Prediction of solid-liquid interface shape during CZ Si crystal growth using experimental and global simulation

摘要

In the Czochralski process, the prediction of characteristics of point defects in growing crystal strongly depends on the geometry of the solid-liquid interface. As the shape of the solid-liquid interface is not known a priori, melt convection and other forms of global simulation are used to determine the interface geometry. Conventional 2D models cannot realize interface shapes precisely, and unsteady 3D models require high large expenditure of both time and money for computer simulation and calculation. To surmount this problem, we found a new parameter (G_(ave)) that could be used to determine the interface height by means of a simple procedure. By deriving G_(ave) from the average axial temperature gradients in the crystals, we can characterize the thermal ambiance surrounding the crystal by hot zone geometry or the process condition. Interface heights h and G_(ave) are formularized as h = a(V/G_(ave)) + b. Fitting parameters a and b were obtained statistically by a number of sets of actual interface heights and calculated temperature distributions with various hot-zone types and process conditions. Pairs of fitting parameters were classified according to the crystal diameter and magnetic field strength.