Quantum Wells, Wires and Dots

Preface Acknowledgements About the author(s) About the book Introduction 1 Semiconductors and heterostructures 1.1 The mechanics of waves 1.2 Crystal structure 1.3 The effective mass approximation 1.4 Band theory 1.5 Heterojunctions 1.6 Heterostructures 1.7 The envelope function approximation 1.8 The reciprocal lattice 2 Solutions to Schrodinger's equation 2.1 The infinite well 2.2 In-plane dispersion 2.3 Density of states 2.4 Subband populations 2.5 Finite well with constant mass 2.6 Effective mass mismatch at heterojunctions 2.7 The infinite barrier height and mass limits 2.8 Hermiticity and the kinetic energy operator 2.9 Alternative kinetic energy operators 2.10 Extension to multiple-well systems 2.11 The asymmetric single quantum well 2.12 Addition of an electric field 2.13 The infinite superlattice 2.14 The single barrier 2.15 The double barrier 2.16 Extension to include electric field 2.17 Magnetic fields and Landau quantisation 2.18 In summary 3 Numerical solutions 3.1 Shooting method 3.2 Generalised initial conditions 3.3 Practical implementation of the shooting method 3.4 Heterojunction boundary conditions 3.5 The parabolic potential well 3.6 The Poschl-Teller potential hole 3.7 Convergence tests 3.8 Extension to variable effective mass 3.9 The double quantum well 3.10 Multiple quantum wells and finite superlattices 3.11 Addition of electric field 3.12 Quantum confined Stark effect 3.13 Field-induced anti-crossings 3.14 Symmetry and selection rules 3.15 The Heisenberg uncertainty principle 3.16 Extension to include band non-parabolicity 3.17 Poisson's equation 3.18 Self-consistent Schrodinger-Poisson solution 3.19 Computational implementation 3.20 Modulation doping 3.21 The high-electron-mobility transistor 3.22 Band filling 4 Diffusion 4.1 Introduction 4.2 Theory 4.3 Boundary conditions 4.4 Convergence tests 4.5 Constant diffusion coefficients 4.6 Concentration dependent diffusion coefficient 4.7 Depth dependent diffusion coefficient 4.8 Time dependent diffusion coefficient 4.9 !-doped quantum wells 4.10 Extension to higher dimensions 5 Impurities 5.1 Donors and acceptors in bulk material 5.2 Binding energy in a heterostructure 5.3 Two-dimensional trial wave function 5.4 Three-dimensional trial wave function 5.5 Variable-symmetry trial wave function 5.6 Inclusion of a central cell correction 5.7 Special considerations for acceptors 5.8 Effective mass and dielectric mismatch 5.9 Band non-parabolicity 5.10 Excited states 5.11 Application to spin-flip Raman spectroscopy 5.12 Alternative approach to excited impurity states 5.13 The ground state 5.14 Position dependence 5.15 Excited States 5.16 Impurity occupancy statistics 6 Excitons 6.1 Excitons in bulk 6.2 Excitons in heterostructures 6.3 Exciton binding energies 6.4 1s exciton 6.5 The two-dimensional and three-dimensional limits 6.6 Excitons in single quantum wells 6.7 Excitons in multiple quantum wells 6.8 Stark Ladders 6.9 Self-consistent effects 6.10 Spontaneous symmetry breaking 6.11 2s exciton 7 Strained quantum wells, V. D. Jovanovic 7.1 Stress and strain in bulk crystals 7.2 Strain in quantum wells 7.3 Strain balancing 7.4 Effect on the band profile of quantum wells 7.5 The piezoelectric effect 7.6 Induced piezoelectric fields in quantum wells 7.7 Effect of piezoelectric fields on quantum wells 8 Simple models of quantum wires and dots 8.1 Further confinement 8.2 Schrodinger's equation in quantum wires 8.3 Infinitely deep rectangular wires 8.4 Simple approximation to a finite rectangular wire 8.5 Circular cross-section wire 8.6 Quantum boxes 8.7 Spherical quantum dots 8.8 Non-zero angular momentum states 8.9 Approaches to pyramidal dots 8.10 Matrix approaches 8.11 Finite difference expansions 8.12 Density of states 9 Quantum dots, M. Califano 9.1 0-dimensional systems and their experimental realisation 9.2 Cuboidal dots 9.3 Dots of