Quantum Wells, Wires and Dots

Paul Harrison is currently working in the Institute of Microwaves and Photonics (IMP), which is a research institute within the school of Electronic and Electrical Engineering t the University of Leeds in the United Kingdom. He can always be found on the web, at the time of writing, at: www.ee.leeds.ac.uk/homes/ph/ and always answers e-mail. Currently he can be reached at: P.harrison@leeds.ac.uk or p.harrison@physics.org Paul is working on a wide variety of Projects, most of which centre around exploiting quantum mechanics for the creation of novel opto-electronic devices, largely, but not exclusively, in semiconductor Quantum Wells, Wires and Dots. Up to date information can be found on his web page. He is always looking for exceptionally well qualified and motivated students to study for a PhD degree with him-if interested, please don't hesitate to contact him.

Preface. Acknowledgements. About the author. About the book. Introduction. 1 Semiconductors and heterostructors. 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 Schrodingers 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 operator. 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 fields. 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 peizoelectric fields in quantum wells. 7.7 Effect of piezoelectric fields on quantum wells. 8 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 Carrier scattering. 9.1 Fermi's Golden Rule. 9.2 Phonons. 9.3 Longitudinal optic phonon scattering of bulk carriers. 9.4 LO phonon scattering of two-dimensional carriers. 9.5 Application to conduction subbands. 9.6 Averaging over carrier distributions. 9.7 Ratio of emission to absorption. 9.8 Screening of the LO phonon interaction. 9.9 Acoustic deformation potential scattering. 9.10 Application to conduction subbands. 9.11 Optical deformation potential scattering. 9.12 Confined and interface phonon modes. 9.13 Carrier-carrier scattering. 9.14 Addition of screening. 9.15 Averaging over an initial state population. 9.16 Intrasubband versus intersubband. 9.17 Thermalised distributions. 9.18 Auger-type intersubband processes. 9.19 Asymmetric intrasubband processes. 9.20 Empirical relationships. 9.21 Carrier-Photon scattering. 9.22 Quantum cascade lasers. 9.23 Carrier scattering in quantum wires and dots. 10 Multiband envelope function (k.p) method, Z. Ikonic. 10.1 Symmetry, basis states and band structure. 10.2 Valence band structure and the 6 X 6 Hamiltonian. 10.3 4 X 4 Valence band Hamiltonian. 10.4 Complex band structure. 10.5 Block-diagonalisation of the Hamiltonian. 10.6 The valence band in strained cubic semiconductors. 10.7 Hole subbands in heterostructures. 10.8 Valence band offset. 10.9 The layer (transfer matrix) method. 10.10 Quantum well subbands. 10.11 The influence of strain. 10.12 Strained quantum well subbands. 10.13 Direct numerical methods. 11 Empirical pseudopotential theory. 11.1 Principles and approximations. 11.2 Elemental band structure calculation. 11.3 Spin-orbit coupling. 11.4 Compound Semiconductors. 11.5 Charge densities. 11.6 Calculating the effective mass. 11.7 Alloys. 11.8 Atomic form factors. 11.9 Generalisation to a large basis. 11.10 Spin-orbit coupling within the large basis approach. 11.11 Computational implementation. 11.12 Deducing the parameters and application. 11.13 Isoelectronic impurities in bulk. 11.14 The electronic structure around point defects. 12 Microscopic electronic properties of heterostructures. 12.1 The superlattice unit cell. 12.2 Application of large basis method to superlattices. 12.3 Comparison with envelope-function approximation. 12.4 In-plane dispersion. 12.5 Interface coordination. 12.6 Strain-layered superlattices. 12.7 The superlattice as a perturbation. 12.8 Application to GaAs/AIAs superlattices. 12.9 Inclusion of remote bands. 12.10 The valence band. 12.11 Computational effort. 12.12 Superlattice dispersion and the interminiband laser. 12.13 Addition of electric field. 13 Application to quantum wires and dots. 13.1 Recent progress. 13.2 The quantum-wire unit cell. 13.3 Confined states. 13.4 V-grooved quantum wires. 13.5 Along-axis dispersion. 13.6 Tiny quantum dots. 13.7 Pyramidal quantum dots. 13.8 Transport through dot arrays. 13.9 Anti-wires and anti-dots. Concluding Remarks. Appendix A: Materials parameters. References. Topic Index.