamics, although useful, merely correlates bulk properties of the sys-tem. The microscopic content of the theory is introduced through statistical mechanics,which relates the thermodynamic functions to the hamiltonian. For a many particle sys-tem that is free to exchange particles with its environment, it is useful to work with thegrand canonical ensemble at chemical potential µ and temperature T = 1/kBβ. The grandpartition function ZG is defined:

ZG ≡∑N

∑j

e−β(Ej−µN) (2.20)

=∑N

∑j

〈Nj|e−β(H−µN)|Nj〉 (2.21)

Here, states are labelled by particle number N , with j running over all energy eigenstatesfor each fixed N , allowing the partition function to be written in terms of the quantumoperators H and N .1 A common shorthand is the use of the trace operator, where Tr(A)denotes the sum over all states of the quantum expectation value 〈A〉. In this notation, thegrand partition function is

ZG = Tr(

e−β(H−µN)). (2.22)

A fundamental result from statistical mechanics relates the grand partition function to thethermodynamic potential Ω:

Ω(T, V, µ) = −kBT lnZG =⇒ Z−1G = eβΩ. (2.23)

1This contains a subtle but important point. Although we are about to specialize to the case of nonin-teracting particles where the energies of the single particle states are additive — i.e. they simply add up togive the total energy Ej that appears in the partition function — this is usually not the case. Any realistichamiltonian contains interactions between the particles, requiring us to solve a different problem for each Nin order to find a new set of eigenstates |Nj〉 and a new set of eigen-energies Ej. This is very difficult todo in practice.


2.4. STATISTICAL MECHANICS OF IDEAL QUANTUM GASES 11

This result allows equilibrium thermodynamic quantities to be calculated from the grandpartition function.

It is also useful to define the statistical operator ρG, whose expectation value in any state isthe probability, in thermal equilibrium, of finding the system in that state.

ρG = Z−1G e−β(H−µN) (2.24)

= eβ(Ω−H+µN) (2.25)

For any operator O, the ensemble average 〈O〉 can be obtained in the following way:

〈O〉 = Tr(ρGO

)(2.26)

= Tr(

eβ(Ω−H+µN)O)

(2.27)

=Tr(

e−β(H−µN)O)

Tr(

e−β(H−µN)) (2.28)

2.4 Statistical mechanics of ideal quantum gases

We now apply these ideas to noninteracting Bose and Fermi gases, in which the total energyof the many-particle states can be computed by adding up the energies of the occupiedone-particle states.2 For this calculation it is convenient to work in the occupation numberrepresentation, where |Nj〉 → |n1...n∞〉. Here, the ni denote the number of particles ineach single-particle state, and

∑i ni = N . There is an important difference in the labelling

scheme: previously j ranged over the true N -particle eigenstates, but now i runs over allthe single-particle states. While the set |n1...n∞〉 is a valid basis set for the many-particleproblem, only in the noninteracting case are these the eigenstates of the system. We cannow calculate the grand partition function for the noninteracting hamiltonian H0:

ZG = Tr(

e−β(H0−µN)) (

= e−βΩ0(T,V,µ))

(2.30)

=∑

n1...n∞

〈n1...n∞|eβ(µN−H0)|n1...n∞〉 (2.31)

=∑

n1...n∞

〈n1...n∞| exp

[β

(µ∑i

ni −∑i

εini

)]|n1...n∞〉, (2.32)

where we have used the fact that the |n1...n∞〉 are eigenstates of H0 and N , and replacedthem by their eigenvalues. The exponential is now a c number and can be factored into a

2Remember, this is not possible in the presence of interactions. In that case, adding a particle to anN -particle system does not in general increase the total energy by the energy of the single particle state ε.

EN+1 6= EN + ε (2.29)



product of exponentials, and the partition function can be factored into a product of traces,one for each mode:

ZG =∑n1

〈n1|eβ(µn1−ε1n1)|n1〉...∑n∞

〈n∞|eβ(µn∞−ε∞n∞)|n∞〉 (2.33)

=∞∏i=1

Tri(e−β(εi−µ)ni

)(2.34)

For bosons the sum over ni runs over all nonnegative integers:

ZG =∞∏i=1

∞∑n=0

(eβ(µ−εi)

)n(2.35)

=∞∏i=1

(1− eβ(µ−εi)

)−1

(∞∑n=0

xn = (1− x)−1, |x| < 1

)(2.36)

From this we immediately obtain the thermodynamic potential:

Ω0(T, V, µ) = −kBT ln∞∏i=1

(1− eβ(µ−εi)

)−1(2.37)

= kBT∞∑i=1

ln(1− eβ(µ−εi)

)Bose–Einstein (2.38)

We now calculate the mean number of particles:

〈N〉 = −(∂Ω0

∂µ

)TV

(2.39)

= kBT∞∑i=1

βeβ(µ−εi)

1− eβ(µ−εi)(2.40)

=∞∑i=1

1

eβ(εi−µ) − 1Bose–Einstein (2.41)

≡∞∑i=1

n0i , (2.42)

where n0i is the mean occupation number in the ith state.

For fermions, the Pauli exclusion principle limits the occupation numbers to be either zeroor one. We have

ZG =∞∏i=1

1∑n=0

(eβ(µ−εi)

)n(2.43)

=∞∏i=1

(1 + eβ(µ−εi)

)(2.44)



and

Ω0(T, V, µ) = −kBT∞∑i=1

ln(1 + eβ(µ−εi)

). Fermi–Dirac (2.45)

The mean number of particles is

〈N〉 ≡∞∑i=1

n0i =

∞∑i=1

1

eβ(εi−µ) + 1. Fermi–Dirac (2.46)

2.4.1 Example: statistics of an impurity in a semiconductor

We will look at the simplest example of correlation beyond the independent electron ap-proximation, and show how it modifies the Fermi–Dirac distribution. Imagine we have asingle donor impurity atom in a semiconductor, and that this atom has available a singleorbital, which can be empty, contain one electron of either spin, or be doubly occupied. Inthis example, the two singly occupied states are degenerate, and we label their energy ε1.This energy will lie just below the bottom of the conduction band. In the first instance, weignore interactions between the electrons, so that the energy of the doubly occupied state isjust 2ε1, as shown in Fig. 2.2.

We now calculate the mean occupation number:

ZG =∑N

∑j

〈Nj|e−β(H−µN)|Nj〉 (2.47)

= 1 + 2 · e−β(ε1−1·µ) + e−β(2·ε1−2·µ) (2.48)

=(1 + e−β(ε1−µ)

)2(2.49)

Ω = −kBT lnZG (2.50)

= −2kBT ln(1 + e−β(ε1−µ)

)(2.51)

〈N〉 = −(∂Ω

∂µ

)TV

(2.52)

= 2kBTβe−β(ε1−µ)

1 + e−β(ε1−µ)(2.53)

= 21

eβ(ε1−µ) + 1(2.54)

This is what we expect: twice the mean occupation number of a single state at energy ε1.

What happens when we include the Coulomb repulsion between electrons? If we call thisenergy U , we get a new energy level diagram, with the energy of the doubly occupied stateshifted to 2ε1+U . For donor impurities, it is often the case that the doubly occupied state lieswell above the bottom of the conduction band, preventing two electrons from ever occupyingthe impurity orbital at the same time. We must therefore exclude the doubly occupied state



ε

ConductionBand

Donor Level

ValenceBand

ε

N = 0E = 0

N = 1E = ε1

N = 2E = 2 ε1

ε

N = 0E = 0

N = 1E = ε1

N = 2E = 2 ε1+U

Noninteracting electrons Interacting electrons

Figure 2.2: Donor impurities in a semiconductor give rise to donor levels just below thebottom of the conduction band. In the absence of Coulomb repulsion the energy of thedoubly occupied donor level is just 2ε1. In reality, the Coulomb interaction contributes anadditional energy U , lifting the energy of the doubly occupied level well above the bottomof the conduction band. This greatly modifies the statistics of the impurity level.

from our set of eigenstates |Nj〉. We calculate the average occupation number again:

ZG =∑N

∑j

〈Nj|e−β(H−µN)|Nj〉 (2.55)

= 1 + 2 · e−β(ε1−µ) (2.56)

Ω = −kBT lnZG (2.57)

= −kBT ln(1 + 2e−β(ε1−µ)

)(2.58)

〈N〉 = −(∂Ω

∂µ

)TV

(2.59)

= kBT2βe−β(ε1−µ)

1 + 2e−β(ε1−µ)(2.60)

= 21

eβ(ε1−µ) + 2(2.61)

The two situations, interacting and noninteracting, are plotted in Fig. 2.3 and we can seeright away that correlation effects make them quite different.



-5 0 5 10

1

2

(ε − µ)/kBT

nnoninteracting

ninteracting

-10

Figure 2.3: The average occupation number of the donor level is shown in the interactingand noninteracting cases. Interactions substantially modify the statistics, with Coulombrepulsion blocking double occupancy.

2.4.2 Example: interacting electrons in a metal

Another example of how interactions modify quantum statistics is provided by electrons in ametal. Figure 2.4 shows what happens when interactions are switched back on in the Fermigas. Parts (a) and (b) show the result we derived above for the Fermi function. At finitetemperature, the distribution function ne smoothly drops from 1 to 0 in an energy range ofa few kBT about the chemical potential. At zero temperature the drop is abrupt. Whathappens in a real metal, where there is a strong Coulomb repulsion between the electrons?Something like what is shown in Fig. 2.4(c). Note that we are still at zero temperature, butthe distribution function has substantial rounding in the vicinity of the chemical potential.An abrupt drop remains, but is reduced in magnitude. The zero-temperature distributionfunction represents the lowest energy configuration of the electrons and takes into accountboth the electron–electron repulsion and the electron kinetic energy. The system is clearlynot described by Fermi–Dirac statistics. Why then do we spend so much time talking aboutmetals as if they were noninteracting electron systems, and why do we use Fermi statisticsto calculate their thermal properties? The answer is subtle, and lies in the fact that formany systems the true eigenmodes retain many of the properties of noninteracting electrons.That is, they are weakly interacting, have definite charge, and well-defined quantum statis-tics. Such states are called electron quasiparticles and in many cases (for systems where thebalance between kinetic and potential energy is not too delicate) resemble electrons withmodest mass renormalization. Figure 2.4 (d) sketches nqp, the distribution function of theinteracting electrons in the quasiparticle basis. It is a zero-temperature Fermi–Dirac distri-bution. We will go into more detail about this when we get to Landau’s Fermi-liquid theoryin Lecture 28.



(a)

1ne

0 µ ε

T > 0noninteracting

(b)

1ne

0 µ ε

T = 0noninteracting

(c)

1ne

0 µ ε

T = 0interacting

(d)

1nqp

0 µ ε

T = 0interacting

Figure 2.4: The Fermi function, which gives the average occupation number of noninter-acting electron states, is shown at finite temperature in (a) and at T = 0 in (b). The T = 0electron occupation number is shown in (c) in the presence of interactions, and is nothinglike the Fermi function. However, in many cases an interacting electron system has fermionicnormal modes that closely resemble electrons. These are called quasiparticles, and at lowtemperatures have a Fermi distribution, like that shown in (d).


Lecture 3

Thermal properties of the Fermi gas

References: Most solid-state text books include a chapter on the thermal properties of the free Fermi gas.

For instance, see: Ashcroft and Mermim [4], Chapter 2; Reif [2], Chapter 9; or Marder [5], Chapter 6.

3.1 Specific heat of an electron gas

We would like to calculate the specific heat at constant volume, cV , of a system of Nnoninteracting electrons in a volume V . The specific heat is defined to be:

cV =1

V

(∂E

∂T

)V N

(3.1)

=T

V

(∂S

∂T

)V N

, (3.2)

where E is the internal energy. In the independent electron approximation the single particleenergies are additive, and the average internal energy 〈E〉 is the sum over all occupied levelsof the single particle energy:

〈E〉 = 2∑k

εkf(εk), (3.3)

where the factor of 2 comes from the spin degeneracy. In our problem, we have the additonalconstraint of maintaining, on average, a fixed number of electrons.

〈N〉 = 2∑k

f(εk), (3.4)

which we impose by adjusting the chemical potential µ. (This is the reason the chemicalpotential was introduced in the first place.) In the expressions for 〈E〉 and 〈N〉, the quantitybeing summed depends on k only through the single-particle energy εk. It therefore makessense to introduce the density of states D(ε), which is defined so that

D(ε)dε ≡ number of states per unit volume with energy between ε and ε+ dε. (3.5)


18 LECTURE 3. THERMAL PROPERTIES OF THE FERMI GAS

D(ε)

ε ε+dε

D(ε)dε

Figure 3.1: The density of states (D.O.S.) is the number of states per unit energy, per unitvolume.

We will give a precise definition of D(ε) shortly. In the meantime, let’s write down expres-sions, per unit volume, for the two thermally averaged quantities we need:

n ≡ 〈N〉V

=2

V

∑k

f(εk) → 2

V

V

(2π)3

∫dk f(ε(k)) =

∫dk

4π3f(ε(k)) (3.6)

u ≡ 〈E〉V

=2

V

∑k

εkf(εk) → 2

V

V

(2π)3

∫dk ε(k)f(ε(k)) =

∫dk

4π3ε(k)f(ε(k)) (3.7)

These are both integrals of the form∫dk

4π3F (ε(k)) =

∫ 2π

0

dφ

∫ 1

−1

d(cos θ)

∫ ∞0

k2dk

4π3F (ε(k)) =

∫ ∞0

k2dk

π2F (ε(k)), (3.8)

where the integrand F (ε(k)) depends on k only through the energy and, therefore, only onthe magnitude of k. Let’s make the definition:∫ ∞

0

k2dk

π2F (ε(k)) ≡

∫ ∞−∞

dεD(ε)F (ε), (3.9)

so that

D(ε)dε =k2dk

π2. (3.10)

Since ε ∝ k2, we have (taking the logarithmic derivative)

dε

ε= 2

dk

k⇒ dε

dk= 2

ε

k. (3.11)


3.2. THE SOMMERFELD EXPANSION 19

Rearranging for D(ε), we get

D(ε) =k2

π2

dk

dε=

k2

π2

k

2ε(3.12)

=2m

~2π2

~2k2

2m

k

2ε(3.13)

=m

~2π2k. (D.O.S. at Fermi surface =

m

~2π2kF.) (3.14)

Using the free electron dispersion relation ε(k) = ~2k2

2m, this can also be written:

D(ε) =m

~2π2

√2mε

~2. (3.15)

The final form for our thermal averages is:

electron density: n =

∫ ∞−∞

dεD(ε)f(ε) (3.16)

energy density: u =

∫ ∞−∞

dεD(ε)εf(ε) (3.17)

These integrals are quite complicated to evaluate, but we can make a great simplificationin the low temperature limit. (And remember, low temperature in a metal means T TF, where the Fermi temperature TF is typically tens of thousands of kelvin.) In the zerotemperature limit, the Fermi function becomes a step function, and the integrals are verysimple:

n0 =

∫ εF

−∞dεD(ε) (3.18)

u0 =

∫ εF

−∞dεD(ε)ε, (3.19)

where εF = µ(T = 0). At low temperatures T TF, the Fermi function differs from a stepfunction only in the vicinity of the chemical potential. We can then make an expansion inpowers of T . This procedure is called the Sommerfeld expansion.

3.2 The Sommerfeld expansion

Say that we want to integrate ∫ ∞−∞

H(ε)f(ε)dε, (3.20)

where H(ε) vanishes as ε→ −∞ and is polynomial in ε as ε→ +∞.

Ideally, we want to work with the derivative of the Fermi function, because the derivativeis localized near the Fermi surface (ε = µ), and we know that this is where all the actionoccurs. Therefore, let

K(ε) =

∫ ε

−∞H(ε′)dε′, so that

dK(ε)

dε= H(ε). (3.21)



-30 -20 -10 0

1

f(x)

-df(x)/dx

x = (ε−µ)/kBT

Figure 3.2: The Fermi function f(x) = 1/ (1 + ex) and its derivative. df/dx is sharplypeaked within a thermal energy of x = 0.

Now integrate the original expression by parts:∫ ∞−∞

H(ε)f(ε)dε = [K(ε)f(ε)]+∞−∞ +

∫ ∞−∞

K(ε)

(−∂f∂ε

)dε. (3.22)

The term in brackets vanishes: K(ε) → 0 as ε → −∞, and f(ε) goes to zero exponentiallyquickly as ε→ +∞.

Now expand K(ε) about the chemical potential:

K(ε) = K(µ) +∞∑n=1

(ε− µ)n

n!

dnK(ε)

dεn

∣∣∣∣ε=µ

(3.23)

Substituting into the original expression, we get:∫ ∞−∞

H(ε)f(ε)dε = K(µ)

∫ ∞−∞

(−∂f∂ε

)dε+

∞∑n=1

∫ ∞−∞

(ε− µ)n

n!

dnK(ε)

dεn

∣∣∣∣ε=µ

(−∂f∂ε

)dε.

(3.24)Now

K(µ)

∫ ∞−∞

(−∂f∂ε

)dε = K(µ) f(−∞)− f(+∞) = K(µ) =

∫ µ

−∞H(ε)f(ε)dε. (3.25)

Also, since(−∂f

∂ε

)is an even function of (ε−µ), only terms with even n survive the integration.

Finally,

dnK(ε)

dεn

∣∣∣∣ε=µ

=dn

dεn

[∫ ε

−∞H(ε′)dε′

]∣∣∣∣ε=µ

=dn−1

dεn−1H(ε)|ε=µ ≡ H(n−1)(µ). (3.26)

We can therefore write:∫ ∞−∞

H(ε)f(ε)dε =

∫ µ

−∞H(ε)dε+

∞∑n=1

H(2n−1)(µ)

∫ ∞−∞

(ε− µ)2n

(2n)!

(−∂f∂ε

)dε (3.27)

=

∫ µ

−∞H(ε)dε+

∞∑n=1

H(2n−1)(µ)(kBT )2n

∫ ∞−∞

x2n

(2n)!

(−∂f∂x

)dx (3.28)


3.3. SPECIFIC HEAT OF THE IDEAL FERMI GAS 21

This is a rapidly converging power series in even powers of T . The coefficients of the firstfew terms are: ∫ ∞

−∞

x2

2!

(−∂f∂x

)dx =

π2

6, (3.29)∫ ∞

−∞

x4

4!

(−∂f∂x

)dx =

7π4

360. (3.30)

The final result, up to fourth order in T is:∫ ∞−∞

H(ε)f(ε)dε =

∫ µ

−∞H(ε)dε+

π2

6(kBT )2H ′(µ) +

7π4

360(kBT )4H ′′′(µ) +O

(T

TF

)6

(3.31)

3.3 Specific heat of the ideal Fermi gas

To begin, let’s calculate the temperature dependence of the chemical potential, and showthat it is weak, even at room temperature.

n =

∫ ∞−∞

D(ε)f(ε)dε ⇒ H(ε) = D(ε). (3.32)

Using the Sommerfeld expansion:

n =

∫ µ

−∞D(ε)dε+

π2

6(kBT )2D′(µ) +O

(T

TF

)4

(3.33)

=

∫ εF

0

D(ε)dε+

∫ µ

εF

D(ε)dε+π2

6(kBT )2D′(µ) +O

(T

TF

)4

(3.34)

≈∫ εF

0

D(ε)dε+ (µ− εF)D(εF) +π2

6(kBT )2D′(εF), (3.35)

where we are assuming that the temperature dependence of µ is weak (we must verify thisbelow) in order to make the approximation µ ≈ εF in the higher order terms. The first termis just the electron density at T = 0, n0. Rearranging, we get

n− n0 = (µ− εF)D(εF) +π2

6(kBT )2D′(εF). (3.36)

We want to keep the average electron density fixed, so n− n0 = 0, implying

µ = εF −π2

6(kBT )2D

′(εF)

D(εF). (3.37)

For electrons in 3 D, D(ε) ∝ ε1/2, so that

D′(εF)

D(εF)=

d lnD(ε)

dε

∣∣∣∣ε=εF

=1

2εF. (3.38)



Therefore

µ = εF −π2

6(kBT )2 1

2εF(3.39)

= εF

(1− π2

12

(T

TF

)2), (3.40)

which is a very small shift in chemical potential at room temperature.

Now to calculate the internal energy:

u =

∫ ∞−∞

εD(ε)f(ε)dε ⇒ H(ε) = εD(ε). (3.41)

Using the Sommerfeld expansion again:

u =

∫ µ

−∞εD(ε)dε+

π2

6(kBT )2 [µD′(µ) +D(µ)] +O

(T

TF

)4

(3.42)

≈∫ εF

0

εD(ε)dε+ (µ− εF)εFD(εF) +π2

6(kBT )2 [εFD

′(εF) +D(εF)] (3.43)

= u0 + εF

(µ− εF)D(εF) +

π2

6(kBT )2D′(εF)

+π2

6(kBT )2D(εF), (3.44)

where u0 is the energy density at zero temperature. We have already shown that the termin braces is just n− n0, which must vanish. Therefore,

u = u0 +π2

6(kBT )2D(εF). (3.45)

We obtain the specific heat from the temperature derivative

cV =

(∂u

∂T

)NV

=π2

3k2

BTD(εF). (3.46)

We can write down a useful expression for the density of states, just by knowing how thetotal energy depends on the density of electrons. Since ε ∝ k2, and n ∝ k3 (in 3-D), weknow that n ∝ ε3/2. This implies that

dn

n=

3

2

dε

εor

dn

dε≡ D(ε) =

3n

2ε. (3.47)

Therefore, the density of states at the Fermi level, which determines many of the importantlow temperature properties such as specific heat and Pauli susceptibility, is given by:

D(εF) =dn

dε

∣∣∣∣ε=εF

=3n

2εF. (3.48)

We have then that

cV =π2

3k2

BT3n

2εF(3.49)

=π2

3

3kBn

2

T

TF

. (3.50)


3.3. SPECIFIC HEAT OF THE IDEAL FERMI GAS 23

Compare this to the specific heat of a classical gas:

ucl =3

2kBTn, from the equipartition theorem (3.51)

cclV =

(∂ucl

∂T

)NV

=3

2kBn. (3.52)

The specific heat of the Fermi gas,

cV = cclV

π2

3

T

TF

, (3.53)

is reduced by a factor π2

3TTF

, emphasizing that in degenerate Fermi systems, all the actionindeed takes place within a thermal energy kBT of the Fermi surface.


Lecture 4

Review of single-particle quantummechanics

References: One of the best books on single-particle quantum mechanics is ‘Principles of Quantum Mechan-

ics’, by R. Shankar [6]. This develops the abstract approach in a very clear way and has a good mathematical

introduction on linear algebra. Chapter 2 of Taylor and Heinonen [7] also has a good review of single-particle

quantum mechanics and Dirac notation, done in a way that leads naturally to the second quantized formalism

used for many-particle systems.

4.1 Single-particle quantum mechanics

The next few lectures will set up the mathematical framework necessary for describing thequantum mechanics of many-particle systems. In particular we will show that all physicalinformation can be obtained using operators instead of wavefunctions. It is a good idea tobegin this task with a review of single-particle quantum mechanics, as a foundation for themany-particle formalism and to see precisely how the two differ.

Quantum mechanics cannot be derived, but begins from a set of postulates. These arewritten here using Dirac notation:

(i) Complete information about the state of a particle is provided by the state function |ψ〉,a vector in a Hilbert space.

(ii) For every observable Ω there exists a corresponding Hermitian operator Ω. When thatobservable is measured we will obtain one of the eigenvalues ω of Ω.

(iii) If the particle is in state |ψ〉 then a measurement of Ω will give one of the eigenvaluesω with probability P (ω) ∝ |〈ω|ψ〉|2 and leave the system in the eigenstate |ω〉.

(iv) The expectation value of an observable Ω in a state |ψ〉 is 〈Ω〉 = 〈ψ|Ω|ψ〉.

(v) In the absence of a measurement, the time dependence of the state function is given by

H|ψ〉 = i~∂

∂t|ψ〉. (4.1)


26 LECTURE 4. REVIEW OF SINGLE-PARTICLE QUANTUM MECHANICS

In single-particle quantum mechanics, the hamiltonian has the form

H =p2

2m+ U(r) (4.2)

(Later on, the general many-body hamiltonian will be expanded

H =∑i

p2i

2m+∑i

U(ri) +1

2

∑i6=j

V (ri − rj) + three-body terms + ... (4.3)

Along with the added complexity of interactions between the particles, the number of particleswill in general not be fixed.)

In quantum mechanics we often work in the position representation (or r-rep). More formally,this is the basis set formed by the eigenstates of the position operator r.

r|r0〉 = r0|r0〉 (4.4)

This satisfies the necessary requirements of a basis set:

• it spans the Hilbert space ∫dr|r〉〈r| = 1 completeness (4.5)

• and it forms an orthonormal set

〈r|r′〉 = δ(r− r′). orthonormality (4.6)

This is a perfectly good way of doing single-particle quantum mechanics. We can take anyabstract state |ψ〉, and project it onto the position basis:

|ψ〉 → 〈r|ψ〉 = ψ(r), (4.7)

showing that the wavefunction is the probability amplitude of finding a particle at a givenpoint. In the position basis the hamiltonian takes the form of the Schrodinger equation

H =p2

2m+ U(r)→ − ~2

2m∇2 + U(r), (4.8)

where r is now just a coordinate, not an operator.

But this is not the only way of representing a statefunction and the hamiltonian. Wecan equally well work in the momentum representation, where the statefunctions ψ(k) areFourier transforms of the Schrodinger wavefunctions ψ(r). The basis functions |k〉 are theplane waves

〈r|k〉 =1√V

eik·r. (4.9)


4.1. SINGLE-PARTICLE QUANTUM MECHANICS 27

Because we are working with a discrete set of allowed wavevectors in a finite volume V , theconditions for completeness and orthonormality take a slightly different form:∑

k

|k〉〈k| = 1 completeness (4.10)

〈k|k′〉 = δk,k′ . orthonormality (4.11)

We can also choose to work in the natural representation of the Hamiltonian — the energybasis. The basis states |α〉 are the energy eigenstates of the hamiltonian.

H|α〉 = Eα|α〉. (4.12)

For this basis, we have ∑α

|α〉〈α| = 1 completeness (4.13)

〈α|α′〉 = δα,α′ . orthonormality (4.14)

Any state in the Hilbert space can be expanded in terms of the set |α〉:

|ψ〉 =∑α

|α〉〈α|ψ〉 =∑α

cα|α〉, cα = 〈α|ψ〉. (4.15)

This can be projected onto the coordinate basis 〈r| to give:

ψ(r) =∑α

cαφα(r), where φα(r) ≡ 〈r|α〉. (4.16)

Operators can be expressed in terms of their matrix elements Vαα′ ≡ 〈α|V |α′〉:

V =

(∑α′

|α′〉〈α′|

)V

(∑α′′

|α′′〉〈α′′|

)(4.17)

=∑α′α′′

Vα′α′′ |α′〉〈α′′|. (4.18)

The operator |α′〉〈α′′| gives zero when it acts on any state |α〉 except when α = α′′ — itthen gives |α′〉. In other words |α′〉〈α′′| removes or annihilates a particle from state |α′′〉 andputs it into or creates state |α′〉. We can also define an empty or vacuum state |0〉, that isnormalized but contains no particles. We can then write

|α′〉〈α′′| = |α′〉〈0|0〉〈α′′| (4.19)

= (|α′〉〈0|) (|0〉〈α′′|) (4.20)

where we now have explicit expressions for the creation operator c†α′ ≡ |α′〉〈0| and theannihilation operator cα′′ ≡ |0〉〈α′′|. Using these, the original operator V becomes

V =∑α′α′′

Vα′α′′c†α′cα′′ . (4.21)

This way of representing operators comes into its own when we work with many-particlesystems — in that case it is the only feasible approach.



|0>|0>

|1>|1>

|2>|2>

|3>|3>

|4>|4>

V = V = 1/21/2 k x k x22

Figure 4.1: Energy eigenstates of the harmonic oscillator.

4.2 The harmonic oscillator

Any system fluctuating by a small amount near a configuration of stable equilibrium can bedescribed by a collection of oscillators, which can be decoupled by making a transformation tothe system’s normal coordinates. (The dynamics of a collection of noninteracting oscillatorsis no more complicated than that of a single oscillator, so the problem we look at here isvery relevant to many-particle systems.) In one dimension the hamiltonian and Schrodingerequation are

H =p2

2m+

1

2mω2x2 (4.22)

H|ψ〉 = E|ψ〉. (4.23)

In the coordinate basis this becomes(− ~2

2m∇2 +

1

2mω2x2

)ψ(x) = Eψ(x), (4.24)

which has solution

ψn(x) = A(n) exp

(−mωx

2

2~

)Hn

(√mω

~x

), En =

(n+

1

2

)~ω. (4.25)


4.2. THE HARMONIC OSCILLATOR 29

Here A(n) is a normalization constant that depends on n and Hn(y) are the Hermitepolynomials:

H0(y) = 1 (4.26)

H1(y) = 2y (4.27)

H2(y) = 4y2 − 2 (4.28)

H3(y) = 8y3 − 12y (4.29)

Hn+1(y) = 2yHn(y)− 2nHn−1(y). (4.30)

As we stated at the beginning, all we can know about a quantum system is the statisticaldistribution of the eigenvalues of our measurement operators, which we obtain by evaluatingmatrix elements. In the coordinate basis, this requires us to form inner products by sand-wiching a differential operator between two wavefunctions and integrating, something thatrapidly becomes tedious. There must be a better way!

Fortunately there is, using a method due to Dirac. With this technique we can not only findthe energy spectrum, but evaluate the matrix elements of any operator, as long as it can beexpressed as a function of x and p. All we need is the canonical commutation relation

[x, p] = i~. (4.31)

Let’s introduce the operator

a =

√mω

2~x+ i

√1

2mω~p, (4.32)

and its adjoint

a† =

√mω

2~x− i

√1

2mω~p. (4.33)

Now evaluate the commutation relation [a, a†]. This has the form

[A+ iB,A− iB] = [A,−iB] + [iB,A] = −2i[A,B] (4.34)

Therefore

[a, a†] = −2i

√mω

2~

√1

2mω~[x, p] (4.35)

= − 2i

2~i~ (4.36)

= 1. (4.37)

Next, write down a†a:

a†a = (A− iB)(A+ iB) (4.38)

= A2 +B2 + i[A,B] (4.39)

=1

~ω

(mω2

2x2 +

1

2mp2

)− 1

2(4.40)

=H

~ω− 1

2. (4.41)



Therefore

H =

(a†a+

1

2

)~ω. (4.42)

(One interesting observation apparent from this derivation is that the extra term ~ω/2 —the zero point energy of the oscillator — comes from the noncommuting nature of x and p.)

We want to solveH|ε〉 = ε|ε〉. (4.43)

Scale the hamiltonian to get a dimensionless operator:

H → H ′ =H

~ω=

(a†a+

1

2

). (4.44)

Now evaluate the commutators with a and a†:

[a, H ′] = [a, a†a+1

2] = [a, a†a] = [a, a†]a = a, (4.45)

and similarly[a†, H ′] = −a†. (4.46)

a and a† are useful because given an eigenstate of H ′ they generate others. To see this,assume |ε〉 is a eigenstate. What is the energy of a|ε〉?

H ′ (a|ε〉) =(aH ′ − [a, H ′]

)|ε〉 (4.47)

=(aH ′ − a

)|ε〉 (4.48)

= (ε− 1)a|ε〉. (4.49)

a|ε〉 is an eigenstate with energy (ε− 1). Similarly, a†|ε〉 is a eigenstate with energy (ε+ 1).a† is a creation or raising operator, and a an annihilation or lowering operator.

It can easily be shown that the eigenvalues of the harmonic oscillator are nonnegative (thisis because the hamiltonian is quadratic in both the hermitian operators x and p), so theremust be a lowest eigenstate |ε0〉. Acting on |ε0〉 with the lowering operator a must thereforegive zero.

a|ε0〉 = 0 (4.50)

Operating on this with a† must also give zero:

a†a|ε0〉 = 0 (4.51)

⇒(H ′ − 1

2

)|ε0〉 = 0 (4.52)

⇒ H ′|ε0〉 =1

2|ε0〉 ≡

1

2|0〉. (4.53)

The ground state energy is therefore ε0 = 12~ω.

The excited states can be obtained from the ground state by repeated application of a†.

|n〉 ∝ (a†)n|0〉. (4.54)


4.2. THE HARMONIC OSCILLATOR 31

The excited states form an evenly spaced ladder of levels, so εn = (n+ 1/2)~ω. (There mustbe only one family of levels, because there is no degeneracy in one dimension.)

Let’s now determine the normalization constant. Assume that |n − 1〉 is normalized andwork out the normalization factor for |n〉.

a|n〉 = cn|n− 1〉, (〈n|a† = 〈n− 1|c∗n) (4.55)

〈n|a†a|n〉 = 〈n− 1|n− 1〉|cn|2 = |cn|2 (4.56)

⇒ 〈n|H ′ − 1

2|n〉 = 〈n|n|n〉 (εn = (n+

1

2)~ω) (4.57)

so |cn|2 = n ⇒ cn =√neiφ, (4.58)

where we usually set the phase φ to zero. This and a similar derivation for a† give us thetwo important equations

a|n〉 =√n|n− 1〉, (4.59)

a†|n〉 =√n+ 1|n+ 1〉. (4.60)

Using these two equations we can show that

a†a|n〉 = a†√n|n− 1〉 =

√n√n|n〉 = n|n〉. (4.61)

In other words, n = a†a is the number operator, which when acting on an eigenstate givesthe number of quanta. This is very useful.

Equations 4.59 and 4.60 are also useful for evaluating matrix elements, and therefore physicalobservables.

〈n′|a|n〉 =√n〈n′|n− 1〉 =

√n δn′,n−1 (4.62)

〈n′|a†|n〉 =√n+ 1〈n′|n− 1〉 =

√n+ 1 δn′,n+1 (4.63)

These equations can be used to find any physical observable that is a function of x and p,as these can be written, in turn, in terms of a and a†:

x =

√~

2mω(a+ a†) (4.64)

p = i

√mω~

2(a† − a). (4.65)

Finally, the nth excited state, properly normalized, is generated by

|n〉 =a†√n|n− 1〉 =

a†√n

a†√n− 1

|n− 2〉 = ... =(a†)n√n!|0〉. (4.66)

A general feature of this approach is that the hamiltonian is diagonal in the energy basis.

H → ~ω

1/2 0 0 . . .0 3/2 00 0 5/2...

(4.67)



This can also be the case in a many-particle system. If we can diagonalize the many-particlehamiltonian, that is put it in the form

H =∑i

εia†iai + const =

∑i

εini + const, (4.68)

then we have solved the problem. The total energy is then the sum of the energy of theindividual oscillators, which are independent and are the true quasiparticles of the system.Mean field theory is all about making approximate factorizations of the interaction terms inorder that the hamiltonian can be transformed to a basis in which it is diagonal.


Lecture 5

Many-particle quantum systems

References: Introductions to second quantization can be found in many books, including Chapter 2 of

Taylor and Heinonen [7], ‘Many-Particle Physics’, by G. Mahan [8] and ‘Quantum Theory of Many Particle

Systems’, by Fetter and Walecka [3].

5.1 Quantum mechanics of many-particle systems

This lecture contains an introduction to second quantization, an essential tool for many-particle quantum mechanics. What is important to remember is that second-quantizednotation is used everywhere in the description of many particle quantum systems, and notjust by people who carry out calculations. It is therefore important that theorists andexperimentalists alike be familiar with it.

What happens in the many-particle problem (N 1, usually N ∼ 1020) if we try to workin the coordinate representation? In principle we can still do quantum mechanics this way.We can write down the many-particle wavefunction ψ(r1, σ1; ...; rN , σN ; t), which is now afunction of 4N + 1 variables. (3N coordinates, N spins, and a time variable.) The followingstep would be to write down the Schrodinger equation

Hrψ(r1, σ1; ...; rN , σN ; t) = i~∂

∂tψ(r1, σ1; ...; rN , σN ; t), (5.1)

where

Hr =∑i

− ~2

2m∇2i +

∑i

U(ri) +1

2

∑i6=j

V (ri − rj) + ... (5.2)

Forget for the moment that such an equation would be impossible to write down, let alonesolve. Our next step might be to expand the wavefunction in a convenient basis set, builtfor instance from the one-particle complete basis φα(r), in terms of which any function inthe single-particle Hilbert space can be expanded:

f(r) =∑α

cαφα(r). (5.3)


34 LECTURE 5. MANY-PARTICLE QUANTUM SYSTEMS

In principle, the many-particle wavefunction can be expanded in a set of products of thesingle-particle basis functions

ψ(r1, ..., rN , t) =∑

α1...αN

cα1...αN (t)φα1(r1)...φαN (rN). (5.4)

Provided the original single-particle basis is orthonormal, we now have a complete and or-thonormal basis for theN -particle Hilbert space: φα1...αN (r1, ..., rN) = φα1(r1)...φαN (rN).But, in the case of a system of electrons, we are dealing with fermions. The many-particlewavefunction must therefore be antisymmetric under the interchange of the coordinates ofany pair of particles. As a result, we need to work in a much smaller subspace of the originalN -particle Hilbert space that contains only properly antisymmetrized wavefunctions.

φα1...αN (r1, ..., rN)→ φAα1...αN(r1, ..., rN) =

1√N !

∑P

(−1)Pφα1(rP1)...φαN (rPN ), (5.5)

where P runs over all possible permuations of (1, 2, ..., N). This can be written compactlyas a Slater determinant:

φAα1...αN(r1, ..., rN) =

1√N !

∣∣∣∣∣∣∣φα1(r1) . . . φαN (r1)

......

φα1(rN) . . . φαN (rN)

∣∣∣∣∣∣∣ (5.6)

Each Slater determinant hasN ! terms. We now have a way of expanding any proper fermionicwavefunction:

ψ(r1, ..., rN , t) =∑

α1...αN

cα1...αN (t)φAα1...αN(r1, ..., rN). (5.7)

We showed in the previous lecture on single particle quantum mechanics that completeinformation about the state of a quantum system can be obtained by computing matrixelements. How difficult will this task be in the N -particle case? In our hamiltonian we havetwo types of operators:

• Single particle operators:

A =N∑i=1

Ai → N terms (5.8)

• Two particle operators:

B =1

2

N∑i6=j=1

Bij →N(N − 1)

2terms (5.9)

A general matrix element will have the following form:

〈ψ|O|ψ〉 =

∫dr1

∑σ1

. . .

∫drN

∑σN

ψ∗(r1, σ1; . . . ; rN , σN ; t)Oψ(r1, σ1; . . . ; rN , σN ; t) (5.10)


5.2. SECOND QUANTIZATION 35

In the best possible case, our wavefunction will be a single Slater determinant. Then

〈ψ|O|ψ〉 → (N integrals +N spin sums )×N !×(

NN(N−1)

2

)×N ! (5.11)

This is clearly impossible to carry out, even in the simplest case. Fortunately, these difficul-ties are due entirely to the fact that we insisted on working in the coordinate representation.In what follows we will see that all these problems go away when we use second quantizationand work in the number representation.

5.2 Second quantization

In the previous section we showed that wavefunctions become extremely complicated whenwe work with large numbers of identical interacting particles and the task of evaluatingmatrix elements between them is essentially impossible. Interactions, and the requirementsof the Pauli exclusion principle, introduce strong correlations between the positions of theparticles.1 In the presence of these correlations, it is no longer possible to say which single-particle states are occupied, independently of the motion of the rest of the particles in thesystem. However, we can always use the single-particle states to form a basis set for themany-particle Hilbert space — the set of antisymmetrized product functions:

φAα1...αN(r1, ..., rN) =

1√N !

∑P

(−1)Pφα1(rP1)...φαN (rPN ). (5.12)

These states clearly have the proper symmetry under particle exchange. However, we showedthat we cannot work with these Slater determinants explicitly — they are impossible to con-ceive of using for a system of N ∼ 1020 particles. Also, the Schrodinger wavefunctioncontains a feature that makes no sense at all — we label the particles as though they weredistinguishable, and then promptly remove that distinguishability by antisymmetrizing thewavefunction. It is the process of antisymmetrizing that introduces all the algebraic com-plexity. Taking a hint from the single-particle quantum harmonic oscillator, we look for adifferent representation that focuses on operators instead of coordinate wavefunctions. Wenow introduce the occupation number representation, which never labels the indistinguish-able particles, and has the added virtue that it treats states with different total numbers ofparticles on exactly the same footing.

5.2.1 Occupation number representation

Since the particles are identical, all we need to know is which single-particle states areoccupied and which are empty.

First step: Choose the single particle states |α〉 ≡ φα(r). These must be eigenstates of aseparable (i.e. noninteracting) hamiltonian. For example:

1Hence the name ‘strongly correlated electrons’, given to this general area of research. It is importantto note that correlation 6= interaction. Identical particles are correlated by virtue of the Pauli exclusionprinciple, even in the absence of interaction.



• Eigenstates of momentum and spin: |α〉 → |kσ〉• Eigenstates of position (and spin): |α〉 → |rσ〉, 〈rσ|r′σ′〉 = δ(r− r′)δσ,σ′

• Atomic-like orbitals centred on a particular atom in the crystal. (The only differ-ence from true atomic orbitals is that they must have been made orthogonal tothe orbitals centred on the other lattice sites. We will discuss this more when wecome to the tight-binding model.): |α〉 → |riσ〉

Second step: Order the states by index α. For example, if α is the energy, then orderthe states by increasing single-particle energy: E1 < E2 < ... < Eα < ... It doesn’tmatter what ordering scheme we use — all physical observables we calculate will beindependent of the ordering scheme — as long as we are consistent. This must includethe ordering scheme for the spin labels as well.

Third step: Use abstract occupation-number vectors |n1, n2, ..., nα, ...〉 as the basis vec-tors of the many-particle Hilbert space. (i.e. state 1 has n1 particles, state 2 has n2

particles, ... and so on.) For fermions, nα = 0, 1. For bosons, nα = 0, 1, 2, ...

Example: |0, 1, 0, 0, 1, 0, 1, 0, 0, ...〉 represents a state with identical fermions in states φ2, φ5

and φ7. This is equivalent to the Slater determinant

|0, 1, 0, 0, 1, 0, 1, 0, 0, ...〉 ≡ 1√3!

∣∣∣∣∣∣φ7(a) φ7(b) φ7(c)φ5(a) φ5(b) φ5(c)φ2(a) φ2(b) φ2(c)

∣∣∣∣∣∣ . (5.13)

The rows of the determinant are written in reverse order. The reason for doing this will be-come clear below when we compare the Schrodinger and occupation number representations.

A very compact way of ensuring that our states have the proper fermion antisymmetry underparticle exchange is to use the Jordan–Wigner algebra. For each state |α〉, define a creationoperator c†α and an annihilation operator cα. These satisfy the anticommutation relations:

cα, c†α′

= cαc

†α′ + c†α′cα = δα,α′ , (5.14)

cα, cα′

= 0, (5.15)c†α, c

†α′

= 0. (5.16)

It follows from this that:

(cα)2 =(c†α)2

= 0 (Pauli principle) (5.17)(c†αcα

)2= c†αcαc

†αcα = c†α

(1− c†αcα

)cα = c†αcα (5.18)

Let nα = c†αcα. Therefore n2α = nα, so nα must have eigenvalues 0 and 1, as expected for

fermions.

We also need to define phase factors. Dropping the index α, let

c†|0〉 = |1〉 (5.19)

c†|1〉 = 0 (5.20)

c|0〉 = 0 (5.21)

c|1〉 = |0〉. (5.22)



A general antisymmetric basis state in the occupation number representation can then bewritten:

|n1, n2, ..., nα, ...〉 = (c†1)n1(c†2)n2 ...(c†α)nα ...|0〉, (5.23)

where |0〉 is the vacuum state or empty state.

Because the c†α anticommute, order is important!

5.2.2 Representation of states in second quantization

Here are some examples of many particle states in second quantized notation:

Filled Fermi sea:|FS〉 =

∏|k|<kF,σ

c†kσ|0〉. (5.24)

Bardeen–Cooper–Schrieffer superconducting state:

|BCS〉 =∏k

(uk + vkc

†k↑c†−k↓

)|0〉. (5.25)

This demonstrates that in the occupation number representation we can easily workwith states of mixed particle number, something we can not conveniently do in theSchrodinger representation.

Occupation number representation vs. Slater determinants:Let |2, 5, 7〉 = c†2c

†5c†7|0〉 be the state we wrote down before:

|2, 5, 7〉 ≡ 1√3!


∣∣∣∣∣∣ . (5.26)

c1 acting on |2, 5, 7〉 is clearly zero, because state 1 is unoccupied.

c1c†2c†5c†7|0〉 = (−1)3c†2c

†5c†7c1|0〉 = 0. (5.27)

(The phase factor (−1)3 comes from moving c1 through the rest of the operators.) Nowlook at the effect of c5 on the same state:

c5c†2c†5c†7|0〉 = (−1)c†2c5c

†5c†7|0〉 (5.28)

= (−1)c†2

(1− c†5c5

)c†7|0〉 (5.29)

= (−1)c†2c†7|0〉. (5.30)

Let’s look at the same operation carried out on the Slater determinant. First we haveto ‘bubble’ the row corresponding to the state we want to destroy to the bottom of thedeterminant.

|2, 5, 7〉 ≡ 1√3!


∣∣∣∣∣∣ =(−1)√

3!


∣∣∣∣∣∣ . (5.31)



(Swapping rows in the determinant introduces a minus sign each time.) The bottomrow and right column can then be removed:

|2, 7〉 =(−1)√

2!

∣∣∣∣ φ7(a) φ7(b)φ2(a) φ2(b)

∣∣∣∣ . (5.32)

General states:In general, the annihilation operator cα acting on a state |n1, n2, ..., nα, ...〉 removes aparticle and gives:

cα|n1, n2, ..., nα, ...〉 = (−1)Sαnα|n1, n2, ..., nα − 1, ...〉, (5.33)

where Sα =∑α−1

i=1 ni is the number of states that come before α in the ordering.Likewise, we can show that when the creation operator adds a particle we get:

c†α|n1, n2, ..., nα, ...〉 = (−1)Sα(1− nα)|n1, n2, ..., nα + 1, ...〉. (5.34)

For example, the number operator nα = c†αcα:

c†αcα|n1, n2, ..., nα, ...〉 = (−1)Sαnαc†α|n1, n2, ..., nα − 1, ...〉 (5.35)

= (−1)Sα(−1)Sαnα(1− (nα − 1))|n1, n2, ..., nα, ...〉 (5.36)

= (2nα − n2α)|n1, n2, ..., nα, ...〉 (5.37)

2nα − n2α = 1, nα = 1 (5.38)

= 0, nα = 0. (5.39)

5.2.3 Representation of operators in second quantization

One and two particle operators are represented in the following ways in second quantization:

One particle operators:

U =N∑i=1

U(ri) (first quantization) (5.40)

→∑αα′

〈α|U |α′〉c†αcα′ , (second quantization) (5.41)

where

〈α|U |α′〉 =∑σ

∫dr φ∗α(r)U(r)φα′(r). (5.42)

Two particle operators:

V =1

2

N∑i6=j=1

V (ri, rj) (first quantization) (5.43)

→ 1

2

∑αβα′β′

〈αβ|V |α′β′〉c†αc†βcβ′cα′ , (second quantization) (5.44)



where

〈αβ|V |α′β′〉 =∑σσ′

∫dr

∫dr′ φ∗α(r)φ∗β(r′)V (r, r′)φα′(r)φβ′(r

′). (5.45)

Note carefully the order of the electron operators.

5.2.4 Field operators

Now introduce the field operators:

ψ†(r) =∑α

φ∗α(r)c†α (5.46)

ψ(r) =∑α

φα(r)cα (5.47)

where the functions φα(r) form an orthonormal set. Using the anticommutation relationsfor cα and c†α, it is straightforward to show that

ψ(r), ψ†(r′)

= δ(r− r′) (5.48)ψ(r), ψ(r′)

= 0 (5.49)

ψ†(r), ψ†(r′)

= 0. (5.50)

For the most commonly encountered case, where the single particle basis |α〉 is the set ofmomentum eigenstates of spin projection σ, we have

ψ†σ(r) =1√V

∑k

e−ik·rc†kσ (5.51)

ψσ(r) =1√V

∑k

eik·rckσ. (5.52)

In terms of these the one- and two-particle operators become

U(r) →∑σ

∫dr ψ†σ(r)U(r)ψσ(r) (5.53)

V (r, r′) → 1

2

∑σσ′

∫dr

∫dr′ ψ†σ(r)ψ†σ′(r

′)V (r, r′)ψσ′(r′)ψσ(r), (5.54)

where the order of the operators is very important. (For one thing, this ordering preventsself interactions.) What we see is that the second quantized operators look very similar toexpectation values in Schrodinger quantum mechanics, with the state ψ and its conjugate ψ∗

replaced by the operators ψ and ψ†. The localized operators are what we get if we quantizethe Schrodinger field — hence the name second quantization, as the eigenvalues of particlenumber are now integers.



5.2.5 Examples of second quantized operators

Kinetic energy

T =N∑i=1

p2i

2m→ T =

∑σ

∫dr ψ†σ(r)

(− ~2

2m∇2

)ψσ(r) (5.55)

Density operator

n(r) =N∑i=1

δ(r−ri)→ n(r) =∑σ

∫dr′ ψ†σ(r′)δ(r−r′)ψσ(r′) =

∑σ

ψ†σ(r)ψσ(r) (5.56)

Total number of particles

N =

∫dr n(r) =

∑σ

∫dr ψ†σ(r)ψσ(r) =

∑ασ

c†ασcασ (5.57)

Spin operator

~S =N∑i=1

~Si =~2

N∑i=1

~σi →~2

∑σ,σ′

∫dr ψ†σ(r)~σσ,σ′ψσ′(r), (5.58)

where ~σ is the vector of Pauli matrices

~σ = (σx, σy, σz) , σx =

(0 11 0

), σy =

(0 −ii 0

), σz =

(1 00 −1

). (5.59)

In particular,

Sz =~2

∫drψ†↑(r)ψ↑(r)− ψ†↓(r)ψ↓(r)

=

~2

∫dr n↑(r)− n↓(r) , (5.60)

as expected.

Hamiltonian

H =∑σ

∫drψ†σ(r)

(− ~2

2m∇2 + U(r)

)ψσ(r)+

1

2

∑σσ′

∫dr

∫dr′ψ†σ(r)ψ†σ′(r

′)V (r−r′)ψσ′(r′)ψσ(r)

(5.61)

Note that the second quantized operators have the same form regardless of the number ofparticles, which is very convenient for many particle problems where we are often interestedin systems where the total number of particles is fixed only on average. The number ofparticles is contained in the wavefunctions, which we have already shown are very flexible inthis regard and are even capable of having mixed particle number.


