Simulation of Transport Phenomena and Interfacial Dynamics During Czochralski Growth of Oxide Crystals

Abstract

The quality of crystal grown in a Czochralski apparatus depends on a well organized flow and thermal field in the vicinity of the melt-crystal interface. A mathematical model that explores transport phenomena and the formation, evaluation and dynamics of the interface in a Czochralski apparatus is presented. The numerical formulation allows for the consideration of conduction in the crystal, conduction and convection in the melt and the gas phase, surface tension gradients, bulk and surface radiation and crystal rotation. Low temperature experiments using liquid crystal thermography have been conducted for validating the numerical codes. The control parameters for the Czochralski crystal growth process are crystal rotation, crucible wall and the enclosure wall temperature. Numerical simulations are carried out to establish the influence of these control parameters on the quality of oxide crystals grown in a Czochralski apparatus. Rare earth garnet YAG is considered as representative oxide material for the purpose of modeling and numerical simulation. Quantities of interest are the shape of the melt-crystal interface and the variation of pull velocity for the growth of constant diameter crystal. Steady state simulations carried out for the full Czochralski domain reveals the possibility of superheating of the crystal beyond its melting point, thus leading to the grown crystal returning to the melt. Similarly, the possibility of subcooling of the melt below the melting point of YAG at locations away from the crystal edge is indicated. The quasi-steady simulation of the growth process establishes the need for simultaneous control of crystal rotation and the crucible and enclosure wall temperature, along with the pull velocity for growth of high quality constant diameter YAG crystals.