Thermal Modeling of Crystal Growth by the Czochralski Method Including Radius Control

Abstract

Analysis of thermal phenomena in the crystal grown by the Czochralski process has been addressed. The model has been employed for deriving conditions under which the crystal radius during growth can be kept a constant. Two parameters that influence the size of the crystal are the pull velocity and the base temperature of the crucible. The present work explores the sensitivity of the crystal radius to these parameters. The mathematical model is based on one dimensional unsteady heat conduction in the solid phase with radiation to the enclosure. The model can predict the time-wise variation of the pull velocity and the base temperature so that the radius of the crystal can be maintained constant to within a prescribed tolerance. Materials considered for analysis are silicon and Nd:YAG. The analysis can account for variation in transport properties with temperature and differences in properties between the liquid melt and the crystalline solid. Results derived in the present work show that the crystal radius exhibits greater sensitivity to the pull velocity as compared to the crucible base temperature.