Lecture 6

Applications of second quantization

6.1 The tight-binding model

A common starting point for the discussion of electrons in solids is the tight-binding model.(We will come back to the physics of this model in more detail later in the course, but for nowit serves as a good example.) Here we focus attention on a set of atomic orbitals arrangedon the sites of a lattice, with each orbital able to hold one electron of either spin. We havelocalized creation operators c†iσ that create an electron of spin projection σ at the site i. Atthe typical densities found in solids, the atomic orbitals have substantial overlap with theirneighbours, resulting in a finite amplitude for the electron to tunnel, or ‘hop’, from one siteto the next. The typical arrangement is shown below, for a simple square lattice of side a.

aa

j

i

Figure 6.1: An electron hopping from site j to site i on a simple square lattice of side a.

The time evolution depicted in the figure would be driven by a term in the hamiltonian ofthe form −tc†i↑cj↑, where t is the overlap between orbitals on neighbouring sites i and j. Notethat spin is conserved in this process. Including all such terms results in the hamiltonian

H = −t∑〈i,j〉,σ

c†iσcjσ, (6.1)

where the brackets 〈i, j〉 denote a sum over all sites and nearest neighbour combinations of i


42 LECTURE 6. APPLICATIONS OF SECOND QUANTIZATION

and j. With periodic boundary conditions, this sum guarantees the hamiltonian is hermitian,as it must be.

We now want to manipulate the hamiltonian into diagonal form — that is, we want to findthe noninteracting eigenmodes and their energies. We begin by noting that the intersitehopping and the spatial periodicity of the problem should result in running-wave solutions.We look for these by transforming into momentum space. Suppressing the spin index, wedefine creation and annihilation operators for electrons of wavevector k:

c†k =1√N

∑ri

eik·ric†i (6.2)

ck =1√N

∑ri

e−ik·rici. (6.3)

The inverse transform is defined:

c†i =1√N

∑k

e−ik·ric†k. (6.4)

We now substitute into the original hamiltonian.

H = −t∑〈i,j〉

c†icj (6.5)

= −t∑〈i,j〉

1√N

∑k

e−ik·ric†k1√N

∑q

eiq·rjcq (6.6)

A convenient way to do the nearest neighbour sum is to define a set of vectors δ such thatrj = ri + δ. For the 2D square lattice δ ∈ ±ax,±ay, where x and y are unit vectors.

H = − t

N

∑i

∑δ

∑k

∑q

e−i(k−q)·rieiq·δc†kcq (6.7)

The sum over the phase factors involving ri oscillates rapidly and vanishes unless k = q:∑ri

e−i(k−q)·ri = Nδk,q. (6.8)

We use this to perform two of the summations and put H into diagonal form:

H = −t∑δ

∑k

∑q

δk,qeiq·δc†kcq (6.9)

= −t∑δ

∑k

eik·δc†kck. (6.10)

The sum over δ we write out, and call εk:

εk ≡ −t∑δ

eik·δ (6.11)

= −2t (cos(kxa) + cos(kya)) . (6.12)

Finally,

H =∑k

εknk, nk = c†kck. (6.13)


6.2. THE JELLIUM MODEL 43

6.2 The jellium model

As another good example, we now consider the model problem of N electrons interacting witha fixed uniform background of positive charge, (hence ‘jellium’.) We will initially assume theCoulomb interaction is screened, with a Yukawa-type form. The hamiltonian of the problemcan be broken into three parts:

H = Hel + Hback + Hel-back. (6.14)

Only the electrons have any interesting dynamics — the background charge is just there tomaintain overall electrical neutrality.

Hel =N∑i=1

p2i

2m+

e2

4πε0

1

2

N∑i6=j=1

e−µ|ri−rj |

|ri − rj|(6.15)

µ−1 is the electrostatic screening length, needed here as a convergence factor in the integrals.We also have:

Hback =e2

4πε0

1

2

∫dr

∫dr′

n(r)n(r′)e−µ|r−r′|

|r− r′|, (6.16)

where n(r) is the density of positive charges = N/V = constant, from overall charge neu-trality. The interaction between the electrons and the positive background is given by

Hel-back = − e2

4πε0

N∑i=1

∫dr n(r)

e−µ|r−ri|

|r− ri|. (6.17)

By making the change of variables ρ = r− r′, we can immediately evaluate the energy of thepositive background.

Hback =e2

4πε0

1

2

∫dr

∫dr′

n(r)n(r′)e−µ|r−r′|

|r− r′|(6.18)

=

(N

V

)2e2

4πε0

1

2

∫dr

∫d3ρ

e−µρ

ρ(6.19)

=

(N

V

)2e2

4πε0

1

2V 4π

∫ ∞0

dρ ρe−µρ (6.20)

=N2

V

e2

2ε0

1

µ2. (6.21)

We can also evaluate the attractive potential between electrons and the background

Hel-back = − e2

4πε0

N∑i=1

∫dr n(r)

e−µ|r−ri|

|r− ri|(6.22)

= − e2

4πε0

N

V

N∑i=1

∫d3ρ

e−µρ

ρ(6.23)

= − e2

4πε0

N

VN4π

1

µ2(6.24)

= −e2

ε0

N2

V

1

µ2, (6.25)



where we have taken advantage of translational invariance. The jellium hamiltonian canthen be written

H = Hel −e2

2ε0

N2

V

1

µ2. (6.26)

Since this is a translationally invariant problem, we should use the |kσ〉 basis.

T →∑

kσ,k′σ′

〈k′σ′|T |kσ〉c†k′σ′ckσ, (6.27)

where

〈k′σ′|T |kσ〉 =

∫dr

e−ik′·r√V

(− ~2

2m∇2

)eik·r√Vδσ,σ′ (6.28)

= δσ,σ′~2k2

2m

1

V

∫dr ei(k−k′)·r (6.29)

= δσ,σ′~2k2

2mδk,k′ . (6.30)

The kinetic energy is therefore diagonal in wavevector

T =∑kσ

~2k2

2mc†kσckσ. (6.31)

Using the rule for two-particle operators, we can write the electron–electron interaction interms of the plane wave states.

V → 1

2

∑k1σ1,k2σ2

∑k3σ3,k4σ4

〈k1σ1,k2σ2|V |k3σ3,k4σ4〉c†k1σ1c†k2σ2

ck4σ4ck3σ3 , (6.32)

where

〈k1σ1,k2σ2|V |k3σ3,k4σ4〉 =

∫dr

∫dr′

e−ik1·r√V

e−ik2·r′

√V

V (r− r′)δσ1,σ3δσ2,σ4

eik3·r√V

eik4·r′

√V

(6.33)

= δσ1,σ3δσ2,σ4

e2

4πε0V 2

∫dr

∫dr′ ei(k3−k1)·rei(k4−k2)·r′ e

−µ|r−r′|

|r− r′|.(6.34)

Let ρ = r− r′ ⇒ r′ = r− ρ.

〈k1σ1,k2σ2|V |k3σ3,k4σ4〉 = δσ1,σ3δσ2,σ4

e2

4πε0V 2

∫dr ei(k3−k1+k4−k2)·r

∫d3ρ e−i(k4−k2)·ρ e−µρ

ρ.

(6.35)The first integral is a momentum-conserving delta function, and the second is the Fouriertransform of the Yukawa potential with respect to the momentum transfer, so

〈k1σ1,k2σ2|V |k3σ3,k4σ4〉 = δσ1,σ3δσ2,σ4

e2

4πε0Vδk1+k2,k3+k4

4π

|k4 − k2|2 + µ2. (6.36)


6.2. THE JELLIUM MODEL 45

We introduce q = k4 − k2 and redefine the momenta in terms of it:

k1 = k + q (6.37)

k2 = k′ − q (6.38)

k3 = k (6.39)

k4 = k′, (6.40)

so that the momentum-conserving delta function will automatically be satisfied. Then

V =e2

8πε0V

∑kk′q

∑σσ′

4π

q2 + µ2c†k+q,σc

†k′−q,σ′ck′σ′ckσ.

We treat the q = 0 term separately:

c†kσc†k′σ′ck′σ′ckσ = −c†kσc

†k′σ′ckσck′σ′ (6.41)

= −c†kσ[δk,k′δσ,σ′ − ckσc†k′σ′

]ck′σ′ (6.42)

= nkσnk′σ′ − δk,k′δσ,σ′nkσ. (6.43)

Therefore

V (q = 0) =e2

8πε0V

∑kk′

∑σσ′

4π

µ2(nkσnk′σ′ − δk,k′δσ,σ′nkσ) (6.44)

=e2

2ε0

1

µ2

[N2

V− N

V

]. (6.45)

This cancels the constant we already had, provided µ−1 L, where V = L3. Since theenergy no longer depends on the screening length µ−1, we can set µ to zero, and the jelliummodel can finally be written

H =∑kσ

~2k2

2mc†kσckσ +

e2

2ε0V

∑kk′

∑q6=0

∑σσ′

1

q2c†k+q,σc

†k′−q,σ′ck′σ′ckσ. (6.46)


Lecture 7

The Hartree–Fock approximation

References: The Hartree–Fock approximation originated in atomic physics, but is just as easy to understand

in the context of electrons in solids. It is discussed in most solid–state textbooks.

7.1 A model two-electron system.

A very good example of the physics of interacting electrons is provided by a simple modelsystem of an atom or molecule with two single-particle orbitals and two electrons. Let theorthogonal single particle orbitals be φ1(r) and φ2(r) and assume, to begin with, that thesesingle-particle orbitals are unchanged by the interaction. An important simplification arisesbecause the Coulomb interaction is independent of the spin of the particles. The operatorsfor total spin S and spin projection Sz will therefore commute with the hamiltonian, implyingthat the two-electron wavefunctions will factorize into separate spatial and spin parts, andthat s and sz will be good quantum numbers for the two-electron states.1 There are fourpossible spin states:

Triplet S = 1

| ↑↑〉1√2

(| ↑↓〉+ | ↓↑〉)| ↓↓〉

Sz = 1Sz = 0Sz = −1

(7.1)

Singlet S = 0 1√2

(| ↑↓〉 − | ↓↑〉) Sz = 0 (7.2)

The notation | ↑↓〉 means spin up for electron one and spin down for electron two. Thesinglet state is odd under particle exchange, and the triplet states are even. As a result, thesinglet state must be paired with an even spatial wavefunction, and the triplet state with aspatially odd wavefunction, in order that the total wavefunction have overall antisymmetryunder exchange of particle coordinates. Since we must construct our states from only two

1This would not be the case if there were a spin–orbit term αL.S in the hamiltonian. The states wouldthen be eigenstates of total angular momentum J and projection Jz.


48 LECTURE 7. THE HARTREE–FOCK APPROXIMATION

single-particle states, the only possibilities are:

ψT(r1, r2) =1√2

(φ1(r1)φ2(r2)− φ2(r1)φ1(r2)) (7.3)

ψS(r1, r2) =1√2

(φ1(r1)φ2(r2) + φ2(r1)φ1(r2)) (7.4)

The subscripts T and S label triplet and singlet wavefunctions respectively.

The requirement of antisymmetry under particle exchange introduces an important differencebetween the singlet and triplet wavefunctions. The electrons in the triplet state keep furtherapart than they would if they had been independent, and the electrons in the singlet statestay closer together. We therefore expect the triplet state to have a lower Coulomb energythan the singlet, which we can shown explicitly by evaluating the expectation value of thehamiltonian. When we do this we find that

〈H〉S,T = E1 + E2 + EHartree ± Eexchange, (7.5)

where the +/− signs are for singlet and triplet respectively. E1 and E2 are the single particleenergies. The last two terms come from the matrix element of the Coulomb interaction. Thefirst term is the Hartree energy, and is just the interaction energy between the charge densitiesof the single particle states.

EHartree =

∫dr

∫dr′ |φ1(r′)|2 e2

4πε0|r− r′||φ2(r)|2 (7.6)

The second term has no classical analogue. It is called the exchange energy.

Eexchange =

∫dr

∫dr′ φ∗2(r)φ∗1(r′)

e2

4πε0|r− r′|φ1(r)φ2(r′) (7.7)

Despite the fact that the Coulomb interaction is independent of the spin of the electrons, weend up with a spin-dependent term in the energy due solely to the requirement of antisym-metry in the wavefunction. This is the origin of magnetism in solids.

7.2 The Hartree–Fock approximation

It is generally impossible to find the true N -particle ground state, so we need to comeup with an approximation scheme. One way is to use a variational approach. We choosea simpler, approximate form for the many-electron wavefunction ψ, and then optimize itso that it minimizes 〈ψ|H|ψ〉, the total energy, subject to the constraints imposed on thewavefunction by its approximate form.

In the Hartree–Fock method, we work with the simplest possible many-body wavefunctionfor fermions — a single Slater determinant. Although this is a drastic approximation — thetrue wavefunction is a linear combination of many different Slater determinants — it has thevirtue that it will allow us to carry out calculations.


7.2. THE HARTREE–FOCK APPROXIMATION 49

Having decided to throw away all but a single Slater determinant, we are now faced with thechoice of which single-particle orbitals to construct it from. While it may seem obvious thatthese should be the single-particle states with the lowest kinetic energy, this is often not thebest choice. We actually have the freedom to choose any basis set φα(r), and we shouldmake the choice that minimizes 〈ψ|H|ψ〉. 2

Solving a Hartree–Fock problem is a variational procedure, and the Hartree–Fock energywill always be greater than the true ground-state energy. The usual approach is to use aniterative method, repeating until a self-consistent solution is found. It is quite possible thatduring this process stationary states of 〈ψ|H|ψ〉 will be found that do not give the minimumHartree–Fock energy. The procedure should therefore include a way of testing that thesolution is a global minimum of 〈ψ|H|ψ〉. Unfortunately, these other stationary states havelittle to do with the true excited states of the system. We will show below that the problemcan be cast into a set of Hartree–Fock equations, which have the form of coupled nonlocal,nonlinear Schrodinger equations.

The easiest and most honest way to derive the Hartree–Fock equations is to use secondquantization. Assume that we already know which single-particle states minimize the totalenergy, and that these are the first N states drawn from a complete, orthonormal basis |α〉.The corresponding electron operators are cα and c†α. The Hartree–Fock wavefunction|HF〉 is the Slater determinant

|HF〉 = c†1c†2...c

†N |0〉. (7.8)

The hamiltonian, in general, will have a one-body part H1 and a two-body part H2.

H = H1 + H2 (7.9)

=∑αα′

〈α|T + U |α′〉c†αcα′ +1

2

∑αβα′β′

〈αβ|V |α′β′〉c†αc†βcβ′cα′ (7.10)

The one-body energy is

〈HF|H1|HF〉 =∑αα′

〈α|T + U |α′〉〈HF|c†αcα′|HF〉 (7.11)

=∑

occupied αα′

〈α|T + U |α′〉δαα′ (7.12)

=N∑α=1

〈α|T + U |α〉. (7.13)

〈HF|c†αcα′|HF〉 = δαα′ , if |α〉 is occupied, because |HF〉 is a single Slater determinant.

The interaction term is

〈HF|H2|HF〉 =1

2

∑αβα′β′

〈αβ|V |α′β′〉〈HF|c†αc†βcβ′cα′ |HF〉 (7.14)

2We need to be very careful at this point that we don’t overlook any possibilities. The states φα(r) neednot be the single-particle states with the lowest kinetic energy, and the resulting many-body wavefunctionneed not have the full symmetry of the hamiltonian.



Initial guess for the single particle states |α>

Compute VH, VE

Solve the Hartree-Fock equationsFind new single particle states |α>

Is the solution self-consistent?

Done

No

Yes

Figure 7.1: The iterative self-consistency procedure for Hartree–Fock calculation.

There are two possibilities:

α = α′, β = β′ : (7.15)

〈HF|c†αc†βcβcα|HF〉 = 1 |α〉, |β〉 occupied (7.16)

α = β′, β = α′ : (7.17)

〈HF|c†αc†βcαcβ|HF〉 = −〈HF|c†αc

†βcβcα|HF〉 = −1 |α〉, |β〉 occupied (7.18)

The interaction term is therefore

〈HF|H2|HF〉 =1

2

N∑α6=β=1

〈αβ|V |αβ〉 − 〈αβ|V |βα〉

(7.19)

If α = β, c2α = 0. We don’t need to worry about this explicitly, because the direct and

exchange terms will cancel. Writing the total energy in terms of the single particle orbitals,we have

〈H〉 =N∑α=1

∫dr φ∗α(r)

[− ~2

2m∇2 + U(r)

]φα(r) (7.20)

+1

2

N∑α,β=1

∫dr

∫dr′ φ∗α(r)φ∗β(r′)V (r− r′) [φα(r)φβ(r′)− φα(r′)φβ(r)] (7.21)


7.3. HARTREE–FOCK THEORY FOR JELLIUM 51

If |ψ〉 = |HF〉 is the best single Slater determinant, then it will correspond to a minimum ofthe total energy. Using a Lagrange multiplier E to enforce the normalization constraint, weneed to minimize 〈ψ|H|ψ〉 − E〈ψ|ψ〉 with respect to all the single particle orbitals φα(r):

δ

δφ∗α(r)

[〈ψ|H|ψ〉 − E〈ψ|ψ〉

]= 0. (7.22)

This leads directly to the Hartree–Fock equations. For each α, we have:− ~2

2m∇2 + U(r)

φα(r) + VH(r)φα(r)−

∫dr′ VE(r, r′)φα(r′) = Eφα(r), where (7.23)

VH(r) =N∑α=1

∫dr′ V (r− r′)|φα(r′)|2 Hartree potential (7.24)

VE(r) =N∑α=1

V (r− r′)φ∗α(r′)φα(r) exchange potential (7.25)

These equations are solved iteratively.

7.3 Hartree–Fock theory for jellium

In some cases we can use symmetries of the hamiltonian to guess the correct single-particlestates φα(r). For instance, if the hamiltonian doesn’t depend explicitly on spin, we mightexpect a paramagnetic state

|HF〉 = c†1↑c†1↓c†2↑c†2↓...c

†N2↑c†N2↓|0〉. (7.26)

However, this can be dangerous, as the lowest energy state can be a ‘broken-symmetry’ state,such as the ferromagnetic state

|FM〉 = c†1↑c†2↑...c

†N↑|0〉. (7.27)

We now want to apply the Hartree–Fock method to the jellium model. From translationalsymmetry, we can guess that

|HF〉 =∏

|k|<kF,σ

c†kσ|0〉. (7.28)

From the last lecture, the jellium hamiltonian is

H =∑kσ

~2k2

2mc†kσckσ +

e2

2ε0V

∑kk′

∑q6=0

∑σσ′

1

q2c†k+qσc

†k′−qσ′ck′σ′ckσ (7.29)

The kinetic energy is already diagonal in (kσ), so

〈HF|T |HF〉 =∑kσ

~2k2

2m〈HF|c†kσckσ|HF〉 (7.30)

=∑kσ

~2k2

2mΘ(kF − |k|), (7.31)



0.2 0.4 0.6 0.8 1 1.2 1.4

0.2

0.4

0.6

0.8

1

F(x)

x

logarithmicsingularity

Figure 7.2: Plot of the function F (x) = 12

+ 1−x2

4xln∣∣1+x

1−x

∣∣.where Θ(x) is the unit step function.

In the interaction term there is only one possibility that will give a nonzero contribution to〈HF|V |HF〉:

k + q = k′ ⇒ q = k′ − k and σ = σ′. (7.32)

The direct term, q = 0, has already been used to balance the interaction of the electronswith the positive background. This implies that k 6= k′. We therefore have

〈HF|V |HF〉 =e2

2ε0V

∑k6=k′

∑σ

1

|k− k′|2〈HF|c†k′σc

†kσck′σckσ|HF〉 (7.33)

=e2

2ε0V

∑k6=k′

∑σ

1

|k− k′|2〈HF|c†k′σ

[δk,k′ − ck′σc†kσ

]ckσ|HF〉 (7.34)

= − e2

2ε0V

∑k6=k′

∑σ

1

|k− k′|2〈HF|nk′σnkσ|HF〉 (7.35)

= −2e2

2ε0V

∑|k|<kF

∑|k′|<kF

1

|k− k′|2(7.36)

Convert one of the sums to an integral:

〈HF|V |HF〉 = −∑|k|<kF

e2

ε0V

V

(2π)32π

∫ 1

−1

d(cos θ)

∫ kF

0

k′2dk′

k2 + k′2 − 2kk′ cos θ(7.37)

Evaluating this integral allows us to write the single particle energy ε(k) as

ε(k) =~2k2

2m− e2

2π2ε0kF F

(k

kF

), (7.38)

where

F (x) =1

2+

1− x2

4xln

∣∣∣∣1 + x

1− x

∣∣∣∣ . (7.39)


7.3. HARTREE–FOCK THEORY FOR JELLIUM 53

0.5 1 1.5

-2

-1

1

2

ε/εF

k/kF

kinetic energytotal energy

Figure 7.3: Plots of the single particle energy ε(k) with and without the exchange energy.The exchange term substantially lowers the energy and increases the bandwidth. Neitherthe increased bandwidth nor the singular form of the energy dispersion are supported byexperiments such as photoemission spectroscopy, electrical transport and thermodynamics.

In the Hartree–Fock approximation, the one-electron levels of the interacting system are stillplane waves, but their energy is lowered by an exchange interaction that acts only betweenelectrons of the same spin projection. If the single particle energies are summed over alloccupied states, we get the total energy of the N electron system. This can be shown to be

Etotal = N

[3

5εF −

3

16

e2kF

π2ε0

](7.40)

The energy dispersion ε(k) has a logarithmic singularity in the slope at k = kF. This leadsto a depression of the density of states at the Fermi level to zero, called a Coulomb gap. Forelectron systems at metallic densities, this behaviour is unphysical, with no experimentalevidence to support it. The origin of the problem can be traced back to the long-rangenature of the Coulomb interaction, which causes divergences in the interaction strength atsmall momentum transfer. We will see in the next lecture that charge screening by theelectron gas changes the bare Coulomb interaction to a short-range repulsive potential. Thesingularities in the theory then disappear.


Lecture 8

Screening and the random phaseapproximation

References: The Thomas–Fermi theory of screening is discussed in most solid-state textbooks. This presen-

tation of the random phase approximation follows Chapter 2 of Taylor and Heinonen [7].

8.1 Screening

Screening is one of the simplest and most important examples of electron–electron inter-action. Imagine introducing a positive test charge, with charge density ρext(r), into a freeelectron gas, and holding it fixed in position. It will attract the nearby electrons, creatingan excess of negative charge in its vicinity, compensating its own charge. How effective isthis, and over what range does it occur?

Introduce two potentials: φext(r), due solely to the external charge, and φtot(r), due to all thecharges in the metal, including the external charge. Both potentials obey Poisson’s equation:

−∇2φext(r) =ρext(r)

ε0, (8.1)

−∇2φtot(r) =ρtot(r)

ε0. (8.2)

Let ρtot(r) = ρext(r) + ρind(r), where ρind(r), the induced charge density, is what we wantto calculate. φext(r) and φtot(r) are related by the dielectric function εr(r − r′), which in ahomogeneous system is just a function of separation.

φext(r) =

∫dr′εr(r− r′)φtot(r′) (8.3)

In real space the situation is quite complex. The dielectric function εr(r − r′) is a nonlocalresponse function, requiring a convolution integral. The situation becomes much simpler ifwe work in reciprocal space. We define the Fourier transform

fq ≡∫

dr e−iq·rf(r). (8.4)


56 LECTURE 8. SCREENING AND THE RANDOM PHASE APPROXIMATION

After Fourier transformation Eq. 8.3 becomes an algebraic equation.

φextq = εr(q)φtot

q (8.5)

Now introduce the charge susceptibility χq, which is defined by

ρindq = χqφ

totq . (8.6)

In Fourier space, the Poisson equations become

q2φextq =

ρextq

ε0, (8.7)

q2φtotq =

ρtotq

ε0. (8.8)

Rearranging we get:

ρindq = χqφ

totq = ε0q

2(φtotq − φext

q

)(8.9)

⇒ φtotq = φext

q

1

1− χq/ε0q2(8.10)

⇒ εr(q) = 1− χq

ε0q2. (8.11)

8.2 Thomas–Fermi theory

We can make a simple approximation to calculate the induced charge ρind(r) from the po-tential φtot(r), which will be accurate as long as φtot(r) varies slowly in space — that is, aslong as q kF. We assume that the effect of the total potential is to shift all the singleelectron energies by −eφtot(r). This is completely equivalent to a local shift in the chemicalpotential

µ→ µ(r) = µ+ eφtot(r). (8.12)

The chemical potential determines the electron density, with all single particle states filledto the chemical potential at zero temperature. A local shift in the chemical potential hasthe effect:

n(r)→ n(r) + δn(r), (8.13)

where

δn(r) =dn

dµδµ(r) =

dn

dµe φtot(r). (8.14)

The induced charge is

ρind(r) = −e δn(r) (8.15)

= −e2 dn

dµφtot(r). (8.16)


8.2. THOMAS–FERMI THEORY 57

µ0

δn(r)

φext(r)

φtot(r)µ(r)

Figure 8.1: The response of the electron gas to an external charge.

dn/dµ is just the density of states at the Fermi level D(εF).

ρind(r) = −e2D(εF)φtot(r) (8.17)

⇒ χq = −e2D(εF), (8.18)

independent of q. Therefore, the q-dependent dielectric function is

εr(q) = 1 +e2

ε0q2D(εF) (8.19)

= 1 +k2

0

q2, (8.20)

k20 =

e2

ε0D(εF). (8.21)

When the external charge is a point charge

φext(r) =Q

4πε0r(8.22)

φextq =

Q

ε0q2(8.23)

φtotq =

1

εr(q)φextq (8.24)

=Q

ε0

1

q2 + k20

. (8.25)

When Fourier transformed back to real space, this gives a screened, short-ranged, Yukawa-type potential:

φtot(r) =Q

4πε0re−k0r. (8.26)

k0 is typically of the order of kF. At metallic densities the Coulomb interaction is thereforestrongly screened for distances greater than 1 A.



8.3 The density operator

In the following, spin does not play a role in the interactions, so the spin index will besuppressed. Were we to consider a magnetic system, spin could be included and the sameapproach used to discuss spin density waves.

We saw in Lecture 5 that we can define fermion field operators in terms of momentum spaceoperators

ψ†σ(r) =1√V

∑k

e−ik·rc†kσ (8.27)

ψσ(r) =1√V

∑k

eik·rckσ (8.28)

and that these can be used to measure the particle density at point r

ρ(r) ≡ n(r) = ψ†σ(r)ψσ(r). (8.29)

The density operator is defined to be proportional to the Fourier transform of the particledensity.

ρq ≡1

V

∫e−iq·rρ(r)dr (8.30)

=1

V

∑k

c†kck+q. (8.31)

The normalization is chosen so that ρ0 gives the average particle density. The adjoint oper-ator is ρ†q = ρ−q. For the electron gas, the density operators commute: [ρq, ρq′ ] = 0.1

The density operators are useful because we can always express the electron–electron inter-action in terms of them. In Lecture 6 we derived the jellium hamiltonian. With spin indicessuppressed this is

H =∑k

εkc†kck +

e2

2ε0V

∑kk′

∑q6=0

1

q2c†k+qc

†k′−qck′ck, (8.32)

where εk = ~2k2/2m is the kinetic energy. Using the density operators the jellium hamilto-nian can be rewritten[7]

H =∑k

~2k2

2mc†kck +

e2

2ε0V

∑q6=0

1

q2

(V 2ρ†qρq −N

), (8.33)

with N the total particle number. In general, the interaction term of any hamiltonian willbe diagonal in the Fourier components of the particle density. Only in rare circumstances,however, will density fluctuations of well-defined wavenumber also diagonalize the kineticenergy. We will return to this point in the context of spin–charge separation, which occursin one-dimensional metals. In that case both the kinetic and potential energy are diagonalin the basis of spin and charge density waves.

1In Lecture 25 we will study the Luttinger–Tomonaga model of spin–charge separation in a one-dimensional metal. There the density operators do not always commute. In that case, the noncommutationarises from the somewhat artificial assumption that the one-dimensional energy band consists of occupiedstates extending down to infinite negative energy.


8.4. THE RANDOM PHASE APPROXIMATION 59

8.4 The random phase approximation

In the noninteracting case, the excited states of the electron gas are particle–hole excitationsabout the Fermi surface. We would like to know what the excited states of the interactingsystem are. We assume they exist and that they are created by the operator b†.

We start with the many-body system in its ground state |ψ〉. The ground state energy ε0 isgiven by

H|ψ〉 = ε0|ψ〉. (8.34)

If b† creates an excitation, then

Hb†|ψ〉 = (ε0 + εb)b†|ψ〉 (8.35)

or(Hb† − b†H)|ψ〉 = εbb

†|ψ〉. (8.36)

If this holds in general (not just for the ground state) then

[H, b†] = εbb†. (8.37)

This is the algebra for a bosonic raising operator, which we saw earlier in the context of theharmonic oscillator.

What is b†? A first guess might be that

b† = c†p+qcp. (8.38)

This takes an electron from state |p〉 and places it into state |p + q〉. For the noninteractinghamiltonian

H0 =∑k

εkc†kck (8.39)

this choice works. In that case

[H0, c†p+qcp] = (εp+q − εp)c†p+qcp. (8.40)

The situation changes somewhat when we include the Coulomb interaction. At large mo-mentum transfers the kinetic energy dominates the Coulomb energy, and particle–hole exci-tations are still good approximate excited states. However, at small q the interaction termbecomes too important to ignore. We then have to try a different approach, by summingthe particle–hole operator c†p+qcp over all p. This is just the density operator

ρ†q =∑p

c†p+qcp (8.41)

Now when we evaluate the commutator of ρ†q with the hamiltonian, the interaction termmakes no contribution because density operators commute. The commutator is just the sumover all momenta of Eq. 8.40.

[H, ρ†q] =∑p

(εp+q − εp)c†p+qcp. (8.42)



It is still not clear we are on the right track, even approximately. We are looking for somethingof the form

[H, ρ†q] ≈ ~ωρ†q. (8.43)

If indeed that is approximately what we have, then

[H, [H, ρ†q]] ≈ (~ω)2ρ†q. (8.44)

The double commutator is evaluated in Taylor and Heinonen, with the final result that, forthe jellium model,

[H, [H, ρ†q]] =1

V

∑p

(~2

2m(2p · q + q2)

)2

c†p+qcp +V

2

∑q′ 6=0

Vq′~2q · q′

m(ρq′−qρ

†q′ − ρ

†−q′ ρ−q−q′),

(8.45)where Vq ≡ e2/ε0V q

2 are the matrix elements of the Coulomb potential.

All the physics is contained in this expression. We are primarily interested in small q, as thisis where the particle–hole excitations cease to be good eigenstates. The first term goes tozero like q2 so is unimportant in the long wavelength limit. In the second term, one Fouriercomponent of the particle density stands out from the rest. This is the zeroth component, ρ0,which is just the average density of particles. Therefore terms containing ρ0 will dominatethe sum. Because q′ = 0 is omitted from the sum (this term was used to cancel the energyof the positive background in the jellium model) ρ0 only appears when q′ = ±q. Omittingthe rest of the terms is called the ‘random phase approximation’ or RPA. We are left with

[H, [H, ρ†q]] ≈ e2

2ε0q2

~2q2

m(ρ0ρ

†q + ρ†qρ0) (8.46)

=~2e2

ε0mρ0ρ†q (8.47)

= (~ω)2ρ†q (8.48)

The frequency of the charge density excitations is

ω2 =e2ρ0

ε0m, (8.49)

which is just the classical plasma frequency ωp.

There is another way of looking at what we have just done. In the ‘equation of motionmethod’ we make use of the fact that the time dependence of an operator is generated bythe hamiltonian. We then have that

i~∂

∂tO = [O, H]. (8.50)

What we have shown above is that, at long wavelengths

∂2

∂t2ρ†q + ω2

pρ†q = 0. (8.51)

That is, charge fluctuations of definite wavenumber are oscillator-like solutions to the model.


8.5. COLLECTIVE EXCITATIONS OF THE ELECTRON GAS 61

8.5 Collective excitations of the electron gas

The relevant excitations at long wavelengths are collective motions of the electron gas, calledplasmons. We can include them in an energy–momentum diagram along with electron–holeexcitations, which occupy the hatched region. The plasmons become heavily damped in thatregion.

Figure 8.2: Excitations of the electron gas, from Mahan[8].


Lecture 9

Scattering and periodic structures

In the last few lectures we have concentrated on systems of interacting electrons and haveignored the crystalline structure of real solids. We will now take the opposite approach andfocus on the effect of a periodic potential on the electronic structure. For the time being wewill ignore the Coulomb interaction, which will make the problem entirely tractable. In realsolids we will very often get away with doing this, as in many metals the electron–electroninteraction energy is smaller than we would calculate for a bare Coulomb interaction, dueto the phenomenon of screening. Later in the course we will see another reason for treatingthe electrons as weakly interacting particles — Landau’s Fermi liquid theory — which showsthat in many systems there is a one-to-one mapping of the electrons of a noninteractingsystem onto the weakly interacting quasiparticles of the full many-body hamiltonian.

The hamiltonian of the crystal, in the absence of electron–electron interactions, is:

H =N∑i=1

p2i

2m+ U(ri), (9.1)

where U(ri) is a one body potential with the full periodicity of the crystal lattice. Weimmediately notice that this hamiltonian is separable, which allows the N electron problemto be recast in terms on N completely independent one-electron problems. We saw inLecture 7 that for any Slater determinant the expectation value of the one-body part ofthe hamilonian is just the sum of N single-particle energies, despite the strong correlationsbetween particle positions that are built into the Slater determinant. As a result, fermioncorrelation in the many-body wave function only affects the interaction terms, which we arenow going to ignore. This means that, as far as the one-body terms go, the problem ofN electrons really does split up into N separate single-particle problems, all of which areidentical. At this level, fermion correlations only come into the problem through the Pauliexclusion principle, by determining which single particles levels are occupied and which areempty.


64 LECTURE 9. SCATTERING AND PERIODIC STRUCTURES

9.1 The Born approximation

Before treating electrons moving in a periodic potential, it is useful to look at the problem ofa wave scattering from a distributed system. We will consider the problem of electron waves,but the same approach could be taken for the scattering of light, sound, neutrons, etc. Allthat would be required is a different form of the potential. Here we consider the case of aquantum particle scattering weakly from a potential V (r). The Schrodinger equation is:(

− ~2

2m∇2 + V (r)

)ψ(r) = E ψ(r). (9.2)

This can be rewritten:

(∇2 + k2)ψ(r) =2m

~2V (r)ψ(r), (9.3)

where k =√

2mE/~2. Let the incident wave be A exp(iki · r). It is a solution of

(∇2 + k2)ψ(r) = 0. (9.4)

We can write a formal solution to Eq. 9.3 as

ψ(r) = Aeiki·r +

∫dr′G(r− r′)

2m

~2V (r′)ψ(r′), (9.5)

where G(r) is the Green’s function of Eq. 9.3. That is

(∇2 + k2)G(r) = δ3(r). (9.6)

Although this is formally a solution, it contains ψ(r) on both sides. It does, however, providea useful way of obtaining a perturbative solution to Eq. 9.3 by iteration. This procedure iscalled the Born approximation.

It turns out that G(r) has the form of a spherical wave,

G(r) = − eikr

4πr, (9.7)

which can be seen by substituting into Eq. 9.6 and using

∇2

(1

r

)= −4πδ3(r). (9.8)

(It is a good exercise to verify this result — if you do so, use the spherical coordinateexpressions for div, grad and ∇2 = ∇ · ∇.)

Using this we obtain

ψ(r) = Aeiki·r − m

2π~2

∫dr′

eik|r−r′|

|r− r′|V (r′)ψ(r′). (9.9)


9.2. PERIODIC STRUCTURES 65

|r− r′| =(r2 − 2r · r′ + r′

2) 1

2(9.10)

= r

(1− 2r · r

′

r+r′2

r2

) 12

(r is a unit vector) (9.11)

→ r

(1− r · r

′

r

)as r →∞ (9.12)

We need to retain both these terms in the exponent, but only the first in the denominator:

eik|r−r′|

|r− r′|→

exp(ikr(1− r · r′

r))

r(9.13)

=1

reikre−ikf ·r′ , (9.14)

where kf = kr is the outgoing wavevector.

So

ψ(r)→ Aeiki·r − meikr

2π~2r

∫dr′e−ikf ·r′V (r′)ψ(r′). (9.15)

For weak scattering, the wavefunction in the sample is comprised mainly of the incomingwave. So, to first order in the potential, we can substitute the incoming wave into theintegral.

ψ(r) = Aeiki·r − Am eikr

2π~2r

∫dr′ei(ki−kf)·r′V (r′). (9.16)

That is, the scattering amplitude is proportional to the Fourier transform of the potentialwith respect to the momentum transfer q = ki − kf .

f(θ, φ) ∝∫

dr eiq·rV (r). (9.17)

9.2 Periodic structures

In solid state physics we are interested in taking the Fourier transform of periodic functions,either to calculate diffraction conditions, in which case the periodic function is the electrondensity or spin density, or to find out which plane wave states are mixed when we switch onthe periodic potential.

9.2.1 Convolution as a means of replication

The calculation of Fourier transforms of periodic functions is greatly simplified if we introducethe idea of convolution. The convolution of f1(r) and f2(r) is

F (r) = f1(r)⊗ f2(r) ≡∫

dr′ f1(r′)f2(r− r′). (9.18)



We will always be interested in the case where one of the functions is a set of δ-functions.To see why, let f2(r) = δ(r−R).

F (r) = f1(r)⊗ f2(r) ≡∫

dr′ f1(r′)δ((r−R)− r′) (9.19)

= f1(r−R) (9.20)

Convolution of f(r) with a δ-function replicates f(r) at a different position. We now havea way of representing arbitrary periodic functions: we make a new function, which is finiteonly in the vicinity of one of the lattice points (i.e., within one unit cell of the lattice) andconvolve it with a set of δ-functions centred at each of the lattice points, thereby tiling thefunction periodically over the whole volume of the crystal.

9.2.2 The convolution theorem

We often want to calculate the Fourier transform of a convolution:

G(q) =

∫dr e−iq·rF (r), (9.21)

where F (r) = f1(r)⊗ f2(r).

G(q) =

∫dr e−iq·r

∫dr′ f1(r′)f2(r− r′) (9.22)

Let r′′ = r− r′.

G(q) =

∫dr′′ e−iq.(r′′+r′)

∫dr′ f1(r′)f2(r′′) (9.23)

=

∫dr′′ e−iq·r′′f2(r′′)

∫dr′ e−iq·r′f1(r′) (9.24)

In other words, the transform of a convolution = the product of the transforms.

A similar result can be derived for the transform of a product — it is the convolution of thetransforms. (You should prove this as an exercise.)

9.3 The reciprocal lattice

A Bravais lattice is defined by a set of three fundamental (or primitive) translation vectorsa1, a2 and a3. If we choose as the origin one of our lattice points, then the set of latticepoints are the vectors

R = n1a1 + n2a2 + n3a3, (9.25)

where ni are integers. The most convenient unit cell to use is the Wigner–Seitz cell, thesmallest volume enclosed by the intersection of the perpendicular bisecting planes of thelattice vectors. A function with the same periodicity of the lattice has the property

f(r + Ri) = f(r), ∀ lattice vectors Ri. (9.26)


9.3. THE RECIPROCAL LATTICE 67

It can therefore be writtenf(r) = f0(r)⊗

∑i

δ(r−Ri), (9.27)

where f0(r) = f(r) within the Wigner–Seitz cell and is zero everywhere else.

The Fourier transform g(q) is then the product

g(q) =

∫dr e−iq·rf0(r)×

∫dr′ e−iq·r′

∑i

δ(r′ −Ri) (9.28)

= g0(q)×∑i

e−iq·Ri . (9.29)

g0(q) is called the structure factor. In the summation we are adding together terms withdifferent phases. These will cancel (add to zero) unless the Bragg condition is satisfied:

q ·Ri = 2πm, (9.30)

for all the Ri in the lattice, where m is an integer. The special values of q that satisfy thiscondition lie on a lattice, which is called the reciprocal lattice. (It is a lattice, because if wefind any two vectors that satisfy Eq. 9.30, then their sum also satisfies Eq. 9.30.)

The primitive vectors of the reciprocal lattice are given by

b1 = 2πa2 × a3

a1 · a2 × a3

and cyclic permutations. (9.31)


Lecture 10

The nearly free electron model

0

Gk

k - G

1/2G

0

k

1/2G

k - 1/2G

Figure 10.1: Bragg scattering.

We saw in the last lecture that scattering froma periodic structure results in Bragg reflection.The scattering amplitude, as a function of themomentum transfer q = k−k′, is proportional tothe Fourier transform of the scattering potential,and the Fourier transform of a periodic structureis finite only at special values of q. These reflec-tions form a lattice, called the reciprocal lattice.Vectors in the reciprocal lattice are labelled G.The lattice spacing is determined by the period-icity of the scattering potential, and the strengthof the individual reflections is determined by thedetails of the potential or density within a singleunit cell of the scatterer. In particular, when theprimitive unit cell contains more than one atom,the crystal structure consists of a Bravais lattice+ a basis, and we expect the structure factor (theFourier transform of the basis) to cancel some ofthe reflections completely in some instances.

The conditions for Bragg reflection are:

• energy conservation: k2 = k′2

• conservation of crystal momentum: k =k′ + G, where G is a reciprocal lattice vector.

Together, these conditions imply

k · G =|G|2. (10.1)

That is, the projection of k onto G lies halfway along G. The locus of points satisfying thiscondition therefore form a set of planes — Bragg planes — which cut the reciprocal latticevectors perpendicularly through their midpoints. These planes divide k-space into zones,


70 LECTURE 10. THE NEARLY FREE ELECTRON MODEL

called Brillouin zones. The first Brillouin zone is the Wigner–Seitz cell of the reciprocallattice and is the set of all points in k-space closer to G = 0 than any other reciprocal latticevector.

10.1 Bloch’s theorem

In a crystal, the potential of the ion cores is periodic — it has discrete translational symmetry:

U(r + R) = U(r), (10.2)

where R is any vector in the real-space lattice. As a result, the hamiltonian commutes withthe translation operator TR, which has the effect

TRψ(r) = ψ(r + R). (10.3)

We can define TR in terms of the generator of translations, the momentum P:

TR = e−iP·R/~ (10.4)

T †R = eiP·R/~. (10.5)

Because [TR, H] = 0, TR and H must have common eigenfunctions, but with differenteigenvalues. Therefore, any energy eigenstate |ψ〉 must also be an eigenstate of TR.

T †R|ψ〉 = eiP·R/~|ψ〉 ≡ cR|ψ〉 (10.6)

We define the eigenvalue of translation to be cR. Now

〈r|T †R = 〈r + R|, (10.7)

so that

〈r|T †R|ψ〉 = 〈r + R|ψ〉 = cR〈r|ψ〉 (10.8)

⇒ ψ(r + R) = cRψ(r). (10.9)

Now operate on the plane wave state |k〉:

〈k|T †R = eik·R〈k|. (10.10)

Therefore〈k|T †R|ψ〉 = eik·R〈k|ψ〉 = cR〈k|ψ〉, (10.11)

so either 〈k|ψ〉 = 0 or cR = eik·R. In other words, the energy eigenstates |ψ〉 that havefinite overlap with the plane wave state |k〉 can be labeled by their Bloch wavevector k, andhave eigenvalue eik·R when acted on by the translation operator TR. There are infinitelymany different eigenstates with the same eigenvalue under translation, as k is only definedto within a reciprocal lattice vector G, and eiG·R = 1 . To distinguish between them, weneed to introduce a new quantum number, the band index n. However, because k and k+Gare equivalent, we can specify a unique and complete set of labels by restricting ourselves toa single Brillouin zone. In the following, we will use the convention followed in Ashcroft andMermin [4], where q is an arbitrary wavevector, and k = q + G is a wavevector in the firstBrillouin zone.


10.2. FREE ELECTRON IN 1–D AS A BLOCH WAVE 71

0

2

4

6

8

10

π/a 2π/a 3π/a−π/a−2π/a−3π/a

ε / (h2π2/2ma2)

q

Figure 10.2: One dimensional energy dispersion in the extended zone scheme.

10.2 Free electron in 1–D as a Bloch wave

In the limit where the periodic potential vanishes (U(r)→ 0), we must recover free electronswith wavefunction ψq(r) ∝ eiqr. These can be viewed as a special case of Bloch’s theorem, ifwe remember what our reciprocal lattice vectors G were before we switched off the potential.This will serve to demonstrate the connection between the band index n and the reciprocallattice vectors.

ψq(r) ∝ eiqr (10.12)

= eikreiGr (10.13)

= eikrein( 2πa

)r (10.14)

≡ ψn,k(r) (10.15)

(In higher dimensions it is not possible to define such a direct mapping between reciprocallattice vectors and the band index, but the same principle applies). If we plot the energydispersion ε(q) = ~2q2/2m as a function of q, we get the extended zone scheme shown inFig. 10.2. If we plot the dispersion as a function of k, we have the reduced zone scheme,where the energy bands ε(k,G) = ~2(k−G)2/2m are folded back into a single zone, as shownin Fig. 10.3. A third possibility is to use the repeated zone scheme, good for demonstratingthe periodicity of the energy bands, but then we will over-count the states.



2

4

6

8

10ε / (h2π2/2ma2)

0 π/a−π/aq

Figure 10.3: One dimensional energy dispersion in the reduced zone scheme.


10.3. NEARLY FREE ELECTRONS 73

10.3 Nearly free electrons

We can use the periodicity of the ionic potential to expand U(r) in a set of discrete Fouriercomponents at the reciprocal lattice vectors G.

U(r) =∑G

UGeiG·r (10.16)

U(r + R) =∑G

UGeiG·reiG·R (10.17)

= U(r), since eiG·R = 1. (10.18)

10.3.1 Properties of UG

• U(r) is real:

U(r) = U∗(r) (10.19)∑G

UGeiG·r =∑G

U∗Ge−iG·r (10.20)

=∑G

U∗−GeiG·r (10.21)

⇒ UG = U∗−G. (10.22)

• If the crystal has inversion symmetry:

U(r) = U(−r) (10.23)∑G

UGeiG·r =∑G

UGe−iG·r (10.24)

=∑G

U−GeiG·r (10.25)

⇒ UG = U−G. (10.26)

While the ionic potential is always real, crystals do not always have inversion symmetry. Ifa crystal does, we can make the Fourier components of the potential real by choosing theorigin r = 0 to be a centre of inversion symmetry.

What are the UG?

U(r) =∑G

UGeiG·r (10.27)∫cell

dr U(r)e−iG′·r =∑G

UG

∫cell

dr ei(G−G′)·r (10.28)

=∑G

UG Vcell δG,G′ (10.29)

= U ′GV

N(10.30)

⇒ UG =N

V

∫cell

dr U(r)e−iG·r, (10.31)



where N is the number of unit cells in the crystal and V is the volume of the crystal.

10.4 The Schrodinger equation in momentum space

We can now write the Schrodinger equation[− ~2

2m∇2 + U(r)

]ψ(r) = Eψ(r) (10.32)

in momentum space, for a periodic potential. Let’s first Fourier-expand the wavefunction:

ψ(r) =∑q

cqeiq·r, (10.33)

remembering that q is an arbitrary wavevector. Substituting this into the Schrodingerequation we get ∑

q

cqeiq·r(~2q2

2m− E

)+∑q

cqeiq·r∑G

UGeiG·r = 0. (10.34)

We want to concentrate on each Fourier component separately, so rearrange and grouptogether terms with common plane-wave components:∑

q

cqeiq·r(~2q2

2m− E

)+∑q

∑G

cqUGei(q+G)·r = 0. (10.35)

Let q′ = q + G, and use the fact that the sums over q and G run to infinity:∑q

cqeiq·r(~2q2

2m− E

)+∑q′

∑G

cq′−GUGeiq′·r = 0. (10.36)

But q′ is just an index of summation, so we are free to relabel it q′ → q:

∑q

eiq·r

(~2q2

2m− E

)cq +

∑G

cq−GUG

= 0. (10.37)

The plane waves form an orthogonal set, so each term in the sum must separately be zero:(~2q2

2m− E

)cq +

∑G

cq−GUG = 0. (10.38)

We usually work in the reduced zone scheme, with q = k −K, where K is the reciprocallattice vector that maps q into the first Brillouin zone.(

~2(k−K)2

2m− E

)ck−K +

∑G

UGck−K−G = 0, (10.39)


10.5. THE PERIODIC POTENTIAL AS A WEAK PERTURBATION 75

or, if we redefine G→ G−K,(~2(k−K)2

2m− E

)ck−K +

∑G

UG−Kck−G = 0. (10.40)

The original problem separates into N independent problems, one for each allowed value ofk in the first Brillouin zone. Each problem has an infinite set of solutions constructed fromplane waves of wavevector k, and wavevectors differing from k by a reciprocal lattice vector.

ψk(r) =∑G

ck−Gei(k−G)·r (10.41)

States of this form are Bloch functions:

ψk(r + R) =∑G

ck−Gei(k−G)·reik·Re−iG·R (10.42)

= eik·Rψk(r). (10.43)

10.5 The periodic potential as a weak perturbation

We now imagine switching on the periodic potential as a weak perturbation. To zeroth order,our solutions are just free electron plane waves. All the UG are zero and we can write(

~2(k−K)2

2m− E

)ck−K = 0. (10.44)

For a given k and K, only one coefficient is nonzero, ck−K, and E = ~2

2m(k −K)2 ≡ ε0k−K.

We get one solution for each wavevector k and each band K.

10.5.1 The nondegenerate case

If, for some fixed wavevector k and a band corresponding to a particular reciprocal latticevector K1,

|ε0k−K1− ε0k−K| U,∀ K 6= K1, (10.45)

we can use nondegenerate perturbation theory. We want to solve(ε0k−K1

− E)ck−K1 +

∑G

UG−K1ck−G = 0. (10.46)

To first order, set ck−G = ck−K1δG,K1 .(ε0k−K1

− E)ck−K1 + U0ck−K1 = 0 (10.47)

The energies are just shifted by the average potential U0. We will set U0 = 0, withoutany loss of generality. Now calculate the shift in the ck−K to first order, again assumingck−G = ck−K1δG,K1 in the sum.(

ε0k−K − ε0k−K1

)ck−K +

∑G6=K

UG−Kck−K1δG,K1 = 0 (10.48)

⇒(ε0k−K − ε0k−K1

)ck−K + UK1−Kck−K1 = 0 (10.49)



Therefore

ck−K =UK1−K

ε0k−K1− ε0k−K

ck−K1 +O(U2). (10.50)

We can now substitute in the new coefficients to the original equation to obtain the shift inenergy to second order:(

E − ε0k−K1

)ck−K1 =

∑G6=K1

UG−K1ck−G (10.51)

=∑

G6=K1

UG−K1UK1−G

ε0k−K1− ε0k−G

ck−K1 (10.52)

=∑

G6=K1

|UG−K1|2


ck−K1 . (10.53)

The energy of the Bloch states, to second order, is

εk−K1 = ε0k−K1+∑

G6=K1

|UG−K1|2


. (10.54)

This shows that the perturbed energy levels repel each other. (If ε0k−G > ε0k−K1, then the

contribution to the shift is negative, and vice versa.)


