Finite element method for epitaxial growth with attachment–detachment kinetics

Abstract

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.

A new diffuse-interface model for step flow in epitaxial growth

“…Thereby, we combine the diffuse-interface approximation for motion by surface diffusion by a Cahn-Hilliard equation in [5] with the approach proposed in [13], where diffusion equations with different types of boundary conditions are treated in a diffuse-interface context. In this way, one can interpret the treatment presented here as diffuseinterface counterpart of the numerical sharp-interface treatment in [3]. One advantage of our model is that it incorporates the time derivative in the diffusion equation into a phase-field approximation for a BCF-model with asymmetric boundary conditions.…”

Interface Growth Driven by Surface Kinetics and Convection

“…Models that assume quasi-steadiness have been used to study a wide range of different phenomena including the effects of an applied electric field [23], strain [12] and material deposition [20] on the growth and relaxation of thin films. Models that do not assume quasi-steadiness are much less common and seem to focus mainly on growth by deposition [1,14,21].…”

A Finite Element Framework for Burton-Cabrera-Frank Equation

“…Numerical tests of the algorithms described in Section 3 have been presented in [1,2,27]. In [26], the algorithm is used to simulate Ostwald ripening using 400 islands.…”

“…A phase-field model which includes isotropic edge diffusion is introduced in [20] (this volume). A front-tracking finite element approach for epitaxial growth was developed in [1,2]. This approach will be reviewed and extended to treat anisotropic step free energies, edge adatom mobilities and kinetic coefficients.…”

“…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.…”