Finite Element Method for Epitaxial Growth with Thermodynamic Boundary Conditions

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.