Numerical analysis of quantum dots nanostructures

NUMERICAL SIMULATION OF QUANTUM DOT NANOSTRUCTURES QUEK SIU SIN JERRY (B.Eng. (Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTORATE OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2003 Ph. D. Dissertation ⋅ Quek Siu Sin Jerry Acknowledgements ACKNOWLEDGEMENTS 1. A/Prof. Liu Gui Rong (Supervisor) – for his supervision, guidance and many valuable advice. And for the many learning opportunities given throughout the years. 2. Dr. Han Xu (Manager, Centre for ACES) – for his support while working in the Centre for ACES. 3. Various staff and graduate students of Centre for ACES – for sharing their ideas and knowledge. 4. Staff of NUS IT Unit, CALES – for their support in providing the computing resources. Ph. D. Dissertation ⋅ Quek Siu Sin Jerry Contents CONTENTS SUMMARY . i LIST OF SYMBOLS . iii LIST OF FIGURES v LIST OF TABLES . xi INTRODUCTION 1.1. Literature Review 1.1.1. Development of Quantum Dots in Brief . 1.1.2. Fabrication Techniques for Quantum Dots . 1.1.3. Characterization of Quantum Dots . 11 1.1.4. Self-organized Ordering of Quantum Dot Superlattices . 13 1.1.5. Modeling and Simulation of Quantum Dot Heterostructure . 15 FINITE ELEMENT FORMULATION FOR ANALYSIS OF QUANTUM DOT HETEROSTRUCTURES . 20 2.1. The SK Growth Mode, Stress Relaxation and Vertical Correlation . 20 2.2. Finite Element Modeling 22 2.2.1. 2D Axi-symmetric Model . 24 2.2.2. 3D Model 32 2.3. 3. Strain Energy Density Calculation 38 NUMERICAL FORMULATION OF QUANTUM DOT SURFACE EVOLUTION . 40 3.1. Surface Roughening 40 Ph. D. Dissertation ⋅ Quek Siu Sin Jerry Contents 3.1.1. Moving Least Squares (MLS) Interpolation . 42 3.1.2. Surface Displacement Time Marching . 44 4. FINITE ELEMENT NUMERICAL SIMULATIONS OF QUANTUM DOT HETEROSTRUCTURE . 50 4.1. 2D Axi-symmetric Finite Element Model 51 4.1.1. Model Parameters . 51 4.1.2. Results and Discussions 52 4.2. 3D Finite Element Model 62 4.2.1. Model Parameters . 62 4.2.2. Results and Discussions 64 4.3. Effects of Elastic Anisotropy on the Self Organized Ordering of Quantum Dots . 74 4.3.1. Model Parameters . 74 4.3.2. Results and Discussions 77 5. NUMERICAL SIMULATION OF QUANTUM DOT SURFACE EVOLUTION . 94 5.1. 2D Quantum Dot Island Growth . 94 5.2. Note on Sensitivity of Time Step Used . 101 5.3. 3D Quantum Dot Island Growth . 103 5.3.1. Growth of CdSe Quantum Dots on ZnSe(001) . 106 5.3.2. Growth of CdSe Quantum Dots on ZnSe(111) . 117 6. CONCLUSION . 123 REFERENCES . 130 Ph. D. Dissertation ⋅ Quek Siu Sin Jerry Contents APPENDIX A . A-1 APPENDIX B . B-1 APPENDIX C . C-1 Ph. D. Dissertation ⋅ Quek Siu Sin Jerry Summary SUMMARY Nanotechonology is the key to future’s technology and while researchers build devices and novel materials at the nanoscale, the one major obstacle is the efficient mass-production of these nano-materials that have sizes of just a few atoms. A particularly successful and interesting technique being used in recent years is by selfassembly. Based on theories of lattice mismatch, coherent islands can be formed via self-assembly that are small enough for size-quantization effects to be noticed. Such islands are often called quantum dots and they possess interesting opto-electronic properties with potential for novel device applications. The numerical simulation of quantum dot nanostructures is presented in this thesis. The finite element method is used to analyze the stress and strain fields in the quantum dot heterostructure. The stress fields play a very important role in the lattice arrangement of the heterostructures when quantum dot superlattice is fabricated. Furthermore, the elastostatic fields in the quantum dot heterostructure are vital for determining the opto-electronic properties of these structures. The finite element model