Quantum wells, wires and dots: theoretical and computational physics of semiconductor nanostructures

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 Schrodinger   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 Schrodinger 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 Schrodinger equation 91     3.10 The parabolic potential well 94     3.11 The Poschl   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 Schrodinger   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   -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 Schrodinger   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 x 6 Hamiltonian 466     14.3 4 x 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