3D unsteady and steady modeling of heat and mass transfer during Cz Si crystal growth with a horizontal magnetic field

Abstract

A combined LES/RANS approach is applied to modeling Czochralski (Cz) Si growth with a strong horizontal magnetic field stabilizing the melt flow. A model furnace geometry with a cylindrical crucible is considered, which can be suggested for benchmarking Cz Si melt convection of industrial scale. Grid resolution in boundary layers is carefully adjusted and the 4th approximation order for diffusion terms is shown to provide sufficient computation accuracy even with comparatively rough grids. The computed flow structure, temperature distribution on the melt free surface, and RMS temperature fluctuations of the melt are in good agreement with published experimental data and modeling results for industrial furnaces. The effect of the crystal rotation rate on the interface shape and oxygen incorporation into the crystal is analyzed. Due to high thermal conductivity (or low Pr number) of the melt, the unsteady oscillations do not exert a significant effect on the temperature and heat flux distribution in the melt, which allows quite accurate predictions of the interface shape, even neglecting the flow unsteadiness. In contrast, due to a low diffusivity (or high Sc number) of atomic oxygen in the melt, the unsteady oscillations quite significantly affect its transport through the melt and incorporation into the crystal, which results in remarkable overestimation of the oxygen concentration without consideration of unsteady melt flow evolution. The formulation of the LES/RANS approach used in the benchmark problem was verified on experimental data for an industrial furnace for Magnetic Cz (MCz) growth of 300 mm silicon crystals.