Continuum Relaxation of Interacting Steps on Crystal Surfaces in 2+1 Dimensions

We study the relaxation of crystal surfaces in 2+1 dimensions via the motion of interacting atomic steps. The goal is the rigorous derivation of the continuum limit. The starting point is a discrete scheme from the Burton, Cabrera and Frank (“BCF”) model, which accounts for diffusion of point defects (“adatoms”) on terraces and attachment-detachment of atoms at step edges. It is shown that the macroscopic adatom current involves a tensor mobility, which describes different fluxes in directions parallel and transverse to step edges, although the physics of each terrace is assumed isotropic. When the steps are everywhere parallel (straight or circular) the tensor character of the mobility is unimportant; in the general (2+1)-dimensional setting, however, it is crucial. Our methods consist of: (i) the solution of the diffusion equation for adatoms via the separation of local space variables into “fast” and “slow”; and (ii) the treatment of the step chemical potential for a wide class of step energies and repulsive interactions. Previous works using similar methods were mainly restricted to (1+1)-dimensional or axisymmetric geometries and entirely missed the tensor character of the mobility. The continuum limit of the step flow yields a fourth-order partial differential equation (PDE) for the surface height profile.