Dependence of the Marangoni flow and impurity distribution on the vertical temperature gradient during the growth of aluminum-doped silicon fibers by the EFG technique
by
Liliana Braescu
Department of Computer Science, West University of Timisoara, Romania
Coauthors: T. F. George (University of Missouri–St. Louis, USA)

Demand for single or poly-silicon has increased dramatically due to rapid expansion of the photovoltaic industry. Various growth techniques have been used for producing silicon wafers for PV cells, such as the Czochralski, floating zone, and edge-defined film-fed growth (EFG) techniques. Among these, EFG is the first non-conventional technique for crystalline silicon wafer production to enter into large-scale manufacturing. Molten silicon is known to be an extremely-reactive material, with strong thermal forcing in surface-tension-driven flows being realized during the growth process. The surface tension value and its temperature dependence are essential for describing surface-tension-driven flow (Marangoni flow) on the liquid free surface, i.e., the liquid bridge between the die and the crystal. It is generally accepted that thermal fluctuations are a serious drawback in growing a high-quality crystal. Moreover, the difficulty in the experimental determination of the principal physical constants (heat expansion coefficient, surface tension, temperature coefficient of the surface tension, etc.) requires powerful tools to analyze and optimize crystal growth processes, offering the possibility to change the configuration of the equipment, process parameters and optimize the crystal quality.

In this paper, is proved numerically that controlling one of the most important process parameters – vertical temperature gradient from the EFG furnace – the best homogeneity of the crystal can be obtained over a wide range of values of the surface tension temperature coefficient. Towards this end, the dependence of the Marangoni flow and impurity distribution on the vertical temperature gradient is analyzed for aluminum-doped silicon fibers grown from the melt by the EFG technique with central capillary channel shaper and melt replenishment. For describing the growth process, the incompressible Navier-Stokes equations in the Boussinesq approximation, convection-conduction and conservative convection-diffusion equations are utilized. The Marangoni effect is implemented for the free surface by the weak form-boundary application mode. The computations are carried out in the stationary case in a 2D axisymmetric model by the finite-element numerical technique using COMSOL Multiphysics 3.3 software for surface tension temperature coefficients dγ/dT situated in the range [-7×10^(-4); 0] N/m×K (i.e., for Marangoni numbers Ma between zero and 406.25), and for three representative vertical temperature gradients in the furnace: k1=5,000; k2=50,000; k3=100,000 K/m. The employed mesh is manually refined along the boundaries of the molten liquid-bridge being considered: 17,072 triangular elements and 150,803 degrees of freedom. The computed fluid flows show that there are four possible different behaviors of the dopant concentration, which depend on the length range of Ma and on the value of the vertical temperature gradient as follows:

(i) If the Ma number increases in the ranges [0; 75.4], [0; 7.54] and [0; 3.65] corresponding to k1, k2 and k3, respectively, then the downward flow (dγ/dT<0) on the free liquid surface leads to a decrease of the maximum of the dopant concentration Cmax, which is located at the triple-point.

(ii) If Ma increases in the ranges [75.4; 406.25], [7.54; 39.46) and [3.65; 19.73] corresponding to k1, k2 and k3, respectively, then very small turbulences in the fluid flow take place, leading to an increase of Cmax, which is located at the triple-point.

(iii) If Ma increases in the ranges [39.46; 69.64] and [19.73; 35.4] corresponding to k2 and k3, respectively, then turbulences in the fluid increase, and Cmax is pushed inside at the level of the melt/crystal interface, at a distance on the same order as the meniscus height from the external crystal surface.

(iv) If Ma increases in the ranges [69.64; 406.25] and (35.4; 406.25] corresponding to k2 and k3, respectively, then higher turbulences move Cmax at the triple-point.

It can be concluded that in the case of a small vertical temperature gradient, the best homogeneity of the crystal is obtained for a larger range of the Ma number.

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Date received: January 27, 2008