Lecture 11

The nearly free electron model andtight binding

11.1 Degenerate electrons in a periodic potential

We will always have points in k-space where free electron bands from two or more differentreciprocal lattice vectors come close in energy. The situation is sketched in Fig. 11.1. Wethen have no choice but to consider all free electron bands on the same footing and solve thefull Master equation for the bands concerned:(

ε0k−K − E)ck−K +

∑G

UG−Kck−G = 0. (11.1)

To make the problem easier, we usually just focus on the near-degenerate states, ignoringbands that are further off in energy. (The correct approach is to carry out a consistencycheck — to make sure that the shifts in energy that result from the mixing of bands aresmaller than the separation in energy of any neglected bands.)

Figure 11.1: The shaded region shows the range of k-space where degenerate perturbationtheory must be used. (From Ashcroft and Mermin[4].)

We start by considering a simple example in one dimension — that of two bands mixing atthe zone boundary.


78 LECTURE 11. THE NEARLY FREE ELECTRON MODEL AND TIGHT BINDING

1/2K = π/a k

0

ε

2|U2π/a

|

Figure 11.2: The Fourier component of the periodic potential U 2πa

opens an energy gap atthe Bragg plane equal to twice its magnitude.

At the point k = πa, the bands associated with K = 0 and K = 2π

aare degenerate. We have(

ε0k − E U 2πa

U− 2πa

ε0k− 2π

a

− E

)(ck

ck− 2πa

)= 0 (11.2)

This has a nontrivial solution when(ε0k − E

) (ε0k− 2π

a− E

)−∣∣∣U 2π

a

∣∣∣2 = 0. (11.3)

That is, when

E =1

2

ε0k + ε0

k− 2πa±√(

ε0k − ε0k− 2πa

)2

+ 4∣∣∣U 2π

a

∣∣∣2 . (11.4)

Right at the zone boundary, k = πa, the energies are equal (ε0k = ε0

k− 2πa

) so E = ε0πa±∣∣∣U 2π

a

∣∣∣.That is, we open up an energy gap of magnitude 2

∣∣∣U 2πa

∣∣∣. This is sketched in Fig. 11.2.

If the potential is attractive, U 2πa< 0, the wavefunctions are

ψ−(πa

)∝ cos

(πxa

), (11.5)

ψ+(πa

)∝ sin

(πxa

). (11.6)

The modulus squared of these wavefunctions gives the real-space charge densities of thestates, which are plotted in Fig. 11.3. The results are physically appealing — in the lowerenergy state, the electrons are attracted to the ion cores, and vice versa for the high energystate. At the zone boundary, Bragg reflection makes the states equal mixtures of left- and


11.1. DEGENERATE ELECTRONS IN A PERIODIC POTENTIAL 79

a 2a-a-2a 0

|ψ-(x)|2

|ψ+(x)|2

Ions

Figure 11.3: Charge densities of the zone-boundary states ψ−(πa

)and ψ+

(πa

)

right-moving waves, resulting in standing waves. This is a general property of solutionsat the zone boundary. We will see later that these states have zero velocity and carry nocurrent. This is an important difference from free electrons. Away from the zone boundary,the states quickly become much more plane-wave like, have a more uniform charge density,and have finite velocity and carry currents. Figure 11.4 shows the scheme for constructingenergy bands in a one dimensional periodic potential.

In three dimensions we must consider the most general case. All the action still takes placeat Bragg planes, although now it is possible to have more than two free electrons bandsintersecting at the same point, as we can see in Fig. 11.5 for the face-centred cubic structure.For instance, we could have four near-degenerate bands, resulting from four reciprocal latticevectors K1, K2, K3 and K4. If we write out the nonzero terms from Eq. 11.1, we get thefollowing linear system.

ε0k−K1

− E UK2−K1 UK3−K1 UK4−K1

UK1−K2 ε0k−K2− E UK3−K2 UK4−K2

UK1−K3 UK2−K3 ε0k−K3− E UK4−K3

UK1−K4 UK2−K4 UK3−K4 ε0k−K4− E

ck−K1

ck−K2

ck−K3

ck−K4

= 0 (11.7)

This is a Hermitian matrix, because U−G = U∗G, and the (real) energy eigenvalues are foundby setting the determinant equal to zero.

In two and three dimensions, the periodic potential opens an energy gap all along the Braggplane. This is illustrated in Fig. 11.6, which shows a free electron Fermi circle intersectingthe edge of the first Brillouin zone.



Figure 11.4: Construction of energy bands in a one-dimensional periodic potential. (a) Theparabolic free-electron dispersion. (b) A second free-electron parabola is drawn, shifted bythe reciprocal lattice vector K. (c) The degeneracy of the two parabolas at k = K/2 is splitby the Fourier components of the potential UK and U−K . The bands repel and an energy gapis opened. (d) The portions of (c) corresponding to the original free-electron parabola. (e)The effect of all Bragg planes on the free electron parabola. This is known as the extendedzone scheme. (f) The same energy levels in the reduced zone scheme. (g) The energy levelsin the repeated zone scheme. (From Ashcroft and Mermin[4].)


11.2. THE TIGHT BINDING METHOD 81

Figure 11.5: Free electron bands of the FCC structure. (From Ashcroft and Mermin[4].)

11.2 The tight binding method

So far we have treated the periodic potential of the crystal lattice as a weak perturbationto otherwise free electrons. We now begin from the opposite point of view, by starting froma collection of N widely separated neutral atoms and bringing these together to form thecrystal. Initially, all the electrons will be in highly localized atomic orbitals, and the bandstructure will be trivial — a set of flat bands at the energies of the atomic levels, which willeach be N -fold degenerate. However, the orbitals will eventually start to have significantoverlap as the atoms approach one another. At this point, the atomic wavefunctions willcease to be good approximations to the energy eigenstates. The tight-binding approximationworks in this limit, where the overlap between orbitals on different atoms is small and theresulting eigenstates have strong atomic character.

Many people, most famously Philip Anderson[9], have been dismissive of the tight-bindingmethod. While there are many more powerful techniques available for ab-initio calculationsof electronic structure, the tight-binding method has many advantages of its own. Theseinclude:

• allowing the description, in an intuitive way, of energy bands and bonds in a widevariety of periodic systems.

• having a direct connection to the chemistry of the material, so that the effects of theshape and symmetry of the atomic orbitals can readily be seen.



Figure 11.6: (a) Free electron Fermi circle cutting a Bragg plane at k = K/2. (b) Deformationof the free electron Fermi surface near the Bragg plane. The ‘necking out’ of the Fermi surfaceto touch the Bragg plane will always occur. Note that the Fermi surface intersects the Braggplane perpendicular to it. (From Ashcroft and Mermin[4].)

• easy modification to permit the description of incommensurate systems and disorderedsystems.

• allowing a local description of transport phenomena. (When considered locally, theelectric potential is finite and easy to treat.)

• being the natural framework in which to introduce correlation effects, as these areusually the result of short-range forces.

• providing excellent parameterizations of the electronic structure of a solid, when theoverlap integrals are treated as phenomenological parameters to be taken from moredetailed calculations, or from fits to experimental data.

These last two points bear more comment. In most ‘interesting’ materials currently beingstudied, even a qualitative description of the electronic behaviour eludes a first-principlescalculation. We rely on model building to incorporate the essential physics, and tight-binding models provide a simple and transparent parameterization of the band-structure, towhich interaction terms can readily be added. Tight-binding should be the technique of firstresort for everyone except perhaps the professional band theorist.

11.2.1 Introduction to tight binding

Before getting into the formalism, it is good to start with a physical picture of the tight-binding method. Figure 11.7 shows a schematic model of a one-dimensional metal as acollection of potential wells, which need not be all the same depth, although they will bein a crystalline solid (or at least alternate in a regular pattern). Wells of random depthwould correspond to a highly disordered solid. Figure 11.7 also shows an electron, almost



1 2 3 4 5 6

φ(r)

Potential

Figure 11.7: A model of a one-dimensional metal.

trapped in one of the wells. The tight-binding approach begins by considering the atomiceigenstates, which are localized in individual wells, but, realizing that such states are nolonger eigenstates when the wells are brought together, allows for a finite probability for theelectron to leak into a neighbouring well. The rate at which it does determines the energydispersion of electrons propagating through the extended structure. The depth of the wellsin Fig. 11.7 doesn’t really indicate the magnitude of the potential, but rather the energies ofthe atomic bound states. Therefore, the one-dimensional potential could represent a chainof atoms stuck together. In a random potential, electrons can become trapped in one place.

To find out how electrons move through such a system, we suppose we know the individualatomic binding energies. We then introduce a set of states |i〉, where |i〉 is a state localizedat the ith atom. In order to make the mathematics as simple as possible, we choose theset |i〉 to be orthonormal. Such a choice is always possible by construction, althoughhow localized the states |i〉 are depends on the extent to which the original atomic statesoverlapped with their neighbours to begin with. The procedure is to use Gramm–Schmidtorthogonalization, starting from the original atomic eigenstates |ψi〉. Let’s concentrate onthe state at the origin, |ψ0〉, and its nearest neighbours |ψ1〉 and |ψ−1〉. We modifiy |ψ0〉 bysubtracting off the parts that aren’t orthogonal to the states on neighbouring sites.

|ψ0〉 → |ψ0〉 − |ψ1〉〈ψ1|ψ0〉 − |ψ−1〉〈ψ−1|ψ0〉 (11.8)

This can be extended over all the atoms in the chain, to get the orthonormalized functions|φi〉:

|φi〉 = A

(|ψi〉 −

∑j 6=i

|ψj〉〈ψj|ψi〉

), (11.9)

where A is the new normalization constant. The orthogonalized states are call Wannierorbitals, and are important in the formal development of the tight-binding method. Theprocedure is illustrated in Fig. 11.8. It works best when the original atomic states don’toverlap too strongly, in which case the Wannier functions are very similar to the originalatomic orbitals, and are well localized. If the overlap between neighbouring states becomestoo strong, the Wannier states start to become quite spread out. Another thing to note is



-6 -4 -2 2 4 6

0.2

0.4

0.6

0.8

-6 -4 -2 2 4 6

0.2

0.4

0.6

0.8

-6 -4 -2 2 4 6

0.2

0.4

0.6

0.8

-6 -4 -2 2 4 6

0.2

0.4

0.6

0.8

-6 -4 -2 2 4 6

-0.5-0.25

0.250.5

0.751

-6 -4 -2 2 4 6

-0.5-0.25

0.250.5

0.751

(a)

(b)

(c)

(d)

(e)

(f)

Figure 11.8: The process of orthogonalizing atomic orbitals. a) An atomic orbital φ(x) =A exp(−x2), centered on the origin. b) φ(x) is clearly not orthogonal to its neighbours.c) The orbitals after Gramm–Schmidt orthogonalization. d) A comparison of the orbitalsbefore and after orthogonalizing. e) When the lattice sites get closer together, the originalatomic orbitals overlap more strongly and it becomes difficult to keep the orthogonalizedorbitals well localized. f) The results of orthogonalizing closely spaced orbitals.



that the Wannier states are neither perfect eigenfunctions of the isolated potential wells, noreigenfunctions of the infinite collection of wells.

As a shorthand, let c†i be the electron creation operator for the state |φi〉. An approximatehamiltonian for an electron in the collection of potential wells is

Htb =∑i

Uic†ici, (11.10)

where Ui is the energy of the bound state at site i. It is a good thing that this is justan approximate hamiltonian, because it doesn’t do anything. An electron starting out ina state |φj〉 would just sit there forever. What is missing is the fact that the tails of thewavefunctions extend out into neighbouring wells, giving the electron a finite amplitude to‘leak’ or tunnel into an adjacent site. If we call this amplitude t, then we can allow for itseffect by adding terms like −tc†1c0 to the hamiltonian. (We also have to add the adjoint,−t∗c†0c1, to make the hamiltonian hermitian.) Doing this for all the sites, and consideringonly nearest neighbours, we have the full tight-binding model in one dimension.

Htb =∑i

−tc†ici+1 + c†i+1ci

+∑i

Uic†ici. (11.11)

In the simplest case, in which all the Ui are the same (= U0), we can solve the tight-bindingmodel by taking the Fourier transform of the Wannier orbitals, as we have already seen:

c†k =1√N

∑i

eikric†i . (11.12)

In that case the solution is

H =∑k

εknk, nk = c†kck, (11.13)

whereεk = −2t cos(ka) + U0. (11.14)


Lecture 12

The tight-binding approximation

Let’s put the tight-binding method on a more rigorous footing. Imagine we have a periodiclattice of identical atoms, with atomic hamiltonians Hatom, each with a set of discrete levelswith wavefunctions φn(r) and energies En. Our eigenstates must obey Bloch’s theorem, solet’s take a linear combination of atomic orbitals as our ansatz for the eigenstate of the fullcrystal Hamiltonian:

|k〉 = ψnk(r) =1√N

∑R

eik·Rφn(r−R), (12.1)

where R are the lattice vectors and N is the number of cells in the crystal. Specializing tothe case of a single orbital per lattice site, we can now calculate the energy of this state tofirst order in perturbation theory:

E(k) = 〈k|H|k〉 =1

N

∑Ri,Rj

e−ik.(Ri−Rj)〈φ(r−Ri)|H|φ(r−Rj)〉 (12.2)

=∑R

e−ik·R∫

drφ∗(r−R)Hφ(r). (12.3)

The wavefunctions are weakly overlapping, so we keep only the terms where the orbitals areeither on the same site, or on nearest neighbour sites connected by the vector ρ. The onsiteterm gives a contribution close to the atomic energy, since H ≈ Hatom in the vicinity of theorigin:

ε0 =

∫dr φ∗(r)Hφ(r) ≈ En. (12.4)

The overlap term between neighbouring orbitals is called the hopping integral:

t = −∫

dr φ∗(r− ρ)Hφ(r). (12.5)

As a result, the band energy is:

E(k) = ε0 − t∑ρ

eik·ρ. (12.6)


88 LECTURE 12. THE TIGHT-BINDING APPROXIMATION

For a cubic lattice of spacing a, which has six nearest neighbours, the dispersion relation is

E(k) = ε0 − 2t[cos(kxa) + cos(kya) + cos(kza)]. (12.7)

The bandwidth is 12t. Bandwidth is an important parameter, as it sets the scale of the kineticenergy of the electrons in the solid. The more atomic-like the structure is, the narrower theband, and the less electron kinetic energy. In narrow-band metals the potential energyis proportionally more important, and is why many narrow band systems have interestingmagnetic properties. The tight binding energy dispersion also allows us to say somethingabout the effective mass of the electrons. The top and bottom of the band are parabolic,and the effective mass m∗ is defined:

m∗ =

[1

~2

∂2E(k)

∂k2i

]−1

=~2

2ta2. (12.8)

We will later see that the effective mass defines the response of the Bloch electrons to externalfields. As a rule, the narrower the band, the heavier the mass.

12.1 Degenerate tight-binding theory

The calculation we have just seen is too simple to capture the physics of many real metals.We often need to form trial Bloch states from more than one type of atomic orbital, astwo or more atomic orbitals can be degenerate with one other. Even in the absence ofstrict degeneracy, tunneling of electrons between adjacent sites adds dispersion to the energyspectrum, giving a finite bandwidth equal to the number of neighbours × twice the hoppingintegral t. Any other atomic level that falls within or even close to this energy range mustbe included in the trial wavefunction, unless it is prevented from mixing by symmetry.Fortunately the general case is not much more difficult to set up. For each atomic orbitalφi(r), we construct a Bloch sum of wavevector k:

|Φki〉 =1√N

∑R

eik·R|φi(r−R)〉. (12.9)

Our trial wavefunction for the degenerate case is then the linear combination:

|ψk〉 =∑i

ci(k)|Φki〉, (12.10)

where the coefficients ci(k) are to be determined as part of the eigenvalue problem. Act onthis state with the hamiltonian.

〈ψk|H|ψk〉 = E(k)〈ψk|ψk〉 (12.11)∑i,j

c∗i (k)cj(k)〈Φki|H|Φkj〉 = E(k)∑i,j

c∗i (k)cj(k)〈Φki|Φkj〉 (12.12)


12.1. DEGENERATE TIGHT-BINDING THEORY 89

Call the matrix elements of the crystal hamiltonian Mij(k):

Mij(k) = 〈Φki|H|Φkj〉 (12.13)

=1

N

∑R,R′

e−ik.(R−R′)〈φi(r−R)|H|φj(r−R′)〉 (12.14)

=∑R

eik·R〈φi(r)|H|φj(r−R)〉. (12.15)

The overlap matrix elements Sij(k) = 〈Φki|Φkj〉 are approximately diagonal in the case ofwell localized orbitals. In fact, if the orbitals we began with were Wannier orbitals, thiswould be exactly the case — Sij(k) = δij. From now on we will make that assumption. Theproblem can be written as the matrix equation: M11(k)− E(k) · · · M1N(k)

......

MN1(k) · · · MNN(k)− E(k)

c1(k)

...cN(k)

= 0. (12.16)

We set the determinant to zero to get the energy eigenvalues and the coefficients ci(k) of theeigenstates.

It is common to express the crystal potential as a sum of spherically symmetric atomic-likepotentials Va(r−R) centred at the lattice positions.

H =p2

2m+∑R

Va(r−R) (12.17)

Then

Mij(k) =∑R

eik·R∫φ∗i (r)

[p2

2m+ Va(r) + V ′(r)

]φj(r−R)dr, (12.18)

whereV ′(r) =

∑R6=0

Va(r−R). (12.19)

If we use the fact that φi(r) is an eigenfunction of Hatom = p2/2m + Va(r) with energy Ei,then

Mij(k) = Eiδij +∑R

eik·R∫φ∗i (r)V ′(r)φj(r−R)dr. (12.20)

The R = 0 term gives the crystal field integral

Iij =

∫φ∗i (r)V ′(r)φj(r)dr. (12.21)

If V ′(r) is almost constant near r = 0, then Iij is a constant diagonal matrix and contributesa rigid energy shift of the entire band structure but does not influence the dispersion curves.For this reason it is often neglected. For the R 6= 0 terms, we usually limit the sum toa small number of neighbouring sites. (A more accurate model might require next-nearest



neighbours, and so on.) We also neglect three-center integrals in Mij(k), and limit thecalculation to two-centre integrals, which can be tabulated in terms of a small number ofindependent parameters. (All the quantum chemistry and orbital symmetries are containedin the two-centre integrals — these are explained and tabulated in Figs. 12.1 and 12.2.)Putting all this together we have

Mij(k) ≈ Eiδij +∑R

eik·R∫φ∗i (r)Va(r−R)φj(r−R)dr. (12.22)

Finally, if the unit cell contains more than one atom, which in general may be different, weneed to include Bloch sums centred at each of the atomic sites.

|Φkid〉 =1√N

∑R

eik·R|φi(r−R− d)〉. (12.23)

Note that it is not necessary to include d in the phase factor, as we have a phase degree offreedom in the wavefunction anzatz, thanks to the coefficients cid(k).

|ψk〉 =∑id

cid(k)|Φkid〉 (12.24)

In the following examples we will only consider crystals with a monatomic basis, but theextension to the multi-atom case is straightforward to implement.

In summary, the approximations made in the tight-binding method are:

• to assume we are working with orthogonal Wannier orbitals, not the eigenstates of thewell-separated atoms. In practice, for well-localized orbitals, the differences are small.

• we ignore integrals of the form

Iij =

∫φ∗i (r)V ′(r)φj(r)dr, (12.25)

whereV ′(r) =

∑R6=0

Va(r−R). (12.26)

V ′(r) is only finite away from the origin, where φi(r) should be rapidly becoming small.

• we ignore three centre integrals∫φ∗i (r)Va(r−R1)φj(r−R2)dr, R1 6= R2, (12.27)

as these are typically small.

We usually take the point of view that the tight-binding model is a phenomenological pa-rameterization of experimental results or a more accurate calculation, in which case theseapproximations do not matter. The great benefit of the tight-binding method is its sim-plicity, and the transparency with which we can see the effects of symmetry and chemicalstructure.



-4

-3

-2

-1

-4

-3

-2

-1

0 0Γ L X

E(k)/eV

Figure 12.1: s-like band in a face-centred cubic crystal. Γ = (0, 0, 0), X = (2πa, 0, 0), L =

(πa, πa, πa). Es = −1 eV, V (ssσ) = −0.2 eV.

12.1.1 Example 1: s-like band in a face-centred cubic crystal

We now use the tight-binding method to calculate the energy dispersion of an s-like band ina face-centred cubic crystal. We assume there are no orbitals that are near-degenerate withthe atomic s orbitals φs(r).

We begin by forming a single Bloch sum

|Φks〉 =1√N

∑R

eik·R|φs(r−R)〉. (12.28)

The energy dispersion is given by the expectation value of the hamiltonian in this state.

E(k) =〈Φks|H|Φks〉〈Φks|Φks〉

(12.29)

≈ Es +∑R

eik·R∫φ∗s(r)Va(r−R)φs(r−R)dr, (12.30)

where the sum is restricted to the 12 nearest neighbours, R = a2(±1,±1, 0), a

2(0,±1,±1), a

2(±1, 0,±1).

This involves a two-centre integral between s states, with Va(r) spherically symmetric. Look-ing this up in Fig. 12.1 or 12.2, we see that there is no dependence on bond angle. All 12integrals will be the same (nearest neighbour distance is a constant) and will be denotedV (ssσ). All that remains is to evaluate the phase factors, which we will call F (k).

F (k) =∑R

eik·R (12.31)



This consists of three similar parts, for xy, yz and xz.

Fxy(k) =∑±

e±ikxa/2∑±

e±ikya/2 (12.32)

= 4 coskxa

2cos

kya

2. (12.33)

F (k) = Fxy(k) + Fyz(k) + Fxz(k) (12.34)

= 4

(cos

kxa

2cos

kya

2+ cos

kya

2cos

kza

2+ cos

kxa

2cos

kza

2

). (12.35)

The energy dispersion, plotted in Fig. 12.1, is:

E(k) = Es + V (ssσ)F (k). (12.36)

12.1.2 Example 2: p-like bands in a face-centred cubic crystal

We now use the tight-binding method to calculate the dispersion of p-like bands in an FCCcrystal. The atomic p orbitals are threefold degenerate, so we need to use 3 Bloch sums.

|Φki〉 =1√N

∑R

eik·R|φi(r−R)〉, i = x, y, z (12.37)

According to our prescription, we need to set up a secular equation:∣∣∣∣∣∣Mxx(k)− E(k) Mxy(k) Mxz(k)

M∗xy(k) Myy(k)− E(k) Myz(k)

M∗xz(k) M∗

yz(k) Mzz(k)− E(k)

∣∣∣∣∣∣ = 0, (12.38)

where

Mij(k) = Eiδij +∑R

eik·R∫φ∗i (r)Va(r−R)φj(r−R)dr. (12.39)

The calculation is greatly simplified by the fact that there are only two independent matrixelements, Mxx(k) and Mxy(k), with the rest obtained by cyclic permutation of the indices.

The overlap integrals of p-orbitals have nontrivial angle dependence, expressed in terms ofthe direction cosines `i ≡ cos θi, where θi is the angle between the lattice vector R and thei axis. (i = x, y, z.) Let us first make a table of the direction cosines for the face-centredcubic structure.

R a2(±1,±1, 0) a

2(0,±1,±1) a

2(±1, 0,±1)

cos θx ± 1√2

0 ± 1√2

cos θy ± 1√2

± 1√2

0

cos2 θx12

0 12

sin2 θx12

1 12

cos θx cos θy ±12

0 0



-2

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

-2

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

ΓL X

E(k)/eV

Figure 12.2: p bands in a face-centred cubic crystal. Γ = (0, 0, 0), X = (2πa, 0, 0), L =

(πa, πa, πa). Ep = −1 eV, V (ppσ) = 0.2 eV and V (ppπ) = −0.02 eV.

Mxx(k) = Ep +∑R

eik·R∫φ∗x(r)Va(r−R)φx(r−R)dr. (12.40)

∫φ∗x(r)Va(r−R)φx(r−R)dr = `2

xV (ppσ) + (1− `2x)V (ppπ) (12.41)

= cos2 θxV (ppσ) + sin2 θxV (ppπ) (12.42)

Taking the direction cosines from the table and evaluating the phase factors in a similarway to Example 1, we have.

Mxx(k) = Ep+2 (V (ppσ) + V (ppπ))

cos

kxa

2cos

kya

2+ cos

kxa

2cos

kza

2

+4V (ppπ) cos

kya

2cos

kza

2(12.43)

Similarly,

Mxy(k) =∑R

eik·R∫φ∗x(r)Va(r−R)φy(r−R)dr. (12.44)

∫φ∗x(r)Va(r−R)φy(r−R)dr = `x`y (V (ppσ)− V (ppπ)) (12.45)

= cos θx cos θy [V (ppσ)− V (ppπ)] (12.46)

Paying careful attention to signs in the phase factor, we get:

Mxy(k) = −2 [V (ppσ)− V (ppπ)] sinkxa

2sin

kya

2. (12.47)



We use cyclic permutation to find the other matrix elements, and then solve the eigensystemnumerically, as a function of k. The resulting band structure has been plotted along twosymmetry directions in Fig. 12.2. We can see that Ep is just an additive offset, and thequalitative behaviour is controlled by the ratio V (ppπ)/V (ppσ).



Table 12.1: Two centre integrals, from Grosso and Parravicini[10].



Table 12.2: The definitive table of two-centre integrals[11]. l,m and n are the directioncosines for the x, y, and z axes respectively.


Lecture 13

Energy bands and electronic structure

So far we have used two different approaches to calculate the properties of electrons in crys-talline solids. In the first method we started with free electrons, which have plane-waveeigenstates, and then introduced the periodic potential of the ion cores as a weak pertur-bation. This approach focussed explicitly on how the symmetry of the solid modifies theone-electron Schrodinger equation. In the tight-binding method we began at the oppositeextreme, considering electron tunneling between atoms as a perturbation to the highly local-ized atomic eigenstates of a set of well-separated atoms. In this lecture we review what wehave done so far and then focus on the particular problem of understanding why some ma-terials are metals, while others are semiconductors or insulators. We will be able to addressthis issue using a simple geometric recipe for constructing energy bands and Fermi surfaces.

13.1 Review of the nearly free electron model

The main results for a single electron moving in a periodic potential are summarized belowand generalized for two- and three-dimensional problems.

• A periodic potential in real space can be decomposed into a set of discrete Fouriercomponents. These are finite only at special wavevectors G. The vectors G form alattice in reciprocal space.

U(r) =∑G

UGeiG·r (13.1)

• Every reciprocal lattice vector G has a corresponding Bragg plane, defined to be itsperpendicular bisector. That is, k · G = |G|/2.

• Along the Bragg plane defined by G, the free electron energies εk and εk−G are degen-erate:

~2k2

2m=

~2(k−G)2

2m. (13.2)

As a result, states near these planes are strongly mixed by the Fourier components ofthe potential UG and U−G, and an energy gap in k-space is opened at the Bragg plane.The energy gap is proportional to |UG|.


98 LECTURE 13. ENERGY BANDS AND ELECTRONIC STRUCTURE

0

Gk

k - G

1/2G 0

k

1/2G

k - 1/2G

Figure 13.1: Every reciprocal lattice vector G defines a Bragg plane.

(a) (b)

Figure 13.2: (a) The opening of an energy gap at a Bragg plane in one dimension. (b)Constant energy contours for a two-dimensional electron gas with a one dimensional periodicpotential. (From Taylor and Heinonen[7].)

• The Bragg planes break reciprocal space up into zones. The first Brillouin zone is theset of points in k-space that can be reached without crossing any Bragg planes. Thisis the Wigner–Seitz cell of the reciprocal lattice. The nth Brillouin zone is defined asthe set of points that can be reached from the origin by crossing n − 1 Bragg planes,but no fewer. The Brillouin zones of the square lattice are shown in Fig. 13.3.

• Each Brillouin zone contains one state, of each spin, for every unit cell in the real-spacelattice. As a result, if the crystal contains N primitive cells in real space, each Brillouinzone will be able to hold 2N electrons.

• Each Brillouin zone contains one branch, or band, of the energy dispersion εk.

• All higher-order Brillouin zones can be mapped back into the first Brillouin zone bytranslating them by reciprocal lattice vectors and, when this procedure is carried out,each Brillouin zone will exactly fill the first Brillouin zone. This is called the reducedzone scheme. The mapping procedure is shown in Fig. 13.3.


13.1. REVIEW OF THE NEARLY FREE ELECTRON MODEL 99

12 2

2

2 44

44

3 3

3 3

3 3

3 3

55

55

5

5

5

5

6 6

6 6

12 2

2

2

3 3

3 3

3 3

3 31st zone

2nd zone

3rd zone

Figure 13.3: Brillouin zones of the two-dimensional square lattice. Each Brillouin zone inthe extended zone scheme can be mapped back to precisely fill the Wigner–Seitz cell of thereciprocal lattice in the reduced zone scheme.

• Each zone is a primitive cell of the reciprocal lattice, so there exists a simple procedurefor constructing the different branches of the Fermi surface in the reduced zone scheme.

1. Draw the free electron sphere in the extended zone scheme. The radius of thesphere is the Fermi wavevector kF, determined by the density of electrons n.

2. Deform the sphere slightly near the Bragg planes. The effect of the periodic po-tential is to open gaps at the Bragg planes, requiring the constant energy surfacesto intersect Bragg planes at right angles. In the limit of weak potentials this stepcan be skipped.

3. Take the portion of the sphere lying in the nth zone and translate it by reciprocallattice vectors back into the first zone.

4. The resulting surface will be continuous and is a branch, or sheet, of the Fermisurface.



5. The effect of a weak potential is to round the sharp corners of the Fermi surfacesheets.

6. Stronger potentials can cause small pieces of the Fermi surface to disappear. Thesecan be electron-like or hole-like, as defined below.

12 2

2

2

3 3

3 3

3 3

3 3

1st zone 2nd zone

3rd zone

3rd zone

Figure 13.4: The Fermi surface of a square lattice of tetravalent ions. The first Brillouinzone is completely filled. The second Brillouin zone is over half full and contains a pocketof empty states surrounding the zone centre. The third Brillouin zone is only partly filledand consists of four small pieces near the zone corners. A redrawn third zone, centred onthe point (π/a, π/a), consists of a single, connected pocket of electron states.

13.2 Free electron Fermi surfaces in 2D

As an example, we consider a tetravalent metal with a simple square lattice of side a in twodimensions. There are four electrons per primitive cell so, allowing for spin degeneracy, thearea of the Fermi surface will precisely fill two Brillouin zones.

2

(2π

a

)2

= πk2F ⇒ kF =

√8π

a(13.3)

The Fermi surface is sketched in the extended zone scheme in Fig. 13.4. Pieces of the Fermisurface lying in higher-order zones are mapped back into the first Brillouin zone. The firstzone is completely filled and as a result is electrically inert. The second zone is largely filled,but has a small pocket of empty states in the centre. The third zone is only partly filled — itsarea corresponds precisely to the missing area in the second zone. The pieces of the Fermi


13.3. ELECTRONS AND HOLES 101

surface are located in the corners of the Brillouin zone, and appear not to be connected.However, there is nothing to say that when we map the Fermi surface pieces we have to mapthem into a zone centred on the origin of k-space. If we were to view the 3rd zone in therepeated zone scheme we would immediately see a better way to represent this piece of theFermi surface. We can choose a new zone, of the same area and shape as the first Brillouinzone, centred on the zone corner. With this choice, the 3rd zone Fermi surface is continuousand has a shape called a rosette.

2nd zone 3rd zone

Figure 13.5: Second zone hole pocket and third zone electron pocket for the square lattice.Fermi surface sheets are constant energy surfaces. The gradient of the energy is perpendicularto the Fermi surface, and points from filled to empty states.

Now consider the action of a weak periodic potential on the Fermi surface. This will opensmall energy gaps at the Bragg planes and cause the sharp corners of the Fermi surface piecesin the second and third zones to be rounded, as shown in Fig. 13.5.

A simple way of constructing the different Fermi surface pieces is to use the Harrison con-struction, shown in Fig. 13.6. This is a series of intersecting spheres of radius kF, eachcentred on a different point in the reciprocal lattice. The surfaces of the intersecting spheresdefine the different Fermi surface sheets.

13.3 Electrons and holes

We have seen above that there are two distinct types of Fermi surface sheets — those thatclose around filled states and those that close around empty states. There is a third kind aswell — open Fermi surface sheets, which do not close within the Brillouin zone. A surfacethat closes around empty states is called a hole surface. In this case the gradient of the energypoints into the surface. Surfaces that close around filled states are called electron surfaces.When we come to discuss electrical transport, we will see that hole surfaces respond toapplied fields as though they contain a small density of positively charged particles. OpenFermi surfaces have mixed character.



Figure 13.6: The Harrison construction in three dimensions.

13.4 Metals, semimetals, insulators and semiconduc-

tors

We are now in a position to see why some materials are metals and some are insulators.Depending on the number of electrons in the unit cell, and the size of the energy gapsopened along the Bragg planes by the periodic potential, it is possible to have situations inwhich all the bands (all the Brillouin zones) in a solid are either completely filled or empty.A filled Brillouin zone is electrically inert — there are no low energy excited states to maketransitions into, so the electron assembly cannot respond to low frequency fields by carryinga current. It is therefore an insulator. The different types of solid are outlined below andare sketched in Fig. 13.7.

• Metals. These are solids with at least one partially filled band. This can occur becausethere is an odd number of electrons per unit cell, making it impossible to completelyfill all occupied bands, or, when there is an even number of electrons per cell, becausethe energy gaps are small enough that there are several partially occupied bands. Thelatter case is called a compensated metal. It must be pointed out that this prescriptiondoes not always work — there are solids, such as NiO, with an odd number of electronsper unit cell, that form Mott insulating states due to Coulomb repulsion between theelectrons. The Mott insulator is one of the best examples of the breakdown of theindependent electron approximation.

• Insulators. To the extent that one-electron band structure provides a good descriptionof the states, these are solids with an even number of electrons per primitive cell,


13.4. METALS, SEMIMETALS, INSULATORS AND SEMICONDUCTORS 103

Insulator Metal Semimetal Semiconductor

Energy

Density of States

Figure 13.7: A schematic view of the density of states of various types of electronic solid.

and band gaps large enough that every Brillouin zone is either completely filled orcompletely empty.

• Semiconductors. These are small-gap insulators, in which the energy gap is lowenough (typically ≤ 1 − 2 eV) that a small but finite proportion of the electrons arethermally excited from filled to empty bands, creating a low density of mobile chargecarriers.

• Semimetals. These are compensated metals, with an even number of electrons perunit cell, that verge on being insulators. That is, they have a very small densityof holes in an otherwise filled band, and the same density of electrons in the next,almost empty, Brillouin zone. An example is bismuth metal. Carrier densities aretypically 3 to 4 orders of magnitude lower than in metals, with Fermi wavelengths thatare correspondingly larger. This makes these materials of interest for studying theclassical to quantum crossover that occurs when the thermal de Broglie wavelengthbecomes of the order of the interparticle spacing.

We close this lecture by looking at electronic structure from the tight-binding viewpoint.It is useful to imagine what would happen if the atoms of a solid were brought together ina controlled way, so that the lattice spacing a was a parameter which could be varied. Atlarge lattice spacings there would be very little overlap and the electronic structure wouldbe identical to that of the isolated atoms, although each atomic level would have an N -folddegeneracy. On reducing the lattice spacing, the atomic levels would broaden into bands.



Figure 13.8: (a) For large lattice spacing a the eigenstates are discrete atomic levels. Asthe lattice spacing is reduced the atomic levels broaden into bands. (b) Monovalent metalshave half-filled bands and are metallic. (c) Divalent atoms have a filled band at large latticespacings and are insulating. At smaller lattice spacings, band overlap leads to the formationof a compensated metal. (From Taylor and Heinonen[7].)

Fig. 13.8 illustrates this for monovalent atoms, which have a half-filled band and are usuallymetallic as a result, and for divalent atoms, which form insulators when the bandwidth issmall (at large a) but become metallic when the bands overlap at smaller lattice spacings.


Lecture 14

Electronic structure calculations

Nearly free electrons and tight binding are not accurate methods of electronic structure cal-culation, but both are good for illustrating the basic principles. They are commonly usedto provide simple parameterisations of energy bands, with the parameters determined bymore accurate, first-principles calculations or by comparison with experiment. The nexttwo lectures will look at some of the more accurate techniques used for calculating electron-ics structure, and point out other considerations, such as relativistic effects, that must beincluded in real calculations.

14.1 Orthogonalized plane waves and pseudopotentials

Experiments such as de Haas–van Alphen spectroscopy reveal the Fermi surface of gold to beroughly spherical. Taken with other experiments such as heat capacity, we can view a pieceof gold as a nearly empty box containing one electron per atom. But the atomic numberof gold is 79. The other 78 electrons seem somehow to just crowd around the nucleus andscreen its charge from the outside. The problem of how to treat these other electrons, and themuch-reduced core potential they leave behind, was solved by Philips and Kleinman in 1959,when they introduced the concept of pseudopotentials. Pseudopotentials do two things: theyshow how the real Schrodinger equation of a solid can be mapped on to a much simpler one,providing a justification of the nearly free electron model; and they substantially improvecomputational techniques, allowing the band structure of very complicated materials withmany electrons to be calculated using smooth potentials and far fewer basis states.

14.1.1 Example: the need for pseudopotentials

Imaging using a brute force approach to solve the Schrodinger equation for silver in thenearly free electron approximation(

ε0k−K − E)ck−K +

∑G

UG−Kck−G = 0. (14.1)

How many plane wave states would be needed to represent the valence electrons, which aredrawn from atomic 5s states? The 5s state has 4 nodes, in a distance of about 0.5 A. This


106 LECTURE 14. ELECTRONIC STRUCTURE CALCULATIONS

implies that k ≡ 2π/λ = 2π/0.25 A ≈ 24 A−1. The lattice constant of silver is a = 4 A, sothe smallest reciprocal lattice vector is G0 = 4π/a ≈ 3 A−1. In order to Fourier expand the5s state, we need Fourier components ranging from −8G0...8G0, in all three directions. Thisis a total of 173 ≈ 5000 plane wave states!

By introducing pseudopotentials, we will get pseuodwavefunctions with far fewer nodes.Away from the ion core these can be chosen to be the same as the real 5s valence states.The result is that we will need a much smaller basis set of plane waves.

14.2 Orthogonalized plane waves

While not particularly useful in band-structure calculations, orthogonalized plane wavesprovide a clear illustration of idea of the pseudopotential method. The approximation relieson our being able to divide the electrons into two groups: core electrons and conductionelectrons. The core electrons are localized around particular atomic sites, and we assumewe know their wavefunctions |ψc〉. The basic idea is that a pure plane wave is not a verygood approximation to the conduction electron wavefunction because it is not orthogonal tothe core states. As it makes its way past each ion core, the wavefunction of the conductionelectrons must wiggle rapidly in order not to overlap with any of the electrons in the core.However, we know that plane waves must be reasonably good solutions away from the ioncores, because they do a good job of describing the properties of many solids. Realizingthis, we introduce new basis states that retain plane-wave-like characteristics away from theion cores, but have been made orthogonal to the core states using the Gramm–Schmidtprocedure.

|kps〉 = |k〉 −∑

c

|ψc〉〈ψc|k〉, (14.2)

where ‘ps’ stands for pseudo k-state, and the sum is over all occupied core levels |ψc〉. Nowwrite the Schrodinger equation in the |kps〉 basis.

(H − ε)|kps〉 =

(p2

2m+ U − ε

)|kps〉 (14.3)

=

(p2

2m+ U − ε

)|k〉 −

∑c

|ψc〉〈ψc|k〉

(14.4)

=

(p2

2m− ε)|k〉+

(U −

∑c

(εc − ε)|ψc〉〈ψc|

)|k〉 (14.5)

=

(p2

2m+ Ups − ε

)|k〉, (14.6)

where we have used the fact that(p2

2m+ U

)|ψc〉 = εc|ψc〉 (14.7)


14.2. ORTHOGONALIZED PLANE WAVES 107

U(r)

Ups

(r)

Φ(r)

Φps

(r)

Figure 14.1: The true potential U(r) has a wavefunction Φ(r) for the valence electrons thatoscillates rapidly near the core. The pseudopotential Ups(r) has a wavefunction Φps(r) thatis smooth near the core but approximates the true wavefunction far from the core region.

to define the pseudopotential

Ups = U −∑

c

(εc − ε)|ψc〉〈ψc|. (14.8)

The final result is that the full Schrodinger equation in the pseudo plane-wave basis mapsonto a much less singular Schrodinger equation in a pure plane-wave basis.

(H − ε)|kps〉 = (Hps − ε)|k〉. (14.9)

The pseudopotential is useful because it justifies working with a small number of plane wavesin the nearly free electron model. Ups is smooth, so its Fourier components are small exceptfor the first few reciprocal lattice vectors. A minor disadvantage is that the pseudopotentialin nonlocal but, to a good approximation, Ups can be replaced in calculations by a localpotential. It also depends on the unknown eigenenergy ε, requiring a self-consistent solution.

In actual electronic structure calculations, each atomic potential is replaced by a model pseu-dopotential that has the same scattering amplitude for valence electrons as the real potential.This is shown schematically in Fig. 14.1. The real- and pseudopotentials are identical out-side the core, so the radial parts of the wavefunction are proportional. A norm-conservingpseudopotential results in pseudowavefunctions that are identical to the real wavefunctionsoutside the core, corresponding to the same charge density. This constraint is sometimesrelaxed in calculations in favour of a so-called ultrasoft pseudopotential. Pseudopotentialsand wavefunctions for silver are shown in Fig. 14.2.



Figure 14.2: Real and pseudo potentials (top) and wavefunctions (bottom) for the 5s, 5pand 4d states of silver. (From Marder[5].)

14.3 Relativistic effects

We will now look at one of the effects that are important in accurate electronic structurecalculations — relativistic corrections. At first sight it seems as though relativistic correctionswould not be significant in solids. The Fermi energy of a metal (the energy difference betweenthe highest and lowest filled conduction band states) is usually just a few electron volts, avery small fraction of the 0.5 MeV rest mass energy of the electron. But for electronsmoving in the deep nuclear potentials of heavy ions, relativistic effects can be substantial.More importantly, relativistic effects can lift some of the degeneracies that occur at highsymmetry points in k-space.

We have been describing the quantum behaviour of electrons in solids by using the Schrodingerequation and putting the electron spin in by hand. Electrons are actually described by the


14.3. RELATIVISTIC EFFECTS 109

Dirac equation

i~∂

∂t|ψ〉 =

(cα · p + βmc2

)|ψ〉, (14.10)

where |ψ〉 is a four-component spinor. In the nonrelativistic limit the Dirac equation canbe split into separate equations for the electron and positron. The hamiltonian, which nowacts on a two-component electron wavefunction, is

H =p2

2m+ V (r)− p4

8m3c2+

~4m2c2

~σ · [∇V (r)× p] +~2

8m2c2∇2V (r) + ... (14.11)

We will focus on the second correction to the Schrodinger equation, the spin–orbit term

~4m2c2

~σ · [∇V (r)× p], (14.12)

which is a measure of the coupling of the electron’s magnetic moment to the magnetic fieldproduced by its orbital motion.1 Of all the the relativistic effects, the spin–orbit couplingproduces the most observable effects, not because it is the largest in magnitude, but becauseit does not have the same symmetry as the periodic potential. As a result, it can liftdegeneracies, in some cases causing large changes in Fermi surface topology. These effectsare particularly important in hexagonal-closest-packed metals, in which the crystal structureis not a Bravais lattice, but a lattice convolved with a basis of two identical atoms withinthe unit cell. The Fourier transform of the real-space potential has a structure factor thatcancels some of the reflections exactly, resulting in degeneracies along the missing Braggplanes. The spin–orbit interaction does not have the same symmetry and is able to liftthese degeneracies. These effects are less important in the light elements but become moreimportant the heavier the nuclei are.

We will illustrate these effects with a simple example from Section 4.5 of Taylor and Heinonen[7].

1The spin–orbit correction is the only term in the hamiltonian that explicitly depends on the spin of theelectron. However, it is not the origin of magnetism in solids. We have already seen that the Pauli exclu-sion principle, coupled with the spin-independent Coulomb repulsion, results in a spin-dependent exchangepotential. The exchange interaction is always much larger than the spin–orbit coupling.


Lecture 15

Density functional theory

In the course so far we have taken two approaches to calculating and understanding theproperties of electrons in solids. We initially focussed on electrons in a uniform system,and looked at the effect of interactions between the electrons. This was a hopelessly difficultproblem to solve exactly and required approximate methods to make any headway at all. Wethen discarded the interactions and considered the problem of electrons moving in a periodicone-body potential, a difficult problem in its own right but one that could in principlebe solved exactly. In real materials both effects are important. How do we include themin electronic structure calculations in any sort of tractable way? For materials in whichcorrelation effects are not too strong, the answer is to use density functional theory.

15.1 The Hohenberg–Kohn theorem

Hohenberg and Kohn observed that, as far as calculation of the ground state energy goes,the electron density n(r) contains, in principle, all the information of the many-body wave-function ψ(r1, σ1; ...; rN , σN , t). That this is true is quite astonishing. The many body wave-function of an N -particle system is a function of 3N + 1 variables and N spin coordinates.The density, on the other hand, is just a function of 3 space variables (and time). Not onlydoes n(r) determine the ground state energy, but many other important physical propertiesas well. These include all the linear response functions, the spectrum of excited states, andmost transport properties.

Hohenberg and Kohn showed that if we know the ground state density n(r), we can deducefrom it the external potential Vext(r) in which the electrons move, up to an overall constant.This is very important, because there are only two ways in which distinct many-electronproblems can differ:

• in the external potential Vext(r). (i.e. the nuclear potentials)

• and in the number of electrons

N =

∫n(r)dr. (15.1)


112 LECTURE 15. DENSITY FUNCTIONAL THEORY

According to Hohenberg and Kohn, both of these are determined purely by the electrondensity. In the case of the total number of electrons this is plain to see. That n(r) alsodetermines Vext(r) is far from clear.

15.1.1 Proof of the Hohenberg–Kohn theorem

(This proof is appropriate for nondegenerate ground states. Other proofs are possible forthe more general case.) We suppose that the theorem is false. Therefore there exist twoexternal potentials V1(r) and V2(r) that result in the same ground-state density n(r). Callthe hamiltonians H1 and H2, and let the normalized ground-state wavefunctions be ψ1 andψ2. Here

H1 = T + V + V1 and H2 = T + V + V2. (15.2)

The kinetic energy T and the two-body electron–electron interaction V are the same for allsystems of interacting electrons.

From the Rayleigh–Ritz variational theorem,

ε1 = 〈ψ1|H1|ψ1〉 < 〈ψ2|H1|ψ2〉, (15.3)

because ψ1 is the only wavefunction that minimises H1 and the states are nondegenerate.Then

ε1 < 〈ψ2|H2|ψ2〉+ 〈ψ2|H1 − H2|ψ2〉 (15.4)

= ε2 + 〈ψ2|V1 − V2|ψ2〉 (15.5)

because H1 and H2 differ only in the external potential. The expectation value of the one-body potential can be expressed directly in terms of the ground-state density n(r). Therefore

ε1 < ε2 +

∫n(r) (V1(r)− V2(r)) dr. (15.6)

But we can switch indices and find that

ε2 < ε1 +

∫n(r) (V2(r)− V1(r)) dr. (15.7)

Adding the two inequalities, we have

ε1 + ε2 < ε2 + ε1, (15.8)

which is a contradiction. As a result, the external potential corresponding to n(r) must beunique. In other words, the external potential is a functional of the density:

Vext = Vext[n(r)]. (15.9)

The ground state wavefunction is a functional of the external potential Vext, and thereforeso is the energy E [Vext], the kinetic energy T [Vext] and the interaction energy V [Vext]. But


15.2. APPLICATION OF THE HOHENBERG–KOHN THEOREM 113

since Vext is a functional of the density n(r), all of these quantities are functionals of theground-state density, including the ground-state wavefunction.

We now come to the second Hohenberg–Kohn theorem, or density functional theorem. Thisstates that the ground-state density n0(r) minimizes the energy functional

E [n(r);Vext] = T [n(r)] + V [n(r)] +

∫n(r)Vext(r)dr. (15.10)

15.1.2 Proof of the second Hohenberg–Kohn theorem

Start with the wrong density n′(r). This corresponds to a different external potential and adifferent hamiltonian, H ′, with ground state ψ′. Therefore

〈ψ′|H|ψ′〉 = E [n′(r)] > 〈ψ|H|ψ〉 = E [n0(r)]. (15.11)

15.2 Application of the Hohenberg–Kohn theorem

The Hohenberg–Kohn result has important implications for the many-body problem. Wedefine the Hohenberg–Kohn functional FHK[n(r)]:

FHK[n(r)] = T [n(r)] + V [n(r)], so that (15.12)

E [n(r);Vext] = FHK[n(r)] +

∫n(r)Vext(r)dr. (15.13)

FHK is independent of Vext, and is therefore a unique functional, the same for all interactingN -electron systems. In principle if we knew FHK we could solve all many-body problems interms of their density. In practice FHK is not known, and never will be known. People havedeveloped various approximations to it over the years, but do not have a single functionalthat works for all types of materials.

15.3 The Kohn–Sham equations

The Hohenberg–Kohn theorems show that the ground state density n0(r) on its own uniquelydetermines the energy and ground-state properties of an interacting many-body system.However, it doesn’t give us any way of calculating n0(r). The Kohn–Sham procedure is onepossible way of carrying out the calculation. The first density functional theory calculationswere made in this way, and it remains appealing to us for the physical insight it gives.

We want to minimize the energy as a function of the density n(r). A convenient way toproceed is to decompose n(r) in to a sum of fictitious independent orbital contributions

n(r) =N∑i=1

φ∗i (r)φi(r) (15.14)

where the φi(r) are orthonormal. It is very important to note that these are not the wave-functions of the interacting many-body hamiltonian. However, this form for the density is



exactly what we would find for a noninteracting system, in which the many-body wavefunc-tion is a Slater determinant of the N lowest-lying single-particle orbitals.

The Hohenberg–Kohn theorem guarantees that such a noninteracting system exists. Thatis, there is a noninteracting system whose ground-state density n0(r) = n(r) and, by theHohenberg–Kohn theorem, n0(r) uniquely determines the external potential Vs in the non-interacting hamiltonian.

