Comparison of thermal stress computations in Czochralski and Kyropoulos growth of sapphire crystals

Thermal stress computations during sapphire growth are compared between a resistive Czochralski furnace and a Kyropoulos inductive furnace. 2D - axisymmetric global simulations are performed to compute the thermal field in the furnace and the convection in the melt. Temperatures carried out from global modeling at a given stage of the growth process, are used for thermal stress computations in the crystal. Three-dimensional stress analysis, which takes into account the anisotropic elastic constants of sapphire, shows nearly axisymmetric von Mises stress distribution in the crystal. It is shown that applying 2D - axisymmetric modeling of thermal stress may result in significant errors. Computations performed for a crystal of 10 cm in diameter grown in a Kyropoulos furnace show only a thin region of 2-3 mm with high thermal stress located at the crystal periphery. The model predicts very low thermal stresses in the central part of the crystal. Simulations of an ingot grown by Czochralski technique, show higher thermal stresses in almost the whole volume of the crystal. Numerical computations are in agreement with our previous measurements of dislocation density in sapphire crystals grown by the Kyropoulos method. The present numerical results can explain the experimental observations showing that sapphire crystals grown by Czochralski technique have a much higher dislocation density than Kyropoulos grown ingots.