Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-20T01:00:04.971Z Has data issue: false hasContentIssue false
  • English
  • Français

Melt motion in a Czochralski crystal puller with a non-uniform axisymmetric magnetic field: isothermal motion

Published online by Cambridge University Press:  26 April 2006

L. N. Hjellming ,
P. A. Tolley  and
J. S. Walker
L. N. Hjellming
Affiliation:
Department of Mechanical Engineering and Engineering Science, University of North Carolina-Charlotte, Charlotte, NC 28223, USA
P. A. Tolley
Affiliation:
Department of Mechanical Engineering and Engineering Science, University of North Carolina-Charlotte, Charlotte, NC 28223, USA
J. S. Walker
Affiliation:
Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, IL 61801, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The use of magnetic fields during the growth of semiconductor crystals from the melt in a Czochralski (CZ) crystal puller has shown promise in controlling the heat and mass transport to the growth interface. The magnetic field suppresses turbulence and thermal convection in the melt in which large thermal gradients are present, thus improving the quality of the crystal. In this paper, analytical solutions are presented for the isothermal melt motion and electric current density driven by the differential rotation of the crystal and crucible about their common vertical axis. There is an applied, non-uniform, axisymmetric magnetic field with only radial and axial components which are independent of the azimuthal coordinate. The melt motion with a uniform axial magnetic field represents a singular limit of the flow considered here: as the radial magnetic field component goes to zero, the radial and axial (meridional) velocity components decrease in magnitude by a factor of M-1, where M is the large Hartmann number. The uniform axial field is a singular limit because the centrifugal acceleration due to the azimuthal velocity is exactly perpendicular to the magnetic field. Since the radial isothermal motion near the growth interface controls the radial distributions of dopants and impurities in the crystals, a non-uniform axisymmetric magnetic field is better than the uniform axial field. In addition, the axisymmetric field avoids the detrimental deviations from axisymmetric heat and mass transport associated with a uniform transverse (horizontal) magnetic field.

Two classes of shaped fields are considered, with only one class leading to the presence of the large meridional flow driven by differential rotation. The small electrical conductivity of the crystal plays an important role in determining the behaviour of the melt's angular velocity, which is constant along each magnetic field line. Results for two simple field configurations are presented in order to illustrate the effect of the field configuration on the nature of the meridional circulation and the potential for flow tailoring with the shaped field.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 249 , April 1993 , pp. 1 - 34
Copyright
© 1993 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. & Stegun, I. A. 1970 Handbook of Mathematical Functions. Dover.
Hicks, T. W., Organ, A. E. & Riley, N. 1989 Oxygen transport in magnetic Czochralski growth of silicon with a non-uniform magnetic field. J. Cryst. Growth 94, 213– 228.Google Scholar
Hicks, T. W. & Riley, N. 1989 Boundary layers in magnetic Czochralski crystal growth. J. Cryst. Growth 96, 957– 968.Google Scholar
Hirata, H. & Hoshikawa, K. 1989 Silicon crystal growth in a cusp magnetic field. J. Cryst. Growth 96, 747– 755.Google Scholar
Hjellming, L. N. 1990 A thermal model for Czochralski silicon crystal growth with an axial magnetic field. J. Cryst. Growth 104, 327– 344.Google Scholar
Hjellming, L. N. & Walker, J. S. 1986 Melt motion in a Czochralski crystal puller with an axial magnetic field: isothermal motion. J. Fluid Mech. 164, 237– 273 (referred to herein as HWI.)Google Scholar
Hjellming, L. N. & Walker, J. S. 1987 Melt motion in a Czochralski crystal puller with an axial magnetic field: motion due to buoyancy and thermocapillarity. J. Fluid Mech. 182, 335– 368.Google Scholar
Hjellming, L. N. & Walker, J. S. 1988a Melt motion in a Czochralski crystal puller with an axial magnetic field: uncertainty in the thermal constants. J. Cryst. Growth 87, 18– 32.Google Scholar
Hjellming, L. N. & Walker, J. S. 1988b Mass transport in a Czochralski crystal puller with an axial magnetic field: melt motion due to crystal growth and buoyancy. J. Cryst. Growth 92, 371– 389.Google Scholar
Kuroda, E., Kozuka, H. & Takano, Y. 1984 The effect of temperature oscillations at the growth interface on crystal perfection. J. Cryst. Growth 68, 613– 623.Google Scholar
Langlois, W. E. 1984 Computer simulation of Czochralski melt convection in a magnetic field. J. Cryst. Growth 70, 73– 77.Google Scholar
Langlois, W. E., Hjellming, L. N. & Walker, J. S. 1987 Effects of finite electrical conductivity of the crystal on hydromagnetic Czochralski flow. J. Cryst. Growth 83, 51– 61.Google Scholar
Langlois, W. E., Kim, K. M. & Walker, J. S. 1992 Hydromagnetic crystal growth. J. Cryst. Growth (to appear.)Google Scholar
Morse, P. M. & Feshback, H. 1953 Methods in Theoretical Physics: Part I. McGraw-Hill.
Ravishankar, P. S., Braggins, T. T. & Thomas, R. N. 1990 Impurities in commercial-scale magnetic Czochralski silicon: axial versus transverse fields. J. Cryst. Growth 104, 617– 628.Google Scholar
Thomas, R. N., Hobgood, H. M., Ravishankar, P. S. & Braggins, T. T. 1990 Melt growth of large diameter semiconductors: part II. Solid State Technol. 33, 163– 167.Google Scholar
Tolley, P. A. 1991 The use of non-uniform axisymmetric magnetic fields in silicon crystal growth. M.S.M.E. Dissertation, Department of Mechanical Engineering and Engineering Science, University of North Carolina at Charlotte, Charlotte, NC.
Williams, M. G. 1989 Melt flow in a Czochralski crystal puller with a transverse magnetic field. Ph.D. dissertation, Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, IL.
Williams, M. G., Walker, J. S. & Langlois, W. E. 1990 Melt motion in a Czochralski crystal puller with-a weak transverse magnetic field. J. Cryst. Growth 100, 233– 253.Google Scholar