We start with this noninteracting system. Its hamiltonian is

Hs = T + Vs (15.15)

By the Hohenberg–Kohn theorem there exists a unique energy functional

Es[n] = Ts +

∫Vs(r)n(r)dr. (15.16)

Note that, while the kinetic energy operator is the same for interacting and noninteractingsystems, the kinetic energy functional for the noninteracting system, Ts[n], will be differentfrom that of interacting systems T [n] due to the effects of electron correlation. Actually,unlike the interacting case, we can write down Ts[n]:

Ts[n] =N∑i=1

〈i| − ~2∇2

2m|i〉, (15.17)

because there are no correlations in the wavefunction beyond those of Pauli exclusion.

Since this is a noninteracting system we can easily determine the ground state density. Wesolve the Schrodinger equation for the single particle orbitals(

− ~2

2m∇2 + Vs(r)

)φi(r) = εiφi(r) (15.18)

and use Eq. 15.14 to construct the ground state density from the N lowest lying levels.

We are actually interested in a system of N interacting electrons in an external potentialVext(r). We look for a potential Vs(r) for the noninteracting system such that the ground-state density of the noninteracting system is the same as that of the interacting system.

We start by taking the energy functional of the interacting system and trying to cast it intoa noninteracting form.

E [n] = T [n] + V [n] +

∫n(r)Vext(r)dr (15.19)

= Ts[n] + (T [n]− Ts[n]) +

(V [n]− e2

4πε0

1

2

∫∫n(r)n(r′)

|r− r′|drdr′

)(15.20)

+e2

4πε0

1

2

∫∫n(r)n(r′)

|r− r′|drdr′ +

∫n(r)Vext(r)dr (15.21)

= Ts[n] +e2

4πε0

1

2

∫∫n(r)n(r′)

|r− r′|drdr′ +

∫n(r)Vext(r)dr + Exc[n(r)] (15.22)


15.4. THE LOCAL DENSITY APPROXIMATION 115

This looks much simpler. We have the kinetic energy of the noninteracting system, Ts[n], aHartree-like interaction term, an external potential, and an exchange–correlation energy

Exc[n] = FHK[n]−Ts[n] +

e2

4πε0

1

2

∫∫n(r)n(r′)

|r− r′|drdr′

(15.23)

that contains all our ignorance about electron–electron interactions beyond the Hartree term.

Taking the functional derivative δE [n]/δn(r), we see that we have what appears to be inde-pendent electrons moving in an external potential

Vs(r) =e2

4πε0

∫n(r′)

|r− r′|dr′ + Vext(r) + Vxc[n(r)] (15.24)

where

Vxc[n(r)] =δExc[n]

δn(r). (15.25)

The only catch is that Vs(r) depends on the density n(r), requiring us to solve the singleparticle Schrodinger equation interatively, until a self-consistent determination of n(r) isachieved.

The advantages of the Kohn–Sham approach is that it is formally exact, as long as weknow Vxc(r). Of course, in practice, we don’t, but good approximations exist and in weaklycorrelated materials, such as most metals and semiconductors, the effects of exchange andcorrelation are small.

The Kohn–Sham equations resemble self-consistent Hartree equations with an added exchange–correlation potential. They are much easier to solve than the Hartree–Fock equations wesaw in an earlier lecture, because the effective potential is the same for each orbital.

Finally, the energies εi in the Kohn–Sham equations are just a formal device. However, ifVxc(r) is small, they are often a good approximation to the single particle energies.

15.4 The local density approximation

To use the Kohn–Sham equations we need an approximation for the exchange–correlationpotential Vxc(r). A common choice, used in many band-structure calculations, is the localdensity approximation or LDA. In LDA, the nonlocal nature of Vxc(r) is ignored and theexchange correlation energy written

Exc[n] =

∫εxc[n]n(r)dr. (15.26)

Various approximations for εxc[n] exist, interpolating between limits that can be solvedexactly.

15.5 Thomas–Fermi as a density functional theory

We have already seen one example of a density functional theory — the Thomas–Fermitheory. We used this to find the charge response of an electron gas to an external potential.



We made the approximation that the external potential was in the long wavelength limitwhere it simply caused a rigid shift of the single particle energies. We then used it to calculatethe charge density induced around a point-like object, which really was overstretching thelimits of its validity. Using the Hartree–Fock approximation in the jellium model, we canconstruct an explicity density functional for the total energy E, as a sum of the followingterms:

• kinetic energy

V~2

2m

3

5

(3π2)2/3

n5/3 (15.27)

• external potential ∫Vext(r)n(r)dr (15.28)

• Coulomb energye2

4πε0

1

2

∫n(r)n(r′)

|r− r′|drdr′ (15.29)

• exchange energy

− V 3

4

(3

π

)1/3e2

4πε0n4/3 (15.30)

(This is called the Thomas–Fermi–Dirac theory if we include the exchange term.)

We need to constrain the total number of particles to be N , which we do by introducing aLagrange multiplier µ, the chemical potential, defined to be

δE

δN≡ µ (15.31)

We use µ as a dial with which to set the electron density. Subject to this constraint, theenergy is minimized by taking the functional derivative with respect to n(r).

~2

2m

(3π2)2/3

n2/3 + Vext(r) +e2

4πε0

∫n(r′)

|r− r′|dr′ −

(3

π

)1/3e2

4πε0n1/3 = µ (15.32)

Unfortunately, the Thomas–Fermi–Dirac model is not very accurate.

15.6 What can be calculated with density functional

theory?

Density functional theory is a theory for the ground-state energy of an interacting electronsystem. Quantities that depend on the ground state energy and density can be calculatedmost accurately. These include:

• the total energy and structural phase transitions. The total energy is calculated for avariety of different crystal structures and lattice spacings, allowing the crystal structureto be predicted at different densities.


15.6. WHAT CAN BE CALCULATED WITH DENSITY FUNCTIONAL THEORY? 117

• surface reconstructions. The total energy of a surface can be calculated very accuratelyto predict how it will differ structurally from the bulk.

• phonon energy dispersions. Using the Born–Oppenheimer approximation (i.e. thatelectrons move much faster than ion cores), so called ‘frozen phonon’ calculationsdirectly obtain the energy cost of passing an elastic wave through a solid.

• quantum chemistry. Properties of molecules such as bond length, electron density, vi-brational frequencies and ionization energies can be calculated using LDA, or the localspin density approximation (LSDA). Chemists also commonly use another approxima-tion to Vxc[n] — the generalized gradient approximation (GGA).

• band structures. The single-particle energy eigenvalues in the Kohn–Sham equationscan be interpreted as energy bands and, although this has no formal justification, itworks well in many materials.

• linear response functions. It can be shown formally that linear response functions suchas the electrical conductivity are properties of the ground state. There exist variantsof density functional theory for calculating them. As we will see later in the course,poles in linear response functions correspond to the excited states of the system.

Density functional theory has some notable failures, too. It is not able to calculate band-gapenergies in semiconductors and insulators, and breaks down entirely in correlated electronsystems, where the exchange–correlation energy is large.


Lecture 16

The dynamics of Bloch electrons I

The most important way in which a crystalline solid differs from a box of free electronsis in the existence of bands of allowed energies. The eigenstates of a periodic solid areBloch electrons. Like free electrons, Bloch electrons for the most part propagate through aregular array of ion cores as though it were not there, although with modified energies thatwe describe by an energy band structure. Given any crystal, we can now calculate theseenergy bands, in principle quite accurately for the many materials in which the effects ofelectron–electron correlation are weak.

Now, with a view to understanding and predicting the transport properties of solids, we wouldlike to develop a model for the dynamics of electrons in crystalline materials, in particulartheir response to external fields. We will see that Bloch electrons can be understood with asimple set of rules, which we call the semiclassical model of electron dynamics.

16.1 Energy bands and group velocity

The correct velocity to use for a Bloch electron is the group velocity

vnk =1

~∇kεnk. (16.1)

In order to show this, we need to simultaneously specify the position and wavevector of theelectron. This is the semiclassical approach, which can be made rigorous by setting up awavepacket, centred on r with width ∆r, constructed from a range of wavevectors of width∆k, centred on k as shown in Fig. 16.1. This will be subject to the constraint ∆r∆k > 1.

The Bloch electron wavepacket has the following form

W (r,k, t) =

∫dk′ w(k′ − k)ψk′e

−iεk′ t/~, (16.2)

where w(k′ − k) is a function sharply peaked at the origin and

ψk(r) = eik·ruk(r), (16.3)

uk(r) = uk(r + R). (16.4)


120 LECTURE 16. THE DYNAMICS OF BLOCH ELECTRONS I

lattice constant

spread of wave packet

wavelength of applied field

Figure 16.1: The hierarchy of length scales in a Bloch wavepacket.

W (r,k, t) =

∫dk′ w(k′ − k)eik′·ruk′(r)e−iεk′ t/~ (16.5)

= eik·r−iεkt/~∫

dk′ w(k′ − k)ei(k′−k)·ruk′(r)e−i(εk′−εk)t/~ (16.6)

Taylor expand the energy about k:

εk′ = εk + (k′ − k) · ∇kεk. (16.7)

Let k′′ = k′ − k, then

W (r,k, t) = eik·r−iεkt/~∫

dk′′ w(k′′)eik′′·(r−∇kεkt/~)uk+k′′(r) (16.8)

≈ eik·r−iεkt/~ F (r−∇kεkt/~). (16.9)

The wavefronts move with phase velocity εk/~k. The centre of the envelope function Fmoves at the group velocity

r =1

~∇kεk. (16.10)

This forms the basis of the semiclassical model, in which a classical treatment of the electrondynamics allows us to obtain deep insights into how the electrons will behave in transportand thermodynamic measurements. By constructing the wavepacket from Bloch functions wehave ensured that the electron will be able to propagate through the periodic lattice withoutattenuation. The only scattering will be caused by defects in the periodicity, includingstatic defects such as vacancies and impurities, and dynamical defects such as the thermalvibrations of the ions.

One subtlety of the semiclassical model is that the periodic potential varies on a scale smallcompared to the size of the wavepacket. The semiclassical model is actually only partiallyin the classical limit — the response to external, long-wavelength fields is treated classically,but the periodic field of the ions is treated quantum mechanically in the calculation of theBloch wavefunctions.


16.2. RULES OF THE SEMICLASSICAL MODEL 121

16.2 Rules of the semiclassical model

The semiclassical model predicts how the position r and wavevector k of each electron evolvein the presence of externally applied electric and magnetic fields, in the absence of collisions.This can be achieved based entirely on a knowledge of the bandstructure, εn(k), with noother information about the periodic potential of the ions necessary. The end result of thisprocess is that we can calculate the response of the electrons to applied fields or temperaturegradients. That is, we can predict the solid’s transport properties. Very importantly, theprocedure can be run backwards, to infer the form of εn(k) by analyzing data from transportexperiments.

Given the functions εn(k), the model associates a position r, a wavevector k, and a bandindex n for each electron. The external electric and magnetic fields in general depend onspace and time.

E = E(r, t) (16.11)

B = B(r, t) (16.12)

The electron dynamics are described by a set of rules, which we will justify further below:

• The band index n is a constant of the motion.

• The time evolution of r and k is determined by the equations of motion

r = vn(k) =1

~∇kεn(k) (16.13)

~k = −e[E(r, t) + vn(k)×B(r, t)

]. (16.14)

• Wavevector is defined only to within an additive reciprocal lattice vector G. Thismeans that we cannot have two distinct electrons with the same band index n andposition r, whose wavevectors k and k′ differ by a reciprocal lattice vector G. As aresult, n, r,k and n, r,k + G are equivalent ways of describing the same electronand therefore all distinct wavevectors lie within the first Brillouin zone.

• In thermal equilibrium, the occupation number of the volume element dk of k-space,working in unit volume and including a factor of 2 for spin, is the usual Fermi distri-bution:

f(εn(k))dk

4π3=

dk/4π3

exp((εn(k)− µ)/kBT

)+ 1

. (16.15)



Comments on the rules of semiclassical electron dynamics.

• The external fields are not allowed to cause interband transitions. Each band thereforecontains a fixed number of electrons of a particular type.

• Completely filled bands are inert. They do not respond to external fields and thereforecarry no current. As a result, we only need to consider partially filled bands — i.e.bands within a few kBT of the Fermi energy.

• Crystal momentum is not momentum! Within each band, the equations of motion arethe same as for free electrons, but with ~2k2/2m → εn(k). ~k is not the momentumof a Bloch electron. While p equals the total force, ~k equals the force exerted by theexternal fields, not by the periodic field of the lattice. We have already taken accountof the electrostatic fields of the lattice quantum mechanically, in deriving the Blochstates.

16.3 The k·P method

Before discussing the consequences and limits of validity of the semiclassical model, we makea brief digression and introduce the k·P method, which underlies the equations of motion ofBloch electrons.

In a finite sample, there is a conceptual difficulty that arises from the fact that the allowedwavevectors are a discrete set of points in k-space. At any one of these points the energyeigenstates in the presence of a periodic potential are formed by combining plane waves thathave wavevector differing from the original k by a reciprocal lattice vector G. The energydispersion εk is then a discrete set of points in k-space, not a set of continuous curves. Butwe need continuous curves in order to calculate the semiclassical dynamics. In other words,given the eigen-energies at some wavevector k, we need to be able to track continuously howthe energies evolve when we move to some nearby wavevector k + δk. The approach wetake is to recast the original hamiltonian into a form that depends explicitly on k, and thentreat the shift in wavevector δk as a small change to be handled in perturbation theory. Bysetting up the problem in this way we get an expression for the energies at wavevector k+δkentirely in terms of the wavefunctions at the original wavevector k.

Bloch functions have the form

ψnk(r) = eik·runk(r), (16.16)

where unk(r) has the full periodicity of the crystal lattice. If we substitute this into theSchrodinger equation [

− ~2

2m∇2 + U(r)

]ψnk(r) = ε ψnk(r) (16.17)

it is straightforward to show that unk(r) obeys the equation[~2

2m

(−∇2 − 2ik · ∇+ k2

)+ U(r)

]unk(r) = ε unk(r). (16.18)


16.3. THE K·P METHOD 123

Because unk(r) is periodic, Eq. 16.18 can be solved in a single unit cell of the crystal.

We now have a hamiltonian that depends explicitly on k, so we are ready to carry out ourpertubation procedure. Before we start, we expect that the energy dispersion εn(k) will beTaylor expandable around the point k:

εn(k + δk) = εn(k) +∑i

∂εn∂ki

δki +1

2

∑ij

∂2εn∂ki∂kj

δkiδkj +O(δk)3. (16.19)

The shift in wavevector, k→ k+δk, can be treated as a perturbation term in the hamiltonian:

Hk → Hk+δk = Hk +~2

mδk · (−i∇+ k) +

~2δk2

2m≡ Hk + Vδk. (16.20)

We will make use of the result from time-independent perturbation theory:

En = E0n + 〈n|V |n〉+

∑n′ 6=n

∣∣〈n|V |n′〉∣∣2E0n − E0

n′+O(V )3. (16.21)

16.3.1 First order terms:

Keep terms linear in δk:

∑i

∂εn∂ki

δki =∑i

〈unk|~2

mδki(−i∇+ k)i|unk〉 (16.22)

⇒ ∇kεn =~2

m〈unk|(k− i∇)|unk〉 (16.23)

But

unk = e−ik·rψnk (16.24)

and

(k− i∇)e−ik·r = −ie−ik·r∇, (16.25)

so

∇kεn =~m〈ψnk| − i~∇|ψnk〉 (16.26)

=~m〈ψnk|P |ψnk〉 (16.27)

= ~〈v〉, (16.28)

defining v ≡ P /m. This confirms what we earlier deduced — that the average velocity isthe group velocity of the wavepacket.



16.3.2 Second order terms:

Now isolate the second order terms.

1

2

∑ij

∂2εn∂ki∂kj

δkiδkj =~2δk2

2m+∑n′ 6=n

∑ij

∣∣∣〈unk|~2

mδki(k− i∇)i|un′k〉〈unk|~

2

mδkj(k− i∇)j|un′k〉

∣∣∣εnk − εn′k

(16.29)We can use the same trick as Eqs. 16.24 and 16.25 to recast this in terms of the originalBloch functions ψnk.

1

2

∑ij

∂2εn∂ki∂kj

δkiδkj =~2δk2

2m+

1

2

(~m

)2 ∑n′ 6=n

∑ij

〈ψnk| − i~ ∂∂xiδki|ψn′k〉〈ψnk| − i~ ∂

∂xjδkj|ψn′k〉+ c.c.

εnk − εn′k(16.30)

That is∂2εn∂ki∂kj

=~2

mδij +

(~m

)2 ∑n′ 6=n

〈ψnk|Pi|ψn′k〉〈ψnk|Pj|ψn′k〉+ c.c.

εnk − εn′k(16.31)

To see what the curvature of the energy dispersion has to do with the dynamics of Blochelectrons, calculate the acceleration.

d

dt〈vi〉 =

∑j

∂〈vi〉∂kj

dkjdt

=1

~∑j

∂2εn∂ki∂kj

dkjdt

= ~ M−1 · k (16.32)

where

(M−1

)ij

=1

~2

∂2εn∂ki∂kj

(16.33)

=δijm

+1

m2

∑n′ 6=n

〈ψnk|Pi|ψn′k〉〈ψnk|Pj|ψn′k〉+ c.c.

εnk − εn′k(16.34)

is the inverse effective mass tensor, and controls how Bloch electrons accelerate in responseto external fields. We can interpret the effective mass equation (Eq. 16.34) as saying thatthe band mass arises from virtual transitions between bands. The closer the adjacent band,the bigger the deviation from the free electron mass.

• nearby band of higher energy → negative contribution to 1/m

• nearby band of lower energy → positive contribution to 1/m

At the top of the band, the overall curvature is negative. These negative mass Bloch electronsaccelerate in the opposite direction to the Lorentz force. (Actually, the fact that we have aneffective mass tensor means that the acceleration does not need to be parallel to the force.)The usual way to think of these negative-mass electrons is as positively charged holes.


16.4. CONSEQUENCES OF THE SEMICLASSICAL MODEL 125

16.4 Consequences of the semiclassical model

We now look at some of the consequences of the semiclassical model of electron dynamics.For simplicity we work with a single band and a zero temperature Fermi distribution nk:

nk = 1, ε(k) < EF (16.35)

nk = 0, ε(k) > EF. (16.36)

16.4.1 Electrical current

The current density of a single electron in the state k will be −evk/V , where V is the volumeof the system.

The total current density carried by the band will be

j =2

V

∑k

(−e)nkvk (with a factor of 2 for spin) (16.37)

→ 2

V

V

(2π)3

∫(−e)nkvkdk (16.38)

= −e∫

occupied

dk

4π3

1

~∇kεk, (16.39)

where the integral is over the occupied states. The current density is independent of samplevolume, as it should be. In equilibrium j = 0, as for every occupied state k there is anoccupied state −k that makes an opposite contribution to the electrical current. In thepresence of external fields this balance can be broken, resulting in a finite current.

16.4.2 Thermal current

The flow of heat can be treated in a similar way. A state k carries an energy currentε(k)vk/V . The total energy current density is

jε =2

V

∑k

ε(k)nkvk (16.40)

→ 2

V

V

(2π)3

∫ε(k)nkvkdk (16.41)

=1

2

∫occupied

dk

4π3

1

~∇k(ε2k). (16.42)

As with the electrical current, jε = 0 in equilibrium.

16.4.3 Filled bands

For a filled band, the integrals in Eqs. 16.39 and 16.42 are over an entire Brillouin zone— i.e. over whole unit cells of the reciprocal lattice. The integrals therefore vanish, by



the theorem that the integral over any primitive cell of the gradient of a periodic functionmust vanish. Filled bands are therefore inert, and make no contribution to electrical orthermal currents. This justifies our earlier assertion — that we focus attention on partiallyfilled bands when calculating the dynamical properties of Bloch electrons — and shows whyelectronic transport plays no role in insulators, as insulators consist solely of completely filledand completely empty bands.

16.4.4 Electrons and holes

The contribution of all the electrons in a given band to the current density is

j = (−e)∫

occupied

dk

4π3vk. (16.43)

A completely filled band carries no current, so

0 =

∫zone

dk

4π3vk =

∫occupied

dk

4π3vk +

∫unoccupied

dk

4π3vk (16.44)

We can therefore write

j = (+e)

∫unoccupied

dk

4π3vk. (16.45)

That is, the current carried by a band is the same as if the states occupied by electrons wereempty, and the unoccupied states were filled with fictitious particles of positive charge +ecalled holes.

This can be very convenient — instead of working with an almost filled band of electrons, wecan work with a small density of holes, for which the energy dispersion at the top of the bandwill be almost perfectly parabolic. That is, we can use the effective-mass approximation. Aquick work of warning though — do not think of a band as consisting of electrons and holesat the same time!

16.5 Semiclassical motion in a uniform dc electric field

In a static, uniform electric field the semiclassical equation of motion is

~k = −eE. (16.46)

This can be integrated to get

k(t) = k(0)− eEt

~. (16.47)

That is, in a time t the wavevector of every electron changes by the same amount, −eEt/~.For a metal (a solid with a Fermi surface) this corresponds to a sideways displacement ofthe Fermi sea, in the opposite direction to the electric field, that grows linearly in time. Theresulting current density also grows linearly in time. Such a situation is unsustainable and ina real material scattering processes intervene to restrict the displacement of the Fermi surface


16.5. SEMICLASSICAL MOTION IN A UNIFORM DC ELECTRIC FIELD 127

−eτE/h

Figure 16.2: A uniform electric field shifts the Fermi surface in k-space at a constant rate.Scattering processes limit the acceleration to a finite displacement.

to an amount δk = −eEτ/~, where τ is a characteristic relaxation time of the electrons.Later in the course we will develop transport theories to study this situation in more detail.

For an insulator the situation is less intuitive. The electrons still in principle obey an acceler-ation equation, with wavevectors that increase linearly in time. However, in an insulator allthe occupied Brillouin zones are completely filled. We then have to examine what happens atthe edge of the Brillouin zone, at a Bragg plane. In the semiclassical model our wavevectorsare only defined modulo a reciprocal lattice vector G. As a result, when the electric fieldcauses the wavevector of an electron to cross a Bragg plane, it will be Bragg reflected backto the other side of the Brillouin zone, where it will make the opposite contribution to thecurrent density.

Another way to see this is in the extended zone scheme, as shown in Fig. 2. There ε(k) itsa periodic function, as is v(k). If an electron could be accelerated for an extended period oftime without scattering, its velocity would be a periodic function of time:

v(k(t)) = v(k(0)− eEt/~). (16.48)

The resulting contribution to current would also be oscillatory in time. Effects of this sortare called ‘Bloch oscillations’, and have been observed in very clean semiconductors and,very recently, in laser-cooled atoms trapped in optical lattices.



(k) v(k)

k

zone boundary zone boundary

Figure 16.3: ε(k) and v(k) are periodic functions in the extended zone scheme.


Lecture 17

The dynamics of Bloch electrons II

We investigate the dynamics of Bloch electrons further by studying their motion in a uniformmagnetic field. Along with the simple case of the response to a uniform static electric field,which determines the electrical resistance, the motion of Bloch electrons in a magnetic fieldis perhaps the most important experimental arrangement, being responsible for most of theexperimental data we have on the structure of Fermi surfaces of metals. These experimentshave at the same time provided detailed tests of the theory of Bloch electrons and extensionsto it such as Landau’s Fermi liquid theory. The agreement with theory is very satisfying andthe experiments themselves are beautiful, so it is well worth taking the time to study howelectrons in solids respond to magnetic fields.

17.1 Semiclassical motion in a uniform magnetic field

In a static, uniform magnetic field B, the equations of motion for Bloch electrons are:

r ≡ v(k) =1

~∇kε(k), (17.1)

~k = (−e)v(k)×B (17.2)

= − e~∇kε(k)×B. (17.3)

The constants of the motion are the energy

dε

dt= ∇kε(k)

dk

dt(17.4)

= − e

~2∇kε(k) ·

(∇kε(k)×B

)(17.5)

= − e

~2B.(∇kε(k)×∇kε(k)

)(17.6)

= 0 (17.7)


130 LECTURE 17. THE DYNAMICS OF BLOCH ELECTRONS II

Figure 17.1: In an applied magnetic field, the electrons undergo cyclotron motion, in aplane perpendicular to the field and along contours of constant energy. (From Ashcroft andMermin[4].)

and the projection of the wavevector along the field direction

B · dk = − e~

B ·(v ×B

)dt (17.8)

= − e~

v ·(B×B

)dt (17.9)

= 0. (17.10)

The resulting motion is shown in Fig. 17.1. The trajectories lie in planes perpendicular tothe magnetic field, and follow contours of constant energy. In particular, an electron at εFwill trace out an orbit on the Fermi surface.

It is useful to calculate the real-space motion r(t) corresponding to the k-space orbit. Letr‖ be the projection of r along the field direction, and r⊥ be the component normal to B.

Define a unit vector B in the direction of the magnetic field:

r‖ = B(r · B) (17.11)

r⊥ = r− r‖ = r− B(r · B) (17.12)

r⊥ = r− B(r · B). (17.13)

Taking the vector product of B with the equation for the acceleration, Eq. 17.2, and usingthe vector identity (A×B×C) = B(A ·C)−C(A ·B), we have

B× ~k = (−e)B× r×B (17.14)

= (−e)(r− B(B · r)

)|B| (17.15)

= (−e)|B| r⊥. (17.16)


17.1. SEMICLASSICAL MOTION IN A UNIFORM MAGNETIC FIELD 131

Figure 17.2: The k-space orbit, shown on the left, is rotated by 90 and scaled by ~/eB togive the perpendicular component of the real-space orbit. (From Ashcroft and Mermin[4].)

If we integrate the last equation, we have

r⊥(t)− r⊥(0) = − ~e|B|

B×(k(t)− k(0)

). (17.17)

This shows that the projection of the real-space orbit onto a plane perpendicular to themagnetic field is just the k-space orbit, rotated by 90, and scaled by ~/eB. Figure 17.2shows the idea. This is a very useful result when trying to visualize the effect of a magneticfield on the electrons. The real-space motion of a Bloch electron along the magnetic field isaffected by the field, but not in a simple way. In general the velocity along the field directionwill be periodically modulated in time.

17.1.1 The cyclotron frequency

We now calculate the time taken to traverse a closed orbit in k-space. Let t2− t1 be the timetaken for the electron to go from k1 to k2. This can be written

∆t = t2 − t1 =

∫ t2

t1

dt (17.18)

=

∫ k2

k1

1

|k|dk. (17.19)



From Eq. 17.17 we know that the real-space and k-space trajectories are related.

|k| =eB

~|r⊥| (17.20)

=eB

~2|(∇kεk)⊥| (17.21)

So the time taken to traverse the trajectory is

∆t =~2

eB

∫ k2

k1

1

|(∇kεk)⊥|dk. (17.22)

Figure 17.3: Diagram showing the construction used to calculate the cyclotron period. (FromAshcroft and Mermin[4].)

It turns out that Eq. 17.22 has a simple geometric interpretation. To see this, let ∆(k)be a vector in the plane of the orbit, perpendicular to the orbit at k and joining k to aneighbouring orbit with energy ε+ ∆ε. We then have

∆ε = ∇kεk ·∆(k) (17.23)

= |(∇kεk)⊥|∆(k). (The vectors are parallel.) (17.24)

As a result

|(∇kεk)⊥|−1 =∆(k)

∆ε(17.25)


17.2. LIMITS OF VALIDITY OF THE SEMICLASSICAL MODEL 133

and

∆t =~2

eB

1

∆ε

∫ k2

k1

∆(k)dk. (17.26)

∫ k2

k1∆(k)dk is just the area between two neighbouring orbits with energy difference ∆ε, which

we call ∆A12. Then

t2 − t1 =~2

eB

∆A12

∆ε→ ~2

eB

∂A12

∂ε(17.27)

in the limit ∆ε→ 0. We are usually interested in the traversal time of closed orbits, in whichcase k1 = k2. Then t2 − t1 = T , the cyclotron period, which will be a function of the energy(usually the Fermi energy) and kz, the projection of the wavevector onto the field axis.

T (ε, kz) =~2

eB

∂

∂εA(ε, kz), (17.28)

where A(ε, kz) is the cross sectional area of an orbit of energy ε and projection kz. We usuallyfocus on the cyclotron frequency ωc = 2π/T and, in analogy to the free-electron result, definea cyclotron mass m∗ by

ωc =eB

m∗. (17.29)

The cyclotron mass is

m∗ =~2

2π

∂

∂εA(ε, kz). (17.30)

Again we see that an important dynamical property of the Bloch electrons is determined bythe geometry of the constant energy surfaces, which we can extract from the band structureεn(k).

The cyclotron frequency is directly accessible in experiments. One typically applies a largemagnetic field and then looks for the cyclotron motion of the electrons to come into resonancewith an applied microwave-frequency field. There are several experimental geometries inwhich the resonance condition corresponds to an enhanced absorption of microwaves by theorbiting electrons. The width of the resonant absorption in frequency is ∼ 1/τ — that is,inversely proportional to the relaxation time of the electrons. For this reason, high puritysamples and low temperatures are usually required to see a sharp resonance.

17.2 Limits of validity of the semiclassical model

We know that in the limit of vanishingly small periodic potential the semiclassical modelmust break down. In this limit, the Bloch electron becomes a free electron, and free electronscan accelerate without limit, breaking our requirement that the band index be a constantof the motion. As a result, there must be some minimum strength of the periodic potential,relative to the strength of the external fields, before the semiclassical model applies. It turnsout that approximate constraints can be obtained using simple arguments. The conditions



for the semiclassical model to apply are:

eEa (εgap(k)

)2

εF(17.31)

~ωc (εgap(k)

)2

εF, (17.32)

where a is the lattice constant, ωc = eB/m∗ is the cyclotron frequency, and εgap(k) =min

∣∣εn′(k)− εn(k)∣∣ is the minimum energy separation to the nearest band.

Equation 17.31 is never violated in a metal, but can be exceeded in insulators and homoge-neous semiconductors. When electric breakdown occurs, it is called Zener tunneling betweenbands.

Equation 17.32 is relatively easy to violate in a metal, and is called magnetic breakdown(or breakthrough). When it occurs, electrons can jump across Bragg planes to follow afree-electron-like trajectory.

Other conditions that must be satisfied in the semiclassical model are that the frequency ofthe external fields be low compared to the energy gap, ~ω εgap(k), and that the wavelengthof the external fields be long compared to the lattice spacing, λ a.

We can use simple arguments to arrive at the conditions for electric and magnetic breakdown.When U(r) is vanishly small, electric breakdown occurs whenever an electron crosses a Braggplane, as this causes it to make an interband transition. When U(r) is weak but finite, wecan estimate how large an electric field would be needed to cause breakdown by focussingon electrons in the vicinity of a Bragg plane.

Near Bragg planes εgap(k) has large curvature. A small spread in wavevector (an unavoidableconsequence of forming a wavepacket) causes a large spread in velocity.

∆v(k) =

∣∣∣∣∂v∂k

∣∣∣∣∆k ≈ 1

~

(∂2ε

∂k2

)∆k (17.33)

For the semiclassical picture to hold, the spread in velocity must be much smaller than theFermi velocity.

∆v vF (17.34)

⇒ ∆k ~vF

(∂2ε/∂k2)(17.35)

Since the potential is weak we can use the result derived for nearly free electrons near aBragg plane, sketched in Fig. 17.4.

In the vicinity of the Bragg plane, the energy dispersion is just the sum in quadrature ofthe unperturbed free electron energy and the relevant Fourier component of the periodicpotential. (An identical expression results for the energy dispersion of a superconductor,and is a handy form to remember.)

(∆εk)2 = (∆ε0k)2 + U2G, (17.36)


17.2. LIMITS OF VALIDITY OF THE SEMICLASSICAL MODEL 135

ε0(k)

ε0(k-G)

ε+(k)

ε-(k)

UG

UG

Figure 17.4: Plot of the energy dispersion near a Bragg plane. The Fourier component ofthe potential UG mixes the free-electron curves and opens a gap in the spectrum. This hasbeen approximated by the form (∆εk)2 = (∆ε0k)2 + U2

G, which is a general form for energygaps.

where

∆ε0k = ε0k − ε0G/2 ≈~2G

2m(k− G

2) ≡ ~2G

2m∆k ≈ vF~∆k. (17.37)

Therefore

∆εk =√U2G + (∆ε0k)2 (17.38)

≈ UG +(∆ε0k)2

2UG

(17.39)

= UG +~2v2

F(∆k)2

2UG

. (17.40)

We need the curvature of the energy dispersion in the vicinity of the Bragg plane (where itis largest). This is

∂2∆εk∂k2

=∂2∆εk∂(∆k)2

=~2v2

F

UG

. (17.41)

We require that

∆k ~vFUG

~2v2F

=UG

~vF

≈ εgap

~vF

. (17.42)

This requires that

∆x ∼ 1

∆k ~vF

εgap

, (17.43)



or that

∆φ ≡ eE∆x eE~vF

εgap

. (17.44)

When the uncertainty in potential, ∆φ, becomes larger than the energy gap, the electroncan easily make a transition between bands. To prevent this from happening, we requirethat εgap ∆φ. That is

εgap eE~vF

εgap

. (17.45)

Since the Fermi energy εF ≈ ~vF/a, where a is the lattice spacing, the condition can berewritten

eEa(εgap(k)

)2

εF. (17.46)

17.3 Magnetic breakdown

In the magnetic case, energy cannot be gained from the magnetic field, which we showedabove. In order for breakdown to occur, the electron must therefore jump from one bandto the next while conserving energy. The situation in which this can occur is depicted inFig. 17.5, and is common in many metals. To avoid magnetic breakdown, we require that the

ε(k)

ε-(k)

εgap

k

δk

ε+(k)

Figure 17.5: Plot of the energy dispersions required for magnetic breakdown. In this case,the electron cannot take energy from the field, so it must pass horizontally from one bandto the other. In a magnetic field B, the characteristic spread of the wavepacket goes as∆k ∼

√B. At high enough fields ∆k > δk, and magnetic breakdown occurs.

spread in wavevector of our wavepacket, ∆k, be much smaller than the k-space separationbetween bands, δk ∼ εgap/~vF. We showed above that the semiclassical orbit in real spaceis the same as the k-space orbit, but rotated by 90 and scaled by ~/eB. Therefore, if the


17.3. MAGNETIC BREAKDOWN 137

Figure 17.6: The figure on the left shows a free electron orbit, and the same orbit shifted bya reciprocal lattice vector. Where the orbits cross the Bragg plane they are degenerate. Inthe presence of a lattice potential the degeneracy is lifted and the k-space orbits disconnectto form two new orbits. These are shown on the right. At low fields the electrons followthe perturbed orbits. At high fields magnetic breakdown causes the electrons to follow theoriginal free electron orbits. (From Ziman[12].)

wavepacket has a width ∆kx in the x direction in k-space, it will have a width ∆y = ∆kx ~/eBin the y direction in real space. The wave nature of the packet requires that

∆ky∆y ∼ 1⇒ ∆ky∆kx ∼eB

~. (17.47)

If we assume the wavepacket is isotropic, we can estimate the size of the wavepacket in agiven magnetic field.

∆k ≈√

∆kx∆ky ∼√eB

~. (17.48)

To avoid magnetic breakdown, we require

∆k ∼√eB

~ εgap

~vF

. (17.49)

⇒ eB

~

ε2gap

~2v2F

(17.50)

⇒ ~eBm

2ε2gap

12mv2

F

(17.51)

⇒ ~ωc ε2gap

εF(17.52)

It turns out that this condition is quite easy to violate in a metal, in laboratory fields, andmust always be considered as a possibility when interpreting magnetotransport data takenunder high field.


Lecture 18

Quantum oscillatory phenomena

In this lecture we treat quantum mechanically the cyclotron motion of electrons in a highmagnetic field. The results are quite surprising — at sufficiently low temperatures, in highpurity samples, all physical quantities oscillate periodically in inverse field. These effectshave been central to the development of our current understanding of electrons in solids.

18.1 Quantum mechanics of the orbital motion

The hamiltonian of a (nonrelativistic) charged particle in a magnetic field is

H =1

2m(p− eA)2. (18.1)

A very important property of this Hamiltonian is gauge invariance — it is invariant under asimultaneous transformation of the potential

A→ A +∇χ (18.2)

and the wavefunctionψ → ψeieχ/~. (18.3)

In the following we will work in a particular gauge, the Coulomb gauge, in which ∇ ·A = 0,although all physical quantities will of course be independent of this choice.

Let A = (0, Bx, 0), then

∇×A = (0, 0,∂Ay∂x

) = (0, 0, B), (18.4)

corresponding to a magnetic field in the z direction.

Substituting this into the hamiltonian gives the following Schrodinger equation:

− ~2

2m

[∂2

∂x2+ (

∂

∂y− ieB

~x)2 +

∂2

∂z2

]ψ(r) = εψ(r). (18.5)

We expect the solution to have the form

ψ(x, y, z) = exp [i(βy + kzz)]u(x). (18.6)


140 LECTURE 18. QUANTUM OSCILLATORY PHENOMENA

That is, we expect the motion along the field direction to be unmodified, we still expectplane-wave-like behaviour in the y direction with wavenumber β, but we will have to solvefor the detailed shape of the wavefunction in the x direction.

The next step is to substitute the trial solution into the Schrodinger equation in order tofind u(x) and ε. We get

− ~2

2m

[∂2

∂x2+ (iβ − ieB

~x)2 − k2

z

]u(x) = εu(x) (18.7)

On rearranging this becomes− ~2

2m

∂2

∂x2+

1

2mω2

c (x− x0)2

u(x) = ε′u(x), (18.8)

where

ε′ = ε− ~2k2z

2m, (18.9)

ωc =eB

m, and (18.10)

x0 =~βeB

=1

ωc

~βm. (18.11)

That is, the problem decouples into the translational motion of a free particle along the zdirection and, in the plane perpendicular to the field, quantized harmonic motion, centredon x0 and oscillating at the cyclotron frequency. We have already solved the problem of theharmonic oscillator and can write down its energy spectrum directly.

ε′ = (n+1

2)~ωc (18.12)

ε = (n+1

2)~ωc +

~2k2z

2m. (18.13)

The eigenstates of quantized cycltron motion are called ‘Landau levels’ and are central to theunderstanding of all electronic phenomena in high magnetic fields, for example the quantumHall effects, and quantum oscillatory phenomena.

18.2 Degeneracy of the Landau levels

The significance of the oscillator’s position becomes apparent when we calculate the degen-eracy of the Landau levels. In the presence of a magnetic field, how do we count states?Previously we imposed the Born–von Karmann periodic boundary conditions to a sample offinite volume LxLyLz by requiring that the wavefunction be single valued. This of coursecontinues to hold, requiring that kz be quantized in units of 2π/Lz and, from our wavefunc-tion ansatz, that β be quantized in units of 2π/Ly. But there is an additional restriction on


18.3. LANDAU LEVELS IN A PERIODIC POTENTIAL 141

β — it cannot just take any value. The centre of each cyclotron orbit in real space is relatedto β

x0 =1

ωc

~βm

(18.14)

and must be somewhere within the sample, i.e. 0 ≤ x0 ≤ Lx, which implies that

0 ≤ β ≤ mωc

~Lx (18.15)

The number of allowed values of β is the degeneracy p.

p =mωc

2π~LxLy (18.16)

=e

hBLxLy (18.17)

=φ

φ0

, (18.18)

where φ = BLxLy is the magnetic flux through the sample and φ0 = h/e is the quantumof magnetic flux for a particle of charge e. Note that p counts the number of states for onespin only.

We can now check to see if this correctly counts the number of levels in a two-dimensionalslice of k-space.

In the last lecture we showed that two-dimensional areas Ak in k-space can be related to thecyclotron mass m∗.

∂Ak∂ε

=2πm∗

~2(18.19)

⇒ δAk =2πm∗

~2δε (18.20)

If we suppose that δε = ~ωc, the quantum of cyclotron frequency, the number of statesenclosed between adjacent cyclotron orbits in k-space will be

LxLy(2π)2

δAk =LxLy(2π)2

2πm∗

~2~ωc (18.21)

=m∗ωc

2π~LxLy (18.22)

= p. (18.23)

We get the same result.

18.3 Landau levels in a periodic potential

The results we derived above were for free electrons of mass m∗. What happens when weswitch on the periodic potential of the ion cores? The potential will alter the shape ofthe Fermi surface — drastically in some cases — but we still expect the electrons to trace



out closed orbits in k-space, in planes perpendicular to the magnetic field. What is thequantization condition?

In the last lecture we showed that cyclotron frequency corresponded to geometric propertiesof the constant-energy surfaces. Specifically, in the kz plane

εn+1 − εn = ~ωc =h

T (εn, kz), where (18.24)

T (εn, kz) =~2

eB

∂A(ε, kz)

∂ε. (18.25)

(18.26)

Therefore

(εn+1 − εn)∂A(εn, kz)

∂ε=

2πeB

~

⇒ A(εn+1)− A(εn) =2πeB

~,

∆A =2πeB

~.

Adjacent cyclotron orbits in k-space differ by a fixed area, so the area enclosed by the nthorbit is (n+ λ)2πeB/~, where λ is some (small) constant.

We could have arrived at this result a different way, by appealling to the Bohr–Sommerfeldquantization condition. This states that the integral of the momentum around closed, semi-classical orbits must be quantized in units of Planck’s constant:∮

p · dr = (n+ γ)h, (18.27)

where 0 ≤ γ ≤ 1. For the case of free electrons, γ = 1/2. (Bohr–Sommerfeld quantizationfollows from the requirement that phase be quantized in units of 2π, up to some additivefraction of 2π, which is equivalent to saying that the number of nodes in the wavefunction is agood quantum number.) We will take the semiclassical motion to be in a plane perpendicularto the magnetic field (which is along the z direction) and use our previous results for thesemiclassical trajectory.

dr =~eB

dk× z (18.28)∮p · dr = ~

∮k · dr (18.29)

=~2

eB

∮k · (dk× z) (18.30)∮

k · (dk× z) is just the area of k-space enclosed by the semiclassical orbit. Our quantizationcondition is then

An =2πeB

~(n+ γ), (18.31)

where An is the area of k-space enclosed by the nth orbit.


18.4. VISUALIZING LANDAU QUANTIZATION 143

18.4 Visualizing Landau quantization

In zero field, our quantum states are a set of points in k-space, laid out on a three-dimensionalgrid with a spacing inversely proportional to the size of the sample. When a strong field isswitched on along the z direction, kx and ky cease to be good quantum numbers. We nowvisualize the allowed states as ‘Landau’ tubes or rings, which have quantized cross sectionalarea and energy spacing that grows in proportion to the applied field. In zero field eachk-state could hold one electron of each spin. For a macroscopic sample, the Landau levelsare highly degenerate, as shown above. Quantization along the field direction is unaffected.

k

k

k

k

k

k

filled states

= 0B

empty states

filled states

empty states

z

y

x/= 0B

y

zB

x

Figure 18.1: In the field-free case (top), the electronic eigenstates have well-defined crystalmomenta in a coarse-grained k-space. If a magnetic field Bz is applied, kx and ky cease tobe good quantum numbers, and the new states can be visualized as Landau tubes. (AfterChristoph Bergemann[13].)



18.5 Quantum oscillations as a Fermi surface probe

The transition to Landau quantization in high magnetic fields has profound experimentalconsequences, and leads to a number of effects that have the power to reveal in detail thestructure of the Fermi surface. For these effects to be observable, several rather stringentexperimental conditions must be satisfied. To begin with, Landau quantization, which stemsfrom the Bohr–Sommerfeld requirement of single-valued wavefunctions, will only occur if theelectron can complete a k-space orbit coherently, with a small probability of scattering. Thisrequires that the mean time between collisions that randomize the phase of the electronwavefunction, τ , be much greater than the cyclotron period: 1

τ ωc. The second condition

relates to the temperature. We know that the Fermi distribution changes from 1 to 0 inan energy range of order kBT . In order that thermal smearing not wipe out the effect ofthe Landau quantization, we require the Fermi distribution to be sharp on the scale of theLandau level spacing. This requires that kBT ~ωc.

Fortunately, in many materials these conditions can be met, although very high magneticfields and temperatures much smaller than 1 K are often required. When these requirementsare satisfied, the effect of sweeping the magnetic field is quite dramatic. The spacing betweenthe Landau levels grows in proportion to magnetic field. As the field is swept, the detailedstructure of k-space and the degeneracy of the Landau levels change. The Fermi sea muststill hold N electrons so, as field increases, Landau tubes pop through the Fermi surfaceand are emptied. The thermodynamic potential changes abruptly at this point, and a sharpjump occurs in all thermal and transport properties. The oscillations show up most stronglyin the magnetization. This is called the de Haas–van Alphen effect and historically has beenvery important in surveying the Fermi surfaces of metals. Properties that exhibit quantumoscillations include the resistivity (the Shubnikov–de Haas effect) and even the size of thesample.

We now calculate the thermodynamic potential Ω0(T, V, µ), of a two-dimensional slice ofarea LxLy, for a noninteracting metal in a strong magnetic field. We will work in the grandcanonical ensemble. Although we focus attention on a single slice through the Fermi surface,this will give us great insight into the behaviour of a three-dimensional metal. While theoscillatory contributions from different Fermi-surface slices will in general add out of phase,the regions near extremal cross sections of the Fermi surface will add coherently and willresemble what we calculate for a single slice.

Ω0(T, V, µ) = −kBT∞∑n=0

ln(1 + eβ(µ−εn)

)(18.32)

To demonstrate the effect we carry out the calculation in the zero temperature limit. Ifεn < µ,


)≈ β(µ− εn) as β →∞. (18.33)

On the other hand if εn > µ,

limβ→∞


)= ln 1 = 0. (18.34)


18.5. QUANTUM OSCILLATIONS AS A FERMI SURFACE PROBE 145

As a result,

Ω0(T = 0, V, µ) =n′∑n=0

(εn − µ), (18.35)

where the sum runs over all levels of energy less than the chemical potential µ. In a highmagnetic field the energy levels are Landau levels εn = (n + 1/2)~ωc and these are p-folddegenerate, where p = φ/φ0. p can be made arbitrarily close to an integer by going to thelimit of large sample size. The number of Landau levels required to hold N electrons of onespin will be ≈ N/p.

Ω0(T = 0, V, µ) = p

n′∑n=0

((n+

1

2)~ωc − µ

). (18.36)

n′ =

⌊N

p

⌋− 1, (18.37)

µ = (n′ + 1 +1

2)~ωc. (18.38)

The highest Landau level is only partly occupied, but makes no contribution to the thermo-dynamic potential. In any real situation we will be at a finite temperature, and will be ableto tune this partial occupancy by placing µ just below the highest filled Landau level.

Ω0(T = 0, V, µ) = pn′∑n=0

[(n+

1

2)− (n′ + 1 +

1

2)

]~ωc (18.39)

= −p~ωc

2(n′ + 1)(n′ + 2) (18.40)

= − e2

4πm∗LxLyB

2

⌊N

p

⌋(⌊N

p

⌋+ 1

). (18.41)

N

p=Nφ0

φ=B0

B,B0 =

Akh

4π2e(18.42)

Ak is the k-space area of the two-dimensional slice. Putting this together, the thermodynamicpotential per unit area for one spin is

Ω0(T = 0, V, µ)

LxLy= − e2

4πm∗

(Akh

4π2e

)2

b2

⌊1

b

⌋⌊1

b+ 1

⌋, where b = B/B0. (18.43)

Ω0(T = 0, V, µ) is plotted in Fig. 18.2 as a function of B. The function is periodic in 1/B,which can be seen clearly in Fig. 18.3. Figure 18.4 shows de Haas–van Alphen data on thelayered oxide metal SrRu2O4, from experiments by Mackenzie and coworkers[14]. These datahave been used by Bergemann et al.[15] to constuct the Fermi surface shown in Fig. 18.5.



0.2 0.4 0.6 0.8 1 1.2 1.4

-2

-1

0B/B0

Ω (a

rb. u

nits

)

Figure 18.2: Thermodynamic potential Ω plotted vs B in units of B0 = AkΦ0/8π2. (Ω(B) ∝

−b2b1/bcb1 + 1/bc, b = B/B0).

5 10 15 20 25

-2

-1

0

Ω (a

rb. u

nits

)

B0/B

Figure 18.3: Thermodynamic potential Ω plotted vs 1/B, revealing that the cusps are evenlyspaced in inverse field.


18.5. QUANTUM OSCILLATIONS AS A FERMI SURFACE PROBE 147

0 5 10 15 20DHvA Frequency (kT)

0.00

0.01

0.02

0.03

0.04

Am

plitu

de S

pect

rum

(a.u

.)

2.9

3 kT

3.0

9 kT

6.0

9 kT

12.

58 k

T 12.

88 k

T

18.

64 k

T

15 16 17 18Field (T)

Figure 18.4: Quantum oscillations of the magnetization in SrRu2O4, showing the Fouriertransform with respect to inverse field. (After Mackenzie et al.[14])

Figure 18.5: The Fermi surface of SrRu2O4 inferred from the angle dependence of the deHaas–van Alphen amplitudes. (After Bergemann et al.[15])


Lecture 19

Electronic structure of selected metals

In some materials, electron correlations effects are weak and the electronic structure is welldescribed by the nearly free-electron model. This occurs when the electron density is high,for two reasons. First, the electrostatic screening length is short, so the screened potentialsof the ion cores are weak. Second, at high electron densities the kinetic energy is muchmore important than the electron–electron interaction energy. We now make a brief surveyof the periodic table and look at metallic elements in which the electrons can be treated asindependent particles.

19.1 Construction of free-electron Fermi surfaces

We begin with a recapitulation of the construction of free-electron Fermi surfaces in twodimensions.

19.1.1 Free-electron Fermi surfaces in two dimensions

The recipe for creating free electron Fermi surfaces in 2D is as follows:

• Begin with the reciprocal lattice vectors G. Their perpendicular bisectors define theBragg planes, which in turn divide reciprocal space into Brillouin zones, as shown inFig. 19.1.

• Draw a free-electron Fermi circle of radius kF corresponding to the desired electrondensity n, as shown in Fig. 19.2a. The effects of a weak periodic potential can betaken into account by curving the sections of arc to meet the Bragg planes at rightangles.

• Filled zones are inert.

• Pieces of the higher Brillouin zones are mapped back into the first Brillouin zone ontranslation by reciprocal lattice vectors. This is shown in Figs. 19.2b and 19.2c for thesecond and third zones respectively.


150 LECTURE 19. ELECTRONIC STRUCTURE OF SELECTED METALS

1

23

3

3

3

3

3

2

2 2

(A) (B)

Figure 19.1: Bragg planes and Brillouin zones in two dimensions. (From M. P. Marder[5].)

