Abstract

In optical scatterometry, the least squares (LSQ) function is usually used as the objective function to quantify the difference between the calculated and measured signatures, which is based on the belief that the actual measurement errors are normally distributed with zero mean. However, in practice the normal distribution assumption of measurement errors is oversimplified since these errors come from the superimposed effect of different error sources. Biased or inaccurate results may be induced when the traditional LSQ function based Gauss-Newton (GN) method is used in optical scatterometry. In this paper, we propose a robust method based on the principle of robust estimation to deal with the abnormal distributed errors. An additional robust regression procedure is used at the end of each iteration of the GN method to obtain the more accurate parameter departure vector. Simulations and experiments have demonstrated the feasibility of our proposed method.

© 2014 Optical Society of America

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References

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2014 (2)

2013 (5)

X. G. Chen, S. Y. Liu, C. W. Zhang, and H. Jiang, “Improved measurement accuracy in optical scatterometry using correction-based library search,” Appl. Opt. 52(27), 6726–6734 (2013).
[Crossref] [PubMed]

J. L. Zhu, S. Y. Liu, C. W. Zhang, X. G. Chen, and Z. Q. Dong, “Identification and reconstruction of diffraction structures in optical scatterometry using support vector machine method,” J. Micro/Nanolith. 12(1), 013004 (2013).
[Crossref]

X. G. Chen, S. Y. Liu, C. W. Zhang, and J. L. Zhu, “Improved measurement accuracy in optical scatterometry using fitting error interpolation based library search,” Measurement 46(8), 2638–2646 (2013).
[Crossref]

X. G. Chen, C. W. Zhang, and S. Y. Liu, “Depolarization effects from nanoimprinted grating structures as measured by Mueller matrix polarimetry,” Appl. Phys. Lett. 103(15), 151605 (2013).
[Crossref]

X. G. Chen, S. Y. Liu, C. W. Zhang, and H. Jiang, “Measurement configuration optimization for accurate grating reconstruction by Mueller matrix polarimetry,” J. Micro/Nanolith. 12(3), 033013 (2013).
[Crossref]

2012 (2)

S. Y. Liu, Y. Ma, X. G. Chen, and C. W. Zhang, “Estimation of the convergence order of rigorous coupled-wave analysis for binary gratings in optical critical dimension metrology,” Opt. Eng. 51(8), 081504 (2012).
[Crossref]

M. A. Henn, H. Gross, F. Scholze, M. Wurm, C. Elster, and M. Bär, “A maximum likelihood approach to the inverse problem of scatterometry,” Opt. Express 20(12), 12771–12786 (2012).
[Crossref] [PubMed]

2011 (1)

M. G. Faruk, S. Zangooie, M. Angyal, D. K. Watts, M. Sendelbach, L. Economikos, P. Herrera, and R. Wilkins, “Enabling scatterometry as an in-line measurement technique for 32nm BEOL application,” IEEE Trans. Semicond. Manuf. 24(4), 499–512 (2011).
[Crossref]

2010 (1)

2009 (3)

2008 (1)

H. J. Patrick, T. A. Gemer, Y. F. Ding, H. W. Ro, L. J. Richter, and C. L. Soles, “Scatterometry for in situ measurement of pattern flow in nanoimprinted polymers,” Appl. Phys. Lett. 93(23), 233105 (2008).
[Crossref]

2007 (1)

2006 (1)

2004 (1)

H. Huang and F. Terry., “Spectroscopic ellipsometry and reflectometry from gratings (scatterometry) for critical dimension measurement and in situ, real-time process monitoring,” Thin Solid Films 455-456, 828–836 (2004).
[Crossref]

2001 (1)

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14(2), 97–111 (2001).
[Crossref]

1998 (2)

C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83(6), 3323–3336 (1998).
[Crossref]

H. Ichikawa, “Electromagnetic analysis of diffraction gratings by the finite-difference time-domain method,” J. Opt. Soc. Am. A 15(1), 152–157 (1998).
[Crossref]

1997 (1)

C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. W. Hosch, and J. R. McNeil, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. 15(2), 361–368 (1997).
[Crossref]

1996 (2)

1995 (1)

1990 (1)

1989 (1)

