Model-predictive control of the Czochralski crystallization process .1. Conduction-dominated melt

Two model-predictive controllers (MPCs), operating on different time and length scales, coupled through constraints, are introduced for the Czochralski crystallization process. 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, 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. The main advantage of the coupled MPCs, compared with conventional, proportional-integral-derivative (PID) control, is that the microscopic properties in the crystal, such as the distributions of oxygen, dopant, and dislocations, are controlled indirectly by the addition of constraints to their nonlinear programs (NLPs).