A new diffuse-interface model for step flow in epitaxial growth
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Date
2012-03-19
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Abstract
In this work, we consider epitaxial growth of thin crystalline films. Thereby, we propose
a new diffuse-interface approximation of a semi-continuous model resolving atomic distances in the
growth direction but being coarse-grained in the lateral directions. Mathematically, this leads to a free
boundary problem proposed by Burton, Cabrera and Frank for steps separating terraces of different
atomic heights. The evolution of the steps is coupled to a diffusion equation for the adatom (adsorbed
atom) concentration fulfilling Robin-type boundary conditions at the steps. Our approach allows to
incorporate an Ehrlich-Schwoebel barrier as well as diffusion along step edges into a diffuse-interface
model.
This model results in a Cahn-Hilliard equation with a degenerate mobility coupled to diffusion
equations on the terraces with a diffuse-interface description of the boundary conditions at the steps.
We provide a justification by matched asymptotic expansions formally showing the convergence of the
diffuse-interface model towards the sharp-interface model as the interface width shrinks to zero. The
results of the asymptotic analysis are numerically reproduced by a finite element discretisation.