Quantum Wells, Wires and DotsTheoretical and Computational Physics of Semiconductor Nanostructures
Quantum Wells, Wires and Dots Second Edition: Theoretical and Computational Physics of Semiconductor Nanostructures provides all the essential information, both theoretical and computational, for complete beginners to develop an understanding of how the electronic, optical and transport properties of quantum wells, wires and dots are calculated. Readers are lead through a series of simple theoretical and computational examples giving solid foundations from which they will gain the confidence to initiate theoretical investigations or explanations of their own. Emphasis on combining the analysis and interpretation of experimental data with the development of theoretical ideas Complementary to the more standard texts Aimed at the physics community at large, rather than just the low-dimensional semiconductor expert The text present solutions for a large number of real situations Presented in a lucid style with easy to follow steps related to accompanying illustrative examples
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.
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.email@example.com or firstname.lastname@example.orgPaul 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.
This book is aimed at providing all of the essential information, both theoretical and computational, in order that the reader can, starting from essentially nothing, understand how the electronic, optical and transport properties of semiconductor heterostructures are calculated. However, perhaps more importantly, starting from this low common denominator, this text is designed to lead the reader through a series of simple example theoretical and computational implementations, and slowly build from solid foundations, to a level where the reader can begin to initiate theoretical investigations or explanations of their own. The author believes that there are two aspects to theoretical work, with the first being to analyse and interpret experimental data, while the second is to advance new ideas. His hope is that this book will certainly facilitate the former and will at least provide the knowledge and skills base from which quantified predictions can be developed from the beginnings of an idea. Written in the style of a mathematics course text, it is hoped that this book will appeal to readers from within as well as outside the low dimensional semiconductor community. Some of the examples developed are relevant to the semiconductor community at large, while the microscopic calculations presented could be of interest to other areas of condensed matter, such as carbon nanostructures, high-temperature superconductors, etc. New material in this second edition includes: sections on effects of magnetic fields on quantum wells excited impurity levels screening of the optical phonon interaction acoustic and optical deformation potential scattering spin-orbit coupling in the pseudopotential calculation and New chapters on strained quantum wells and k.p theory. Aimed at postgraduate students of semiconductor and condensed matter physics, the book will be invaluable to all those researching in academic and industrial laboratories worldwide.
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