Continuum Approach to Self-Similarity and Scaling in Nanostructure Decay

Abstract

The morphological relaxation of axisymmetric crystal surfaces with a single facet below the roughening transition temperature is studied analytically for diffusion-limited (DL) and attachment-detachment-limited (ADL) kinetics with inclusion of the Ehrlich-Schwoebel barrier. The slope profile F(r,t) , where r is the polar distance and t is time, is described via a nonlinear, fourth-order partial differential equation (PDE) that accounts for step line-tension energy g1 and step-step repulsive interaction energy g3 ; for ADL kinetics, an effective surface diffusivity that depends on the step density is included. The PDE is derived directly from the step-flow equations and, alternatively, via a continuum surface free energy. The facet evolution is treated as a free-boundary problem where the interplay between g1 and g3 gives rise to a region of rapid variations of F , a boundary layer, near the expanding facet. For long times and g3∕g1<O(1) singular perturbation theory is applied for self-similar shapes close to the facet. For DL kinetics and a class of axisymmetric shapes, (a) the boundary-layer width varies as (g3∕g1)1∕3 , (b) a universal ordinary differential equation (ODE) is derived for F , and (c) a one-parameter family of solutions of the ODE are found; furthermore, for a conical initial shape, (d) distinct solutions of the ODE are identified for different g3∕g1 via effective boundary conditions at the facet edge, (e) the profile peak scales as (g3∕g1)−1∕6 , and (f) the change of the facet radius from its limit as g3∕g1→0 scales as (g3∕g1)1∕3 . For ADL kinetics a boundary layer can still be defined, with thickness that varies as (g3∕g1)3∕8 . Our scaling results are in excellent agreement with kinetic simulations.