Modelling of the temperature changes by adequate changes of the pulling rate in the case of sheets and filaments grown from the melt in a vacuum by EFG method

Abstract

The main purpose of this paper is to give a model-based proof of the fact that the effect of the variations of the melt temperature at the meniscus basis can be compensated by an adequate variation of the pulling rate in the case of sheets and filaments grown from the melt in a vacuum by EFG method. For that, we find the range of the pulling rate and the melt temperature couples for which the system of differential equations which governs the evolution of the sheet half-thickness (or filament radius) x= x( t) and the meniscus height h= h( t) has asymptotically stable steady states. Computation is made in a nonlinear model for silicon sheets and filaments for a die of half-thickness x 0s =0.03 cm and radius x 0f=0.2 cm, respectively. The asymptotically stable steady states are determined and the region of attraction of each steady state is estimated. Using these regions of attractions we show what happens if during the growth the pulling rate v or/and the melt temperature T m at the meniscus basis are changed. For a given variation of the melt temperature at the meniscus basis during the growth, we find an adequate variation of the pulling rate v in order to obtain a single crystal sheet or filament with constant half-thickness or radius, respectively.