Model reduction for optimization of rapid thermal chemical vapor deposition systems

A model of a three-zone rapid thermal chemical vapor deposition (RTCVD) system is developed to study the effects of spatial wafer temperature patterns on polysilicon deposition uniformity. A sequence of simulated runs is performed, varying the lamp power profiles so that different wafer temperature modes are excited. The dominant spatial wafer thermal modes are extracted via proper orthogonal decomposition and subsequently used as a set of trial functions to represent both the wafer temperature and deposition thickness. A collocation formulation of Galerkin's method is used to discretize the original modeling equations, giving a low-order model which loses little of the original, high order model's fidelity. We make use of the excellent predictive capabilities of the reduced model to optimize power inputs to the lamp banks to achieve a desired polysilicon deposition thickness at the end of a run with minimal deposition spatial nonuniformity. Since the results illustrate that the optimization procedure benefits from the use of the reduced-order model, our future goal is to integrate the model reduction methodology into real-time and run-to-run control algorithms. While developed in the context of optimizing a specific RTP process, the model reduction techniques presented in this paper are applicable to other materials processing systems.