A study was made of the continuum limit, in 2+1 dimensions, of nanoscale anisotropic diffusion processes on crystal surfaces which relaxed to become flat below roughening. The main result was a continuum law for the surface flux in terms of a new continuum-scale tensor mobility. The starting point was the Burton-Cabrera-Frank theory, which offered a discrete scheme for atomic steps whose motion drove surface evolution. The derivation was based upon the separation of local space variables into fast and slow. The model included the anisotropic diffusion of adsorbed atoms (adatoms) on terraces separating steps, the diffusion of atoms along step edges, and the attachment–detachment of atoms at step edges. A parabolic fourth-order non-linear partial differential equation was derived for the continuum surface height profile. A factor in this partial differential equation was the surface mobility of the adatom flux, which was a non-trivial extension of the tensor mobility for isotropic terrace diffusion derived previously by Margetis and Kohn. Approximate separable solutions of the partial differential equation were found.

Anisotropic Diffusion in Continuum Relaxation of Stepped Crystal Surfaces. J.Quah, D.Margetis: Journal of Physics A, 2008, 41[23], 235004 (18pp)