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Mass and Heat Transport in BS and EFG Systems

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Springer Handbook of Crystal Growth

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Abstract

In this chapter several mathematical models describing processes which take place in the Bridgman–Stockbarger (BS) and edge-defined film-fed growth (EFG) systems are presented. Predictions are made concerning the impurity repartition in the crystal in the framework of each of the models. First, a short description of the real processes which are modeled is given, along with the equations, boundary conditions, and initial values defining the mathematical model. After that, numerical results obtained by computations in the framework of the model are provided, making a comparison between the computed results and those obtained in other models, and with the experimental data.

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Abbreviations

ACC:

annular capillary channel

ALE:

arbitrary Lagrangian Eulerian

ALE:

atomic layer epitaxy

BS:

Bridgman–Stockbarger

CCC:

central capillary channel

CD:

convection diffusion

EFG:

edge-defined film-fed growth

MIT:

Massachusetts Institute of Technology

MNSM:

modified nonstationary model

MQSSM:

modified quasi-steady-state model

MR:

melt replenishment

MRM:

melt replenishment model

MWRM:

melt without replenishment model

NS:

Navier–Stokes

NSM:

nonstationary model

PDE:

partial differential equation

PSSM:

pseudo-steady-state model

QSSM:

quasi-steady-state model

TSSM:

Tatarchenko steady-state model

UDLM:

uniform-diffusion-layer model

YAG:

yttrium aluminum garnet

References

  1. J.A. Burton, R.C. Prim, W.P. Slichter: The distribution of solute on crystals grown from the melt, J. Chem. Phys. 21, 1987–1996 (1953)

    Article  ADS  Google Scholar 

  2. C. Wagner: Theoretical analysis of diffusion of solutes during the solidification of alloys, J. Met. 6, 154–160 (1954)

    Google Scholar 

  3. K.M. Kim, A.F. Witt, M. Lichtensteiger, H.C. Gatos: Quantitative analysis of the thermo-hydrodynamic effects on crystal growth and segregation under destabilizing vertical thermal gradients: Ga-doped germanium, J. Electrochem. Soc. 125, 475–480 (1974)

    Article  Google Scholar 

  4. J.J. Favier: Macrosegregation-I: Unified analysis during non-steady state. Solidification, Acta Metall. 29, 197–204 (1981)

    Article  Google Scholar 

  5. J.J. Favier: Macrosegregation-II: A comparative study of theories, Acta Metall. 29, 205–214 (1981)

    Article  Google Scholar 

  6. D.E. Holmes, H.C. Gatos: Convective interference and "effective" diffusion-controlled segregation during directional solidification under stabilizing vertical thermal gradients; Ge, J. Electrochem. Soc. 128, 429–437 (1981)

    Article  Google Scholar 

  7. C.J. Chang, R.A. Brown: Radial segregation induced by natural convection and melt/solid interface shape in vertical Bridgman growth, J. Cryst. Growth 63, 343–364 (1983)

    Article  ADS  Google Scholar 

  8. W.A. Tiller, K.A. Jackson, J.W. Rutter, B. Chalmers: The redistribution of solute atoms during the solidification of metals, Acta Metall. 1, 428–437 (1953)

    Article  Google Scholar 

  9. P.R. Griffin, S. Motakef: Influence of non-steady gravity on natural convection during micro-gravity solidification of semiconductors. Part I. Time Scale Analysis, J. Appl. Microgravity II 3, 121–127 (1989)

    Google Scholar 

  10. P.R. Griffin, S. Motakef: Influence of non-steady gravity on natural convection during micro-gravity solidification of semiconductors. Part II. Implications for crystal growth experiments, J. Appl. Microgravity II 3, 128–132 (1989)

    Google Scholar 

  11. A.F. Witt, H.C. Gatos, M. Lichtensteiger, M.C. Lavine, C.J. Herman: Crystal growth and steady-state segregation under zero gravity, J. Electrochem. Soc. 122, 276–283 (1975)

    Article  ADS  Google Scholar 

  12. P.M. Adornato, R.A. Brown: Convection and segregation in directional solidification of dilute and non-dilute binary alloys, J. Cryst. Growth 80, 155–190 (1987)

    Article  ADS  Google Scholar 

  13. P.A. Clark, W.R. Wilcox: Influence of gravity on thermocapillary convection in floating zone melting of silicon, J. Cryst. Growth 50, 461–469 (1980)

    Article  ADS  Google Scholar 

  14. S.A.I. Nikitin, V. Polezhayev, A.I. Fedyushkin: Mathematical simulation of impurity distribution in crystals prepared under microgravity conditions, J. Cryst. Growth 52, 471–477 (1981)

