Dedication iii

List of Contributors xiii

Preface xv

Acknowledgements xix

Introduction xxiii

References xxiv

1 Semiconductors and heterostructures 1

1.1 The mechanics of waves 1

1.2 Crystal structure 3

1.3 The effective mass approximation 5

1.4 Band theory 5

1.5 Heterojunctions 7

1.6 Heterostructures 7

1.7 The envelope function approximation 10

1.8 Band non-parabolicity 11

1.9 The reciprocal lattice 13

Exercises 16

References 17

2 Solutions to Schrödinger's equation 19

2.1 The infinite well 19

2.2 In-plane dispersion 22

2.3 Extension to include band non-parabolicity 24

2.4 Density of states 26

2.4.1 Density-of-states effective mass 28

2.4.2 Two-dimensional systems 29

2.5 Subband populations 31

2.5.1 Populations in non-parabolic subbands 33

2.5.2 Calculation of quasi-Fermi energy 35

2.6 Thermalised distributions 36

2.7 Finite well with constant mass 37

2.7.1 Unbound states 43

2.7.2 Effective mass mismatch at heterojunctions 45

2.7.3 The infinite barrier height and mass limits 49

2.8 Extension to multiple-well systems 50

2.9 The asymmetric single quantum well 53

2.10 Addition of an electric field 54

2.11 The infinite superlattice 57

2.12 The single barrier 63

2.13 The double barrier 65

2.14 Extension to include electric field 71

2.15 Magnetic fields and Landau quantisation 72

2.16 In summary 74

Exercises 74

References 76

3 Numerical solutions 79

3.1 Bisection root-finding 79

3.2 Newton-Raphson root finding 81

3.3 Numerical differentiation 83

3.4 Discretised Schrödinger equation 84

3.5 Shooting method 84

3.6 Generalized initial conditions 86

3.7 Practical implementation of the shooting method 88

3.8 Heterojunction boundary conditions 90

3.9 Matrix solutions of the discretised Schrödinger equation 91

3.10 The parabolic potential well 94

3.11 The Pöschl-Teller potential hole 98

3.12 Convergence tests 98

3.13 Extension to variable effective mass 99

3.14 The double quantum well 103

3.15 Multiple quantum wells and finite superlattices 104

3.16 Addition of electric field 106

3.17 Extension to include variable permittivity 106

3.18 Quantum confined Stark effect 108

3.19 Field-induced anti-crossings 108

3.20 Symmetry and selection rules 110

3.21 The Heisenberg uncertainty principle 110

3.22 Extension to include band non-parabolicity 113

3.23 Poisson's equation 114

3.24 Matrix solution of Poisson's equation 118

3.25 Self-consistent Schrödinger-Poisson solution 119

3.26 Modulation doping 121

3.27 The high-electron-mobility transistor 122

3.28 Band filling 123

Exercises 124

References 125

4 Diffusion 127

4.1 Introduction 127

4.2 Theory 129

4.3 Boundary conditions 130

4.4 Convergence tests 131

4.5 Numerical stability 133

4.6 Constant diffusion coefficients 133

4.7 Concentration dependent diffusion coefficient 135

4.8 Depth dependent diffusion coefficient 136

4.9 Time dependent diffusion coefficient 138

4.10 delta-doped quantum wells 138

4.11 Extension to higher dimensions 141

Exercises 142

References 142

5 Impurities 145

5.1 Donors and acceptors in bulk material 145

5.2 Binding energy in a heterostructure 147

5.3 Two-dimensional trial wave function 152

5.4 Three-dimensional trial wave function 158

5.5 Variable-symmetry trial wave function 164

5.6 Inclusion of a central cell correction 170

5.7 Special considerations for acceptors 171

5.8 Effective mass and dielectric mismatch 172

5.9 Band non-parabolicity 173

5.10 Excited states 173

5.11 Application to spin-flip Raman spectroscopy 174

5.11.1 Diluted magnetic semiconductors 174

5.11.2 Spin-flip Raman spectroscopy 176

5.12 Alternative approach to excited impurity states 178

5.13 The ground state 180

5.14 Position dependence 181

5.15 Excited states 181

5.16 Impurity occupancy statistics 184

Exercises 188

References 189

6 Excitons 191

6.1 Excitons in bulk 191

6.2 Excitons in heterostructures 193

6.3 Exciton binding energies 193

6.4 1s exciton 198

6.5 The two-dimensional and three-dimensional limits 202

6.6 Excitons in single quantum wells 206

6.7 Excitons in multiple quantum wells 208

6.8 Stark ladders 210

6.9 Self-consistent effects 211

6.10 2s exciton 212

Exercises 214

References 215

7 Strained quantum wells 217

7.1 Stress and strain in bulk crystals 217

