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Bänsch, Eberhard; Haußer, Frank; Lakkis, Omar; Li, Bo; Voigt, Axel

Finite element method for epitaxial growth with attachment–detachment kinetics. (English) Zbl 1100.82024

J. Comput. Phys. 194, No. 2, 409-434 (2004).
Summary: An adaptive finite element method is developed for a class of free or moving boundary problems modeling island dynamics in epitaxial growth. Such problems consist of an adatom (adsorbed atom) diffusion equation on terraces of different height; boundary conditions on terrace boundaries including the kinetic asymmetry in the adatom attachment and detachment; and the normal velocity law for the motion of such boundaries determined by a two-sided flux, together with the one-dimensional ”surface” diffusion. The problem is solved using two independent meshes: a two-dimensional mesh for the adatom diffusion and a one-dimensional mesh for the boundary evolution. The diffusion equation is discretized by the first-order implicit scheme in time and the linear finite element method in space. A technique of extension is used to avoid the complexity in the spatial discretization near boundaries. All the elements are marked, and the marking is updated in each time step, to trace the terrace height. The evolution of the terrace boundaries includes both the mean curvature flow and the surface diffusion. Its governing equation is solved by a semi-implicit front-tracking method using parametric finite elements. Simple adaptive techniques are employed in solving the adatom diffusion as well as the boundary motion problem. Numerical tests on pure geometrical motion, mass balance, and the stability of a growing circular island demonstrate that the method is stable, efficient, and accurate enough to simulate the growing of epitaxial islands over a sufficiently long time period.

Cited in 1 Review
Cited in 13 Documents

MSC:

82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs

Keywords:

Epitaxial growth; Island dynamics; Free or moving; boundary problem; Adatom diffusion; Attachment–detachment kinetics; Surface diffusion; Mean curvature flow; Finite elements; Adaptivity; Front tracking

Software:

ALBERT
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Full Text: DOI

References:

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