The Numerical Analyses of the Solid-Liquid Interface Shapes during the Crystal Growth by the Czochralski Method

Abstract

The problem of thermal conduction during the crystal growth process is solved computationally using a model simplified to involve the essential features of the Czochralski crystal growth system. In the crystal the Laplace equation is solved, while in the melt the modified Laplace equation taking account of the fluid motion and the Navier-Stokes equations with the bouyant force being considered are solved simultaneously. The solid-liquid interface shapes are calculated as a function of the crystal rotation rate. Without the crystal rotation, the solid-liquid interface shape is concave to the melt and it becomes more concave as the crystal rotation rate increases. This dependence of the solid-liquid interface shape on the crystal rotation rate is explained by taking the fluid motion into account.