    Article  ADS  Google Scholar 

  15. S.R. Coriell, R.F. Sekerka: Lateral solute segregation during unidirectional solidification of a binary alloy with a curved solid-liquid interface, J. Cryst. Growth 46, 479–482 (1979)

    Article  ADS  Google Scholar 

  16. S.R. Coriell, R.F. Boisvert, R.G. Rehm, R.F. Sekerka: Lateral solute segregation during unidirectional solidification of a binary alloy with a curved solid-liquid interface II. Large departures from planarity, J. Cryst. Growth 54, 167–175 (1981)

    Article  ADS  Google Scholar 

  17. H.M. Ettouney, R.A. Brown: Effect of heat transfer on melt/solid interface shape and solute segregation in edge-defined film-fed growth: Finite element analysis, J. Cryst. Growth 58, 313–329 (1982)

    Article  ADS  Google Scholar 

  18. J.P. Kalejs, L.Y. Chin, F.M. Carlson: Interface shape studies for silicon ribbon growth by the EFG technique I. Transport phenomena modeling, J. Cryst. Growth 61, 473–484 (1983)

    Article  ADS  Google Scholar 

  19. C.E. Chang, W.R. Wilcox: Control of interface shape in the vertical Bridgman-Stockbarger technique, J. Cryst. Growth 21, 135–140 (1974)

    Article  ADS  Google Scholar 

  20. T.W. Fu, W.R. Wilcox: Influence of insulation on stability of interface shape and position in the vertical Bridgman-Stockbarger technique, J. Cryst. Growth 48, 416–424 (1980)

    Article  ADS  Google Scholar 

  21. T.W. Clyne: Heat flow in controlled directional solidification of metals I. Experimental investigation, J. Cryst. Growth 50, 684–690 (1980)

    Article  ADS  Google Scholar 

  22. T.W. Clyne: Heat flow in controlled directional solidification of metals II. Mathematical model, J. Cryst. Growth 50, 691–700 (1980)

    Article  ADS  Google Scholar 

  23. P.C. Sukanek: Deviation of freezing rate from translation rate in the Bridgman-Stockbarger technique I. Very low translation rates, J. Cryst. Growth 58, 208–218 (1982)

    Article  ADS  Google Scholar 

  24. P.C. Sukanek: Deviation of freezing rate from translation rate in the Bridgman-Stockbarger technique II. Moderate translation rates, J. Cryst. Growth 58, 219–228 (1982)

    Article  ADS  Google Scholar 

  25. T. Jasinski, W.M. Rohsenow, A.F. Witt: Heat transfer analysis of the Bridgman-Stockbarger configuration for crystal growth I. Analytical treatment of the axial temperature profile, J. Cryst. Growth 61, 339–354 (1983)

    Article  ADS  Google Scholar 

  26. L.R. Morris, W.C. Winegard: The development of cells during the solidification dilute Pb-Sb alloy, J. Cryst. Growth 5, 361–375 (1969)

    Article  ADS  Google Scholar 

  27. A.F. Witt, H.C. Gatos, M. Lichtensteiger, C.J. Herman: Crystal growth and segregation under zero gravity, J. Electrochem. Soc. 125, 1832–1840 (1978)

    Article  Google Scholar 

  28. A.M. Balint, D.G. Baltean, T. Levy, M. Mihailovici, A. Neculae, S. Balint: The dopant fields in uniform-diffusion-layer, global-thermal-convection and precrystallization-zone models, Mater. Sci. Semicond. Process. 3, 115–121 (2000)

    Article  Google Scholar 

  29. D.T.J. Hurle, E. Jakeman, C.P. Johnson: Convective temperature oscillations in molten gallium, J. Fluid Mech. 64, 565–576 (1974)

    Article  ADS  Google Scholar 

  30. C.A. Wang, A.F. Witt: Annual Report Material Processing Center (Massachusetts Institute of Technology, Massachusetts 1984)

    Google Scholar 

  31. M.M. Mihailovici, A.M. Balint, S. Balint: The axial and radial segregation due to the thermo-convection, the decrease of the melt in the ampoule and the effect of the precrystallization-zone in the semiconductor crystals grown in a Bridgman-Stockbarger system in a low gravity environment, J. Cryst. Growth 237–239, 1752–1756 (2002)

    Article  Google Scholar 

  32. M.M. Mihailovici, A.M. Balint, S. Balint: On the controllability of the level of the dopant concentration and of the compositional uniformity of a doped crystal, grown in a low gravity environment by Bridgman-Stockbarger method, Int. J. Theor. Physics, Group Theory Nonlin. Opt. 10, 425–436 (2003)

