Quantum Wells, Wires and Dots

Professor Paul HarrisonAfter 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 iiiList of Contributors xiiiPreface xvAcknowledgements xixIntroduction xxiiiReferences xxiv1 Semiconductors and heterostructures 11.1 The mechanics of waves 11.2 Crystal structure 31.3 The effective mass approximation 51.4 Band theory 51.5 Heterojunctions 71.6 Heterostructures 71.7 The envelope function approximation 101.8 Band non-parabolicity 111.9 The reciprocal lattice 13Exercises 16References 172 Solutions to Schrödinger’s equation 192.1 The infinite well 192.2 In-plane dispersion 222.3 Extension to include band non-parabolicity 242.4 Density of states 262.4.1 Density-of-states effective mass 282.4.2 Two-dimensional systems 292.5 Subband populations 312.5.1 Populations in non-parabolic subbands 332.5.2 Calculation of quasi-Fermi energy 352.6 Thermalised distributions 362.7 Finite well with constant mass 372.7.1 Unbound states 432.7.2 Effective mass mismatch at heterojunctions 452.7.3 The infinite barrier height and mass limits 492.8 Extension to multiple-well systems 502.9 The asymmetric single quantum well 532.10 Addition of an electric field 542.11 The infinite superlattice 572.12 The single barrier 632.13 The double barrier 652.14 Extension to include electric field 712.15 Magnetic fields and Landau quantisation 722.16 In summary 74Exercises 74References 763 Numerical solutions 793.1 Bisection root-finding 793.2 Newton–Raphson root finding 813.3 Numerical differentiation 833.4 Discretised Schrödinger equation 843.5 Shooting method 843.6 Generalized initial conditions 863.7 Practical implementation of the shooting method 883.8 Heterojunction boundary conditions 903.9 Matrix solutions of the discretised Schrödinger equation 913.10 The parabolic potential well 943.11 The Pöschl–Teller potential hole 983.12 Convergence tests 983.13 Extension to variable effective mass 993.14 The double quantum well 1033.15 Multiple quantum wells and finite superlattices 1043.16 Addition of electric field 1063.17 Extension to include variable permittivity 1063.18 Quantum confined Stark effect 1083.19 Field–induced anti-crossings 1083.20 Symmetry and selection rules 1103.21 The Heisenberg uncertainty principle 1103.22 Extension to include band non-parabolicity 1133.23 Poisson’s equation 1143.24 Matrix solution of Poisson’s equation 1183.25 Self-consistent Schrödinger–Poisson solution 1193.26 Modulation doping 1213.27 The high-electron-mobility transistor 1223.28 Band filling 123Exercises 124References 1254 Diffusion 1274.1 Introduction 1274.2 Theory 1294.3 Boundary conditions 1304.4 Convergence tests 1314.5 Numerical stability 1334.6 Constant diffusion coefficients 1334.7 Concentration dependent diffusion coefficient 1354.8 Depth dependent diffusion coefficient 1364.9 Time dependent diffusion coefficient 1384.10 δ-doped quantum wells 1384.11 Extension to higher dimensions 141Exercises 142References 1425 Impurities 1455.1 Donors and acceptors in bulk material 1455.2 Binding energy in a heterostructure 1475.3 Two-dimensional trial wave function 1525.4 Three-dimensional trial wave function 1585.5 Variable-symmetry trial wave function 1645.6 Inclusion of a central cell correction 1705.7 Special considerations for acceptors 1715.8 Effective mass and dielectric mismatch 1725.9 Band non-parabolicity 1735.10 Excited states 1735.11 Application to spin-flip Raman spectroscopy 1745.11.1 Diluted magnetic semiconductors 1745.11.2 Spin-flip Raman spectroscopy 1765.12 Alternative approach to excited impurity states 1785.13 The ground state 1805.14 Position dependence 1815.15 Excited states 1815.16 Impurity occupancy statistics 184Exercises 188References 1896 Excitons 1916.1 Excitons in bulk 1916.2 Excitons in heterostructures 1936.3 Exciton binding energies 1936.4 1s exciton 1986.5 The two-dimensional and three-dimensional limits 2026.6 Excitons in single quantum wells 2066.7 Excitons in multiple quantum wells 2086.8 Stark ladders 2106.9 Self-consistent effects 2116.10 2s exciton 212Exercises 214References 2157 Strained quantum wells 2177.1 Stress and strain in bulk crystals 2177.2 Strain in quantum wells 2217.3 Critical thickness of layers 2247.4 Strain balancing 2267.5 Effect on the band profile of quantum wells 2287.6 The piezoelectric effect 2317.7 Induced piezoelectric fields in quantum wells 2347.8 Effect of piezoelectric fields on quantum wells 236Exercises 239References 2408 Simple models of quantum wires and dots 2418.1 Further confinement 2418.2 Schrödinger’s equation in quantum wires 2438.3 Infinitely deep rectangular wires 2458.4 Simple approximation to a finite rectangular wire 2478.5 Circular cross-section wire 2518.6 Quantum boxes 2558.7 Spherical quantum dots 2568.8 Non-zero angular momentum states 2598.9 Approaches to pyramidal dots 2628.10 Matrix approaches 2638.11 Finite difference expansions 2638.12 Density of states 265Exercises 267References 2689 Quantum dots 2699.1 0-dimensional systems and their experimental realization 2699.2 Cuboidal dots 2719.3 Dots of arbitrary shape 2729.3.1 Convergence tests 2779.3.2 Efficiency 2799.3.3 Optimization 2819.4 Application to real problems 2829.4.1 InAs/GaAs self-assembled quantum dots 2829.4.2 Working assumptions 2829.4.3 Results 2839.4.4 Concluding remarks 2869.5 A more complex model is not always a better model 288Exercises 289References 29010 Carrier scattering 29310.1 Introduction 29310.2 Fermi’s Golden Rule 29410.3 Extension to sinusoidal perturbations 