An analytical approach to thermal modeling of bridgman-type crystal growth: II. Two-dimensional analysis

Abstract

This paper develops a hybrid analytical/numerical model for two-dimensional heat flow in a Bridgman-type crystal growth configuration. The growth system is approximated by a stationary infinite cylinder in a three-zone furnace (hot zone, adiabatic zone, and cold zone). This is a reasonable approximation for systems with long length to diameter ratios and very slow translation rates. A form of a collocation method is employed in which an analytical solution is obtained for the temperature field in each zone and the Fourier-Bessel coefficients are obtained numerically by solving a system of linear equations determined from the boundary conditions at the zone interfaces. The model can accommodate different heat transfer coefficients in the hot and cold zone as well as different thermal properties in the solid and liquid phases and determines the position and the shape of the solidification isotherm. The effect of the ampoule can be included by adjusting the heat transfer coefficients provided the axial heat transfer in the ampoule is small compared to the sample. For cases where thick-walled conductive ampoules are required, a hybrid approximation is developed in which a one-dimensional analysis is used to obtain the thermal profile of the ampoule, which in turn establishes the outer boundary conditions for the sample. Finally, special cases involving high gradients are considered. The limiting value of the axial thermal gradient in Bridgman-type solidification is found, and the effect of the length of adiabatic zone on the gradient and curvature of the isotherms is analyzed. Also, approximate estimates are made for the distortion in isotherms from sample motion and for the power conducted through the sample.