Czochralski growth of crystals - Simple models for growth rate and interface shape

A simple model for the crystal growth by the Czochralski (CZ) process has been proposed based on semiquantitative arguments. The model provides empirical relationships for the dependence of the pulling rate and the interface shape on the important process variables such as crystal radius, crucible temperature, height of the melt level, and the height of the exposed portion of the crucible wall. The parameters of the model can be evaluated by matching the results obtained from a detailed mathematical model of the CZ process or from extensive experimental data. The model has, therefore, the potential application for determining the best process conditions and for on-line control and optimization of the crystal puller to grow crystals with constant diameter and nearly planar interface.