Connection of Kinetic Monte Carlo Model for Surfaces to One-Step Flow Theory in 1+1 Dimensions

The Burton–Cabrera–Frank (BCF) theory of step flow has been recognized as a valuable tool for describing nanoscale evolution of crystal surfaces. We formally derive a single-step BCF-type model from an atomistic, kinetic Monte Carlo (KMC) model of a crystal surface in 1+1 dimensions without external material deposition. Through an averaging procedure, consistent with Boltzmann statistics, we introduce definitions of the continuous quantities of the BCF theory, i.e., the step position and density of adsorbed atoms (adatoms), as expectation values taken over the discrete probabilities of the KMC model. The equations of our BCF-type model describe the time evolution of these expectation values. A central idea of our approach is to exploit the observation that the number of adatoms on a surface is typically small at experimentally relevant temperatures. Accordingly, we (i) show that our BCF-type theory arises from a KMC model in which only one adatom is allowed to move, and (ii) characterize corrections to the theory, which come from correlations between two or more atoms. We derive (via a discrete maximum principle) a criterion on the initial conditions under which such corrections are negligible for all times; this allows us to interpret our BCF-type model as a near-equilibrium theory. Our approach reveals the atomistic origins of the material parameters entering the BCF model.