The finite element method is used to solve a detailed model of heat and momentum transport in the vertical float-zone refinement of thin silicon sheets. The model formulation is much like that used to study float-zone refinement of cylindrical ingots, but the dominant physical mechanisms differ because of the much smaller length scale. The curvature of the meniscus remains nearly constant under all conditions due to the dominance of surface tension. The solid-liquid interface deviates considerably from a planar shape, contrary to the assumption of previous studies. The release and uptake of latent heat appear to play only minor roles in determining this shape, which results primarily from the sharp decrease of silicon emissivity upon melting. Strong flow in the melt due to the Marangoni effect is driven by large temperature gradients (O(100 K/cm)) at the melt surface, whereas buoyancy effects are negligible. Effective Reynolds numbers exceeding 103 are calculated. Multiple solutions are found under some circumstances. The different solution branches show little difference in the temperature field or free surface shape, but show a large difference in the flow field, which is likely to affect the redistribution of impurities. Transient calculations are used to determine the thickness variation of the sheet during the approach to steady state.
Bibliographical noteFunding Information:
This work was supported by the National Science Foundation under grant number CTS-9315980 and by SEMATECH under contract number 033006801. Partial support was also provided by the Minnesota Supercomputer Institute and the University of Minnesota Army High Performance Computing Research Center (under the auspices of Army Research Office contract number DAAL03-89-C-0038). The authors express their gratitude for the encouragement provided by ET. Geyling.