**Growth from melt and solutions**

We have been working in modeling and optimization of crystal growth from the
melt for over 10 years. During this period of time, we worked with industries
on the optimization of the growth technology and participated in a number of
projects with research laboratories. The experience we have gained covers
modeling and optimization of such growth techniques as Czochralski (CZ),
Liquid Encapsulated Czochralski (LEC), and Bridgman. In each of these areas,
numerical simulation proved to be a reliable and effective tool helping
to solve practical problems. It provides insight into such aspects of the
technological process that are extremely difficult to monitor experimentally.
Modeling can provide valuable results fast and at a low cost as compared to a
typical experimental run. By modeling, we can help you to analyze and
optimize the following aspects of a particular growth process:

- significant increase of the crystallization rate with keeping high crystal quality;
- temperature and thermal gradient distributions in the whole growth system including the melt and the crystal;
- characteristics of convection in the melt, gas, and encapsulant;
- 3D unsteady turbulent convection in the melt and in the gas;
- heat flux balance and optimization of the hot zone geometry for reduction of the heater power;
- the geometry of the crystallization front and thermal stresses in the crystal;
- defect characteristics in the growing crystal;
- impurity and species concentrations and the melt and crystal homogeneity.

The computational tool we use is the program package Crystal Growth Simulator (CGSim)
developed at STR, Inc. There is a basic version with simple and powerful graphic
interface and a number of add-ons for specific applications.

Capabilities of the CGSim package are illustrated through detailed
application examples:

**Numerical models applied to bulk crystal growth analysis**

The mathematical models are based on the Navier-Stokes equations and the heat &
mass transport equations in the 2D and 3D approximations. Special models are
available for crystallization phenomena and chemical reactions.

The mathematical aspects of the models used for the analysis of bulk
crystal growth are described in [1-17]. The global heat transfer calculations
[1-3, 10, 15] are the basis for a detailed analysis of flows, mass transfer,
and crystallization. Usually, the view-factor (surface-to-surface) method is
applied to simulate radiative heat exchange; a heat transport conservation
equation is used to predict the temperature distribution. To simulate
radiative heat transfer in semitransparent media, we apply an original model
based on a combination of the ray-tracing and discrete ordinate approach
[29-30].

To calculate the melt and encapsulant convection, we use the Boussinesq
approximation of the Navier-Stokes equations. The gas flow is simulated by
the approximation for slow subsonic gas motion, and the turbulent mixing
within the 2D or axisymmetric approximations involves a Reynolds-averaging
(RANS) procedure [1,3,6,10,15]. In the 3D unsteady approximation, there are
options for Direct Numerical Simulation (DNS) or for Large Eddy Simulation
(LES) [4-9,11-14,16-17]. Original combined LES/RANS models have been
developed for CZ Si growth to reduce the computational time.

The moving grid approach is used to find the geometry of the melt/crystal
interface. The grid is reconstructed after each step of finding the
crystallization front.

References:

[1] "Modelling analysis of oxygen transport during Czochralski growth of silicon crystals", Yu.E. Egorov, Yu.N. Makarov, E.A. Rudinsky, E.M. Smirnov, A.I. Zhmakin, in: S.T. Dunham, J.S. Nelson (Eds.), Semiconductor Process and Device Performance Modelling, Mat. Res. Soc. Symp. Proc., vol.490, MRS, Pennsylvania, 1998, p.181.

[2] " Global model of Czochralski silicon growth to predict oxygen content and thermal fluctuations at the melt-crystal interface", I.Yu. Evstratov, V.V. Kalaev, V.N. Nabokov, A.I. Zhmakin, Yu.N. Makarov, A.G.Abramov, N.G. Ivanov, E.A. Rudinsky, E.M. Smirnov, S.A. Lowry, E. Dornberger, J. Virbulis, E. Tomzig, W. v.Ammon, Microelectronic Engineering, 56/1-2 (2001) pp. 139-142

[3] "Numerical modeling of Czochralski silicon crystal growth", V.V. Kalaev, I.Yu. Evstratov, Yu.N. Makarov, E.M. Smirnov, A.I. Zhmakin, ECCOMAS-2000, September 11-14, 2000, Barcelona, Spain, CD-ROM Proceeding, N677 (9 pages)

