Challenges to mathematical modeling of heat and mass transfer in silicon Czochralski growth are overviewed. The mathematical modeling of heat transfer has been extensively progressed in a past decade, and becomes highly reliable to predict the thermal history of crystals through the development of “global” models. The melt convection has been understood more deeply through mathematical modeling as well as experimental works in a past decade; it is of a turbulent nature, or time dependent and three dimensional at least. Incorporation of a turbulent model into a global model is successful for predicting the shape of melt-crystal interface. However, the melt convection, coupling of forced and natural convections, is still not well defined because of its exceeding complexity. Mass transfer is more closely related to the convection in both melt and ambient, and consequently ill defined at present. A current state of oxygen transport model is discussed.
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