Scaling behavior in submonolayer film growth: A one-dimensional model

The theory of submonolayer thin-film growth has focused on the use of mean-field rate equations as analytic support to Monte Carlo simulations. While these give an excellent account of certain properties, they are totally inadequate for dealing with the distribution of island sizes and the related scaling properties. The reason is the neglect of spatial fluctuations in the mean-field theory. A capture zone construction has recently been introduced by the present authors [Philos. Mag. Lett. 72, 55 (1995); Phys. Rev. B 53, 10 261 (1996)] as a way of overcoming this deficiency. The present paper explores and develops this idea in a one-dimensional ''point island'' model. Note that the rate of capture of diffusing monomers is independent of the spatial extent of an island in one dimension. A. complete theory for the dynamical and spatial features of the model is developed, including the derivation of a scale-invariant analytic expression for the island size distribution function which agrees very well with the results of Monte Carlo simulations. The implications for two-dimensional systems are discussed.