## 3 Results and Discussion
### Improvement of Silicon Crystal Growth  
As mentioned above, the top and side boundaries are adiabatic. Figure 8 shows the silicon crystal growth process with a different heat transfer coefficient at the bottom boundary. Controlling the morphology of the crystal-melt interface during silicon growth is crucial for obtaining high-quality crystals, which affects the macrostructure and microstructure of the products. In the end, the mechanical, optical and electrical properties of materials are also affected. Therefore, the heat transfer coefficient at the bottom boundary is adjusted in the simulations. As shown in Figure 8a, at 10 s, the number of nuclei with a lower heat transfer coefficient of 750 W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\) at the bottom boundaries is less than those with a higher heat transfer coefficient. Because the boundary condition is adiabatic on the top and side, the heat exchange via the bottom boundary is the main source for heat transfer. It is not desirable for nucleation at the bottom boundary due to poor heat transfer ability. Therefore, the energy for nucleation is inadequate and a small number of nuclei are formed. Based on this, the heat transfer coefficient at the bottom boundary is increased in order to obtain more nuclei. It can be seen that the number of nuclei increases significantly when the heat transfer coefficient is adjusted to 1000 W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\) and 1250 W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\). At 100 s, the silicon grains grow into columnar zone and they propel stably to silicon melt as evidenced in Figure 8a-c. The convexity of the crystal-melt interface and the length of the small columnar crystal zone increase with the increase of heat transfer coefficient at the bottom boundary because the growth energy is mainly provided by larger undercooling through crucible bottom. Furthermore, it increases the crystal growth velocity of the silicon melt. In practice, we not only want to improve the velocity of crystal growth, but also to get a high-quality mc-Si ingot. To do so, we can adjust the heat transfer coefficient at the bottom boundary. A small number of nuclei were obtained and the velocity of crystal growth was slow with lower heat transfer coefficient of 750 W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\). Also, the length of small columnar crystal is short and crystal-melt interface convex to silicon melt is small during the silicon grains growth. When increasing the bottom heat transfer coefficient to 1250 W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\), although the crystal growth velocity increases with large columnar crystal formed in the top zone, most part of the mc-Si ingot is undesirable. This is because the fine-grain crystals are formed easily due to a great number of nuclei that are generated at the bottom boundary with large undercooling. In addition, the crystal-melt interface is more convex to silicon melt and large temperature gradient in horizontal direction is generated with the increase of crystal length. Also, fast growth leads to inhomogeneous crystal. Therefore, it is suitable for silicon grain growth with the heat transfer coefficient of 1000 W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\) at the bottom boundary. This guarantees that reasonable number of silicon nuclei are formed at early stage and it may provide enough energy for crystal growth and relatively improvement in uniformity. As a result, a relatively perfect mc-Si ingot will be achieved where the impurities and dislocation are released with the moderate convexity of crystal-melt interface. According to Figure 9, the number of silicon grains decreases with increasing crystal length. It also affects the process of grains competition and the change of grain number with different heat transfer coefficient at bottom boundary. At the final stage, only three grains exist with heat exchange coefficient of 1000 W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\). The size of the grains is in the range of 1-20 mm. Therefore, the results are reasonable, and it is similar with the theories of crystal growth.