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Model Predictive Control of Thin Film Porosity Using Stochastic Differential Equations

Gangshi Hu1, Yiming Lou2, Gerassimos Orkoulas1, and Panagiotis D. Christofides3. (1) Department of Chemical and Biomolecular Engineering, University of California, Los Angeles, Los Angeles, CA 90095, (2) Advanced Projects Research, Inc, 1925 McKinley Ave., Suite B, La Verne, CA 91750, (3) Department of Chemical and Biomolecular Engineering, Department of Electrical Engineering, University of California, Los Angeles, Los Angeles, CA 90095

Thin film porosity strongly affects device performance and should be carefully controlled. For example, low-k dielectric films of high porosity are being used in current interconnect technologies to meet resistive capacitive (RC) delay goals and minimize cross-talk. However, increased porosity negatively affects the mechanical properties of dielectric films, increasing the risk of thermo-mechanical failures [1]. Furthermore, in the case of gate dielectrics, it is important to reduce thin film porosity as much as possible and eliminate the development of holes close to the interface.

Two different approaches, kinetic Monte Carlo (kMC) methods and stochastic differential equation (SDE) models, are employed to describe the evolution of film microscopic configurations and design feedback control laws. KMC simulation methods are widely used in microscopic modeling due to their efficiency in large temporal/spatial scale microscopic simulations. SDEs arise naturally in the modeling of surface morphology of thin film growth in a variety of material preparation processes [2, 3]. SDE models are effective as the basis for the design of control systems for regulating high-order moments of the microscopic distributions. Both linear and nonlinear controllers were developed based on SDE models to control surface roughness in various thin film growth processes, including deposition and sputtering process [4, 5]. However, there are no works that have focused on feedback control of film porosity.

Motivated by these considerations, a model predictive controller is designed using a linear SDE model of a porous film deposition process to regulate the film density to a desired value and minimize run-to-run fluctuations. The SDE model is constructed from the kMC model of a deposition process, which includes two types of microscopic processes, deposition and diffusion. Vacancies and overhangs are allowed in the kMC model to introduce porosity, which is mathematically represented by the thin film density. The predictive controller is formulated to minimize a cost function including the distance between the predicted density and the desired value. The manipulated input in the closed-loop system is the substrate temperature. Finally, the predictive controller is applied to the kMC simulation of the deposition process. Simulation results demonstrate the effectiveness of the proposed control method.

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2. Edwards, S. F. and Wilkinson, D. R. The surface statistics of a granular aggregate. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 1982;381:17-31.

3. Lauritsen, K. B., Cuerno, R. and Makse, H. A. Noisy Kuramoto-Sivashinsky equation for an erosion model. Physical Review E 1996; 54:3577-3580.

4. Lou, Y. and Christofides, P. D. Feedback control of surface roughness using stochastic PDEs. AIChE Journal 2005;51:345-352.

5. Lou, Y. and Christofides, P. D. Nonlinear feedback control of surface roughness using a stochastic PDE: Design and application to a sputtering process. Industrial & Engineering Chemistry Research 2006;45:7177-7189.