H. Ekblom and K. Madsen, “Algorithms for non-linear Huber estimation,” BIT Numer. Math. 29(1), 60–76 (1989).
[Crossref]

1985 (1)

J. Hald and K. Madsen, “Combined LP and Quasi-Newton methods for nonlinear l1 optimization,” J. Numer. Anal. 22(1), 68–80 (1985).
[Crossref]

1979 (1)

R. A. El-Attar, M. Vidyasagar, and S. R. K. Dutta, “An algorithm for l1-norm minimization with application to nonlinear l1-approximation,” J. Numer. Anal. 16(1), 70–86 (1979).
[Crossref]

1974 (1)

A. E. Beaton and J. W. Tukey, “The fitting of power series, meaning polynomials, illustrated on band-spectroscopic data,” Technometrics 16(2), 147–185 (1974).
[Crossref]

1973 (1)

E. J. Schlossmacher, “An iterative technique for absolute deviations curve fitting,” J. Am. Stat. Assoc. 68(344), 857–859 (1973).
[Crossref]

1947 (1)

R. C. Geary, “Testing for normality,” Biometrika 34(3-4), 209–242 (1947).
[Crossref] [PubMed]

Al-Assaad, R. M.

Angyal, M.

M. G. Faruk, S. Zangooie, M. Angyal, D. K. Watts, M. Sendelbach, L. Economikos, P. Herrera, and R. Wilkins, “Enabling scatterometry as an in-line measurement technique for 32nm BEOL application,” IEEE Trans. Semicond. Manuf. 24(4), 499–512 (2011).
[Crossref]

Bao, J.

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14(2), 97–111 (2001).
[Crossref]

Bar, M.

H. Gross, A. Rathsfeld, F. Scholze, and M. Bar, “Profile reconstruction in extreme ultraviolet (EUV) scatterometry modeling and uncertainty estimates,” Meas. Sci. Technol. 20(10), 105102 (2009).
[Crossref]

Bär, M.

Beaton, A. E.

A. E. Beaton and J. W. Tukey, “The fitting of power series, meaning polynomials, illustrated on band-spectroscopic data,” Technometrics 16(2), 147–185 (1974).
[Crossref]

Ben Hatit, S.

Byrne, D. M.

Chen, X. G.

X. G. Chen, S. Y. Liu, C. W. Zhang, H. Jiang, Z. C. Ma, T. Y. Sun, and Z. M. Xu, “Accurate characterization of nanoimprinted resist patterns using Mueller matrix ellipsometry,” Opt. Express 22(12), 15165–15177 (2014).
[Crossref] [PubMed]

S. Y. Liu, X. G. Chen, and C. W. Zhang, “Mueller matrix polarimetry: A powerful tool for nanostructure metrology,” ECS Trans. 60(1), 237–242 (2014).
[Crossref]

X. G. Chen, S. Y. Liu, C. W. Zhang, and H. Jiang, “Measurement configuration optimization for accurate grating reconstruction by Mueller matrix polarimetry,” J. Micro/Nanolith. 12(3), 033013 (2013).
[Crossref]

X. G. Chen, C. W. Zhang, and S. Y. Liu, “Depolarization effects from nanoimprinted grating structures as measured by Mueller matrix polarimetry,” Appl. Phys. Lett. 103(15), 151605 (2013).
[Crossref]

X. G. Chen, S. Y. Liu, C. W. Zhang, and J. L. Zhu, “Improved measurement accuracy in optical scatterometry using fitting error interpolation based library search,” Measurement 46(8), 2638–2646 (2013).
[Crossref]

J. L. Zhu, S. Y. Liu, C. W. Zhang, X. G. Chen, and Z. Q. Dong, “Identification and reconstruction of diffraction structures in optical scatterometry using support vector machine method,” J. Micro/Nanolith. 12(1), 013004 (2013).
[Crossref]

X. G. Chen, S. Y. Liu, C. W. Zhang, and H. Jiang, “Improved measurement accuracy in optical scatterometry using correction-based library search,” Appl. Opt. 52(27), 6726–6734 (2013).
[Crossref] [PubMed]

S. Y. Liu, Y. Ma, X. G. Chen, and C. W. Zhang, “Estimation of the convergence order of rigorous coupled-wave analysis for binary gratings in optical critical dimension metrology,” Opt. Eng. 51(8), 081504 (2012).
[Crossref]

De Martino, A.

