Nucleation, adatom capture, and island size distributions: Unified scaling analysis for submonolayer deposition

We consider the irreversible nucleation and growth of two-dimensional islands during submonolayer deposition in the regime of large island sizes. A quasihydrodynamic analysis of rate equations for island densities yields an ordinary differential equation (ODE) for the scaling function describing the island size distribution. This ODE involves the scaling function for the dependence on island size of ''capture numbers'' describing the aggregation of diffusing adatoms. The latter is determined via a quasihydrodynamic analysis of rate equations for the areas of "capture zones" surrounding islands. Alternatively, a more complicated analysis yields a partial differential equation (PDE) for the scaling function describing the joint probability distribution for island sizes and capture zone areas. Then, applying a moment analysis to this PDE, we obtain refined versions of the above ODE's, together with a third equation for the variance of the cell area distribution (for islands of a given size). The key nontrivial input to the above equations is a detailed characterization of nucleation. We analyze these equations for a general formulation of nucleation, as well as for an idealized picture considered previously, wherein nucleated islands have capture zones lying completely within those of existing islands.