Numerical analysis of Oxygen control during growth of Czochralski silicon single crystals
In the present work, a set of 2D global furnace simulations accounting for oxygen dissolution and transport have been used to predict Oi content along the produced Silicon crystal. It is shown that Oi content is sensitive to several growth parameters (melt level, crucible rotation, etc.). Results indicate Oxygen tends to increase at the end of the crystal due to a combination of weakening of turbulence and planar velocity in addition to decrease of the melt free surface. Crucible rotation is the most influencing parameter at low melt level. Argon gas pressure and flow have a limited impact at low melt level.
Effect of crystal and crucible rotations on the interface shape of Czochralski grown silicon single crystals
A set of global heat transfer simulations in Czochralski (Cz) furnace for producing silicon single crystals have been performed to investigate the effect of crystal and crucible rotations on melt convection and crystal/melt interface shape. The 2D axisymmetric simulations are carried out taking into account radiative and conductive heat transfers between furnace components, melt convection including thermocapillary forces (Marangoni) and gas flow. Melt flow pattern and temperature distribution have been studied for several combinations of crystal and crucible rotations and for different crystal heights. The result shows that crystal/melt interface shape and melt flow regime are strongly sensitive to the rate of rotations of both crystal and crucible. Interfaces with low deflection can be achieved for certain combinations of crystal and crucible rotation rates.
The parallel waveform relaxation stochastic Runge–Kutta method for stochastic differential equations
For large-scale non-autonomous Stratonovich stochastic differential equations, we study a very general parallel waveform relaxation process which is on the basis of stochastic Runge–Kutta (SRK) method of mean-square order 1.0 in this literature. The convergence of the whole parallel numerical iterative scheme can be guaranteed and the scheme provides better properties in terms of decreasing the load of the computation and operating speed. At the same time, the related limit method is also introduced as the continuous approximation derived from the iterative scheme. In the approximation interval, it is worth noting that the mean-square order of the parallel numerical iterative scheme can be kept consistent with the previous SRK method at any arbitrary time point, not just at discrete points. Some numerical simulations are presented to elaborate the computing efficiency of the parallel numerical iterative scheme.