Epitaxial Growth and Shape Transitions of Quantum Dots
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In this dissertation, we construct mathematical models and use mathematical tools to analyze the models to study the phenomenon of epitaxial growth and shape transitions of quantum dots. We mainly focus on two key problems of this phenomenon: (1) the energetics of shape transitions from pyramid shape to multifaceted dome shape for a single strained quantum dot during epitaxial growth; (2) the dynamics of evolution of the distribution (shape and size) for an array of quantum dots during the process of epitaxial growth and shape transitions. To understand the first problem, we construct a two-dimensional continuum model to describe the energetics of shape transitions in fully-faceted epitaxial quantum dots (strained islands) via minimization of elastic energy and surface energy at fixed volume. The elastic energy of the island is based on a third-order approximation, enabling us to consider shape transitions between pyramids, domes, multifaceted domes and asymmetric intermediate states. The energetics of the shape transitions are determined by numerically calculating the facet lengths that minimize the energy of a given island type of prescribed island volume. By comparing the energy of different island types with the same volume and analyzing the energy surface as a function of the island shape parameters, we determine the bifurcation diagram of equilibrium solutions and their stability, as well as the lowest-barrier transition pathway for the island shape as a function of increasing volume. The main result is that the shape transition from pyramid to dome to multifaceted dome occurs through sequential nucleation of facets and involves asymmetric metastable transition shapes. We also explicitly determine the effect of corner energy (facet edge energy) on shape transitions and interpret the results in terms of the relative stability of asymmetric island shapes as observed in experiment. For the second problem, we construct a Fokker-Planck reaction model to investigate the dynamics of the coupled epitaxial growth and shape transition process of an array of quantum dots. The Fokker-Planck reaction model is based on a coupled system of Fokker-Planck equations wherein the distribution of each island type is governed by its own Fokker-Planck equation for growth, with reaction terms describing the shape transitions between islands of different types including asymmetric shapes. The reaction terms for the shape transitions depend on the island size and are determined from explicit calculations of the lowest-barrier pathway for each shape transition from the results of the first problem. This mean-field model enables us to consider the kinetics of asymmetric shape transitions and study the evolution of island shape distributions during the coupled growth and transition process. Through numerical simulations over a range of growth parameters, we find multimodal and unimodal evolution modes of the shape distribution of island arrays, which depend on the external deposition flux rate and temperature rather than the shape transition rate. However, the shape transition rate governs the kinetics of shape transitions and determines the fraction of islands that form via asymmetric states, which has implications for the development of asymmetric composition profiles within alloy islands.