Mathematical modelling of the industrial growth of large silicon crystals by CZ And FZ process

The present paper gives an overview of the complex mathematical modelling of industrial Czochralski (CZ) and floating-zone (FZ) processes for the growth of large silicon single crystals from melt. Extensive numerical investigations of turbulent Si-melt flows in large diameter CZ crucibles, global thermal calculations in growth facilities and analysis of the influence of various electromagnetic fields on CZ process are presented. For FZ process, a complex system of coupled 2D and 3D mathematical models is presented to show the possibilities of modelling from the calculation of the molten zone shape till the resistivity distribution in the grown crystal. A special developed program code is presented that is used to calculate the temperature field in the crystal including radiation exchange with reflectors, stress field due to thermal expansion and shape of the dislocated zone in the case of dislocation generation. Besides the macroscopic modelling of crystal growth processes, the crystallisation model on the atomistic level in the mean field approximation is also presented.