• The same result can be achieved using the Harrison construction, shown in Figs. 19.2d.The Fermi surface pieces are intersections of circles of radius kF.

19.1.2 Free-electron Fermi surfaces in three dimensions

Similar constructions are used to obtain free-electron Fermi surfaces in three dimensions. Inthis case the separate pieces are formed by the intersections of spheres. The convention isto plot each Fermi-surface sheet separately. This is usually carried out in the reduced zonescheme, as shown in Fig. 19.3 for the face-centred cubic structure.

Figure 19.4 shows the free-electron Fermi surfaces for the face-centred cubic structure, formaterials with one, two and three electrons per unit cell. Figure 19.5 does the same for thebody-centred cubic structure. Figure 19.6 shows the Fermi surfaces for the hexagonal closestpacked structure for two and four electrons per unit cell. In the hcp structure, in the absenceof spin–orbit coupling, there is no first-order band gap along the (001) Bragg planes. For thelighter elements, it is therefore customary to draw the Fermi surface in an enlarged Brillouinzone with the (001) planes removed.


19.1. CONSTRUCTION OF FREE-ELECTRON FERMI SURFACES 151

(A) (B)

(C) (D)

Figure 19.2: Constructing the two dimensional Fermi surface. The free-electron circle (A)extends into the second (B) and third (C) Brillouin zones. (D) The Harrison construction.(From M. P. Marder[5].)



Brillouin zone Extended zone scheme Reduced zone scheme

FirstEmpty Empty

Second

Third

Figure 19.3: Free-electron Fermi spheres for the face-centred cubic structure in the extended(left) and reduced (right) zone scheme. The Fermi surface shown here holds three electronsper unit cell. (From M. P. Marder[5].)

Brillouin 1 electron/cell 2 electrons/cell 3 electrons/cell

zone

First

Second

Third

Figure 19.4: Free electron Fermi surfaces for the face-centred cubic structure for one, twoand three electrons per unit cell. (From M. P. Marder[5].)



Brillouin 1 electron/cell 2 electrons/cell 3 electrons/cell

zone

First

Second

Third

Fourth

Figure 19.5: Free electron Fermi surfaces for the body-centred cubic structure for one, twoand three electrons per unit cell. (From M. P. Marder[5].)

Brillouin 2 electrons/cell 4 electrons/cell 4 electrons/cell

zone with hcp extinction

First

Second

Third

Fourth

Figure 19.6: Fermi surfaces for the hexagonal closest-packed structure for one, two, three andfour electrons per unit cell, and with the (001) Bragg plane removed. (From M. P. Marder[5].)



LITHIUM

SODIUM

POTASSIUM

RUBIDIUM

CESIUM

BERYLLIUM

MAGNESIUM

CALCIUM

STRONTIUM

BARIUM

SCANDIUM TITANIUM VANADIUM CHROMIUM MANGANESE

YTTRIUM ZIRCONIUM NIOBIUM MOLYBDENUM TECHNETIUM

LANTHANUM

HAFNIUM TANTALUM TUNGSTEN RHENIUM

IRON COBALT NICKEL

RUTHENIUM

OSMIUM

RHODIUM

IRIDIUM

PALLADIUM

PLATINUM

CERIUM

THORIUM

PRASEODYMIUM

PROTACTINIUM

NEODYMIUM

URANIUM

PROMETHIUM SAMARIUM

NEPTUNIUM PLUTONIUM

EUROPIUM

AMERICIUM

LUTETIUM

LAWRENCIUMFRANCIUM RADIUM

ACTINIUM

3 Li 4 Be

11 Na 12 Mg

19 K 20 Ca 21 Sc

37 Rb

55 Cs

38 Sr

56 Ba

39 Y

22 Ti

40 Zr

57 La

72 Hf

23 V

41 Nb

73 Ta

24 Cr 25 Mn 26 Fe 27 Co 28 Ni

42 Mo 43 Tc 44 Ru 45 Rh 46 Pd

74 W 75 Re 76 Os 77 Ir 78 Pt

87 Fr 88 Ra

89 Ac

58 Ce 59 Pr 60 Nd

90 Th 91 Pa 92 U

61 Pm

93 Np

62 Sm

94 Pu

63 Eu

95 Am

71 Lu

103 Lr

1s22s1 1s22s2

[Ne]3s1 [Ne]3s2

[Ar]4s1 [Ar]4s2 [Ar]3d14s2

[Kr]5s1 [Kr]5s2 [Kr]4d15s2 [Kr]4d25s2

[Xe]6s1 [Xe]6s2

[Xe]5d16s2

[Xe]4 f 145d26s2 [Xe]4 f 145d36s2

[Rn]7s1 [Rn]7s2

[Rn]6d17s2

[Ar]3d24s2 [Ar]3d34s2 [Ar]3d54s1 [Ar]3d54s2

[Kr]4d45s1 [Kr]4d55s1 [Kr]4d55s2

[Xe]4 f 145d46s2 [Xe]4 f 145d56s2

[Ar]3d64s2 [Ar]3d74s2 [Ar]3d84s2

[Kr]4d75s1 [Kr]4d85s1 [Kr]4d105s0

[Xe]4 f 145d66s2 [Xe]4 f 145d76s2 [Xe]4 f 145d106s0

[WOLFRAM]

[Xe]4 f 25d06s2 [Xe]4 f 35d06s2 [Xe]4 f 45d06s2 [Xe]4 f 55d06s2 [Xe]4 f 65d06s2 [Xe]4 f 75d06s2

[Rn]5 f 06d27s2 [Rn]5 f 26d17s2 [Rn]5 f 36d17s2 [Rn]5 f 46d17s2 [Rn]5 f 66d07s2 [Rn]5 f 76d07s2

[Xe]4 f 145d16s2

[Rn]5 f 146d17s2

a 4 38

u 6 94

n 4 63

T 454 8 55

a 2 29

c 3 58

u 9 01

n 12 36

T 1551 4 0

a 3 77

c 6 15

u 22 99

n 2 54

T 371 4 2a 3 21

c 5 21

u 24 31

n 4 31

T 922 4 45

a 5 33

u 39 10

n 1 33

T 337 6 15

a 5 59

u 40 08

n 2 33

T 1112 3 43

a 3 31

c 5 27

u 44 96

n 4 00

T 1814 61 0

a 5 62

u 85 47

n 1 08

T 312 12 5a 6 08

u 87 62

n 1 75

T 1042 23 0a 3 65

c 5 73

u 88 91

n 3 03

T 1795 57 0a 3 23

c 5 15

u 91 22

n 4 30

T 2125 42 1

a 2 95

c 4 68

u 47 88

n 5 70

T 1933 42 0

a 6 14

u 132 91

n 0 85

T 302 20 0a 5 03

u 137 33

n 1 57

T 1002 50

a 3 77

c 1 22

u 138 91

n 2 67

T 1194 57

a 3 19

c 5 05

u 178 49

n 4 49

T 2503 35 1

u 223

n

T 300

a 5 15

u 226 03

n 1 33

T 973 100

a 5 31

u 227 03

n 2 67

T 1320

a 3 02

u 50 94

n 7 22

T 2160 24 8

a 3 30

u 92 91

n 5 55

T 2741 12 5

a 3 30

u 180 95

n 5 54

T 3269 12 45

a 3 17

u 183 85

n 6 32

T 3680 5 65

a 3 15

u 95 94

n 6 41

T 2890 5 2

a 2 88

u 52 00

n 8 32

T 2130 12 7a 8 91

u 54 94

n 8 15

T 1517 185 0

a 2 74

c 4 40

u 98 91

n 7 00

T 2445 22 6

a 2 76

c 4 46

u 186 20

n 6 80

T 3453 19 3

a 2 87

u 55 85

n 8 48

T 1808 9 71

a 2 71

c 4 28

u 101 07

n 7 37

T 2583 7 6

a 2 73

c 4 32

u 190 2n 7 15

T 3327 8 12

a 3 54

u 58 93

n 9 09

T 1768 6 24

a 3 52

u 58 69

n 9 13

T 1726 6 84

a 3 89

u 106 42

n 6 80

T 1825 10 8a 3 80

u 102 91

n 7 26

T 2239 4 51

a 3 84

u 192 22

n 7 07

T 2683 5 3a 3 92

u 195 08

n 6 62

T 2045 10 6

a 4 85

u 140 12

n 3 54

T 1072 73

a 3 67

c 11 83

u 140 91

n 2 89

T 1204 68

a 3 66

c 11 80

u 144 24

n 2 93

T 1294 64 0a

u 145

n 3 00

T 1441 50

a 9 00 23 13

u 150 36

n 3 01

T 1350 94 0

a 6 18

b 4 82

c 10 96 101 48

u 244

n 4 89

T 914 146

a 4 72

b 4 89

c 6 66

u 237 05

n 5 14

T 913 122

a 2 85

b 5 86

c 4 95

u 238 03

n 4 79

T 1406 30 8a 3 93

c 3 24

u 231 04

n 4 34

T 2113 17 7a 5 08

u 232 04

n 3 04

T 2023 13 0

a 4 58

u 151 97

n 2 08

T 1095 90 0

a 3 47

c 11 24

u 243

n 3 39

T 1267 68

a 3 50

c 5 55

u 174 97

n k3 39

T 1936 79 0

u 260

n

T

? ?

RUTHERFORDIUM

104 Rf

DUBNIUM

105 Db

SEABORGIUM

106 SgBOHRIUM

107 Bh

HASSIUM

108 Hs

MEITNERIUM

109 Mt

Alk

aliM

etal

s

Transition Metals

Actinides

Lanthanides

[Rare Earths]

IIIb IVb Vb VIb VIIb VIIIb

Ia IIa Metal Insulator Semiconductor Semi-metal

Atomic name

Ground state electron configuration

Melting temperature in K

Crystal structure, either at 293 K, or at

melting if liquid at 293 K

Atomic weight (12C=12)

Density in 1022 atomscm 3, at 293K or at melting

Electrical resistivity in -cm at 298 K

Lattice parameters

Atomic number and symbol

SILICON

14 Si

[Ne]3s23p2

a 5 43

u 28 09

n 4 99

T 1683 1 105

Figure 19.7: The periodic table showing crystal structures at room temperature. (FromM. P. Marder[5].)



COPPER ZINC

SILVER

GOLD

CADMIUM

MERCURY

BORON CARBON NITROGEN OXYGEN FLUORINE

HELIUM

NEON

ALUMINUM SILICON PHOSPHORUS SULFUR CHLORINE ARGON

GALLIUM

INDIUM

THALLIUM

TIN

LEAD

ARSENIC

ANTIMONY

BISMUTH

SELENIUM

TELLURIUM

POLONIUM

BROMINE KRYPTON

IODINE

ASTATINE

XENON

RADON

GADOLINIUM

CURIUM

TERBIUM

BERKLIUM

DYSPROSIUM

CALIFORNIUM EINSTEINIUM

HOLMIUM ERBIUM THULIUM

FERMIUM MENDELEVIUM

YTTERBIUM

NOBELIUM

86 Rn

[Xe]4 f 145d106s26p6

29 Cu

47 Ag

79 Au

30 Zn

48 Cd

80 Hg

31 Ga

49 In

81 Tl

50 Sn 51 Sb

82 Pb 83 Bi

33 As

13 Al

5 B 6 C 7 N 8 O 9 F

14 Si 15 P 16 S 17 Cl

2 He

10 Ne

34 Se 35 Br

18 Ar

36 Kr

52 Te 53 I 54 Xe

84 Po 85 At

64 Gd

96 Cm

65 Tb

97 Bk

66 Dy 67 Ho 68 Er 69 Tm

98 Cf 99 Es 100 Fm 101 Md

70 Yb

102 No

[Ar]3d104s1 [Ar]3d104s2

[Kr]4d105s1 [Kr]4d105s2

[Xe]4 f 145d106s1 [Xe]4 f 145d106s2

[Kr]4d105s25p1

1s22s22p1 1s22s22p2 1s22s22p3 1s22s22p4 1s22s22p5

[Ne]3s23p1 [Ne]3s23p2 [Ne]3s23p3 [Ne]3s23p4 [Ne]3s23p5 [Ne]3s23p6

[Ar]3d104s24p1 [Ar]3d104s24p3 [Ar]3d104s24p4 [Ar]3d104s24p5

[Kr]4d105s25p2 [Kr]4d105s25p3 [Kr]4d105s25p4 [Kr]4d105s25p5

[Ar]3d104s24p6

[Kr]4d105s25p6

1s22s22p6

1s2

[Xe]4 f 145d106s26p5[Xe]4 f 145d106s26p4[Xe]4 f 145d106s26p3[Xe]4 f 145d106s26p2[Xe]4 f 145d106s26p1

[Xe]4 f 75d16s2 [Xe]4 f 95d06s2 [Xe]4 f 105d06s2

[Rn]5 f 76d17s2 [Rn]5 f 96d07s2 [Rn]5 f 106d07s2

[Xe]4 f 115d06s2 [Xe]4 f 125d06s2 [Xe]4 f 135d06s2 [Xe]4 f 145d06s2

[Rn]5 f 116d07s2 [Rn]5 f 126d07s2 [Rn]5 f 136d07s2 [Rn]5 f 146d07s2

a 3 61

u 63 55

n 8 49

T 1357 1 67

a 2 66

c 4 95

u 65 39

n 6 56

T 693 5 92

a 4 52

b 7 66

c 4 53

u 69 72

n 5 10

T 303 27

a 3 23

c 4 94

u 114 82

n 3 83

T 429 8 37

a 2 98

c 5 62

u 112 41

n 4 63

T 594 6 83

a 4 09

u 107 87

n 5 86

T 1235 1 59

a 4 08

u 196 97

n 5 90

T 1338 2 35

a 2 99 70 45

u 100 59

n 4 07

T 234 94 1a 3 46

c 5 53

u 204 38

n 3 49

T 577 18 0

a 4 05

u 26 98

n 6 02

T 934 2 65

a 8 74

c 5 06

u 10 81

n 13 03

T 2573 1 8 1012

a 3 57

u 12 01

n 17 59

T 3820 1019

a 5 64

u 14 01

n 4 43

T 63 29

a 6 83

u 16 00

n 7 53

T 54 8

a 6 67

u 19 00

n

T 53 5

a 5 43

u 28 09

n 4 99

T 1683 1 105

a 18 51

u 30 97

n 3 54

T 317 1 1017

a 10 46

b 12 87

c 24 49

u 32 07

n 3 89

T 386 2 1023

a 6 24

b 4 48

c 8 26

u 35 45

n 3 45

T 172

a 4 13 54 10

u 74 92

n 4 64

T 1090 26

a 4 37

c 4 96

u 78 96

n 3 65

T 490 1 106

a 6 74

b 4 55

c 8 76

u 79 90

n 3 05

T 266

a 5 83

c 3 18

u 118 71

n 3 71

T 505 11 0a 4 51 57 7

u 121 75

n 3 31

T 904 39 0a 4 46

c 5 93

u 127 60

n 2 94

T 723 4 36 105

a 7 26

b 4 79

c 9 79

u 126 91

n 2 34

T 387 1 3 1015

a 4 95

u 207 2n 3 30

T 601 20 65

a 4 75 57 14

u 208 98

n 2 81

T 545 106 8a 3 35

u 209

n 2 68

T 527 140

u 210

n

T 575

a

u 222

n

T 202

a 6 19

u 131 29

n 1 62

T 161

a 5 72

u 83 80

n 2 03

T 117

a 5 31

u 39 95

n 2 50

T 83 8

a 4 45

u 20 18

n 4 30

T 24 5

a 3 53

c 4 24

u 4 003

n 3 11

At 2 K, 26 atm

HYDROGEN

1 H

1s1

a 3 77

c 6 16

u 1 008

n 4 54

T 14 01

a 3 64

u 157 25

n 3 02

T 1586 134

a 3 59

b 6 26

c 5 72

u 158 93

n 3 12

T 1629 114

a 3 59

c 5 65

u 162 50

n 3 17

T 1685 57 0a 3 58

c 5 62

u 164 93

n 3 21

T 1747 87 0a 3 56

c 5 59

u 167 27

n 3 26

T 1802 87

u 254

n

T

a

u 251

n

T

u 247

n 3 60

T

u 247

n 3 24

T 1610

a 3 54

c 5 55

u 168 93

n 3 32

T 1818 79 0a 5 49

u 173 04

n 2 42

T 1097 29 0

u 259

n

T

u 258

n

T

u 257

n

T

GERMANIUM

32 Ge

[Ar]3d104s24p2

a 5 66

u 72 61

n 4 41

T 1211 4 6 107

?

? ? ? ? ? ?

Hal

ogen

s

Nob

leG

ases

Nob

lem

etal

s

VIIIaVIIaVIaVaIVaIIIa

Ib IIb

fcc:

hcp:

bcc:

cubic: orthorhombic: monoclinic:

hexagonal:

diamond:

rhombohedral: tetragonal:

Figure 19.8: The periodic table showing crystal structures at room temperature. (FromM. P. Marder[5].)



19.2 The alkali metals

The alkali metals crystallize in the body-centred cubic structure, although lithium andsodium undergo low-temperature structural phase transitions. The alkali atoms contributeone outer electron to the conduction band, with the remaining electrons tightly bound in theion cores. The core electrons give rise to flat bands that can be ignored when calculating theelectronic structure and Fermi surface. With one electron per atom, the Fermi wavevectoris given by

k3F

3π2= n =

2

a3(19.1)

where the factor of 2 is due to the fact that a is the side of the conventional cubic unit cell,which holds two atoms. kF = 0.620(2π/a), which is less than the distance from the originto the nearest Bragg plane, ∆kmin = 0.707(2π/a). The Fermi sphere is confined to the firstBrillouin zone. Figure 19.9 shows a pseudopotential band-structure for sodium. (No corestates are included.) The Fermi energy is the dashed line, and falls entirely within the firstband. The Fermi surface, shown in Fig. 19.10, is spherical to within 0.1%.

Figure 19.9: Band structure of sodium. (From D. A. Papaconstantopoulos[16].)

The effective masses m∗ of the alkali atoms Li, Na, K, Rb and Cs are 1.33, 0.97, 0.86, 0.78 and0.73 electron masses respectively, very close to the free electron value. We can understandthis progression from the second-order perturbation theory result

m

m∗= 1 +

2

m

∑n′ 6=n

|〈n|P |n′〉|2

E0n − E0

n′. (19.2)


19.3. THE NOBLE METALS 157

Figure 19.10: First Brillouin zone of the body-centred cubic structure, showing a monovalentFermi surface. (From Ashcroft and Mermin[4].)

Mixing with states higher in energy acts to increase the effect mass, and coupling to lowerenergy states has the opposite effect. For Li there are no lower energy states, so the massmust increase.

Figure 19.11: The Fermi surface of copper showing neck and belly extremal orbits on theleft, and the dog-bone orbit on the right. (From Ashcroft and Mermin[4].)

19.3 The noble metals

The noble metals are Cu, Ag and Au. The electronic structure of copper ([Ar]3d104s1) differsfrom that of potassium ([Ar]4s1) by a filled d shell. The closed-shell argon configuration



Figure 19.12: Band structure of copper[17].

leads to filled inert bands in both Cu and K. In the case of potassium the extra electron isaccommodated by half filling a free-electron like band. In copper (and silver and gold) thed electrons are not so tightly bound to the atom that they can be considered as part of thecore — six bands are required to accommodate the eleven additional electrons. We expectto find a broad s band and a narrow d band, and these bands overlap and hybridize. Themixing can be seen in the band structure of copper shown in Fig. 19.12. The result of thehybridization is to lower the energy of the s band near the L point in the Brillouin zone.This has an important effect on the Fermi surface topology, which now ‘necks out’ to touchthe edge of the Brilluoin zone. The open Fermi surface can be seen in Fig. 19.11.

19.4 Divalent metals

The alkaline earth metals (column two of the periodic table) are divalent. There are twoelectrons in the outer shell, which form the conduction band. The Fermi surface musttherefore have the same volume as the first Brillouin zone. Since the materials are metals,the Fermi surface occupies the first and second Brillouin zones and is now quite complicated.The band structure of calcium is shown in Fig. 19.14.

Be, Mg, Zn and Cd crystallize in the hexagonal closest packed structure. Beryllium is thelightest and has the weakest spin–orbit coupling. It has the enlarged Brillouin zone shownin Fig. 19.6 and the Fermi surface shown in Fig. 19.13.


19.4. DIVALENT METALS 159

Figure 19.13: Fermi surface of beryllium[18].

Figure 19.14: Band structure of calcium. (From D. A. Papaconstantopoulos[16].)



19.5 The trivalent and tetravalent metals

The simplest trivalent metal is aluminium. It crystallizes in the face centred cubic structureand has the Fermi surface shown in Fig. 19.15. The simplest tetravalent metal, lead, alsocrystallizes in the fcc structure, shown in Fig. 19.16.

Figure 19.15: Fermi surface of aluminium[19].


19.6. TRANSITION METALS 161

Figure 19.16: Free electron Fermi surfaces for the face-centred cubic structure, with the thirdzone Fermi surface sheet of Pb shown on the right.

19.6 Transition metals

The transition metals cannot be described by the nearly free electron model due to strongcorrelations in the flat, partially filled d bands. We will discuss these electron correlationeffects in the next lecture. A periodic table showing Fermi surfaces for the metallic elementscan be downloaded from http://www.phys.ufl.edu/fermisurface . These Fermi surfaceswere calculated from tight-binding parameterizations of more accurate band structure cal-culations.


Lecture 20

Strongly correlated electron systems& semiconductors

In the last lecture we discussed the elemental metals that can be described by the nearlyfree electron theory. In the materials in which it applies, the success of the theory is near-miraculous, with many Fermi surfaces instantly recognizable as reconstructed pieces of afree-electron sphere.

We begin this lecture by looking at a number of materials in which the independent electronapproximation breaks down. These are materials in which the energy of electron–electroninteraction is comparable to or greater than the kinetic energy1. We call these materialsstrongly correlated electron systems. The consequences of the more delicate balance betweenkinetic and potential energy are to form magnetic states, or for a ‘band-metal’2 to becomean insulator.

20.1 Interactions and the Hubbard U

A discussion of strongly correlated electron systems requires a localized tight-binding de-scription of the electronic structure. The kinetic energy is of order t, the nearest neighbouroverlap integral. The dominant term in the electron–electron repulsion comes from twoelectrons occupying the same site. This is called the Hubbard ‘U ’, where

U =1

2

∫dr

∫dr′|φ(r)|2|φ(r′)|2 1

4πε0

e2

|r− r′|. (20.1)

Kinetic and potential terms work against each other. Imagine starting with a simple solidof N sites that has one electron and one orbital per site. We form extended states by

1By kinetic energy we mean the bandwidth of the electronic states, which is proportional to the overlapintegral t between neighbouring atoms. The kinetic energy is then measured from the bottom of the bandand does not include the contribution from the electron’s motion in the nuclear potential.

2A band metal is a material that independent electron band theory predicts to be a metal. In particular,band theory predicts that any material with an odd number of electrons per unit cell should be a metal,because it will be unable to completely fill all the occupied Brillouin zones in order to become an insulator.


164 LECTURE 20. STRONGLY CORRELATED SYSTEMS & SEMICONDUCTORS

UHB

U

µ µ ε ε

D(ε) D(ε)

LHB

Figure 20.1: The density of states D(ε) is sketched for the case t U (left) and t U(right). When the Hubbard U is strong enough, it splits the conduction band into upper(UHB) and lower (LHB) Hubbard bands, driving a metal–insulator transition.

making linear combinations of the atomic orbitals, which lowers the kinetic energy by ordert. The potential energy increases by ∼ U , because the probability of double occupancy inthe extended state is 1

2.3 If t U , the extended state will have the lower energy. In the

opposite limit, t U , it is energetically favourable to localize the electrons, because thiswill make the probability of double occupancy zero. The resulting state is an insulator, withan energy gap U for charge excitations. The transition from extended to localized states iscalled the Mott transition. Figure 20.1 shows a schematic of the Mott–Hubbard density ofstates.

20.2 Measuring the Hubbard U

−e→ +e

−e

+e I1 ≈1

4 0

e2

a0− initial kinetic energy

−e+2e→

−e

+2e I2 ≈1

4 0

2e2

a0− initial kinetic energy

Figure 20.2: An illustration of the ionization processes that occur when measuring the firstand second ionization potentials. The ionization energy is the Coloumb energy required toseparate the electron from the positive ion, minus the initial kinetic energy of the electron.

Imagine carrying out the following experiment, depicted in Fig 20.2. Take a neutral atom.

3To calculate the probability of double occupancy, note that in the extended state the probability of anyone electron being at a particular site is 1/N . There are N ways to choose the first electron at a site but,once it has been chosen, Pauli exclusion requires that the second electron have opposite spin. There are thenonly N/2 ways of choosing the second electron.


20.3. 3D TRANSITION METALS 165

Then measure the energy required to excite a bound electron and move it to infinity, leavingbehind a singly charged positive ion. (This experiment is carried out using photo-electronspectroscopy at x-ray wavelengths.) The excitation energy is the first ionization potential I1,and is equal to the Coulomb energy required to separate an electron from the positive ion,minus the initial kinetic energy of the electron. Now remove a second electron and move it toinfinite separation. This is the second ionization potential I2, given by the energy requiredto separate an electron from a doubly charged ion, minus the initial kinetic energy. Dueto the double charge on the ion, I2 is larger than I1. In fact, neglecting details about thekinetic energy, the difference I2 − I1 is just the Coulomb energy of two electrons separatedby distance a0.

We have seen this problem before in a different guise. Imagine a highly simplified model ofan atom, in which we have one orbital that can be at most doubly occupied.

Hatom = ε0 +∑σ

ε1nσ + Un↑n↓, (20.2)

where nσ is the number operator and σ is the spin projection. The energies for zero, singleand double occupancy are

E(0) = ε0 (20.3)

E(1) = ε0 + ε1 (20.4)

E(2) = ε0 + 2ε1 + U (20.5)

Rearranging, we see that

U = E(0) + E(2)− 2E(1) (20.6)

=(E(2)− E(1)

)−(E(1)− E(0)

)(20.7)

If we identify E(2)− E(1) with I2 and E(1)− E(0) with I1, we see that the Hubbard U isjust the difference between the first and second ionization potentials. The quantum particlesin the hamiltonian above are really holes, which are the elementary excitations of chargeremoval.

20.3 3d transition metals

The 3d transition metals range from Sc to Zn. They are characterized by the gradual fillingof the d shell, going from scandium ([Ar]3d14s2) to zinc ([Ar]3d104s2). The argon-like core ishighly localized and plays no role in determining the physical properties of the solid. The 4sorbitals are filled before the 3d orbitals, because the 4s states have a larger amplitude at thenucleus and therefore experience a higher effective charge. However, for Cr ([Ar]3d54s1) andCu ([Ar]3d104s1) the stability of a filled or half-filled d shell partially empties the s state.

The 4s electrons extend away from the atomic core, in order to maintain orthogonality withthe 1s, 2s and 3s electrons. The 3d electrons have no such need and are much more highlylocalized. They therefore overlap very little with d electrons on neighbouring atoms and theeffects of Coulomb repulsion are of much greater importance relative to the bandwidth. The



predominant effect of Coulomb repulsion in the 3d elements is the ferromagnetic alignmentof spins on the same atom (Hund’s rule). In principle, the Coulomb repulsion could alsoforce the d electrons to localize and the d band to split into as many as 10 separate Hubbardbands. Instead, the effective Coulomb repulsion between the 3d electrons is reduced by thescreening effects of electrons in other bands, especially the 4s. To see how this works, let’stry to estimate U for the Cu2+ or Ni+ configurations, which have the electronic configuration3d9s0.

U = I2 − I1 (20.8)

=(E(3d8)− E(3d9)

)−(E(3d9)− E(3d10)

)(20.9)

=(E(3d10) + E(3d8)

)− 2E(3d9) (20.10)

This is the energy of the transition

2(3d9)→ 3d10 + 3d8. (20.11)

This works out to be about 13.5 eV, most of which is the Coulomb cost of a doubly-chargedion versus two singly charged ions. In a solid this is an overestimate, because the s electronscan relax to compensate for the charging of the ions. We should really consider the transition

2(3d94s1)→ 3d104s0 + 3d84s2, (20.12)

which has an energy of only 1.8 eV. In transition metal oxides, however, the transition metalion is in a M+ or M2+ state, the 4s state is empty and, as a result, interatomic screening isabsent. Materials such as NiO or V2O3 are the archetypical Mott–Hubbard insulators.

20.4 Semiconductor structures

The semiconducting elements Si and Ge have the diamond structure, shown in Fig. 20.3.This is a face-centred cubic (fcc) lattice, except that there are two atoms per primitive cell.The first is at a corner of the fcc lattice, and the second is located 1

4the way along the

main diagonal. Carbon, and one form of tin (grey tin) also adopt the diamond structure.In the case of C (diamond), the energy gap is very large, approximately 5.5 eV, and thestructure is an excellent electrical insulator. In the case of grey tin, conduction and valencebands touch at the centre of the Brillouin zone, making the material a semi-metal, althoughthe average energy gap is small and positive. In addition, tin can take two different crystalstructures, depending on the temperature. This reflects the fact that the diamond structurebecomes increasingly energetically unfavourable as we go down the fourth column of theperiodic table, for reasons we will discuss below. (Pb is a metal, and forms in a close-packedstructure.)

In contrast to metals, semiconductors are covalently bonded, with only four nearest neigh-bours at the corners of a tetrahedron, rather than the twelve of a close-packed metal. Insteadof lowering its energy by forming a high-density metallic state, cohesion in semiconductingmaterials comes from directed bonding. In fact, we can think of a semiconductor crystal asa giant molecule. This is borne out by detailed electronic structure calculations, which show


20.5. SEMICONDUCTOR CHEMISTRY 167

Figure 20.3: The diamond structure. (From Phillips[20].)

the valence charge concentrated along the bond directions. A number of binary compoundswith eight valence electrons per atom are also formed in tetrahedrally coordinated struc-tures. These include the III–V materials InSb and GaAs. Here the group III and group Velements sit on separate, interpenetrating fcc lattices, in an arrangement called the zincblendestructure. If you look carefully at Fig. 20.3 you will notice that such a structure lacks in-version symmetry. II–VI compounds are also possible — they start to have substantial ioniccharacter and are called polar semiconductors.

20.5 Semiconductor chemistry

For the group IV elements, the valence shell electronic structure is ns2np2, giving 4 electronsper atom and 8 per primitive cell. We therefore need to fill 4 bands. (Each band holds 2electrons per primitive cell.) Adopting a tight-binding approach, we can see how these bandscome from the four orbitals of the free atoms: the 3s and three 3p orbitals of Si, and the4s and three 4p orbitals of Ge. Since we have two electrons per unit cell, our wavefunctionansatz will consist of Bloch sums of 8 orbitals, giving a total of eight bands. We thereforeneed to fill half the bands to accommodate the valence electrons.

In the theory of chemical bonds, the tetrahedral structure is well known. The s and p orbitalscan be combined to form a new basis set, the sp3 hybrids shown in Fig. 20.4, which point inthe direction of the vertices of a tetrahedron. (In the case of the diamond structure, pointingin (half of) the 〈111〉 directions.) The lobes of these orbitals are directional and extend inthe direction of the nearest neighbours.



Figure 20.4: sp3 hybridization.

In a tight-binding description, our effective hamiltonian consists of atomic energies and two-centre overlap integrals of the form

tij =

∫dr φi(r)Va(r−R)φj(r−R) (20.13)

In the case of the tetrahedral diamond structure, the largest overlap integrals by far arethose between pairs of sp3 hybrids in which the lobe of each orbital points directly alongthe bond to its neighbour. If we neglect all other terms in the tight-binding model, wewould have a hamiltonian that factors into pairs of noninteracting diatomic molecules. Theband structure would reflect the presence of these 2 × 2 sub-units in the hamiltonian: themixing of degenerate orbitals would give a splitting into bonding and antibonding orbitals.There would then be four occupied bonding orbitals, split by a hybridization gap fromthe four empty antibonding orbitals. However, there would be no dispersion because wehave neglected two things: the energy cost of ‘promoting’ an s electron into a p state4

and the further hybridization or hopping between non-colinear orbitals. These extra effectswill broaden the bonding and antibonding orbitals into bands in the real material, but thefundamental origin of the gap is chemical.


20.6. BONDING, NONBONDING AND ANTIBONDING STATES 169

Figure 20.5: Bonding and antibonding orbitals. (From Phillips[20].)

20.6 Bonding, nonbonding and antibonding states

This terminology comes up all the time in discussions of the electronic structure of realmaterials, but is never properly explained in physics textbooks. It is therefore worth sayinga few words about it here.

• A bonding orbital ψb(r) consists of a linear combination of two directed orbitalsφi(r−R) and φj(r − R′) associated with nearest neighbour atoms at sites R andR′, combined in phase in such a way that ψb(r) is large in the bonding region betweenatoms.

• An antibonding orbital ψa(r) is similar to a bonding orbital except the phase betweenthe two hybrid states has been reversed. Therefore ψa(r) has a node in the bondingregion.

• A nonbonding orbital is usually centred on one atom and has little directional character.

Rules for bonding:

• The number of independent hybridized orbitals is equal to the number of atomic or-bitals.

4As we will see below, the sp3 hybrids are not quite the right description. The energy of an s-state isalways lower than that of a p state: Ens < Enp. The s electrons penetrate closer to the nucleus than thep electrons, and therefore feel the attractive core potential more. This energy separation increases as we godown the periodic table, eventually destabilizing the sp3 hybrids.



Figure 20.6: The band structure of silicon. (From Phillips[20].)

• In order of increasing energy one has bonding, nonbonding, and antibonding states.Usually the first two groups are occupied, and the third group is not. The bondingelectrons have the lowest energy because they benefit most from the large attractivepotential produced by the overlap of the Coulomb potentials of the neighbouring atoms.

20.7 Electronic structure

The band structure of silicon is shown in Fig. 20.6. Silicon is an indirect-gap material. Thevalence band maximum occurs at the Γ point, k = 0. The conduction band minima lie alongthe [100] directions, about 80% of the way to the zone boundary.

Because the properties of semiconductors are determined by relatively small number of elec-trons and holes, it is usual to focus attention on the levels near the bottom of the conductionband, which hold electrons, and the levels near the top of the valence band, which will


20.7. ELECTRONIC STRUCTURE 171

Figure 20.7: Constant energy surfaces for the conduction band in a) germanium and b)silicon. (From M. P. Marder[5].)

contain holes. The energy dispersions are well approximated by quadratic forms

ε(k) = εc +~2

2

∑µν

kµ(M−1

)µνkν electrons (20.14)

ε(k) = εv −~2

2

∑µν

kµ(M−1

)µνkν holes, (20.15)

where kµ = (k − k0)µ, and k0 is the location in k-space of the conduction band minimumor valence band maximum. The effective mass tensor is real and symmetric, so it is alwayspossible to transform to orthogonal coordinates in which it is diagonal.

ε(k) = εc + ~2

(k2

1

2m1

+k2

2

2m2

+k2

3

2m3

)electrons (20.16)

ε(k) = εv − ~2

(k2

1

2m1

+k2

2

2m2

+k2

3

2m3

)holes (20.17)

The constant energy surfaces are ellipsoidal in shape, with three different effective massesin general. At the centre of the zone, cubic symmetry maps the different directions into oneanother, so the mass tensor is isotropic and the constant energy surfaces spherical. Alongaxes of high symmetry, the constant energy surfaces are typically cigar shaped, as shown forsilicon and germanium in Fig. 20.7.

20.7.1 Spin–orbit coupling

In heavy atoms there is a strong spin–orbit coupling interaction — an operator of the formλL · S. In free atoms this removes the degeneracy of some states with the same space



Figure 20.8: Sketch of the valence bands of diamond and zincblende structures showing thedispersions with and without the spin–orbit interaction. (From Phillips[20].)

wavefunction but opposite electron spin. For example, the 6 atomic p states are split into afour-fold degenerate P3/2 multiplet and a P1/2 doublet.

When constructing energy bands we usually assume that spin is irrelevant. We just usea wavefunction in space and for each ψk assume we can put in two electrons of oppositespin. However, the spin–orbit interaction can resolve this degeneracy. This is particularlyimportant at points in the Brillouin zone with high symmetry. In the fcc structure, the Γpoint, k = 0, is a point of cubic symmetry. In the tight-binding model our wavefunctionswould consist of linear combinations of p orbitals, giving these bands a six-fold degeneracyat the Γ point, including spin degeneracy. In the presence of a spin–orbit interaction, weshould have started with the four-fold degenerate P3/2 states and the two-fold degenerateP1/2 states when constructing Bloch sums.

What happens as we move away from k = 0 depends on whether or not our crystal hasa centre of inversion symmetry. It it does, as in silicon and germanium, the spin–orbitinteraction will not separate states of opposite spin because, in effect, reversing the spin in astate ψk(r) is equivalent to looking at ψk(−r), which has the same energy. The j = 3

2band

therefore splits into a band with mj = ±32, and a band with mj = ±1

2. In a crystal that

lacks inversion symmetry, such as InSb, the bands split again as we move away from k = 0.

20.8 Other semiconductors

The band structures for germanium, another indirect-gap material, and GaAs, a direct-gapsemiconductor are shown in Figs. 20.9 and 20.10.


20.8. OTHER SEMICONDUCTORS 173

Figure 20.9: The band structure of germanium. (From Phillips[20].)

Figure 20.10: The band structure of GaAs. (From Phillips[20].)


Lecture 21

Semiconductors

21.1 Homogeneous semiconductors

The technological importance of semiconductors is due to the flexibility of their electricalproperties. For instance, the resistivity can be tuned from 10−5 Ω.m to 107 Ω.m by makingsmall changes to the dopant density. (These numbers can be compared to the resistivityof typical metals, ρ ∼ 10−8 Ω.m and of insulators, where ρ ∼ 1020 Ω.m.) The high degreeof tunability is due to the extreme sensitivity of semiconductors to charged defects, andthis initially delayed a proper understanding of semiconductor materials until high-quality,low-impurity single crystals could be produced. In good quality samples the resistivity isstrongly temperature dependent. This is a unique consequence of the way that charge carriersin semiconductors are formed — by thermally exciting electrons from the valence band intothe conduction band, which leaves behind holes in the valence band.

21.1.1 Carrier density in thermal equilibrium

Calculating the carrier density of a semiconductor in thermal equilibrium requires a straight-forward application of Fermi statistics. Let nc be the number of electrons in the conductionband and pv be the number of holes in the valence band. Although the density of carriers isvery sensitive to the presence of impurities, we start by considering the so-called ‘intrinsic’regime, appropriate to highly pure material. It will turn out that at temperatures not toofar below room temperature, impurities just affect the position of the chemical potential µ.

Let Dc(ε) be the conduction band density of states and Dv(ε) be the valence band densityof states.

nc(T ) =

∫ ∞εc

dε Dc(ε)f(ε) (21.1)

=

∫ ∞εc

dε Dc(ε)1

exp((ε− µ)/kBT

)+ 1

(21.2)


176 LECTURE 21. SEMICONDUCTORS

acceptor states

D ( )

conduction bandvalence bandband gap

donor states

Figure 21.1: The density of states of a doped semiconductor.

pv(T ) =

∫ εv

−∞dε Dv(ε) [1− f(ε)] (21.3)

=

∫ εv

−∞dε Dv(ε)

[1− 1

exp((ε− µ)/kBT

)+ 1

](21.4)

=

∫ εv

−∞dε Dv(ε)

1

exp((µ− ε)/kBT

)+ 1

(21.5)

Impurities affect this picture by introducing defect states in the gap, as shown in Fig. 21.1.Their presence can be used to tune the position of the chemical potential, which in mostcases still falls within the band gap, allowing us to write:

εc − µ kBT (21.6)

µ− εv kBT. (21.7)

When these assumptions hold, we have a nondegenerate semiconductor. When the assump-tions break down, the semiconductor is said to be degenerate, and we need to solve Eqs. 21.2and 21.5 numerically.

21.1.2 The nondegenerate case

In the nondegnerate case, the Fermi factors in Eqs. 21.2 and 21.5 can be approximated byclassical Boltzmann factors:

1

exp((ε− µ)/kBT

)+ 1

≈ e−(ε−µ)/kBT (21.8)

1

exp((µ− ε)/kBT

)+ 1

≈ e−(µ−ε)/kBT . (21.9)

We can then write

nc(T ) = Nc(T )e−(εc−µ)/kBT (21.10)

pv(T ) = Pv(T )e−(µ−εv)/kBT , (21.11)


21.1. HOMOGENEOUS SEMICONDUCTORS 177

where

Nc(T ) =

∫ ∞εc

dε Dc(ε)e−(ε−εc)/kBT and (21.12)

Pv(T ) =

∫ εv

−∞dε Dv(ε)e−(εv−ε)/kBT (21.13)

are slowly varying functions of T . In the effective mass approximation these can be evaluateddirectly. Using

Dc,v(ε) =√

2|ε− εc,v|m

3/2c,v

~3π2, (21.14)

where m3c = det(Mc) and m3

v = det(Mv), we have

Nc(T ) =1

4

(2mckBT

π~2

)3/2

(21.15)

Pv(T ) =1

4

(2mvkBT

π~2

)3/2

. (21.16)

Although the actual carrier densities are determined by µ, the product ncpv is independentof µ:

ncpv = NcPve−(εc−εv)/kBT (21.17)

= NcPve−Eg/kBT . (21.18)

In the language of chemistry, this defines a law of mass action, in which the carriers areformed by dissociation of electrons and holes. One consequence of Eq. 21.18 is that if weknow the density of one carrier type, we automatically know the other. In addition, as soonas one carrier type begins to dominate, it will completely overwhelm the other.

21.1.3 The intrinsic case

In the intrinsic case the impurity concentration is neglible. Therefore

nc(T ) = pv(T ) = ni(T ), (21.19)

whereni(T ) =

√NcPve−Eg/2kBT . (21.20)

For this to hold, we must have

µ = µi = εv +1

2Eg +

1

2kBT ln

(Pv

Nc

)(21.21)

= εv +1

2Eg +

3

4kBT ln

(mv

mc

). (21.22)

We can see that as T → 0, µ goes to the middle of the gap. Since ln(mv/mc) ∼ O(1), atfinite temperatures µ is typically located within several kBT of the middle of the gap, makingthe degeneracy condition easy to satisfy.



21.1.4 The extrinsic case

In the presence of charged defects,

nc − pv ≡ ∆n 6= 0. (21.23)

The law of mass action, Eq. 21.18, still holds, so

ncpv = n2i (21.24)

= nc(nc −∆n). (21.25)

Therefore

nc =∆n

2+

1

2

√∆n2 + 4n2

i (21.26)

pv = −∆n

2+

1

2

√∆n2 + 4n2

i (21.27)

(21.28)

This can be written in another way:

nc = eβ(µ−µi)ni (21.29)

pv = e−β(µ−µi)ni (21.30)

so that∆n

ni

= 2 sinh β(µ− µi). (21.31)

When ∆n is large compared to ni, the density of one type of carrier is ≈ ∆n, and the densityof the other carrier type is smaller by a factor (ni/∆n)2. One type of carrier dominates, andwe have either n-type or p-type material.

21.1.5 Impurities

Two types of impurity are essential to semiconductor electronics:

• Donors — these are impurities that are easily ionized and supply electrons to theconduction band. In group IV semiconductors such as silicon and germanium, anyelement from group V of the periodic table, such as phosphorous or arsenic, acts asdonor.

• Acceptors — these are impurities that readily capture an electron, effectively supplyingholes to the valence band. For silicon and germanium, group III elements such as boronand gallium act as acceptors.

in a crystal, the binding energy of Bloch electrons at impurities is enormously reduced overthe binding energy of electrons in the free atom. This reduction occurs for two reasons:


21.1. HOMOGENEOUS SEMICONDUCTORS 179

• The Coulomb attraction is reduced by the static dielectric constant εr, which is typically10–20 in a semiconductor. (Large dielectric constants occur as a result of the smallenergy gaps.)

• An electron moving in a semiconductor medium has semiclassical dynamics, describedby the appropriate effective mass tensor. This is because the impurity level is built outof Bloch states near the conduction band minimum or valence band maximum. To afirst approximation, m → m∗, where m∗ is an appropriate average of the componentsof the effective mass tensor.

For a hydrogen atom in free space, the binding energy is

ε = me4/32π2ε20~2. (21.32)

In a crystal, the Bohr radius is increased:

r0 ≈m

m∗εra0 ∼ 100 A (21.33)

and the binding energy is reduced:

εb =m∗

m

1

ε2r× 13.6 eV, (21.34)

lowered by a factor of 1000 or more. The energy of a donor impurity is measured relative tothe energy of the conduction band levels from which it is formed:

εd = εc − εb. (21.35)

A similar argument holds for acceptors, and the binding energies of both types of impurityare small compared to the energy gap.

21.1.6 Population of impurity levels in thermal equilibrium

To calculate the effect of impurities on the density of carriers, we ignore the interaction ofelectrons bound at different impurity sites.1 As discussed in Lecture 2, we usually assumethe each impurity has a single one-electron orbital that can hold electrons of either spin,but not two electrons, due to their mutual Coulomb repulsion. In this case, the impuritystatistics have a modified Fermi–Dirac form:

nd =Nd

1 + 12

exp β(εd − µ), (21.36)

where Nd is the density of donor atoms. The same is true for acceptors, which have anaverage occupation number

na =Na

1 + 12

exp β(µ− εa), (21.37)

where Na is the density of acceptor impurities.

1These interactions can be important at low temperatures where, for example, they can lead to theimpurity levels forming an impurity band through which electrical conduction occurs.



21.1.7 Thermal equilibrium carrier density

Consider the electronic configuration at T = 0. If Nd ≥ Na, then Na of the donated electronsdrop down into acceptor levels, leaving Nd −Na electrons behind in the donor levels.

nc = 0 (21.38)

pv = 0 (21.39)

nd = Nd −Na (21.40)

pa = 0 (21.41)

At finite T , the electrons are redistributed among the levels, but the total number remainsthe same.

nc + nd − pv − pa = Nd −Na = constant (21.42)

We can use the expressions for nc, nd, pv and pa to find µ in terms of Nd and Na. Thegeneral case is complicated, but the following simple case is quite important:

εd − µ kBT (21.43)

µ− εa kBT . (21.44)

In this case, the impurities are fully ionized, and ∆n = nc−pv = Nd−Na. This approximationwill hold from the predominantly intrinsic regime to well into the extrinsic regime. Usingthe expressions we found earlier, we get the following results in the two cases of n-type andp-type doping.

• n-type: Nd > Na

nc ≈ Nd −Na (21.45)

pv ≈n2

i

Nd −Na

(21.46)

• p-type: Na > Nd

nc ≈n2

i

Na −Nd

(21.47)

pv ≈ Na −Nd (21.48)

If the temperature is too low, or the impurity concentration too high, inequalities 21.43 and21.44 fail, and one of the impurity types is no longer completely ionized.

21.1.8 Transport in nondegenerate semiconductors

The velocity distribution for carriers in the conduction or valence band of a semiconductoris:

f(v) = ndet(M)1/2

(2πkBT )3/2exp

−β

2

∑µν

vµMµνvν

. (21.49)


21.2. INHOMOGENEOUS SEMICONDUCTORS 181

Compare this to the velocity distribution in a classical gas:

f(v) = n

(m

2πkBT

)3/2

exp(−mv2/2kBT

). (21.50)

The two are very similar except that:

• in a classical gas the density of molecules n is fixed and independent of temperature;

• in a classical gas the effective mass is isotropic.

Other than that, transport in a nondegenerate semiconductor is the same as for a chargedclassical gas with several components.

21.2 Inhomogeneous semiconductors

The sensitivity of semiconductor properties to defects permits the electronic structure to betuned on a small spatial scale, allowing the fabrication of devices such as quantum wells and,using interfaces between n-type and p-type material, nonlinear devices such as diodes andtransistors. Here we are usually talking about single crystals in which the dopant densitiesare spatially varying.

21.2.1 The p-n junction

We consider an abrupt junction, defined to be narrow compared to the depletion layer, inwhich the carrier densities are nonuniform.

The doping profile is given by Nd(x) and Na(x). In the model p-n junction we considerbelow, these take a simple form.

Nd(x) =

Nd, x > 00, x < 0

(21.51)

Na(x) =

0, x > 0Na, x < 0

(21.52)

We assume we can treat the junction using the semiclassical model, where the effect of anelectric potential is to shift the one-electron levels. This is just what we assumed in theThomas–Fermi theory of screening. Here the approximation is much better justified becausethe length scales, typically 100 A to 1000 A, are much larger than atomic dimensions.

In the general the change in eφ in crossing the junction is ∼ Eg but, as we will see below, thisoccurs over at least 100 lattice constants. This will satisfy the electric breakdown condition

eEa(εgap(k)

)2

EF

. (21.53)

derived in Lecture 17. It is convenient to do the same thing we did for the Thomas–Fermi



d ( x )

v ( x ) a ( x )

µ c ( x )

d

a

c µ e ( x )

v

Figure 21.2: Two views of the spatially varying electronic structure in a p-n junction.

theory and define a position-dependent electrochemical potential,

µe(x) = µ+ eφ(x). (21.54)

Using the results from earlier, the equilibrium carrier densities are

nc(x) = Nc(T ) exp

[−εc − µe(x)

kBT

](21.55)

pv(x) = Pv(T ) exp

[−µe(x)− εv

kBT

]. (21.56)

As with the Thomas–Fermi theory, the potential φ(x) must be determined self-consistentlyby solving Poisson’s equation, which we will now proceed to do.

When we are a long distance from the junction, uniform extrinsic conditions apply — allimpurities are therefore fully ionized. The change in potential on crossing the junction is

e∆φ = µe(∞)− µe(−∞). (21.57)

The two equivalent ways of representing the effect of the potential are shown in Fig. 21.2.By looking at the position of µ or µe(x) with respect to the band edges, we can see that theimpurities will remain fully ionized at all x.

