Temperature control of filament growth

Abstract

In this paper for a given pulling rate we find the range of the melt temperature at the meniscus basis for which the system of differential equations governing the evolution of the crystal radius r and the meniscus height h in the case of silicon filaments grown from the melt in a vacuum by edge-defined film-fed growth method (EFG method), has asymptotically stable steady states. Computation is made in a nonlinear model for a die of radius r0=20(cm×10−2) in the case when the meniscus weight is ignored. For the pulling rate v=4((cm×10−2)/s) we find that the computed range of the melt temperature at the meniscus basis Tm is 1674–1752 (K). For the melt temperature Tm in this range the computed radius r of the filament is in the range 10.269–19.966 (cm×10−2) and the meniscus height h is in the range 0.245–11.235 (cm×10−2). For each asymptotically stable steady state we estimate the region of attraction and using these regions we give a model based numerical proof of the fact that it is possible to control the diameter of a single crystal filament by changing the melt temperature at the meniscus basis, i.e. it is possible to obtain a desired piece-wise constant output with an adequate piece-wise constant input.