## 3 Results and Discussion
### Temperature Distribution
In shown in Figure 4, the temperature field is more suitable when the lower heat transfer coefficient is dropped to 50 W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\). In this case, not only does the isothermal surface become more uniform, but the larger temperature gradient also appears at the bottom boundary. Compared with the bottom boundary as presented in Figure 3, the heat transport is also balanced on the side boundaries. Moreover, the undercooling in the horizontal direction is slower than that at the bottom boundary. The isotherm is flatter than that with a heat transfer coefficient of 100 W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\) as shown in Figure 3. The evolution of the crystal-melt interface from the central axis to the right sidewall with a different heat transfer coefficient on the side boundaries during mc-Si ingot growth can be seen from Figure 5. Based on this, we predict that a high-quality mc-Si ingot will be obtained with lower heat transfer coefficients on the side boundaries. Figure 3: Temperature distribution and crystal-melt interface position with heat transfer coefficient of 100 W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\) at different times: (**a**) \(t=0\) s; (**b**) \(t=100\) s; (**c**) \(t=200\) s; and (**d**) \(t=300\) s.  
The ideal mc-Si ingot consists of uniform vertical columnar crystal. The temperature gradient is the driving force for crystal growth and it is desirable to have vertical growth instead of horizontal. Hence, the vertical direction needs to maintain an appropriate temperature gradient and the temperature gradient in a horizontal direction needs to constantly decrease. Figure 6 shows the temperature gradient in a horizontal direction at different stages during the mc-Si ingot solidification. The vertical axis represents the temperature gradient and the horizontal one is the distance from central axis to the side boundaries along the crystal-melt interface (the pink line in Figure 6a). Figure 4: Temperature distribution and crystal-melt interface position with heat transfer coefficient of 50 W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\) at different time: (**a**) \(t\) = 0 s; (**b**) \(t\) = 100 s; (**c**) \(t\) = 200 s; and (**d**) \(t\) = 300 s.  
Figure 5: The evolution of the crystal-melt interface with different heat transfer coefficients on the side boundaries during mc-Si ingot growth. It can be seen that the maximum negative temperature gradient in the horizontal direction usually appears at the side boundaries with the higher heat transfer coefficient of 100 W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\). The growth orientation is perpendicular to the crystal-melt interface as usual, and the interface is concave with a higher heat transfer coefficient because the ability of heat transport on the side boundaries is strong. The temperature gradient in the vertical direction is positive and the crystal grows along this direction. However, the temperature gradient in the horizontal direction is negative because the heat flux transports effectively to the side boundaries with higher heat transfer coefficients and the crystal grows faster than that in the central region. The temperature of silicon melt in the central region is higher than that near side boundaries and the heat transports from the central to the side region. Based on this condition, the negative temperature gradient is formed and its value increases constantly with the growth of the mc-Si ingot. Moreover, the crystal-melt interface becomes more and more concave to the silicon melt with higher heat transfer coefficients on the side boundaries as shown in Figure 3.  
In order to get a suitable interface shape that helps to obtain a perfect silicon crystal, the heat transfer coefficient is decreased. The heat transfer becomes more balanced between central melt and side boundaries with an adjusted heat transfer coefficient where the temperature gradient in the horizontal direction is around 0.5 K/mm and the crystal-melt interface is close to flat.