The spatially varying charge density is given by

ρ(x) = e[Nd(x)−Na(x)− nc(x) + pv(x)

]. (21.58)

In this case, the Poisson equation is a nonlinear differential equation that must be solved


21.2. INHOMOGENEOUS SEMICONDUCTORS 183

+ + + + +

+ + + + +

+ + +

- - - - - - - - -

- - - - - - - - -

carrier density

N a

x

−⊥d p d n

x

potential

−⊥d p d n

N d n c ( x )

depletion layer

+ + + + +

+ + + + +

+ + +

- - - - - - - - -

- - - - - - - - - x

charge density

−⊥eN a

−⊥d p d n

eN d

p – type n – type

p v ( x )

Figure 21.3: The spatially varying carrier density, charge density and potential in a p-njunction.



numerically. We can find an approximate solution by realizing that except near the edges ofthe depletion layer (x = dn and x = −dp) the conduction- and valence-band carrier densitiesare negligible compared to Nd and Na. The charge density is therefore

ρ(x) ≈ e[Nd(x)−Na(x)

]. (21.59)

in the depletion region and zero elsewhere. This sets the curvature of the potential throughthe depletion region:

d2φ

dx2=

0 x > dn

−eNd/ε0εr 0 < x < dn

eNa/ε0εr −dp < x < 00 x < −dp

. (21.60)

This has solution

φ(x) =

φ(∞) x > dn

φ(∞)− (eNd/2ε0εr)(x− dn)2 0 < x < dn

φ(−∞) + (eNa/2ε0εr)(x− dp)2 −dp < x < 0φ(−∞) x < −dp

. (21.61)

A unique solution is obtained by requiring continuity of φ(x) and φ′(x). This leads toexpressions for dp and dn, giving values in the range 102 A to 104 A for the thickness of thedepletion layer.

21.3 The p-n junction as a rectifier

We now consider the effect of applying an external bias to the p-n junction. Because theconduction- and valence-band charge density is very low in the depletion layer, the resistanceof the p-n junction is high, and all the voltage drop will occur across the p-n junction. Wedefine a voltage V so that positive voltage corresponds to a raising of the potential of thep-type region with respect to the n-type, thereby reducing the total change in potential.

∆φ = (∆φ)0 − V (21.62)

This will be accompanied by changes in the thickness of the depletion layer, but these haveno important effect on the magnitude of the current that flows. The reason we get diodeaction from a p-n junction is that, for each type of carrier, there are two types of currentthat flow when the junction is biased, one of which depends nonlinearly on voltage.

The two types of current (for holes) are:

• Generation current. A very small density of holes, as the minority carrier, is generatedat the edge of the n-type depletion layer. If these can diffuse into the depletion region,they are immediately swept over to the p-type region by the strong electric fields inthe depletion layer. This current is Jgen

h .

• Recombination current. Holes are continuously being formed at the boundary of thep-type region, but the proportion of these that have enough kinetic energy to surmountthe potential barrier ∆φ is exponentially small, J rec

h ∝ exp(− e[(∆φ)0 − V ]/kBT

).


21.3. THE P-N JUNCTION AS A RECTIFIER 185

I

V

Figure 21.4: The current–voltage characteristic of a p-n junction, showing diode action.

At zero bias, the generation and recombination currents must balance, implying

J rech = Jgen

h eeV/kBT (21.63)

The net current in the forward direction is

Jh = J rech − J

genh = Jgen

h

(eeV/kBT − 1

). (21.64)

A similar contribution, of the same sign, comes from the electrons. The current–voltagecharacteristic of a diode is plotted in Fig. 21.4.


Lecture 22

The Boltzmann transport equation

Our aim in this lecture is to develop a framework for calculating the transport of electricityand heat by Bloch electrons, starting with the main results from the semiclassical theoryof electron dynamics developed in Lectures 16 and 17. As before, we specify the position rand wavevector k of the Bloch electron simultaneously. This can be done without violatingthe uncertainty principle by using wavepackets. That is, our definition of position andmomentum is sufficiently fuzzy that ∆r∆k > 1. According to the semiclassical equations ofmotion, the position and wavevector evolve in time as:

r = vn(k) =1

~∇kεn(k) (22.1)

~k = −e[E(r, t) + vn(k)×B(r, t)

]. (22.2)

For a system of electrons in a real sample, under the application of external fields, at atemperature T that in general varies in space and time, and in the presence of static anddynamic disorder, we have to give up any hope of calculating the states of definite occupationnumber. Instead, we describe the electrons in terms of a distribution function f(r,k, t) thatmeasures the average occupancy of the Bloch state k as a function of space and time. f(r,k, t)is a function of seven variables, describing the time evolution of electrons in a six-dimensionalphase space. This sounds complicated, but is actually fairly simple if the problem is tackledsystematically. The approach we in this lecture is intended to be somewhat complementaryto that given in Taylor and Heinonen.

22.1 The distribution function

To start with, let’s imagine f(r,k, t) in the simplest situation: at zero-temperature, in theabsence of applied fields. Although six dimensions are impossible to visualize, the structureof f(r,k, t) in the r–k phase space is clear enough. f(r,k, t) = 1 within the Fermi sea, anddrops abruptly to zero outside it. In real space f(r,k, t) is constant. The integral∫

drdk

4π3f(r,k, t) = N (22.3)

where N is the number of electrons in the sample.


188 LECTURE 22. THE BOLTZMANN TRANSPORT EQUATION

If we now heat the sample uniformly to a temperature T , the effect on f(r,k, t) is to broadenthe sharp step in k-space at the Fermi surface. It is also possible to impose a nonuniformtemperature distribution on the system. The nonuniformity will almost always be in realspace1 and very often be composed of a uniform temperature gradient.

Figure 22.1: Schematic illustration of the time evolution of electron states in the six-dimensional semiclassical phase space. (After Ashcroft and Mermin[4].)

The effect of external fields is to cause electrons to move along well-defined trajectories inthe r–k phase space, as sketched in Fig. 22.1. These trajectories obey the Liouville theorem— they sweep out phase space volume at a constant rate. As a result, electrons do not pileup at particular parts in phase space under the action of the external fields, which wouldviolate the exclusion principle.

Finally, scattering from impurities, defects, phonons and other excitiations causes electronsto make random transitions from one point in k-space to another. (The scattering eventsare always local in real space.) The scattering probability from state k′ to k is representedby the function Q(k,k′). The net rate of scattering into state k is

fk =∑k′

[fk′Q(k,k′)(1− fk)− fkQ(k′,k)(1− fk′)

], (22.4)

showing how allowance is made for the Pauli exclusion principle. We will not go into anymore detail about the function Q(k,k′) — much of the complexity of transport theory iscontained within it.

1It is possible to create a temperature nonuniformity in k-space, by pumping energy into the system atparticular wavevectors. Situations like this are rare, but interesting when they occur. For an example seeN. Gedik et al. [21].


22.2. THE CONTINUITY EQUATION 189

22.2 The continuity equation

Before deriving the Boltzmann transport equation for electrons, it is useful to review how aconservation law results in a continuity equation. For any conserved quantity, such as fluidvolume or electric charge, the net flux through a surface must be equal to rate at which theamount of material within the surfaces decreases. Thus conservation of charge, for instance,results in continuity of electrical current:∫

S

j · dA = − ∂

∂t

∫V

ρdV. (22.5)

Using Gauss’ theorem this can be expressed in differential form as

∂ρ

∂t+∇ · j = 0. (22.6)

The current is related to charge density by

j = ρu (22.7)

where u is the velocity. Therefore

∂ρ

∂t+∇ · (ρu) = 0 (22.8)

⇒ ∂ρ

∂t+ u · ∇ρ+ ρ∇ · u = 0. (22.9)

22.3 The Boltzmann equation

We begin by deriving the equation of continuity of the electron distribution function f(r,k, t)in the absence of collisions, which is equivalent to the statement that the rate at which thedensity of electrons increases in a given volume of phase space is equal to the net rate atwhich electrons enter the volume. That is, in the absence of scattering, electrons cannotspontaneously appear in or disappear from a volume of the phase space. Generalizing thecontinuity equation derived above to six dimensions, we have

∂f

∂t+∇r · (f r) +∇k · (f k) = 0. (22.10)

Carrying out the differentiation gives

∂f

∂t= −v · ∇rf − k · ∇kf − f

[∇r · v +∇k · k

]. (22.11)

For Bloch electrons, the term in brackets vanishes, which can be shown using Eqs. 1 and 2.(The net flow into any volume of phase space is equal to the net flow out, by the Liouvilletheorem.)



Finally, we include a collision term of the form of Eq. 22.4 on the right hand side of theequation.

∂f

∂t= −v · ∇rf −

1

~F · ∇kf +

(∂f

∂t

)collisions

, (22.12)

where F = −e(E + v × B) is the Lorentz force. This is the Boltzmann transport equationfor electrons in a periodic solid.

If we note that the spatial variation of f is due solely to variation in the temperature, wecan write

∂f

∂r=∂f

∂T∇T, (22.13)

leading, in the steady state (∂f/∂T = 0), to(∂f

∂t

)collisions

=1

~∇kε · ∇T

(∂f

∂T

)− e

~(E + v ×B

)· ∇kf. (22.14)

22.4 The linearized Boltzmann equation

In most cases we are interested in calculating the electrical and thermal currents that flow inresponse to small electric fields and temperature gradients. It then suffices to make a seriesexpansion of f(r,k, t) in powers of the applied fields:

f = f 0 + f 1. (22.15)

To zeroth order, f = f 0, the equilibrium Fermi–Dirac distribution. f 1 represents the localdeparture from equilibrium. If the applied fields are small, f 1 will be everywhere much smallerthan 1.

The main role of collisions is to return the system to equilibrium, so(∂f 0

∂t

)collisions

= 0. (22.16)

Also, since f 0(k) = f 0(ε(k)

)∇kf

0 =∂f 0

∂ε∇kε. (22.17)

Putting all this into the Boltzmann equation and keeping only terms linear in the temperaturegradient and electric field we get(

∂f 1

∂t

)collisions

= − e~

vk ×B · ∇kf1 + vk ·

(∇T ∂f

0

∂T− eE∂f

0

∂ε

)(22.18)

= − e~

vk ×B · ∇kf1 − vk ·

((ε− µ)∇T/T + eE

)∂f 0

∂ε. (22.19)

This is the linearized Boltzmann equation for f 1, the departure from equilibrium.


22.5. THE RELAXATION TIME APPROXIMATION 191

22.5 The relaxation time approximation

The main difficulty in solving the linearized Boltzmann equation is in accurately modelingthe collision term. This is discussed in some detail in Taylor and Heinonen. We will insteadfocus on a simple but important limit, the relaxation time approximation, in which thedistribution function relaxes to equilibrium with a characteristic relaxation time τ . In thiscase (

∂f 1

∂t

)collisions

= −f1

τ. (22.20)

Although simple, the approximation is quite often valid, especially for impurity scattering.One simple extension is to let the relaxation time depend on wavevector. In the following,however, we will assume that τ is a constant.

Figure 22.2: Solutions of the linearized Boltzmann equation for a free electron metal plottedfor an electrical current (left) and a thermal current (right). Note that the effect of anelectrical current is to displace the Fermi sea, while the electrons are hotter on one side ofthe Fermi sea than the other for a thermal current. (After J. R. Waldram[22].)

In the relaxation time approximation, in the absence of magnetic field, the linearized Boltz-mann equation is

− f 1

τ= −vk ·

((ε− µ)∇T/T + eE

)∂f 0

∂ε, (22.21)

with solution

f 1k = vk ·

((ε− µ)∇T/T + eE

)∂f 0

∂ετ. (22.22)



(∂f 0/∂ε) peaks sharply near the Fermi level,

∂f 0

∂ε= −f

0(1− f 0)

kBT(22.23)

→ −δ(ε− εF) as T → 0. (22.24)

so the deviation of the distribution function f from the equilibrium Fermi–Dirac distributionis localized near the Fermi surface. f 1 is plotted in Fig. 22.2 for situations correspondingto an electrical current and a thermal current. Note that the main difference is due to thefactor (ε − µ), which makes f 1 antisymmetric about the Fermi level for thermal currents.This acts to either broaden or sharpen the step in the distribution function at the Fermisurface. Note that f 1 is always antisymmetric about the origin of k-space and, for a uniformelectrical or thermal current density, is independent of position.

22.6 Transport properties of metals

We can now derive expressions for the transport coefficients of metals. These relate electricaland thermal currents to electric fields and temperature gradients. The electrical currentdensity is

j =1

V

∑k

(−e)vkfk (22.25)

=1

V

∑k

(−e)vkf1k, (22.26)

because j = 0 in equilibrium.

Similarly, the heat current density can be written

jε =1

V

∑k

(ε− µ)vkf1k. (22.27)

(It is necessary to subtract a way a term proportional to the chemical potential, becausethat contribution to the energy flux is purely electrical, not thermal.)

We now evaluate these in the relaxation time approximation.

22.6.1 Electrical conductivity

To obtain the electrical conductivity we set the thermal gradient to zero and calculate theelectrical current in the presence of an electric field.

j = −e2τ

V

∑k

vk(vk · E)∂f 0

∂ε(22.28)

j = σ · E, so

σ = −e2τ

V

∑k

vk ⊗ vk∂f 0

∂ε. (22.29)


22.7. THE WIEDEMANN–FRANZ LAW 193

In the simple case of metal with a spherical Fermi surface, the conductivity is a diagonaltensor and

σxx =1

3Tr(σ) (22.30)

= −e2τ

3V

∑k

|vk|2∂f 0

∂ε(22.31)

→ e2v2Fτ

3V

∑k

δ(εk − εF) as T → 0 (22.32)

=2e2εFτ

3mVD(εF), (22.33)

where D(εF) is the density of states at the Fermi level. We previously showed that

D(εF) =3N

2εF, (22.34)

so

σxx =Ne2τ

mV=ne2τ

m, (22.35)

where n = N/V is the density of electrons.

22.6.2 Thermal conductivity

The heat current density is

jε = − τ

V T

∑k

vk(vk · ∇T )(ε− µ)2∂f0

∂ε. (22.36)

The thermal conductivity κ is defined

jε = −κ · ∇T. (22.37)

The (ε−µ)2(∂f 0/∂ε) term has a double peaked structure about εF. However, the summationis straightforward to evaluate by converting it to an integral and using the Sommerfeldexpansion that we saw in Lecture 3. The result is

κ =π2

3

nk2Bτ

mT (22.38)

for the isotropic metal.

22.7 The Wiedemann–Franz law

For isotropic scattering (either by impurities or phonons) the ratio of the electrical andthermal conductivities takes on a universal value, independent of the band structure of themetal. The Lorenz ratio is

κ

σT=π2k2

B

3e2. (22.39)



The electrical and thermal conductivities and Lorenz ratio of copper are plotted in Fig. 22.3.The Lorentz ratio becomes small at intermediate temperatures because there small-anglescattering by phonons dominates.

Figure 22.3: Electrical and thermal conductivities and Lorenz ratio κ/σT for copper, mea-sured by Berman and MacDonald[23].


Lecture 23

Time-dependent perturbation theory

References: The introduction to time-dependent perturbation theory is based on the excellent treatment

given by Shankar[6].

23.1 Time-dependent perturbation theory

In the time independent case, solving the Schrodinger equation

i~∂

∂t|ψ〉 = H0|ψ〉 (23.1)

reduces to solving the eigenvalue problem of H0 to find the stationary states. We assume wecan do this, and label the eigenstates and eigenenergies of the time-independent hamiltonianby |n〉 and En.

We now introduce a time-dependent potential U(t) to the hamiltonian as a perturbation.

H0 → H(t) = H0 + U(t). (23.2)

As with all perturbative methods, we require the perturbation U(t) to be small, but delayour definition of ‘small’ until later.

In what follows we will only be interested in first order perturbation theory.

The states |n〉 form a complete set, so the wavefunction at time t can be expanded in termsof them.

|ψ(t)〉 =∑n

cn(t)|n〉 (23.3)

The process of calculating the cn(t) given cn(0) is equivalent to finding |ψ(t)〉 given |ψ(0)〉.cn(t) changes with time due to the action of both the time-independent hamiltonian H0 andthe time-dependent perturbation U(t). Had U(t) been absent, we would have

cn(t) = cn(0)e−iEnt/~. (23.4)

Let’s take this into account explicitly by carrying out a slightly different expansion.

|ψ(t)〉 =∑n

dn(t)e−iEnt/~|n〉 (23.5)


196 LECTURE 23. TIME-DEPENDENT PERTURBATION THEORY

Now, if dn changes with time it is due solely to the action of U(t).

The next step is to operate on both sides of Eq. 23.5 with(i~ ∂/∂t−H0 − U(t)

). Because

|ψ(t)〉 is an eigenstate of H(t), we then have

0 =∑n

(i~dn(t) + Endn(t)− Endn(t)− U(t)dn(t)

)e−iEnt/~|n〉 (23.6)

=∑n

(i~dn(t)− U(t)dn(t)

)e−iEnt/~|n〉. (23.7)

We now project out one term from the last equation by multiplying by 〈f| exp(iEft/~), whichgives

i~df(t) =∑n

〈f|U(t)|n〉eiωfntdn(t), (23.8)

where ~ωfn = Ef − En.

We imagine the system starts at time t = 0 in one of its eigenstates, which we call the initialstate and label it |i〉. Therefore

dn(t = 0) = δni. (23.9)

We want to know how the system evolves at later times under the action of the time-dependent perturbation. That is, we want to know df(t).

To zeroth order we ignore the perturbation and find that df = 0, ⇒ dn(t) = δni.

To first order in the perturbation, we use the zeroth-order dn, resulting in the equation

df = − i

~〈f|U(t)|i〉eiωfit. (23.10)

This can be integrated to obtain the solution

df(t) = δfi −i

~

∫ t

−∞〈f|U(t′)|i〉eiωfit

′dt′. (23.11)

The first order calculation will be reliable if |df(t)| 1 for all f 6= i.

Substituting back into the original expansion we obtain

|ψ(t)〉 = N∑

f

df(t)e−iEft/~|f〉 (23.12)

= N∑

f

(δfi −

i

~

∫ t


′dt′)

e−iEft/~|f〉 (23.13)

= e−iEit/~|i〉 − i

~∑f6=i

(∫ t


′dt′)

e−iEft/~|f〉. (23.14)

In the last line, the normalization factor N has been taken into account, to first order, byexcluding |i〉 from the sum over final states |f〉.


23.2. SUDDEN PERTURBATIONS 197

23.2 Sudden perturbations

In many cases our perturbation amounts to a sudden change in potential at time t = 0. Wecan solve this by integrating the Schrodinger equation between t = −ε/2 and t = ε/2.

|ψ( ε2)〉 − |ψ(− ε

2)〉 = |ψafter〉 − |ψbefore〉 (23.15)

= − i

~

∫ ε2

− ε2

H(t)|ψ(t))〉dt. (23.16)

If H(t) is finite, then there is no change in |ψ(t)〉 in the limit ε→ 0.

|ψafter〉 = |ψbefore〉 (23.17)

However, the wavefunction is no longer an eigenstate of the hamiltonian, as the hamiltonianitself has changed. Sudden perturbations are very important in the theory of angle-resolvedphotoemission spectroscopy, which we will discuss in Lecture 27.

23.3 Adiabatic perturbations

The adiabatic theorem states that if the rate of change of the hamiltonian H(t) is slowenough, the system, starting at t = 0 in an eigenstate |n(0)〉 of H(0), will end up in thecorresponding eigenstate |n(τ)〉 of H(τ) at a later time τ .

This result is true to all orders in perturbation theory, but places stringent requirementson how slowly the perturbation must be switched on. The condition is that τ T , whereT = ~/Emin and Emin is the minimum energy separation between the initial state and allother eigenstates. For any system that is near degenerate, the required rate is very slow. Fordegenerate systems τ → ∞, and it becomes impossible to make an adiabatic perturbation.This result is very important in Landau’s Fermi liquid theory, where it allows a connectionto be made between the excited states of an interacting electron system and those of a modelnoninteracting system.

23.4 Periodic perturbations

Let the time dependent potential have the form

U(t) = U0e−iωt, (23.18)

where ω has a small imaginary part that causes the perturbation to be switched on slowlyfrom some time in the distance past:

ω → ω + iη. (23.19)

Also, let’s define the frequencies ωn = En/~.



Then, substituting into Eq. 23.14, our main perturbation theory result, we obtain

|ψ(t)〉 = e−iωit|i〉 − i

~∑f6=i

(∫ t

−∞〈f|U0|i〉e−iωt′ei(ωf−ωi)t

′dt′)

e−iωft|f〉. (23.20)

The integral over t′ can be evaluated:∫ t

−∞ei(ωf−ωi−ω−iη)t′dt′ =

i

ωi − ωf + ω + iηei(ωf−ωi−ω−iη)t. (23.21)

The expression for the wavefunction is then

|ψ(t)〉 = e−iωit|i〉+1

~∑f6=i

〈f|U0|i〉ωi − ωf + ω + iη

e−i(ωi+ω+iη)t|f〉. (23.22)

The adjoint equation, which we will need later, is

〈ψ(t)| = eiωit〈i|+ 1

~∑f6=i

〈i|U0|f〉ω∗i − ω∗f + ω − iη

ei(ω∗i +ω−iη)t〈f|. (23.23)

If the perturbation is U(t) = U0 exp(iωt), then we require ω to have a small negative imagi-nary part. The corresponding expression for the wavefunction is

|ψ(t)〉 = e−iωit|i〉+1

~∑f6=i

〈f|U0|i〉ωi − ωf − ω + iη

e−i(ωi−ω+iη)t|f〉. (23.24)

This last point is important. Any linearly polarized field applied to the system can bedecomposed into equal amplitudes of left- and right-circularly polarized radiation, so theperturbation must consist of negative and positive frequencies.

23.5 Fermi’s golden rule

As well as following the time evolution of the wavefunction, we can also calculate the transi-tion rate between the initial state |i〉 and a particular final state |f〉 under the influence of aperiodic perturbation U(t) = U0e−iωt. Let’s assume we start in state |i〉 at time t = −T/2 andintegrate Eq. 23.10 until a later time t = T/2. We wish to calculate the average transitionrate in the limit T →∞. Then

df(t) = limT→∞

− i

~

∫ T/2

−T/2〈f|U0|i〉ei(ωfi−ω)t′dt′ (23.25)

= −2πi

~〈f|U0|i〉δ(ωfi − ω). (23.26)

The transition probability is

Pi→f = |df|2 (23.27)

=4π2

~2|〈f|U0|i〉|2δ(ωfi − ω)δ(ωfi − ω). (23.28)


23.6. THE KUBO FORMULA 199

The double delta function requires special care:

δ(ωfi − ω)δ(ωfi − ω) = limT→∞

δ(ωfi − ω)1

2π

∫ T/2

−T/2ei(ωfi−ω)tdt. (23.29)

Set ω = ωfi in the integral:

δ(ωfi − ω)δ(ωfi − ω) = δ(ωfi − ω) limT→∞

T

2π. (23.30)

The average transition rate is

Ri→f =Pi→f

T(23.31)

=2π

~|〈f|U0|i〉|2δ(Ef − Ei − ~ω). (23.32)

This is Fermi’s golden rule. The delta function explicitly enforces conservation of energy. Inthe context of solid state physics, we are usually trying to calculate the net transition rateout of a particular Bloch state. We then integrate over all final states of the same energy andobtain the result that the transition rate or ‘scattering rate’ is proportional to the densityof final states.

23.6 The Kubo formula

We now use time dependent perturbation theory to calculate the electrical conductivity. Theelectromagnetic field is introduced through the vector potential A.

E = −∂A

∂t(23.33)

if E = E0e−iωt + complex conjugate (23.34)

⇒ A =E0

iωe−iωt + complex conjugate (23.35)

How does the vector potential couple to the electron? In the presence of a field the canonicalmomentum is p = mv − eA, which implies that kinematic momentum is mv = p + eA.That is, the kinematic momentum acquires a kick from the field when the field is switchedon. It is the kinematic momentum that goes into the Schrodinger equation, so p is replacedby p + eA.

p2

2m→ (p + eA)2

2m=

p2

2m+

e

2m

[A · p + p ·A

]+ ... (23.36)

If we specialize to the transverse gauge ∇ · A = 0, we can show that p · A = A · p, andtherefore

(p + eA)2

2m≈ p2

2m+

e

mA · p. (23.37)

To linear order, the perturbation to the hamiltonian is

U(t) =e

miωE0 · p e−iωt − e

miω∗E0 · p eiω∗t. (23.38)



We can now use the first order perturbation theory results to calculate the current.

〈J〉 = V j = − e

m〈ψ(t)|mv|ψ(t)〉 (23.39)

= − e

m〈ψ(t)|p + eA|ψ(t)〉, (23.40)

where V is the volume of the sample.

The current carried by the perturbed state, evaluated to first order in E0, is

J = − e

m〈i|p|i〉 − 〈i|e

2E0

imωe−iωt − e2E0

imω∗eiω∗t|i〉 (23.41)

− e2

i~m2

∑f6=i

〈i|p|f〉〈f|E0 · p|i〉

e−iωt

ω(ωi − ωf + ω + iη)− eiω∗t

ω∗(ωi − ωf − ω + iη)

(23.42)

− e2

i~m2

∑f6=i

〈i|E0 · p|f〉〈f|p|i〉

e−iωt

ω(ω∗i − ω∗f − ω − iη)− eiω∗t

ω∗(ω∗i − ω∗f + ω − iη)

.(23.43)

We assume that the current in the ground state, J0 = em〈i|p|i〉, vanishes.

The electrical conductivity σ(ω) is the coefficient of exp(−iωt) that relates j to the appliedelectric field and can be isolated from this expression.

jα(t) = jαe−iωt + complex conjugate (23.44)

=∑β

σαβ E0,β e−iωt + complex conjugate (23.45)

We assume that the initial state is occupied with probability fi, and sum over all initialstates to get the Kubo formula for the conductivity:

σαβ = − e2

imωV

∑i

fi

[δαβ +

∑f6=i

1

~m

〈i|pα|f〉〈f|pβ|i〉ωi − ωf + ω + iη

+〈i|pβ|f〉〈f|pα|i〉ω∗i − ω∗f − ω − iη

]. (23.46)

If all the ωn are real (the imaginary part of ωn represents the lifetime of the state |n〉) thenwe can exchange f and i in the final term.

σαβ = − e2

imωV

∑i

[fiδαβ +

∑f6=i

fi − ff

~m〈i|pα|f〉〈f|pβ|i〉ωi − ωf + ω + iη

]. (23.47)

Absorption of energy is given by the real part of the conductivity, which we can obtain usingthe identity

limη→0

1

x+ iη= P

(1

x

)− iπδ(x), (23.48)

where P denotes the principle part of the integral.

Re[σαβ] =πe2

ωm2V

∑i,f

(fi − ff)〈i|pα|f〉〈f|pβ|i〉δ(~ωf − ~ωi − ~ω). (23.49)


23.6. THE KUBO FORMULA 201

For small values of ω (and with a finite lifetime for the excited states) this reduces to theBoltzmann result for a single band. For high frequencies, where the semiclassical approachfails due to interband transitions, we get transitions between levels whenever the resonancecondition ωf − ωi = ω is satisfied.

Eq. 23.49 also applies if the system is prepared out of equilibrium, as in a laser.


Lecture 24

Density response of the electron gas

In Lecture 8 we used the Thomas–Fermi theory of screening to calculate the response ofthe electron gas to an external potential. The nature of the approximation meant that thetheory would only be accurate for long wavelength, low frequency perturbations. However,we went ahead anyway and used the theory to calculate how the potential of a point chargeis screened. We will now develop a more accurate method for calculating the response of theelectron gas to time-dependent potentials of arbitrary wavelength.

We define the density response function χ(q, ω) by

δρ(q, ω) = χ(q, ω)V (q, ω), (24.1)

where δρ(q, ω) is the charge density induced by the total potential V (q, ω).

We will carry out the calculation for a noninteracting, free Fermi gas. The calculation willremain valid for Fermi liquids, which we will meet in Lecture 28.

24.1 Time and space dependent perturbations

We again use time dependent perturbation theory to calculate solutions of the Schrodingerequation in the presence of a time and space dependent potential U(r, t).

i~∂ψ(r, t)

∂t=[H0 + U(r, t)

]ψ(r, t) (24.2)

As before, we assume that we can solve H0, which has eigenfunctions φn(r) and eigenvaluesεn. (We will soon specialize to the case of free-electron plane waves.) We look for a generalsolution of the form

ψ(r, t) =∑n

an(t)e−iεnt/~φn(r) (24.3)

to first order in the perturbation. Using the result from the previous lecture,

am(t) = δm,n −i

~

∫ t

−∞〈m|U(r, t′)|n〉eiωmnt′dt′. (24.4)


204 LECTURE 24. DENSITY RESPONSE OF THE ELECTRON GAS

To calculate the effect of fields that vary in time and space, we apply a perturbation of theform

U(r, t) = U0eiq·re−i(ω+iη)t, (24.5)

where the infinitesimal η causes the perturbation to be switched on slowly from some timein the distant past. In all our calculations we will set η to zero in the final result.

As basis states for our problem we choose the eigenstates of the time-independent part ofthe hamiltonian, which in this case will be the free-electron plane waves

|k〉 =1√V

eik·r. (24.6)

To begin with, we assume our initial state at time t = −∞ is a particular plane wave state|k〉 and calculate the wavefunction |ψ(r, t)〉 at a later time t. We will eventually extend thisto a sum over all occupied initial states.

|ψ(r, t)〉 =∑k′

ak′(t)e−iεk′ t/~|k′〉, (24.7)

where

ak′(t) = δk′,k −i

~

∫ t

−∞〈k′|U(r, t)|k〉ei(εk′−εk)t′/~dt′. (24.8)

The matrix element between plane wave states can be evaluated:

〈k′|U(r, t)|k〉 = U0e−i(ω+iη)t 1

V

∫e−ik′·reiq·reik·rdr (24.9)

= U0e−i(ω+iη)tδk′,k+q. (24.10)

Therefore, the amplitude to be in state |k′〉 at a later time t is

ak′(t) = δk′,k −i

~U0δk′,k+q

∫ t

−∞ei(εk′−εk−~(ω+iη))t′/~dt′ (24.11)

= δk′,k + U0δk′,k+qei(εk+q−εk−~ω)t/~

εk+q − εk − ~(ω + iη). (24.12)

On inserting this into the wavefunction ansatz we obtain

|ψk(r, t)〉 = |k〉e−iεkt/~ + |k + q〉 U0

εk+q − εk − ~(ω + iη)e−iωte−iεkt/~ (24.13)

= a(t)|k〉+ b(t)|k + q〉. (24.14)

This is a simple and physically appealing result. A time dependent perturbation of wavevec-tor q couples the state |k〉 to state |k + q〉. The transition between states is not sudden —when the perturbation is applied the initial state starts to evolve into a quantum superposi-tion of |k〉 and |k + q〉. (Remember, as we saw in the last lecture, the states are normalizedto first order in the perturbation.)


24.2. DENSITY RESPONSE 205

24.2 Density response

We are now in a position to calculate the density response of the electron gas. We need totake all the k-states that were initially occupied (these will be occupied with probability nk,where nk is the Fermi–Dirac function at temperature T ) and use Eq. 24.13 to follow theirtime evolution under the action of the applied field. We then evaluate the density at thelater time. The change in the density is

δρ(r, t) =∑k

|ψk(r, t)|2 − 1

V

, (24.15)

where the sum is over initially occupied k-states. Working to first order, we have

|ψk(r, t)|2 =1

V

|a(t)|2e−ik·reik·r+a(t)∗b(t)e−ik·rei(k+q)·r+b(t)∗a(t)e−i(k+q)·reik·r+...

(24.16)

The first term on the right hand side is the unperturbed density. The phase factor exp(−iεkt/~)is common to a(t) and b(t) and makes no contribution, so

δρ(r, t) =U0

V

∑k

eiq·re−iωt

εk+q − εk − ~(ω + iη)+ complex conjugate

. (24.17)

In an actual experiment, the potentials we apply are real functions of space and time, so weadd to the potential its complex conjugate

U∗(r, t) = U0e−iq·rei(ω−iη)t (24.18)

This is equivalent to q→ −q and ω + iη → −ω + iη, which gives

δρ∗(r, t) =U0

V

∑k

e−iq·reiωt

εk−q − εk + ~(ω − iη)+ complex conjugate

. (24.19)

We collect everything together and include a factor of 2 for spin.

δρtot = 2(δρ(r, t) + δρ∗(r, t)

)(24.20)

=2U0

V

∑k

eiq·re−iωt

[1

εk+q − εk − ~(ω + iη)+

1

εk−q − εk + ~(ω + iη)

](24.21)

+ e−iq·reiωt

[1

εk+q − εk − ~(ω − iη)+

1

εk−q − εk + ~(ω − iη)

]. (24.22)

Implicit in this expression is that the sum is over occupied initial states. We now includethe occupation numbers nk explicitly, and carry out the sum over all k-states.

δρtot =2U0

V

∑k

eiq·re−iωt

[nk

εk+q − εk − ~(ω + iη)+

nk

εk−q − εk + ~(ω + iη)

](24.23)

+ e−iq·reiωt

[nk

εk+q − εk − ~(ω − iη)+

nk

εk−q − εk + ~(ω − iη)

]. (24.24)



Because we are now summing over all k-states, we are free to redefine the summation indexk. We let k→ k + q in the right hand fractions.

δρtot =2U0

V

∑k

eiq·re−iωt

[nk

εk+q − εk − ~(ω + iη)+

nk+q

εk − εk+q + ~(ω + iη)

](24.25)

+ e−iq·reiωt

[nk

εk+q − εk − ~(ω − iη)+

nk+q

εk − εk+q + ~(ω − iη)

](24.26)

=2U0

V

∑k

nk − nk+q

εk+q − εk − ~(ω + iη)eiq·re−iωt + complex conjugate

(24.27)

We define the density response function χ(q, ω) to be the coefficient of U0 exp(iq · r− iωt).

χ(q, ω) =2

V

∑k

nk − nk+q

εk+q − εk − ~(ω + iη)(24.28)

24.3 Energy loss

Using the identity

limη→0

1

x+ iη= P

(1

x

)− iπδ(x) (24.29)

we obtain the imaginary part of χ(q, ω).

Imχ(q, ω)

=

2π

V

∑k

(nk − nk+q

)δ(εk+q − εk − ~ω). (24.30)

We will now show that this is associated with energy loss.

We use Fermi’s golden rule

Ri→f =2π

~|〈f|U0|i〉|2δ(Ef − Ei − ~ω) (24.31)

to calculate the power absorption. In our case the perturbing potential is

U(r, t) = U0eiq·re−iωt + U0e−iq·reiωt. (24.32)

The relevant matrix elements are

〈k′|e±iq·r|k〉 = δk′,k±q. (24.33)

The transition rate from state |k〉 to |k′〉 is

Wkk′ =2π

~U2

0

δk′,k+qδ(εk′ − εk − ~ω) + δk′,k−qδ(εk′ − εk + ~ω)

. (24.34)

The total transition rate out of state k is

Wk =∑k′

Wkk′ (24.35)

=2π

~U2

0

δ(εk+q − εk − ~ω) + δ(εk−q − εk + ~ω)

. (24.36)


24.4. SCREENING AND THE DIELECTRIC FUNCTION 207

In each transition the system absorbs or emits a quantum of energy ~ω. The mean energyabsorbed per unit time by the state |k〉 is

Qk =2π

~U2

0~ωδ(εk+q − εk − ~ω)− δ(εk−q − εk + ~ω)

. (24.37)

Note the difference in sign for emission and absorption. If we now sum this over all occupiedstates, we have the total rate of energy absorption P for a perturbation of wavevector q andfrequency ω. Normalizing to unit volume, we have

P

V= 2πωU2

0

2

V

∑k

nk

δ(εk+q − εk − ~ω)− δ(εk−q − εk + ~ω)

(24.38)

= 2πωU20

2

V

∑k

(nk − nk+q

)δ(εk+q − εk − ~ω) (24.39)

= 2ωU20 Im

χ(q, ω)

, (24.40)

confirming that energy loss is proportional to the imaginary part of the density responsefunction.

24.4 Screening and the dielectric function

We now have a much more accurate model of the density response of the electron gas thanthe Thomas–Fermi theory we used in Lecture 8. Let’s use it to examine the issue of screeningagain.

A q and ω dependent external potential V ext(q, ω) will induce a density response that par-tially screens the applied (external) potential, to give a screened potential V scr(q, ω).

V scr(q, ω) = V ext(q, ω)− δρ(q, ω)V coul(q). (24.41)

where

V coul(q) =e2

ε0q2(24.42)

is the spatial Fourier transform of the Coulomb potential.

From the calculation above

δρ(q, ω) = χ(q, ω)V scr(q, ω). (24.43)

Therefore

V scr(q, ω) = V ext(q, ω)− χ(q, ω)V coul(q)V scr(q, ω) (24.44)

V scr(q, ω)(

1 + χ(q, ω)V coul(q))

= V ext(q, ω) (24.45)

δρ(q, ω) = χ(q, ω)V scr(q, ω) (24.46)

=χ(q, ω)V ext(q, ω)

1 + χ(q, ω)V coul(q)(24.47)



Another way to write this is in terms of the dielectric function:

V scr(q, ω)

V ext(q, ω)≡ 1

εr(q, ω)(24.48)

=1

1 + V coul(q)χ(q, ω)(24.49)

This defines the dielectric function in the random phase approximation.

εRPAr (q, ω) = 1 +

2e2

ε0q2

1

V

∑k

nk − nk+q

εk+q − εk − ~(ω + iη). (24.50)

This is the dynamical generalization of Thomas–Fermi theory.

24.5 Properties of the RPA dielectric function

In the static limit ω → 0

εRPAr (q, 0) = 1 +

q2TF

q2F (q/2kF), (24.51)

where F (x)→ 1 as x→ 0.

F (x) is plotted for 1, 2 and 3 dimensions in Fig. 24.1. At q = 2kF, χ0 is singular in 1D,and has discontinuous derivatives in 2D and 3D. The singular response leads to the Peierl’sdistortion, which we will discuss in the next lecture. In higher dimensions the singularderivative means that δρ(q, ω) oscillates in real space as a function of distance. Near adefect in a metal of charge Q, the induced charge density has the following form

ρind(r) ∝ Q

r3cos(2kFr), (24.52)

which is plotted in Fig. 24.2. These variations of the charge density are called ‘Friedel oscil-lations’ and can nowadays been seen directly in scanning tunneling microscopy experiments.

Similar oscillations occur in the spin density response when a local exchange potential (inother words a spin-dependent potential) is applied. This is usually due to the presence of alocal magnetic moment, such as those found in transition-metal and rare earth ions.

The spin-density oscillations mediate the magnetic interaction between neighbouring ions.The details of this were worked out by Ruderman, Kittel, Kasuya and Yoshida, so the effectis called the RKKY interaction. The very interesting thing is that the sign of the magneticinteraction oscillates between ferromagnetic and antiferromagnetic as a function of distance.In modern magnetic multilayer devices such as those use in hard-disk media, accurate controlof layer thicknesses is used to tune the sign and strength of the magnetic interaction betweenneighbouring ferromagnetic layers.


24.5. PROPERTIES OF THE RPA DIELECTRIC FUNCTION 209

Figure 24.1: Static susceptibility χ(q, ω → 0) in one, two and three dimensions. (FromKagoshima et al.[24])

Figure 24.2: The charge density induced by a point-like potential, showing Freidel oscilla-tions. (From Kagoshima et al.[24])


Lecture 25

Electrons in one dimension

A characteristic property of electronic systems in one dimension is their instability to per-turbations of wavevector 2kF. The static density response function χ(q, ω = 0) diverges atq = 2kF, leading to an instability towards a collective state of the electrons called a chargedensity wave. This instability opens an energy gap at the Fermi level and leads to a metal–insulator transition called the Peierls transition. This is not just exotic physics confined toa small class of materials. In real 3D systems, such as chromium, part of the Fermi surfacebehaves as though it were one dimensional, due to a process called nesting.

25.1 One dimensional conductors

In Lecture 24 we derived the density response function

χ(q, ω) =2

V

∑k

nk − nk+q

εk+q − εk − ~(ω + iη). (25.1)

We now want to evaluate this for a one dimensional metal in the static limit, at zero tem-perature.

The energy dispersion for free electrons, ε = ~2k2/2m, is plotted in Fig. 25.1. At zerotemperature, states with k in the range −kF < k < kF are occupied. Other states are empty.χ(q, ω = 0) has contributions when nk = 1, nk+q = 0 and when nk = 0, nk+q = 1. Thecalculation is straightforward. We convert the sum to an integral, and the bounds on theintegral are set by the requirement that only one of |k〉 or |k + q〉 be occupied.

χ(q, ω = 0) =2

L

L

2π

[∫ kF

kF−q

dk

εk+q − εk−∫ −kF

−kF−q

dk

εk+q − εk

](25.2)

=1

π

m

~2q

[∫ kF

kF−q

dk

k + q/2−∫ −kF

−kF−q

dk

k + q/2

](25.3)

=m

π~2q2 ln

∣∣∣∣q + 2kF

q − 2kF

∣∣∣∣ (25.4)

This result is plotted in Fig. 24.1, showing the strong divergence at q = 2kF in one dimension.


212 LECTURE 25. ELECTRONS IN ONE DIMENSION

Figure 25.1: The energy dispersion for free electrons in one dimension. (From Kagoshima etal.[24])

In higher dimensions, the divergence is replaced by a discontinuous derivative, leading toFriedel oscillations but no instability.

25.2 The Peierls instability

We now switch to a lattice system in one dimension. The band structure and crystal structureare shown in Figs. 25.2a and 25.2b. The energy bands are free-electron like apart from energygaps at the Bragg planes. We consider a one-dimensional conductor, with the first Brillouinzone partially filled.

25.2.1 Static lattice distortions

Consider a static lattice distortion of wavenumber Q,

u(x) = uQ cos(Qx), (25.5)

where u(x) measures the displacement from equilibrium of each atom in the lattice. Anexample is sketched in Fig. 25.2d. This distortion produces a potential V that acts on theelectronic system,

V (x) ≡ VQ cos(Qx) = guQ cos(Qx), (25.6)

where g is a coupling constant that measures the strength of the electron–lattice interaction.


25.2. THE PEIERLS INSTABILITY 213

The electron system responds to the external potential by forming a charge density wave

δρQ = VQχ(q). (25.7)

This must reduce the energy of the electron system, or it would not occur. The periodiclattice distortion increases the energy — it costs an elastic energy

δU = 12cLu2

Q, (25.8)

where c is the elastic stiffness. If the decrease in the electronic energy is greater than theincrease in elastic energy, a spontaneous lattice distortion of wavevector Q will form andbe frozen into the system. In one dimensional metals this phenomenon readily occurs forQ = 2kF.

25.2.2 The energy balance

A potential of wavevector Q opens a band gap at k = Q/2, as shown in Fig. 25.2c. Themagnitude of the gap is Eg = 2|VQ| = 2g|uQ|. The bands in the vicinity of k = Q/2 bendapart to form the gap, and the perturbed energy dispersons are

ε±k =ε0k + ε0k+Q

2±

√(ε0k − ε0k+Q)2 + 4|VQ|2

2. (25.9)

The change in electron energy is

δK = 2∑k

ε±k n(ε±k )− 2∑k

ε0kn(ε0k). (25.10)

If VQ EF then

δK = −2|VQ|2∑k

nk − nk+Q

εk+Q − εk(25.11)

= −L|VQ|2χ(Q, 0). (25.12)

The total change in energy is

δK + δU = −|VQ|2Lχ(Q, 0) + 12cLu2

Q (25.13)

δK + δU

L= −|VQ|2

χ(Q, 0)− c

2g2

. (25.14)

At zero temperature, χ(Q, 0) diverges at Q = 2kF, while the elastic stiffness remains finite.The system will therefore always be unstable to a charge density wave of wavevector 2kF.The peak in χ(Q, 0) weakens as temperature increases and the Fermi-functions in the densityresponse function broaden. As a result, there is an upper bound on the temperature at whichthe Peierls instability can occur.

Because the energy gap is opened right at k = kF, we end up with an energy band that iscompletely filled, corresponding to an insulating state. The Peierls instability is an exampleof a metal–insulator transition.



Figure 25.2: (a) Electron dispersion for a 1D lattice. (b) The crystal structure of such alattice — it is a chain of atoms or other structural units. (c) and (d) Dimerization leads toa doubling of the unit cell size and opens up energy gaps at new Bragg planes following aPeierls transition. (From Kagoshima et al.[24])


25.3. KOHN ANOMALIES 215

(a)

(b)

Figure 25.3: (a) Softening of the phonon frequency as the Peierls transistion is approached.(b) In one dimension, the phonon at Q = 2kF is pulled down to zero frequency at thePeierls transition. In two and three dimensions the effect is present but much weaker. (FromKagoshima et al.[24])

25.3 Kohn anomalies

The onset of the Peierls transition changes the restoring force of the mechanical oscillationof the ions. The phonon frequencies at wavevector Q = 2kF start to decrease or ‘soften’well above the Peierls transition temperature, giving an early sign of what is about to occur.The phonon frequency goes to zero right at the Peierls transition, at which point the phononquite literally ‘freezes’ into the system as a static lattice distortion. This is illustrated inFig. 25.3. The depression of phonon frequency at Q = 2kF is called the Kohn anomaly. Theeffect is much weaker in two and three dimensions.



Figure 25.4: (a) The Fermi surface in 1D is a set of two isolated points. (b) We canimagine extending these points in perpendicular directions to get a Fermi ‘surface’. (c) MostFermi surfaces in two and three dimensions do not show nesting. (d) Nesting occurs whena substantial fraction of the Fermi surface is spanned by one wavevector. Equivalently, alarge section of the Fermi surface is mapped onto itself by translation in k-space. (FromKagoshima et al.[24])


25.4. NESTING OF THE FERMI SURFACE 217

25.4 Nesting of the Fermi surface

Can the Peierls instability occur in two or three dimensional metals? The instability arisesfrom the singular nature of χ(q, ω), which for a free electron Fermi surface is much weakerin two and three dimensions. It turns out that some higher-dimensional Fermi surfaces havestrong one-dimensional character and retain some of the singular nature of χ(q, ω). For thisto occur, large sections of the Fermi surface must be spanned by the same wavevector, Q0.Fig. 25.4(d) shows an example: translating the Fermi surface by wavevector Q0 maps a largesection of the Fermi surface on to itself. The density response will be very strong near Q0,with χ(Q0, 0) showing a divergence. This is called nesting.

Nesting is perfect in one dimension, where the Fermi surface consists of two isolated points.Nesting in higher dimensions leads to spin and charge density wave instabilities, but doesnot give rise to a metal–insulator transition, because nesting only opens a gap along theparts of the Fermi surface spanned by the nesting vector. In this case it is common to referto parts of the Fermi surface being ‘consumed’ by the spin or charge density wave instability.Metallic antiferromagnetism in chromium is an example of a spin density wave instability.

25.5 Spin–charge separation

A remarkable phenomenon occurs in interacting electron systems in one dimension. This isspin–charge separation, in which the electron quasiparticles are spinons and holons, particlesthat respectively carry the electrons spin and charge, separately. This electron fractionaliza-tion really does occur in one dimension, and has been confirmed experimentally.

What is special about 1D? In one dimension all excitations of wavevector q have the sameenergy ωq = vFq. This is not the case in two or three dimensions.

The Hamiltonian for a general interacting electron system in 1D is

H =∑k,σ

εkc†k,σck,σ +

1

2L

∑k,k′

∑q

∑σ,σ′

v(q)c†k−q,σc†k′+q,σ′ck′,σ′ck,σ. (25.15)

As we saw, in Lecture 8, the interaction term can be rewritten in terms of the densityoperators ρqσ =

∑k c†k,σck+q,σ:

V →∑q,σ

1

LV (q)ρq,σρ−q,σ. (25.16)

This shows that the eigenmodes of interaction are density waves. The peculiar thing about1D is that the eigenmodes of kinetic energy are also density waves.

25.6 The Luttinger–Tomonaga model

The Luttinger–Tomonaga model begins by replacing the quadratic energy spectrum by alinearized spectrum that extends to infinite negative energies. We will focus on the positive



k side of the spectrum. These electrons are called ‘right movers’. We assume here thatthey do not couple to the ‘left movers’, but that is not a necessary condition for spin–chargeseparation to occur.

The commutation relations for bosons are:[bq′ , b

†q

]= δq,q′ (25.17)[

bq′ , bq]

= 0 (25.18)[b†q′ , b

†q

]= 0. (25.19)

The anticommutation relations for fermions are:cq′ , c

†q

= δq,q′ (25.20)

cq′ , cq

= 0 (25.21)c†q′ , c

†q

= 0. (25.22)

Using these, it is straightforward to show that the commutator of two density operators is[ρq′ , ρq

]=∑p

(c†p+q+q′cp − c

†p+qcp−q′

). (25.23)

This must be evaluated separately for two cases:

q 6= −q′Let p′ = p+ q′, then [

ρq′ , ρq]

=∑p′

c†p′+qcp′−q′ −∑p

c†p+qcp−q′ . (25.24)

For the linearized spectrum the sums run from −∞ to +∞. p′ is just a dummy index,so the two sums are the same and cancel.[

ρq′ , ρq]

= 0, q 6= −q′. (25.25)

q = −q′

[ρ−q, ρq

]=[ρ†q, ρq

]=

∑p

(c†pcp − c

†p+qcp+q

)(25.26)

=∑p

(np − np+q

)(25.27)

= number of states in interval q (25.28)

=Lq

2π(25.29)

In this last step we have used the fact that the energy dispersion continues to −∞:it has only one Fermi point, and the difference in occupation number is due to statesaround that point.


25.6. THE LUTTINGER–TOMONAGA MODEL 219

[ρq′ , ρq

]= δq,−q′

Lq

2π, (25.30)

which means our boson operators are

b†q =

√2π

Lqρq (25.31)

bq =

√2π

Lqρ−q. (25.32)

(25.33)

The eigenstates of kinetic energy in one dimension are density waves. The linearized kineticenergy spectrum is

H0 =∑k

~vFkc†kck. (25.34)

We can use an equation of motion approach to rewrite this in terms of density operators. Itis straightforward to show that [

H0, ρq]

= ~vFqρq, (25.35)

which implies that [H0, b

†q

]= ~vFqb

†q. (25.36)

ThereforeH0 =

∑q

~vFq b†qbq. (25.37)

The interaction term is already diagonal in the density operators:

HI =1

L

∑q

V (q)ρqρ−q (25.38)

=1

L

∑q

V (q)Lq

2πb†qbq (25.39)

=1

2π

∑q

V (q)q b†qbq. (25.40)

At this point we reintroduce spin,

H =∑q,σ

~vFq b†q,σbq,σ +

1

2π

∑q,σ

V (q, σ)q b†q,σbq,σ. (25.41)

and split the interaction into spin-symmetric and spin-antisymmetric parts:

spin symmetric : S(q)(ρq↑ρ−q↑ + ρq↓ρ−q↓

)(25.42)

spin antisymmetric : A(q)(ρq↑ρ−q↑ − ρq↓ρ−q↓

)(25.43)

Then the hamiltonian is

H =∑q,σ

~vFq b†q,σbq,σ +

1

2π

∑q

S(q)q(bq↑b−q↑ + bq↓b−q↓

)+

1

2π

∑q

A(q)q(bq↑b−q↑ − bq↓b−q↓

).

(25.44)



The hamiltonian can be diagonalized by a bosonic Bogoliubov transformation.

