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Year 2011, Volume 01, Issue 2, 127 - 149, 01.12.2011

Abstract

References

  • [1] Dost, S. and Lent, B., (2007), Single Crystal Growth of Semiconductors from Metallic Solutions, Elsevier, Amsterdam, the Netherlands.
  • [2] Hauke, G., (2002), A simple subgrid scale stabilized method for the advection-diffusion-reaction equation, Comput. Methods Appl. Mech. Engrg. 191, 2925-2947.
  • [3] Muller, G. and Ostrogorsky, A.G., (1994), Convection in melt growth. In Handbook of Crystal Growth. (ed. D.T.J. Hurle), North-Holland, Amsterdam, vol. 2, p. 711-814.
  • [4] Deal, A., (2004), Enhanced morphological stability in Sb-doped Ge single crystals, PhD thesis, Florida University, USA.
  • [5] Hurle, D. T. J., (1973), Melt Growth chapter in Crystal Growth, edited by P. Hartman, North-Holland, Amsterdam.
  • [6] F. Mechighel (2010): Mod´elisation de la convection au cours de changement de phase liquide-solide: effet d’un champ magn´etique, PhD thesis University of Limoges, France.
  • [7] Ozoe, H., Szmyd, J. S. and Tagawa,T., (2007), Magnetic Fields in Semiconductor Crystal Growth, S. Molokov et al. (eds.), Magnetohydrodynamics - Historical Evolution and Trends, 375-390, Springer.
  • [8] Armour, N. and Dost, S., (2009), Effect of an applied static magnetic field on silicon dissolution into a germanium melt, Journal of Crystal Growth, 311, 780-782.
  • [9] Sistek, J., (2008), The finite element method in fluids: stabilization and domain decomposition Ph.D. thesis, Czech Technical University in Prague.
  • [10] Donea, J. and Huerta, A., (2003), Finite Element Methods for Flow Problems, John Wiley and Sons, Ltd.
  • [11] Polner, M., (2005), Galerkin Least-Squares Stabilization Operators for the Navier-Stokes Equations A Unified Approach, PhD thesis, University of Twente, The Netherlands.
  • [12] Franca, L. P., Frey, S. L. and Hughes, T. J. R., (1990), Stabilization Finite Element Methods: I. Applications to the advective-diffusive model, INRA research report (N. 1300), France.
  • [13] Brooks, A. N. and Hughes, T. J. R., (1982), Streamline Upwind/Petrov-Galerkin formulations for convective dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 32, 199-259.
  • [14] Tezduyar, T. E. and Ganjoo, D. K., (1986), Petrov-Galerkin formulations with weighting functions dependent upon spatial and temporal discretization: Applications to transient convection-diffusuion problmes. Computer Methods in Applied Mechanics and Engineering, 59, 49-71.
  • [15] Hughes, T. J. R., Franca, L. P. and Hulbert, G. M., (1989), A new finite element formulation for computational fluid dynamics: VIII. The Galerkin-least-squares method for advective-diffusive equations, Comput. Methods Appl. Mech. Engrg. 73, 173-189.
  • [16] Feby, A., Behr, M. and Heinkenschloss, M., (2004), The effect of stabilization in finite element methods for the optimal boundary control of the Oseen equations, Finite Elements in Analysis and Design 41, 229-251.
  • [17] Hughes, T. J. R., (1995), Multiscale phenomena: Green’s function, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized formulations, Comput. Methods Appl. Mech. Engrg. 127, 387-401.
  • [18] Franca, L. P. and Farhat, C., (1994), Bubble functions prompt unusual stabilized finite element methods, Comput. Methods Appl. Mech. Engrg. 123, 299-308.
  • [19] Donea, J., (1984), A Taylor-Galerkin method for convection transport problems, Int. J. Numer. Methods Engrg. 20, 101-l19.
  • [20] Codina, R., (1998), Comparison of some finite element methods for solving the diffusion-convectionreaction equation, Comput. Methods Appl. Mech. Engrg. 156, 185-210.
  • [21] Yildiz, M., Dost, S. and Lent, B., (2005), Growth of bulk SiGe single crystals by liquid phase diffusion, Journal of Crystal Growth 280, 151-160.
  • [22] Yildiz, M., (2005), A Combined Experimental and Modeling Study for the Growth of SixGe1-x Single Crystals by Liquid Phase Diffusion (LPD), PhD thesis, University of Victoria, Canada.
  • [23] Yildiz, M. and Dost, S., (2005), A continuum model for the Liquid Phase Diffusion growth of bulk SiGe single crystals, International Journal of Engineering Science 43, 1059-1080.
  • [24] Yildiz, E., Dost, S. and Yildiz, M., (2006), A numerical simulation study for the effect of magnetic fields in liquid phase diffusion growth of SiGe single crystals, Journal of Crystal Growth, 291, 497-511.
  • [25] Yildiz, E. and Dost, S., (2007), A numerical simulation study for the combined effect of static and rotating magnetic fields in liquid phase diffusion growth of SiGe, Journal of Crystal Growth, 303, 279-283.
  • [26] Thess, A., Votyakov, E., Knaepen, B. and Zikanov, O., (2007), Theory of the Lorentz force flowmeter, New Journal of Physics, 9, 299-325.
  • [27] Tezduyar, T. E., (1992), Stabilized Finite Element Formulations for Incompressible Flow Computations, Advances in Applied Mechanics, vol. 28, 1-44.
  • [28] Tezduyar, T. E., Mittal, S. and Shih, R., (1991), Time-accurate incompressible flow computations with Quadrilateral velocity-pressure elements, computer methods in applied mechanics and engineering, 87, 363-384.
  • [29] Tezduyar, T. E., Mittal, S., Ray, S. E. and Shih, R., (1992), Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements, Computer Methods in Applied Mechanics and Engineering, 95, 221-242.
  • [30] Shakib, F., (1988), Finite element analysis of the compressible Euler and Navier-Stokes equations, Ph.D. Thesis, Department of Mechanical Engineering, Stanford University, Stanford, California.
  • [31] COMSOL Multiphysics Modeling Guide, COMSOL AB. (2007), SE-111 40 Stockholm.
  • [32] Shakib, F. and Hughes, T. J. R., (1991), A new finite element formulation for computational fluid dynamics. IX. Fourier analysis of space-time Galerkin/least-squares algorithms, Comput. Methods Appl. Mech. Eng. 87(1), 35-58.
  • [33] Saad, Y. and Schultz, M. H., (1983), GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. Yale University Research Report YALEU/DCS/ RR-254.
  • [34] Akin, E. J. and Tezduyar, T. E., (2004), Calculation of the advective limit of the SUPG stabilization parameter for linear and higher-order elements, Comput. Methods Appl. Mech. Engrg., 193, 1909- 1922.
  • [35] Tezduyar, T. E. and Osawa, Y., (2000), Finite element stabilization parameters computed from element matrices and vectors, Comput. Methods Appl. Mech. Engrg., 190, 411-430.

