Optimal Experiment Design for Improved Parameter Estimation in Thermo-Mechanical Feedforward Models
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Abstract
ASML's wafer scanners are crucial machines in the production of integrated circuits (ICs). An important performance parameter in these photo-lithographic machines is the so-called stacking precision, which is a measure for the accurate stacking of multiple layers during the photo-lithography process in the xy-plane. The stacking precision is, amongst others, hampered by disturbances. Currently, a model is used to predict the disturbances and, subseqently, this information is used to compensate for stacking misalignment by active control. Some of the model parameters, used in the model, are not accurately known which result in inaccurate stacking compensation. The model parameter accuracy can be improved by using experiments subjected to model parameter calibration. Unfortunately, model parameter calibration may be hampered when choosing an insufficient experimental set-up. For example, it can lead to severe model parameter correlation. One way to improve the stacking precision is to calibrate the model parameters of the model by means of an optimal conducted experiment. Two experimental cases of the model are optimized using two different objective functions. Case 1 contains model parameter 1 and 2, and case 2 contains model parameter 1 and 3. A D-optimality objective function, which is the determinant of the information matrix, and an ACE1-optimality objective function, which focusses on the correlation between specified model parameter and the eigenvalues of the information matrix associated to the same model parameters. An optimization algorithm is used to perform the complex experiment optimization problem that originates by the characteristics of the model. Eventually, it is possible to acquire an optimal experiment which can increase model parameter accuracy and, therefore, give a reduction of worst-case stacking misalignment error by 54.4% and 9.4% when considering a 95% confidence bounce for model parameter case 1 and 2, respectively.