Computer simulations of the growth of binary alloy thin films in two and three dimensions were performed using an extension of the Solid on Solid model. Snapshots for a range of interactions and diffusion rates are presented and critically compared to experiment. A wide variety of distinct phases is identified and their growth conditions analyzed. These findings are summarized in a phase diagram. In addition, a fractal analysis of the domains is performed. It is found that for negative interactions the islands are two-dimensional, while for positive interactions, regardless of diffusion rate, a fractal dimension of 1.78 is obtained.