Finite element method for epitaxial growth with thermodynamic boundary conditions

• Published in: SIAM journal on scientific computing Vol. 26; no. 6; pp. 2029 - 2046
• Main Authors: BÄNSCH, Eberhard, HAUSSER, Frank, VOIGT, Axel
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 2005
• Subjects: Boundary conditions; Exact sciences and technology; Finite element analysis; Fundamental areas of phenomenology (including applications); Islands; Mathematical analysis; Mathematics; Methods; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations; Partial differential equations, boundary value problems; Physics; Sciences and techniques of general use; Simulation; Solid mechanics; Structural and continuum mechanics; Theory and numerical methods; Adaptivity; 74S05; Gibbs-Thomson; Moving-boundary conditions; Adaptive method; Dynamic method; Penalty method; Parametric method; Finite element method; Thermodynamics; 65Z05; Flow models; Diffusion equation; mean curvature flow; 65N30; front tracking 35Q99; Gibbs Thomson; Grid pattern; Surface diffusion; Free boundary problem; free or moving boundary problem; Numerical analysis; Scientific computation; Adatom diffusion; 35R35; Epitaxial growth; finite elements; island dynamics
• ISSN: 1064-8275; 1095-7197
• DOI: 10.1137/030601028

We develop an adaptive finite element method for island dynamics in epitaxial growth. We study a step-flow model, which consists of an adatom (adsorbed atom) diffusion equation on terraces of different height; thermodynamic boundary conditions on terrace boundaries including anisotropic line tension; and the normal velocity law for the motion of such boundaries determined by a two-sided flux, together with the one-dimensional anisotropic ``surface' diffusion (edge diffusion) of edge adatoms along the step edges. The problem is solved using independent meshes: a two-dimensional mesh for the adatom diffusion and one-dimensional meshes for the boundary evolution. A penalty method is used to incorporate the boundary conditions. The evolution of the terrace boundaries includes both the weighted/anisotropic mean curvature flow and the weighted/anisotropic edge diffusion. Its governing equation is solved by a semi-implicit front-tracking method using parametric finite elements.