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K · P THEORY of SEMICONDUCTOR NANOSTRUCTURES k p THEORY OF SEMICONDUCTOR · NANOSTRUCTURES by CALIN GALERIU, B.S., M.S., M.A. A Dissertation Submitted to the Faculty of the WORCESTER POLYTECHNIC INSTITUTE in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in Physics November 30, 2005 APPROVED: Professor Lok C. Lew Yan Voon, Dissertation Advisor Professor Richard S. Quimby, Committee Member Professor Florin Catrina, Committee Member Abstract The objective of this project was to extend fundamentally the current k p theory by applying · the Burt-Foreman formalism, rather than the conventional Luttinger-Kohn formalism, to a number of novel nanostructure geometries. The theory itself was extended in two ways. First in the application of the Burt-Foreman theory to computing the momentum matrix elements. Second in the development of a new formulation of the multiband k p Hamiltonian describing · cylindrical quantum dots. A number of new and interesting results have been obtained. The computational imple- mentation using the finite difference method of the Burt-Foreman theory for two dimensional nanostructures has confirmed that a non-uniform grid is much more efficient, as had been ob- tained by others in one dimensional nanostructures. In addition we have demonstrated that the multiband problem can be very effectively and efficiently solved with commercial software (FEMLAB). Two of the most important physical results obtained and discussed in the dissertation are the following. One is the first ab initio demonstration of possible electron localization in a nanowire superlattice in a barrier material, using a full numerical solution to the one band k p · equation. The second is the demonstration of the exactness of the Sercel-Vahala transformation for cylindrical wurtzite nanostructures. Comparison of the subsequent calculations to experi- mental data on CdSe nanorods revealed the important role of the linear spin splitting term in the wurtzite valence band. ii Acknowledgments I would like to thank my advisor, Professor Lok C. Lew Yan Voon, for his patient guidance and constant encouragement during the course of my research. I would like to thank all members of our research group, and especially Professor Morten Willatzen, Professor Roderik Melnik, Dr. Smagul Karazhanov, and Benny Lassen. I would like to thank all faculty members, staff persons, and fellow students at Worcester Polytechnic Institute. I would like to thank my family, and especially my wife Luminit¸a, for her patience and support during my years of graduate studies. I would like to thank my cat Pisoanc˘a, who has kept me company during many hours of typing and programming. The research was supported through grants from the National Science Foundation, Grant No. DMR-9984059 and Grant No. DMR-0454849, and by the Department of Physics at Worces- ter Polytechnic Institute. Copyright c 2005 by C˘alin Galeriu, all rights reserved. iii Contents Abstract . ii Acknowledgments . iii List of Figures . viii List of Tables . xi 1 Introduction 1 2 Symmetry and the calculation of matrix elements 5 2.1 Introduction . 5 2.2 Selection rules for matrix elements . 5 2.3 Wigner-Eckart theorem . 6 2.4 Momentum matrix elements at the Γ point in ZB structures . 7 2.4.1 Γ Γ Γ momentum matrix elements . 8 h 15j 15j 1i 2.4.2 Γ Γ Γ momentum matrix elements . 