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

The Burton-Cabrera-Frank (BCF) model of step flow has been recognized as a valuable tool for describing nanoscale surface evolution of crystals. In this work, we formally derive the BCF theory from an atomistic, kinetic Monte Carlo model of a crystal surface in 1+1 dimensions with a single step, in the absence of external material deposition. In order to reconcile the discrete nature of the kMC model with the notions of a continuous density of adsorbed atoms (adatoms) and step edge in the BCF theory, we define an averaging procedure that is consistent with Boltzmann statistics. A central idea of our approach is to exploit the observation that the number of adatoms on a surface can be small for experimentally relevant temperatures. Accordingly, we (i) show that the BCF theory arises from a kMC model in which only one adatom is allowed to hop, and (ii) characterize corrections to the theory, which arise from correlations between two or more atoms. We determine (via a discrete maximum principle) initial conditions under which such corrections are negligible for all times; this allows us to interpret the BCF model as a near-equilibrium theory. Our approach reveals the atomistic origins of the material parameters entering the BCF model.