used consists of a pseudo thermal expansion of the buried quantum dot and the wetting layer so as to simulate the mismatch strain caused by the difference in the lattice parameters of the dissimilar semiconductor materials. A tied contact condition is also modeled at the interface of the dissimilar materials, which basically ties the nodes belonging to the island to the matrix. Using results obtained from the finite element method, the strain energy density of a new wetting layer surface can be calculated. The strain energy density is a major i Ph. D. Dissertation ⋅ Quek Siu Sin Jerry Summary contributor to the surface chemical potential and by analyzing the distribution, one can predict the preferential site whereby new quantum dot(s) will be formed on the surface. It is shown that consideration of elastic anisotropy of the semiconductor materials has a significant effect on the vertical and lateral correlation of quantum dots in the heterostructure. Various parameters like the variation in the thickness of the cap layer was also shown to affect the correlation. The author hypothesized that at small cap layer thickness, quantum dot islands will tend to coalesce directly above the buried island. When the thickness of the cap layer increases, the islands separate and move further away from the vertically correlated position. The author also developed a code to simulate the surface evolution of the quantum dot-growing layer. A mesh-free approach is employed by using the moving least squares (MLS) interpolation method together with a forward marching finite difference method. This method provides the approximation for solving the surface diffusion equation typical of most epitaxy processes. The normal surface velocity and hence the displacements can be deduced from the surface flux. The problem then becomes an initial-value problem and is then marched forward in time using a forward marching finite difference method. This method will enable analysts to simulate real-time quantum dot formation and therefore able to predict the island shape, size and vertical and lateral ordering. ii Ph. D. Dissertation ⋅ Quek Siu Sin Jerry List of Symbols LIST OF SYMBOLS ε Strain Ω Atomic volume δ Number of atoms per unit area υ Poisson’s ratio σ Stress µ Surface chemical potential κ Surface curvature γ Surface energy µo Initial bulk chemical potential εo Lattice mismatch strain σo Lattice mismatch stress εT Thermal strain αT Thermal expansion coefficient A Anisotropic ratio Lattice parameter of island material am Lattice parameter of matrix material cij Components of material elastic matrix, c D Surface diffusivity Dm Dissipating function E Young’s modulus f Global force vector fe Element force vector G Shear modulus iii Ph. D. Dissertation ⋅ Quek Siu Sin Jerry List of Symbols H Cap layer thickness hi Quantum dot island height hw Wetting layer thickness v J Surface flux k Global stiffness matrix kB Boltzmann constant ke Element stiffness matrix Ni Finite element shape functions qi Generalized displacements T Temperature t Time TK Kinetic energy U Strain energy density Ue Strain energy Uh Approximation of global variables/generalized displacements ui x displacement component of node i Uo Initial strain energy density vi y displacement component of node i Surface normal velocity w(r) Weight function wi z displacement component of node i iv Ph. D. Dissertation ⋅ Quek Siu Sin Jerry List of Figures LIST OF FIGURES Figure 1.1 Nature of electronic states in bulk material, quantum wells and quantum dots. Top row: schematic morphology, bottom row: density of electronic states. . Figure 1.2 Schematic representation of different nanostructures fabrication processes . Figure 2.1 Schematic diagram of epitaxial strain relaxation via island formation 21 Figure 2.2 Schematic representation of basic unit of quantum dot heterostructure . 