Advertisement

Modeling Semiconductor Crystal Growth Under Electromagnetic Fields

  • Sadik DostEmail author
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 90)

Abstract

Growth of semiconductor single crystals under electric and magnetic fields is of interest to increase and better control of crystal growth rate, to suppress and control the adverse effect of natural convection and to obtain better mixing in the growth melt (liquid solution) for better crystal uniformity, which all are favorable conditions for a prolonged growth of high quality crystals. To this end, in parallel to well-designed experiments, modeling is essential to shed light on various aspects of these growth processes and also to better understand the transport phenomena involved. In this article the models developed over the years, mostly based on Professor Gerard Maugin’s well-known contributions to “electromagnetic interactions”, are briefly presented for “solution growth” conducted under electric and magnetic fields. Basic and constitutive equations of a binary electromagnetic continuum mixture are specialized for two important solution growth techniques—Liquid Phase Electroepitaxy (LPEE) and Travelling Heater Method (THM). As an application, an LPEE growth of GaAs bulk crystals under a strong static magnetic field is considered. Experimental results, that have shown that the growth rate under an applied static magnetic field is also proportional to the applied magnetic field and increases with the field intensity level, are predicted from these models. The contribution of a third-order material constant in LPEE is also predicted from these models. The prediction of increasing growth rate in THM growth under rotating magnetic fields from modeling was verified by experiments.

Notes

Acknowledgements

The financial support provided by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Canada Research Chairs (CRC) Program is gratefully acknowledged.

