Improved Methods for Lithography Model Calibration

摘要

Lithography models, including rigorous first principle models and fast approximate models used for OPC, require calibration using measured linewidth data. For models that predict process window behavior, the basic calibration data is linewidth versus focus and exposure over a range of feature sizes and types. The most common numerical method of finding the best fit model parameters is standard least-squares regression. While simple, this approach suffers from a number of well known problems. First, least-squares regression in not robust, meaning that even one bad data point can make the fit meaningless. Thus, outlier rejection becomes an important part of this approach. Both outlier rejection strategies and the use of robust fitting methods will be discussed. Second, standard least-squares may weight the data using the uncertainty in the measured linewidths, but uncertainty in the input variables, focus and exposure, is ignored. Often, at the extremes of focus and dose, errors in focus and dose actually dominate the resulting uncertainty in the measured linewidth. This can be accounted for using total least-squares regression. While often computationally difficult, in this paper an extremely fast and simple method for total least-squares regression will be presented for focus-exposure linewidth data. Finally, uncertainty in nominally fixed parameters, such as the linewidths of the features on the photomask used in the calibration, can lead to significant uncertainty in the resulting model parameters. The two standard approaches for dealing with this would be to leave these parameters fixed, or allow them to 'float' and be adjusted for best fit. Neither approach is satisfying. A better solution is to use Bayesian fitting, where a priori estimates of the mask feature widths and their uncertainties are used in the fitting merit function.