Modeling the temperature field dynamics during the Czochralski single crystal growth in a non-stationary approximation

The transient process of growing axisymmetric crystals by the liquid-encapsulated Czochralski method is considered. The mathematical model accounts for heat transfer in the crucible-crystal-melt-encapsulant, formation of the melt/encapsulant meniscus, crystallization interface movement, and changes in crystal radius. A new algorithm was developed to determine the lateral surface shape of the crystal during the process. The proposed numerical approach utilizes a geometrically conservative difference scheme that guarantees the fulfillment of conservation laws of energy and mass. The special splitting technique is used to solve the corresponding set of finite difference equations. The proposed approach ensures the consistency of crystal shape evolution with conservation laws. The designed numerical procedure is used to evaluate the impact of external thermal regime on the shape of the growing crystal. When the heater temperature is maintained constant, the crystal radius gradually decreases over time. To study the influence of the external temperature field on the shape of the lateral surface, the mathematical model is supplemented with a proportional-integral temperature controller equation that links the change in heater temperature to the radius of the growing crystal. In a general case, the application of an integral temperature controller leads to fluctuations in the crystal radius around a set value, with both the frequency and amplitude of fluctuations increasing progressively. Based on the results of the numerical experiments, the parameter values for a proportional-integral temperature controller that ensure the growth of crystals with a nearly constant radius are determined. The results of transient numerical simulations are compared with the results obtained using the quasi-steady-state model of crystal growth.