From Microscopic to Macroscopic: The Dynamics of Crystal Surface Diffusion
Anya Katsevich, New York University
In the production of thin crystalline films, which are widely used in semiconductors, photovoltaic cells and batteries, atoms are deposited onto a substrate in a process called epitaxy. The atoms subsequently diffuse across the substrate, deforming the crystal surface. Studying the dynamics of crystal surface diffusion is important because the efficiency of thin-film devices depends on the film's surface configuration.
Since the crystal surface is composed of discrete particles, a mathematical model faithful to the physical process should prescribe microscopic, discrete dynamics. As is characteristic of large microscopic systems, however, insight into the diffusion is easier to achieve by analyzing the macroscopic, continuum-level dynamics, which are governed by a partial differential equation that lends itself more readily to mathematical analysis.
In this work, we specify a microscopic model of a crystal surface whose dynamics are governed by a continuous-time Markov jump process. By taking a scaling limit as the number of particles approaches infinity, we derive the associated deterministic macroscopic dynamics while preserving the microscopic signature of the dynamics. We rely on the assumption of local equilibration, which allows us to "smooth out" random fluctuations in the limit. This means that as the number of particles increases, their configuration should adhere increasingly well to the local equilibrium probability distribution. Under this distribution, particles in small but macroscopic regions are distributed homogeneously, with a density that varies smoothly across space. However, specifying this local equilibrium distribution precisely and rigorously justifying that the microscopic process adheres to it is challenging. The distribution is tied to a certain local surface tension function, which is particularly sensitive to the way the microscopic model is defined. We show numerically and analytically that the accuracy of the local equilibration assumption depends on the transition rates of the Markov jump process.
Abstract Author(s): Anya Katsevich