For vertical Bridgman growth of thermally anisotropic semiconductors, we present a detailed model accounting for heat transfer, flow driven by thermal buoyancy and solidification shrinkage, and interface deformation. The model allows for anisotropic solid-phase thermal conductivity, characteristic of nonlinear optical materials, as well as conduction in the ampoule wall, and conduction and convection in the liquid. The interface shape is determined as part of the solution of a moving boundary problem. For the nonlinear optical material gallium selenide and a range of growth conditions of practical interest, we present steady axisymmetric computations of the isotherms, flow, and interface shape. For ampoule-wall temperature profiles typical of three-zone Bridgman furnaces, the strength of the flow and deflection of the interface increase considerably with increasing growth rate, while the temperature distribution is relatively insensitive, except near the interface. Interface deflection decreases as the maximum ampoule-wall temperature gradient increases. The flow depends significantly on whether the melting temperature is “centered” between the high and low temperatures. The 23°C uncertainty in the melting temperature of GaSe is shown to have little effect on the flow and interface shape over the entire range of growth conditions. We show that properly accounting for thermal anisotropy is critical to predicting the flow and interface shape, both of which are relatively insensitive to the temperature dependence of the viscosity and thermal conductivities. We also show that localized heating along the ampoule wall can both reverse the direction of flow along the interface, which is expected to significantly influence distribution of dopants or impurities in the solid phase, as well as reduce interfacial curvature. When GaSe is grown under zero gravity conditions, the only flow is due to solidification shrinkage, and is essentially normal to the interface, whose shape is similar to those computed at normal gravity. Comparison of results for GaSe to previous work for benzene, a surrogate for organic nonlinear optical materials, shows that the qualitatively different results are associated with differences in the anisotropy of the thermal conductivity.
Interface shape control using localized heating during Bridgman growth