arbitrary shape 9.4 Application to real problems 9.5 A more complex model is not always a better model 10 Carrier scattering 10.1 Fermi's Golden Rule 10.2 Phonons 10.3 Longitudinal optic phonon scattering of bulk carriers 10.4 LO phonon scattering of two-dimensional carriers 10.5 Application to conduction subbands 10.6 Averaging over carrier distributions 10.7 Ratio of emission to absorption 10.8 Screening of the LO phonon interaction 10.9 Acoustic deformation potential scattering 10.10 Application to conduction subbands 10.11 Optical deformation potential scattering 10.12 Confined and interface phonon modes 10.13 Carrier-carrier scattering 10.14 Addition of screening 10.15 Averaging over an initial state population 10.16 Intrasubband versus intersubband 10.17 Thermalised distributions 10.18 Auger-type intersubband processes 10.19 Asymmetric intrasubband processes 10.20 Empirical relationships 10.21 Carrier-photon scattering 10.22 Carrier scattering in quantum wires and dots 11 Electron transport 11.1 Introduction 11.2 Mid-infrared quantum cascade lasers 11.3 Realistic quantum cascade laser 11.4 Rate equations 11.5 Self-consistent solution of the rate equations 11.6 Calculation of the current density 11.7 Phonon and carrier-carrier scattering transport 11.8 Electron temperature 11.9 Calculation of the gain 11.10 QCLs, QWIPs, QDIPs and other methods 12 Optical properties of quantum wells, D. Indjin 12.1 Intersubband absorption in quantum wells 12.2 Bound-bound transitions 12.3 Bound-free transitions 12.4 Fermi level 12.5 Rectangular quantum well 12.6 Intersubband optical non-linearities 12.7 Electric polarisation 12.8 Intersubband second harmonic generation 12.9 Maximization of resonant susceptibility 13 Optical waveguides, C. A. Evans 13.1 Introduction to optical waveguides 13.2 Optical waveguide analysis 13.3 Optical properties of materials 13.4 Application to waveguides of laser devices 14 Multiband envelope function (k.p) method, Z. Ikonic 14.1 Symmetry, basis states and band structure 14.2 Valence band structure and the 6 x 6 Hamiltonian 14.3 4 x 4 valence band Hamiltonian 14.4 Complex band structure 14.5 Block-diagonalisation of the Hamiltonian 14.6 The valence band in strained cubic semiconductors 14.7 Hole subbands in heterostructures 14.8 Valence band offset 14.9 The layer (transfer matrix) method 14.10 Quantum well subbands 14.11 The influence of strain 14.12 Strained quantum well subbands 14.13 Direct numerical methods 15 Empirical pseudopotential theory 15.1 Principles and Approximations 15.2 Elemental Band Structure Calculation 15.3 Spin-orbit coupling 15.4 Compound Semiconductors 15.5 Charge densities 15.6 Calculating the effective mass 15.7 Alloys 15.8 Atomic form factors 15.9 Generalisation to a large basis 15.10 Spin-orbit coupling within the large basis approach 15.11 Computational implementation 15.12 Deducing the parameters and application 15.13 Isoelectronic impurities in bulk 15.14 The electronic structure around point defects 16 Microscopic electronic properties of heterostructures 16.1 The superlattice unit cell 16.2 Application of large basis method to superlattices 16.3 Comparison with envelope-function approximation 16.4 In-plane dispersion 16.5 Interface coordination 16.6 Strain-layered superlattices 16.7 The superlattice as a perturbation 16.8 Application to GaAs/AlAs superlattices 16.9 Inclusion of remote bands 16.10 The valence band 16.11 Computational effort 16.12 Superlattice dispersion and the interminiband laser 16.13 Addition of electric field 17 Application to quantum wires and dots 17.1 Recent progress 17.2 The quantum-wire unit cell 17.3 Confined states 17.4 V-grooved quantum wires 17.5 Along-axis dispersion 17.6 Tiny quantum dots 17.7 Pyramidal quantum dots 17.8 Transport through dot arrays 17.9 Anti-wires and anti-dots Concluding Remarks Appendix A: Materials parameters References Topic Index