Analytical Approximations for the Distributions of Intrinsic Point Defects in Grown Silicon Crystals

Abstract

Analytical approximations for the distributions of vacancies and self-interstitials in Si crystals are derived in a one-dimensional point defect model. In the case of an unlimited high recombination rate all results depend only on the growth parameter P = V/G with V being the growth rate and G the temperature gradient at the melt/solid interface. For the critical growth parameter P crit , which separates the vacancy-dominated region from the interstitial-dominated region, the well-known relation of Voronkov is derived. This relation and also the first correction to P crit are in excellent agreement with recent experimental results published in literature. In the case of a limited recombination rate the transport equations were solved first of all by neglecting diffusion. This solution is completed by small correction terms using a special series expansion, which depend on the diffusivities. With this approximation important features such as the influence of the recombination and cooling rate on the distributions of intrinsic point defects can be analyzed.