Theoretical study of the flow and temperature fields in cz single crystal growth

For the CZ crystal growth of silicon and sapphire, the flow and temperature fields with non-flat melt/crystal and melt/gas interfaces were studied theoretically by use of the finite element method. The theoretical method used here can predict the temperature distribution and the flow pattern (forced convection, free convection, Marangoni convection and their combined flow) in the melt and crystal, including the shapes of the melt/crystal and melt/gas interfaces and the crystal radius, although calculation was limited to the case of small CZ apparatus. It is found that the melt/gas interface shape affects the flow pattern in the melt, and that the melt/crystal interface shape for a system of small Pr such as silicon is not sensitive to the flow field. But for a system of larger Pr such as sapphire, the melt/crystal interface shape is strongly dependent on the flow field. It is also found that the Marangoni effect, if it operates, plays the most important roll in promoting the flow in the melt.