Roughness in the Kolmogorov-Johnson-Mehl-Avrami framework: extension to (2+1)D of the Trofimov-Park model

In this paper the reformulation of Trofimov-Park (TP) model, [V.I. Trofimov, Appl. Surf. Sci. 219 (2003) 93), of thin film roughness evolution during nucleation and growth of islands in case of simultaneous nucleation is presented. The calculation of TP restricted to one-dimensional triangular islands has been extended to both the one-dimensional elliptical case and to the pyramidal two-dimensional one. The kinetics of the interface width, w, and the height-height autocorrelation function G, through which the correlation length ξ has been defined, have been estimated. Moreover, w(Θ) and ξ(Θ), where Θ  is the fraction of the covered substrate, if properly rescaled to the density of nuclei N and to the aspect ratio of islands, are universal functions that, for a conspicuous range of Θ, obey a power law with the exponent depending upon island shape.