25 Figure 2.3 Geometry for 2D axi-symmetric finite element analysis . 26 Figure 2.4 2D axi-symmetric finite element mesh . 27 Figure 2.5 Tied contact in abaqus 31 Figure 2.6 3D finite element quarter model of heterostructure . 33 Figure 2.7 3D finite element model of heterostructure with part of cap layer cut away to reveal the pyramidal shaped quantum dot . 35 Figure 3.1 Schematic of surface re-construction by adding blocks to an initially planar surface 45 Figure 3.2 Flow chart of quantum dot surface evolution numerical procedure . 49 Figure 4.1 2D stress (σxx) distribution of matrix in the x direction (hi = 3nm, H = 5nm) 53 Figure 4.2 2D stress (σxx) distribution of island in the x direction (hi = 3nm, H = 5nm) 53 Figure 4.3 Strain (εxx) distribution through centre of heterostructure model (hi = 3nm, H = 5nm) 56 v Ph.D. Dissertation · Quek Siu Sin Jerry Chapter 63. Stern, M.B., Craighead, H.G., Liao, P.F. and Mankiewich, P.M., Fabrication of 20nm structures in GaAs, Applied Physics Letters 45(4), pp.410-412. 1984. 64. Strassburg, M., Kutzer, V., Pohl, U. W., Hoffmann, A., Broser, I., Ledentsov, N. N., Bimberg, D., Rosenauer, A., Fischer, U., Gerthsen, D., Krestnikov, I. L., Maximov, M. V., Kop’ev, P. S. and Alferov, Zh. I., Gain studies of (Cd, Zn)Se quantum islands in a ZnSe matrix, Applied Physics Letters 72(8), pp.942-944. 1998. 65. Tersoff, J., Teichert, C. and Lagally, M. G., Self-organization in growth of quantum dot superlattices, Physical Review Letter 76(10), pp.1675-1678. 1996. 66. Warren, A.C., Plotnik, I., Anderson, E.H., Schattenburg, M.L., Antoniadis, D.A. and Smith, H.I., Fabrication of sub-100nm linewidth periodic structures for study of quantum effects from interference and confinement in Si inversion layers, Journal of Vacuum Science & Technology B 4(1), pp.365-368. 1986 67. Werner, J., Kapon, E., Stoffel, N.G., Colas, E., Schwarz, S.A., Schwartz, C.L. and Andreadakis, N., Integrated external cavity GaAs/AlGaAs lasers using selective quantum well disordering, Applied Physics Letters 55(6), pp.540-542. 1989. 68. Wu, W., Tucker, J.R., Solomon, G.S. and Harris, Jr., J.S., Atom-resolved scanning tunneling microscopy of vertically ordered InAs quantum dots, Applied Physics Letters 71(8), pp.1083-1085. 1997. 69. Xie, Q., Madhukar, A., Chen, P. and Kobayashi, N. P., Vertically self-organized InAs quantum box islands on GaAs(100), Physical Review Letter 75(13), pp.2542-2545. 1995. 138 Ph.D. Dissertation · Quek Siu Sin Jerry Chapter 70. Yao, J. Y., Andersson, T. G. and Dunlop, G. L., The interfacial morphology of strained epitaxial InxGa1-xAs/GaAs, Journal of Applied Physics 69(4), pp.2224-2230. 1991. 71. Zhang, Y.W. and Bower, A.F., Numerical simulation of island formation in a coherent strained epitaxial thin film system, Journal of the Mechanics and Physics of Solids 47, 2273-2297. 1999. 72. Zhang, Y.W., Bower, A.F., Xia, L. and Shih, C.F., Three dimensional finite element analysis of the evolution of voids and thin films by strain and electromigration induced surface diffusion, Journal of the Mechanics and Physics of Solids 47, pp.173199. 1999. 139 Ph.D. Dissertation · Quek Siu Sin Jerry Appendix A APPENDIX A FINITE ELEMENT PROCEDURE AND FORMULATION Standard finite element method usually begins with discretizing or meshing of the domain of interest into elements. For complicated geometry, this is more often than not done with pre-processors, which at the same time allocate unique nodal and element identifications to create the connectivity of the element. This connectivity information is crucial in the assembly of the elemental matrices. For each element, the displacement (or any other global variable to be calculated) within the element is