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Open Physics

formerly Central European Journal of Physics

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Volume 2, Issue 1


Volume 13 (2015)

Industrial challenges for numerical simulation of crystal growth

M. Bogdanov / D. Ofengeim / A. Zhmakin
  • Softimpact Ltd., P.O. 83, 194156, St. Petersburg, Russia
  • A.F.Ioffe Physical Technical Institute, Russian Academy of Sciences, 194021, St. Petersburg, Russia
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Published Online: 2004-03-01 | DOI: https://doi.org/10.2478/BF02476280


Numerical simulation of industrial crystal growth is difficult due to its multidisciplinary nature and the complex geometry of the real-life growth equipment. An attempt is made to itemize physical phenomena dominant in the different methods for growth of bulk crystals from the melt and the vapor phase as well as to review corresponding numerical approaches. Academic research and industrial applications are compared. Development of a computational engine and a graphic user interface of the industry-oriented codes is discussed. A simulator for the entire growth process of bulk crystals by sublimation method is described.

Keywords: Crystal growth; CFD; heat transfer; materials science

Keywords: 44; 47.11; 61.72; 81; 89.20.Bb

  • [1] D.M. Anderson, G.B. McFadden, A.A. Wheeler: “A phase-field model of solidification with convection”, Physica D, Vol. 135, (2000), pp. 175–194. http://dx.doi.org/10.1016/S0167-2789(99)00109-8CrossrefGoogle Scholar

  • [2] C. Barat, T. Duffar, P. Dusserre, J.P. Garandet: “Chemical segregation in vertical Bridgman growth of GaInSb alloys”, Crystal Res. Technol., Vol. 34, (1999), pp. 449–456. http://dx.doi.org/10.1002/(SICI)1521-4079(199904)34:4<449::AID-CRAT449>3.0.CO;2-2CrossrefGoogle Scholar

  • [3] S.V. Batill, J.E. Renand, X. Gu: “Modeling and simulation uncertainty in multidisciplinary design optimization”, In: AIAA Multidisciplinary Analysis and Optimization Conference and Exhibit, Long Beach (USA), 2000, AIAA Paper 2000-4803 pp. 11. Google Scholar

  • [4] M. Beneś: “Mathematical and computational aspects of solidification of pure substances”, Acta Math. Univ. Comenianae, Vol. LXX, (2001), pp. 123–151. Google Scholar

  • [5] M.V. Bogdanov, A.O. Galyukov, S.Yu. Karpov, A.V. Kulik, S.K. Kochuguev, D.Kh. Ofengeim, A.V. Tsirulnikov, I.A. Zhmakin, A.E. Komissarov, O.V. Bord, M.S. Ramm, A.I. Zhmakin, Yu.N. Makarov: “Virtual reactor: a new tool for SiC bulk crystal growth study and optimization”, Mat. Sci. Forum, Vol. 353–356, (2001), pp. 57–60. http://dx.doi.org/10.4028/www.scientific.net/MSF.353-356.57CrossrefGoogle Scholar

  • [6] Yu.A. Borovlev, N.V. Ivannikova, V.N. Shlegel, Ya.V. Vasiliev, V.A. Gusev: “Progress in growth of large sized BGO crystals by the low-thermal-gradient Czochralski technique”, J. Crystal Growth, Vol. 229, (2001), pp. 305–311. http://dx.doi.org/10.1016/S0022-0248(01)01162-9CrossrefGoogle Scholar

  • [7] G. Butlin, C. Stops: “CAD data repair”, 5th International Meshing Roundtable, Sandia National Laboratories, (1996), pp. 7–12. Google Scholar

  • [8] Cape Simulation, Engineering Solutions Through Computer Simulations, 2002, http://www.capesim.com/simulators.htm. Google Scholar

  • [9] W. Christopher, CFD codes list, ICEM CFD Engineering, 2002, http://www.icemcfd.com/cfd/CFD-codes.html. Google Scholar

  • [10] Welcome to the Crystal Growth Laboratory, Crystal Growth Laboratory, 2003, http://www.cgl-erlangen.com/. Google Scholar

  • [11] PHOENICS Overview, CHAM Technical Report TR 001, http://www.cham.co.uk/phoenics/d-polis/d-info/phover.htm. Google Scholar

