Critical island size for layer-by-layer growth

We show that in epitaxial growth, there is a critical island size ${\mathit{R}}_{\mathit{c}}$ at which a second layer nucleates on top of the island. This provides a simple perspective on the conditions for smooth versus rough growth: For island spacing ${\mathit{L}}_{\mathit{n}}$, if ${\mathit{R}}_{\mathit{c}}$<${\mathit{L}}_{\mathit{n}}$ the islands will nucleate a second layer before coalescence, giving multilayer growth. This occurs when there is a sufficiently large diffusion barrier (the ‘‘Schwoebel barrier’’) at the island edge. We demonstrate that surfactants can enforce layer-by-layer growth by reducing the island spacing. The dependence upon temperature is also explained.