We studied the effects of cooling process on the generation of dislocations in multicrystalline silicon grown by the vertical Bridgman process. From the temperature field obtained by a global model, the stress relaxation and multiplication of dislocations were calculated using the Haasen-Alexander-Sumino model. It was found that the multiplication of dislocations is higher in fast cooling processes. It was confirmed that residual stress is low at high temperatures because the movement of the dislocations relaxes the thermal strain, while the residual stress increases with decreasing temperature, because of reduced motion of dislocations and formation of a strain field at lower temperatures.
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The multiplication of dislocations is mainly caused by thermal stress. Thermal stresses are generated in the silicon ingot by nonuniform temperature distribution, which is mainly affected by furnace design [
A global model [
Configuration inside the furnace (1: melt; 2: crystal; 3-4: crucibles; 5-6: pedestals; 7–11: heat shields; 12–15: heaters).
Position of the crucible (a) at the beginning of solidification and (b) at the end of solidification.
We considered conductive heat transfer in all the furnace components, heat radiation exchange between all the diffuse surfaces in the furnace, and heat convection for melt flow during heat transfer calculations for the entire furnace. We coupled these elements and solved the temperature field in them by using a transient global model. In addition, the melt-solid interface shape was obtained by a dynamic tracking method [
where
Based on the temperature distributions in the crystal, stress relaxation, time-dependent creep deformation, and the multiplication of the dislocations were calculated from the beginning of the solidification process to the end of the cooling process at room temperature. We assumed that
where
where
where
where
The displacement boundary conditions are derived from the stress boundary conditions, that is, Give temperature field, and obtain thermal strain Give initial dislocation density Update dislocation density By substituting If there is no convergence, return to step
Figure
Analysis conditions for heater power (solid lines) and crucible positions (dashed lines).
Average crystal temperature histories.
Figures
Average dislocation density versus average temperature during the cooling process.
Dislocation density distributions at room temperature (1, 3, and 5).
Average residual stress of plastic body (solid lines) and elastic body (dashed lines) versus the average temperature during the cooling process.
Figures
Residual stress distributions at room temperature (1, 3, and 5).
We calculated the average residual stresses based on the plastic and elastic body models, as shown in Figure
In addition, the residual stress distributions at room temperature varied with the cooling rates. At the bottom of the crystal, higher cooling rates resulted in higher residual stress levels. In contrast, the residual stress of the elastic body model decreased with decreasing temperature and reached zero at room temperature at all cooling rates.
Figure
Model of the relationship between the thermal strain and the residual stress. (a) Variation of the residual stress in the elastic body from high temperature to room temperature during cooling process. (b) Variation of the residual stress in the viscoplastic body from high temperature to room temperature during cooling process.
The relationship between the relaxation stress and dislocations. (a) The model of the stress-strain curve at the yield point. (b) Comparison between the multiplication of the dislocation density and the relaxation stress in 1.
Based on the transient global model and the HAS model, we have studied the effects of the cooling process on the dislocation density in multicrystalline silicon grown by the vertical Bridgman process. We found that the dislocation multiplication becomes high at fast cooling rates because residual stress increases dramatically due to the multiplication of dislocations. Also, the residual stress is low at high temperatures because of the movement of dislocations. The residual stress then increases with decreasing temperature because the fixed dislocations produce a strain field. These results indicated that maintaining homogeneous temperature distributions reduces the dislocation density and the residual stress in the crystal.