Analytical studies on the crystal-melt interface shape in the Czochralski process for oxide single crystals

Effects of the operating conditions on the crystal-melt interface shape are analytically investigated for the Czochralski process of the oxide single crystals. The ideas, which were used for the silicon single-crystal growth by Jeong et al. (J. Crystal Growth 177 (1997) 157), are extended to the oxide single-crystal growth problem by considering the internal radiation in the crystal phase and the melt phase heat transfer with the high Prandtl number. The interface shape is approximated in the simplest form as a quadratic function of radial position and an expression for the deviation from the flat interface shape is derived as a function of operating conditions. The radiative heat transfer rate between the interface and the ambient is computed by calculating the view factors for the curved interface shape with the assumption that the crystal phase is completely transparent. For the melt phase, the well-known results from the thermal boundary layer analysis are applied for the asymptotic case of high Prandtl number based on the idea that the flow field near the crystal-melt interface can be modeled as either a uniaxial or a biaxial flow. Through this work, essential information on the interface shape deformation and the effects of operating conditions are brought out for the oxide single-crystal growth. (C) 2000 Elsevier Science B.V. AII rights reserved.