Transient global modeling for the pulling process of Czochralski silicon crystal growth. I. Principles, formulation, and implementation of the model

A transient global model for the crystal pulling process was developed for magnetic-field-applied Czochralski silicon (CZ-Si) growth. Heat transfer by solid conduction, melt convection, and diffuse gray radiation is taken into account for the crystal, melt, and other components in the furnace. The mesh adaption and view factor updating for the dynamic pulling process were realized by the structured grid deformations for different domains. By imposing the thermal boundary conditions in the vicinity of the triple point, the inverse control by the virtual proportional integral derivative controller and heat flux is acceptable for the transient global simulation of the pulling process in CZ-Si crystal growth. The applied cusp-shaped magnetic field (CMF) suppressed the turbulent melt flow and stabilized the heat and mass transport. CMF with different zero-Gauss-plane locations resulted in different flow patterns, which could affect the impurity transport during the pulling process. This developed transient global model can be applied for the segregation predictions of impurities (oxygen and carbon) and dopants in the CZ-Si growing process.