(453e) Recent Advances in Uncertainty Analysis and Control in Multiscale Process Systems with an Application to Thin Film Deposition

Thin film deposition is a key batch process in semiconductor manufacturing that is characterized by phenomena that evolve at different time and length scales. To capture the multiscale phenomena occurring in this process, the evolution of the thin film is often modeled using nonlinear partial differential equations (PDEs) embedded with lattice-based kinetic Monte Carlo (KMC) simulations. Strong dependence of the electrical and mechanical properties of thin films on their microstructure, e.g. surface roughness and film thickness, has motivated research on control and optimization in thin film growth. Despite the extensive body of research, there is still a significant discrepancy between the expected performance and the actual yield that can be accomplished employing the current control methodologies. This gap is mainly related to the complexities associated with the multiscale phenomena in the thin film deposition process, lack of practical online in-situ sensors at the fine-scale level, and uncertainties in the mechanisms and parameters of the system¹. This research presents recent approaches that have been developed that make the control and optimization of multiscale processes more realistic by partially addressing these issues.

Although the recently introduced optical in-situ sensors have motivated the development and implementation of feedback control in the thin film deposition process, their application is still limited in practice since most of the measurements are only available at the end of the batch. This has motivated the development of suitable optimization and control approaches that do not have access to online fine-scale measurements. An approach to perform offline optimization of the deposition process will be presented based on desired product quality specification in the presence of model-plant mismatch^2-3. The uncertainty propagation is performed employing power series expansion (PSE). To provide a computationally tractable optimization, the sensitivities in PSEs are evaluated using reduced-order lattices in the KMC models. The results from this research have demonstrated the need to develop robust strategies for this process and have shown that optimal control trajectories obtained from robust optimization formulations can produce thin films under stringent product constraints.

One key challenge in multiscale modelling analysis is the lack of closed-form models, which are essential for model-based control and optimization applications. To address this issue, closed-form models that can predict the control objectives in the presence of uncertainty have been developed⁴. The robust performance is quantified by estimates of the distribution of the controlled variables employing PSE. Since these models can predict the controlled outputs efficiently, i.e. the film’s roughness and growth rate, they can either be used as estimators for feedback control purposes in the lack of sensors or as a basis of Model predictive control (MPC) framework. Thus, a multivariable robust estimator that predicts the surface roughness and growth rate under uncertainty has been designed. To ensure that the control objectives are met at the end of the batch, the robust estimator is designed such that it computes bounds on the process outputs. The estimator has been coupled with traditional feedback controllers to evaluate the performance of the system in the lack of online measurements and under uncertainty in the multiscale model parameters.

MPC is an effective advanced control framework widely used in the industry. A robust nonlinear model predictive control (NMPC) application for the thin film deposition process will be presented⁵. The NMPC makes use of closed-form models, which were identified offline and are able to predict the thin film’s roughness and thickness at a predefined probability of satisfaction. The NMPC framework aims to minimize the final surface roughness while satisfying constraints on the control actions and film thickness at the end of the deposition process. Since the identification is performed at a specific probability limit, hard constraints were specified in that NMPC formulation. Accordingly, conservative control actions are predicted by the NMPC algorithm. To improve the robust performance of NMPC using soft constraints, closed-form models were developed to estimate the first and second-order statistical moments of the thin film properties under uncertainty in the multiscale model parameters. The results show that NMPC is an efficient technique that can be implemented to address controllability in thin film deposition while explicitly considering uncertainty in the analysis.

The outcomes derived from this study have shown that parameter uncertainty plays a key role in the design of efficient control and optimization techniques in thin film deposition and that must be explicitly considered in the multiscale modelling analysis of these systems to produce thin films with specific high-quality requirements.

[1] Ricardez-Sandoval, L.A., 2011. Current challenges in the design and control of multiscale systems. Can. J. Chem. Eng. 89, 1324–1341.

[2] Rasoulian, S., Ricardez-Sandoval, L.A., 2014. Uncertainty analysis and robust optimization of multiscale process systems with application to epitaxial thin film growth. Chem. Eng. Sci. 116, 590–600.

[3] Rasoulian, S., Ricardez-Sandoval, 2015. Worst-case and distributional robustness analysis of a thin film deposition process. Proceedings of the 9th International Symposium on Advanced Control of Chemical Processes, ADCHEM, Vancouver, Canada.

[4] Rasoulian, S., Ricardez-Sandoval, L.A.. Robust multivariable estimation and control in an epitaxial thin film growth process under uncertainty. Submitted to Journal of Process Control (2014).

[5] Rasoulian, S., Ricardez-Sandoval, L.A., A robust nonlinear model predictive controller for a multiscale thin film deposition process. Chem. Eng. Sci. (2015), http://dx.doi.org/10.1016/j.ces.2015.02.002