Ding, Y. F.

H. J. Patrick, T. A. Gemer, Y. F. Ding, H. W. Ro, L. J. Richter, and C. L. Soles, “Scatterometry for in situ measurement of pattern flow in nanoimprinted polymers,” Appl. Phys. Lett. 93(23), 233105 (2008).
[Crossref]

Dong, Z. Q.

J. L. Zhu, S. Y. Liu, C. W. Zhang, X. G. Chen, and Z. Q. Dong, “Identification and reconstruction of diffraction structures in optical scatterometry using support vector machine method,” J. Micro/Nanolith. 12(1), 013004 (2013).
[Crossref]

Drévillon, B.

Dutta, S. R. K.

R. A. El-Attar, M. Vidyasagar, and S. R. K. Dutta, “An algorithm for l1-norm minimization with application to nonlinear l1-approximation,” J. Numer. Anal. 16(1), 70–86 (1979).
[Crossref]

Economikos, L.

M. G. Faruk, S. Zangooie, M. Angyal, D. K. Watts, M. Sendelbach, L. Economikos, P. Herrera, and R. Wilkins, “Enabling scatterometry as an in-line measurement technique for 32nm BEOL application,” IEEE Trans. Semicond. Manuf. 24(4), 499–512 (2011).
[Crossref]

Ekblom, H.

H. Ekblom and K. Madsen, “Algorithms for non-linear Huber estimation,” BIT Numer. Math. 29(1), 60–76 (1989).
[Crossref]

El-Attar, R. A.

R. A. El-Attar, M. Vidyasagar, and S. R. K. Dutta, “An algorithm for l1-norm minimization with application to nonlinear l1-approximation,” J. Numer. Anal. 16(1), 70–86 (1979).
[Crossref]

Elster, C.

Faruk, M. G.

M. G. Faruk, S. Zangooie, M. Angyal, D. K. Watts, M. Sendelbach, L. Economikos, P. Herrera, and R. Wilkins, “Enabling scatterometry as an in-line measurement technique for 32nm BEOL application,” IEEE Trans. Semicond. Manuf. 24(4), 499–512 (2011).
[Crossref]

Gaylord, T. K.

Geary, R. C.

R. C. Geary, “Testing for normality,” Biometrika 34(3-4), 209–242 (1947).
[Crossref] [PubMed]

Gemer, T. A.

H. J. Patrick, T. A. Gemer, Y. F. Ding, H. W. Ro, L. J. Richter, and C. L. Soles, “Scatterometry for in situ measurement of pattern flow in nanoimprinted polymers,” Appl. Phys. Lett. 93(23), 233105 (2008).
[Crossref]

Grann, E. B.

Gross, H.

M. A. Henn, H. Gross, F. Scholze, M. Wurm, C. Elster, and M. Bär, “A maximum likelihood approach to the inverse problem of scatterometry,” Opt. Express 20(12), 12771–12786 (2012).
[Crossref] [PubMed]

H. Gross, A. Rathsfeld, F. Scholze, and M. Bar, “Profile reconstruction in extreme ultraviolet (EUV) scatterometry modeling and uncertainty estimates,” Meas. Sci. Technol. 20(10), 105102 (2009).
[Crossref]

Hald, J.

J. Hald and K. Madsen, “Combined LP and Quasi-Newton methods for nonlinear l1 optimization,” J. Numer. Anal. 22(1), 68–80 (1985).
[Crossref]

Hao, Y. D.

Henn, M. A.

Herrera, P.

M. G. Faruk, S. Zangooie, M. Angyal, D. K. Watts, M. Sendelbach, L. Economikos, P. Herrera, and R. Wilkins, “Enabling scatterometry as an in-line measurement technique for 32nm BEOL application,” IEEE Trans. Semicond. Manuf. 24(4), 499–512 (2011).
[Crossref]

Herzinger, C. M.

C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83(6), 3323–3336 (1998).
[Crossref]

Hosch, J. W.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. W. Hosch, and J. R. McNeil, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. 15(2), 361–368 (1997).
[Crossref]

Hu, J. T.

Huang, H.

