An analytical approach to thermal modeling of Bridgman-type crystal growth: I. One-dimensional analysis

Abstract

The analysis of heat flow in directional solidification and crystal growth by Bridgman-type processes has received considerable attention in recent years. Since the problem is complex, particularly when detailed descriptions of furnace and sample properties are included, numerical techniques are often employed. While such techniques are indispensable for modeling of real systems, it is valuable to have available analytical methods - albeit simplified - to check and to guide the numerical analysis, to perform sensitivity analyses, and to perform trade studies on furnace design. This work develops a one-dimensional analytical description of the heat flow in a translating rod that is applicable for Biot numbers less than unity. The model can accomodate different properties of the sample in the solid and molten state, different heat transfer coefficients in the hot and cold zone, finite length effects in one zone, and approximates ampoule effects by adjusting the Biot numbers to account for heat transfer in the ampoule. Also, the model contains an adiabatic zone and a booster heater zone that can be used to increase the thermal gradient near the solidification interface or to provide an independent control over the position of the interface. Trade studies are performed on optimizing the length of the booster heater zone to obtain the maximum gradient in the sample without exceeding a specified maximum sample temperature.