Prof. Markus Bär Profile

Prof. Markus Bär

Head of Dept 8.4 at Physikalisch-Technische Bundesanstalt

SPIE Involvement:

Author

SPIE Journal Paper | 5 May 2020

Efficient Bayesian inversion for shape reconstruction of lithography masks

Nando Farchmin, Martin Hammerschmidt, Philipp-Immanuel Schneider, Matthias Wurm, Bernd Bodermann, Markus Bär, Sebastian Heidenreich

JM3, Vol. 19, Issue 02, 024001, (May 2020) https://doi.org/10.1117/12.10.1117/1.JMM.19.2.024001

KEYWORDS: Photomasks, Scatterometry, Lithography, Chaos, Inverse problems, Stochastic processes, Bayesian inference, Silicon, Error analysis, Oxides

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Proceedings Article | 21 June 2019 Presentation + Paper

Efficient global sensitivity analysis for silicon line gratings using polynomial chaos

Nando Farchmin, Martin Hammerschmidt, Philipp-Immanuel Schneider, Matthias Wurm, Bernd Bodermann, Markus Bär, Sebastian Heidenreich

Proceedings Volume 11057, 110570J (2019) https://doi.org/10.1117/12.2525978

KEYWORDS: Scatterometry, Polarization, Chaos, Silicon, Inverse problems, Systems modeling, Oxides, Photomasks, Scatter measurement, Data modeling

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Proceedings Article | 21 June 2015 Paper

The statistical inverse problem of scatterometry: Bayesian inference and the effect of different priors

Sebastian Heidenreich, Hermann Gross, Matthias Wurm, Bernd Bodermann, Markus Bär

Proceedings Volume 9526, 95260U (2015) https://doi.org/10.1117/12.2185707

KEYWORDS: Scatterometry, Inverse problems, Bayesian inference, Photomasks, Chaos, Atomic force microscopy, Scatter measurement, Finite element methods, Inverse optics, Metrology

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Proceedings Article | 13 May 2013 Paper

The effect of line roughness on DUV scatterometry

Mark-Alexander Henn, Sebastian Heidenreich, Hermann Gross, Bernd Bodermann, Markus Bär

Proceedings Volume 8789, 87890U (2013) https://doi.org/10.1117/12.2020761

KEYWORDS: Diffraction, Scatterometry, Extreme ultraviolet, Deep ultraviolet, Line edge roughness, Line width roughness, Finite element methods, Photomasks, Numerical simulations, Mathematical modeling

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Proceedings Article | 13 May 2013 Paper

Alternative methods for uncertainty evaluation in EUV scatterometry

Sebastian Heidenreich, Mark-Alexander Henn, Hermann Gross, Bernd Bodermann, Markus Bär

Proceedings Volume 8789, 87890T (2013) https://doi.org/10.1117/12.2020677

KEYWORDS: Data modeling, Scatterometry, Monte Carlo methods, Mathematical modeling, Extreme ultraviolet, Inverse problems, Photomasks, Scatter measurement, Inverse optics, Statistical modeling

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Proceedings Article | 18 December 2012 Paper

Impact of line edge and line width roughness on diffraction intensities in scatterometry

H. Gross, M.-A. Henn, S. Heidenreich, A. Rathsfeld, M. Bär

Proceedings Volume 8550, 85503R (2012) https://doi.org/10.1117/12.981327

KEYWORDS: Diffraction, Line width roughness, Line edge roughness, Extreme ultraviolet, Finite element methods, Scatterometry, Critical dimension metrology, Edge roughness, Optical testing, Cadmium

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Proceedings Article | 25 October 2012 Paper

First steps towards a scatterometry reference standard

Bernd Bodermann, Poul-Erik Hansen, Sven Burger, Mark-Alexander Henn, Hermann Gross, Markus Bär, Frank Scholze, Johannes Endres, Matthias Wurm

Proceedings Volume 8466, 84660E (2012) https://doi.org/10.1117/12.929903

KEYWORDS: Scatterometry, Metrology, Standards development, Scatter measurement, Scanning electron microscopy, Atomic force microscopy, Data analysis, Calibration, Diffraction, Modeling

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Proceedings Article | 23 May 2011 Paper

Improved geometry reconstruction and uncertainty evaluation for extreme ultraviolet (EUV) scatterometry based on maximum likelyhood estimation

M.-A. Henn, H. Gross, F. Scholze, C. Elster, M. Bär

Proceedings Volume 8083, 80830N (2011) https://doi.org/10.1117/12.889479

KEYWORDS: Extreme ultraviolet, Scatterometry, Inverse problems, Data modeling, Diffraction, Photomasks, Statistical analysis, Inverse optics, Lithography, Finite element methods

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Proceedings Article | 23 May 2011 Paper

Geometry reconstruction for scatterometry on a MoSi photo mask based on maximum likelihood estimation

M.-A. Henn, H. Gross, M. Bär, M. Wurm, B. Bodermann

Proceedings Volume 8083, 808306 (2011) https://doi.org/10.1117/12.895027

KEYWORDS: Scatterometry, Photomasks, Diffraction, Inverse problems, Data modeling, Deep ultraviolet, Phase shifts, Inverse optics, Error analysis, Modeling

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Proceedings Article | 17 June 2009 Paper

Evaluation of measurement uncertainties in EUV scatterometry

H. Gross, F. Scholze, A. Rathsfeld, M. Bär

Proceedings Volume 7390, 73900T (2009) https://doi.org/10.1117/12.827018

KEYWORDS: Extreme ultraviolet, Photomasks, Scatterometry, Inverse problems, Reconstruction algorithms, Diffraction, Monte Carlo methods, Reflectometry, Data modeling, Finite element methods