H. Huang and F. Terry., “Spectroscopic ellipsometry and reflectometry from gratings (scatterometry) for critical dimension measurement and in situ, real-time process monitoring,” Thin Solid Films 455-456, 828–836 (2004).
[Crossref]

Ichikawa, H.

Jakatdar, N.

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14(2), 97–111 (2001).
[Crossref]

Jiang, H.

Johs, B.

C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83(6), 3323–3336 (1998).
[Crossref]

Kato, A.

Kim, Y. N.

Koshiba, M.

Lee, S.

Li, L.

Liu, S. Y.

X. G. Chen, S. Y. Liu, C. W. Zhang, H. Jiang, Z. C. Ma, T. Y. Sun, and Z. M. Xu, “Accurate characterization of nanoimprinted resist patterns using Mueller matrix ellipsometry,” Opt. Express 22(12), 15165–15177 (2014).
[Crossref] [PubMed]

S. Y. Liu, X. G. Chen, and C. W. Zhang, “Mueller matrix polarimetry: A powerful tool for nanostructure metrology,” ECS Trans. 60(1), 237–242 (2014).
[Crossref]

X. G. Chen, C. W. Zhang, and S. Y. Liu, “Depolarization effects from nanoimprinted grating structures as measured by Mueller matrix polarimetry,” Appl. Phys. Lett. 103(15), 151605 (2013).
[Crossref]

X. G. Chen, S. Y. Liu, C. W. Zhang, and H. Jiang, “Measurement configuration optimization for accurate grating reconstruction by Mueller matrix polarimetry,” J. Micro/Nanolith. 12(3), 033013 (2013).
[Crossref]

X. G. Chen, S. Y. Liu, C. W. Zhang, and J. L. Zhu, “Improved measurement accuracy in optical scatterometry using fitting error interpolation based library search,” Measurement 46(8), 2638–2646 (2013).
[Crossref]

J. L. Zhu, S. Y. Liu, C. W. Zhang, X. G. Chen, and Z. Q. Dong, “Identification and reconstruction of diffraction structures in optical scatterometry using support vector machine method,” J. Micro/Nanolith. 12(1), 013004 (2013).
[Crossref]

X. G. Chen, S. Y. Liu, C. W. Zhang, and H. Jiang, “Improved measurement accuracy in optical scatterometry using correction-based library search,” Appl. Opt. 52(27), 6726–6734 (2013).
[Crossref] [PubMed]

S. Y. Liu, Y. Ma, X. G. Chen, and C. W. Zhang, “Estimation of the convergence order of rigorous coupled-wave analysis for binary gratings in optical critical dimension metrology,” Opt. Eng. 51(8), 081504 (2012).
[Crossref]

C. W. Zhang, S. Y. Liu, T. L. Shi, and Z. R. Tang, “Improved model-based infrared reflectrometry for measuring deep trench structures,” J. Opt. Soc. Am. A 26(11), 2327–2335 (2009).
[Crossref] [PubMed]

Liu, Z.

Ma, Y.

S. Y. Liu, Y. Ma, X. G. Chen, and C. W. Zhang, “Estimation of the convergence order of rigorous coupled-wave analysis for binary gratings in optical critical dimension metrology,” Opt. Eng. 51(8), 081504 (2012).
[Crossref]

Ma, Z. C.

Madsen, K.

H. Ekblom and K. Madsen, “Algorithms for non-linear Huber estimation,” BIT Numer. Math. 29(1), 60–76 (1989).
[Crossref]

J. Hald and K. Madsen, “Combined LP and Quasi-Newton methods for nonlinear l1 optimization,” J. Numer. Anal. 22(1), 68–80 (1985).
[Crossref]

McGahan, W.

McGahan, W. A.

C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83(6), 3323–3336 (1998).
[Crossref]

McNeil, J. R.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. W. Hosch, and J. R. McNeil, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. 15(2), 361–368 (1997).
[Crossref]

Moharam, M. G.

Murnane, M. R.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. W. Hosch, and J. R. McNeil, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. 15(2), 361–368 (1997).
[Crossref]

Nakata, Y.

Naqvi, S. S. H.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. W. Hosch, and J. R. McNeil, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. 15(2), 361–368 (1997).
[Crossref]

Niu, X.

