Boundary Conditions for Burton–Cabrera–Frank Type Step-Flow Models: Coarse-Graining of Discrete 2D Deposition-Diffusion Equations

We analyze discrete two-dimensional (2D) deposition-diffusion equations for the density of adatoms deposited at a periodic array of adsorption sites on a vicinal crystalline surface with kinked steps. Our analysis provides insight into the appropriate boundary conditions (BC) at steps for a coarse-grained Burton-Cabrera-Frank (BCF) type treatment involving continuum 2D deposition-diffusion equations. Such a BCF type treatment should describe step flow on vicinal surfaces under nonequilibrium growth conditions. We focus on cases where there is no additional activation barrier inhibiting to attachment at steps beyond that for terrace diffusion. Then, the classical BCF treatment simply imposes a Dirichlet BC equating the limiting value of the terrace adatom density to its equilibrium value at the step edge. Our analysis replaces this BC with one incorporating finite kinetic coefficients, K&plusmn, measuring inhibited diffusion-limited attachment at kinks. We determine the dependence of K± on key parameters such as the kink separation and terrace width, and on the width of nearby terraces. Our formulation provides a framework within which to describe step pairing phenomena observed on so-called AB-vicinal surfaces without attachment barriers, a feature not captured by the classical BCF treatment. [PUBLICATION ABSTRACT]