Mound formation and slope selection in irreversible fcc (111) growth

Abstract

An analytic calculation of the surface current and selected mound angle is presented for the case of irreversible epitaxial growth on an fcc(111) surface with a finite Ehrlich-Schwoebel (ES) barrier. The special cases of short terraces and combinations of short terraces with facets which lead to large mound slopes are also discussed. We find that for both $A$ and $B$ steps the surface current and selected mound slope are determined by two key parameters---the Ehrlich-Schwoebel barrier and the degree of uphill funneling due to short-range attraction. However, the presence or absence of a small-slope instability is exclusively determined by the value of the ES barrier. In particular there exists a critical value of the parameter $\ensuremath{\rho}$ (where $\ensuremath{\rho}=({\ensuremath{\nu}}_{\mathrm{ES}}∕{\ensuremath{\nu}}_{0}){e}^{\ensuremath{-}{E}_{\mathrm{ES}}∕{k}_{B}T}$ and ${E}_{\mathrm{ES}}$ is the ES barrier) such that for $\ensuremath{\rho}<{\ensuremath{\rho}}_{c}$, the flat surface is unstable to mound formation while for $\ensuremath{\rho}>{\ensuremath{\rho}}_{c}$ there is no such instability. The critical value ${\ensuremath{\rho}}_{c}\ensuremath{\simeq}0.21$ is the same for both $A$ and $B$ steps and independent of the degree of uphill funneling due to short-range attraction. When the uphill funneling is not too large, the selected slope decreases continuously with increasing $\ensuremath{\rho}$, reaching zero at ${\ensuremath{\rho}}_{c}$. However, in the presence of sufficiently large uphill funneling, the selected slope is independent of $\ensuremath{\rho}$ for $\ensuremath{\rho}<{\ensuremath{\rho}}_{c}$. In this case a new phenomenon which we refer to as fluctuation-induced instability also occurs. In particular, while the surface remains stable for $\ensuremath{\rho}>{\ensuremath{\rho}}_{c}$ for small slopes, for larger slopes the surface current may become positive due to uphill funneling. Thus, even in the presence of a small ES barrier, mound formation may still occur. Finally, we present typical results for the dependence of the mound slope on $\ensuremath{\rho}$ for both $A$ and $B$ steps.