(698b) Modeling and Control of the Gradient Freeze Method of Crystal Growth

In this paper, we discuss our efforts to model gradient freeze furnaces used to grow large crystals of cadmium zinc telluride (CZT), at eV Products, Inc., and at Pacific Northwest National Laboratories, and application of control theory to improve melt-crystal interface shape. Specifically, we present simulations from detailed analyses in which a global model, CrysVUn developed by the Fraunhofer Institute, that computes high-temperature, furnace heat transfer, is coupled with a local model, Cats2D developed by our group, which solves for heat transfer, incompressible melt flow, and melt-crystal interface shape. The global model includes all of the details of the furnace and ampoule assembly, while the local model is chosen to consist only of the melt, interface, and a portion of the crystal. These models are solved using an iterative scheme to compute a self-consistent solution.

We have demonstrated that quantitative modeling of a complete system, ranging from global furnace modeling to detailed modeling of convective heat and mass transport at the growth interface, is possible using these computational tools. Furthermore, we have conducted parametric sensitivity studies to assess furnace design features that can be used to engineer interface shape during growth.

With a self-consistent model of the crystal growth apparatus and the growing crystal, we are able to explore methods of improving the crystal growth process. Specifically, we will apply control theory to improve the melt-crystal interface during the growth process. The most obvious method of affecting the interface is through the applied thermal gradient. Gradient freeze furnaces are characterized by vertically stacked heaters that produce a thermal profile, usually a monotonic function of height. The heater powers are varied during a growth cycle to mimic the action of pulling an ampoule through a static thermal gradient. Varying these heater powers in a smart manner can result in a more favorable interface shape. As a feasibility test, a control algorithm will be applied to the local subsystem using a measure of the interface curvature as the controlled output. The control actuators are the temperature set-points of the thermocouples corresponding to the heaters. At discrete time points, the global and local models will be coupled using a quasi-steady-state assumption.