    Google Scholar 

  33. A.M. Balint, M.M. Mihailovici, D.G. Baltean, S. Balint: A modified Chang-Brown model for the determination of the dopant distribution in a Bridgman-Stockbarger semiconductor crystal growth system, J. Cryst. Growth 230, 195–201 (2001)

    Article  ADS  Google Scholar 

  34. A.M. Balint, M.M. Mihailovici, D.G. Baltean, S. Balint: Interface structure in the growth of semiconductor crystals using the Bridgman-Stockbarger method, Thin Solid Films 380, 108–110 (2000)

    Article  ADS  Google Scholar 

  35. M.M. Mihailovici, A.M. Balint, S. Balint: The dopant distribution computed in the modified Chang-Brown model using quasi-steady state approximation, Comput. Mater. Sci. 24, 262–267 (2002)

    Article  Google Scholar 

  36. D.G. Baltean, T. Levy, S. Balint: Transport de masse par convection et diffusion dans un milieu multiporeux, C. r. Acad. Sci. Paris, Ser. IIB 326, 821–826 (1998)

    ADS  MATH  Google Scholar 

  37. K. Moutsopoulos, S. Bories: Dispersion en milieux poreux hétérogènes, C. R. Acad. Sci. Paris, Ser. IIB 316, 1667–1672 (1993), in French

    MATH  Google Scholar 

  38. J.C. Maxwell: Electricity and Magnetism (Clarendon, Oxford 1873)

    MATH  Google Scholar 

  39. V.I. Avetisov, Mendeleev Institute Moscow (personal communication)

    Google Scholar 

  40. M.M. Mihailovici, A.M. Balint, S. Balint: Way to improve the compositional uniformity of doped crystals grown by Bridgman-Stockbarger method in a low gravity environment, Int. J. Theor. Phys. Group Theory Nonlinear Opt. 11, 109–119 (2004)

    MATH  Google Scholar 

  41. D.J. Larson, J. Bethin, B.S. Dressler: Shuttle Mission 51-G, Experiment MRS77F055, Flight Sample Characterization (Grumman Corporate Research Center, Bethpage 1988), NASA Report RE-753

    Google Scholar 

  42. P.S. Dutta, A.G. Ostrogorsky: Segregation of Ga in Ge and InSb in GaSb, J. Cryst. Growth 217, 360–365 (2000)

    Article  ADS  Google Scholar 

  43. L.L. Zheng, D.J. Larson Jr., H. Zhang: Role of thermotransport (Sorret effect) in macrosegregation during eutectic/off-eutectic directional solidification, J. Cryst. Growth 191, 243–251 (1998)

    Article  ADS  Google Scholar 

  44. L. Braescu: The dependence of the dopant distribution on the pulling rate and on the capillary channel radius in the case of a Nd:YVO4 cylindrical bar grown from the melt by the EFG method, Mater. Sci. Eng. B 146, 41–44 (2008)

    Article  Google Scholar 

  45. E. Tulcan-Paulescu, A.M. Balint, S. Balint: The effect of the initial dopant distribution in the melt on the axial compositional uniformity of a thin doped crystal grown in strictly zero-gravity environment by Bridgman-Stockbarger method, J. Cryst. Growth 247, 313–319 (2003)

    Article  ADS  MATH  Google Scholar 

  46. V.A. Tatarchenko: Shaped Crystal Growth (Kluwer, Dordecht 1993)

    Book  Google Scholar 

  47. H.E. LaBelle Jr., A.I. Mlavsky, B. Chalmers: Growth of controlled profile crystals from the melt: Part I – Sapphire filaments, Mater. Res. Bull 6, 571–579 (1971)

    Article  Google Scholar 

  48. H.E. LaBelle Jr., A.I. Mlavsky, B. Chalmers: Growth of controlled profile crystals from the melt: Part II – Edge-defined, film-fed growth (EFG), Mater. Res. Bull. 6, 581 (1971)

    Article  Google Scholar 

  49. B. Chalmers, H.E. LaBelle Jr., A.I. Mlavsky: Edge-defined, film-fed crystal growth, J. Cryst. Growth 13/14, 84–87 (1972)

    Article  ADS  Google Scholar 

  50. J.P. Kalejs: Impurity redistribution in EFG, J. Cryst. Growth 44, 329–344 (1978)

    Article  ADS  Google Scholar 

  51. J.P. Kalejs, G.M. Freedman, F.V. Wald: Aluminium redistribution in EFG silicon ribbon, J. Cryst. Growth 48, 74–84 (1980)