  • [12] F. Dupret, P. Nicodeme, Y. Ryckmans, M.J. Crochet: “Global modeling of heat transfer in crystal growth furnaces”, Int. J. Heat Mass Transfer, Vol. 33, (1990), pp. 1849–1871. http://dx.doi.org/10.1016/0017-9310(90)90218-JCrossrefGoogle Scholar

  • [13] P.S. Dutta, A.G. Ostrogorski: “Suppression of cracks inIn x Ga 1−x Sb crystals through forced convection in the melt”, J. Crystal Growth, Vol. 194, (1998), pp. 1–7. http://dx.doi.org/10.1016/S0022-0248(98)00621-6CrossrefGoogle Scholar

  • [14] Yu.E. Egorov, Yu.N. Makarov, E.A. Rudinsky, E.M. Smirnov, A.I. Zhmakin: “Modelling analysis of oxygen transport during Czochralski growth of silicon crystals”, Mat. Res. Soc. Proc., Vol. 490, (1998), pp. 181–186. Google Scholar

  • [15] Yu.E. Egorov, A.I. Zhmakin: “Numerical simulation of low-Mach number gas mixture flows with heat and mass transfer using unstructured grid”, Comput. Mater. Sci., Vol. 11, (1998), pp. 204–220. http://dx.doi.org/10.1016/S0927-0256(98)00005-6CrossrefGoogle Scholar

  • [16] M.S. Eldred, W.E. Hart, W.J. Bohnhoff, V.J. Romero, S.A. Hutchinson, A.G. Salinger: “Utilizing Object-Oriented Design to Build Advanced Optimization Strategies with Generic Implementation”, (1996), pp. 16. Google Scholar

  • [17] 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. von Ammon: “Global model of Czochralski silicon growth to predict oxygen content and thermal fluctuations at the melt-crystal interface”, Microelectronic Engineering, Vol. 56, (2001), pp. 139–142. http://dx.doi.org/10.1016/S0167-9317(00)00516-5CrossrefGoogle Scholar

  • [18] 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. von Ammon: “Modeling analysis of unsteady three-dimensional turbulent melt flow during Czochralski growth of Si crystals”, J. Crystal Growth, Vol. 230, (2001), pp. 22–29. http://dx.doi.org/10.1016/S0022-0248(01)01314-8CrossrefGoogle Scholar

  • [19] F. Dupret: FEMAG. The Software for Bulk Crystal Growth Simulation, http://www.meca.ucl.ac.be/femag. Google Scholar

  • [20] J.P. Giesing, J.-F.M. Barthelemy: “A Summary of Industry MDO Applications and Needs”, In: AIAA 98-4737, 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, St. Louis (USA), 1998, pp. 20. Google Scholar

  • [21] John W. Slater: Glossary of Verification and Validation Terms, http://www.grc.nasa.gov/WWW/wind/valid/tutorial/glossary.html. Google Scholar

  • [22] V.D. Golyshev, M.A. Gonik, V.B. Tsvetovsky: “In situ measurement ofBi 4Ge 3O 12 interface supercooling during melt crystal growth”, J. Crystal Growth, Vol. 237–239, (2002), pp. 735–739. http://dx.doi.org/10.1016/S0022-0248(01)01990-XCrossrefGoogle Scholar

  • [23] Guide for the Vertification and Validation of Computational Fluid Dynamixcs (G-077-1998), AIAA Standards Series, 1998. Google Scholar

  • [24] Ch. Hirsh: “The QNET-CFD project”, Network Bulletin, Vol. 1, (2001), pp. 4–5. (http://www.qnet-cfd.net/newsletter/1st/newsletter-1.pdf) Google Scholar

  • [25] I. Hooks: “Writing Good Requirements”, In: NCOSE-93, 3rd Int. Symposium NCOSE, Washington, (USA), 1993, INCOSE, 1993, http://www.incose.org./rwg/writing.html. Google Scholar

  • [26] D.T.J. Hurle (Ed.): Handbook of Crystal Growth. Bulk Crystal Growth, Elsevier, North-Holland, 1994. Google Scholar

