Abstract
We study a simple model of chemical vapor deposition on a silicon wafer. The control is the flux of chemical species, and the objective is to grow the semiconductor film so that its surface attains a prescribed profile as nearly as possible. The surface is spatially fast oscillating due to the small feature scale, and therefore the problem is formulated in terms of its homogenized approximation. We prove that the optimal control is bang-bang, and we use this information to develop a numerical scheme for computing the optimal control.
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Friedman, A., and Hu, B., A Nonstationary Multi-Scale Oscillating Free Boundary for the Laplace Equation, Journal of Differential Equations, Vol. 137, pp. 119–165, 1997.
Meza, J. C., and Plantenga, T. D., Optimal Control of a CVD Reactor for Prescribed Temperature Behavior, SANDIA Report 95-8224, UC-406, 1995.
Moen, C. D., Spence, P. A., and Meza, J. C., Optimal Heat Transfer Design of Chemical Vapor Deposition Reactors, SANDIA Report 95-8223, UC-404, 1995.
Belyaev, A. G., Average of the Third Boundary-Value Problem for the Poisson Equation with Rapidly Oscillating Boundary, Vestnik Moscow University (Seriya I, Matematika i Mekhanika), Vol. 6, pp. 63–66, 1988.
Belyaev, A. G., On Singular Perturbations of Boundary Problems, PhD Thesis, Moscow State University, 1990 (in Russian).
Oleinik, O. A., Shamayev, A. S., and Yosifian, G. A., Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, Holland, 1992.
Gobbert, M. K., Cale, T. S., and Ringhofer, C., One Approach to Combining Equipment Scale and Feature Scale Models, Proceedings of the 187th Meeting of the Electrochemical Society, Reno, Nevada, pp. 553–563, 1995.
Gobbert, M. K., and Ringhofer, C., Mesoscopic Scale Modeling of Microloading during LPCVD, Journal of the Electrochemical Society (submitted).
Gobbert, M. K., and Ringhofer, C., An Asymptotic Analysis for a Model of Chemical Vapor Deposition on a Micro-Structured Surface, SIAM Journal on Applied Mathematics (submitted).
Friedman, A., Mathematics in Industrial Problems, Part 9, Institute for Mathematics and Its Applications, Springer Verlag, New York, New York, Vol. 88, 1997.
Hell, J. P., Jensen, R. F., and Egan, E. W., The Potential Effect of Multilayer Patterns on Temperature Uniformity during Rapid Thermal Processing, Proceedings of the Spring Materials Research Society Meeting, Vol. 287, 1995.
Friedman, A., Hu, B., and Liu, Y., A Boundary-Value Problem for the Poisson Equation with Multi-Scale Oscillating Boundary, Journal of Differential Equations, Vol. 137, pp. 54–93, 1997.
Friedman, A., Optimal Control for Variational Inequalities, SIAM Journal on Control and Optimization, Vol. 24, pp. 439–451, 1986.
Friedman, A., Optimal Control for Parabolic Variational Inequalities, SIAM Journal on Control and Optimization, Vol. 25, pp. 482–497, 1987.
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Friedman, A., Hu, B. Optimal Control of a Chemical Vapor Deposition Reactor. Journal of Optimization Theory and Applications 97, 623–644 (1998). https://doi.org/10.1023/A:1022694210246
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DOI: https://doi.org/10.1023/A:1022694210246