From atoms to steps: the microscopic origins of crystal growth

The Burton-Cabrera-Frank (BCF) theory of crystal growth has been successful in describing a wide range of phenomena in surface physics. Typical crystal surfaces are slightly mis-oriented with respect to a facet plane; thus, the BCF theory views such systems as composed of staircaselike structures of steps separating terraces. A continuous density of adsorbed atoms (adatoms) diffuses on the terraces, and steps move by absorbing or emitting flux of adatoms. In this paper, we shed light on the microscopic origins of the BCF theory by deriving a simple, 1D version of the theory from an atomistic, stochastic lattice-gas (SLG) model. We define the time-dependent adatom density and step position as appropriate ensemble averages in the SLG model, thereby exposing the non-equilibrium statistical mechanics origins of the BCF theory. Our analysis reveals that the BCF theory is valid in a low adatom-density regime, much in the same way that an ideal gas approximation applies to dilute gasses. We find conditions under which the surface remains in a low-density regime and discuss the microscopic origin of corrections to the BCF model.