Effect of Rotating Magnetic Field on Thermal Convection and Dopant Transport in Floating-Zone Crystal Growth

Abstract

A three-dimensional numerical study was carried out to understand the effects of rotating magnetic field on thermocapillary convection and impurity concentration distribution in a floating full zone growth process of doped Si crystals under zero gravity. Even though the applied temperature profile was axisymmetric, the simulation results showed that the flow structures and concentration distribution in the molten zone exhibited three-dimensional disordered patterns in the absence of a rotating magnetic field. For the application of rotating magnetic field, Lorentz force forced convection to rotate in the same direction of the magnetic field and made the tangential velocity in the melt increase with the growing radial distance. Under a relatively low magnetic field, the thermocapillary flow became an oscillatory three-dimensional convection. However, two dimensional axisymmetric distributions of both the melt flow and the impurity concentration were presented under the magnetic field with sufficiently strong intensity. Meanwhile, the thermocapillary convection in the molten zone formed a back flow region with flow in the direction from the high temperature to low temperature along the free surface and through intermediate section. The concentration contours became uniform, and thus the isolines of concentration formed a series of concentric circles in the growth interface. Therefore, the rotating magnetic field was effective for ameliorating the stability of the melt flow and the uniformity of the concentration distribution which was beneficial to the growth of crystals with a radial uniform crystal.

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Abbreviations

a :

Gaussian distribution scale parameter [m]

B :

applied external RMF

B 0 :

amplitude of RMF [mT]

C :

impurity concentration

C 0 :

impurity concentration in the multicrystal

C p :

specific heat [J·kg-1·K-1]

D :

diffusion coefficient [ m2·s-1]

er,eθ, ez :

unit vectors in the r, θ, z direction

f rot :

Lorentz force

f :

rotating frequency [Hz]

K :

impurity segregation coefficient

L :

half-length of float zone [m]

m :

azimuthal wave number

Ma :

Marangoni number

n :

unit normal vector

P :

pressure [Pa]

Pr :

Prandtl number

Q:

convergence criteria

R :

radius of float zone [m]

r,θ,z :

radial, azimuthal and axial coordinates

Re :

Reynolds number

Sc :

Schmidt number

t :

time [s]

T :

temperature [K]

Ta :

ambient temperature [K]

Tm :

melting temperature [K]

ΔT :

temperature difference [K]

u,v, w :

velocity components in the r, z and θ directions [m·s-1]

Vp :

growth velocity [m·s-1]

V :

velocity vector

α :

thermal diffusion coefficient [m2·s-1]

γT :

surface tension gradient [N·m-1·K-1]

ε :

emissivity

κ :

thermal conductivity [W·m-1·K-1]

μ :

dynamic viscosity [kg·m-1·s-1]

ν :

kinematic viscosity [m2·s-1]

ρ :

density [kg·m-3]

σ :

surface tension [N·m-1]

σ0 :

Stefan Boltzmann constant [W·m-2·K-4]

σe :

electric conductivity [Ω-1·m-1]

σm :

initial surface tension at T = Tm

ω :

rotating angular frequency

av :

average

max :

maximum

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Acknowledgements

One of the authors Zou would like to thank Fawad Ahmed of the Nanjing University of Aeronautics and Astronautics for the technical help on this paper. This work is supported by the National Natural Science Foundation of China (Grant Nos. 51276089, 11474003 and 11702134) and Key projects of Natural Science Research of Anhui Provincial Department of Education (Grant Nos. KJ2018A0051 and KJ2018A0048).

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Zou, Y., Huang, H., Zhu, G. et al. Effect of Rotating Magnetic Field on Thermal Convection and Dopant Transport in Floating-Zone Crystal Growth. Microgravity Sci. Technol. 32, 349–361 (2020). https://doi.org/10.1007/s12217-019-09776-w

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Keywords

  • Numerical simulation
  • Convection
  • Doping
  • Rotating magnetic fields
  • Floating zone technique