A finite element model for heat transfer, melt convection, and interface position is employed to analyze the role of support materials and ampoule geometry in influencing initial solidification during the vertical Bridgman growth of CdZnTe. Simulations show that the shape of the solid-liquid interface is determined by a delicate balance between axial and radial heat flows and that this balance is controlled during the growth through the cone region of the ampoule by the configuration of the ampoule and support. A design which features an ampoule with a shallow cone sitting upon a composite support made of a highly conducting core and a less conducting outer sheath promotes axial heat transfer while inhibiting radial thermal flow, thus resulting in an interface shape which is convex toward the melt. Such an interface shape may be favorable for increasing process yields and may also facilitate seeded growth. For the system studied here, a hollow core, which acts as a radiation channel to enhance axial heat transfer, is not found to be as effective as a highly conductive core in enabling convex interface shapes.