References

  1. 1.
    Eringen, A.C., Maugin, G.A.: Electrodynamics of Continua, vol. I and II. Springer, New York (1989)Google Scholar
  2. 2.
    Bowen, R.M.: Theory of mixtures. In: Eringen, A.C. (ed.) Continuum Physics, vol. 3, pp. 1–127. Academic Press, New YorkGoogle Scholar
  3. 3.
    Dost, S., Erbay, H.A.: A continuum model for liquid-phase electroepitaxy. Int. J. Eng. Sci. 33, 1385–1402 (1995)CrossRefGoogle Scholar
  4. 4.
    Dost, S., Qin, Z.: A model for liquid phase electroepitaxy under an external magnetic field I. Theory. J. Cryst. Growth 153, 123–130 (1995)CrossRefGoogle Scholar
  5. 5.
    Series, R.W., Hurle, D.T.J.: The use of magnetic fields in semiconductor crystal-growth. J. Cryst. Growth 113, 305–328 (1991)CrossRefGoogle Scholar
  6. 6.
    Dost, S., Sheibani, H.: A mathematical model for solution growth of bulk crystals under electric and magnetic fields. Philos. Mag. 85(33–35), 4331–4351 (2005)CrossRefGoogle Scholar
  7. 7.
    Dost, S., Lent, B.: Single crystal growth of semiconductors from metallic solutions. Elsevier, Amsterdam, The Netherlands (2007). ISBN: 0 444 52232CrossRefGoogle Scholar
  8. 8.
    Kim, D.H., Adornato, P.M., Brown, R.A.: Effect of vertical magnetic field on convection and segregation in vertical Bridgman crystal growth. J. Cryst. Growth 89, 339–356 (1988)CrossRefGoogle Scholar
  9. 9.
    Hirata, H., Hoshikawa, K.: 3-dimensional numerical analyses of the effects of a cusp magnetic field on the flows, oxygen transport and heat transfer in a Czochralski silicon melt. J. Cryst. Growth 125, 181–207 (1992)CrossRefGoogle Scholar
  10. 10.
    Baumgartl, J., Muller, G.: Calculation of the effects of magnetic field damping on fluid flow: comparison of magnetohydrodynamic models of different complexity. In: Proceedings of the VIIIth Enropean Symposium on Materials and Fluid Sciences in Microgravity, Noordwijk, The Netherlands, pp. 161–164 (1992)Google Scholar
  11. 11.
    Baumgartl, J., Hubert, A., Muller, G.: The use of magnetohydrodynamic effects to investigate fluid flow in electrically conducting melts. Phys. Fluids A 5, 3280–3289 (1993)CrossRefGoogle Scholar
  12. 12.
    Salk, M., Lexow, B., Benz, K.W., et al.: CdTe crystal growth in the soviet facility ZONA 4. Microgravity Sci. Technol. 6, 88 (1993)Google Scholar
  13. 13.
    Hurle, D.T.J. (ed.): Handbook of Crystal Growth 2: Bulk crystal growth, Part B: Growth Mechanisms and Dynamics, North-Holland (1994)Google Scholar
  14. 14.
    Oshima, M., Taniguchi, N., Kobayashi, T.: Numerical investigation of 3-dimensional melt convection with the magnetic Czochralski method. J. Crystal Growth 137, 48–53 (1994)Google Scholar
  15. 15.
    Salk, M., Fiederle, M., Benz, K.W., Senchenkov, A.S., Egorov, A.V., Matioukhin, D.G.: CdTe and CdTe0.9Se0.1 crystal grown by the traveling heater method using a rotating magnetic field. J. Cryst. Growth 138, 161–167 (1994)CrossRefGoogle Scholar
  16. 16.
    Price, M.W., Andrews, R.N., Su, C.H., Lehoczky, S.L., Szofran, F.R.: The effect of a transverse magnetic field on the microstructure of directionally solidified CdTe. J. Cryst. Growth 137, 201–207 (1994)CrossRefGoogle Scholar
  17. 17.
    Qin, Z., Dost, S., Djilali, N., Tabarrok, B.: A model for liquid phase electroepitaxy under an external magnetic field II. Application. J. Cryst. Growth 153, 131–139 (1995)CrossRefGoogle Scholar
  18. 18.
    Ben Hadid, H., Henry, D.: Numerical study of convection in the horizontal Bridgman configuration under the action of a constant magnetic field. Part 1. Two dimensional flow. J. Fluid Mech. 333, 23–56 (1996)CrossRefGoogle Scholar
  19. 19.
    Ben Hadid, H., Henry, D.: Numerical study of convection in the horizontal Bridgman configuration under the action of a constant magnetic field. Part 2. Three-dimensional flow. J. Fluid Mech. 333, 57–83 (1996)CrossRefGoogle Scholar
  20. 20.
    Kakimoto, K., Yi, K.W., Eguchi, M.: Oxygen transfer during single silicon growth in Czochralski system with vertical magnetic fields. J. Cryst. Growth 163, 238–242 (1996)CrossRefGoogle Scholar
  21. 21.
    Fiederle, M., Eiche, C., Joerger, W., Salk, M., Senchenkov, A.S., Egorov, A.V., Ebling, D.G., Benz, K.W.: Radiation detector properties of CdTe0.9Se0.1Cl crystals grown under microgravity in a rotating magnetic field. J. Cryst. Growth 166, 256–260 (1996)CrossRefGoogle Scholar
  22. 22.