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14(2), 97–111 (2001).
[Crossref]

Novikova, T.

Ohkawa, Y.

Paek, J. S.

Patrick, H. J.

H. J. Patrick, T. A. Gemer, Y. F. Ding, H. W. Ro, L. J. Richter, and C. L. Soles, “Scatterometry for in situ measurement of pattern flow in nanoimprinted polymers,” Appl. Phys. Lett. 93(23), 233105 (2008).
[Crossref]

Paulson, W.

C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83(6), 3323–3336 (1998).
[Crossref]

Pommet, D. A.

Prins, S. L.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. W. Hosch, and J. R. McNeil, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. 15(2), 361–368 (1997).
[Crossref]

Rabello, S.

Rathsfeld, A.

H. Gross, A. Rathsfeld, F. Scholze, and M. Bar, “Profile reconstruction in extreme ultraviolet (EUV) scatterometry modeling and uncertainty estimates,” Meas. Sci. Technol. 20(10), 105102 (2009).
[Crossref]

Raymond, C. J.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. W. Hosch, and J. R. McNeil, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. 15(2), 361–368 (1997).
[Crossref]

Richter, L. J.

H. J. Patrick, T. A. Gemer, Y. F. Ding, H. W. Ro, L. J. Richter, and C. L. Soles, “Scatterometry for in situ measurement of pattern flow in nanoimprinted polymers,” Appl. Phys. Lett. 93(23), 233105 (2008).
[Crossref]

Ro, H. W.

H. J. Patrick, T. A. Gemer, Y. F. Ding, H. W. Ro, L. J. Richter, and C. L. Soles, “Scatterometry for in situ measurement of pattern flow in nanoimprinted polymers,” Appl. Phys. Lett. 93(23), 233105 (2008).
[Crossref]

Schlossmacher, E. J.

E. J. Schlossmacher, “An iterative technique for absolute deviations curve fitting,” J. Am. Stat. Assoc. 68(344), 857–859 (1973).
[Crossref]

Scholze, F.

Sendelbach, M.

M. G. Faruk, S. Zangooie, M. Angyal, D. K. Watts, M. Sendelbach, L. Economikos, P. Herrera, and R. Wilkins, “Enabling scatterometry as an in-line measurement technique for 32nm BEOL application,” IEEE Trans. Semicond. Manuf. 24(4), 499–512 (2011).
[Crossref]

Shi, T. L.

Soles, C. L.

H. J. Patrick, T. A. Gemer, Y. F. Ding, H. W. Ro, L. J. Richter, and C. L. Soles, “Scatterometry for in situ measurement of pattern flow in nanoimprinted polymers,” Appl. Phys. Lett. 93(23), 233105 (2008).
[Crossref]

Spanos, C. J.

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14(2), 97–111 (2001).
[Crossref]

Sun, T. Y.

Tang, Z. R.

Terry, F.

H. Huang and F. Terry., “Spectroscopic ellipsometry and reflectometry from gratings (scatterometry) for critical dimension measurement and in situ, real-time process monitoring,” Thin Solid Films 455-456, 828–836 (2004).
[Crossref]

Tsuji, Y.

Tukey, J. W.

A. E. Beaton and J. W. Tukey, “The fitting of power series, meaning polynomials, illustrated on band-spectroscopic data,” Technometrics 16(2), 147–185 (1974).
[Crossref]

Vidyasagar, M.

R. A. El-Attar, M. Vidyasagar, and S. R. K. Dutta, “An algorithm for l1-norm minimization with application to nonlinear l1-approximation,” J. Numer. Anal. 16(1), 70–86 (1979).
[Crossref]

Watts, D. K.

M. G. Faruk, S. Zangooie, M. Angyal, D. K. Watts, M. Sendelbach, L. Economikos, P. Herrera, and R. Wilkins, “Enabling scatterometry as an in-line measurement technique for 32nm BEOL application,” IEEE Trans. Semicond. Manuf. 24(4), 499–512 (2011).
[Crossref]

Wilkins, R.

