Analytical studies on the crystal-melt interface shape in the Czochralski process

Abstract

Analytical studies are performed to understand the effects of various operating conditions on the crystal-melt interface shape in the Czochralski process. The temperature field in the crystal region with curved bottom is obtained by using the domain perturbation method. For the temperature field in the melt phase, the well-known results from the thermal boundary layer analysis are applied based on the idea that the flow held near the crystal-melt interface can be modeled as either a biaxial or a uniaxial flow with azimuthal velocity component. 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 analytical predictions agree qualitatively with the experimental and numerical results reported elsewhere.