A one-dimensional model of interacting line defects (steps) fluctuating on a vicinal crystal was studied analytically and numerically. The goal was to formulate and validate analytical techniques for approximately solving systems of coupled non-linear stochastic differential equations governing fluctuations in surface motion. In this analytical approach, the starting point was the Burton-Cabrera-Frank model in which step motion was driven by the diffusion of adsorbed atoms on terraces and atom attachment-detachment at steps. The step energy accounted for entropic and nearest-neighbor elastic-dipole interactions. By incorporating Gaussian white noise into the equations of motion for terrace widths, large systems of stochastic differential equations were formulated under different choices of diffusion coefficients for the noise. This description was simplified via perturbation theory and linearization of the step interactions or, alternatively, via a mean-field approximation in which the widths of adjacent terraces were replaced by a self-consistent field while nonlinearities in step interaction were retained. Simplified formulas were derived for the time-dependent terrace-width distribution and its steady-state limit. The mean-field analytical predictions for the terrace-width distribution compared favorably with kinetic Monte Carlo simulations with the addition of suitably conservative white noise to the Burton-Cabrera-Frank equations.

One-Dimensional Model of Interacting-Step Fluctuations on Vicinal Surfaces: Analytical Formulas and Kinetic Monte Carlo Simulations. P.N.Patrone, T.L.Einstein, D.Margetis: Physical Review E, 2010, 82[6], 061601