ANALYTICAL STUDIES ON THE CRYSTAL-MELT INTERFACE SHAPE IN THE CZOCHRALSKI PROCESS

Analytical studies are performed to understand the effects of various operating conditions on the crystal-melt interface shape in the Czochr alski process. The temperature field in the crystal region with curved bottom is obtained by using the domain perturbation method. For the t emperature field in the melt phase, the well-known results from the th ermal boundary layer analysis are applied based on the idea that the f low held near the crystal-melt interface can be modeled as either a bi axial or a uniaxial flow with azimuthal velocity component. The interf ace 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 analytical predictions agree qualitatively with the experimental and n umerical results reported elsewhere.