Island sizes and capture zone areas in submonolayer deposition: Scaling and factorization of the joint probability distribution

The joint probability distribution (JPD) for island sizes, s, and capture zone areas, A, provides extensive information on the distribution of islands formed during submonolayer deposition. For irreversible island formation via homogeneous nucleation, this JPD is shown to display scaling of the type F(s/s(av),A/A(av)), where "av" denotes average values. The form of F reflects both a broad monomodal distribution of island sizes, and a significant spread of capture zone areas for each island size. A key ingredient determining this scaling behavior is the impact of each nucleation event on existing capture zone areas, which we quantify by kinetic Monte Carlo simulations. Combining this characterization of the spatial aspects of nucleation with a simplified but realistic factorization ansatz for the JPD, we provide a concise rate equation formulation for the variation of both the capture zone area and the island density with island size. This is achieved by analysis of the first two moments of the evolution equations for the JPD.