Relationship between temperature gradient and growth rate during CZ silicon crystal growth

Temperature distribution in the growing crystal is the most important parameter that determines the grown-in- defects, growth rate, etc. There is a discussion either higher growth rate leads to larger thermal gradient or smaller thermal gradient. In this study, in order to make clear the reason of this discrepancy, the effects of growth rate on the shape of melt/crystal interface, and temperature distribution in growing crystal were investigated by numerical modeling. Firstly, as increasing the growth rate, the shape of melt/crystal interface becomes more concave. And temperature gradient along center axis on growing crystal increases as increasing the growth rate. On the other hand, temperature gradient along surface of growing crystal decreases as increasing the growth rate. To obtain higher growth rate, heat transfer should be enhanced. Along the center axis, heat transfer in vertical direction by heat conduction is dominant. Then concave interface shape and larger thermal gradient along center axis were obtained. In the periphery of grown crystal near the triple points, because of concave interface shape, heat transfer in radial direction, and radiative heat transfer from growing crystal become more important than heat transfer in vertical direction. Then smaller thermal gradient along the growing crystal surface was obtained. This surface temperature profile agrees well with Abe's measurement results. It is cleared that higher growth rate leads to the higher heat transfer, and melt/crystal interface shape, and temperature distribution in growing crystal are determined by the balance of growth rate and heat transfer between heat conduction in vertical direction and heat conduction in radial direction combined with radiation heat transfer from crystal surface.