Modeling of Sapphire Crystal Growth by the Kyropoulos Technique
Below we show the results of numerical simulation of single leucosapphire crystal growth by the Kyropoulos method using CGSim software. The computations in the crystallization zone include the turbulent flow of the sapphire melt, laminar gas flow, radiative heat exchange in the semi-transparent crystal including specular reflectivity at the boundaries, internal absorption and scattering. Moreover, attempts to apply a simplified model, ignoring semitransparency of the crystal and the melt, give physically unrealistic results for the crystallization front geometry and temperature gradients at the metl-crystal interface.
Numerical simulation comes very helpful in the analysis and optimization of growth systems since it gives us an insight into the processes that are otherwise extremely difficult to observe or measure. For example, temperature distribution inside the melt and the structure of the flow are practically inaccessible for direct measurement.
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 at the side, top, and bottom of the furnace (right). They help to maintain optimal temperature distribution around the crucible and growing crystal that, in turn, affects melt flow structure and the crystallization front geometry. Temperature of elements surrounding the crystal will also affect thermal stresses in the crystal.
Global heat transfer in the Ky crystal growth furnace with molybdenum heat shields
The numerical model in CGSim and recommended for modeling of Ky sapphire growth was verified using experimental data. As one can see (right), simulations successfully reproduce the spoke pattern observed in experiment. A good agreement between the crystallization shapes predicted via computations and those observed experimentally, indicates that the model provides an adequate prediction of the temperature and heat fluxes in the crystal and in the melt. This, in turn, also indicates that we can expect accurate numerical prediction of the thermal stresses generating dislocations in the crystal.
Above: Temperature distribution over the melt free surface predicted in 3D unsteady modeling and a photograph of the melt surface (courtesy of Crystal Development company, Moscow). Right: Computed and observed crystal shapes
Industrial application: recipe optimization
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. Such flow pattern provided direct delivery of the hot melt to the crystallization front, resulting in high temperature gradients along the melt/crystal interface.
Right: Distributions of the temperature gradient in the crystal, the temperature in the melt and the crucible, and the flow pattern in the melt for Modification 1 (a) and Modification 2 (b)
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 (right ), see  for more details.
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 below.
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 (right), see  for details.
The von Mises norm stress distributions in the Standard and Modified Cases
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. Animated images below show rapid transitions of the melt flow after start of seeding and at shouldering stage.
Right: 3D unsteady modeling of heat transfer, melt flow, and crystallization before seeding, at seeding stage, and at shouldering stage
Temperature distribution over the melt free surface before seeding
Temperature distribution over the melt free surface at the seeding stage
Temperature distribution over the melt free surface at the shouldering stage
“Effect of heating conditions on flow patterns during the seeding stage of Kyropoulos sapphire crystal growth”, Vladimir V.Timofeev, Vladimir V.Kalaev, Vadim G.Ivanov, J. of Crystal Growth 445 (2016) 47-52, https://doi.org/10.1016/j.jcrysgro.2016.04.016
“Study on crystal-melt interface shape of sapphire crystal growth by the KY method”, Weina Liu, Jijun Lu, Hongjian Chen, Wenbo Yan, Chunhua Min, Qingqing Lian, Yunman Wang, Peng Cheng, Caichi Liu, Yongliang Xu, J. of Crystal Growth 431 (2015) 15-23, http://dx.doi.org/10.1016/j.jcrysgro.2015.08.018
“3D melt convection in sapphire crystal growth: Evaluation of physical properties”, Vladimir V. Timofeev, Vladimir V. Kalaev, Vadim G. Ivanov, International Journal of Heat and Mass Transfer 87 (2015) 42–48, http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.03.058
“Effect of crucible shape on heat transport and melt–crystal interface during the Kyropoulos sapphire crystal growth”, C. Chen, H. J. Chen, W. B. Yan, C. H. Min, H. Q. Yu, Y. M. Wang, P. Cheng, C. C. Liu, J. of Crystal Growth 388 (2014), 29-34, https://doi.org/10.1016/j.jcrysgro.2013.11.002
“3D unsteady computer modeling of industrial scale Ky and Cz sapphire crystal growth”, S.E. Demina, V.V. Kalaev, J. of Crystal Growth 320 (2011) 23-27, https://doi.org/10.1016/j.jcrysgro.2011.01.101
(*) “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.
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