MATHEMATICAL MODELING OF THE DISSOLUTION PROCESS OF SILICON INTO GERMANIUM MELT

Year 2011, Volume 01, Issue 2, 127 - 149, 01.12.2011

Abstract

Numerical simulations were carried out to study the thermosolutal and flow structures observed in the dissolution experiments of silicon into a germanium melt. The dissolution experiments utilized a material configuration similar to that used in the Liquid Phase Diffusion LPD and Melt-Replenishment Czochralski Cz crystal growth systems. In the present model, the computational domain was assumed axisymmetric. Governing equations of the liquid phase Si-Ge mixture , namely the equations of conservation of mass, momentum balance, energy balance, and solute species transport balance were solved using the Stabilized Finite Element Methods ST-GLS for fluid flow, SUPG for heat and solute transport . Measured concentration profiles and dissolution height from the samples processed with and without the application of magnetic field show that the amount of silicon transported into the melt is slightly higher in the samples processed under magnetic field, and there is a difference in dissolution interface shape indicating a change in the flow structure during the dissolution process. The present mathematical model predicts this difference in the flow structure. In the absence of magnetic field, a flat stable interface is observed. In the presence of an applied field, however, the dissolution interface remains flat in the center but curves back into the source material near the edge of the wall. This indicates a far higher dissolution rate at the edge of the silicon source

