Coarsening of step bunches in step flow growth: a reaction–diffusion model and its travelling wave solutions

Abstract

We consider a model of the Frank impurity mechanism in step flow growth based on reaction–diffusion equations. The equations, based on the BCF model of crystal growth, are designed to capture the physics of both the diffusion of adatoms, and the appearance of impurities on the crystal surface following the model proposed by Frank. The model was originally considered by Kandel and Weeks [Physica D 66 (1993) 78; Phys. Rev. B 49 (1994) 5554; Phys. Rev. B 52 (1995) 2154]. The model is a fundamentally two-dimensional one. It exhibits quick formation of step bunches, followed by a much longer period of coarsening, in the direction lateral to the step flow direction. Using tools from the theory of reaction–diffusion equations we are able to reduce the complicated pattern formation of the model to a simple dynamical picture. In most regions, step bunches form quickly, and these step bunches are equilibrium solutions of the equations. As the step bunches form, transitions form between different step bunches. These transitions are travelling wave solutions of the equations. These travelling waves govern the coarsening process. The spatial patterns coarsen as the different travelling waves march across the surface of the crystal and encounter and annihilate each other. By classifying the equilibrium states and travelling waves of the equations we can collapse the number of travelling waves to a small reduced set. This reduction of the dynamics to travelling waves from a small set of equivalent classes, is the basis of a simple reduced model which, despite its simplicity, can capture in entirety the coarsening process arising from the original equations. Using this reduced model, we can investigate large scale nature of the coarsening process, and behaviour of the coarsening process, and reconsider some of the issues raised by Kandel and Weeks in their original analysis.