Scaling and a Possible Phase Transition in Models for Thin Film Growth

Abstract

In recent years there has been a significant interest in simple models for the growth of random rough surfaces. In this note we will report on some progress we have made in a particular sort of model. It is most easily understood as a model for thin film growth. Particles launched from random points bombard a substrate and stick to it and to other particles. To fix our ideas, we give the exact rule for this model (the ballistic aggregation model) written on a lattice: Choose a column at random, and let it grow according to:

$$h(r,t + \tau ) = Max\{ h(r,t) + A,h(r + \delta ,t)\}$$

(1)

where h(r,t) is the height of the the surface at time t and substrate position r, A is the particle diameter, and δ runs over the nearest neighbors of the column in question. The term involving the neighbors means that the particles can stick on the side of the columns. We will discuss several variants of this kind of growth.