Silicon single crystals produced by the Czochralski (Cz) process provide a majority of silicon substrates for the fabrication of microelectronic devices. The quality of a Cz crystal, determined from its oxygen content, defect distribution and dislocations, is affected by the temperature and flow field in the silicon melt from which the crystal is grown. Temperature difference across the melt is responsible for generation of buoyancy flow which is turbulent in nature for large size crystals (300mm). Most of the previous work was focused on simulation of smaller diameter crystals (>200mm). In large diameter crucibles (300mm crystal growth process, Grashof number increases beyond critical number (Gr > 10⁸), which indicates flow is turbulent in nature. Anticipation of turbulence is possible by applying external forces such as centrifugal force (by rotation of crystal and crucible) and Lorentz force (by applying external magnetic field). Combination of these external forces leads to complex flow inside the crucible which is associated with various flow instabilities such as Rayleigh-Bènard, baroclinic and Küppers-Lorz instabilities. This leads to time dependent three dimensional flow which influences the heat and mass transfer rates, impurity transport and interface shape and thus quality of crystal. Various theoretical as well as experimental efforts are underway to understand these complexities (1-3). Here the objective of this work is to study the complexities associated with flow numerically, which will be useful to gain better control over operational parameters.

For this purpose, we use two dimensional as well as three dimensional CFD based models to simulate the Silicon melt flow which is subjected to rotational speed and magnetic field. In these models, we consider natural convection due to temperature difference, forced convection due to crystal/crucible rotation, and the Lorentz force due externally applied magnetic filed. Turbulence in the system is modeled using low Reynolds number k-e model. The marangoni stresses generated due to thermocapillarity are considered at gas-melt interface. Resultant model equations, i.e., the conservation of mass, energy, and momentum along with phase change are solved using CFD code FLUENT 6.3. The temperature profile along the boundaries of the crucible and crystal were implemented from the global simulation in which convection and radiation heat loss from crystal and crucible were taken into account.

A numerical parameter study is performed to investigate the effect of applied magnetic field on local flow and turbulence parameters. Simulation results indicate that strong vertically upward flow dominates near crystallization zone (Fig. 1a) which increases temperature gradient in the same region (Fig.1b). 3D model results show completely non-symmetric temperature as well as flow profiles in the melt region (Fig. 2). Model results presented here are found to be useful for optimization of operational parameters for producing better quality crystals.

References:

1) Kumar V., Basu B., S. Enger, G. Brenner and F. Durst, Jou. of Crystal Growth, 255, 27 (2003).

2) Kalaev, V.V., D. P. Lukanin, V. A. Zabelin, Yu. N. Makarov, J. Virbulis, E. Dornberger, W. von Ammon, Jou. Crystal Growth, 250, 203 (2003).

3) Th. Wetzel, J. Virbulis, A. Muiznieks, W. von Ammon, E. Tomzig, G. Raming, M. Weber.. Jou. of Crystal Growth, 266, 34 (2004).