Effective thermal conductivity of poly-silicon chunks and its size dependence in a melting process of silicon Czochralski crystal growth: Part 2. Numerical simulations and comparison with measurements

Abstract

In the article Part 1, experimental relation of a temperature distribution and a poly-silicon size in a melting process of silicon Czochralski (CZ) method is investigated. In this work, numerical simulations are performed to obtain temperature distributions in a crucible of a silicon CZ melting process. And applicability of International Atomic Energy Agency (IAEA) formula of effective thermal conductivity to poly-silicon chunks is examined. Calculations are done with a two-dimensional symmetric axis system and an explicit finite difference method. Samples are poly-silicon chunks of two sizes and small cylindrical graphite pellets of three sizes. Temperatures at 6 points in a crucible have been simulated. Simulations using IAEA formula give a good agreement with measured temperatures in the article Part 1. The experimental relation of temperature and a sample size is reproduced by simulations. Though IAEA formula is constructed for sphere particles, it serves in predicting an effective thermal conductivity of poly-silicon chunks. When applying the formula to chunks, a measured value of the porosity and an average size are used as parameters. The method of estimating an effective thermal conductivity is hoped to be applied to a large diameter silicon CZ growth.