The casting technique is widely used for producing multi-crystalline silicon for solar cells (SCs). Crystallization front dynamics during casting of silicon ingots (CSi) and flow characteristics in the melt affect the quality of the multi-crystal, in particular, the dislocation density in crystal grains, growth shape of grains, the content and uniformity of impurities in silicon ingot. Below, 3D unsteady analysis of melt convection in a squared mould, coupled with the calculation of the crystallization front shape is illustrated.

(a)	(b)
Fig. 1. Detailed 2D (a) and 3D (b) analysis of global heat transfer during the growth of 300x300mm Si ingot by the casting technique.

Fig. 2. Temperature and velocity distributions in vertical cross section of the crystallization zone

The melt flow can be three dimensional and unsteady due to natural convection, Marangoni stress tension on the melt free surface, or the Lorenz force induced in the melt by electro-magnetic effects.

3D features of the flow in CSi are due to moderate temperature gradients along the melt free surface and in the melt, which appeared to be sufficient to generate velocity fluctuations of about 1 cm/s in the large melt volume. It has been found that the flow in squared moulds can be essentially asymmetric, which is illustrated in Fig.2. In a cross section, a stable large vortex can appear for a long time changing the geometry of the crystallization front. However, it is mixing by intermediate scale flow structures that is mostly observed in the melt and contributes to better uniformity of impurities.

(a)	(b)
Fig. 3. Instantaneous distribution of the crystallization rate (Vcr) over the crystallization front (a) and the computed 3D shape of melt/crystal interface (b)

• for the squared mould with dominating side heating, it has been found that the melt flow is turbulent with pronounced stable asymmetric features affecting the crystallization front geometry due to a back action of the melt flow on the temperature distribution along the side walls of the crucible;
• melt turbulence contributes to better mixing of impurities along the crystallization front, which may increase the yield;
• neglecting melt turbulence results in poor predictions of the crystallization front geometry.

(a)	(b)
Fig. 4. Comparison of crystallization front geometries (a) and the distributions of the heat flux from the melt into the crystallization front (b) computed by different approximations

Advanced 2D and 3D modeling of Directional Solidification of multi-crystalline silicon ingots (UNDER CONSTRUCTION)

Detailed 3D heat transfer modeling of a Directional Solidification System is discussed in consulting section of the site.

Figure 1 (P = 50000 Pa, Vinlet = 0.3 m/s, Process time: 25 h, Vcryst about 15 mm/h)

Figure 2 Four large vortices are observed in the melt as the combined effect of buoyancy and Marangoni surface tension

Figure ... Crystallization front profiles: comparison of experiment by SAS (left) and simulation results (right). Vertical cut of the ingot is 250x700x700 mm. Data is published in the paper: Y. Y. Teng et al., Proceedings of PVSEC-18 conference

Figure ... 3D Ar gas flow above the melt

Figure ... Animation of Carbon concentration in the crystal

Figure ... From “The carbon distribution of growing multicrystalline silicon ingot during the directional solidification process” by Ying-Yang Teng, Jyh-Chen Chen, Chung-Wei Lu, Chi-Yung Chen; IWMCG-6, Poster Number PVM 3