Modeling, optimization and control of the Bridgman and Czochralski crystallization processes

Roberto Irizarry-Rivera, University of Pennsylvania


This thesis is concerned with strategies for the modeling, design, and control of the Czochralski (CZ) and Bridgman (BR) crystallization processes. During the last decade, these processes have been studied extensively, both experimentally and theoretically. Thus far, control and design strategies are lagging behind since the processes have very complex hydrodynamics, heat transfer and mass transfer. An optimal-design strategy is developed for the Bridgman crystallization process through the formulation of a nonlinear program. The objective function and constraints are designed to optimize the conditions for the growth of crystals with neatly flat melt/crystal interfaces. The equations for momentum and energy transport in the melt, for conduction in the crystal and crucible, and for energy transfer at the melt/crystal interface are solved using finite elements. The nonlinear program is solved using the controlled-random-search (CRS) technique. Results are presented for the growth of GaAs crystals. An optimal control strategy is formulated for the Czochralski crystallization process through the solution of a nonlinear program (NLP). A controlled thermal environment is studied with an extended set of heat pipes to improve the control of the temperature profile in the crucible and crystal. The decision variables are the temperatures of the heat pipes surrounding the crucible and crystal and the rotation rates of the crucible and crystal. The controlled-random search (CRS) optimization technique is utilized to solve the NLP. To reject disturbances, proportional-integral-derivative (PID) controllers are used commonly. In this work, model-predictive controllers (MPCs) are introduced. The controller design for the CZ process consists of two MPCs, operating on different time and length scales, coupled through constraints. The first, a capillary controller, controls the radius of the crystal by manipulating the pulling velocity. The second, a bulk controller, manipulates the power inputs to the heaters (and heat pipes) to control the pulling velocity, the thermal stresses, and the oxygen and dopant distributions under the crystal. The latter employs a distributed-parameter, conduction-dominated model for heat transfer in the melt and crystal, which is discretized using the boundary-element method to reduce the order of the system. A model for the melt/crystal interface is developed to represent the radius dynamics as a function of the pulling velocity for the capillary controller. To extend the MPC to apply when convection is important, a reduced-order model (ROM) is developed to account for convective heat and mass transfer in the melt. The model is based upon the assumption of an idealized geometry for flow consisting of horizontal donuts and vertical tubes with radial and axial dispersion, respectively. Boundary and bulk layers are assumed to separate the core fluid from the crystal and crucible. Scale analysis and integral boundary-layer theory are utilized to estimate the boundary-layer thicknesses and the maximum stream function in the core fluid. A strategy is presented for adjusting the imperfect model online to reduce the process/model mismatch during MPC.

Subject Area

Chemical engineering|Electrical engineering

Recommended Citation

Irizarry-Rivera, Roberto, "Modeling, optimization and control of the Bridgman and Czochralski crystallization processes" (1995). Dissertations available from ProQuest. AAI9615057.