Markov Models of Steady Crystal Growth

Abstract

We consider solid-on-solid models of two- and three-dimensional crystal growth described by Markov rate processes whose states are representations of the entire crystal surface and whose transitions are captures of single atoms by this surface or escapes of atoms. Under natural conditions on the transition probability rates, we prove ergodicity of the Markov process. This implies the existence of states of steady growth and statistically stationary surface structure. Microscopic surface instabilities do not therefore occur under such conditions. For two dimensions, our conditions are very general; for three dimensions, they are less general.