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Scatterometry, the analysis of light diffracted from a periodic structure, is a versatile metrology tool for characterizing periodic surface structures, regarding the critical dimension (CD) and other properties of the surface profile. For extreme ultraviolet (EUV) masks, only EUV radiation provides direct information on the mask performance comparable to the operating regime in an EUV lithography tool. With respect to the small feature dimensions on EUV masks, the short wavelength of EUV is also advantageous since it provides a large number of diffraction orders from the periodic structures irradiated. We present measurements at a prototype EUV mask with large fields of periodic lines-space structures using an EUV reflectometer at the Berlin storage ring BESSY II and discuss the corresponding reconstruction results with respect to their measurement uncertainties. As a non-imaging indirect optical method scatterometry requires the solution of the inverse problem, i.e., the determination of the geometry parameters describing the surface profile from the measured light diffraction patterns. In the time-harmonic case the numerical simulation of the diffraction process for periodic 2D structures can be realized by the finite element solution of the two-dimensional Helmholtz equation. Restricting the solutions to a class of surface profiles and fixing the set of measurements, the inverse problem can be formulated as a nonlinear operator equation in Euclidean space. The operator maps the profile parameters to special efficiencies of diffracted plane wave modes. We employ a Gauss-Newton type iterative method to solve this operator equation, i.e., we minimize the deviation of the calculated efficiencies from the measured ones by variation of the geometry parameters. The uncertainties of the reconstructed geometry parameters depend on the uncertainties of the input data and can be estimated by statistical methods like Monte Carlo or the covariance method applied to the reconstruction algorithm. The input data of the reconstruction are very complex, i.e., they consists not only of the measured efficiencies, but furthermore of fixed and presumed model parameters such as the widths of the layers in the Mo/Si multilayer mirror beneath the line-space structure. Beside the impact of the uncertainties on the measured efficiencies, we analyze the influence of deviations in the thickness and periodicity of the multilayer stack on the measurement uncertainties of the critical dimensions.

Proceedings Article | 17 June 2009 Paper

On numerical reconstructions of lithographic masks in DUV scatterometry

M.-A. Henn, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, H. Gross

Proceedings Volume 7390, 73900Q (2009) https://doi.org/10.1117/12.827547

KEYWORDS: Extreme ultraviolet, Deep ultraviolet, Scatterometry, Photomasks, Inverse problems, Diffraction, Scanning electron microscopy, Diffraction gratings, Monte Carlo methods, Inverse optics

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The solution of the inverse problem in scatterometry employing deep ultraviolet light (DUV) is discussed, i.e. we consider the determination of periodic surface structures from light diffraction patterns. With decreasing dimensions of the structures on photo lithography masks and wafers, increasing demands on the required metrology techniques arise. Scatterometry as a non-imaging indirect optical method is applied to periodic line structures in order to determine the sidewall angles, heights, and critical dimensions (CD), i.e., the top and bottom widths. The latter quantities are typically in the range of tens of nanometers. All these angles, heights, and CDs are the fundamental figures in order to evaluate the quality of the manufacturing process. To measure those quantities a DUV scatterometer is used, which typically operates at a wavelength of 193 nm. The diffraction of light by periodic 2D structures can be simulated using the finite element method for the Helmholtz equation. The corresponding inverse problem seeks to reconstruct the grating geometry from measured diffraction patterns. Fixing the class of gratings and the set of measurements, this inverse problem reduces to a finite dimensional nonlinear operator equation. Reformulating the problem as an optimization problem, a vast number of numerical schemes can be applied. Our tool is a sequential quadratic programing (SQP) variant of the Gauss-Newton iteration. In a first step, in which we use a simulated data set, we investigate how accurate the geometrical parameters of an EUV mask can be reconstructed, using light in the DUV range. We then determine the expected uncertainties of geometric parameters by reconstructing from simulated input data perturbed by noise representing the estimated uncertainties of input data. In the last step, we use the measurement data obtained from the new DUV scatterometer at PTB to determine the geometrical parameters of a typical EUV mask with our reconstruction algorithm. The results are compared to the outcome of investigations with two alternative methods namely EUV scatterometry and SEM measurements.

Proceedings Article | 25 April 2008 Paper

Computational methods estimating uncertainties for profile reconstruction in scatterometry

H. Gross, A. Rathsfeld, F. Scholze, R. Model, M. Bär

Proceedings Volume 6995, 69950T (2008) https://doi.org/10.1117/12.781006

KEYWORDS: Extreme ultraviolet, Scatterometry, Phase shifts, Inverse problems, Diffraction gratings, Monte Carlo methods, Diffraction, Reconstruction algorithms, Finite element methods, Photomasks

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Proceedings Article | 18 June 2007 Paper

Optimal sets of measurement data for profile reconstruction in scatterometry

H. Gross, A. Rathsfeld, F. Scholze, M. Bär, U. Dersch

Proceedings Volume 6617, 66171B (2007) https://doi.org/10.1117/12.726070

KEYWORDS: Reconstruction algorithms, Photomasks, Inverse problems, Diffraction gratings, Extreme ultraviolet, Scatterometry, Condition numbers, Diffraction, Finite element methods, Glasses

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Showing 5 of 13 publications

Conference Committee Involvement (9)

Modeling Aspects in Optical Metrology IX

26 June 2023 | Munich, Germany

Modeling Aspects in Optical Metrology VIII

21 June 2021 | Online Only, Germany

Modeling Aspects in Optical Metrology VII

24 June 2019 | Munich, Germany

Modeling Aspects in Optical Metrology

26 June 2017 | Munich, Germany

Modeling Aspects in Optical Metrology V

23 June 2015 | Munich, Germany

Showing 5 of 9 Conference Committees

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