Continuum relaxation of interacting steps on crystal surfaces in 2+1 dimensions

Dionisios Margetis, Robert V. Kohn

Research output: Contribution to journalArticlepeer-review


We study the relaxation of crystal surfaces in 2+1 dimensions via the motion of interacting atomic steps. The goal is the rigorous derivation of the continuum limit. The starting point is a discrete scheme from the Burton, Cabrera, and Frank model, which accounts for diffusion of point defects ("adatoms") on terraces and attachment-detachment of atoms at step edges. It is shown that the macroscopic adatom current involves a tensor mobility, which describes different fluxes in directions parallel and transverse to step edges, although the physics of each terrace is assumed isotropic. When the steps are everywhere parallel (straight or circular) the tensor character of the mobility is unimportant; in the general (2+1)-dimensional setting, however, it is crucial. Our methods consist of (i) the solution of the diffusion equation for adatoms via the separation of local space variables into "fast" and "slow," and (ii) the treatment of the step chemical potential for a wide class of step energies and repulsive interactions. Previous works using similar methods were mainly restricted to (1+1)-dimensional or axisymmetric geometries and entirely missed the tensor character of the mobility. The continuum limit of the step flow yields a fourth-order partial differential equation for the surface height profile.

Original languageEnglish (US)
Pages (from-to)729-758
Number of pages30
JournalMultiscale Modeling and Simulation
Issue number3
StatePublished - Sep 2006


  • Burton-Cabrera-Frank model
  • Continuum limit
  • Crystal surface
  • Ehrlich-Schwoebel barrier
  • Elastic dipole interactions
  • Entropic interactions
  • Epitaxial growth
  • Morphological evolution
  • Step chemical potential
  • Surface mobility

ASJC Scopus subject areas

  • General Chemistry
  • Modeling and Simulation
  • Ecological Modeling
  • General Physics and Astronomy
  • Computer Science Applications


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