Polynuclear Growth of Square Crystallites on a Flat Substrate

Abstract

We study a polynuclear growth model in which the crystallites are aligned squares, as observed in micrographs of epitaxial thin films. The expected volumes of lower layers are calculated by series expansion methods. The coefficients are calculated exactly up to the 4th power in the intensity of the nucleation process or the 12th power in the time. The method is based on exact integral expressions recently obtained by the author. The resulting instantaneous growth rate or surface speed has an initial oscillation, consistent with long-standing experimental observations. The method is also applied to 1-dimensional rod crystallites and d-dimensional cubic crystallites. For large \(d\) the ultimate \({\text{(time}} \to \infty )\) growth rate and oscillating growth profile are obtained. The coefficients in the series are derived from basis functions, which involve only 1-dimensional spatial integrals, and which are common to all dimensions. For the second layer, the series is derived by a cluster expansion method, analogous to methods in equilibrium statistical mechanics. For higher layers, the integrands are broken down into products of pairs of nested crystallites.