Scaling regimes for second layer nucleation

Nucleation on top of two-dimensional islands with step edge barriers is investigated using scaling arguments. The nucleation rate is expressed in terms of three basic time scales: The time interval between deposition events, the residence time of atoms on the island, and the encounter time required for $i^{\ast }+1$ atoms forming a stable nucleus to meet. Application to the problem of second-layer nucleation on growing first layer islands yields a sequence of scaling regimes with different scaling exponents relating the critical island size, at which nucleation takes place, to the diffusion and deposition rates. Second layer nucleation is fluctuation-dominated, in the sense that the typical number of atoms on the island is small compared to $i^{\ast }+1$, when the first layer island density exponent $\chi$ satisfies $\chi (i^{\ast }+1)<2$. The upper critical nucleus size, above which the conventional mean-field theory of second layer nucleation is valid, increases with decreasing dimensionality. In the related case of nucleation on top of multilayer mounds fluctuation-dominated and mean-field like regimes coexist for arbitrary values of the critical nucleus size $i^{\ast }$.