α†q = b†q↑ + b†q↓ charge density operator (25.45)

β†q = b†q↑ − b†q↓ spin density operator (25.46)

H =∑q

[(~vF +

S(q)

2π+A(q)

2π

)q α†qαq +

(~vF +

S(q)

2π− A(q)

2π

)q β†qβq

](25.47)

The velocities of the charge and spin density excitations are

vα = vF +1

2π~∂

∂q

(q(S(q) + A(q)

))(25.48)

vβ = vF +1

2π~∂

∂q

(q(S(q)− A(q)

)). (25.49)

The separation of spin and charge is complete — the original electrons are replaced bybosonic quasiparticles that carry the spin and charge separately. That this occurs is firmlyestablished, both theoretically and experimentally, for one-dimensional electron systems. Anopen question is whether it occurs in two dimensions and, if so, whether it is responsible forthe very unusual physical properties of the cuprate superconductors.


Lecture 26

Collective modes and responsefunctions

26.1 Response functions

Very generally, we define a linear response function as a susceptibility χαβ(r, t; r′, t′) thatrelates a response uα(r, t) at point r and time t to a disturbance due to an external fieldFβ(r′, t′) at point r′ and time t′.

uα(r, t) =

∫dr′∫

dt′χαβ(r, t; r′, t′)Fβ(r′, t′) (26.1)

If the system is translationally invariant (we are usually concerned with wavelengths muchgreater than the interatomic spacing) then χ is a function of the relative distance r− r′. Ifthe system is already in equilibrium then χ is a function of t− t′.

uα(r, t) =

∫dr′∫

dt′χαβ(r− r′, t− t′)Fβ(r′, t′). (26.2)

This is a convolution integral and the usual theorems to do with convolutions apply. Inparticular, the convolution becomes a simple product in Fourier space.

uα(q, ω) = χαβ(q, ω)Fβ(q, ω) (26.3)

While the susceptibility is clearly useful for describing the response of a system to a generalstimulus, what is less obvious is that it contains information about the collective modes ofthe system.

26.2 Collective modes

There is fundamental relationship between response functions and collective modes. If aresponse function χ(q, ω) is divergent at certain wavevectors q and frequencies ω0(q), thenapplying a driving force at the correct ω and q produces an infinite response. Equivalently,


222 LECTURE 26. COLLECTIVE MODES AND RESPONSE FUNCTIONS

an infinitesimal driving force at the same frequencies and wavevectors will produce a finiteresponse. The frequencies ω0(q) describe modes of free oscillation of the system, which wecall collective modes.

In many senses, the collective modes of a system are particles that are just as real as theatoms and nuclei which the solid was built from in the first place. This is in spite of the factthat the collective modes only exist inside the solid. The spectrum of collective excitationsoften represents the most we can know about many systems, and so a system for which wecan accurately predict the observed spectrum of collective modes is a system we can regardas understood. This is one of the reasons that low temperature physics has played such acentral role in the development of solid state physics — weakly excited collective modes areusually much simpler to understand.

Collective modes are directly related to the pole structure of the response function χ(q, ω).Poles close to the real frequency axis correspond to sharp resonances, while a continuum ofexcitations gives broad features in the spectrum. (A continuum of poles can be representedby a branch cut.)

Examples of collective modes in solids include

• phonons (elastic waves)

• plasmons (charge fluctuations with longitudinal polarization)

• exitons (bound electron–hole pairs)

• magnons (spin density waves)

• spinons and holons (fractionalized electrons)

• Landau quasiparticles (electron-like quasiparticles)

• Laughlin quasiparticles (charge e/3 carriers in the fractional quantum Hall effect)

• Bogoliubov quasiparticles (linear combinations of particles and holes that are the fun-damental excitations of a superconductor)

26.3 Inelastic scattering

Collective modes may be directly visible in an inelastic scattering experiment. The idea isto send in a beam of radiation or particles of momentum q and frequency ω, and measurethe frequency and momentum of the scattered particles or radiation. The energy absorptionis proportional to Imχ(q, ω). Experiments of this sort include:

• Raman and Brillouin scattering (inelastic scattering of visible light)

• Inelastic x-ray scattering

• Angle resolved photoemission (the ejection of electrons from solids using UV light)


26.4. CAUSALITY 22380 CHAPTER 4. COLLECTIVE PHENOMENA

Figure 4.2: Generic description of an inelastic scattering experiment

both in frequency and energy. By comparing the incident and outgoingenergy and momenta, one can then deduce the spectrum of excitations. Ifthe excitations are underdamped (poles close to the real frequency axis) thenthe features will appear as sharp resonances; continuum excitations will leadto broad features. Very generally , the probability of making an excitationof momentum q and energy ω is proportional to

=χ(q, ω) (4.33)

of course multiplied by matrix elements and selection rules appropriate forthe probe at hand. 4

The collective modes are just waves propagating through the solid; inthe examples above they are waves of atomic vibration (phonons), of longi-tudinal polarisation (plasmons),or of transverse electric field (light). Lateron we shall see other examples, e.g. spin waves.

4.1.6 Causality and Kramers-Kronig relations

The response functions we are using must be causal or retarded, so thatthere is no possibility of a response before the force is applied, i.e. a causalresponse function κ must satisfy

κ(t− t′) = 0 if t′ > t . (4.34)

The principle of causality imposes conditions on the behaviour of κ(ω) inFourier space (we shall drop the momentum or space coordinate for themoment): κ(ω) must be an analytic function of ω in the upper half plane,

4We shall use the symbols = (Imaginary part), < (Real part) and ℘ (Principal value).

Figure 26.1: Inelastic scattering probes excite collective excitations of the solid.

• Neutron scattering (inelastic scattering of thermal neutron beams)

• Electron energy-loss spectroscopy (inelastic scattering of an incident electron beam)

In all cases a direct measure of the spectrum of excitations is obtained by comparing theincident and outgoing energy and momenta.

26.4 Causality

The response functions of physical systems must be causal. Causality requires that

χ(t− t′) = 0 if t < t′. (26.4)

(We neglect the position dependence of the response function for now and concentrate onthe time dependence.) That is, the state of the system cannot depend on things that havenot yet happened. This sounds obvious and simple, but it has important consequences andis extremely useful for experimentalists, allowing them to infer things about the system thatare not easily or directly accessible by experiment.

To see the utility of the causal nature of response functions, we need to turn to the complexfrequency plane. χ(t − t′) is the inverse Fourier transform of χ(ω), and for it to be zerofor t < t′, χ(ω) must be analytic in the upper half plane. This analyticity in turn enforcesstrict relationships, called Kramers–Kronig relations, between the real and imaginary partsof χ(ω), which we respectively label χ′(ω) and χ′′(ω). (i.e. χ(ω) = χ′(ω) + iχ′′(ω).)

We will derive the Kramers–Kronig relations in two ways. First, using complex function the-ory, and second, using a physical argument for the specific case of the electrical conductivity.



We start by defining the Fourier-transform pair:

χ(ω) =

∫ ∞−∞

χ(t)eiωtdt (26.5)

χ(t) =1

2π

∫ ∞−∞

χ(ω)e−iωtdω. (26.6)

>

Figure 26.2: Contours for evaluating the Fourier transform.

We can obtain χ(t) by evaluating the inverse Fourier transform as a contour integral, usingone of two contours depending on whether t > 0 or t < 0: if t < 0 we close the contour in theupper half plane; and if t > 0 in the lower half plane. Provided |χ(ω)| falls off faster than1/ω, the closing contour at infinity doesn’t contribute to the integral. Using the calculus ofresidues

χ(t) =1

2π

∫ ∞−∞

χ(ω)e−iωtdω (26.7)

=1

2π

∮χ(ω)e−iωtdω (26.8)

= 2πi∑

residues. (26.9)

By causality χ(t) = 0 for t < 0 so there can be no poles in the upper half plane. Thereforeχ(ω) is analytic for Imω > 0. In order to show that this places strict constraints on thereal and imaginary parts of χ(ω), we need a way of relating χ′(ω) and χ′′(ω). We do thisusing Cauchy’s theorem, another result from the calculus of residues.

χ(ω) ≈ χ(ω + iη) =

∮dω′

2πi

χ(ω′)

ω′ − ω − iη, (26.10)

where, because χ(ω) is analytic in the upper half plane, the contour enclosing the pole atω + iη can be deformed to run along the entire length of the real axis.


26.5. OPTICAL CONDUCTIVITY 225

ω'=ω+iη

ω'=ω+iη

Figure 26.3: Contours used for evaluating the Cauchy integral.

Setting η to zero,

χ(ω) =

∮dω′

2πi

χ(ω′)

ω′ − ω(26.11)

= P∫ ∞−∞

dω′

2πi

χ(ω′)

ω′ − ω+

1

2

∮dω′

2πi

χ(ω′)

ω′ − ω(26.12)

= P∫ ∞−∞

dω′

2πi

χ(ω′)

ω′ − ω+ 1

2χ(ω). (26.13)

Here P denotes the principle-part integral (i.e. the integral along the real axis, taking sym-metric limits when approaching any poles on the axis.) Therefore

χ(ω) = P∫ ∞−∞

dω′

πi

χ(ω′)

ω′ − ω. (26.14)

We can now use this to relate χ′(ω) to χ′′(ω) and vice versa.

χ′(ω) = P∫ ∞−∞

dω′

π

χ′′(ω′)

ω′ − ω(26.15)

χ′′(ω) = −P∫ ∞−∞

dω′

π

χ′(ω′)

ω′ − ω(26.16)

(26.17)

These are the Kramers–Kronig relations for causal response functions. We usually writeKramers–Kronig relations in terms of positive frequency, but this requires us to know thesymmetry of χ′(ω) and χ′′(ω) under the tranformation ω → −ω, which is different for thevarious response functions. We will now look at how this applies in the specific case of theelectrical conductivity.

26.5 Optical conductivity

The electrical conductivity or ‘optical conductivity’ σ relates the current to the electric fieldat earlier times.

J(t) =

∫ ∞−∞

σ(t− t′)E(t′)dt′ (26.18)



By causality σ(t − t′) ≡ σ(τ) = 0 for τ < 0. We can therefore multiply σ(τ) by θ(τ), theunit step function, without changing anything.

σ(τ) = σ(τ)θ(τ) (26.19)

The convolution theorem states that the Fourier transform of a product is the convolutionof the Fourier transforms, so

σ(ω) =

∫ ∞−∞

σ(ω′)θ(ω − ω′)dω′

2π, (26.20)

where θ(ω) is the transform of the step function:

θ(ω) =

∫ ∞0

eiωt−ηtdt. (26.21)

Here η is an infinitesimal that makes the integral converge. For ω 6= 0,

θ(ω) = limη→0

[ 1

iω − ηeiωt−ηt

]∞0

(26.22)

=i

ω. (26.23)

For ω = 0, we can use the definition of the Dirac delta function,

δ(ω) =1

2π

∫ ∞−∞

eiωtdt. (26.24)

This can all be brought together as

θ(ω) = πδ(ω) + P(

i

ω

), (26.25)

Both δ() and P() are well defined only within an integral.

Using this result

σ(ω) =

∫ ∞−∞

σ(ω′)θ(ω − ω′)dω′

2π(26.26)

=

∫ ∞−∞

σ(ω′)

πδ(ω − ω′) + P

(i

ω − ω′

)dω′

2π(26.27)

= 12σ(ω) + P

∫ ∞−∞

dω′

2πσ(ω′)

i

ω − ω′. (26.28)

This implies that

σ(ω) = P∫ ∞−∞

dω′

πi

σ(ω′)

ω′ − ω, (26.29)

the same result we obtained using Cauchy’s theorem.


26.6. OSCILLATOR STRENGTH SUM RULE 227

Let σ(ω) = σ1(ω) + iσ2(ω), where σ1 and σ2 are real. Then the Kramers–Kronig relationsfor the optical conductivity are

σ1(ω) =1

πP∫ ∞−∞

σ2(ω′)

ω′ − ωdω′, (26.30)

σ2(ω) = − 1

πP∫ ∞−∞

σ1(ω′)

ω′ − ωdω′. (26.31)

One consequence of this is that, for a superconductor, which has an infinite dc conductivity

σ1(ω) = Cδ(ω), (26.32)

there must be a corresponding term in the imaginary part of the conductivity,

σ2(ω) =C

π

1

ω, (26.33)

which is purely inductive and correponds to the kinetic energy of the superelectrons.

We have already seen, in the context of the periodic potential, that the Fourier transform ofa real function, in this case σ(τ), has the property

σ(−ω) = σ∗(ω). (26.34)

Thereforeσ(−ω) = σ1(−ω) + iσ2(−ω) = σ1(ω)− iσ2(ω). (26.35)

This implies that

σ1(ω) = σ1(−ω), (26.36)

σ2(ω) = −σ2(−ω). (26.37)

We can therefore write the Kramers–Kronig relations as integrals over positive frequencies.

σ1(ω) =2

πP∫ ∞

0

ω′σ2(ω′)

ω′2 − ω2dω′ (26.38)

σ2(ω) = −2ω

πP∫ ∞

0

σ1(ω′)

ω′2 − ω2dω′ (26.39)

26.6 Oscillator strength sum rule

If we evaluate the electrical conductivity in the relaxation time approximation, we find that

σ(ω) =ne2τ

m

1

1− iωτ, (26.40)

where τ is the relaxation time of the system. We now imagine hitting the system withan impulse of electric field, so that the time over which the field is applied is so short thesystem cannot relax to equilibrium. This is equivalent to going to very high frequencies



in the conductivity. The details of the relaxation processes are then unimportant and thefollowing discussion is completely general.1 In the high frequency limit,

σ(ω) → ine2

ωm(26.41)

⇒ σ2(ω) =ne2

mω(26.42)

⇒ limω→∞

ωσ2(ω) =ne2

m. (26.43)

We now bring in the second Kramers–Kronig relation, and find that

limω→∞

ωσ2(ω) = limω→∞

2ω2

πP∫ ∞

0

σ1(ω′)

ω2 − ω′2dω′ (26.44)

=2

πP∫ ∞

0

σ1(ω′)dω′ (26.45)

=ne2

m. (26.46)

This is usually written in the form

P∫ ∞

0

σ1(ω′)dω′ =π

2

ne2

m, (26.47)

where it is called the oscillator-strength sum rule or f-sum rule. Oscillator strength refers tothe frequency integral of the conductivity. What the rule says is that as the temperature of asystem is changed, or as the effective interaction strength is tuned, the oscillator strength canbe redistributed in frequency subject to the constraint that its frequency integral be fixed.As an example, consider a superconductor. As it is cooled through the superconductingtransition temperature, the onset of superconductivity suppresses all electronic transitionsbelow an energy 2∆. This opens a gap in the optical conductivity, up to a frequency ~ω = 2∆.Any conductivity spectral weight that was in this frequency range above the transitiontemperature must now condense into a zero-frequency delta function, and becomes associatedwith the reactive response of the superfluid.

1Not only are scattering processes unimportant but, if the applied impulse is short enough, so are theinteractions of the electrons with the nuclei and the other electrons. For the purposes of the sum ruleargument, the electron density is the total electron density, and the mass the bare electron mass. In practicethere is a hierarchy of time and frequency scales, corresponding to interaction effects at low energies, bandformation, and atomic energies, and a corresponding hierarchy of effective masses.


Lecture 27

The electron spectral function

27.1 The Schrodinger representation

The most familiar representation of quantum mechanics is the Schrodinger representation.In it, the wavefunction |ψS(t)〉 evolves in time according to the Schrodinger equation

i~∂|ψS(t)〉∂t

= H|ψS(t)〉. (27.1)

This has a formal solution

|ψ(t)〉 = e−iH(t−t0)/~|ψ(t0)〉. (27.2)

Physical observables are represented by time-independent operators OS. Measurements ofthe observable involve calculation of matrix elements and expectation values.

〈OS〉 = 〈ψS|OS|ψS〉 (27.3)

In many body systems it is more convenient to factor the time dependence out of the wave-functions.

27.2 The Heisenberg representation

In the Heisenberg representation, the state vectors |ψH〉 are time independent. All the timedependence goes into the operators OH(t).

We can go between Schrodinger and Heisenberg representations by a unitary transformation:

|ψH〉 = eiH(t−t0)/~|ψS(t)〉 = |ψS(t0)〉 (27.4)

OH(t) = eiH(t−t0)/~OSe−iH(t−t0)/~. (27.5)

This transformation leaves all matrix elements (and all physical observables) unchanged.The hamiltonian is also unchanged by the transformation, because it commutes with itselfin the exponential factors.


230 LECTURE 27. THE ELECTRON SPECTRAL FUNCTION

The time dependence of the operators (instead of the state functions) is computed using thefollowing equation of motion:

i~∂OH(t)

∂t= OHH −HOH = [OH, H]. (27.6)

This is called the Heisenberg equation of motion.

27.3 Particles and quasiparticles

We usually think of electrons as fundamental particles. But this no longer makes sense ina solid, because an individual electron cannot be separated from the rest of the interactingelectron liquid. The best we can do is think of experiments we might perform to study theelectron-like excitations. One of the most direct experiments is to insert an electron intoa state |p〉 at time t = 0, and remove an electron from a state |p′〉 at a later time t. Theprobability of propagating from one state to the other is related to the expectation value

G(p′, t; p, 0) = −i〈ψG|T [cp′(t)c†p(0)]|ψG〉. (27.7)

Here ψG is the ground state wavefunction of the many-body system, and T is the time-ordering operator. That is,

T [c(t)c†(t′)] = c(t)c†(t′), t > t′ (27.8)

= −c†(t′)c(t), t′ > t. (27.9)

This orders the operators according to which one is earlier and is necessary because wewant to calculate a causal response function. G(p′, t; p, 0) is called the Green’s function orpropagator.

These operators are in the Heisenberg representation, so

c(t) = eiHt/~ce−iHt/~, (27.10)

where c(t = 0) ≡ c.

We will now look at some examples.

27.4 A single free particle

A single free particle has the hamiltonian

H0 =∑p

εpc†pcp. (27.11)


27.4. A SINGLE FREE PARTICLE 231

The Heisenberg equation of motion is

i~∂cp(t)

∂t= [cp(t), H] (27.12)

= eiHt/~εp[cp, c†pcp]e−iHt/~ (27.13)

= eiHt/~εp(cpc†pcp − c†pcpcp)e−iHt/~ (27.14)

= eiHt/~εp(1− c†pcp)cpe−iHt/~ (27.15)

= εpcp(t). (27.16)

This has solution

cp(t) = cpe−iεpt/~. (27.17)

Similarly

i~∂c†p(t)

∂t= −εpc†p(t), (27.18)

and

c†p(t) = c†peiεpt/~. (27.19)

We can now evaluate the free particle propagator or Green’s function, by taking the expec-tation value in the vacuum state.

G0(p′, t; p, 0) = −i〈0|T [cp′(t)c†p(0)|0〉 (27.20)

For t > 0

G0(p′, t; p, 0) = −ie−iεpt/~〈0|cp′(0)c†p(0)|0〉 (27.21)

= −ie−iεpt/~〈0|(1− c†pcp|0〉δp′p (27.22)

= −ie−iεpt/~δp′p. (27.23)

For t < 0, G0 = 0. (cp′ acts directly on the vacuum.) The time-ordering operator enforcescausality. Momentum is also conserved.

(In a solid, G(p, t < 0) represents the propagation of holes. In a solid the vaccum is a filledFermi sea, and acting on it with an annhilation operator just creates a hole.)

We are often interested in the Fourier transform of G, which gives us its energy dependence.

G(p, ω) =

∫ ∞−∞

G(p, t)eiωtdt (27.24)

G(p, t) =1

2π

∫ ∞−∞

G(p, ω)e−iωtdω (27.25)



For a free particle

G0(p, ω) =

∫ ∞−∞

G0(p, t)eiωtdt (27.26)

= −i

∫ ∞0

ei(ω−εp/~+iη)tdt (27.27)

= − 1

ω − εp/~ + iη

[ei(ω−εp/~+iη)t

]∞0

(27.28)

=1

ω − εp/~ + iη. (27.29)

Here η is a positive infinitesimal convergence factor. In a solid we have the possibility ofpropagating holes and would in that case use a negative infinitesimal η = 0−.

Using the identity

limη→0+

1

x+ iη= P

(1

x

)− iπδ(x) (27.30)

we can write the free-particle Green’s function as

G0(p, ω) = P(

1

ω − εp/~

)− iπδ(ω − εp/~). (27.31)

27.5 The spectral function

The spectral function A(p, ω) is defined to be the imaginary part of the Green’s function.

A(p, ω) = − 1

πImG(p, ω) (27.32)

For noninteracting particles

A(p, ω) = δ(ω − εp/~). (27.33)

The spectral function has a straightforward interpretation — it is the probability of findinga quasiparticle excitation of momentum p with energy ~ω. For noninteracting particles, weonly find excitations at the band energy εp, hence the delta function. This is illustrated inFig. 27.1.

The Green’s function as we have defined it above is a causal response function. As a result,its imaginary part — the spectral function — contains all the information of the full Green’sfunction. This is made explicit using the Lehmann representation:

G(p, ω) =

∫ ∞−∞

A(p, ω′)

ω − ω′ + iηdω′. (27.34)

The causal Green’s function is also known as the retarded Green’s function, because it de-scribes how a particle propagates to a distant point at a later time. In a solid, the Green’sfunction for holes is an advanced Green’s function.


27.6. INTERACTING SYSTEMS 233

Figure 27.1: The spectral function of a noninteracting electron system. (From Damascelli,Hussain and Shen[25].)

27.6 Interacting systems

The Green’s function for an interacting system must be different from a noninteractingGreen’s function. But if the interactions are weak, as they are in many real systems, theGreen’s function should not be too different from a free particle propagator. We can guesswhat the answer would look like — a single state interacting with a continuum of levels willdecay in time. In the Schrodinger representation we would expect our wavefunction to looklike

ψS(t) ∝ e−iεpt/~−Γpt, (27.35)

where Γ is the (momentum-dependent) decay rate. If Γ εp/~ the quasiparticle will stillbe well defined. We would then expect for the Green’s function

G(p, t) = −iZpe−iεpt/~e−Γptθ(t), (27.36)

where θ(t) is the unit step function and the amplitude Zp is the so-called ‘quasiparticlerenormalization factor’. Zp ≤ 1. In frequency space this is

G(p, ω) =Zp

ω − εp/~ + iΓp

. (27.37)

The corresponding spectral function is

A(p, ω) =1

π

ZpΓp

(ω − εp/~)2 + Γ2p

. (27.38)

This is a Lorentzian with total weight∫ ∞−∞

A(p, ω)dω = Zp. (27.39)



It can be shown rigorously that ∫ ∞−∞

A(p, ω)dω = 1. (27.40)

That is, the probability of finding the excitation of momentum p at some energy is 1, evenin the presence of interactions. The effect of interactions is to move some of the spectralweight from the quasiparticle peak to a broad, incoherent part of the Green’s function.

Figure 27.2: The spectral function for an interacting electron system. (From Damascelli,Hussain and Shen[25].)

In general, the exact Green’s function for the interacting system can be written

G(p, ω) =1

ω − εp/~− Σ(p, ω). (27.41)

Σ(p, ω) is called the self energy. Its real part, Σ′(p, ω), shifts the excitation energies anddescribes the effect of interactions on the velocity of the quasiparticle. Its imaginary part,Σ′′(p, ω), is related to damping. The Green’s function can also be written

G(p, ω) =1

ω − εp/~ + iΓp

+Gincoh(p, ω), (27.42)

where εp is the renormalized quasiparticle energy dispersion (frequency independent) andGincoh(p, ω) is the incoherent part of the Green’s function, which usually contributes a broadbackground to the spectral function that continues up to high energies. The spectral functionfor an interacting system is plotted in Fig. 27.2.


27.7. ANGLE RESOLVED PHOTOEMISSION SPECTROSCOPY 235

27.7 Angle resolved photoemission spectroscopy

The most direct way to measure the electron spectral function is by angle resolved photo-emission spectroscopy, or ARPES. Photons are incident on a single crystal sample, and causetransitions from low-energy occupied states to empty, plane-wave-like states well above thevacuum energy. The excited photoelectron leaves the sample and is collected in a detectorthat measures both the energy and momentum of the photoelectron. The photons typicallyhave ultraviolet wavelengths, with energies of about 20 eV, and carry very little momentumcompared to electrons. The transitions are therefore vertical (i.e. do not appreciably changethe momentum of the electron) so the component of the momentum of the photo-emittedelectron parallel to the sample surface is the same as the parallel component of the momentumof the low-energy initial state. The perpendicular component of the momentum is notconserved, because the presence of the sample surface breaks translational symmetry. Theprocess is sketched in Fig. 27.3 and the geometry is shown in Fig. 27.4.

Figure 27.3: The energetics of photoemission. (From Damascelli, Hussain and Shen[25].)



Figure 27.4: The experimental geometry for angle-resolved photoemission. (From Damas-celli, Hussain and Shen[25].)

In angle resolved photoemission, an analyzer measures the kinetic energy of the emittedelectrons, Ekin. This also sets the magnitude of the momentum |p| =

√2mEkin. The com-

ponents of the momentum perpendicular and parallel to the surface are determined by thepolar angle θ and azimuthal angle φ.

p⊥|p|

= cos θ (27.43)

px|p|

= sin θ cosφ (27.44)

py|p|

= sin θ sinφ (27.45)

The energy ε of the electron before being photoexcited can be calculated by energy conser-vation. This quantity is also called the binding energy EB, measured down in energy fromthe chemical potential µ.

EB = µ− ε. (27.46)

This is obtained from the kinetic energy of the emitted electron:

Ekin = hν − φ− |EB|, (27.47)

where φ is the work function of the surface.

A typical experimental set up is shown in Fig. 27.5. Because only parallel components ofthe momentum are conserved, ARPES works best on materials with two-dimensional orone-dimensional electronic structure. As an example, the ARPES-measured Fermi surfaceof Sr2RuO4 is shown in Fig. 27.6.


27.7. ANGLE RESOLVED PHOTOEMISSION SPECTROSCOPY 237

Figure 27.5: The experimental configuration of a synchrotron-based photoemission experi-ment. (From Damascelli, Hussain and Shen[25].)



Figure 27.6: ARPES spectra for Sr2RuO4, with the ARPES-derived Fermi surface and atheoretical Fermi surface. (From Damascelli et al. [26].)


Lecture 28

Landau’s Fermi Liquid theory

Fermi liquid theory was originally developed by Lev Landau to describe the physics of 3He.It has been useful far beyond its initial application, being generally applicable to degenerateinteracting Fermi systems as diverse as neutron stars and atomic nuclei. In addition, it hasbecome the ‘Standard Model’ of the metallic state of solids. Indeed, in analogy to particlephysics, much current research is devoted to looking for physics beyond this standard model.

Landau’s Fermi liquid theory explains why a system of strongly interacting particles canbehave to a very high degree like a set of independent particles, and provides a way ofincorporating the effects of interactions in a phenomenological way. The basic idea is tofocus on the excitations of the interacting system without worrying about the precise formof the ground state. The elementary excitations act like particles, and are therefore called‘quasiparticles’. The energies of the quasiparticles are nearly additive.1 Complicated states,such as conventional superconducting states, can be formed by adding together many quasi-particles. The quasiparticles interact with each other, but far less strongly than the originalparticles of the theory from which they are constructed.

28.1 The noninteracting Fermi gas

In the limit that T → 0, the Fermi–Dirac distribution becomes a step function, and theelectron system is said to be fully degenerate. In this regime, the number of excited statesavailable to the system is greatly reduced, with striking consequences for the physical proper-ties of the system: the specific heat becomes proportional to T , instead of T -independent; andthe spin susceptibility becomes temperature independent instead of varying with a Curie-like1/T power law.

In a real fermion system, particle interactions and the exclusion principle are both important,leading to the study of degenerate Fermi liquids. Sometimes the properties of the interactingsystem are drastically modified by interactions, such as in superconductors and Mott insu-lators. In many cases, however, the interacting liquid retains many properties of the gas. Itis then called a normal Fermi liquid.

1Remember, our definition of a noninteracting system is one in which the total energy can be found byadding up the single-particle energies.


240 LECTURE 28. LANDAU’S FERMI LIQUID THEORY

1n0

0 pFp

Figure 28.1: The ground state distribution function.

Consider a system of N noninteracting free fermions, each of mass m, enclosed in a volume V .Eigenstates are antisymmetrized combinations of N different single particle states, with eachparticle represented by is momentum p and spin σ = ±1

2and corresponding wavefunction

ψp(r) =1√V

eip·r/~. (28.1)

The eigenstates of the system can be characterized by the distribution function npσ:

npσ = 1 if |p, σ〉 is occupied, (28.2)

= 0 otherwise. (28.3)

The eigenenergies are

εp =p2

2m, (28.4)

and the total energy is additive:

E =∑p,σ

npσp2

2m. (28.5)

The states are filled according to the exclusion principle, up to the Fermi momentum

pF = ~(3π2N/V )13 . (28.6)

The Fermi surface is a sphere of radius pF.

Now add a single particle. In the ground state of the N + 1 particle system, the additionalparticle occupies a state on the Fermi surface, and adds an energy equal to the chemicalpotential µ. The change in ground state energy is

µ = E0(N + 1)− E0(N) =∂E0

∂N=

p2F

2m. (28.7)


28.2. EXCITED STATES 241

28.2 Excited states

Excited states are specified with respect to the ground state. A given excited state is obtainedby ‘exciting’ a certain number of particles across the Fermi surface. This creates an equalnumber of particles outside the Fermi surface and holes inside the Fermi surface. Particlesand holes are therefore the elementary excitations of the system, as illustrated in Fig. 28.2.The amount of excitation is characterized by the departure of the distribution function from

Figure 28.2: Particle and hole excitations near the Fermi surface. (From Schofield[1].)

its ground-state value. (We will ignore spin for now.)

δnp = np − n0p. (28.8)

If we have a particle excitation of momentum p′ > pF,

δnp = δpp′ . (28.9)

A hole excitation with p′ < pF corresponds to

δnp = −δpp′ . (28.10)

For a noninteracting system

E − E0 =∑p

p2

2mδnp. (28.11)

At low temperature, δnp is O(1) only in the immediate vicinity of the Fermi surface.

In an isolated system, the total number of particles is conserved, which requires that thenumber of excited particles equals the number of excited holes. This restriction can beinconvenient, so we switch to the grand canonical ensemble. The free energy is

F = E − µN (28.12)

F − F0 =∑p

(p2

2m− µ

)δnp. (28.13)



The free energy associated with particles is p2/2m−µ. Inside the Fermi surface, δnp < 0, sothe free energy for holes is µ−p2/2m. The free energy for elementary excitations if therefore∣∣∣∣ p2

2m− µ

∣∣∣∣ . (28.14)

The excitation energy is always positive, ensuring the stability of the ground state.

28.2.1 Heat capacity

We calculated the heat capacity of a noninteracting Fermi gas in Lecture 3 using the Som-merfeld expansion. The result was

cV =π2

3k2

BTD(εF), (28.15)

where D(εF) is the density of states at the Fermi level.

28.2.2 Pauli spin susceptibility

We now calculate the Pauli spin suscepstibility, ignoring orbital effects. In that case, theapplication of a magnetic field leads to Zeeman splitting of the energy levels. The field willraise the energy of the up-spin electrons by an amount

∆ε↑ = gµBσB (28.16)

≈ µBB. (28.17)

The down-spin electrons are lowered in energy by

∆ε↓ = −µBB. (28.18)

The two spin sub-bands will come into equilibrium with equal chemical potentials, as shownin Fig. 28.3. The number of up spins is

N↑ =N

2− 1

2µBBD(εF)V. (28.19)

Similarly, the number of down spins is

N↓ =N

2+ 1

2µBBD(εF)V. (28.20)

The spin polarization isN↑−N↓ = −µBBVD(εF) and the net magnetizationM = µB(N↓ −N↑).The magnetization per unit volume is m = M/V . The susceptibility is

χ ≡ dm

dH≈ µ0

dm

dB(28.21)

= µ0µ2BD(εF). (28.22)

The Pauli spin susceptibility is temperature independent, because the number of excitations(∼ T ) goes oppositely to the Curie susceptibility (∼ 1/T ).


28.3. LANDAU QUASIPARTICLES 243

D(ε)

ε

D(ε)

ε

2µBB

D(ε)

ε

2µBB

Figure 28.3: Partial spin polarization of a Fermi gas by the application of a magnetic field.

28.3 Landau quasiparticles

Landau considered a fictitious model in which the interactions between particles could beswitched on and off at will. He then imagined starting in the distant past with a nonin-teracting system and slowly switching the inter-particle interactions on. According to theadiabatic principle of quantum mechanics (see Lecture 23), if the interaction is switchedon slowly enough a system starting in an eigenstate of the noninteracting system will re-main in the corresponding eigenstate of the interacting system at a later time t. That is,the particles of the noninteracting system will smoothly become the quasiparticles of theinteracting system, with a rigorous one-to-one correspondence. We can imagine carryingout a procedure in which we map out the spectrum of the excited quasiparticle states bystarting off in an excited state of the noninteracting system and following its developmentas we switch on interactions.2 This exposes the chief limitation of Fermi liquid theory — in

2There is no guarantee that this will generate all eigenstates and, in particular, the ground state. We willassume, however, that is does, thereby restricting ourselves to ‘normal’ Fermi liquids.



order for the adiabatic theorem to apply, the time taken to switch on the interactions mustbe large compared to ~/∆ε, where ∆ε is the minimum spacing between the energy levels ofthe system. If a degeneracy occurs while we are switching on the interaction it will becomeimpossible to proceed any further. We will then have found an instability of the Fermi liquid.In addition, in order to map out the excitation spectrum of the Fermi liquid, we will needto be able to switch the interaction on in a time that is short compared to the decay timeof the excited state, 1/Γ(ε). A calculation of the decay rate shows that there should exist arange of energies for which this is possible.

Figure 28.4: Illustration of the adiabatic principle using an example from single particlequantum mechanics. A harmonic potential is gradually added to the square-well poten-tial, smoothing perturbing the eigenstates and energies. The states of the perturbed andunperturbed systems exist in one-to-one correspondence. (From Schofield[1].)

28.4 Quasiparticle decay rate

Imagine a single particle of momentum |p1| > pF above a filled Fermi sea. It interacts witha particle of momentum |p2| < pF within the Fermi sea. As a result, the particles scatterand two particles appear above the Fermi sea, with momenta p3 and p4 = p1 + p2 − p3.|p3|, |p4| > pF. Another way of looking at the same process is to say the particle withmomentum p1 has ‘decayed’ into two particles, of momenta p3 and p4 respectively, and ahole with momentum p2.

The total probability of this process is given by Fermi’s golden rule and is proportional to∫δ(ε1 + ε2 − ε3 − ε4)dp2dp3, (28.23)


28.4. QUASIPARTICLE DECAY RATE 245

p1

p4

p3

p2

Figure 28.5: Kinematics places tight constraints on the decay of quasiparticles near the Fermisurface.

subject to the constraint that |p2| < pF, |p3| > pF and |p4| = |p1 + p2 − p3| > pF. For|p1| − pF pF,

2pF − |p1| < |p2| < pF (28.24)

pF < |p3| < |p1|+ |p2| − pF (28.25)

The angle between p1 and p2 is arbitrary. The angle between p3 and p1 + p2 is fixed byenergy conservation. This leaves integrals over the magnitudes of p2 and p3. The decay rateis then proportional to ∫ pF

2pF−p1dp2

∫ p1+p2−pF

pF

dp3 =1

2(p1 − pF)2 (28.26)

∝ (ε1 − εF)2. (28.27)

As the particle gets closer and closer to the Fermi surface, it becomes increasingly difficultto satisfy the kinematic constraints and find final states for it to recoil into after scattering.At finite temperature, T plays the role of energy, setting the characteristic spread of thethermal distribution about the Fermi surface. The Fermi liquid decay rate is then

1

τ= a(ε1 − εF)2 + b(kBT )2, (28.28)

where a and b are constants. The larger of temperature or energy dominates. For statesnear the Fermi surface, τ becomes arbitrarily long, and we can apply the adiabatic principle.The kinematic constraints on the decay rate lead to a characteristic form for the electricalresistivity

ρ(T ) ∝ 1

τ∝ T 2. (28.29)

This is often taken as an identifying signature of Fermi liquid behaviour.



28.5 Landau’s Fermi liquid theory

The total energy of the system is a functional of the departure from the ground state distri-bution.

E[np] = E0 +∑p

εpδnp +O(δn)2 (28.30)

The functional derivative δE/δ(δnp) is the quasiparticle excitation energy εqpp . The quasi-

particle velocity is defined in terms of this energy:

vp = ∇pεqpp . (28.31)

We can also define an effective mass m∗ through

vF =pF

m∗. (28.32)

Mass renormalization is not the only difference between the an electron gas and a Fermiliquid!

28.5.1 Interactions between quasiparticles

For the calculation of most physical properties it is necessary to extend the expansion ofthe energy to second order in the distribution function. We will now include spin explicitlyagain.

E − E0 =∑p,σ

ε0pδnp,σ +1

2V

∑p,σ;p′,σ′

f(p, σ; p′, σ′)δnp,σδnp′,σ′ +O(δn)3 (28.33)

Here ε0p = vF(p−pF). The quasiparticle energies are again given by the functional derivative

εqppσ =

δE

δ(δnp,σ)= vF(p− pF) +

1

V

∑p′,σ′

f(p, σ; p′, σ′)δnp′,σ′ . (28.34)

The energy of an individual quasiparticle excitation now depends on the occupation of otherquasiparticle states! This is a new feature of the Fermi liquid theory and represents in-teraction between the quasiparticles. All details of the interactions (which are weak) arecontained in the Landau f -function f(p, σ; p′, σ′).

At low temperatures, where the Landau Fermi liquid theory holds, δnp,σ is O(1) only in theimmediate vicinity of the Fermi surface. It is therefore only necessary to define f on the Fermisurface. (That is, the interaction function f(p, σ; p′, σ′) won’t depend on the magnitude ofthe momenta, only on the angles). For the problem Landau first considered, that of 3He,the system is spherically symmetric. This rotational invariance means that the interactionenergy only depends on the angle between p and p′. In addition, it must be symmetric underinterchange of p and p′, so it can be expanded as a function of cos θ = p · p′/p2

F. The usualmethod is to use a set of orthogonal polynomials called the Legendre polynomials P`(cos θ).

P0(cos θ) = 1 (28.35)

P1(cos θ) = cos θ (28.36)

P2(cos θ) = 12(3 cos2 θ − 1) etc. (28.37)


28.6. EXPERIMENTAL CONSEQUENCES OF FERMI LIQUID THEORY 247

The most important spin-dependent part of the interaction is the exchange interaction, whichis invariant under a global spin rotation and therefore only depends on the relative angle ofthe spins σ · σ′. In the Russian literature the f function is written in a compact way as

f(p, σ; p′, σ′)→ φpp′ + σ · σ′ψpp′ . (28.38)

Another way to do the same thing is to break the interaction up into spin-symmetric andspin-antisymmetric parts. This is more transparent but less compact. The up-spin anddown-spin quasiparticle energies are then

εqpp↑ = vF(p− pF) +

1

V

∑p′,`

fS` P`(cos θ)

(δnp′↑+ δnp′↓

)+ fA

` P`(cos θ)(δnp′↑− δnp′↓

)(28.39)

εqpp↓ = vF(p−pF)+

1

V

∑p′,`

fS` P`(cos θ)

(δnp′↑+δnp′↓

)−fA

` P`(cos θ)(δnp′↑−δnp′↓

), (28.40)

where S and A stand for spin-symmetric and spin-antisymmetric parts of the interactionrespectively.

28.6 Experimental consequences of Fermi liquid theory

Different experimental probes polarize the Fermi surface in different ways, and thereforecouple to different angular harmonics of the Landau f function. Examples include:

• Heat capacityBy changing the temperature we broaden the Fermi–Dirac distribution as shown inFig. 28.6(a). We don’t couple to any angular harmonics, because the broadening leadsto a change in the distribution function that averages to zero about the Fermi energy.The heat capacity is therefore unrenormalized by Fermi liquid interactions.

• CompressibilityIn a compressibility experiment we shrink or expand the Fermi distribution symmet-rically in k-space, as shown in Fig. 28.6(b). The change in pF is uniform around theFermi surface and is achieved by changing the density of the particles. A compres-sive polarization couples to the spin-symmetric, ` = 0 (monopolar) component of theinteraction. The speed of sound will be renormalized by the fS

0 Landau parameter.

• SusceptibilityIn an applied magnetic field the density of spin-up particles decreases with a cor-responding increase in the density of spin-down particles. The distribution func-tions change as in Fig. 28.6(c). Like a hydrostatic compression, this is an ` = 0(monopolar) polarization of the distribution function, but in this case the change isspin-antisymmetric. A magnetic polarization therefore couples to the fA

0 Landau pa-rameter.

• Charge currents. A charge current displaces the Fermi surface in k-space, as shownin Fig. 28.6(d). This is a spin-symmetric, dipolar perturbation that couples to the fS

1

Landau parameter.



(a) (b) (c) (d)

Figure 28.6: Different polarizations of the distribution function: (a) heating the distribution;(b) compression; (c) spin polarization; and (d) a charge current.

28.6.1 Heat capacity

Because heating the Fermi liquid doesn’t couple to any angular harmonic, the heat capacitydoes not depend on the Landau interaction parameters and retains the free electron form

cV =π2

3k2

BTD(εF). (28.41)

However, the Fermi velocity is renormalized in a Fermi liquid, which affects the density ofstates D(εF).

D(εF) =m∗kF

π2~2, (28.42)

so

cV =m∗kF

3~2k2

BT (28.43)

for the interacting Fermi liquid. Measurements of heat capacity therefore have the abilityto directly probe the effective mass enhancement, without additional affects from Landauparameters.

28.6.2 Spin susceptibility

A magnetic field B spin polarizes the quasiparticles, as we saw above for the noninteractingsystem. However, the energetics are changed by the presence of the Landau parameters. Aspin polarization will couple to the spin-antisymmetric, monopolar parameter fA

0 . In thepresence of a magnetic field, the change in energy of the up-spin electrons will be

∆E = µBB +1

V

∑p

fS0 (δnp↑ + δnp↓) + fA

0 (δnp↑ − δnp↓) (28.44)

The magnetization is given by

M = −µB

∑p

(δnp↑ − δnp↓) (28.45)


28.6. EXPERIMENTAL CONSEQUENCES OF FERMI LIQUID THEORY 249

and there is an overall constraint of fixed particle number∑p

(δnp↑ + δnp↓) = 0. (28.46)

We can therefore write the energy shift as

∆E = µBB −MfA

0

µBV. (28.47)

This lets us determine the degree of spin polarization.

N↑ =N

2− 1

2∆ED(εF)V (28.48)

N↓ =N

2+ 1

2∆ED(εF)V (28.49)

The net magnetization is then

M = −µB(N↑ −N↓) (28.50)

= µBD(εF)V

µBB −

MfA0

µBV

(28.51)

= µ2BD(εF)V B −MD(εF)fA

0 (28.52)

=µ2

BD(εF)V B

1 +D(εF)fA0

(28.53)

=µ2

BD(εF)V B

1 + FA0

, (28.54)

where FA0 is a dimensionless Landau parameter, obtained by scaling the interaction param-

eters by the density of states.

The susceptibility χ is

χ =µ0

V

dM

dB(28.55)

=µ0µ

2BD(εF)

1 + FA0

. (28.56)

This is modified from the free electron result by D(εF) ∝ m∗ and by the presence of FA0 .

This is not as innocuous a change as for the heat capacity — when FA0 = −1 there is a

transition to a ferromagnetic state. This is called the Stoner instability.


Lecture 29

Superconductivity I —phenomenology

References: the two books giving the best introduction to superconductivity are those by Tinkham[27] and

Waldram[28]. Taylor and Heinonen[7] give good, succinct explanations of some of the key phenomena.

29.1 Superconductivity

We finish this lecture course with a short survey of superconductivity, a subject that hasbeen at the forefront of experimental and theoretical physics research for almost one hun-dred years. With no disrespect for theory intended, superconductivity is an example of aphenomenon that could only have been discovered by experiment. The physical effects as-sociated with superconductivity are so diverse and fantastic that had they been predictedahead of time the theory would not have been believed. As it was, it took almost 50 yearsand a complete quantum theory of solids before a satisfactory explanation of superconduc-tivity was obtained. Every detail of the resulting Bardeen–Cooper–Schrieffer (BCS) theoryhas since been confirmed experimentally. The microscopic theory in turn led to many newdiscoveries and applications, most notable among them the Josephson effect. Superconduc-tivity continues to be a vibrant field with whole new classes of superconductors discoveredevery decade or so. The most famous of these discoveries was high temperature supercon-ductivity in copper oxide or ‘cuprate’ materials, in 1986. Over 20 years later, this remainsone of the most important unsolved problems in correlated electron physics. Other classesof superconductors include organic superconductors and the so-called ‘heavy fermion’ super-conductors, in which the electron quasiparticles have effective masses approaching that of aproton or a neutron.

29.2 Perfect conductivity

Shortly after liquefying helium in Leiden, Kammerlingh Onnes started studying the electricalresistivity of various metals in the zero-temperature limit, with a view to testing how far the


252 LECTURE 29. SUPERCONDUCTIVITY I — PHENOMENOLOGY

Figure 29.1: The superconducting transition of mercury[29].

electrical resistivity could be reduced. He began with mercury, which could easily be purifiedby distillation. What he discovered was nothing short of amazing — at a temperature of4.2 K, the electrical resistance of mercury dropped abruptly to zero. Other common metals,such as lead and tin, showed a similar transition. The phenomenon was named superconduc-tivity. Increasingly sophisticated measurements to quantify the electrical resistance have allobtained the same result — that the resistance is zero to within experimental error. Theseexperiments take the form of a persistent current flowing in a loop and use sensitive mag-netometers to detect small changes in the magnetic field produced by the loop. Theoreticalestimates have shown that such a configuration, if kept cooled below the superconductingtransition temperature, should be stable for longer than the age of the universe.

29.3 The Meissner–Ochsenfeld effect

The next experimental signature to be discovered was perfect diamagnetism, observed in1933 by Meissner and Ochsenfeld. In a perfect conductor, we expect Lenz’ law to applyperfectly. That is, any attempt to change the magnetic flux linking the perfect conductorwould be met with a self-generated flux, induced by the Faraday electric fields, that perfectlycancels the change. (In a perfect conductor, even in the presence of a large, time-varyingcurrent, the electric field is everywhere zero. Therefore∮

E · d` = 0 = − d

dt

∫B · dA = −dΦ

dt. (29.1)

Magnetic flux is therefore trapped in place.)


29.4. THE LONDON THEORY 253

Figure 29.2: The Meissner–Ochsenfeld effect.

Meissner and Ochsenfeld discovered that not only is a magnetic field prevented from enteringa superconductor, as would be expected from the perfect conductivity, but when a purespecimen is cooled in the presence of a small field, the field is expelled as the sample coolsthrough Tc. This phenomenon cannot be explained by perfect conductivity, and gives afar more important clue as to the nature of superconductivity than the lack of electricalresistance, because it indicates that the onset of superconductivity is a transition to a newthermodynamic ground state. Zero resistance, on the other hand, could be explained by asudden switching off of scattering processes in the conductor.

29.4 The London theory

The first successful theory of superconductivity was the phenomenological model put forwardby Fritz and Heinz London in 1935. The Londons proposed the existence of a superfluidwavefunction ψ(r) that extended coherently throughout the superconductor. We now knowthat superconductors are described in microscopic detail by the Bardeen–Cooper–Schrieffer(BCS) pairing theory, which we will look at in the next lecture. The London model survives ifwe identify the superfluid wavefunction with the centre-of-mass motion of the Cooper pairs.In that case, a wavefunction ψ(r) of the form exp(is · r) corresponds to a state in whichevery Cooper pair has the same centre-of-mass momentum ~s and pair velocity ~s/2me. Wewill carry on using this ‘pairing’ language while discussing the London theory, although itshould be remembered that the Londons had no inkling that their theory actually referredto electron pairs.

The London wavefunction is usually normalized to the effective density of Cooper pairs ns.

ψ(r) =√nse

iθ(r) (29.2)



The supercurrent density is js = −2ensvs = −2eψ∗ψvs or, more generally,

js =ie~2me

(ψ∗∇ψ − ψ∇ψ∗

), (29.3)

which is the usual quantum mechanical form for a current density.

We will now show how the London theory leads to perfect conductivity and the expulsion ofmagnetic flux. In the presence of a magnetic field, the quantum-mechanical current operatortakes the gauge invariant form

js =ie~2me

(ψ∗∇ψ − ψ∇ψ∗

)− 2e2

me

ψ∗ψA, (29.4)

where A is the vector potential. If we write ψ as√nse

iθ(r), then this becomes

Λjs = −( ~

2e∇θ + A

), (29.5)

where Λ is the London paramater Λ = me/2nse2. ~∇θ is the local canonical pair momentum.

The rate of change of the wavefunction gives the local pair energy 2µ. That is

i~∂

∂t

√nse

iθ(r) = 2µ√nse

iθ(r) (29.6)

⇒ ~∂θ

∂t= −2µ. (29.7)

Taking the time derivative of Eq. 29.5 leads to the first London equation:

∂(Λjs)

∂t= −∂A

∂t+∇µe

(29.8)

= Eeff , (29.9)

where Eeff is the effective electric field. This is an acceleration equation for the superelectrons,representing dissipationless motion.

Taking the curl of Eq. 29.5 gives∇× (Λjs) = −B, (29.10)

since the gradient of a scalar is always irrotational. Writing Ampere’s law as ∇×B = µ0js,we have

∇×∇×B = µ0∇× js (29.11)

= −µ0

ΛB. (29.12)

Using ∇×∇×B = ∇(∇ ·B)−∇2B and ∇ ·B = 0, we have

∇2B =B

λ2L

, (29.13)

where λL =√

Λ/µ0 =√me/2µ0nse2 is the London penetration depth. This is a screening

equation for the magnetic field whose solutions decay exponentially into the superconductorwith depth z as exp(−z/λL). λL is typically of the order of 10−7 m in superconductors so,for all intents and purposes, the expulsion of magnetic flux from a macroscopic sample isperfect.


29.5. FLUX TRAPPING AND QUANTIZATION 255

29.5 Flux trapping and quantization

It is very common for a superonducting sample to have a geometry that is not simply con-nected. (A simply connected geometry is one in which any contour within the superconductorcan be continuously deformed into any other contour, without any part of the contour leavingthe sample.) The simplest such case is a ring of superconductor. According to the Londontheory, the phase of the superfluid wavefunction is related to the supercurrent density

−∇θ =2e

~(A + Λjs

). (29.14)

The superfluid wavefunction must be single valued, so the change in θ around the loop mustbe an integer multiple of 2π.

