Bayesian approach to determine critical dimensions from scatterometric measurements

摘要

Supplement one of the 'Guide to the Expression of Uncertainties in Measurements (GUM)' recommends to estimate measurement uncertainties by Monte Carlo calculations. For computationally expensive models, Monte Carlo sampling can be often only realized for a small number of samples. This may cause inaccurate or even useless results for problems with a slow convergence rate. In this paper we apply a recently developed method that enables the Bayesian approach to determine the geometry of photo masks from measurements in extreme ultraviolet (EUV) scatterometry. The key idea of this approach is the construction of an efficient surrogate model which reduces the computational costs of sampling steps by several orders of magnitude. This allows us to use Markov Chain Monte Carlo sampling to determine the posterior distribution of desired critical dimensions of the photo mask geometry. The Bayesian approach leads to substantially larger uncertainties of the critical dimension than the previously employed maximum likelihood approach. The results for critical dimensions and line edge roughness are consistent with independent atomic force measurements of a test photo mask arrays from which scatterometric data were used.