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
Heat exchange in the furnace
Figure 1 Global heat transfer in the reactor
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.
Figure 2 Temperature distribution over the melt free surface
predicted in 3D unsteady computer modeling of melt
convection (right) and a photograph of the melt surface
(left, courtesy of Crystal Development company, Moscow)
Figure 3 Computed and observed crystal shapes
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
Figure 4 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)
Figure 5(a) Crystal shape evolution for modification 1
Figure 5(b) Crystal shape evolution for modification 2
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 .
Figure 6 Crystal shape before and after modifications
of the growth technology (courtesy of Monocrystal Inc.,) see  for 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 in Fig. 7
Figure 7 The von Mises norm stress distributions in
the Standard and Modified Cases.
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.
Figure 8 The dislocation density in morphological
R-plane before and after modifications
Figure 9 The optical
nonuniformity in the polarized light, the plane (0001)
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 .
Figure 10 3D unsteady modeling of heat transfer, melt flow,
and crystallization before seeding, at seeding stage, and at shouldering stage
Figure 11(a) Animation of the temperature distribution over the melt free surface before seeding
Figure 11(b) Animation of the temperature distribution over the melt free surface at seeding stage
Figure 11(c) Animation of the temperature distribution over the melt free surface at shouldering stage
1. "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
2. "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)
3. "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)
4. "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
5. "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.
6. "Numerical solution of problems with radiation transfer in axisymmetric
areas of a complex shape with specular Fresnel's", V.M.Mamedov, S.À. Rukolaine,
Math. Modeling, vol. 16, 10 (2004) pp.15-28