−∮∇θ · d` =

2e

~

∮(A + Λjs) · d` = n2π. (29.15)

The line integral of the vector potential is the magnetic flux linking the ring:∮A · d` =

∫B · dA = Φ (29.16)

so

Φ +

∮Λjs · d` = Φ− me

e

∮vs · d` = n

h

2e. (29.17)

If we take the integral over a contour deep inside the superconductor where js and vs vanish,then the flux through the loop is quantized in units of the flux quantum Φ0 = h/2e. The pairnature of the superfluid wavefunction shows up directly in the value of the flux quantum. Theargument for the quantization of flux breaks down when the thickness of the superconductingring is small compared to the penetration depth. However, the fluxoid,

Φ− me

e

∮vs · d` = nΦ0, (29.18)

remains quantized.

29.6 The Josephson effect

We are accustomed to the idea of quantum mechanical tunneling where, for example, anelectron can quantum mechanically hop through a thin insulating barrier separating twoconducting electrodes. The amplitude for this process is small, and decreases exponentiallywith the barrier thickness. Early on, electron tunneling became a very important experimen-tal probe of superconductivity because it permitted a direct measure of the energy-dependentdensity of states. In these experiments it was single electrons that were transferred acrossthe tunneling barrier. Following the development of the BCS theory it became clear thatsuperconductivity is associated with electron pairs, but no one gave much serious consider-ation to the possibility that these may tunnel through an insulating barrier, because it was



thought that such a process, involving two electrons, would be second order in the tunnelingamplitude and therefore be immeasurably small. This is not the case. While a graduatestudent at Cambridge, Brian Josephson carried out a difficult calculation that showed thatpair tunneling was a coherent process, and as such had an amplitude comparable to singleelectron tunneling. After establishing that a measurable supercurrent could tunnel throughan insulating barrier, Josephson explored the experimental consequences of such a device,and discovered several remarkable phenomena that now go by the name of the Josephsoneffects[30]. For this work, described in a short paper in 1962, he won the Nobel prize.

Figure 29.3: Geometry of a superconducting ring containing a Josephson junction. (FromTaylor and Heinonen[7].)

Consider a superconducting ring that contains a thin, insulating tunneling barrier, like theone shown in Fig. 29.3. This small slit now allows us to move magnetic flux in and outof the ring without having to warm it through Tc, but is small enough that is contains noappreciable flux itself. Given Josephson’s result that a supercurrent can still circulate in thering, we expect the fluxoid equation, Eq. 29.18, to remain valid.

Φ− me

e

∮vs · d` = nΦ0, (29.19)

We are assuming that our ring has cylindrical symmetry, which allows us to carry out theline integral of the velocity. We do so and find that

vs =e

2πRme

(Φ− nΦ0

). (29.20)

This is the speed of a Cooper pair that moves on a circle of radius R enclosing a flux Φ.The Cooper pairs can only move with certain quantized speeds, corresponding to differentintegers n. We will usually be concerned with the lowest speed. The pair kinetic energy is

122mev

2s =

e2

4π2R2me

(Φ− nΦ0

)2

. (29.21)


29.6. THE JOSEPHSON EFFECT 257

This energy–flux relation is plotted in Fig. 29.4. As a magnetic flux is applied to the ring,the kinetic energy increases quadratically. The energy relation has a different branch foreach n, and when the flux linking the ring is equal to a half odd integer, the kinetic energybecomes degenerate with the energy of a different branch. The Cooper pairs will generallystay in the lowest kinetic energy state so, when the degeneracy point is crossed, a phase slipof 2π occurs in the phase difference across the junction and the system jumps to the nextenergy branch. The ground-state energy is therefore periodic in the applied magnetic flux.

Figure 29.4: The multiple quadratic branches of the Cooper pair kinetic energy as a functionof the applied magnetic flux, in units of a flux quantum. (From Taylor and Heinonen[7].)

The total energy εtot is a sum of similar contributions from all the Cooper pairs. The totalcurrent flowing in the ring is proportional to dεtot/dΦ. It has the sawtooth form shown inFig. 29.5. If the flux is maintained at a value different from nΦ0, a current flows throughthe junction and a difference in the phase of the superfluid wavefunction across the junctionforms. Because the flux through the loop is held constant, there is no electromotive forcearound the ring and the current flows at zero voltage — it is therefore a supercurrent. Thisis the dc-Josephson effect.

Figure 29.5: The supercurrent that flows in the ground state of a superconducting ring is aperiodic function of the applied flux. (From Taylor and Heinonen[7].)

If we now apply a flux that increases linearly in time there will be an electromotive forceV = −dΦ/dt. The current through the junction will then alternate in sign with frequency

ω =2eV

~. (29.22)

This is the ac Josephson effect. The standard volt is now defined in terms of it.



The beauty of the Josephson effect is that it leads to experimentally observable effects thatdepend on the phase of the superfluid wavefunction. This is a quantity that is not normallyexperimentally accessible. The effect has been put to great use in building superconductinginterferometers that are exquisitely sensitive to magnetic flux. These go by the name ofSQUIDs (Superconducting QUantum Interference Devices) and can detect magnetic flux(and any other physical quantity that can be converted to a flux) with a sensitivity of about10−6 Φ0/

√Hz (about 10−21 Wb/

√Hz).


Lecture 30

Superconductivity II — pairingtheory

30.1 The Cooper problem

The ground state of the electron gas is a filled Fermi sea. The Fermi surface of a real solidmay have a complicated shape, and the Fermi sea may only show a sharp surface whenviewed in the Landau quasiparticle basis, but a metallic state can none the less be identifiedwith the presence of a special surface in momentum space that holds weakly interacting,electron-like states.

In 1957 Cooper made a breakthrough, and showed that in the presence of an attractiveinteraction, no matter how weak, the Fermi sea is unstable to the formation of a two-electronbound state that has since been named the Cooper pair[31]. Although Cooper’s calculationstops well short of explaining superconductivity, it provided the necessary toehold to begindevelopment of the final theory, by demonstrating the existence of a state with lower energythan the filled Fermi sea.

Cooper’s calculation focuses on the two electrons of the pair, with coordinates r1 and r2.The other electrons are inert and simply serve to block occupation of k-states inside theFermi surface by the exclusion principle. The calculation further specializes to the case ofpair states with zero centre-of-mass momentum,

ψ(r1, r2) = ψ(r1 − r2). (30.1)

A general state of this form can be expanded in plane waves:

ψ(r1 − r2) =∑k

gkeik·(r1−r2). (30.2)

gk is the amplitude to find one electron in the plane wave state |k〉 and the other in state| − k〉. Pauli exclusion requires that gk = 0 for |k| < kF.

The two-particle Schrodinger equation is

− ~2

2m

(∇2

1 +∇22

)ψ(r1, r2) + V (r1, r2) =

(E + 2

~2k2F

2m

)ψ(r1, r2), (30.3)


260 LECTURE 30. SUPERCONDUCTIVITY II — PAIRING THEORY

where E is the energy of the pair measured with respect to 2EF. Inserting the plane waveexpansion of the wavefunction, Eq. 30.2, we can recast the Schrodinger equation as analgebraic equation in k-space.

2~2

2mk2gk +

∑k′

gk′Vkk′ = (E + 2EF)gk, (30.4)

where the matrix elements of the interaction between states |k〉 and |k′〉 are

Vkk′ =1

L3

∫V (r)ei(k−k′)·rdr. (30.5)

We are looking for bound state solutions (E < 0), when Vkk′ is attractive. We postponefor now any discussion of why an attractive interaction between electrons might exist, andjust examine what its consequences would be if it did. Cooper used the simplest possibleapproximation — he assumed the matrix element was attractive (negative) if both states |k〉and |k′〉 were within an energy cut-off ~ωD of the Fermi surface. Later on we will see thatωD is a characteristic frequency of the phonons.

Vkk′ = − VL3,

~2k2

2m< EF + ~ωD and

~2k′2

2m< EF + ~ωD (30.6)

= 0 otherwise. (30.7)

On rearranging Eq. 30.4, we have(−2

~2k2

2m+ E + 2EF

)gk = − V

L3

∑k′

gk′ , (30.8)

where the sum over k′ is for EF < ~2k′2/2m < EF + ~ωD. Let’s now measure excitationenergy from the Fermi energy, by defining

ξk =~2k2

2m− EF. (30.9)

Then

(E − 2ξk)gk = − VL3

∑k′>kF

gk′ , (30.10)

and

gk =V

L3

1

2ξk − E∑k′>kF

gk′ . (30.11)

We can deal with the sum over k′ by summing both sides over k,∑k>kF

gk =V

L3

∑k>kF

1

2ξk − E∑k′>kF

gk′ , (30.12)

and canceling the sums:

1 =V

L3

∑k>kF

1

2ξk − E. (30.13)


30.2. THE ORIGIN OF THE ATTRACTIVE INTERACTION 261

The sum now only depends on k through the energy ξk, so we can rewrite it as an integral,by introducing the density of states D(ξ) in the usual way.

1 = V

∫ ~ωD

0

D(ξ)dξ

2ξ − E(30.14)

1

V≈ 1

2D(0)

∫ ~ωD

0

dξ

ξ − E/2(30.15)

=1

2D(0) ln

∣∣∣∣E − 2~ωD

E

∣∣∣∣ (30.16)

In the weak coupling limit, D(0)V 1, then E ~ωD, and the solution becomes

E ≈ −2~ωD exp

(−2

D(0)V

). (30.17)

Although exponentially small, there is a bound state solution (E < 0) for arbitrarily weakattractive interactions. The importance of Cooper’s result, derived for a single pair of elec-trons, was to show that the Fermi surface was unstable to the formation of Cooper pairs. Itwas realized straight away that pair formation would continue until a new equilibrium wasreached, but further work was required before this could be described mathematically.

30.2 The origin of the attractive interaction

We spent some time at the beginning of the course calculating the interaction betweenelectrons in solids. The dominant force between two electrons is the repulsive Coulombinteraction

V (r1, r2) =1

4πε0

e2

|r1 − r2|. (30.18)

This has matrix element

V (q) =e2

ε0q2. (30.19)

We also saw that in the presence of a medium the matrix element becomes screened by thedielectric function εr(q, ω):

V (q)→ V (q, ω) =e2

εr(q, ω)ε0q2. (30.20)

Without going into the details, a polarizable lattice has long wavelength resonances at thefrequencies of the optical phonons. This leads to a dielectric function of the form

εr(q, ω) = εelec +ω2

p

Ω2 − ω2, (30.21)

where ωp =√ne2/mε0 is the unbound plasma frequency of the ions and Ω is the frequency

of the optical phonon.



Ve(q,ω)

ωωq0

Figure 30.1: Veff(q, ω) = VCoulomb(q) + Vphonon(q, ω) is repulsive at low energy transfer butbecomes attractive over a range of energies below the phonon frequency.

Figure 30.2: Polarization of the lattice by electron 1 leads to dynamical overscreening, anda net attractive interaction with electron 2.

The effective electron–electron interaction is plotted as a function of frequency in the long-wavelength limit in Fig. 30.1. It is repulsive at low frequencies but becomes attractive for arange of energies below the phonon frequency.

The phonon mediated interaction between electrons is effective because the time scales forelectrons and phonons are so different. An electron propagating through the solid polarizesthe lattice around it. The electron moves on a time scale of the order of ~/EF. The latticerelaxes much more slowly, on a time scale ∼ 1/Ω, leading to a residual polarization thatsurvives long after the original electron has left. A second electron moving through thelattice sees this as an excess of positive charge and is attracted to it. This process, knownas dynamical overscreening, is illustrated in Fig. 30.2.


30.3. BARDEEN–COOPER–SCHRIEFFER THEORY 263

30.3 Bardeen–Cooper–Schrieffer theory

In the presence of an attractive interaction, Cooper showed that the Fermi sea was unstableto the formation of zero centre-of-mass-momentum Cooper pairs. The pairs are formed fromelectrons above the Fermi sea, so the electrons in a Cooper pair have a kinetic energy thatis greater than the Fermi energy. It is this that sets a limit on the Cooper-pairing process— pairing will proceed until the kinetic energy cost of forming another pair outweighs thepotential energy gain from the attractive interaction, which is only available between k-states with opposite wavevector. The problem of describing the system when there is morethan one Cooper pair present is a difficult one, and it was the great insight of Schrieffer, agraduate student at the time, to guess a very good approximate form for the many-bodywavefunction.

This time using second quantization, we can write a state |ψN〉 with N/2 Cooper pairs, usinga form similar to the single pair wavefunction:

|ψN〉 =∑

g(k1, ...,kN/2)c†k1↑c†−k1↓...c

†kN/2↑c

†−kN/2↓|φ0〉, (30.22)

where |φ0〉 is the vacuum state. g(k1, ...,kN/2) is the weighting coefficient of each term inthe expansion. We have specialized to the case where the centre-of-mass momentum is zero,although we would relax this constraint if we wanted to represent the flow of a supercurrent.This wavefunction is horribly complicated. The sum is over all choices of N/2 wavevectorsfrom M possible wavevectors, where M is of the order of the number of electrons in thesystem. There are

M !

(M − N2

)!(N2

)!≈ 101020

(30.23)

different terms and as many g(k1, ...,kN/2) to determine. Part of the complication stemsfrom the fact that we constrained the number of pairs to a fixed value. This is not onlymathematically inconvenient — we would also find it impossible to describe the Josephsoneffect with such a wavefunction.

Bardeen, Cooper and Schrieffer (BCS) found a much simpler wavefunction that captured theessential physics[32]. They used a mean-field approach that depended solely on the averageoccupancy of the pair states, which, however, works very well in the limit of large N . TheBCS ground state is

|BCS〉 =∏k

(uk + vkc

†k↑c†−k↓)|φ0〉, (30.24)

subject to the constraint that |uk|2 + |vk|2 = 1. (That is, any particular pair of electrons iseither occupied or unoccupied.) The number of pairs in the BCS ground state is not fixed —if we multiply out the product we will find terms with different numbers of pairs — but thedistribution of terms is sharply peaked around the average number of pairs. By relaxing thenumber constraint we can also work with a wavefunction with a well-defined phase, whichis essential to the physics of the Josephson effect. The BCS state is a wavefunction of spin-singlet pairs. Triplet states are also possible but were considered later, in the context ofsuperfluid 3He.



BCS also made the simplification of using a reduced Hamiltonian

H =∑kσ

εknkσ +∑k`

Vk`c†k↑c†−k↓c−`↓c`↑. (30.25)

The interaction term in the BCS Hamiltonian includes only terms that scatter one electronpair (`,−`) into another (k,−k). This is a gross simplification of the actual electron–electroninteraction but captures the essential physics of the phonon-mediated pairing interaction.

BCS employed a variational method to optimize the choice of uk and vk: they used a Lagrangemultiplier (the chemical potential µ) to fix the average particle number and minimized theexpectation value of H − µN in the BCS ground state by adjusting uk and vk. We will usean algebraically simpler method and solve the problem by canonical transformation.

30.4 The Bogoliubov transformation

The Bogoliubov method works by diagonalizing a mean-field approximation to the Hamil-tonian and in the process finds the operators that generate the excited states of the system.It is therefore better than the BCS method at dealing with excitations, which are necessaryin order to understand experiments such as spectroscopies, tunneling and thermodynamics.

The BCS wavefunction |BCS〉 is a phase-coherent superposition of pairs, without fixed pairnumber. Operators such as c−k↓ck↑ can therefore have nonzero expectation value. Let

bk = 〈BCS|c−k↓ck↑|BCS〉. (30.26)

This would average to zero in a normal metal. We can write

c−k↓ck↑ = bk +(c−k↓ck↑ − bk

), (30.27)

where the term in brackets is a small fluctuation term, and use this to rewrite the BCShamiltonian in the following approximate form:

H =∑kσ

ξknkσ +∑k`

Vk`

(c†k↑c

†−k↓b` + b†kc−`↓c`↑ − b

†kb`

), (30.28)

where ξk = εk − µ. This mean-field approximation effectively turns the quartic interactionterm into a quadratic term, making the hamiltonian bilinear. It can therefore be diagonalizedby a canonical transformation. To see this, let

∆k = −∑`

Vk`b` (30.29)

= −∑`

Vk`〈c−`↓c`↑〉. (30.30)

The hamiltonian can then be rewritten

H =∑kσ

ξknkσ −∑k

(∆kc

†k↑c†−k↓ + ∆†kc−k↓ck↑ −∆kb

†k

). (30.31)


30.4. THE BOGOLIUBOV TRANSFORMATION 265

Bogoliubov and Valatin independently showed that this could be diagonalized by the trans-formation

ck↑ = u∗kγk0 + vkγ†k1 (30.32)

c†−k↓ = −v∗kγk0 + ukγ†k1. (30.33)

γk0 participates in destroying an electron |k ↑〉 and creating an electron | − k ↓〉. In bothcases the momentum is reduced by k and Sz is reduced by ~/2. γk0 on the other handincreases momentum by k and Sz by ~/2.

The procedure is to substitute the new operators into the hamiltonian. The constraint onuk and vk (|uk|2 + |vk|2 = 1) makes this a canonical transformation — the canonical com-mutation relations of the electron operators are preserved. This means that the Bogoliubovquasiparticles are fermions.

(a) -6 -4 -2 2 4 6

0.2

0.4

0.6

0.8

1

vk

2 uk

2

ξ/Δ (b) -4 -2 2 4

0.2

0.4

0.6

0.8

1

ξ/Δ

Fermi function at Tc

vk

2 at T = 0

Figure 30.3: (a) Coefficients of the variational wavefunction, u2k and v2

k. (b) v2k at T = 0 and

the Fermi function at Tc.

The condition that the hamiltonian be diagonal places additional constraints on uk and vk:

|vk|2 = 1− |uk|2 =1

2

(1− ξk

Ek

), (30.34)

where

Ek =√ξ2k + |∆k|2. (30.35)

uk and vk are plotted in Fig. 30.3. The occupation of pair states at T = 0 is not verydifferent from the Fermi–Dirac function at T = Tc.

The hamiltonian, in diagonal form, is

H = const +∑k

Ek

(γ†k0γk0 + γ†k1γk1

). (30.36)

The γk are the elementary excitations, with energy Ek. ∆k is the minimum excitation energy,and is referred to as the ‘energy gap’. The energy gap is best seen in the density of stateswhich can be calculated as follows. The states of the superconducting phase and the normalphase must exist in one to one correspondence. Therefore

Ds(E)dE = Dn(ξ)dξ. (30.37)



The density of states in the normal state is constant, Dn(ξ) = D(0), so

D(E) = D(0)dξ

dE=

E√E2 −∆2

. (30.38)

The energy gap shows up in many experiments, giving an activated temperature dependenceto thermodynamic properties, and a gap in the optical conductivity and tunneling densityof states.

-4 -2 2 4

1

2

3

4

ξ/Δ

|ξ|

Ε = (ξ2 + Δ2)1/2

Figure 30.4: Excitation energies of the normal and superconducting states.

-2 -1 1 2

1

2

3

4

0

Ds(Ε)/D(0)

Ε/∆

Figure 30.5: The density of states Ds(E) of a BCS superconductor. There is gap 2∆ widethat appears in tunneling measurements and the optical conductivity.


Lecture 31

Superconductivity III — exoticpairing

31.1 Conventional superconductors

BCS theory has been fantastically successful at describing the physical properties of thevast number of materials that were known to superconduct at the time of its inception.Most of these are elemental superconductors, but the theory, in a form modified to allow forstrong spatial variations of the pairing field, also describes the alloy superconductors thatbecame technologically important with the advent of high field superconducting magnets.The success of the BCS theory is owed to two things: the inspired guess, by Schrieffer,of a good approximate form for the mean-field, ground-state wavefunction; and the cleanseparation of phonon and electronic energy scales (the Debye energy ~ωD and the Fermienergy εF, respectively) that allows for a perturbative treatment of the phonon-mediatedinteraction.

This phonon mechanism lies at the heart of the pairing theory of conventional supercon-ductors. It leads to an attractive interaction that is approximately isotropic in k-space, andopens a correspondingly isotropic energy gap on the Fermi surface.1

31.2 Pairing glue

While important and ubiquitous, phonons are not the only bosonic mode that emerges whenatoms are combined to form an electronic solid. It is therefore interesting to ask if otherexcitations can play the role of pairing glue, giving rise to exotic types of superconductivity.Some of the collective modes one might consider include:

1Even in the elemental superconductors, strong coupling effects and crystalline anisotropy lead to somevariation of the energy gap magnitude over the Fermi surface. The energy gap in tin, for instance, hasone of the stronger variations of all classic elemental superconductors, with a modulation in magnitude ofabout 30%. Transition metal superconductors, such iridium, which have multiple Fermi surface sheets forthe partially filled s and d bands, also show strong variation in |∆| from sheet to sheet.


268 LECTURE 31. SUPERCONDUCTIVITY III — EXOTIC PAIRING

Strachey, J.; ed. Strachey, J.) (Hogarth, London, 1957).

2. Freud, S. The Standard Edition of the Complete Psychological

Works of Sigmund Freud Vol. 6 (transl. Strachey, J.; ed. Strachey,

J.) (Hogarth, London, 1960).

3. Anderson, M. C. & Green, C. Nature 410, 366–369 (2001).

4. Bjork, R. A., Bjork, E. L. & Anderson, M. C. in Intentional

Forgetting: Interdisciplinary Approaches (eds Golding, J. M. &

MacLeod, C. M.) 103–137 (Lawrence Erlbaum, Mahwah, NJ,

1998).

5. Houghton, G. & Tipper, S. T. Brain Cognition 30, 20–42 (1996).

6. Freud, S. The Standard Edition of the Complete Psychological

Works of Sigmund Freud Vol. 1 (transl. Strachey, J.; ed. Strachey,

J.) (Hogarth, London, 1957).

7. Anderson, M. C. J. Aggression Maltreatment Trauma (in the press).

the electrons in a Cooper pair produces tinyharmonics — wiggles and bumps — in theenergy distribution of the electrons that canbe detected using electron-tunnelling spec-troscopy3. In the late 1960s, McMillan andRowell4 confirmed the long suspected role ofphonons in conventional superconductors.They showed that the phonon energy spec-trum measured by neutron scattering fromsuperconducting mercury and lead agreedwith the spectrum that they had deducedfrom their precise electron-tunnellingmeasurements. In so doing, they also con-firmed that the Eliashberg refinement ofBCS theory was accurate to about 1%.

The 1970s and 1980s led to the discoveryof superconductivity in two new, unconven-tional classes of material, the heavy-fermion5,6

and high-temperature7 superconductors.Conventional superconductivity is sup-pressed by the tiniest concentration of magnetic atoms, but the unconventionalsuperconductors contain a dense array ofmagnetic atoms, which appear to be activelyinvolved in electron pairing. This led physi-cists to consider a new kind of ‘magneticallymediated’ superconductivity in which thequanta that glue electrons into pairs arederived from magnetic fluctuations8–12.

Like electrons in atoms, the character ofthe electron waves in Cooper pairs is labelleds, p or d, depending on the angular momen-tum of the electrons. Phonon-mediatedpairing favours the formation of s-wave electron pairs, but magnetic fluctuationsfavour p- or d-wave pairs (Fig. 1). This isbecause magnetic fluctuations develop inmetals with strongly interacting electrons:p- or d-wave pairs form in direct response,lowering the mutual repulsion between theelectrons by keeping them further apart. Inthe unconventional superconductors, neu-tron scattering experiments have revealedlarge fluctuations in magnetic spin at lowtemperatures, which could mediate electronpairing, and several other experiments pro-vide good evidence for the existence of p- ord-wave electron pairs in these materials.

But despite strong circumstantial evi-dence for magnetically mediated super-conductivity, definitive proof that spin fluctuations are the glue for p- or d-wave pair-ing has been hard to come by. This is partlybecause magnetic fluctuations are far moredifficult to characterize than phonons, butalso because, unlike phonons, magnetic fluc-tuations interact with each other, making thetheoretical description of magnetically medi-ated pairing a far more complex problem.

Sato et al.1 provide new evidence for this magnetic glue. They studied the heavy-fermion superconductor UPd2Al3, so calledbecause the conducting electrons in itacquire very large effective masses, oftenhundreds of times that of a free electron.UPd2Al3 is one of a handful of heavy-fermionsuperconductors in which both antiferro-

news and views

320 NATURE | VOL 410 | 15 MARCH 2001 | www.nature.com

Physicists are fascinated by the ‘glue’ thatholds matter together. According toquantum mechanics, forces are not

instantaneous, but are transmitted by theexchange of tiny packets of energy, or ‘quan-ta’, between particles. So electromagneticforces are produced by the exchange of pho-tons, and the strong forces that keep quarkstightly bound inside protons are driven bythe exchange of quanta suggestively calledgluons.

But inside metals, different sorts of quanta become possible, and electrons thatexchange these quanta can experience newkinds of attractive forces that profoundlychange their properties. The most celebratedexample of this is superconductivity —which occurs when electrons at low temper-ature pair up and flow without resistance.Now, on page 340 of this issue, Sato et al.1

reveal a magnetic origin for the ‘glue’

between electrons in an unconventionalsuperconductor.

In the late 1950s, Bardeen, Cooper andSchrieffer (BCS) showed that superconduc-tivity involves the formation of bound pairsof electrons, named Cooper pairs. BCSargued that the electron pairs were ‘gluedtogether’ by tiny deformations in the crystallattice, called phonons, that accompany theelectrons’ motion. But although phononswere implicated in superconductivity manyyears before the BCS theory, it was not untilthe 1960s that it became possible to defini-tively identify them as the glue in conven-tional superconductivity.

In the early 1960s, the Russian physicistEliashberg2 showed how the electron-pairingforces created by phonons could be elegantlyincorporated into BCS theory using a set ofequations that now bear his name. It turnsout that the exchange of phonons between

Superconductivity

Magnetic glue exposedPiers Coleman

Overcoming electrons’ mutual repulsion is the key to superconductivity. In simple metals, the forces that bind these charged particles in pairs arewell understood, but what is the glue in other superconductors?

Figure 1 Two ways to make a metallic superconductor. a, In conventional superconductivity the glueresponsible for binding electrons into superconducting pairs is derived from the exchange of ‘quanta’of lattice vibrations — phonons, shown here as a wave passing through the lattice atoms — that bind electrons into ‘s-wave’ pairs. b, Sato et al.1 provide new evidence that, in the unconventionalsuperconductor UPd2Al3, the glue responsible for the electron pairing derives from the exchange ofmagnetic quanta — magnetic excitons — that bind the electrons into ‘d-wave’ pairs (blue arrowsindicate direction of spin).

s-wavepairs

d-wavepairs

a

b

Phonon wave

Magneticexciton wave

Electron

Electron

© 2001 Macmillan Magazines Ltd

Figure 31.1: In conventional superconductors the exchange of phonons binds electrons intos-wave Cooper pairs. Magnetic excitations provide a possible route to exotic pairing states,with antiferromagnetic spin waves giving rise to d-wave Cooper pairs. (After Coleman[33].)

• ferromagnetic spin waves. In a metal, direct exchange causes the spin of an electron tolocally polarize the electron gas surrounding it. In a ferromagnet this effect is strongand leads to a state with a uniform, ordered magnetic moment — a spontaneousbreaking of time reversal symmetry. The Goldstone boson2 of such a broken symmetryis a ferromagnetic magnon that takes the form of a wavelike variation in the directionof the magnetization. The lowest energy spin waves in this case are located nearq = 0, like phonons, but disperse quadratically in wavevector, due to the softness ofthe exchange coupling, ∼ σ · σ′, which goes like the cosine of the angle between thespins. In a nascent ferromagnet (e.g. near a ferromagnetic quantum critical point)there is no spontaneous magnetic order, but the exchange coupling of the electronspin to the surrounding electron fluid is still strong. If this induced polarization decaysslowly, it can mediate an interaction with a second electron. In this case the interactionfavours parallel alignment of the electron spins, leading to spin-triplet pairing. Overallfermionic antisymmetry then requires the orbital wavefunction to have odd angularmomentum. This type of pairing is important in superfluid 3He.

• antiferromagnetic spin waves. An antiferromagnet consists of an alternating ordering

2Goldstone’s theorem states that whenever a continuous symmetry is broken, a new massless scalarparticle appears in the spectrum of excitations. Liquids break Galilean symmetry, and the Goldstone modeis a longitudinal acoustic phonon, dispersing like ω = cq for small q, with vanishingly small energy asq → 0. Solids also break Galilean symmetry, but the presence of a lattice introduces shear stiffness andgives separate transverse and longitudinal phonons as the Goldstone bosons. Magnets break spin-rotationsymmetry, leading to the emergence of magnons as the new, low energy bosonic mode.


31.3. ANDERSON’S THEOREM 269

of magnetic moments, with the spatial modulation occurring at some wavevector Q.The antiferromagnetic interaction favours antiparallel alignment of spins, binding theelectrons of the Cooper pair into a spin singlet, as in conventional superconductors.Overall fermion antisymmetry then requires a spatially even wavefunction, leading topairing in a even angular momentum channel. However, the antiferromagnetic interac-tion is strongly anisotropic in k-space, with the static spin susceptibility (analogous tothe ω → 0 limit of the electron density response function χ(q, ω)) strongly peaked nearq = Q. Moreover, the interaction is fundamentally repulsive, but we will see belowhow this can be turned to the superconductor’s benefit by pairing in higher angularmomentum channels.

• other excitations. Excitations such as electric quadrupolar fluctuations of orbital orderand unscreened current–current fluctuations have been proposed as very exotic pairingmechanisms.

31.3 Anderson’s theorem

In the presence of disorder, Bloch waves of well-defined momentum cease to be good eigen-states. We can think of disorder as giving a finite range to the wavefunction, imposing anexponential envelope with a characteristic decay length equal to the mean free path `. Usinguncertainty arguments, we see that this imparts a finite spread in wavevector, ∆k = 1/`, tothe electron state in k-space. The stronger the disorder, the more the electron state spreadsout around the Fermi surface.

Surprisingly, this smearing has little effect on conventional superconductivity. Substitutionalimpurity studies show that after an initial decline in Tc, addition of further impurities givesno further suppression of transition temperature. In fact, superconductivity is known tooccur in materials such amorphous MoGe, in which the atoms themselves have glass-like,not crystalline, arrangements.

The explanation of how this can occur was given by P. W. Anderson in 1959[34]. In oneof his many strokes of genius, he realized that the (k↑,−k↓) pairing of BCS theory wasjust a special case of a more general pairing of time-reversed states. In the absence ofperturbations from magnetic fields and magnetic impurities, the hamiltonian of a solid hastime reversal symmetry, irrespective of the amount of disorder. That is, for every eigenstateψn(r), the conjugate wavefunction ψ∗n(r) will also be an eigenstate, and will have the sameenergy. The time reversed states will have opposite spin, and are the states that combine toform Cooper pairs in the presence of strong disorder. These states are inherently smeared-out in k-space. For conventional materials this is not harmful to the superconductivity: thephonon-mediated interaction is quite isotropic, and the effect of disorder is simply to averagethe superconducting energy gap over the Fermi surface. However, as we will see below,unconventional superconductors form Cooper pairs in finite angular momentum channels.These states have the property that the energy gap changes sign over the Fermi surface.Anderson’s results still apply, but the smearing of the energy gap rapidly causes it to averageto zero, destroying the superconductivity. This makes unconventional superconductivityexceedingly delicate, and helps explain why it took so long to be observed.



Figure 31.2: The doping–temperature phase diagram of the cuprate superconductors. (AfterBonn[35].)

31.4 Unconventional pairing in specific materials

Rather than survey the full range of possible exotic superconducting states, which is vastand has been thoroughly explored by theorists, we will look at a number of specific examplesof materials that exhibit unconventional superconductivity. Our definition of unconventionalsuperconductivity will be taken to mean pairing states that break symmetries in additionto the broken gauge symmetry found in all superconductors. We will begin with the hightemperature superconductors, not because their transition temperatures are highest, butbecause their form of unconventionality, the d-wave pairing state, is simplest to understandand lays the groundwork for similar pairing states in the heavy fermion materials. We willthen look at heavy fermion superconductivity, which is remarkable because it occurs in thevicinity of magnetic quantum critical points.

31.4.1 Cuprate superconductivity

Discovered in 1986, the cuprate superconductors hold the record for highest Tc, with su-perconductivity observed under pressure at temperatures up to 160 K. Many aspects of thecuprates, particularly their strange normal-state properties, remain poorly understood. Wewill focus here on the superconductivity. The details of the mechanism are still a matter ofgreat contention and debate, but what is clear is that superconductivity emerges from thephysics of strong Coulomb repulsion.


31.4. UNCONVENTIONAL PAIRING IN SPECIFIC MATERIALS 271

La(22x)SrxCuO4 system, for example.For doping levels above xc, various

forms of local or incommensurate magne-tism survive the loss of commensurate an-tiferromagnetic order. At intermediate lev-els, the dynamical properties are those of aspin glass. As the doping increases, thedegree of local magnetic order appears todepend on the material system and on sam-ple purity. At doping levels greater thanoptimal for Tc, magnetic correlations final-ly become negligible. The extraordinarypersistence of antiferromagnetism into thesuperconducting phase has been a long-standing puzzle. However, in the last fewyears a startling new picture has emerged.The revelations came largely from neutronscattering results, made possible by thesynthesis of large single crystals with con-trolled values of x. The new phenomenon isinhomogeneous spin and charge ordering,more colloquially known as “stripes.”

Neutron scattering in the Neel orderedstate at x 5 0 is relatively simple: Thetwo-sublattice structure (Fig. 1) leads toantiferromagnetic Bragg peaks at wavevectors Q 5 61⁄2,61⁄2 (in units of 2p/a,where a is the lattice constant). The loca-tion of scattering peaks in momentum spaceis illustrated in Fig. 3A. The evolution ofthe spin dynamics with doping was consid-erably more difficult to understand. Mea-surements in 1989 and the early 1990s,primarily in La(22x)SrxCuO4, found thatbroadened versions of the antiferromag-netic Bragg peaks persist in samples withthe highest Tc’s (6 –9). Further, the spincorrelations were found to be incommensu-

rate: The single peak found in the insulatorsplits into four (7–9), each displaced fromQ by a small amount d (Fig. 3A). Manyworkers had assumed that the spin fluctua-tions responsible for the scattering could bedescribed by linear response theory andcorresponded to a sinusoidal spin densitywave (SDW) fluctuating slowly in spaceand time. However, scattering at Q 6 d canalso arise from a spin wave that is locallycommensurate but whose phase jumps by pat a periodic array of domain walls termed“antiphase boundaries.” Such a spin waveis shown in Fig. 3B.

Tranquada and collaborators found evi-dence for the latter possibility in a closelyrelated material system in which Nd replacessome of the La atoms in La(22x)SrxCuO4 (10,11). The introduction of Nd changes the di-rection of a distortion of the crystal structure(arising from slight rotation of the CuO6 oc-tahedra) from diagonal to parallel to theCu-O bond. This distortion causes the spinfluctuations to condense into a static SDW.The neutron scattering spectrum of theSDW consists of Bragg peaks that could beanalyzed in detail; the analysis confirmsthat the ordered spin structure is the oneshown in Fig. 3B. In this structure thevacant sites introduced by doping reside atantiphase boundaries, forming chargedstripes. If there is one vacancy for everytwo sites along the stripe, then the distancebetween stripes is a/2x (where a is theCu-Cu separation). This distance gives thecharge periodicity; the antiphase propertymeans that the spin period is twice thisvalue, or a/x. This spin periodicity causes

Bragg peaks displaced from Q by d 5 x; thecharge periodicity leads to Bragg peaks dis-placed from fundamental lattice reflectionsby 2x. Both features were observed by Tran-quada et al., and thus charged stripes, the“latest installment in the mystery series enti-tled ‘High-Tc Superconductivity’ ” (12), werediscovered.

The equality of incommensurability anddoping, d 5 x, follows naturally from thequarter-filled stripe model and has not beenexplained in any other way. The work ofTranquada et al. stimulated a closer investi-gation of the inelastic peaks in compounds inwhich the SDW is not static. In 1998 Yamadaet al. reported that d 5 x in underdopedNd-free La(22x)SrxCu2O4, as was found forthe spin Bragg peaks in the Nd-doped system(13). (This relation is deduced from the spinpeak; the charge peak is too weak to see ininelastic scattering measurements.) Thisstrongly suggests that the inelastic incommen-

La3+

Cu2+

O2-

ab

c

A B

Fig. 1. (A) Crystal structure of La2CuO4, the “parent compound” of the La(2–x)SrxCuO4 family ofhigh-temperature superconductors. The crucial structural subunit is the Cu-O2 plane, whichextends in the a-b direction; parts of three CuO2 planes are shown. Electronic couplings in theinterplane (c) direction are very weak. In the La2CuO4 family of materials, doping is achieved bysubstituting Sr ions for some of the La ions indicated, or by adding interstitial oxygen. In otherfamilies of high-Tc materials (e.g., YBa2Cu3O61x) the crystal structure and mechanism of doping areslightly different, but all materials share the feature of CuO2 planes weakly coupled in thetransverse direction. (B) Schematic of CuO2 plane, the crucial structural subunit for high-Tcsuperconductivity. Red arrows indicate a possible alignment of spins in the antiferromagneticground state of La2CuO4. Speckled shading indicates oxygen “ps orbitals”; coupling through theseorbitals leads to superexchange in the insulator and carrier motion in the doped, metallic state.

Fig. 2. Schematic phase diagram of high-tem-perature superconductors. The shaded red areaindicates the region in which long-range com-mensurate antiferromagnetic order (of the typeshown by the red arrows in Fig. 1) occurs. Theshaded blue area indicates the region in whichsuperconducting long-range order occurs. Thecarrier concentration at which the supercon-ducting transition temperature (upper bound-ary of blue region) is maximal is conventionallydefined as optimal doping. Materials with lowerand higher carrier concentrations are referredto as underdoped and overdoped, respectively.In the regime between the commensurate an-tiferromagnetic phase and the superconductingphase, a different, more complicated magneticorder occurs; this is discussed more fully in thetext, and is shown here as the green region.This order is observed to coexist with super-conductivity in some materials, but whetherthe two phases exist in spatially distinct regionsof a sample or coexist in the same region is stillnot fully clear. The shaded lines indicate qual-itatively defined crossover temperatures belowwhich materials, although still thermodynami-cally in the normal phase, exhibit new behav-iors discussed in more detail in the text. (Thisfigure shows the phase-diagram obtained by“hole doping.” A few electron-doped materialshave been made; because of sample prepara-tion difficulties, their properties are less welldetermined than those of the hole-dopedmaterials.)

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Figure 31.3: The superconducting cuprates consist of layers of CuO2 planes. The copperspins of the undoped parent compound order antiferromagnetically[36].

The cuprates are layered materials, consisting of stacks of CuO2 planes, as shown in Fig-ure 31.3. The materials differ in the number of planes per unit cell, and in the types ofcations found between the planes. However, the CuO2 plane is common to all high Tc super-conductors, and there is ample evidence to suggest that it is the seat of superconductivity.As a result, most attempts to understand the superconductivity start with an isolated CuO2

layer.

The parent compounds of the cuprate superconductors are Mott insulators, which orderantiferromagnetically below TN ≈ 300 K. This physics is driven by strong Coulomb repulsion.As we saw in Lecture 20, Coulomb repulsion leads to the splitting of the conduction bandinto two Hubbard bands. At half filling, which applies to the cuprate parent material, thelower Hubbard band is filled and the upper Hubbard band is empty, creating an insulator.An antiferromagnetic interaction between the localized electrons on neighbouring sites arisesfrom superexchange.3 The spin arrangement in the antiferromagnetic state is shown inFigure 31.3. This can be thought of as a spin density wave with Fourier components Q =(±π/a,±π/a).

As mobile holes (or electrons) are doped in the cuprate parent compound, the antiferro-magnetic state weakens and disappears, and is replaced by superconductivity. Signaturesof the antiferromagnetic order persist to superconducting dopings, showing up as strongpeaks in the spin response function near Q = (±π/a,±π/a). Some theories use these ‘spinfluctuations’ as the glue that binds charge carriers into high Tc Cooper pairs.

To see how this works, we look again at BCS theory, generalized to allow for the possibilitythat the energy gap ∆k be a strongly varying function of k. In the last lecture we studied

3Superexchange is due to the small energy lowering that occurs as a fluctuation in second order perturba-tion theory, if the localized electron can tunnel to a neighbouring site and back again. This is only allowedby the Pauli exclusion principle if the electron on the neighbouring site has antiparallel spin.



Figure 31.4: Magnetic excitations at wavevectors Q = (±π/a,±π/a) couple parts of theFermi surface having different sign of the energy gap. This turns a repulsive interaction intoa means of binding finite angular momentum Cooper pairs.

the Bogoliubov approach, in which an average pair field

bk = 〈BCS|c−k↓ck↑|BCS〉 (31.1)

was introduced in order to recast the hamiltonian as a mean field theory. bk is intimatelyconnected to the opening of the energy gap. The BCS gap parameter

∆k = −∑`

Vk`b` (31.2)

expresses the effect of the pairing fields at wavevectors `, acting through the matrix elementsVk`, on the magnitude of the energy gap ∆k. This process must be treated self consistently,leading to the BCS gap equation

∆k = −∑`

∆`

2E`Vk`. (31.3)

Note the overall minus sign. The excitation energy E` =√ξ2` + ∆2

` is inherently positive. Foran attractive interaction (Vk` < 0) the gap equation yields an isotropic solution. Repulsiveinteractions at first sight appear to give ∆k = 0, but this is only the case if the interactions areuniformly repulsive. If the Vk` are sufficiently anisotropic, the gap equation allows solutionsin higher angular momentum channels.

To picture how this applies in the case of the cuprates, we imagine that Vk` is positive(repulsive) and sharply peaked at momentum transfers q = k − ` equal to the dominantFourier components of the antiferromagnetic state, Q = (±π/a,±π/a). We see in Figure 31.4


31.4. UNCONVENTIONAL PAIRING IN SPECIFIC MATERIALS 273REVIEW ARTICLE

8.0

4.0

00 20 40 60 80

Temperature (K)

N (E )

EΔo

1/λ2

(μm

–2)

Figure 3 Measurement of the superfluid density derived from microwave measurements. TheLondon penetration depth can be measured by determining the depth that microwave fieldspenetrate into the surface of a superconductor. The inverse square of this length scale is a measureof superfluid density, or phase stiffness, which sets a superconductor’s resistance to variation in thephase of the superconducting wavefunction. The linear behaviour at low temperatures is aconsequence of the linear density of states N on energy E (inset) of a dx2−y2 superconductor with acylindrical Fermi surface. The energy gap of this state has a maximum 0 for momenta in thedirection of the Cu–O bonds, but nodes in the diagonal directions that give rise to the linear densityof states at low energy.

candidates: the organic superconductors,heavy-fermion superconductors and most recentlystrontium ruthenate. The primary issue that makesthis quest difficult is sample quality. The s-wave,spin-singlet superconductors are in a very specialstate of matter, protected from the influence ofdefects by Anderson’s theorem37. They possess anenergy gap in their excitation spectrum that is littleaffected by non-magnetic impurities because itbreaks no symmetries of the materials’ crystalstructure. Non-s-wave superconductors breakadditional symmetries, possess energy gaps thatchange sign on reflection or rotation, and alsopossess nodes on the Fermi surface. When the sign ofthe superconducting wavefunction can changethrough a scattering event, impurities become highlydisruptive, masking many of the low-temperatureproperties, or even killing superconductivity entirely.

The cuprates offer one of the great successes inthe identification of a new pairing state6. First, NMRmeasurements of the Knight shift showed that thespin susceptibility falls rapidly below Tc, proof thatthe superconducting pairs are not in a spin-tripletstate38,39. This requires the orbital part of the pairingwavefunction to be symmetric, either the l = 0s-wave state or some higher angular momentum suchas l = 2 (d-wave). In 1993, microwave measurementsof the temperature dependence of the Londonpenetration depth, l, showed linear temperaturedependence consistent with a d-wave density ofstates40. Figure 3 shows 1/l2, referred to as the

superfluid density, in analogy to the two-fluid modelof superfluid 4He. More correctly, it is not a densitybut a tensor quantity known as phase stiffness thatdetermines a superconductor’s resistance to variationin the phase of the superconducting order parameter.As the temperature increases, quasiparticleexcitations deplete the superfluid density until it fallsto zero when the normal state is reached. A d-wavestate would possess lines of nodes on the cylindricalFermi surface expected for these layered materials,and there would thus be quasiparticle excitationspossible down to zero energy, with a density of statesvarying linearly with energy. The linear temperaturedependence of 1/l2(T) was masked by impurities inearlier measurements, something subsequentlyconfirmed when it was shown that substituting Znimpurities on the Cu site could change thelow-temperature power laws41–43. It took six years forthe field to produce samples of sufficient quality tofind the d-wave superconducting state — nature hadgiven us a delicate pairing state wrapped in a verydifficult materials problem. The drive to uncover theintrinsic behaviour of these materials has led to thedevelopment of materials up to 99.999% pure, andwith high crystalline perfection44,45. Although not atthe level that has been achieved over decades ofsemiconductor development, this level of control isexceptional in transition metal oxides.

Further identification came from experimentaltechniques far beyond those used in the era of BCS.One of these is angle-resolved photoemissionspectroscopy (ARPES), which uses the photoelectriceffect to map the energy and momentum of filledstates near the Fermi energy46. The technique isparticularly suited to quasi-two-dimensionalmaterials, and had an energy resolution in the early1990s that was close to that necessary to see thesuperconducting energy gap of the copper oxides.The extraordinary development of ARPES wasfuelled by the prize of observing the variation of anunconventional energy gap around a Fermi surface, aquest that led to an improvement in resolution bymore than an order of magnitude in less than adecade. The material for the job wasBi2Sr2CaCu2O8+δ, which possesses a cleavage planebetween two weakly van der Waals-bound BiOlayers, providing an ideal surface for spectroscopy,with no surface states or reconstruction. The energygap mapped out by ARPES near the Fermi energy(Fig. 4) was indeed consistent with nodes in the gapfunction, which ARPES showed lay in the [110]direction, rotated 45 from the orientation of thesquare CuO2 lattice47,48 — a possible dx2−y2 state.

The finish to the identification of the new statewas an ingenious experiment that directly observedthe extra broken symmetry. A film was grown on atricrystal substrate, one where the three crystalorientations of the fused substrate frustrated theorientation of the dx2−y2 state49. In trying to followthe orientation of the substrate lattice, the dx2−y2 statefinds itself changing sign on rotation through 360

about the axis at which the three orientations met. Asuperconducting ring patterned around this axissolves the frustration by spontaneously running asupercurrent that sustains half of a superconducting

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Figure 31.5: The point nodes in a d-wave superconductor give rise to a linear energy depen-dence of the density of quasiparticle states. This can be probed through the temperaturedependence of the superfluid density 1/λ2(T ). (After Bonn[35].)

that these wavevectors connect parts of the Fermi surface near (±π/a, 0) and (0,±π/a). Bychoosing an energy gap that changes sign along the zone diagonals, we can obtain a finitesolution to the gap equation. Such a state is called a dx2−y2 state, and one of its definingcharacteristics is the presence of cone-shaped excitation-energy surfaces that vanish at point-like nodes on the diagonals. These energy surfaces are similar to those in graphene, and have acharacteristic linear energy dependence of the density of states (see Figure 31.5) very differentfrom the gapped density of states of a conventional superconductor. The nodal quasiparticleexcitations dominate the low temperature thermodynamic response, and are seen clearlyin experiments such as penetration depth, heat capacity and thermal conductivity. Otherexperiments, using clever arrangements of Josephson junctions, are sensitive to the phase ofthe Cooper pair wavefunction, and have shown that it really does change sign along the zonediagonals.

As a final point, we can think of the d-wave state as providing an alternative means for thecharge carriers to minimize the effects of Coulomb repulsion. By forming a Cooper pair withfinite angular momentum, the carriers set up a centrifugal barrier, preventing them fromsimultaneously being at the same place.



Figure 31.6: In CePd2Si2, superconductivity appears in a narrow range of pressure whenantiferromagnetism is suppressed[37].

31.4.2 Heavy fermion superconductivity

Heavy fermion metallic states emerge in materials with partially filled f orbitals. These aretypically intermetallic componds containing cerium or uranium, although a number of othermaterials, based on atoms such as praseodymium and ytterbium, are now known. At hightemperatures the f electrons act as localized magnetic moments, which can be inferred fromthe spin susceptibility χ(T ) ∼ 1/T and from the strong, Kondo-like magnetic scattering inthe electrical resistivity. At low temperatures the f electrons hybridize with the surroundingconduction electrons to form a coherent but narrow band of heavy Landau quasiparticles. Itis the narrowness of these bands, with correspondingly large densities of states, that leadsto the term heavy fermion.

Heavy fermion compounds are interesting enough on their own, without the appearanceof superconductivity. (The Kondo-lattice problem, for instance, remains a major, opentheoretical topic.) However, many of these materials are also superconducting, and thesuperconductivity appears in an unusual way, often tied to a magnetic quantum criticalpoint. In the heavy fermion compounds, the narrow-band, nearly localized f electrons makethe balance between kinetic and potential energies very delicate. In many of the compounds,under ambient conditions, this balance tips in favour of potential energy, resulting in theformation of magnetic states at low temperatures. The balance can be tipped back in favourof kinetic energy by the application of pressure. This increases the density, reducing thespacing between atoms and increasing orbital overlap. The tendency to form a magnetic state


31.4. UNCONVENTIONAL PAIRING IN SPECIFIC MATERIALS 275

is reduced, driving the ordering temperature smoothly to zero, as shown in Figure 31.6. Thisis called a quantum phase transition, because we have made a new ground state energeticallyfavourable by tuning a parameter in the hamiltonian. Temperature and entropy are notimportant here — the quantum phase transition occurs at T = 0.

Near the quantum critical point, even in the absence of magnetism, the electronic medium isreadily polarized magnetically. Magnetic fluctuations are strong — their excitation energyis very small. This then provides a new mechanism for mediating superconducting pairing.Each heavy quasiparticle is surrounded by an electronic fluid that acts as a spin liquid. Thespin of the quasiparticle couples to the fluid and polarizes it magnetically. The responseof the fluid is strongest at particular wavevectors — near q = 0 for a nearly ferromagneticmetal, and at finite Q for an incipient antiferromagnet. The quasiparticle then propagatesoff, but the polarization it induces is left behind and diffuses away slowly. A second heavyquasiparticle can then pass through the polarized spin liquid and experience an attraction.

Superconductivity has now been observed in heavy fermion compounds at both antiferromag-netic and ferromagnetic quantum critical points. In the antiferromagnetic case, the pairingsymmetry is thought to be d-wave, as in the cuprates. The ferromagnetic case is particularlyexotic: pairing must be occurring in a spin triplet state, and it is not immediately clear howthe superconductivity coexists spatially with the ferromagnetism.


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