Numerical analysis and optimization of sapphire crystal growth by the Kyropoulos technique
Presented below are the results of numerical simulation of single leucosapphire crystal growth by the Kyropoulos method using CGSim software. It is a unique feature of the suggested approach that the computations in the crystallization zone involve the turbulent flow of the sapphire melt, laminar gas flow, and radiative heat exchange in the semi-transparent crystal including specular reflectivity at the boundaries, internal absorption and scattering.
Numerical simulation comes very helpful in the analysis and optimization of growth systems as it gives us an insight into the processes that are otherwise extremely difficult to observe or measure, such as, the temperature distributions inside the melt and growing crystal. Attempts to apply a simplified model, ignoring semitransparency of the crystal and the melt, gave physically unrealistic results for the crystallization front geometry and temperature gradients at the metl-crystal interface. To account for sapphire semitransparency, we used an approach developed in  in terms of the discrete transfer method for solving problems of the radiation transfer in axisymmetric areas of complex geometry with specular Fresnel’s boundaries.
Heat exchange in the furnace
Global heat transfer in a Ky furnace is strongly affected by a particular number and geometrical parameters of molybdenum heat shields. Sets of heat shields and other insulation blocks are positioned in the side top, and bottom part of a furnace (see Fig. 1). The aim of heat shield design is to develop a hot zone with optimal temperature distribution around the crucible and growing crystal. Temperature distribution in the crucible will define melt flow structure and affect the crystallization front geometry. Temperature of elements surrounding the crystal will as well affect the crystallization front shape and define thermal stresses in the crystal.
The model was verified using available experimental data. Simulations successfully reproduce the spoke pattern observed in experiment, see Fig. 2. A good agreement between the crystallization shapes predicted via computations and those observed experimentally, see Fig. 3, indicates that the model provides an adequate prediction of the temperature and heat fluxes in the crystal and in the melt. This ensures numerical prediction of the thermal stresses generating dislocations in the crystal.
Example of industrial application:
Improving the hot zone of the Kyropoulos furnace to decrease thermal gradients at the crystallization front results in higher yield ratio and better crystal quality. Using the CGSim package, several configurations of the industrial furnace have been considered .
In the initial configuration (Modification 1), the melt flow had a two-vortex structure with a larger vortex occupying the melt core and a vortex of lower intensity located near the melt free surface during the lateral crystal growth and disappearing at the cylindrical growth stage, Fig. 4-5. Such flow pattern provided direct delivery of the hot melt to the crystallization front, resulting in high temperature gradients along the melt/crystal interface. After considering several hot zone modifications, we found a furnace configuration providing one-vortex flow structure in the melt (Modification 2). Such flow pattern results in gradual cool-down of the melt as it approaches the growing crystal, thus, decreasing the temperature gradients in the crystal by up to 30%.
Originally, in the top part of the crystal, there were well defined regions of decreasing crystal diameter, which might be due to remelting or slow crystallization. After modifications of growth technology, the remelting regions have mainly disappeared, Fig. 6 .
Temperature gradients inside the crystal and thermal stress values have also been reduced as a result of the proposed modifications. Comparison of von Mises stress norm distributions before and after the technology optimization is given in Fig. 7
Improvement of the crystal quality has been confirmed experimentally. For instance, morphological and optical investigation of wafer samples obtained close to the region of remelting had shown that the dislocation density in morphological R-plane after modifications dropped from 103 cm-2 to 102 cm-2, see Fig. 8-9 and  for details.
Example: 3D modeling of Ky sapphire growth in 250 mm diameter crucible
Crystal seeding is successful only if there is prevailing downward melt motion in the point of seeding on the melt free surface. Upward melt flow in the seeding point may result in meldown of the seed. 3D unsteady modeling of melt convection and crystallization helps to find optimal heating conditions for smooth and stable seeding and shouldering stages. Figs.10-11 describes rapid transitions of the melt flow after start of seeding and at shouldering stage .
 “Analysis of melt flow and crystallization during large-scale Kyropoulos sapphire growth”, Svetlana Demina, Vladimir Kalaev, Presentation during ACCGE-17, August 9 – 14, 2009, Grand Geneva Resort, Lake Geneva, Wisconsin USA
 “Use of Numerical Simulation for Growing High Quality Sapphire Crystals by the Kyropoulos method”, S.E. Demina, E.N. Bystrova, V.S. Postolov, E.V. Eskov, M.V. Nikolenko, D.A. Marshanin, V.S. Yuferev, V.V. Kalaev Journal of Crystal Growth 310 (2008) 1443–1447 (20)
 “Numerical analysis of sapphire crystal growth by the Kyropoulos technique”, S.E. Demina, E.N. Bystrova, M.A. Lukanina, V.M.Mamedov, V.S. Yuferev, E.V. Eskov, M.V. Nikolenko, V.S. Postolov, V.V. Kalaev Optical Materials 30 (2007) 62–65 (20)
 “Numerical analysis of sapphire crystal growth by the Kyropoulos technique”, S.E. Demina, E.N. Bystrova, M.A. Lukanina, V.V. Kalaev, V.M. Mamedov, V.S. Yuferev, E.V. Eskov, M.V. Nikolenko, V.S. Postolov, Presentation, ICCG15, Salt-Lake City, August 12–17, 2007
 “Numerical analysis of sapphire crystal growth by the Kyropoulos technique”, S.E. Demina, E.N. Bystrova, M.A. Lukanina, V.V. Kalaev, V.M. Mamedov, V.S. Yuferev, E.V. Eskov, M.V. Nikolenko, V.S. Postolov, to be published in Journal of Optical Materials in 2006.
 “Numerical solution of problems with radiation transfer in axisymmetric areas of a complex shape with specular Fresnel’s”, V.M.Mamedov, S.А. Rukolaine, Math. Modeling, vol. 16, 10 (2004) pp.15-28