    Article  ADS  Google Scholar 

  52. B. Chalmers: Transient solute effects in shaped crystal growth of silicon, J. Cryst. Growth 82, 70–73 (1987)

    Article  ADS  Google Scholar 

  53. J. Cao, M. Prince, J.P. Kalejs: Impurity transients in multiple crystal growth from a single crucible for EFG silicon octagons, J. Cryst. Growth 174, 170–175 (1997)

    Article  ADS  Google Scholar 

  54. J.P. Kalejs: Interface shape studies for silicon ribbon growth by the EFG technique II. Effect of die asymmetry, J. Cryst. Growth 61, 485–493 (1983)

    Article  ADS  Google Scholar 

  55. J.P. Kalejs: Modeling contribution in commercialization of silicon ribbon growth from the melt, J. Cryst. Growth 230, 10–21 (2001)

    Article  ADS  Google Scholar 

  56. O. Bunoiu, I. Nicoara, J.L. Santailler, T. Duffar: Fluid flow and solute segregation in EFG crystal growth process, J. Cryst. Growth 275, 799–805 (2005)

    Article  ADS  Google Scholar 

  57. L. Braescu, S. Balint, L. Tanasie: Numerical studies concerning the dependence of the impurity distribution on the pulling rate and on the radius of the capillary channel in the case of a thin rod grown from the melt by edge-defined film-fed growth (EFG) method, J. Cryst. Growth 291, 52–59 (2006)

    Article  ADS  Google Scholar 

  58. L. Braescu, T.F. George, S. Balint: Evaluation and control of the dopant distribution in a Nd:LiNbO3 fiber grown from the melt by the edge-defined film-fed growth (EFG) method, Photonic Fiber and Crystal Devices: Advances in Materials and Innovations in Device Applications I (Optics and Photonics 2007), SPIE Proc. 6698, 669803:1–8 (2007)

    Article  Google Scholar 

  59. L. Braescu, T. Duffar: Effect of buoyancy and Marangoni forces on the dopant distribution in the case of a single crystal fiber grown from the melt by the edge-defined film-fed growth (EFG) method, J. Cryst. Growth 310, 484–489 (2008)

    Article  ADS  Google Scholar 

  60. T.F. George, L. Braescu: Sapphire fibers grown from the melt by the EFG technique: Dependence of the impurity distribution on temperature and surface tension gradients, Photonic Fiber and Crystal Devices: Advances in Materials and Innovations in Device Applications II (Optics and Photonics 2008), SPIE Proc. 7056, 705603–1–705603–10 (2008)

    Article  Google Scholar 

  61. L. Braescu, T.F. George: Arbitrary Lagrangian-Eulerian method for coupled Navier-Stokes and convection-diffusion equations with moving boundaries, Proc. 12nd WSEAS Int. Conf. Appl. Math. – Mathʼ07 (2007) pp.33–36

    Google Scholar 

  62. F. Duarte, R. Gormaz, S. Natesan: Arbitrary Lagrangian-Eulerian method for Naver-Stokes equations with moving boundaries, Comput. Methods Appl. Mech. Eng. 193, 4819–4836 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  63. M. Fernandez, M. Moubachir: Sensitivity analysis for an incompressible aeroelastic system, Math. Models Methods Appl. Sci. 12, 1109–1130 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Center for Nanoscience, Department of Chemistry and Biochemistry, Department of Physics and Astronomy, University of Missouri-St. Louis, One University Boulevard, 63121, St. Louis, MO, USA

    Thomas F. George

  2. Department of Computer Science, West University of Timisoara, Blvd. V. Parvan 4, 300223, Timisoara, Romania

    Stefan Balint

  3. Department of Computer Science, West University of Timisoara, Blvd. V. Parvan 4, 300223, Timisoara, Romania

    Liliana Braescu

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  1. Thomas F. George
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  2. Stefan Balint
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  3. Liliana Braescu
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Corresponding authors

Correspondence to Thomas F. George , Stefan Balint or Liliana Braescu .

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Editors and Affiliations

  1. Advanced Renewable Energy Company, 18 Celina Avenue, 03063, Nashua, NH, USA

    Govindhan Dhanaraj Dr.

  2. Department of Geology, University of Mysore, Manasagangotri, 570 006, Mysore, India

    Kullaiah Byrappa Ph.D.

  3. University of North Texas, 1155 Union Circle, 76203-5017, Denton, TX, USA

    Vishwanath Prasad Dr.

  4. Department of Materials Science and Engineering, Stony Brook University, 11794-2275, Stony Brook, NY, USA

    Michael Dudley Dr.

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George, T.F., Balint, S., Braescu, L. (2010). Mass and Heat Transport in BS and EFG Systems. In: Dhanaraj, G., Byrappa, K., Prasad, V., Dudley, M. (eds) Springer Handbook of Crystal Growth. Springer Handbooks. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74761-1_40

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