Nucleation on top of islands in epitaxial growth

We develop a theory for nucleation on top of islands in epitaxial growth based on the derivation of lifetimes and rates governing individual microscopic processes. These in particular include the encounter rate of j atoms in a state, where in total n greater than or equal to j atoms are present on top of the island, and for the lifetime of this state. The latter depends strongly on the additional step edge barrier Delta E-S for descending atoms. We present two analytical approaches complemented by kinetic Monte Carlo simulations. In the first approach, we employ a simplified stochastic description that allows us to derive the nucleation rate on top of islands explicitly, if the dissociation times of unstable clusters can be neglected. We find that for small critical nuclei of size i less than or equal to 2 the nucleation is governed by fluctuations, during which by chance i+1 atoms are present on the island. For large critical nuclei i greater than or equal to 3 by contrast, the nucleation process can be described in a mean-field type manner, which for large Delta E-S corresponds to the approach developed by Tersoff et al. [Phys. Rev. Lett. 72, 266 (1994)]. In both the fluctuation-dominated and the mean-field case, various scaling regimes are identified, where the typical island size at the onset of nucleation shows a power law in dependence on the adatom diffusion rates, the incoming atom flux, and the step edge crossing probability exp(-Delta E-S/k(B)T). Although it is possible to extend the simplified approach to more general situations, its applicability is limited, if dissociation rates of metastable clusters enter the problem as additional parameters. For such situations the second semianalytical approach becomes superior. This approach is based on novel rate equations, which can easily be solved numerically. Both theoretical approaches yield good agreement with Monte Carlo data. Implications for various applications are pointed out.