References

  • [1] Dost, S. and Lent, B., (2007), Single Crystal Growth of Semiconductors from Metallic Solutions, Elsevier, Amsterdam, the Netherlands.
  • [2] Hauke, G., (2002), A simple subgrid scale stabilized method for the advection-diffusion-reaction equation, Comput. Methods Appl. Mech. Engrg. 191, 2925-2947.
  • [3] Muller, G. and Ostrogorsky, A.G., (1994), Convection in melt growth. In Handbook of Crystal Growth. (ed. D.T.J. Hurle), North-Holland, Amsterdam, vol. 2, p. 711-814.
  • [4] Deal, A., (2004), Enhanced morphological stability in Sb-doped Ge single crystals, PhD thesis, Florida University, USA.
  • [5] Hurle, D. T. J., (1973), Melt Growth chapter in Crystal Growth, edited by P. Hartman, North-Holland, Amsterdam.
  • [6] F. Mechighel (2010): Mod´elisation de la convection au cours de changement de phase liquide-solide: effet d’un champ magn´etique, PhD thesis University of Limoges, France.
  • [7] Ozoe, H., Szmyd, J. S. and Tagawa,T., (2007), Magnetic Fields in Semiconductor Crystal Growth, S. Molokov et al. (eds.), Magnetohydrodynamics - Historical Evolution and Trends, 375-390, Springer.
  • [8] Armour, N. and Dost, S., (2009), Effect of an applied static magnetic field on silicon dissolution into a germanium melt, Journal of Crystal Growth, 311, 780-782.
  • [9] Sistek, J., (2008), The finite element method in fluids: stabilization and domain decomposition Ph.D. thesis, Czech Technical University in Prague.
  • [10] Donea, J. and Huerta, A., (2003), Finite Element Methods for Flow Problems, John Wiley and Sons, Ltd.
  • [11] Polner, M., (2005), Galerkin Least-Squares Stabilization Operators for the Navier-Stokes Equations A Unified Approach, PhD thesis, University of Twente, The Netherlands.
  • [12] Franca, L. P., Frey, S. L. and Hughes, T. J. R., (1990), Stabilization Finite Element Methods: I. Applications to the advective-diffusive model, INRA research report (N. 1300), France.
  • [13] Brooks, A. N. and Hughes, T. J. R., (1982), Streamline Upwind/Petrov-Galerkin formulations for convective dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 32, 199-259.
  • [14] Tezduyar, T. E. and Ganjoo, D. K., (1986), Petrov-Galerkin formulations with weighting functions dependent upon spatial and temporal discretization: Applications to transient convection-diffusuion problmes. Computer Methods in Applied Mechanics and Engineering, 59, 49-71.
  • [15] Hughes, T. J. R., Franca, L. P. and Hulbert, G. M., (1989), A new finite element formulation for computational fluid dynamics: VIII. The Galerkin-least-squares method for advective-diffusive equations, Comput. Methods Appl. Mech. Engrg. 73, 173-189.
  • [16] Feby, A., Behr, M. and Heinkenschloss, M., (2004), The effect of stabilization in finite element methods for the optimal boundary control of the Oseen equations, Finite Elements in Analysis and Design 41, 229-251.
  • [17] Hughes, T. J. R., (1995), Multiscale phenomena: Green’s function, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized formulations, Comput. Methods Appl. Mech. Engrg. 127, 387-401.
  • [18] Franca, L. P. and Farhat, C., (1994), Bubble functions prompt unusual stabilized finite element methods, Comput. Methods Appl. Mech. Engrg. 123, 299-308.
  • [19] Donea, J., (1984), A Taylor-Galerkin method for convection transport problems, Int. J. Numer. Methods Engrg. 20, 101-l19.
  • [20] Codina, R., (1998), Comparison of some finite element methods for solving the diffusion-convectionreaction equation, Comput. Methods Appl. Mech. Engrg. 156, 185-210.
  • [21] Yildiz, M., Dost, S. and Lent, B., (2005), Growth of bulk SiGe single crystals by liquid phase diffusion, Journal of Crystal Growth 280, 151-160.
  • [22] Yildiz, M., (2005), A Combined Experimental and Modeling Study for the Growth of SixGe1-x Single Crystals by Liquid Phase Diffusion (LPD), PhD thesis, University of Victoria, Canada.
  • [23] Yildiz, M. and Dost, S., (2005), A continuum model for the Liquid Phase Diffusion growth of bulk SiGe single crystals, International Journal of Engineering Science 43, 1059-1080.
  • [24] Yildiz, E., Dost, S. and Yildiz, M., (2006), A numerical simulation study for the effect of magnetic fields in liquid phase diffusion growth of SiGe single crystals, Journal of Crystal Growth, 291, 497-511.
  • [25] Yildiz, E. and Dost, S., (2007), A numerical simulation study for the combined effect of static and rotating magnetic fields in liquid phase diffusion growth of SiGe, Journal of Crystal Growth, 303, 279-283.
  • [26] Thess, A., Votyakov, E., Knaepen, B. and Zikanov, O., (2007), Theory of the Lorentz force flowmeter, New Journal of Physics, 9, 299-325.
  • [27] Tezduyar, T. E., (1992), Stabilized Finite Element Formulations for Incompressible Flow Computations, Advances in Applied Mechanics, vol. 28, 1-44.
  • [28] Tezduyar, T. E., Mittal, S. and Shih, R., (1991), Time-accurate incompressible flow computations with Quadrilateral velocity-pressure elements, computer methods in applied mechanics and engineering, 87, 363-384.
  • [29] Tezduyar, T. E., Mittal, S., Ray, S. E. and Shih, R., (1992), Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements, Computer Methods in Applied Mechanics and Engineering, 95, 221-242.
  • [30] Shakib, F., (1988), Finite element analysis of the compressible Euler and Navier-Stokes equations, Ph.D. Thesis, Department of Mechanical Engineering, Stanford University, Stanford, California.
  • [31] COMSOL Multiphysics Modeling Guide, COMSOL AB. (2007), SE-111 40 Stockholm.
  • [32] Shakib, F. and Hughes, T. J. R., (1991), A new finite element formulation for computational fluid dynamics. IX. Fourier analysis of space-time Galerkin/least-squares algorithms, Comput. Methods Appl. Mech. Eng. 87(1), 35-58.
  • [33] Saad, Y. and Schultz, M. H., (1983), GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. Yale University Research Report YALEU/DCS/ RR-254.
  • [34] Akin, E. J. and Tezduyar, T. E., (2004), Calculation of the advective limit of the SUPG stabilization parameter for linear and higher-order elements, Comput. Methods Appl. Mech. Engrg., 193, 1909- 1922.
  • [35] Tezduyar, T. E. and Osawa, Y., (2000), Finite element stabilization parameters computed from element matrices and vectors, Comput. Methods Appl. Mech. Engrg., 190, 411-430.

