Anisotropic diffusion in continuum relaxation of stepped crystal surfaces

TL;DR

This paper derives a new continuum PDE model for anisotropic surface relaxation of crystal surfaces, incorporating a tensor mobility to describe atom diffusion processes on terraces and steps.

Contribution

It introduces a novel continuum law with a tensor mobility for anisotropic diffusion, extending previous isotropic models based on BCF theory.

Findings

Derived a nonlinear PDE for surface height evolution.

Introduced a tensor mobility for anisotropic diffusion.

Discussed approximate solutions of the PDE.

Abstract

We study the continuum limit in 2+1 dimensions of nanoscale anisotropic diffusion processes on crystal surfaces relaxing to become flat below roughening. Our main result is a continuum law for the surface flux in terms of a new continuum-scale tensor mobility. The starting point is the Burton, Cabrera and Frank (BCF) theory, which offers a discrete scheme for atomic steps whose motion drives surface evolution. Our derivation is based on the separation of local space variables into fast and slow. The model includes: (i) anisotropic diffusion of adsorbed atoms (adatoms) on terraces separating steps; (ii) diffusion of atoms along step edges; and (iii) attachment-detachment of atoms at step edges. We derive a parabolic fourth-order, fully nonlinear partial differential equation (PDE) for the continuum surface height profile. An ingredient of this PDE is the surface mobility for the adatom flux, which is a nontrivial extension of the tensor mobility for isotropic terrace diffusion derived previously by Margetis and Kohn. Approximate, separable solutions of the PDE are discussed.