Simulation of temperature distribution in crystals grown by Czochralski method

Abstract

A computer simulation model has been developed for prediction of the temperature distribution and the position of the melt-crystal interface for crystals grown by the Czochralski method. The method accounts for the detailed radiative heat transfer between the various “surfaces” present in the crystal pulling apparatus. An iterative finite element scheme is developed for the solution of the model. This is computationally more effective than previous attempts based on finite differences. The effects of various factors and operating parameters on the interface shape are illustrated. The results are compared to those of the simplified models and the implications of the observed differences are discussed. Some existing misconceptions with regard to the effect of cooling capacity, melt temperature, etc., on interface shape are corrected. It is shown that two parameters: a dimensionless heat flux and Biot number, are the key factors in controlling interface shape. The importance of accounting for both direct and indirect rediation heat transfer is demonstrated. An illustrative application is presented to show the utility of the model.