Continuum approach to self-similarity and scaling in morphological relaxation of a crystal with a facet

Dionisios Margetis, Michael J. Aziz, Howard A. Stone

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30 Scopus citations


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.

Original languageEnglish (US)
Article number165432
JournalPhysical Review B - Condensed Matter and Materials Physics
Issue number16
StatePublished - 2005
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Electronic, Optical and Magnetic Materials
  • Condensed Matter Physics


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