[4] "Modeling analysis of unsteady three-dimensional turbulent melt flow during Czochralski growth of Si crystals", I.Yu. Evstratov, V.V. Kalaev, A.I. Zhmakin, Yu.N.Makarov, A.G. Abramov, N.G. Ivanov , E.M. Smirnov, E. Dornberger, J. Virbulis, E. Tomzig, W. v.Ammon, Journal of Crystal Growth 230 (2001) pp. 22-29

[5] "Modeling of turbulent melt convection during Czochralski bulk crystal growth", V.V. Kalaev, A.I. Zhmakin, E.M. Smirnov, Journal of Turbulence, vol. 3 (2002) 013,
http://jot.iop.org/ (12 pages)

[6] "Modeling of impurity transport and point defect formation during Cz Si crystal growth", V.V. Kalaev, V.A. Zabelin, Yu.N. Makarov, Solid State Phenomena 82-84 (2002) pp. 41-46

[7] "Numerical study of 3D unsteady melt convection during industrial-scale CZ Si-crystal growth", I.Yu. Evstratov, V.V. Kalaev, A.I. Zhmakin, Yu.N. Makarov,A.G. Abramov, N.G. Ivanov, A.B. Korsakov, E.M. Smirnov, E. Dornberger, J. Virbulis, E. Tomzig, W. von Ammon, Journal of Crystal Growth 237-239 (2002) 1757-1761

[8] "Large Eddy Simulation of Melt Convection during Czochralski Crystal Growth", V.V. Kalaev and A. I. Zhmakin, Advances in Turbulence IX, Proceedings of the Ninth European Turbulence Conference, Southampton, U.K., July 2-5, 2002, pp. 207-210

[9] "Hybrid LES/RANS simulation of melt convection during crystal growth", V.V. Kalaev, A.I. Zhmakin, Engineering Turbulence Modelling and Experiments 5, Edited by W. Rodi and N. Fueyo, Elsevier, The 5th International Symposium on Engineering Turbulence Modelling and Measurements, Mallorca, Spain, 16-18 September, 2002, Proceedings, pp. 337-346

[10] "Gas flow effect on global heat transport and melt convection in Czochralski silicon growth", V. V. Kalaev, I. Yu. Evstratov, Yu. N. Makarov, J. Crystal Growth, 249/1-2 (2003) pp. 87-99

[11] "Calculation of bulk defects in CZ Si growth: impact of melt turbulent fluctuations", V.V. Kalaev, D.P. Lukanin, V.A. Zabelin, Yu.N. Makarov, J. Virbulis, E. Dornberger, W. von Ammon, J. Crystal Growth, 250/1-2 (2003) pp. 203-208

[12] "Analysis of magnetic field effect on 3D melt flow in CZ Si growth", N.G. Ivanov, A.B. Korsakov, E.M. Smirnov, K.V. Khodosevitch, V.V. Kalaev, Yu.N. Makarov, E. Dornberger, J. Virbulis, W. von Ammon, J. Crystal Growth, 250/1-2 (2003) pp. 183-188

[13] "Prediction of the melt/crystal interface geometry in liquid encapsulated Czochralski growth of InP bulk crystals", E.N. Bystrova, V.V. Kalaev, O.V. Smirnova, E.V. Yakovlev, Yu.N. Makarov, J. Crystal Growth, 250/1-2 (2003) pp. 189-194

[14] "Modeling analysis of vCZ growth of GaAs bulk crystals using 3D unsteady melt flow simulations", E.V. Yakovlev, O.V. Smirnova, E.N. Bystrova, V.V. Kalaev, Ch. Frank-Rotsch, M. Neubert, P. Rudolph, Yu.N. Makarov, J. Crystal Growth, 250/1-2 (2003) pp. 195-202

[15] "Global heat and mass transfer in vapor pressure controlled Czochralski growth of GaAs crystals", E.V. Yakovlev, V.V. Kalaev, I.Yu. Evstratov, Ch. Frank, M. Neubert, P. Rudolph, Yu.N. Makarov, J. Crystal Growth, 252/1-3 (2003) pp. 26-36

[16] "Prediction of bulk defects in CZ Si crystals using 3D unsteady calculations of melt convection", V.V. Kalaev, D.P. Lukanin, V.A. Zabelin, Yu.N. Makarov, J. Virbulis, E. Dornberger, W. von Ammon, Materials Science in Semiconductor Processing, 5/4-5 (2003) pp. 369-373

[17] "Modeling Analysis of Liquid Encapsulated Czochralski Growth of GaAs and InP Crystals", E.V.Yakovlev, V.V. Kalaev, E.N. Bystrova, O.V. Smirnova, Yu.N. Makarov, Ch. Frank-Rotsch, M. Neubert, P. Rudolph, Crystal Research and Technology, 38, No. 6 (2003) 506-514

