Factors determining the saturation of point defects in growing silicon crystals

Abstract

An analytical calculation on the behavior of point-defects in growing silicon crystals was performed. The calculation concluded that the saturation of point-defect is determined by only two factors, which are the ratio of the normalized temperature gradient to the growth rate ( T′/ TV) and the ratio of the second-order differential of the temperature to the product of the temperature gradient and the growth rate ( T″/ T′ V). The former is same as Voronkov's conclusion except that the temperature gradient is normalized and the latter is newly found in the current calculation. In case of zero or near to zero growth rate, the saturation is determined by only one factor, which the ratio of the second-order differential of the temperature to the square of the temperature gradient ( T″/( T′) 2). In addition, diffusion equations where the heat of transfer takes minus values of the formation enthalpy, which were adopted by most of the models proposed so far, are also discussed. It was concluded that those equations do not show any dependency of point-defect distributions on the growth conditions without assuming such large diffusivity that the transport of point-defects due to Fickian diffusion is not negligible compared with that due to the crystal pulling.