M. G. Faruk, S. Zangooie, M. Angyal, D. K. Watts, M. Sendelbach, L. Economikos, P. Herrera, and R. Wilkins, “Enabling scatterometry as an in-line measurement technique for 32nm BEOL application,” IEEE Trans. Semicond. Manuf. 24(4), 499–512 (2011).
[Crossref]

Woollam, J. A.

C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83(6), 3323–3336 (1998).
[Crossref]

Wurm, M.

Xu, Z. M.

Zangooie, S.

M. G. Faruk, S. Zangooie, M. Angyal, D. K. Watts, M. Sendelbach, L. Economikos, P. Herrera, and R. Wilkins, “Enabling scatterometry as an in-line measurement technique for 32nm BEOL application,” IEEE Trans. Semicond. Manuf. 24(4), 499–512 (2011).
[Crossref]

Zhang, C. W.

S. Y. Liu, X. G. Chen, and C. W. Zhang, “Mueller matrix polarimetry: A powerful tool for nanostructure metrology,” ECS Trans. 60(1), 237–242 (2014).
[Crossref]

X. G. Chen, S. Y. Liu, C. W. Zhang, H. Jiang, Z. C. Ma, T. Y. Sun, and Z. M. Xu, “Accurate characterization of nanoimprinted resist patterns using Mueller matrix ellipsometry,” Opt. Express 22(12), 15165–15177 (2014).
[Crossref] [PubMed]

X. G. Chen, S. Y. Liu, C. W. Zhang, and H. Jiang, “Improved measurement accuracy in optical scatterometry using correction-based library search,” Appl. Opt. 52(27), 6726–6734 (2013).
[Crossref] [PubMed]

X. G. Chen, S. Y. Liu, C. W. Zhang, and H. Jiang, “Measurement configuration optimization for accurate grating reconstruction by Mueller matrix polarimetry,” J. Micro/Nanolith. 12(3), 033013 (2013).
[Crossref]

X. G. Chen, C. W. Zhang, and S. Y. Liu, “Depolarization effects from nanoimprinted grating structures as measured by Mueller matrix polarimetry,” Appl. Phys. Lett. 103(15), 151605 (2013).
[Crossref]

X. G. Chen, S. Y. Liu, C. W. Zhang, and J. L. Zhu, “Improved measurement accuracy in optical scatterometry using fitting error interpolation based library search,” Measurement 46(8), 2638–2646 (2013).
[Crossref]

J. L. Zhu, S. Y. Liu, C. W. Zhang, X. G. Chen, and Z. Q. Dong, “Identification and reconstruction of diffraction structures in optical scatterometry using support vector machine method,” J. Micro/Nanolith. 12(1), 013004 (2013).
[Crossref]

S. Y. Liu, Y. Ma, X. G. Chen, and C. W. Zhang, “Estimation of the convergence order of rigorous coupled-wave analysis for binary gratings in optical critical dimension metrology,” Opt. Eng. 51(8), 081504 (2012).
[Crossref]

C. W. Zhang, S. Y. Liu, T. L. Shi, and Z. R. Tang, “Improved model-based infrared reflectrometry for measuring deep trench structures,” J. Opt. Soc. Am. A 26(11), 2327–2335 (2009).
[Crossref] [PubMed]

Zhu, J. L.

J. L. Zhu, S. Y. Liu, C. W. Zhang, X. G. Chen, and Z. Q. Dong, “Identification and reconstruction of diffraction structures in optical scatterometry using support vector machine method,” J. Micro/Nanolith. 12(1), 013004 (2013).
[Crossref]

X. G. Chen, S. Y. Liu, C. W. Zhang, and J. L. Zhu, “Improved measurement accuracy in optical scatterometry using fitting error interpolation based library search,” Measurement 46(8), 2638–2646 (2013).
[Crossref]

Appl. Opt. (3)

Appl. Phys. Lett. (2)

X. G. Chen, C. W. Zhang, and S. Y. Liu, “Depolarization effects from nanoimprinted grating structures as measured by Mueller matrix polarimetry,” Appl. Phys. Lett. 103(15), 151605 (2013).
[Crossref]

H. J. Patrick, T. A. Gemer, Y. F. Ding, H. W. Ro, L. J. Richter, and C. L. Soles, “Scatterometry for in situ measurement of pattern flow in nanoimprinted polymers,” Appl. Phys. Lett. 93(23), 233105 (2008).
[Crossref]

