A Finite Element Framework for Burton-Cabrera-Frank Equation

Abstract

Abstract.A finite element framework is presented for the Burton-CabreraFrank (BCF) equation. The model is a 2 + 1-dimensional step flow model, discrete in the height but continuous in the lateral directions. The problem consists of adatom diffusion equations on terraces of different atomic height; boundary conditions at steps (terrace boundaries); and a normal velocity law for the motion of such boundaries determined by a two-sided flux, together with one-dimensional edge-diffusion. Two types of boundary conditions, modeling either diffusion limited growth or growth governed by attachment-detachment kinetics at the steps, are considered. We review the basic ideas of the algorithms, already described in [1,2] and extent it to incorporate anisotropy of the step free energy, the edge mobility and the kinetic coefficients (attachment-detachment rates). The problem is solved using two independent meshes: a two-dimensional mesh for the adatom diffusion and a one-dimensional mesh for the step dynamics governed by an anisotropic geometric evolution law. Finally results on the anisotropic growth of single layer islands are presented. Mathematics Subject Classification (2000). Primary 80A22; Secondary 35R35.

How does your crystal grow? A commentary on Burton, Cabrera and Frank (1951) ‘The growth of crystals and the equilibrium structure of their surfaces’

“…Appropriate theories must therefore take account of the movement of the steps during the arrival of new material from the gas phase and the associated diffusion over the surface (e.g. [38][39][40]).…”