7.2 Strain in quantum wells 221

7.3 Critical thickness of layers 224

7.4 Strain balancing 226

7.5 Effect on the band profile of quantum wells 228

7.6 The piezoelectric effect 231

7.7 Induced piezoelectric fields in quantum wells 234

7.8 Effect of piezoelectric fields on quantum wells 236

Exercises 239

References 240

8 Simple models of quantum wires and dots 241

8.1 Further confinement 241

8.2 Schrödinger's equation in quantum wires 243

8.3 Infinitely deep rectangular wires 245

8.4 Simple approximation to a finite rectangular wire 247

8.5 Circular cross-section wire 251

8.6 Quantum boxes 255

8.7 Spherical quantum dots 256

8.8 Non-zero angular momentum states 259

8.9 Approaches to pyramidal dots 262

8.10 Matrix approaches 263

8.11 Finite difference expansions 263

8.12 Density of states 265

Exercises 267

References 268

9 Quantum dots 269

9.1 0-dimensional systems and their experimental realization 269

9.2 Cuboidal dots 271

9.3 Dots of arbitrary shape 272

9.3.1 Convergence tests 277

9.3.2 Efficiency 279

9.3.3 Optimization 281

9.4 Application to real problems 282

9.4.1 InAs/GaAs self-assembled quantum dots 282

9.4.2 Working assumptions 282

9.4.3 Results 283

9.4.4 Concluding remarks 286

9.5 A more complex model is not always a better model 288

Exercises 289

References 290

10 Carrier scattering 293

10.1 Introduction 293

10.2 Fermi's Golden Rule 294

10.3 Extension to sinusoidal perturbations 296

10.4 Averaging over two-dimensional carrier distributions 296

10.5 Phonons 298

10.6 Longitudinal optic phonon scattering of two-dimensional carriers 301

10.7 Application to conduction subbands 313

10.8 Mean intersubband LO phonon scattering rate 315

10.9 Ratio of emission to absorption 316

10.10 Screening of the LO phonon interaction 318

10.11 Acoustic deformation potential scattering 319

10.12 Application to conduction subbands 324

10.13 Optical deformation potential scattering 326

10.14 Confined and interface phonon modes 328

10.15 Carrier-carrier scattering 328

10.16 Addition of screening 336

10.17 Mean intersubband carrier-carrier scattering rate 337

10.18 Computational implementation 339

10.19 Intrasubband versus intersubband 340

10.20 Thermalized distributions 341

10.21 Auger-type intersubband processes 342

10.22 Asymmetric intrasubband processes 343

10.23 Empirical relationships 344

10.24 A generalised expression for scattering of two-dimensional carriers 345

10.25 Impurity scattering 346

10.26 Alloy disorder scattering 351

10.27 Alloy disorder scattering in quantum wells 354

10.28 Interface roughness scattering 355

10.29 Interface roughness scattering in quantum wells 359

10.30 Carrier scattering in quantum wires and dots 362

Exercises 362

References 364

11 Optical properties of quantum wells 367

11.1 Carrier-photon scattering 367

11.2 Spontaneous emission lifetime 372

11.3 Intersubband absorption in quantum wells 374

11.4 Bound-bound transitions 376

11.5 Bound-free transitions 377

11.6 Rectangular quantum well 379

11.7 Intersubband optical non-linearities 382

11.8 Electric polarization 383

11.9 Intersubband second harmonic generation 384

11.10 Maximization of resonant susceptibility 387

Exercises 390

References 391

12 Carrier transport 393

12.1 Introduction 393

12.2 Quantum cascade lasers 393

12.3 Realistic quantum cascade laser 398

12.4 Rate equations 400

12.5 Self-consistent solution of the rate equations 402

12.6 Calculation of the current density 404

12.7 Phonon and carrier-carrier scattering transport 404

12.8 Electron temperature 405

12.9 Calculation of the gain 408

12.10 QCLs, QWIPs, QDIPs and other methods 411

12.11 Density matrix approaches 412

12.11.1 Time evolution of the density matrix 415

12.11.2 Density matrix modelling of terahertz QCLs 416

Exercises 418

References 420

13 Optical waveguides 423

13.1 Introduction to optical waveguides 423

13.2 Optical waveguide analysis 425

13.2.1 The wave equation 425

13.2.2 The transfer matrix method 428

13.2.3 Guided modes in multi-layer waveguides 431

13.3 Optical properties of materials 434

13.3.1 Semiconductors 434

13.3.2 Influence of free-carriers 436

13.3.3 Carrier mobility model 438

13.3.4 Influence of doping 439

13.4 Application to waveguides of...