An analysis of combined surface and volume diffusion processes in crystal growth. II. Asymmetric capture at steps

Abstract

We consider the terrace-ledge-kink model of crystal growth, and formulate coupled volume and surface diffusion equations. These govern the flow of growth units from a supersaturated fluid phase, through a stagnant layer, and across terraces to a series of parallel steps on a crystal surface. We obtain a steady state solution and the growth rate of vicinal surfaces with equidistant steps when each step can exchange growth units with its two adjacent terraces at unequal rates (asymmetric capture). Diffusion to an isolated step is considered in detail since the resulting surface current determines the general form of the growth rate vs supersaturation curve, and gives precise information on the parabolic region of this curve. Three regimes of diffusion are observed depending on the relative surface and volume diffusion rates, and on the rate of exchange between the fluid and the surface. For an isolated step, small surface diffusion rates and slow exchange define a regime where volume density gradients are negligible, and the classical Burton, Cabrera, and Frank expression for the surface diffusion current is valid. Intermediate surface diffusion and exchange rates cause significant volume density gradients, and nearly cylindrical density contours surrounding a step. Very large surface diffusion and exchange rates cause large portions of the surface to act as a catchment area for a step and the volume density contours assume a nearly planar geometry. Asymmetric capture kinetics at a step influence the geometry of the diffusion field only in the latter case of very large surface diffusion rates. For an array of parallel and equidistant steps, the parabolic region of the growth rate versus supersaturation curve is sensitive to the various diffusion and exchange processes. It measures the distance at which neighboring steps can interact. Changes in the shape and extent of this region with the thickness of the stagnant layer (or the rate of stirring) allow an experimental determination of the model parameters. © 1974.