Details

Primary Language English
Journal Section Research Article
Authors

Farid MECHİGHEL This is me
University of Limoges, SPCTS UMR CNRS 6638, Eur. Centre for Ceramics, Limoges, France, University of Constantine, Mechanical Engineering Department, Constantine, Algeria


Neil ARMOUR This is me
Crystal Growth Laboratory, University of Victoria, Victoria, BC, Canada


Sadik DOST This is me
Crystal Growth Laboratory, University of Victoria, Victoria, BC, Canada


Mahfoud KADJA This is me
University of Constantine, Mechanical Engineering Department, Constantine, Algeria

Publication Date December 1, 2011
Published in Issue Year 2011, Volume 01, Issue 2

Cite

Bibtex @ { twmsjaem761788, journal = {TWMS Journal of Applied and Engineering Mathematics}, issn = {2146-1147}, eissn = {2587-1013}, address = {Işık University ŞİLE KAMPÜSÜ Meşrutiyet Mahallesi, Üniversite Sokak No:2 Şile / İstanbul}, publisher = {Turkic World Mathematical Society}, year = {2011}, volume = {01}, number = {2}, pages = {127 - 149}, title = {MATHEMATICAL MODELING OF THE DISSOLUTION PROCESS OF SILICON INTO GERMANIUM MELT}, key = {cite}, author = {Mechighel, Farid and Armour, Neil and Dost, Sadik and Kadja, Mahfoud} }