[18] "Modeling of point defect formation in silicon monocrystals", V. A. Zabelin and V. V. Kalaev, Microelectronic Engineering 69 (2003) pp. 641-645

[19] "Advances in the simulation of heat transfer and prediction of the melt-crystal interface shape in silicon CZ growth", D.P. Lukanin, V.V. Kalaev, Yu. N. Makarov, T. Wetzel, J. Virbulis, and W. von Ammon, J. Crystal Growth, 266/1-3 (2004) pp. 20 - 27

[20] "3D Computations of Melt Convection and Crystallization Front Geometry during VCz GaAs Growth" O.V. Smirnova, V.V. Kalaev, Yu.N. Makarov, Ch. Frank-Rotsch, M. Neubert, P. Rudolph, J. Crystal Growth 266 (2004) pp 67-73

[21] "Simulation of Heat Transfer and Melt Flow in Czochralski Growth of Si1-xGex Crystals", O.V. Smirnova, V.V. Kalaev, Yu.N. Makarov, N.V. Abrosimov, H. Riemann, J. Crystal Growth 266 (2004) pp. 74-80

[22] "Effect of internal radiation on the crystal-melt interface shape in Czochralski oxide growth", O.N. Budenkova, V.M. Mamedov, M.G. Vasiliev, V.S. Yuferev, Yu.N. Makarov, J. Crystal Growth 266 (2004) pp. 96-102

[23] "Simulation of global heat transfer in the Czochralski process for BGO sillenite crystals", O.N. Budenkova, M.G. Vasiliev, E.N. Bystrova, V.V. Kalaev, V.S. Yuferev, V. Bermudez, E. Dieguez, Yu.N. Makarov, J. Crystal Growth 266 (2004) pp. 103-108

[24] "Modeling of Czochralski growth of Si crystals in industrial systems with and without magnetic field", V.V. Kalaev, K. Khodosevitch, Yu.N. Makarov, The 4th International Symposium on Advanced Science and Technology of Silicon Materials, November 22-26, 2004, Kona, Hawaii, The Japan Society for the Promotion of Science, Proceedings, pp. 32-37

[25] "2D simulation of carbon transport at the growth of GaAs crystals by liquid encapsulated Czochralski techniques", E.N. Bystrova, V.V. Kalaev, Yu. Makarov, Ch. Frank-Rotsch, M. Neubert, P. Rudolph, Journal of Crystal Growth 275 (2005), pp. 507-514.

[26] "Numerical investigation of crucible rotation effect on crystallization rate behavior during Czochralski growth of Si1-xGex crystals", O.V. Smirnova, V.V. Kalaev, Yu.N. Makarov, N.V. Abrosimov, H. Riemann, Journal of Crystal Growth 287-2 (2006) 281-286

[27] "3D unsteady analysis of gas turbulent convection during HPLEC InP growth", E.N. Bystrova and V.V. Kalaev, Journal of Crystal Growth 287-2 (2006) 275-280

[28] 3D numerical simulation of heat transfer during horizontal direct crystallization of corundum single crystals, M.A. Lukanina, K.V. Hodosevitch, V.V. Kalaev, V.B. Semenov, V.N. Sytin and V.L. Raevsky, Journal of Crystal Growth 287-2 (2006) 330-334

[29] Thermal conditions for large alkali-halide crystal growth by the continuous feed method, V.V. Vasilyev, V.I. Goriletsky, O.Ts. Sidletskiy, E.N. Bystrova, V.V. Kalaev, Opt. Mater. (2006), doi:10.1016/j.optmat.2006.11.007

[30] 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, Opt. Mater. (2006), doi:10.1016/j.optmat.2006.11.012

[31] Three-dimensional unsteady modeling analysis of silicon transport in melt during Cz growth of Ge1_xSix bulk crystals, O.V. Smirnova, V.V. Kalaev, Yu.N. Makarov, N.V. Abrosimov, H. Riemann, V.N. Kurlov, J. Crystal Growth (2006), doi:10.1016/j.jcrysgro.2006.11.150

[32] Combined effect of DC magnetic fields and free surface stresses on the melt flow and crystallization front formation during 400mm diameter Si Cz crystal growth, V.V. Kalaev, J. Crystal Growth (2007), doi:10.1016/j.jcrysgro.2006.11.345