assumed by polynomial interpolation using the displacements at the nodes as nd U h ( x, y, z ) = ∑ N i ( x, y, z ) d i = N ( x, y , z )d e (A.1) i =1 where the superscript stands for approximation, nd is the number of the nodes forming the element, and di is the nodal displacement at the ith node which is the unknown, and can be expressed in a general form of ⎧ d1 ⎪d ⎪ di = ⎨ ⎪ M ⎪d n f ⎩ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ (A.2) where nf is the number of degrees of freedom (DOFs) at a node. Note that the displacement components can also consist of rotations for structures of beams and plates. The vector de in Eq.(A.1) is the displacement vector for the entire element and has the form of A-1 Ph.D. Dissertation · Quek Siu Sin Jerry Appendix A ⎧ d1 ⎫ → displacements at node ⎪d ⎪ ⎪ ⎪ → displacements at node de = ⎨ ⎬ M ⎪ M ⎪ M ⎪d nd ⎪ → displacements at node nd ⎩ ⎭ (A.3) Therefore the total DOFs for the entire element is nd×nf. In Eq.(A.1), N is a matrix of shape functions for the nodes in the element, which are predefined to assume the shapes of the displacement variations with respect to the local coordinates. For a nodal, 2D quadrilateral, solid element, the shape function is given as Ni = (1 + ξiξ )(1 + ηiη ) (A.4) Similarly for a nodal, 3D hexahedral, solid element, the shape function is given as N i = (1 + ξξ i )(1 + ηη i )(1 + ζζ i ) (A.5) Note that ξ, η and ζ are local coordinates defined in each element. Details on how these shape functions can be obtained is available in most finite element textbooks (For example: Liu and Quek, 2003). To derive discretized system equations, the Hamilton’s principle may be used. Mathematically, the principle states: t2 δ ∫ Ldt = t1 (A.6) where L is the Lagrangian functional obtained using a set of admissible time histories of displacements, and it consists of A-2 Ph.D. Dissertation · Quek Siu Sin Jerry Appendix A L=T−Π+Wf (A.7) where T is the kinetic energy, Π is the potential energy (for our purpose, it is the elastic strain energy), and Wf is the work done by the external forces. Note that an admissible displacement must satisfy the following conditions: (a) the compatibility equations (b) the essential or the kinematic boundary conditions and (c) the conditions at initial (t1) and final time (t2), The kinetic energy of the entire problem domain is defined in the following integral form of T= ρU& T U& dV V∫ (A.8) where V represents the whole volume of the solid, and U is the set of admissible time histories of displacements. The strain energy in the entire domain of elastic solids and structures can be expressed as Π= 1 ε T σdV = ∫ ε T cεdV ∫ 2V 2V (A.9) where ε is the strains that are obtained using the set of admissible time histories of displacements. The work done by the external forces over the set of admissible time histories of displacements can be obtained by W f = ∫ U T f b dV + V ∫ U T f s dS f (A.10) Sf where Sf represents the surface of the solid on which the surface forces are prescribed. A-3 Ph.D. Dissertation · Quek Siu Sin Jerry Appendix A With the shape functions obtained, substitute the interpolation of the nodes, Eq. (A.1), and the usual strain-displacement equation into the strain energy term (Eq. (A.9)) to obtain Π= 1 εT cεdV = ∫ dTe BT cBd e dV = dTe ( ∫ BT cBdV )d e ∫ Ve Ve Ve (A.11) where the subscribt e stands for the element. Note that the volume integration over the global domain has been changed to that over the elements. This can be done because we assume that the displacement field is compatible on all edges between the elements. B is called the strain matrix, defined by B = LN (A.12) where L is the differential operator. For 2D elements, L is defined as ⎤ ⎡ ∂ / ∂x ⎢ L=⎢ ∂ / ∂y ⎥⎥ ⎢⎣∂ / ∂y ∂ / ∂x ⎥⎦ (A.13) 0 ⎤ ⎡ ∂ / ∂x ⎢ 0 ⎥⎥ ∂ / ∂y ⎢ ⎢ 0 ∂ / ∂z ⎥ L=⎢ ⎥ ∂ / ∂z ∂ / ∂y ⎥ ⎢ ⎢ ∂ / ∂z ∂ / ∂x ⎥ ⎢ ⎥ ⎦⎥ ⎣⎢∂ / ∂y ∂ / ∂x (A.14) k e = ∫ BT cBdV (A.15) and for 3D elements, By denoting Ve which are called element stiffness matrix, Eq. (A.11) can be rewritten as Π= T d e kd e A-4 (A.16) Ph.D. Dissertation · Quek Siu Sin Jerry Appendix A By substituting Eq. (A.1) into Eq. (A.8), the kinetic energy can be expressed as T= & TU & dV = ρd& T NT Nd& dV = d& T ( ρ NT NdV )d& ρU e e e ∫ e ∫ Ve V∫e Ve (A.17) By denoting m e = ∫ ρ NT NdV (A.18) Ve which is call the mass matrix of the element, Eq. (A.17) can be rewritten as T= &T & d e m ed e (A.19) Finally to obtain the work done by external forces, Eq. (A.1) is substituted into Eq. (A.10): W f = ∫ dTe NT fb dV + ∫ dTe NT f s dS = dTe ( ∫ NT fb dV ) + dTe ( ∫ NT f s dS ) Ve Se Ve Se (A.20) where the surface integration is performed only for elements on force boundary of the problem domain. By denoting Fb = ∫ NT fb dV Ve (A.21) and Fs = ∫ NT f s dS Se (A.22) Eq. (A.20) can then be rewritten as W f = d Te Fb + d Te Fs = d Te f e (A.23) Fb and Fs are the nodal forces acting on the nodes of the elements, which are equivalent to the body forces and surface forces applied on the element in terms of A-5 Ph.D. Dissertation · Quek Siu Sin Jerry Appendix A the work done on a virtual displacement. These two nodal force vectors can then be added up to form the total nodal force vector fe. fe = Fb + Fs (A.24) Substituting Eqs (A.16), (A.19) and (A.23) into Lagrangian functional, L (Eq.(A.7)), we have 1 L = d& Te m ed& e − dTe k ed e + dTe fe 2 (A.25) Applying the Hamilton's principle (Eq. (A.6)), we have t2 δ ∫ ( d& Te m e d& e − d Te k e d e + d Te f e )dt = t1 (A.26) Note that the variation and the integration operators are interchangeable, hence we obtain ∫ t2 t1 (δd& Te m e d& e − δd Te k e d e + δd Te f e )dt = (A.27) In Eq. (A.27), the variation and the differentiation with time are also interchangeable, that is δd& Te = δ ( dd Te d ) = (δd Te ) dt dt (A.28) Hence, by substituting Eq. (A.28) into Eq. (A.27), and integrating the first term by parts, we obtain ∫ t2 t1 t2 t2 t2 t1 t1 && dt = − δ dT m d δ d& Te m ed& e dt = δ dTe m e d& e t − ∫ δ dTe m ed e ∫ e e&&edt 142431 =0 (A.29) Note that in deriving Eq. (A.29) above, δde = at t1 and t2 have been used, which leads to the vanishing of the first term in the right-hand-side. This is because the initial condition at t1 and final condition at t2 have to be satisfied for any de A-6 Ph.D. Dissertation · Quek Siu Sin Jerry Appendix A (admissible conditions (c) required by the Hamilton’s principle), and no variation at t1 and t2 is allowed. Substituting Eq. (A.29) into Eq. (A.27) leads to ∫ t2 t1 && − kd + f )dt = δd Te (−m e d e e e (A.30) In order to have the integration in Eq. (A.30) to be zero for an arbitrary integrand, the integrand itself has to vanish, i.e., && − kd + f ) = δ dTe (−m ed e e e (A.31) Due to the arbitrary nature of the variation of the displacements, the only insurance for Eq.(A.31) to be satisfied is && = f k ed e + m ed e e (A.32) The above equation, Eq. (A.32) is the FEM equation for an element, while ke and me are the stiffness and mass matrix for the element, and fe is the element force vector of total external forces acting on the nodes of the element. All these element matrices and vectors can be obtained simply by integration for given shape functions of displacements. It should be noted that in this project, only static finite element analysis is carried out and therefore the kinetic energy term and hence the mass matrix does not come into play. After obtaining the element matrices, transformation from the local coordinate system into a global coordinate system is usually necessary by using the following equations: K e = TT k e T A-7 (A.33) Ph.D. Dissertation · Quek Siu Sin Jerry