Modeling aspects to improve the solution of the  inverse problem in scatterometry

Hermann Gross; Sebastian Heidenreich; Mark-Alexander Henn; Markus Bär; Andreas Rathsfeld

doi:10.3934/dcdss.2015.8.497

Article Contents

2015, Volume 8, Issue 3: 497-519. Doi: 10.3934/dcdss.2015.8.497

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Modeling aspects to improve the solution of the inverse problem in scatterometry

1.
Physikalisch-Techn. Bundesanstalt, Berlin, Abbestr. 2-12
2.
Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Mohrenstr. 39

Received: September 30, 2013

Revised: December 31, 2013

Published: June 2015

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Abstract

The precise and accurate determination of critical dimensions (CDs) of photo masks and their uncertainties is relevant to the lithographic process. Scatterometry is a fast, non-destructive optical method for the indirect determination of geometry parameters of periodic surface structures from scattered light intensities. Shorter wavelengths like extreme ultraviolet (EUV) at 13.5 nm ensure that the measured light diffraction pattern has many higher diffraction orders and is sensitive to the structure details. We present a fast non-rigorous method for the analysis of stochastic line edge roughness with amplitudes in the range of a few nanometers, based on a 2D Fourier transform method. The mean scattering light efficiencies of rough line edges reveal an exponential decrease in terms of the diffraction orders and the standard deviation of the roughness amplitude. Former results obtained by rigorous finite element methods (FEM) are confirmed. The implicated extension of the mathematical model of scatterometry by an exponential damping factor is demonstrated by a maximum likelihood method used to reconstruct the geometrical parameters. Approximate uncertainties are determined by employing the Fisher information matrix and additionally a Monte Carlo method with a limited amount of samplings. It turns out that using incomplete mathematical models may lead to underestimated uncertainties calculated by the Fisher matrix approach and to substantially larger uncertainties for the Monte Carlo method.

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Mathematics Subject Classification: Primary: 34L25, 65T50; Secondary: 65C05.

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