Three Dimensional Phase Field Modelling of Grain Boundary Interaction and Evolution during Directional Solidification of Multi-Crystalline Silicon

Abstract

The development of grain structures during directional solidification of multi-crystalline silicon (mc-Si) plays a crucial role in the materials quality for silicon solar cells. Three dimensional (3D) phase field modelling of the grain boundary (GB) interaction and evolution by considering anisotropic GB energy and mobility for mc-Si is thus carried out to elucidate the process. We also describe a method to find the GB planes formed between two neighboring grains. The energy and mobility of GBs are allowed to depend on misorientation and grain boundary plane. To examine the correctness of our method, we run a test to check the grain boundary interaction and evolution verifying the known CSL combinations such as: (Σ a+Σ b → Σ a x b) or (Σ a +Σ b→Σ a/b). We further discuss how the knowledge of GB normal allows characterizing a grain boundary into a Tilt GB or a Twist GB. In this study, we have taken a simple approach using vectors to investigate GB interactions in mc-Si. GB interaction between Tilt and Twist boundaries is also shown. Two experimental scenarios are considered for comparison and the results are in agreement with experimental observations as well as theoretical predictions. We further propose a model to explain the formation mechanism of twin grains at the three-grain tri-junction (3GTJ) on the growth interface during directional solidification of multi-crystalline silicon. We also attempt to confirm its validity by comparing with the experimental results. This model is an extension of the previous model for 2D nucleation at the grain boundaries (GBs). It is found that the energy barriers for faceting and twinning nucleus at the 3GTJ are much smaller than that at GBs. As a result, a higher twinning probability can be obtained at a much lower undercooling. Two types of tri-junctions are considered according to the experiments and the dominant factors which decide the twinning probability on each facets at the 3GTJ are further discussed. We further extend this model to multi-layer twinning which seems to be another route for the twin grain to grow after nucleation, and the undercooling for twinning with different layers is estimated.