Hydrodynamics in Czochralski growth-computer analysis and experiments

Abstract

Probable flows in a melt are reviewed. In a pure melt, four different flows, free convection, surface tension driven flow (Marangoni flow), forced convection caused by crystal rotation, and forced convection caused by crucible rotation, are to be considered. The mixed flow caused both by buoyant force and by crystal rotation is roughly divided into three classes with respect to the ratio Gr Re 2 , where Gr is the Grashof number due to the temperature difference between the crystal and crucible and Re is the Reynolds number of the rotating crystal. For a low Gr Re 2 , forced convection is dominant in the melt, while for a high Gr Re 2 , free convection is dominant. For an intermediate Gr Re 2 , free and forced convection coexist and they are separ ated by a stagnant surface. The temperature field is slightly dependent on the flow for low Prandtl number melts such as molten metals and semiconductors, while it strongly depends on the flow for high Prandtl number melts such as molten oxides. Two distinct types of temperature field are obtained with respect to the dominant convection and forced convection. For a fixed Grashof number, the heat transfer mechanism changes from free convection dominant mode to forced convection dominant mode as the Reynolds number increases. This explains the change of the interface shape with increasing crystal rotation rate and the abrupt inversion of the interface shape with increasing crystal diameter. A formula to predict the critical Reynolds number at which the interface shape changes from convex to concave is presented. The effect of heat dissipation from the melt free surface on the flow is also examined, and a reason for the difficulty encountered in the growth of some oxide crystals is offered. Finally, the effect of the crucible rotation on the flow is briefly described.