  • [27] V.V. Kalaev, I.Yu. Evstratov, Yu.N. Makarov: “Gas flow effect on global heat transport and melt convection in Czochralski silicon growth”, J. Crystal Growth, Vol. 249, (2003), pp. 87–99. http://dx.doi.org/10.1016/S0022-0248(02)02109-7CrossrefGoogle Scholar

  • [28] V.V. Kalaev, A.I. Zhmakin, E.M. Smirnov: Modeling of turbulent melt convection during Czochralski bulk crystal growth, J. of Turbulence, Vol. 3, (2002), pp. 13. (http://jot.iop.org) Google Scholar

  • [29] V.V. Kalaev, A.I. Zhmakin: “Large Eddy Simulation of melt convection during Czochralski crystal growth”, Proc. 9th European Turbulence Conf, (2002), pp. 207–210. Google Scholar

  • [30] Y. Kaneda, T. Ishihara, M. Yokokawa, K. Itakura, A. Uno: “Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box”, Phys. Fluids, Vol. 15, (2003), pp. L21-L24. http://dx.doi.org/10.1063/1.1539855CrossrefGoogle Scholar

  • [31] S. Kochuguev, D. Ofengeim, A. Zhmakin, A. Galyukov: “Ray tracing method for axisymmetrical global heat transfer simulation”, CFD Journal, Vol. II–33, (2001), pp. 440–448. Google Scholar

  • [32] A.V. Kulik, M.V. Bogdanov, D.Kh. Ofengeim, S.K. Kochuguev, S.E. Demina, S.Yu. Karpov, A.I. Zhmakin, M.S. Ramm, Yu.N. Makarov: “Modeling and optimization of SiC bulk crystal growth by sublimation technique”, Proc. 4th Int. Conf. Single Crystal Growth and Heat Mass Transfer, (2001), pp. 698–705. Google Scholar

  • [33] C.W. Lan: “Three-dimensional simulation of floating-zone crystal growth of oxide crystals”, J. Crystal Growth, Vol. 247, (2003), pp. 597–612. http://dx.doi.org/10.1016/S0022-0248(02)02056-0CrossrefGoogle Scholar

  • [34] A. Lipchin, R.A. Brown: “Hybrid finite-volume/finite-element simulation of heat transfer and melt turbulence in Czochralski crystal growth of silicon”, J. Crystal Growth, Vol. 216, (2000), pp. 192–203. http://dx.doi.org/10.1016/S0022-0248(00)00428-0CrossrefGoogle Scholar

  • [35] Yu.N. Makarov, A.I. Zhmakin: “On flow regimes in VPE reactors”, J. Crystal Growth, Vol. 94, (1989), pp. 537–551. http://dx.doi.org/10.1016/0022-0248(89)90032-8CrossrefGoogle Scholar

  • [36] D. Maroudas, R.A. Brown: “On the prediction of dislocation formation in semiconductor crystals grown from the melt: analysis of the haasen model for plastic deformation dynamics”, J. Crystal Growth, Vol. 108, (1991), pp. 399–415. http://dx.doi.org/10.1016/0022-0248(91)90388-LCrossrefGoogle Scholar

  • [37] S. Maruyama, T. Aihira: “Radiation heat transfer of arbitrary three-dimensional absorbing, emitting and scattering media and specular and diffuse surfaces”, J. Heat Transfer, Vol. 119, (1997), pp. 129–136. Google Scholar

  • [38] M. Meyappan (Ed.): Computational Modeling in Semiconductor Processing, The Artech House, Norwood, 1994. Google Scholar

  • [39] N. Miyazaki: “Development of a thermal stress analysis system for anisotropic single crystal growth”, J. Crystal Growth, Vol. 236, (2002), pp. 455–465. http://dx.doi.org/10.1016/S0022-0248(01)02193-5CrossrefGoogle Scholar

  • [40] A. Muehlbauer, A. Muiznieks, G. Raming: “System of mathematical models for the analysis of industrial FZ-Si-Crystal Growth Processes”, Cryst. Res. Technol., Vol. 34, (1999), pp. 217–226. http://dx.doi.org/10.1002/(SICI)1521-4079(199902)34:2<217::AID-CRAT217>3.0.CO;2-1CrossrefGoogle Scholar

