Electromigration in Macroscopic Relaxation of Stepped Surfaces

Abstract

We study the effect of an external electric field $\mathbf{E}$ on macroscopic relaxation laws for crystal surfaces in $2+1$ dimensions. We derive a nonlinear, fourth-order partial differential equation (PDE) for the surface height from the microscale motion of line defects (steps). This PDE contains a linear-in-$\mathbf{E}$ convective contribution and reflects a variety of microscale kinetic processes. A basic ingredient is an extended Fick's law for the surface flux, which accounts for drift of adsorbed atoms (adatoms), isotropic as well as anisotropic diffusion of adatoms on terraces between steps, attachment-detachment of atoms at step edges, edge atom diffusion, and atom desorption into the surrounding vapor. In particular, we discuss conditions that enable the neglect of desorption. By resorting to stationary PDE solutions, we show how $\mathbf{E}$ can possibly influence spatial changes of the slope profile near a macroscopically planar surface region (facet). We start with the Burton–Cabrera–Frank model for the motion of interacting steps, which is viewed as a discrete scheme for the macroscopic description, and apply coarse graining by separating local space variables into fast and slow.