On the dynamics of Czochralski crystal growth

The linear and nonlinear dynamics of Czochralski (CZ) crystal growth are analyzed by a thermal-capillary model which accounts for transients caused by energy transport, by changes in the shapes of the phase boundaries, and by the batchwise decrease of the melt volume in the crucible. Solutions are computed by a finite-element/Newton method and fully implicit time integration. Solutions for a quasi-steady-state model (QSSM) are calculated for long crystals with the melt volume held. Linear stability analysis of the dynamic model shows that the QSSM solutions are stable to random small disturbances, thereby confirming the inherent stability of the CZ process. Proportional and integral feedback control strategies are tested for on-line control of the crystal radius by integration of the dynamic model augmented with a servo-control equation. Integral control leads to oscillations in the crystal radius that are not present when the controller has a proportional element.