29610.4 Averaging over two-dimensional carrier distributions 29610.5 Phonons 29810.6 Longitudinal optic phonon scattering of two-dimensional carriers 30110.7 Application to conduction subbands 31310.8 Mean intersubband LO phonon scattering rate 31510.9 Ratio of emission to absorption 31610.10 Screening of the LO phonon interaction 31810.11 Acoustic deformation potential scattering 31910.12 Application to conduction subbands 32410.13 Optical deformation potential scattering 32610.14 Confined and interface phonon modes 32810.15 Carrier–carrier scattering 32810.16 Addition of screening 33610.17 Mean intersubband carrier–carrier scattering rate 33710.18 Computational implementation 33910.19 Intrasubband versus intersubband 34010.20 Thermalized distributions 34110.21 Auger-type intersubband processes 34210.22 Asymmetric intrasubband processes 34310.23 Empirical relationships 34410.24 A generalised expression for scattering of two-dimensional carriers 34510.25 Impurity scattering 34610.26 Alloy disorder scattering 35110.27 Alloy disorder scattering in quantum wells 35410.28 Interface roughness scattering 35510.29 Interface roughness scattering in quantum wells 35910.30 Carrier scattering in quantum wires and dots 362Exercises 362References 36411 Optical properties of quantum wells 36711.1 Carrier–photon scattering 36711.2 Spontaneous emission lifetime 37211.3 Intersubband absorption in quantum wells 37411.4 Bound–bound transitions 37611.5 Bound–free transitions 37711.6 Rectangular quantum well 37911.7 Intersubband optical non-linearities 38211.8 Electric polarization 38311.9 Intersubband second harmonic generation 38411.10 Maximization of resonant susceptibility 387Exercises 390References 39112 Carrier transport 39312.1 Introduction 39312.2 Quantum cascade lasers 39312.3 Realistic quantum cascade laser 39812.4 Rate equations 40012.5 Self-consistent solution of the rate equations 40212.6 Calculation of the current density 40412.7 Phonon and carrier–carrier scattering transport 40412.8 Electron temperature 40512.9 Calculation of the gain 40812.10 QCLs, QWIPs, QDIPs and other methods 41112.11 Density matrix approaches 41212.11.1 Time evolution of the density matrix 41512.11.2 Density matrix modelling of terahertz QCLs 416Exercises 418References 42013 Optical waveguides 42313.1 Introduction to optical waveguides 42313.2 Optical waveguide analysis 42513.2.1 The wave equation 42513.2.2 The transfer matrix method 42813.2.3 Guided modes in multi-layer waveguides 43113.3 Optical properties of materials 43413.3.1 Semiconductors 43413.3.2 Influence of free-carriers 43613.3.3 Carrier mobility model 43813.3.4 Influence of doping 43913.4 Application to waveguides of laser devices 44013.4.1 Double heterostructure laser waveguide 44113.4.2 Quantum cascade laser waveguides 44313.5 Thermal properties of waveguides 44713.6 The heat equation 44913.7 Material properties 45013.7.1 Thermal conductivity 45013.7.2 Specific heat capacity 45113.8 Finite difference approximation to the heat equation 45313.9 Steady-state solution of the heat equation 45413.10 Time-resolved solution 45713.11 Simplified RC thermal models 458Exercises 461References 46214 Multiband envelope function (k.p) method 46514.1 Symmetry, basis states and band structure 46514.2 Valence band structure and the 6 × 6 Hamiltonian 46614.3 4 × 4 valence band Hamiltonian 47014.4 Complex band structure 47114.5 Block-diagonalization of the Hamiltonian 47214.6 The valence band in strained cubic semiconductors 47414.7 Hole subbands in heterostructures 47614.8 Valence band offset 47814.9 The layer (transfer matrix) method 47914.10 Quantum well subbands 48314.11 The influence of strain 48414.12 Strained quantum well subbands 48414.13 Direct numerical methods 485Exercises 486References 48615 Empirical pseudo-potential bandstructure 48715.1 Principles and approximations 48715.2 Elemental band structure calculation 48815.3 Spin–orbit coupling 49615.4 Compound semiconductors 49815.5 Charge densities 50115.6 Calculating the effective mass 50415.7 Alloys 50415.8 Atomic form factors 50615.9 Generalization to a large basis 50715.10 Spin–orbit coupling within the large basis approach 51015.11 Computational implementation 51115.12 Deducing the parameters and application 51215.13 Isoelectronic impurities in bulk 51515.14 The electronic structure around point defects 520Exercises 520References 52116 Pseudo-potential calculations of nanostructures 52316.1 The superlattice unit cell 52316.2 Application of large basis method to superlattices 52616.3 Comparison with envelope function approximation 53016.4 In-plane dispersion 53116.5 Interface coordination 53216.6 Strain-layered superlattices 53316.7 The superlattice as a perturbation 53416.8 Application to GaAs/AlAs superlattices 53916.9 Inclusion of remote bands 54116.10 The valence band 54216.11 Computational effort 54216.12 Superlattice dispersion and the interminiband laser 54316.13 Addition of electric field 54516.14 Application of the large basis method to quantum wires 54916.15 Confined states 55216.16 Application of the large basis method to tiny quantum dots 55216.17 Pyramidal quantum dots 55416.18 Transport through dot arrays 55516.19 Recent progress 556Exercises 556References 557Concluding remarks 559A Materials parameters 561B Introduction to the simulation tools 563B.1 Documentation and support 564B.2 Installation and dependencies 564B.3 Simulation programs 565B.4 Introduction to scripting 566B.5 Example calculations 567