Scale Invariance in Epitaxial Growth

Abstract

We examine three manifestations of scale invariance during epitaxial growth from the standpoint of a single atomistic model. The first two manifestations are in the submonolayer regime of growth, where clusters are formed but have not yet begun to coalesce. Depending on the material deposited and the substrate, the clusters can exhibit either an effective fractal dimension or a Euclidean dimension. The cluster size distribution function in this regime also exhibits scaling as a function of time. Interestingly, this distribution function and its scaling properties can be accurately described in terms of rate equations that omit entirely the influence of any fluctuations. An altogether different picture emerges in the regime of long deposition times. To analyze the scaling properties of the growing surface in this limit, we use exact discrete Langevin equations that are obtained directly from the rules of our simulation model. A regularization procedure is then used to convert these discrete equations into (infinite-order) continuum partial differential equations. The truncation of these continuum equations is achieved by performing a renormalization-group analysis, which leads asymptotically to a fourth-order nonlinear stochastic differential equation that is identical to that proposed independently by Wolf and Villain and by Lai and Das Sarma. Large-scale Monte Carlo simulations in d = 1, 2, and 3 substrate dimensions show that for all spatial dimensions the observed exponents correspond to those obtained from our continuum equations. This shows that vapor deposition is an example of a driven, nonequilibrium process that evolves to a nontrivial scale-invariant structure under the influence of input noise.