Appendix A M e = TT m e T (A.34) Fe = T T f e (A.35) where T is the transformation matrix. After transformation, the equations for all the individual elements can be assembled together to form the global finite element system equation: && = F KD + MD (A.36) where K and M is the globe stiffness and mass matrix, and D is a vector of all the displacements at all the nodes in the entire problem domain, and F is a vector of all the equivalent nodal force vectors. The process of the assembly is simply by adding up the contributions from all the elements connected at a node. Boundary conditions are then imposed on Eq. (A.36) and the stiffness matrix will then have Symmetric Positive Definite (SPD) property. The nodal displacements can then be retrieved by solving the global finite element equation. A-8 Ph.D. Dissertation · Quek Siu Sin Jerry Appendix B APPENDIX B STRAIN ENERGY DENSITY DISTRIBUTION OF INAS/GAAS(111) WETTING SURFACE Figure B1 Strain energy density distribution of top surface of second InAs wetting layer with the dimensionless spacer thickness, H/h = 2; island base is 20 x 20 nm; h = 10 nm; crystal orientation of [111] for the normal of the top surface of the substrate and spacer-layer B-1 Ph.D. Dissertation · Quek Siu Sin Jerry Appendix B Figure B2 Strain energy density distribution of top surface of second InAs wetting layer with the dimensionless spacer thickness, H/h = 3; island base is 20 x 20 nm; h = 10 nm; crystal orientation of [111] for the normal of the top surface of the substrate and spacer-layer B-2 Ph.D. Dissertation · Quek Siu Sin Jerry Appendix B Figure B3 Strain energy density distribution of top surface of second InAs wetting layer with the dimensionless spacer thickness, H/h = 4; island base is 20 x 20 nm; h = 10 nm; crystal orientation of [111] for the normal of the top surface of the substrate and spacer-layer B-3 Ph.D. Dissertation · Quek Siu Sin Jerry Appendix C APPENDIX C STRAIN ENERGY DENSITY DISTRIBUTION OF INAS/GAAS(113) WETTING SURFACE Figure C1 Strain energy density distribution of top surface of second InAs wetting layer with the dimensionless spacer thickness, H/h = 2; island base is 20 x 20 nm; h = 10 nm; crystal orientation of [113] for the normal of the top surface of the substrate and spacer-layer C-1 Ph.D. Dissertation · Quek Siu Sin Jerry Appendix C Figure C2 Strain energy density distribution of top surface of second InAs wetting layer with the dimensionless spacer thickness, H/h = 3; island base is 20 x 20 nm; h = 10 nm; crystal orientation of [113] for the normal of the top surface of the substrate and spacer-layer C-2 Ph.D. Dissertation · Quek Siu Sin Jerry Appendix C Figure C3 Strain energy density distribution of top surface of second InAs wetting layer with the dimensionless spacer thickness, H/h = 4; island base is 20 x 20 nm; h = 10 nm; crystal orientation of [113] for the normal of the top surface of the substrate and spacer-layer C-3 [...]... properties of quantum dots resemble that of an atom in a cage The study of single quantum dots and ensembles of quantum dots presents a new chapter in fundamental physics A typical size of such a dot is 10nm and it may contain 104 or more atoms 6 Ph.D Dissertation · Quek Siu Sin Jerry Chapter 1 Of course, having discovered and understood all these theories, to fabricate and make use of quantum dots for... the evolution of the quantum dot island with respect to time can be studied which further enhances the understanding of the spatial distribution, shape and size of quantum dots This thesis consists of first a literature review of the various developments in the field of quantum dot nanostructures as well as a review of computational methods typically used in the numerical simulation of such structures... density of electronic states of