  • [41] G. Müller: “Experimental analysis and modeling of melt growth processes”, J. Crystal Growth, Vol. 237–239, (2002), pp. 1628–1637. http://dx.doi.org/10.1016/S0022-0248(01)02356-9CrossrefGoogle Scholar

  • [42] J.T. Oden: “The promise of Computational Engineering and Science: will it be kept?”, IACM Express, Vol. 12, (2002), pp. 12–15. Google Scholar

  • [43] T. Ozawa, Y. Hayakawa, K. Balakrishnan, M. Kumagawa: “Numerical simulation of effect of ampoule rotation for the growth of InGaSb by rotational Bridgman method”, J. Crystal Growth, Vol. 237–239, (2002), pp. 1692–1696. http://dx.doi.org/10.1016/S0022-0248(01)02332-6CrossrefGoogle Scholar

  • [44] P.Y. Papllambros: “Extending the optimization paradigm in engineering design”, In: Proc. 3rd Int. Symp. Tools Meth. Compet. Engineer., Delft (Netherlands), 2000, pp. 14. Google Scholar

  • [45] Welcome to Crosslight Software, Crosslight Software, Inc., 2003, http://www.crosslight.com. Google Scholar

  • [46] S. Prudhomme, J. T. Oden, T. Westermann, J. Bass, M.E. Botkin: “Practical Methods for a posteriori Error Estimation in Engineering Applications”, Int. J. Num. Meth. Engineer., Vol. 56, (2003), pp. 1193–1224. http://dx.doi.org/10.1002/nme.609CrossrefGoogle Scholar

  • [47] D. Reid, B. Lent, T. Bryskiewicz, P. Singer, E. Mortimer, W.A. Bonner: “Cellular structure in LEC ternaryGa x In 1−x As crystals”, J. Crystal Growth, Vol. 174, (1997), pp. 250–255. http://dx.doi.org/10.1016/S0022-0248(96)01113-XCrossrefGoogle Scholar

  • [48] Ch. Renner, J. Peinke, R. Friedrich, O. Chanal, B. Chaubad: “On the universality of small scale turbulence”, arXiv.org e-Print archive, http://arxiv.org/pdf/physics/0109052 Google Scholar

  • [49] P.J. Roache: Verification and Validation in Computational Science and Engineering, Hermosa Publishers, Albuquerque, New Mexico, 1998. Google Scholar

  • [50] S.A. Rukolaine, M.G. Vasilyev, V.S. Yuferev, A.O. Galyukov: “Numerical solution of axisymmetric radiative transfer problems in arbitrary domains using the characteristic method”, J. Quant. Spectr. Radiat. Transfer, Vol. 73, (2002), pp. 205–217. http://dx.doi.org/10.1016/S0022-4073(01)00204-7CrossrefGoogle Scholar

  • [51] W. Schönauer: “Numerical engineering: design of PDE black-box solvers”, Math. Comput. Simul., Vol. 54, (2000), pp. 269–277. http://dx.doi.org/10.1016/S0378-4754(00)00188-9CrossrefGoogle Scholar

  • [52] Products, semiconductor Technology Research, Inc., 2002, http://www.semitech.us/products/. Google Scholar

  • [53] Computational Fluid Dynamics Modeling for Semiconductor Industry, Soft-Impact Ltd., 2002, http://www.softimpact.ru/main-engl.html Google Scholar

  • [54] J.P. Steinbrenner, N.J. Wyman, J.R. Chawner: “Fast Surface Meshing on Imperfect CAD Models”, In: Proceedings, 9th International Meshing Roundtable, Jew Orleans (USA), 2000, Sandia National Laboratories, pp. 33–41. Google Scholar

  • [55] M. Suezawa, K. Sumino, N. Yonenaga: “Dislocation dynamics in the plastic deformation of silicon crystals”, Phys. Stat. Sol., Vol. A 51, (1979), pp. 217–233. Google Scholar

  • [56] A. N. Tihnnov, V.Ya. Arsenin: Solution of Ill-posed problems, Wiley, New York, 1977. Google Scholar

