A Finite Element Framework for Burton-Cabrera-Frank Equation
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A finite element framework is presented for the Burton-Cabrera-Frank (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.
Keywordsstep-flow model anisotropic geometric evolution laws parametric finite elements discrete-continuous coupling
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