Coarsening Dynamics of Faceted Crystal Surfaces

Northwestern University, Evanston IL

Investigators

Abstract

The study of evolving faceted crystal surfaces has received significant impetus from the discovery of nano-scale faceted pyramidal islands (quantum dots) on the surface of epitaxially grown thin crystal films. The physics of such evolutions involves an intricate interplay between the surface energy, the bulk elastic energy, and the deposition. In particular, it is well known from experiment and numerical studies of model equations that the presence of bulk driving forces has dramatic consequences on the coarsening dynamics of faceted crystal growth; e.g., accelerated coarsening, and significant change in surface morphology. We propose a theoretical study of a class of bulk-driven faceting crystal growth models aimed at elucidating this transition. Our equations arise from continuum models based on strongly anisotropic surface energies which yield faceted structures. The dominant mass transport mechanisms considered include either attachment kinetics (evaporation/condensation) or surface diffusion. The resulting equations have the form of a Hamilton-Jacobi equation supplemented by a Cahn-Hilliard-like operator. Of particular note is a new model equation, which incorporates bulk elastic energy into the surface diffusion-controlled evolution of a faceted, epitaxially strained thin film. The PI studies the coarsening dynamics of these model equations for both one- and two-dimensional surface morphologies. The coarsening exponents asociated with the growth of the characteristic length(s) of the faceted surface are sought. Also, the specific surface morphologies which emerge are analyzed. The approach is based on a novel sharp interface theory which has recently been developed by the PI for the Hamilton-Jacobi-Aviles-Giga equation. It is based on a matched asymptotic analysis that takes advantage of the interfacial structure which emerges during faceted evolution; extended facets meet at narrow rounded edges. The resulting theory yields an intrinsic (sharp-interface) characterization of the evolution of the associated edge network. Ordered arrays of quantum dots are envisioned as the basis for the next generation of supercomputer chips, each dot serving as a "bit." An understanding of the dynamic evolution of such quantum dot arrays is critical for such an ambition to be realized. However, the theory of such dynamics is in its infancy; e.g., during molecular beam epitaxy or liquid-phase growth. The scope of research envisioned in the proposal contributes to our understanding and mastery of this technologically important problem. The PI is also undertaking the research training of a graduate and an undergraduate student as part of this proposal, thereby enhancing the mathematical research and education infrastructure.