Cluster Growth and Saturation Island Densities in Thin-Film Growth

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Abstract
An analysis of the appropriate two-dimensional random walk is used in calculating the probability that an adsorbed atom at a given point on a substrate will eventually reach a specified set of capture sites. From this, a formula is derived for the rate at which critical or stable clusters of atoms will grow when atoms are deposited on the substrate at a fixed rate. These results are applied to a calculation of the nucleation rate and of the saturation density of islands when a thin film is grown by the deposition of atoms from a vapor beam, for substrate temperatures appropriate to the region of incomplete condensation. It is found that this saturation density is usually an increasing function of the substrate temperature at low temperatures, but reaches a maximum and then always decreases, the rate of decrease becoming more rapid if the size of the critical nucleus increases. These results, and the calculated dependence of this saturation density on the various binding energies and on the deposition rate, are in good qualitative agreement with experimental observations.