Modeling anisotropic shape evolution during Czochralski growth of oxide single crystals

Abstract

We present a technique for modeling three-dimensional (3D) anisotropic crystal shape evolution in meniscus defined growth systems. In particular we apply this model to a simplified low gradient Czochralski growth process used to grow bismuth–germanium–oxide–like crystals. Our Lattice Boltzmann Model (LBM) based approach involves a two-dimensional unstructured mesh (embedded within the 3D LBM grid) which accurately tracks the crystal/melt and melt/gas interfaces while accounting for anisotropic interface attachment kinetics as well as capillarity, involving the enforcement of either isotropic or anisotropic growth angles (GA) at the crystal/melt/gas triple phase line (GA-TPL). The robustness of the algorithm is demonstrated by a series of tests involving mesh refinement, comparisons with results from an older study and an evaluation of the impact on accuracy of a time-saving artificial pull-rate speedup scheme described within the manuscript. Physically significant results (which do not involve diameter control) include a demonstration of the impact of changes in kinetic parameters on resultant ingot shapes, calculations showing how (for conditions considered here) the diameter of the growing crystal decreases with increasing isotropic growth angle values, and growth simulations demonstrating how anisotropic growth angles can impact the shape of the evolving ingot. The dependence of crystal diameter on isotropic growth angle values is reproduced in a local sense in the case of anisotropic growth angles. A specific point on the GA-TPL will tend to extend inwards or outwards (respectively exhibiting a decreased or increased local radial position value) when the growth angle value is locally (and respectively) increased or decreased. As a result, considering crystals whose non-circular cross-sectional shapes are due to kinetic anisotropy as a baseline, growth angle anisotropy acts to either decrease or increase shape anisotropy when the extended (rough) parts of the GA-TPL respectively support large or small local growth angle values as compared to reduced-radius (faceted) parts of the GA-TPL. It should finally be mentioned that our results, concerning the sensitivity of the (local or average) radial position value of the GA-TPL to the growth angle, are consistent with other computational results exhibited in the literature (concerning solidifying semiconductor drops and floating zone growth of oxides).