  • [57] Transcen Data, transcendata Europe Ltd., 2001–2003, http://www.fegs.co.uk/. Google Scholar

  • [58] C.T. Tsai: “On the finite element modeling of dislocation dynamics during semiconductorcrystal growth”, J. Cryst. Growth, Vol. 113, (1991), pp. 499–507. http://dx.doi.org/10.1016/0022-0248(91)90085-JCrossrefGoogle Scholar

  • [59] A. Virozub, S. Brandon: “Selecting finite element basis functions for computation of partially facetted melt/crystal interfaces appearing during the directional growth of large-scale single crystals”, Model. Simul. Mater. Sci. Eng. Vol. 10, (2002), pp. 57–72. http://dx.doi.org/10.1088/0965-0393/10/1/305CrossrefGoogle Scholar

  • [60] D. Vizman, O. Grabner, G. Müller: “Three-dimensional numerical simulation of thermal convection in an industrial Czochralski melt: comparison to experimental results”, J. Cryst. Growth, Vol. 233, (2001), pp. 687–698. http://dx.doi.org/10.1016/S0022-0248(01)01633-5CrossrefGoogle Scholar

  • [61] V.V. Voronkov, R. Falster: “Intrinsic point defects and impurities in silicon crystal growth”, J. Electrochem. Soc., Vol. 149, (2002), pp. G167-G174. http://dx.doi.org/10.1149/1.1435361CrossrefGoogle Scholar

  • [62] K. Weihe, Th. Willhalm: “Why CAD data repair requires discrete algorithmic techniques”, Konstanzer Schrift. Math. Inform., N 61, (1998), pp. 12. Google Scholar

  • [63] W.R. Wilcox, L.L. Regel, W.A. Arnold: “Convection and segregation during vertical Bridgman growth with centrifugation”, J. Crystal Growth, Vol. 187, (1998), pp. 543–558. http://dx.doi.org/10.1016/S0022-0248(97)00885-3CrossrefGoogle Scholar

  • [64] M. Yokokawa, K. Itakura, A. Uno, T. Ishihara, Y. Kaneda: 16.4 TFlops direct numerical simulation of turbulence by a Fourier spectral method on the Earth simulator, http://www.sc-2002.org/paperpdfs/pap.pap273.pdf Google Scholar

  • [65] V.S. Yuferev, O.N. Budenkova, M.G. Vasiliev, S.A. Rukolaine, V.N. Shlegel, Ya.V. Vasiliev, A.I. Zhmakin: “Variations of solid-liquid interface in BGO low thermal gradients Cz growth for diffuse and specular crystal side surface”, J. Crystal Growth, Vol. 253, (2003), pp. 383–397. http://dx.doi.org/10.1016/S0022-0248(03)01110-2CrossrefGoogle Scholar

  • [66] N. Zabaras: “Adjoint methods for inverse free convection problems with application to solidification processes”, In: J. Borggaard, E. Cliff, S. Schreck and J. Burns (Eds.): Computational Methods for Optimal Design and Control, Birkhauser Series in Progress in Systems and Control Theory, Birkhauser, 1998, pp. 391–426. Google Scholar

  • [67] A.I. Zhmakin: “A memory-efficient unstructured grid refinement algorithm for computation of 3d steady viscous flows”, Comm. Num. Meth. Eng., Vol. 13, (1997), pp. 219–228. http://dx.doi.org/10.1002/(SICI)1099-0887(199704)13:4<219::AID-CNM41>3.0.CO;2-9CrossrefGoogle Scholar

  • [68] I.A. Zhmakin, A.V. Kulik, S. Yu. Karpov, S.E. Demina, M.S. Ramm, Yu. N. Makarov: “Evolution of thermoelastic strain and dislocation density during sublimation growth of silicon carbide”, Dimond and Related Materials, Vol. 9, (2000), pp. 446–451. http://dx.doi.org/10.1016/S0925-9635(99)00307-6CrossrefGoogle Scholar

About the article

Published Online: 2004-03-01

Published in Print: 2004-03-01

Citation Information: Open Physics, Volume 2, Issue 1, Pages 183–203, ISSN (Online) 2391-5471, DOI: https://doi.org/10.2478/BF02476280.

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