1 Department of Informatics and Mathematical Modeling, Technical University of Denmark
This Ph.D thesis encompasses a global numerical simulation of the needle-eye oat zone process, used to grow silicon single crystals. The numerical models includes coupled electromagnetic and free surface models and a global heat transfer model, with moving boundaries. An axisymmetric uidow model, including centrifugal, buoyancy, thermocapillary and electromagnetic forces, is used to determine flow field, after the phase boundaries have been determined, by the heat transfer model. A finite element model for calculating dopant transport, using the calculated unsteady flow field, has been developed within this project. This model has furthermore been expanded to two equations coupled by a non-zero right hand side, for simulating transport of point defects in the crystal during growth. Free surface shapes and induced electric surface current are calculated for t wo different 4'' congurations and a 0.8'' conguration. The heat transfer calculations of the same three congurations, yields the global temperature field, from which temperature gradients are determined. The heat transfer model is furthermore expanded to study convective cooling of the crystal from natural convection in the pressurized surrounding gas, for one of the 4'' congurations. The depth of the lower phase boundaries of both 4'' congurations have with good agreement to the calculations been measured, to provide experimental verication of the heat transfer model. Calculations of melt convection within the foating zone is done for the two 4'' congurations, for four dierent setups. The flow is unsteady, but laminar and is seen to be repressed by increased rates of rotation. Gas doping is simulated by prescribing a flux of dopant through the free surface, resulting in unsteady dopant concentration fields, with distinct concentration boundary layers at the lower phase boundaries. The dopant concentrations, at the lower phase boundaries, are used to determine radial resistivity profies, which with fair agreement are compared to measurements. Simulations of defect transport are conducted for both of the 4'', as well as the 0.8'' conguration, for two different values of the recombination factor. The calculation of the 0.8'' crystal is compared to DLTS measurements, revealing good agreement for one of the recombination factors, which however does not fit the Voronkov theory. Both factors are used in the simulation of the two 4'' congurations, which both have so high growth parameters, that the vacancy domination is relatively unaffected by the recombination.