Critical Separation of Clusters During Physical Vapor Deposition

The critical separation of clusters, corresponding to the maximum density of clusters, affects growth characteristics during physical vapor deposition (PVD). In particular, this separation can affect surface smoothness in growing single-crystalline films, grain size distribution in growing polycrystalline films, and diameter in growing nanorods. This Letter reports a theoretical expression of the critical separation as a function of deposition conditions and accompanying verifications using lattice kinetic Monte Carlo (MC) simulations of PVD on Cu{111}. In contrast to existing theories, the theoretical expression in this Letter is (1) closed-form, (2) in better agreement with the MC simulations than the lattice approximation and (3) in better agreement with the MC simulations than the mean field approximation when the critical separation is large- larger than 25 nm for PVD on Cu{111}.