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This work describes the fabrication facility (FAB) implementation of a multivariable nonlinear model predictive controller (NMPC) for the regulation of critical dimensions (CD) in photolithography. The controller is based on nonlinear empirical models relating the stepper inputs, exposure dose and focus on the isolated and dense CDs measured by scanning electron microscopy. Since the adjustments are made on the basis of the average value of five measured points in each wafer, this is referred to as average mode control. The optimal structure and parameters of these empirical models were determined by genetic programming, to closely match FAB data. The tuning and testing of the NMPC regulator were facilitated by the use of a simulated photolithography track, using the KLA-Tencor-FINLE PROLITH package, suitably calibrated to match FAB conditions. On implementation in the FAB, the NMPC has been demonstrated to consistently maintain the CDs close to their setpoint values, despite unmeasured disturbances such as shifts in uncontrolled inputs. It was also shown that adopting the multivariable feedback regulatory strategy to regulate the CDs results in significant improvements in the die yield.
Date of Publication: Feb. 2005