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Applied Mathematics Colloquium

Friday, April 25, 11:30 am
Cullimore Lecture Hall II
New Jersey Institute of Technology

Facet evolution on supported nanostructures: the effect of finite height

Ruben Rosales

Department of Mathematics

Massachusetts Institute of Technology

Cambridge, MA

Abstract

The surface of a nanostructure relaxing on a substrate consists of a finite number of interacting steps, and often involves the expansion of facets. Prior theoretical studies of facet evolution focused on models with an infinite number of steps which neglect edge effects caused by the presence of the substrate. We show that these edge or finite height effects play an important role in the structure's macroscopic evolution under the assumption of diffusion limited kinetics and a homoepitaxial substrate. Specifically, using data from step simulations and a continuum theory, we demonstrate a switch in the time behavior of the facet position when finite height effects become significant. Our analysis and numerical simulations focus on two model systems where steps repel each other through entropic and elastic dipolar interactions. The first model is a vicinal surface consisting of a finite number of straight steps. The second model is an axisymmetric structure consisting of a finite number of circular steps --- in this last case we also include curvature effects which cause the steps to collapse under the effect of line tension. In the first case, we show that the facet expansion switches from $O(t^{1/4})\/$ behavior to $O(t^{1/5})\/$ (where $t\/$ is time) and in the second, that the behavior switches from $O(t^{1/4})\/$ to $O(t)\/$. For the axisymmetric case, we also predict analytically (through a continuum model) how the individual collapse times are modified by the effects of finite height --- under the assumption that step interactions are weak compared to the step line tension.