The asymptotic scaling limit of point island models for epitaxial growth

Scaling of the island size distribution in the asymptotic scaling limit Θ → ∞ will be discussed. In particular, it is shown that for the simplest implementation of a model for epitaxial growth, the so-called point island model, the island size distribution (ISD) does not scale, and in fact becomes singular. The ultimate reason is that in a point island model there is no distinct nucleation phase followed by an aggregation phase. The same argument might be used to explain the existence of scaling in extended island models: Only when the nucleation phase is infinitesimal short compared to the aggregation phase, and there is no significant post-nucleation, do we expect scaling of the ISD.