8 h 15j 15j 15i 2.4.3 Γ Γ Γ momentum matrix elements . 8 h 15j 15j 25i 2.4.4 Γ Γ Γ momentum matrix elements . 8 h 15j 15j 12i 2.5 Momentum matrix elements at the Γ point in DM structures . 9 2.6 Momentum matrix elements at the Γ point in WZ structures . 9 2.6.1 Γ Γ Γ momentum matrix elements . 10 h 1j 1j 1i 2.6.2 Γ Γ Γ momentum matrix elements . 10 h 1j 6j 6i 3 k p theory - Kane 11 · 3.1 Introduction . 11 3.2 1-band model . 11 3.3 2-band model, CB-VB coupling only . 13 3.3.1 Electron in GaAs . 13 3.3.2 Light hole in GaAs . 13 3.4 4-band model, CB-VB coupling only . 13 3.5 3-band model . 15 iv 3.5.1 k in the [1,0,0] direction . 17 3.5.2 k in the [1,1,1] direction . 18 3.6 6-band model . 19 3.7 8-band model, CB-VB coupling only . 22 3.8 8-band model . 25 3.9 Appendix A. Degenerate perturbation theory . 29 3.10 Appendix B. L¨owdin perturbation theory . 31 4 k p theory - Burt 33 · 4.1 Introduction . 33 4.2 Burt's theory . 34 4.3 Envelope function equations . 35 4.4 Homogeneous semiconductor . 37 4.5 Burt's Hamiltonian . 38 4.6 Burt's Hamiltonian for ZB structures . 38 4.6.1 s = S; γ = Γ15; s0 = S . 39 4.6.2 s = S; γ = Γ15; s0 = X . 39 4.6.3 s = X; γ = Γ15; s0 = X . 40 4.6.4 s = X; γ = Γ25; s0 = X . 41 4.6.5 s = X; γ = Γ1; s0 = X . 41 4.6.6 s = X; γ = Γ12; s0 = X . 42 4.6.7 s = X; γ = Γ15; s0 = Y . 42 4.6.8 s = X; γ = Γ25; s0 = Y . 43 4.6.9 s = X; γ = Γ1; s0 = Y . 43 4.6.10 s = X; γ = Γ12; s0 = Y . 43 4.7 Conclusions . 44 4.7.1 Foreman's notation . 45 4.8 Symmetrization versus Burt's Hamiltonian . 45 v 5 Momentum Matrix Elements 47 5.1 Introduction . 47 5.2 Calculation of Ψ(N)(r) p^ Ψ(M)(r) . 48 h j j i ¯ (N) @H¯ ¯ (M) 5.3 Calculation of Ψ (r) ^ Ψ (r) . 51 h j @k j i 5.4 Normalization of the Wave Function . 51 5.5 The Effect of Symmetrization on the MME . 52 5.6 Appendix. The Projection Operator Method . 54 6 k p Theory under a Change of Basis 59 · 6.1 Introduction . 59 6.2 Mathematical Formalism . 59 6.3 Diagonalization of the Spin-Orbit Interaction . 60 6.3.1 ZB semiconductors . 61 6.3.2 WZ semiconductors . 62 6.4 Rotation of the Cartesian Axes . 62 6.4.1 Hamiltonian with no spin . 64 6.4.2 Hamiltonian with spin . 65 7 Strain in Cylindrical Heterostructures 67 7.1 Elasticity Theory in Cartesian Coordinates . 67 7.2 Elasticity Theory in Cylindrical Coordinates . 69 7.3 Plane Deformation with Cylindrical Symmetry . 70 7.4 Infinite Embedded Cylindrical Wire with Cubic Structure . 71 7.4.1 The Wire . 73 7.4.2 The Substrate . 74 7.4.3 The Shrink Fit . 74 8 One-Band k p Calculations - Quantum Ice Cream Dot 75 · 8.1 Introduction . 75 8.2 Effect of Dirichlet and van Neumann boundary conditions . 76 vi 9 One-Band k p Calculations - Elliptical Dot 81 · 9.1 Introduction . 81 9.2 The Eigenstates of an Elliptical Quantum Dot . 81 10 One-Band k p Calculations - Nanowire Superlattice 85 · 10.1 Introduction . 85 10.2 Theory . 86 10.3 Computational Models . 87 10.4 FDM applied to a unit cell with PBC . 88 10.5 FDM applied to a finite number of unit cells . 92 10.6 Equivalent Kronig-Penney model . 93 10.7 Results and Discussions . 94 10.7.1 Physical Applications: Inversion . 99 10.7.2 Physical Applications: Embedded nanowire . 100 10.8 Conclusions . 101 11 Eight-Band k p Calculations - ZB Quantum Well 103 · 11.1 Introduction . 103 11.2 The Hamiltonian . 104 11.3 The Finite Difference Method . ..
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