One-dimensional model of the equiaxed grain formation in multi-crystalline silicon

During solidification of low purity silicon for photovoltaic (PV) cells, solute rejection at the growth interface leads to an increase of the carbon concentration in the liquid phase and then to the precipitation of silicon carbide (SiC). When the precipitate radius becomes higher than the silicon critical nucleus radius, SiC can act as a refining agent for the Si and Si equiaxed grains appear in the liquid. The grain structure of the ingot changes from columnar to small grains, also known as grits. We developed a one-dimensional analytical model of this series of phenomena, including C segregation, SiC nucleation and growth, Si nucleation on the SiC precipitates and subsequent growth of the Si equiaxed grains. The equations are implemented under Matlab software in order to predict the columnar to equiaxed transition (CET) during the directional solidification of PV Si. We carried out calculations of the position and thickness of the equiaxed areas and of the number and size of Si grits as a function of the main process parameters: thermal gradient and growth velocity. Recommendations in order to adapt the growth process parameters to the initial carbon content are given. It is expected that coupling this model to global 3D numerical simulation codes could help improving the yield of ingot solidification.