Ehrlich-Schwoebel instability in molecular-beam epitaxy: A minimal model

Abstract

The instability of a growing crystal limited by a high-symmetry surface in molecular-beam epitaxy is studied in the limit where terrace size is very large compared to the atomic distance. In that case, everything is deterministic except the nucleation of new terraces. Moreover, exchange of atoms between steps is ignored. If the typical terrace size ${\mathcal{l}}_{\mathit{c}}$ is chosen as length unit, the model depends on a single parameter (${\mathcal{l}}_{\mathit{s}}$/${\mathcal{l}}_{\mathit{c}}$) which characterizes the strength of step-edge barriers (``Ehrlich-Schwoebel effect''). Numerical simulations are supported by nonlocal evolution equations relating the time and space derivatives of the surface height. The first mounds which appear have a radius ${\ensuremath{\lambda}}_{\mathit{c}}^{\mathrm{inf}}$ proportional to ${\mathcal{l}}_{\mathit{c}}$\ensuremath{\surd}${\mathcal{l}}_{\mathit{c}}$/${\mathcal{l}}_{\mathit{s}}$. In contrast with other authors who studied different models, coarsening is found to become extremely slow after the mounds have reached a radius ${\ensuremath{\lambda}}_{\mathit{c}}^{\mathrm{sup}}$ of order ${\mathcal{l}}_{\mathit{c}}^{2}$/${\mathcal{l}}_{\mathit{s}}$. \textcopyright{} 1996 The American Physical Society.