    Dost, S.: Recent developments in modeling of liquid phase electroepitaxy: a continuum approach. Appl. Mech. Rev. 49(12), 477–495 (1996)CrossRefGoogle Scholar
  23. 23.
    Qin, Z., Dost, S.: A model for liquid phase electroepitaxial growth of ternary alloy semiconductors. Int. J. Electromagnet. Mech. 7(2), 129–142 (1996)Google Scholar
  24. 24.
    Dost, S., Qin, Z.: A numerical simulation model for liquid phase electroepitaxial growth of GaInAs. J. Cryst. Growth 187, 51–64 (1998)CrossRefGoogle Scholar
  25. 25.
    Senchenkov, A.S., Barmin, I.V., Tomson, A.S., Krapukhin, V.V.: Seedless THM growth of Cd(x)Hg(1-x)Te (approximately x = 0.2) single crystals within rotating magnetic field. J. Cryst. Growth 197, 552–556 (1999)CrossRefGoogle Scholar
  26. 26.
    Ghaddar, C.K., Lee, C.K., Motakef, S., Gillies, D.C.: Numerical simulation of THM growth of CdTe in presence of rotating magnetic fields (RMF). J. Cryst. Growth 205, 97–111 (1999)CrossRefGoogle Scholar
  27. 27.
    Davoust, L., Cowley, M.D., Moreau, R., Bolcato, R.: Buoyancy-driven convection with an uniform magnetic field. Part 2. Experimental investigation. J. Fluid Mech. 400, 59–90 (1999)CrossRefGoogle Scholar
  28. 28.
    Meric, R.A., Dost, S., Lent, B., Redden, R.F.: A finite element model for the growth of ternary alloy GaInSb by the travelling heater method. Int. J. Electromagnet. Mech. 10, 505–526 (1999)CrossRefGoogle Scholar
  29. 29.
    Dost, S.: Numerical simulation of liquid phase electroepitaxial growth of GaInAs under magnetic field. ARI-the Bull. ITU 51, 235–246 (1999)Google Scholar
  30. 30.
    Jing, C.J., Imaishi, N., Yasuhiro, S., Sato, T., Miyazawa, Y.: Three-dimensional numerical simulation of rotating spoke pattern in an oxide melt under a magnetic field. Inter. J. Heat Mass Transf. 43, 4347–4359 (2000)CrossRefGoogle Scholar
  31. 31.
    Dost, S., Sheibani, H.: In Mechanics of Electromagnetic Materials and Structures in Studies in Appl. Electr. Mech., (Eds. J.S. Yang, G.A. Maugin), 19, pp. 17–29. IOS Press, Amsterdam (2000)Google Scholar
  32. 32.
    Vizman, D., Friedrich, J., Muller, G.: Comparison of the predictions from 3D numerical simulation with temperature distributions measured in Si Czochralski melts under the influence of different magnetic fields. J. Cryst. Growth 230, 73–80 (2001)CrossRefGoogle Scholar
  33. 33.
    Ben Hadid, H., Vaux, Samuel, Kaddeche, Slim: Three dimensional flow transitions under a rotating magnetic field. J. Cryst. Growth 230, 57–62 (2001)CrossRefGoogle Scholar
  34. 34.
    Akamatsu, M., Higano, M., Ozoe, H.: Elliptic temperature contours under a transverse magnetic field computed for a Czochralski melt. Int. J. Heat Mass Transf. 44, 3253–3264 (2001)CrossRefGoogle Scholar
  35. 35.
    Dost, S., Liu, Y.C., Lent, B.: A numerical simulation study for the effect of applied magnetic field in liquid phase electroepitaxy. J. Cryst. Growth 240, 39–51 (2002)CrossRefGoogle Scholar
  36. 36.
    Liu, Y.C., Okano, Y., Dost, S.: The effect of applied magnetic field on flow structures in liquid phase electroepitaxy—a three-dimensional simulation model. J. Cryst. Growth 244, 12–26 (2002)CrossRefGoogle Scholar
  37. 37.
    Okano, Y., Nishino, S.-S., Ohkubo, S.-S., Dost, S.: Numerical study of transport phenomena in the THM growth of compound semiconductor crystal. J. Cryst. Growth 238–239, 1779–1784 (2002)CrossRefGoogle Scholar
  38. 38.
    Liu, Y.C., Sheibani, H., Sakai, S., Okano, Y., Dost, S.: In: Kleijn, C.R., Kawano, S. (eds.) Computational Technologies for Fluid/Thermal/Structural/Chemical Systems with Industrial Applications. ASME Proceedings, New York, PVP-vol. 448-1, pp. 65–72 (2002). ISBN: 0-7918-4659-8Google Scholar
  39. 39.
    Sheibani, H., Dost, S., Sakai, S., Lent, B.: Growth of bulk single crystals under applied magnetic field by liquid phase electroepitaxy. J. Cryst. Growth 258(3–4), 283–295 (2003)CrossRefGoogle Scholar
  40. 40.
    Sheibani, H., Liu, Y.C., Sakai, S., Lent, B., Dost, S.: The effect of applied magnetic field on the growth mechanisms of liquid phase electroepitaxy. Int. J. Eng. Sci. 41, 401–415 (2003)CrossRefGoogle Scholar
  41. 41.
    Okano, Y., Kondo, H., Dost, S.: Control of transport structures in a rotating liquid cylinder by means of an applied magnetic field. Int. J. Electromagnet. Mech. 18(4), 217–226 (2003)CrossRefGoogle Scholar
  42. 42.
