Quantifying parameter uncertainties in optical scatterometry using Bayesian inversion

Hammerschmidt, Martin; Weiser, Martin; Santiago, Xavier García; Zschiedrich, Lin; Bodermann, Bernd; Burger, Sven

doi:10.1117/12.2270596

Cited by 12 publications

(25 citation statements)

References 15 publications

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“…More details of the measurement setup are described in. 5,14 For Bayesian inversion it is necessary to chose prior distributions. In our investigations we have chosen uniform priors on the domains given in Table 1.…”

Section: Resultsmentioning

confidence: 99%

“…All posterior densities are characterized by sharp peaks with mean and standard deviation similar to the previous publication. 5 The mean and standard deviation for each parameter including the hyperparameter (error parameter) b are shown in Table 1. Since the domain sizes of the parameters vary due to their geometrical meaning, we introduce the relative standard derivation (std).…”

Section: Resultsmentioning

confidence: 99%

“…In principle, the propagation of electromagnetic waves is described by Maxwell's equations, but for our simple grating geometry Maxwell's equations reduce to a single second order partial differential equation, 5 ∇× µ −1 ∇× E + ω 2 ε E = 0.…”

Section: Forward Modelmentioning

confidence: 99%

See 2 more Smart Citations

An efficient approach to global sensitivity analysis and parameter estimation for line gratings

Farchmin,

Hammerschmidt,

Schneider

et al. 2019

Preprint

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Scatterometry is a fast, indirect and nondestructive optical method for the quality control in the production of lithography masks. Geometry parameters of line gratings are obtained from diffracted light intensities by solving an inverse problem. To comply with the upcoming need for improved accuracy and precision and thus for the reduction of uncertainties, typically computationally expansive forward models have been used. In this paper we use Bayesian inversion to estimate parameters from scatterometry measurements of a silicon line grating and determine the associated uncertainties. Since the direct application of Bayesian inference using Markov-Chain Monte Carlo methods to physics-based partial differential equation (PDE) model is not feasible due to high computational costs, we use an approximation of the PDE forward model based on a polynomial chaos expansion. The expansion provides not only a surrogate for the PDE forward model, but also Sobol indices for a global sensitivity analysis. Finally, we compare our results for the global sensitivity analysis with the uncertainties of estimated parameters.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

An efficient approach to global sensitivity analysis and parameter estimation for line gratings

Farchmin,

Hammerschmidt,

Schneider

et al. 2019

Preprint

Self Cite

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show abstract

“…To reconstruct likely parameters, we use the same objective function as in. 1 That is, we minimize the conditional probability π(x|y M ) of the parameter vector x given the measurement vector y M , also known as posterior probability. The posterior is given by Bayes' theorem as the product of likelihood π(y M |x) and prior probability π(x).…”

Section: Scatterometric Measurement Configurationmentioning

confidence: 99%

Using Gaussian process regression for efficient parameter reconstruction

Schneider

Hammerschmidt

Zschiedrich

et al. 2019

Metrology, Inspection, and Process Control for Microlithography XXXIII

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Optical scatterometry is a method to measure the size and shape of periodic micro-or nanostructures on surfaces. For this purpose the geometry parameters of the structures are obtained by reproducing experimental measurement results through numerical simulations. We compare the performance of Bayesian optimization to different local minimization algorithms for this numerical optimization problem. Bayesian optimization uses Gaussianprocess regression to find promising parameter values. We examine how pre-computed simulation results can be used to train the Gaussian process and to accelerate the optimization.

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“…Based on the same principle but additionally including some prior knowledge is the maximum posterior approach, which is a state of the art method in parameter reconstructions. 15 In the above frameworks, uncertainties are typically obtained from the Fischer information or covariance matrix , which relies on an assumed shape of the posterior. However, the shape of the posterior is generally unknown, hence this can lead to significant errors in the estimation of uncertainties if the actual posterior shape differs from the assumed one.…”

Section: Introductionmentioning

confidence: 99%

Efficient Bayesian inversion for shape reconstruction of lithography masks

Farchmin,

Hammerschmidt,

Schneider

et al. 2020

Preprint

Self Cite

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Background: Scatterometry is a fast, indirect and non-destructive optical method for quality control in the production of lithography masks. To solve the inverse problem in compliance with the upcoming need for improved accuracy, a computationally expensive forward model has to be defined which maps geometry parameters to diffracted light intensities. Aim: To quantify the uncertainties in the reconstruction of the geometry parameters, a fast to evaluate surrogate for the forward model has to be introduced. Approach: We use a non-intrusive polynomial chaos based approximation of the forward model which increases speed and thus enables the exploration of the posterior through direct Bayesian inference. Additionally, this surrogate allows for a global sensitivity analysis at no additional computational overhead.Results: This approach yields information about the complete distribution of the geometry parameters of a silicon line grating, which in return allows to quantify the reconstruction uncertainties in the form of means, variances and higher order moments of the parameters. Conclusion:The use of a polynomial chaos surrogate allows to quantify both parameter influences and reconstruction uncertainties. This approach is easy to use since no adaptation of the expensive forward model is required.

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Quantifying parameter uncertainties in optical scatterometry using Bayesian inversion

Cited by 12 publications

References 15 publications

An efficient approach to global sensitivity analysis and parameter estimation for line gratings

An efficient approach to global sensitivity analysis and parameter estimation for line gratings

Using Gaussian process regression for efficient parameter reconstruction

Efficient Bayesian inversion for shape reconstruction of lithography masks

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