Numerical simulation of heat transfer processes during single crystal growth by the Bridgman - Stockbarger method in fixed and rotating crucibles

Численно методом конечных элементов исследованы процессы кристаллизации кремния в плоскодонных неподвижных и равномерно вращающихся графитовых цилиндрических тиглях в режимах сопряженного конвективного теплообмена. Процессы кристаллизации кремния изучены при фиксированной скорости опускания тигля в холодную зону и различных скоростях его вращения. Опускание тигля имитировалось перемещением точки излома в распределении температуры на внешней стороне стенок тигля. Точка излома - это граница перехода от нагретого до начальной температуры участка стенки к области с заданным градиентом температуры. Использованы адаптивные сетки на треугольниках, отслеживающие положение фронта кристаллизации на каждом временном шаге. Использован пакет программ собственной разработки. The dependence of both spatial shape and intensity of the convective flow of silicon melt during the growth of a silicon ingot by the Bridgman-Stockbarger method was studied numerically by the finite element method. Stationary and uniformly rotating graphite crucible in conjugate convective heat transfer regines were examined. The simulation was carried out on the basis of dimensionless system of equations for the thermogravitational convection in the Boussinesq approximation using the bipolar approach. In the mixed convection regine, the system of equations was augmented by an equation for the azimuthal velocity. Adaptive grids on triangles were used to track the position of the crystal-melt interface at each time step. The calculations were carried out at a constant rate of lowering the crucible from the hot to the cold zone, equal to 2.81 cm/h, and at a constant temperature gradient equal to 35 K/cm. Lowering the crucible was simulated by moving the inflection point in the temperature distribution on the outside of the crucible walls. The range of angular velocities of crucible rotation from 0 to 10 rpm is considered. It is shown that as the angular velocity of crucible rotation in the axial region increases, both the velocity of the downward flow arising in the regines of thermogravitational convection gradually and the convective heat flux to the crystal-melt interface decrease. As a result, in the range of angular velocities from 2 to 10 rpm, the shape of the crystal-melt interface gradually approaches to the one typical for the thermal conductivity regime. It is shown that at the initial stage of the process at an angular velocity of 10 rpm in the axial region of the cooled crucible bottom, the nucleation of a single crystal is possible.