Solar cell efficiencies are generally governed by the concentration and type of impurities, and the density and electrical activity of extended defects such as grain boundaries and dislocations [1]. Dislocations have been identified to be one of the most efficiency-relevant defect centers in crystal silicon for photovoltaic [2]. The requirement of the increase of solar cell efficiencies necessitates a reduction of the crystal dislocations.
    To reduce dislocation density, it is essential to know the relationships between the generation of dislocations and its influencing factors, such as geometry of crystal (diameter and height), temperature distribution in radial and axial directions. A quantitative study between the generation of dislocations and its influencing factors has been studied.   
    For radial temperature distribution, diameter of crystal and radial gradient of temperature have obvious effects on the generation of dislocations. If the radial gradient of temperature is fixed, the square root of maximum dislocation density in slices is proportional to the diameter of crystal; if the geometry of crystal is fixed, the square root of maximum dislocation density in slices is proportional to the radial gradient of temperature. We proposed a quantitative equation, which stated that the square root of maximum dislocation density in slices is proportional to the maximum value of  rdT/dr in slices. Numerical simulations show that the proposed formulation can accurately predict the location and the value of maximum dislocation density in every slice. Further extension of the proposed equation indicates that the maximum dislocation density in slices is determined by the energy accumulation rate along radial direction.  
    For axial temperature distribution, height of crystal and axial second-gradient of temperature have obvious effects on the generation of dislocations. If the axial second-gradient of temperature is fixed, the square root of maximum dislocation density is proportional to the height of crystal; if the height of crystal is fixed, the square root of maximum dislocation density is proportional to the axial second-gradient of temperature. We proposed a quantitative equation, which stated that the square root of maximum dislocation density is proportional to the integration of  d²T/dz² along axial direction. Numerical simulations show that the proposed equation can accurately predict the value of maximum dislocation density in crystal. The proposed equation also indicates that the maximum dislocation density is determined by the energy accumulation rate along axial direction.
    For mixed radial and axial temperature distribution, a proposed equation, which stated that the square root of maximum dislocation density in slices has a linear relationship with energy accumulation rate along radial direction and energy accumulation rate along axial direction. Therefore, in essence, maximum dislocation density in slices is determined by energy accumulation rate inside crystal.
    The above analysis indicates a simple relationship between maximum dislocation density in slices, geometry of crystal, temperature distribution, which is very useful for crystal growers to control the generation of dislocations and optimize the distribution of dislocations.

Reference
[1] C. Häβler, G. Stollwerck,W. Koch,W. Krumbe, A. Müller, D. Franke, T. Rettelbach, Adv. Mater. 13 (23) 2001.
[2] H. El Ghitani, M. Pasquinelli and S. Martinuzzi, J. Phys. III 3 (1993) 1941.

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