Dynamics of partially faceted melt-crystal interfaces III: Three-dimensional computational approach and calculations

The modeling of partially faceted melt-crystal interfaces in bulk melt growth systems has been addressed in a number of recent publications. In particular, in Weinstein and Brandon [J. Crystal Growth 268(1-2) (2004) 299], a method for self-consistent two-dimensional dynamic analysis of such systems while accounting for both macro- and nano-scale phenomena, which result from the coupling between competing kinetic mechanisms and associated thermal fields, was presented. In this manuscript, we report on an extension of this approach to three-dimensional systems. The method is first described in detail after which it is applied to model processes involving the vertical gradient freeze growth both of silicon and of yttrium aluminum garnet. In axisymmetric situations, results are shown to successfully reproduce calculations obtained using the previous two-dimensional modeling approach. Additional results demonstrate a number of important three-dimensional nano- and macro-scale features of the melt-crystal interface. These include observations of the dominant role of the coldest dislocation step source in the case where more than one such dislocation line intersects an advancing facet, a demonstration of the effect of growth rate on the morphology of a multi-faceted interface, and a simple explicit analysis of step flow on an evolving facet.