Quantum Wells, Wires and Dots

Professor Paul Harrison After his first degree in physics at the University of Hull, Paul decided to pursue an academic career because of his wish to share the love of his subject with students. With that in mind Paul did a a PhD in computational physics at the University of Newcastle-upon-Tyne and then returned to Hull in 1991 to work as a postdoctoral researcher assistant in the Applied Physics department. After four successful years Paul had built his CV up to the level to obtain a five-year research fellowship in 1995 in the School of Electronic & Electrical Engineering at the University of Leeds. This was a strategic move by the University of Leeds to hire early-career researchers and develop them into independent research leaders. The fellowship gave Paul the time to invest heavily in his own research portfolio and Paul built up a group of research students and postdocs, a strong publications track record and an international reputation in theory and design of semiconductor optoelectronic devices. At the end of his fellowship Paul was promoted to Reader in 2000 and subsequently Professor of Quantum Electronics in 2002. The following year Paul was nominated by the staff to be Head of the School of Electronic and Electrical Engineering, rated 5* in the 2001 Research Assessment Exercise. Under his leadership the school continued to develop its research portfolio and worked as a team to push on with its research strategy: subsequently rewarded by being ranked top of its unit of assessment in the 2008 RAE. Paul was promoted to Dean of Postgraduate Research Studies in 2011, a university-wide portfolio to develop postgraduate research as part of the overall university strategy. This included responsibility for developing research funding for postgraduate students, marketing, recruitment, admissions and progression, improving submission and completion rates, harmonising processes and improving the student experience across campus. Paul joined Sheffield Hallam University as Pro Vice-Chancellor for Research and Innovation at the start of 2014. Over his career, Paul has co-authored nearly 300 journal articles, written two books, successfully supervised 16 students to PhD, raised £3m of research funding and Google Scholar gives him a h-index of 33. Dr Alex Valavanis (MIET) received his MEng (Hons) degree in Electronic Engineering from the University of York and his PhD degree in Electronic and Electrical Engineering from the University of Leeds in 2004 and 2009 respectively. From 2004–2005 he worked with STFC Daresbury Laboratories, Cheshire, developing X-ray detector systems. From 2005–2009, his PhD with the Quantum Electronics Group in the School of Electronic and Electrical Engineering, University of Leeds, focused on the development of numerical simulations, in C/C++, of quantum-cascade lasers (QCLs) in the silicon–germanium material system. Since 2009, he has worked in the Terahertz (THz) Photonics Laboratory within the same institute, developing new THz imaging and sensing techniques. Dr Valavanis holds full membership of the IET, and has received awards including the BNFL Peter Wilson Award (2009) for Materials Engineering, the GW Carter prize (2008) for best publication by a PhD student and the FW Carter prize (2009) for best PhD thesis within the School of Electronic and Electrical Engineering.

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 δ-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 laser devices 440

13.4.1 Double heterostructure laser waveguide 441

13.4.2 Quantum cascade laser waveguides 443

13.5 Thermal properties of waveguides 447

13.6 The heat equation 449

13.7 Material properties 450

13.7.1 Thermal conductivity 450

13.7.2 Specific heat capacity 451

13.8 Finite difference approximation to the heat equation 453

13.9 Steady-state solution of the heat equation 454

13.10 Time-resolved solution 457

13.11 Simplified RC thermal models 458

Exercises 461

References 462

14 Multiband envelope function (k.p) method 465

14.1 Symmetry, basis states and band structure 465

14.2 Valence band structure and the 6 × 6 Hamiltonian 466

14.3 4 × 4 valence band Hamiltonian 470

14.4 Complex band structure 471

14.5 Block-diagonalization of the Hamiltonian 472

14.6 The valence band in strained cubic semiconductors 474

14.7 Hole subbands in heterostructures 476

14.8 Valence band offset 478

14.9 The layer (transfer matrix) method 479

14.10 Quantum well subbands 483

14.11 The influence of strain 484

14.12 Strained quantum well subbands 484

14.13 Direct numerical methods 485

Exercises 486

References 486

15 Empirical pseudo-potential bandstructure 487

15.1 Principles and approximations 487

15.2 Elemental band structure calculation 488

15.3 Spin–orbit coupling 496

15.4 Compound semiconductors 498

15.5 Charge densities 501

15.6 Calculating the effective mass 504

15.7 Alloys 504

15.8 Atomic form factors 506

15.9 Generalization to a large basis 507

15.10 Spin–orbit coupling within the large basis approach 510

15.11 Computational implementation 511

15.12 Deducing the parameters and application 512

15.13 Isoelectronic impurities in bulk 515

15.14 The electronic structure around point defects 520

Exercises 520

References 521

16 Pseudo-potential calculations of nanostructures 523

16.1 The superlattice unit cell 523

16.2 Application of large basis method to superlattices 526

16.3 Comparison with envelope function approximation 530

16.4 In-plane dispersion 531

16.5 Interface coordination 532

16.6 Strain-layered superlattices 533

16.7 The superlattice as a perturbation 534

16.8 Application to GaAs/AlAs superlattices 539

16.9 Inclusion of remote bands 541

16.10 The valence band 542

16.11 Computational effort 542

16.12 Superlattice dispersion and the interminiband laser 543

16.13 Addition of electric field 545

16.14 Application of the large basis method to quantum wires 549

16.15 Confined states 552

16.16 Application of the large basis method to tiny quantum dots 552

16.17 Pyramidal quantum dots 554

16.18 Transport through dot arrays 555

16.19 Recent progress 556

Exercises 556

References 557

Concluding remarks 559

A Materials parameters 561

B Introduction to the simulation tools 563

B.1 Documentation and support 564

B.2 Installation and dependencies 564

B.3 Simulation programs 565

B.4 Introduction to scripting 566

B.5 Example calculations 567