An integrated process model for the growth of oxide crystals by the Czochralski method

A comprehensive model for the Czochralski (CZ) growth of oxide crystals is developed which integrates analyses of global heat transfer, transport phenomena in the crystal and melt, and the interfaces of the growth system. Heat transfer at the global scale includes fundamental descriptions of induction heating and diffuse-gray enclosure radiation. In the crystal and the melt, steady-state axisymmetric solutions for heat transfer and fluid mechanics are computed along with a self-consistent description of the free-boundaries of the melt/crystal interface, the melt meniscus, and the crystal diameter. A Galerkin finite-element method is employed to discretize the model equations, and solutions are obtained using a quasi-Newton iterative scheme. Results are presented for the growth of a large-dimension oxide crystal with realistic thermophysical properties similar to those of gadolinium gallium garnet (GGG). Comparison between the results of this model and those of Sackinger et al. [Intern. J. Numer. Methods Fluids 9 (1988) 453] demonstrate the importance of realistic heat transfer boundary conditions. Calculations also show the effects of pedestal heat transfer on the flow in the melt. The effect of changing melt volume is examined to assess the batchwise evolution of a CZ growth run.