Generalized Burton-Cabrera-Frank theory for growth and equilibration on stepped surfaces.

Abstract

The theory of Burton, Cabrera, and Frank [Philos. Trans. R. Soc. London Ser. A 243, 299 (1951)] for crystal growth on stepped surfaces is extended to include adatom interactions that are responsible for incipient island formation. By using a reaction-diffusion formulation of growth, a system of nonlinear diffusion equations is derived for the concentrations of surface clusters containing up to a prescribed number of atoms. By an appropriate choice of cells in the coarse graining of the lattice model, we show that the ostensibly one-dimensional equations obtained in the continuum limit in fact contain two-dimensional information averaged over the lateral length of the terrace. This allows the evolution equation for the averages of the various species to be decoupled from the higher-order correlation functions and simplifies the specification of the boundary conditions at the step edges. We have previously used this formalism to derive a nonlinear term for the formation of diatomic islands, from which we were able to predict quantitatively the transition to a step propagation mode for growth on stepped surfaces. Here, we extend the applicability of the original model away from growth by step advancement by allowing the surface atoms to form up to ten-atom islands. Furthermore, by including the breakup of atoms from the islands and adatom capture and detachment kinetics at the step edges, the model is capable of describing the relaxation of the surface toward equilibrium upon cessation of growth.