    Dost, S., Liu, Y.C., Lent, B.: A numerical simulation study for the effect of applied magnetic field in growth of CdTe single crystals by the traveling heater method. Int. J. Electromagnet. Mech. 17, 271–288 (2003)CrossRefGoogle Scholar
  43. 43.
    Liu, Y.C., Dost, S., Lent, B., Redden, R.F.: A three-dimensional numerical simulation model for the growth of CdTe single crystals by the traveling heater method under magnetic field. J. Cryst. Growth 254, 285–297 (2003)CrossRefGoogle Scholar
  44. 44.
    Roszmann, J., Dost, S., Lent, F.: Crystal growth by the travelling heater method using tapered crucibles and applied rotating magnetic field. Cryst. Res. Technol. 45(8), 785–790 (2010)CrossRefGoogle Scholar
  45. 45.
    Liu, Y.C., Dost, S., Sheibani, H.: A three dimensional numerical simulation for the transport structures in liquid phase electroepitaxy under applied magnetic field. Int. J. Transp. Phenom. 6, 51–62 (2004)Google Scholar
  46. 46.
    Dost, S., Lent, B., Sheibani, H., Liu, Y.C.: Recent developments in liquid phase electroepitaxial growth of bulk crystals under magnetic field. Comptes rendus de mecanique 332(5–6), 413–428 (2004)CrossRefGoogle Scholar
  47. 47.
    Jastrzebski, L., Gatos, H.C., Witt, A.F.: Electromigration in current-controlled LPEE. J. Electrochem. Soc. 123, 1121 (1976)CrossRefGoogle Scholar
  48. 48.
    Jastrzebski, L., Imamura, Y., Gatos, H.C.: Thickness uniformity of GaAs layers grown by electroepitaxy. J. Electrochem. Soc. 125, 1140–1146 (1978)CrossRefGoogle Scholar
  49. 49.
    Okamoto, A., Lakowski, L., Gatos, H.C.: Enhancement of interface stability in liquid-phase electroepitaxy. J. Appl. Phys. 53, 1706–1713 (1982)CrossRefGoogle Scholar
  50. 50.
    Nakajima, K.: Liquid-phase epitaxial-growth of very thick In1-xGaxAs layers with uniform composition by source-current-controlled method. J. Appl. Phys. 61(9), 4626–4634 (1987)CrossRefGoogle Scholar
  51. 51.
    Bryskiewicz, T., Boucher Jr., C.F., Lagowski, J., Gatos, H.C.: Bulk GaAS crystal growth by liquid phase electroepitaxy. J. Cryst. Growth 82, 279–288 (1987)CrossRefGoogle Scholar
  52. 52.
    Nakajima, K.: Layer thickness calculation of In1-vGavAs grown by the source-current-controlled method—diffusion and electromigration limited growth. J. Cryst. Growth 98, 329–340 (1989)CrossRefGoogle Scholar
  53. 53.
    Bryskiewicz, T., Edelman, P., Wasilewski, Z., Coulas, D., Noad, J.: Properties of very uniform InxGa1-xAs single-crystals grown by liquid-phase electroepitaxy. J. Appl. Phys. 68, 3018–3020 (1990)CrossRefGoogle Scholar
  54. 54.
    Nakajima, K., Kusunoki, T., Takenaka, C.: Growth of ternary InxGa1-xAs bulk crystals with a uniform composition through supply of GaAs. J. Cryst. Growth 113, 485–490 (1991)CrossRefGoogle Scholar
  55. 55.
    Bryskiewicz, T., Laferriere, A.: Growth of alloy substrates by liquid phase electroepitaxy—Theoretical considerations. J. Cryst. Growth 129, 429–442 (1993)CrossRefGoogle Scholar
  56. 56.
    Zytkiewicz, Z.R.: Influence of convection on the composition profiles of thick GaAlAs layers grown by liquid-phase electroepitaxy. J. Cryst. Growth 131, 426–430 (1993)CrossRefGoogle Scholar
  57. 57.
    Zytkiewicz, Z.R.: Joule effect as a barrier for unrestricted growth of bulk crystals by liquid phase electroepitaxy. J. Cryst. Growth 172, 259–268 (1996)CrossRefGoogle Scholar
  58. 58.
    Minakuchi, H., Okano, Y., Dost, S.: A three-dimensional numerical simulation study of the Marangoni convection occurring in the crystal growth of SixGe1-x by the Float-zone technique in zero gravity. J. Cryst. Growth 266, 140–144 (2004)CrossRefGoogle Scholar
  59. 59.
    Minakuchi, H., Okano, Y., Dost, S.: A three dimensional numerical study of marangoni convection in a floating full zone. In: Dost, S. (ed.) Crystal Growth of Semiconductor from the Liquid Phase. IJMPT 22(1/2/3), 151–171 (2005)Google Scholar
  60. 60.
    Timchenko, V., Chen, P.Y.P., de Vahl Davis, G., Leonardi, E., Abbaschian, R.: A computational study of transient plane front solidification of alloys in a Bridgman apparatus under microgravity conditions. Int. J. Heat Mass Transf. 43, 963–980 (2000)CrossRefGoogle Scholar
  61. 61.
    Timchenko, V., Chen, P.Y.P., de Vahl Davis, G., Leonardi, E., Abbaschian, R.: A computational study of binary alloy solidification in the Mephisto experiment. Int. J. Heat Mass Transf. 23, 258–268 (2002)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Crystal Growth LaboratoryUniversity of VictoriaVictoriaCanada

Personalised recommendations