Biometrika (1)

R. C. Geary, “Testing for normality,” Biometrika 34(3-4), 209–242 (1947).
[Crossref] [PubMed]

BIT Numer. Math. (1)

H. Ekblom and K. Madsen, “Algorithms for non-linear Huber estimation,” BIT Numer. Math. 29(1), 60–76 (1989).
[Crossref]

ECS Trans. (1)

S. Y. Liu, X. G. Chen, and C. W. Zhang, “Mueller matrix polarimetry: A powerful tool for nanostructure metrology,” ECS Trans. 60(1), 237–242 (2014).
[Crossref]

IEEE Trans. Semicond. Manuf. (2)

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14(2), 97–111 (2001).
[Crossref]

M. G. Faruk, S. Zangooie, M. Angyal, D. K. Watts, M. Sendelbach, L. Economikos, P. Herrera, and R. Wilkins, “Enabling scatterometry as an in-line measurement technique for 32nm BEOL application,” IEEE Trans. Semicond. Manuf. 24(4), 499–512 (2011).
[Crossref]

J. Am. Stat. Assoc. (1)

E. J. Schlossmacher, “An iterative technique for absolute deviations curve fitting,” J. Am. Stat. Assoc. 68(344), 857–859 (1973).
[Crossref]

J. Appl. Phys. (1)

C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83(6), 3323–3336 (1998).
[Crossref]

J. Micro/Nanolith. (2)

X. G. Chen, S. Y. Liu, C. W. Zhang, and H. Jiang, “Measurement configuration optimization for accurate grating reconstruction by Mueller matrix polarimetry,” J. Micro/Nanolith. 12(3), 033013 (2013).
[Crossref]

J. L. Zhu, S. Y. Liu, C. W. Zhang, X. G. Chen, and Z. Q. Dong, “Identification and reconstruction of diffraction structures in optical scatterometry using support vector machine method,” J. Micro/Nanolith. 12(1), 013004 (2013).
[Crossref]

J. Numer. Anal. (2)

R. A. El-Attar, M. Vidyasagar, and S. R. K. Dutta, “An algorithm for l1-norm minimization with application to nonlinear l1-approximation,” J. Numer. Anal. 16(1), 70–86 (1979).
[Crossref]

J. Hald and K. Madsen, “Combined LP and Quasi-Newton methods for nonlinear l1 optimization,” J. Numer. Anal. 22(1), 68–80 (1985).
[Crossref]

J. Opt. Soc. Am. A (7)

J. Vac. Sci. Technol. (1)

C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. W. Hosch, and J. R. McNeil, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. 15(2), 361–368 (1997).
[Crossref]

Meas. Sci. Technol. (1)

H. Gross, A. Rathsfeld, F. Scholze, and M. Bar, “Profile reconstruction in extreme ultraviolet (EUV) scatterometry modeling and uncertainty estimates,” Meas. Sci. Technol. 20(10), 105102 (2009).
[Crossref]

Measurement (1)

X. G. Chen, S. Y. Liu, C. W. Zhang, and J. L. Zhu, “Improved measurement accuracy in optical scatterometry using fitting error interpolation based library search,” Measurement 46(8), 2638–2646 (2013).
[Crossref]

Opt. Eng. (1)

S. Y. Liu, Y. Ma, X. G. Chen, and C. W. Zhang, “Estimation of the convergence order of rigorous coupled-wave analysis for binary gratings in optical critical dimension metrology,” Opt. Eng. 51(8), 081504 (2012).
[Crossref]

Opt. Express (3)

Technometrics (1)

A. E. Beaton and J. W. Tukey, “The fitting of power series, meaning polynomials, illustrated on band-spectroscopic data,” Technometrics 16(2), 147–185 (1974).
[Crossref]

Thin Solid Films (1)

H. Huang and F. Terry., “Spectroscopic ellipsometry and reflectometry from gratings (scatterometry) for critical dimension measurement and in situ, real-time process monitoring,” Thin Solid Films 455-456, 828–836 (2004).
[Crossref]

Other (3)

D. Andrews, F. Bickel, F. Hampel, P. Huber, W. Rogers, and J. Tukey, Robust Estimation of Location: Survey and Advances, 1st ed. (Princeton University, 1972).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes – the art of science computing, 3rd ed. (Cambridge University, 2007).