quantum dots becoming discrete as shown in Figure 1.1 Bulk Film (Quantum well) D(E) D(E) Ec E Quantum dot D(E) E Ec Ec E Figure 1.1 Nature of electronic states in bulk material, quantum wells and quantum dots Top row: schematic morphology, bottom row: density of electronic states against energy of electronic states When the carrier motion in a solid is limited in layer of. .. explained from a kinetic or energetics point of view This leads to the motivation behind using computational simulation to analyze certain aspects of the quantum dot and thus gain a better understanding of the process If one considers just the growth of a single epitaxial layer of quantum dots on the substrate, the positions of growth as well as the size of the quantum dots can be said to be highly statistical... directions of where the minimas are formed 81 Table 5.1 Model parameters for growth simulation of quantum dot layer 96 xi Ph.D Dissertation · Quek Siu Sin Jerry Chapter 1 1 INTRODUCTION This project involves the numerical computation of various aspects of quantum dot nanostructures formation Quantum dots can be considered to be clusters of atoms in the nano-meter scale usually made up of semiconducting... dimensionality – to quantum wires (Kapon et al, 1989) and quantum dots A complete reduction of the remaining infinite extension of a quantum well in two dimensions to atomic values lead to carrier localization in all three dimensions and breakdown of classical band structure energy level model The resulting energy level structure of quantum dots is discrete (Figure 1.1), like in atomic physics, and many of the... 4.24 Detailed fringe plot of the center region of top surface of second CdSe wetting layer with spacer-layer thickness of H/hi = 3 80 Figure 4.25 Schematic diagram of spatial correlation of multiple layers of CdSe/ZnSe(001) quantum dot structure in a tetragonal body centered arrangement 82 Figure 4.26 Fringe plots of strain energy density of top surface of second InAs wetting layer... usually difficult to interpret 1.1.4 Self-organized Ordering of Quantum Dot Superlattices The positions of growth as well as the size of the quantum dots on an epitaxial layer by itself can be said to be highly statistical in nature Without a regular arrangement of uniformly sized quantum dots, the advantage of the discrete energy level the dots possessed cannot be fully utilized to its full potential... strain in quantum dots as well as the simulation of quantum dot island formation Strain Distribution The elastostatic distribution in and around a quantum dot structure is often of interest to many researchers because firstly, the induced stress fields of buried islands in multiple stacks affects the spatial correlation of islands in subsequent layers; secondly, the strain fields in the quantum dots affect... Figure 4.28 Position and depth of minima with various spacer-layer thickness of CdSe/ZnSe(111) 86 Figure 4.29 Schematic diagram of spatial correlation of multiple layers of CdSe/ZnSe(111) quantum dot structure with the quantum dots displaced from the center in the [1-10] direction 87 Figure 4.30 Strain energy density distribution of top surface of second CdSe wetting layer with . understanding of the process. If one considers just the growth of a single epitaxial layer of quantum dots on the substrate, the positions of growth as well as the size of the quantum dots can. Techniques for Quantum Dots 8 1.1.3. Characterization of Quantum Dots 11 1.1.4. Self-organized Ordering of Quantum Dot Superlattices 13 1.1.5. Modeling and Simulation of Quantum Dot Heterostructure. 4.3. Effects of Elastic Anisotropy on the Self Organized Ordering of Quantum Dots 74 4.3.1. Model Parameters 74 4.3.2. Results and Discussions 77 5. NUMERICAL SIMULATION OF QUANTUM DOT SURFACE