Pattern Evolution of Self-Assembled Quantum Dots Under Biaxial Stresses
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0921-T07-08
Pattern Evolution of Self-Assembled Quantum Dots Under Biaxial Stresses Yaoyu Pang, and Rui Huang Department of Aerospace Engineering and Engineering Mechanics, University of Texas, Austin, TX, 78712
ABSTRACT A stressed epitaxial film can undergo surface instability. The stress field and the interface interaction have profound effects on the dynamics of surface evolution that leads to selfassembled quantum dots. In this paper, by using a nonlinear evolution equation, we investigate pattern evolution of self-assembled quantum dots under general biaxial stresses. It is found that the shape of quantum dots and their spatial ordering are strongly influenced by the relative magnitudes of the biaxial stresses. Linear perturbation analysis and nonlinear numerical simulations are conducted to elucidate the effect of stress anisotropy on the process of selfassembly that selects different patterns. INTRODUCTION Self-assembled nanostructures in epitaxial systems are of great interests for both theoretical understanding and practical applications. An epitaxial thin film is inherently stressed due to lattice mismatch between the film and the substrate. The competition between surface energy and strain energy drives surface instability of an initially flat film [1-3]. In addition, the interface between the film and the substrate plays an important role in the later stage of surface evolution [4]. Recently, we developed a nonlinear evolution equation [5] taking into account the effects of the second-order stress field and a nonlinear wetting potential. Numerical simulations showed that the nonlinear stress field alone induces “blow-up” instability, leading to crack-like grooving in 2D and circular pit-like morphology in 3D. With the wetting potential, the blow-up is suppressed, leading to an array of discrete islands on top of a thin wetting layer. Under an equi-biaxial stress, the system is isotropic (material anisotropy was ignored), and the model predicted self-assembly of circular islands with no spatial ordering (as shown in Figure 1).
x2 x1
t=0
t = 50
t = 200
t = 1000
t = 10000
Figure 1. Evolution of surface morphology from a numerical simulation under an equi-biaxial stress ( c = 1 ), starting from a random initial perturbation at t = 0. A bright spot represents a crest of the surface. The time is normalized by a time scale τ , and the length by a length scale L , both defined in text.
It has been shown that material anisotropy (e.g., crystal elasticity, surface energy, and surface mobility) plays a significant role in the processes of self-organization and shape transition of epitaxial quantum dots [6,7]. On the other hand, the effect of anisotropy in the general biaxial stress state has received less attention. Recently, Berger et al. [8] and Paret [9] showed that, during a melting-crystallization process, a biaxially stressed semi-infinite solid can develop into a rich variety of patterns, especially when the applied stress is tensile in one direction and compressive in the orthogonal direction. In
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