R. C. Aster, B. Borchers, and C. H. Thurber, Parameter Estimation and Inverse Problems, 1st ed. (Academic, 2005).

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Figures (7)

Fig. 1
Fig. 1 (a) Measurement setup of the dual-rotating-compensator Mueller matrix ellipsometer, and (b) cross-section SEM image of the one-dimensional trapezoidal etched silicon grating.
Fig. 2
Fig. 2 Iteration results of (a) TCD, (b) Hgt, and (c) BCD. The black dash dotted line, blue solid line with squares, and red solid line with circles represent the true values, the extracted values by our proposed method, and those by the traditional GN method respectively.
Fig. 3
Fig. 3 Weighted “measurement” errors are represented by blue squares. The red and black dotted lines in each sub-figure represent −3 and 3 respectively.
Fig. 4
Fig. 4 Numerically extracted TCDs, Hgts and BCDs by varying (a) the incident angle and (b) the azimuthal angle. Black squares and triangles represent the values obtained by the LSQ-based GN method and our proposed method respectively. Black dash dotted lines represent the true values.
Fig. 5
Fig. 5 Fitting differences between the measured and the best calculated Mueller matrices, and the standard deviations of the measurement errors used in the LSQ function. The best calculated Mueller matrix is obtained by the GN method.
Fig. 6
Fig. 6 Weighted fitting differences of Mueller matrix elements. The blue squares represent the weighted fitting differences at different wavelengths, and the red and black dotted lines in each sub-figure represent −3 and 3 respectively.
Fig. 7
Fig. 7 Experimentally extracted TCDs, Hgts and BCDs by varying (a) the incident angle and (b) the azimuthal angle. Black squares and triangles represent the values obtained by the LSQ-based GN method and our proposed method respectively. Black dash dotted lines represent the SEM measured values.

Tables (1)

Tables Icon

Table 1 Mean values of TCD, Hgt, and BCD obtained by different methods.

Equations (18)

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F(x)= j=1 m w j [ y j f j (x)] 2 = [yf(x)] T w[yf(x)] .
y=f(x)+ε,
x ^ = arg min xΩ {[yf(x )] T w[yf(x)]},
F( x (i+1) )=F( x (i) +Δ x (i) )=F( x (i) )+ 2 F( x (i) )Δ x (i) ,
F( x (i) )=2J ( x (i) ) T wΔ y (i) ,
Δ y (i) =yf( x (i) ).
2 F( x (i) )2J ( x (i) ) T wJ( x (i) ).
J ( x (i) ) T wJ( x (i) )Δ x (i) =J ( x (i) ) T wΔ y (i) .
Δ x (i) = [J ( x (i) ) T wJ( x (i) )] -1 J ( x (i) ) T wΔ y (i) ,
y=f(x)+ε+μ,
Δ y (i) = {[J ( x (i) ) T w ] T J ( x (i) ) T w} -1 [J ( x (i) ) T w] T J ( x (i) ) T wJ( x (i) )Δ x (i) =J( x (i) )Δ x (i) .
Δ x ^ *(i) = arg min Δ x *(i) Ω * { j=1 m ρ[Δ y j (i) +J ( x (i) ) j Δ x *(i) ] }= arg min Δ x *(i) Ω * { j=1 m ρ[ r j (i) ] },
Δ x ^ *(i) = arg min Δ x ˜ Ω * [ j=1 m ρ(Δ y ˜ j + J ˜ j Δ x ˜ ) ]= arg min Δ x ˜ Ω * [ j=1 m ρ( r ˜ j ) ].
Δ x ˜ (k+1) =Δ x ˜ (k) ( J T Q (k) J) -1 J T P (k) .
Q jj =ρ''[ r ˜ j (k) ],
Δ x ˜ (k+1) =Δ x ˜ (k) ( J T S (k) J) -1 J T S (k) (Δ y ˜ +JΔ x ˜ (k) ),
S jj =ω( r ˜ j (k) )=ω(Δ y ˜ j + J j Δ x ˜ (k) ).
ω(p)={ c A p sinp c A ,| p |π c A 0,| p |>π c A ,

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