3.1. Temperature Distribution
The temperature distribution has a strong influence on crystal growth velocity, interface shape and grain boundaries during the crystal growth process. A suitable temperature field may decrease the defects in silicon ingot and increase the quality of silicon wafers. Therefore, it is necessary to discuss the temperature field during the process of silicon crystal growth.
According to the Procast software, the range of the heat transfer coefficient is about 0–2000 W·m
−2·K
−1 between inorganic nonmetallic materials and air. In this part, we analyzed the effect of the side boundary conditions on temperature field within the computational area. The heat transfer coefficients were 0 and 1000 W·m
−2·K
−1 at the top and bottom, respectively. The temperature distribution is shown in
Figure 3 and
Figure 4 with different heat transfer coefficients on the side boundaries, where the temperature isotherms of the melting point represent the crystal-melt interface at different times. The color of the isotherms reflects the temperature value, with red representing the maximum temperature and deep blue representing the minimum temperature.
As clearly shown in
Figure 3a and
Figure 4a, the computational area is full of silicon melt at an early stage of crystal growth. The temperature isotherms are straight and the maximum temperature gradient appears at the bottom boundary, which is attributed to the fact that heat transfer between environment and silicon melt is more intense. It is beneficial for crystal growth in that it releases the latent heat of crystallization with a high temperature gradient. In the columnar crystal zone, heat transfer is the dominant factor for crystal growth and the isothermal surface is likely to reflect the shape of the crystal-melt interface.
As shown in
Figure 3, when the heat transfer coefficient is 100 W·m
−2·K
−1 on the side boundaries, the shape of the crystal-melt interface changes from straight to slightly curved because the heat transport near the side boundaries is faster than that in the bottom boundary with the increase of crystal length. It can be seen that the temperature distribution at the center point of silicon melt is more uniform when the process of solidification is finished. In this case, the quality of the mc-Si ingot in the center position is much better than that near the side and bottom boundaries where the silicon ingot having higher temperature gradient.
In order to get more straight crystal-melt interface, The temperature field was optimized by means of adjusting the heat transfer coefficient of the side boundaries. In shown in
Figure 4, the temperature field is more suitable when the lower heat transfer coefficient is dropped to 50 W·m
−2·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·m
−2·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.
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).
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·m
−2·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. Hence, it is desirable to grow silicon with a suitable temperature field and flat crystal-melt interface with a lower transfer coefficient of 50 W·m−2·K−1.
3.2. Characteristics of Silicon Crystal Growth
Based on above analysis, we know that the shape of the crystal-melt interface mainly depends on the boundary conditions. In this part, we keep adiabatic conditions on the top and side boundaries while the heat transfer coefficient of 1000 W·m−2·K−1 is used for the bottom boundary.
Figure 7 shows the process of microstructure of silicon grain growth. Blue represents the melt, and other different colors are used to represent different crystal grains. Thus, the interface between blue and other colors is the crystal-melt interface. The nuclei usually appear at the bottom boundary. At the beginning, most nuclei grow gradually with different sizes. The shape of the crystal-melt interface is zigzag faceted in the preliminary stage and the heat of crystallization is taken away by the melt and mc-Si ingot. The heat flux transports slowly and the zigzag faceted interface propels into silicon melt with lower velocity during the crystal growth, which is in line with references [
24,
25].
During crystal growth, the sharp corners in front of interfaces disappear gradually and the shape of the crystal interface becomes smoother. The crystal-melt interface is convex to silicon melt and the convexity decreases with the increase of silicon crystal length. Some grain boundaries disappeared because the array of atoms is similar among neighboring columnar crystal zones with similar phase, and the silicon grains with the higher energy may be merged with the lower energy ones. In addition, structure and energy fluctuations exist in the crystal-melt interface during crystal growth. It will release the latent heat in front of the crystal-melt interface during crystallization, resulting in some silicon grains with high energy which will be fused and stop the growth.
The undercooling is large at the bottom of the computational area because heat transfer is intense between the melt and bottom boundary. Therefore, many fine grains are formed at the bottom of silicon melts. The silicon grains grow quickly due to the higher transfer coefficient. Being influenced by the thermal field, the nuclei appear firstly at the center of the bottom and then the nucleation is expanded to the side boundaries until the process of nucleation is finished at the bottom of the silicon melt.
As shown in
Figure 3,
Figure 4 and
Figure 7, in conditions of top adiabatic and bottom constant heat flux, the shape of the crystal-melt interface changes from concave to convex with the decrease of the heat transfer coefficient on the side boundaries. In the industrial production of polycrystalline silicon, the concave crystal-melt interface easily leads to many stresses and dislocations, which reduce the crystal quality. Conversely, the flat or a slightly convex crystal-melt interface can reduce stresses and dislocations [
26], so reducing the heat transfer coefficient on side boundaries is necessary to produce high-quality polycrystalline silicon crystal.
3.3. 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·m
−2·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·m
−2·K
−1 and 1250 W·m
−2·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·m−2·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·m−2·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·m−2·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·m
−2·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.