2D nucleation limited by slow diffusion

We extend a standard theory of second-layer nucleation in the regime of low temperature, where the diffusion is the limiting process. The theory takes into consideration the fluctuations of the adatoms and the distribution of stable clusters on a surface, and yields an expression for the nucleation rate and time evolution of the density of stable clusters. When diffusion is slow, the fluctuations of the monomer distribution play an important role and results differ qualitatively and quantitatively from those obtained using the standard rate-equation approach to nucleation. We argue that in this case the standard theory fails because slow diffusion delays nucleation and the nucleation rate depends on the fluctuations of interatomic distances. It is shown that the final density of stable clusters depends on the fluctuations of the monomer and stable clusters distribution and can be even independent of the deposition rate, as was experimentally observed at low temperatures.