QUANTUM WELLS, WIRES AND DOTS

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1 QUANTUM WELLS, WIRES AND DOTS Theoretical and Computational Physics of Semiconductor Nanostructures Second Edition Paul Harrison The University of Leeds, UK /Cf}\WILEY~ ^INTERSCIENCE JOHN WILEY & SONS, LTD

2 CONTENTS Preface Acknowledgements About the author About the book Introduction xv xix xxi xxiii xxv 1 Semiconductors and heterostructures The mechanics of waves Crystal structure The effective mass approximation Band theory Heterojunctions Heterostructures The envelope function approximation The reciprocal lattice 12 vii

3 VIII CONTENTS Solutions to Schrödinger's equation The infinite well In-plane dispersion Density of states Subband populations Finite well with constant mass Effective mass mismatch at heterojunctions The infinite barrier height and mass limits Hermiticity and the kinetic energy Operator Alternative kinetic energy Operators Extension to multiple-well Systems The asymmetric Single quantum well Addition of an electric field The infinite superlattice The Single barrier The double barrier Extension to include electric field Magnetic fields and Landau quantisation In summary 71 Numerical Solutions Shooting method Generalised initial conditions Practical implementation of the shooting method Heterojunction boundary conditions The parabolic potential well The Pöschl-Teller potential hole _ Convergence tests Extension to variable effective mass The double quantum well Multiple quantum wells and finite superlattices Addition of electric field Quantum confined Stark effect Field-induced anti-crossings Symmetry and selection rules The Heisenberg uncertainty principle Extension to include band non-parabolicity 103

4 3.17 Poisson's equation 3.18 Self-consistent Schrödinger-Poisson Solution 3.19 Computational implementation 3.20 Modulation doping 3.21 The high-electron-mobility transistor 3.22 Band Alling 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 (5-doped quantum wells 4.10 Extension to higher dimensions 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 raass and dielectric mismatch 5.9 Band non-parabolicity 5.10 Excited states 5.11 Application to spin-flip Raman spectroscopy Diluted magnetic semiconductors 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 X CONTENTS 5.16 Impurity occupancy statistics 184 Excitons Excitons in bulk Excitons in heterostructures Exciton binding energies ls exciton The two-dimensional and three-dimensional limits Excitons in single quantum wells Excitons in multiple quantum wells Stark Ladders Self-consistent effects Spontaneous symmetry breaking s exciton 217 Strained quantum wells, V. D. Jovanovic Stress and strain in bulk crystals Strain in quantum wells Strain balancing Effect on the band profile of quantum wells The piezoelectric effect Induced piezoelectric fields in quantum wells Effect of piezoelectric fields on quantum wells 239 Quantum wires and dots Further confinement Schrödinger's equation in quantum wires Infinitely deep rectangular wires Simple approximation to a finite rectangular wire Circular cross-section wire Quantum boxes Spherical quantum dots Non-zero angular momentum states Approaches to pyramidal dots Matrix approaches Finite difference expansions Density of states 267

6 CONTENTS XI 9 Carrier scattering Fermi's Golden Rule Phonons Longitudinal optic phonon scattering of bulk carriers LO phonon scattering of two-dimensional carriers Application to conduction subbands Averaging over carrier distributions Ratio of emission to absorption Screening of the LO phonon interaction Acoustic deformation potential scattering Application to conduction subbands Optical deformation potential scattering Confined and interface phonon modes Carrier-carrier scattering Addition of Screening Averaging over an initial State population Intrasubband versus intersubband Thermalised distributions Auger-type intersubband processes Asymmetrie intrasubband processes Empirical relationships Carrier-photon scattering Quantum cascade lasers Carrier scattering in quantum wires and dots Multiband envelope funetion (k.p) method, Z. Ikonic Symmetry, basis states and band strueture Valence band strueture and the 6x6 Hamiltonian x 4 valence band Hamiltonian Complex band strueture Block-diagonalisation of the Hamiltonian The valence band in strained eubie semiconduetors Hole subbands in heterostruetures Valence band offset The layer (transfer matrix) method Quantum well subbands The influence of strain 367

7 XII CONTENTS Strained quantum well subbands Direct numerical methods Empirical pseudopotential theory Principles and Approximations Elemental Band Structure Calculation Spin-orbit coupling Compound Semiconductors Charge densities Calculating the effective mass Alloys Atomic form factors Generalisation to a large basis Spin-orbit coupling within the large basis approach Computational implementation Deducing the parameters and application Isoelectronic impurities in bulk The electronic structure around point defects Microscopic electronic properties of heterostructures The superlattice unit cell Application of large basis method to superlattices Comparison with envelope-function approximation In-plane dispersion Interface coordination Strain-layered superlattices The superlattice as a perturbation Application to GaAs/AlAs superlattices Inclusion of remote bands The valence band Computational effort Superlattice dispersion and the interminiband laser Addition of electric field Application to quantum wires and dots Recent progress The quantum-wire unit cell 444

8 CONTENTS XIII 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 array s 13.9 Anti-wires and anti-dots Concluding Remarks Appendix A: Materials Parameters 459 References 461 Topic Index 477

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