Unsteady coupled 3D calculations of melt flow, interface shape, and species transport for directional solidification of silicon in a traveling magnetic field

Directional solidification of large multi-crystalline silicon ingots is a distinctly unsteady process with a complex interaction between melt flow, crystallization interface, and species transport. Both the different time-scales and the three-dimensional character make numerical simulations of this process a challenging task. The complexity of such simulations increases further if external magnetic fields are used to enhance the melt flow. In this contribution, several three-dimensional coupled unsteady calculations are carried out for a 22 x 22 x 11 cm(3) silicon melt directionally solidified in a traveling magnetic field. The justification of various approximations in the numerical models is discussed with an emphasis on the frequently used quasi steady-state models for the calculation of the interface shape. It is shown that an upward traveling magnetic field leads to a symmetric concave interface shape while a downward field results in a convex interface with a distinct asymmetry at the current supplies. These results agree in both unsteady and quasi steady-state calculations, but only unsteady calculations reveal the flow-induced local oscillations of the interface. The unsteady segregation process of carbon and oxygen impurities exhibits a non-uniform concentration along the crystallization interface although the bulk concentration is near to the complete mixing limit in the cases with a traveling magnetic field.