  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 processes are strongly affected by the additional step edge
barrier E_S for descending atoms. If the dissociation times of unstable
clusters can be neglected, the theory predicts that for small critical nuclei
of size i<=2 the nucleation is governed by fluctuations, during which by chance
i+1 atoms are present on the island. For large critical nuclei i>=3 by
contrast, the nucleation process can be described in a mean-field type manner,
which for large 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(-E_S/k_B T). For the case where the dissociation rates of
metastable clusters enter the problem as additional parameters, we set up a
semi-analytical approach based on novel rate equations, which can easily be
solved numerically. The results obtained from the theory are in good agreement
with Monte Carlo data. Implications for various applications are pointed out.
