Abstract

Library search is one of the most commonly used methods for solving the inverse problem in optical scatterometry. The final measurement accuracy of the conventional library search method highly depends on the grid interval selected for each parameter in the signature library, and the time cost of the parameter extraction increases dramatically when the grid interval is decreasing. In this paper, we propose a correction-based library search method to improve the measurement accuracy for a pregenerated signature library. We derive a formulation to estimate the error between the expected solution of the inverse problem and the actually searched solution obtained by the conventional library search method. Then we use the estimate of the error as a correction term to correct the actually searched solution to improve the measurement accuracy. Experiments performed on a photoresist grating have demonstrated that the proposed correction-based library search method can achieve much more accurate measurement with negligible computational penalty to the conventional library search method in the parameter extraction. It has also been observed that the correction-based library search method has higher measurement accuracy and less time cost than the interpolation-based library search method. The proposed correction-based library search method is expected to provide a more practical means to solve the inverse problem in state-of-the-art optical scatterometry.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  15. 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, 2327–2335 (2009).
    [CrossRef]
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    [CrossRef]
  19. P. Thony, D. Herisson, D. Henry, E. Severgnini, and M. Vasconi, “Review of CD measurement and scatterometry,” AIP Conf. Proc. 683, 381–388 (2003).
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    [CrossRef]
  27. T. A. Germer, H. J. Patrick, R. M. Silver, and B. Bunday, “Developing an uncertainty analysis for optical scatterometry,” Proc. SPIE 7272, 72720T (2009).
    [CrossRef]
  28. N. F. Zhang, R. M. Silver, H. Zhou, and B. M. Barnes, “Improving optical measurement uncertainty with combined multitool metrology using a Bayesian approach,” Appl. Opt. 51, 6196–6206 (2012).
    [CrossRef]
  29. 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, 3323–3336 (1998).
    [CrossRef]
  30. G. E. Jellison and F. A. Modine, “Parameterization of the optical functions of amorphous materials in the interband region,” Appl. Phys. Lett. 69, 371–373 (1996).
    [CrossRef]

2013

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, 2638–2646 (2013).
[CrossRef]

2012

N. F. Zhang, R. M. Silver, H. Zhou, and B. M. Barnes, “Improving optical measurement uncertainty with combined multitool metrology using a Bayesian approach,” Appl. Opt. 51, 6196–6206 (2012).
[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, 081504 (2012).
[CrossRef]

2010

J. Pomplun and F. Schmidt, “Accelerated a posterior error estimation for the reduced basis method with application to 3D electromagnetic scattering problems,” SIAM J. Sci. Comput. 32, 498–520 (2010).
[CrossRef]

J. Li, J. J. Hwu, Y. Liu, S. Rabello, Z. Liu, and J. Hu, “Mueller matrix measurement of asymmetric gratings,” J. Micro/Nanolith. MEMS MOEMS 9, 041305 (2010).

2009

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, 2327–2335 (2009).
[CrossRef]

T. A. Germer, H. J. Patrick, R. M. Silver, and B. Bunday, “Developing an uncertainty analysis for optical scatterometry,” Proc. SPIE 7272, 72720T (2009).
[CrossRef]

2008

H. J. Patrick, R. Attota, B. M. Barnes, T. A. Germer, R. G. Dixson, M. T. Stocker, R. M. Silver, and M. R. Bishop, “Optical critical dimension measurement of silicon grating targets using focal plane scatterfield microscopy,” J. Micro/Nanolith. MEMS MOEMS 7, 013012 (2008).

2006

2005

C. Raymond, “Overview of scatterometry applications in high volume silicon manufacturing,” AIP Conf. Proc. 788, 394–402 (2005).
[CrossRef]

2004

C. J. Raymond, M. Littau, A. Chuprin, and S. Ward, “Comparison of solutions to the scatterometry inverse problems,” Proc. SPIE 5375, 564–575 (2004).
[CrossRef]

2003

P. Thony, D. Herisson, D. Henry, E. Severgnini, and M. Vasconi, “Review of CD measurement and scatterometry,” AIP Conf. Proc. 683, 381–388 (2003).
[CrossRef]

2002

E. Drége, J. Reed, and D. Byrne, “Linearized inversion of scatterometric data to obtain surface profile information,” Opt. Eng. 41, 225–236 (2002).
[CrossRef]

2001

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

H. T. Huang, W. Kong, and F. L. Terry, “Normal-incidence spectroscopic ellipsometry for critical dimension monitoring,” Appl. Phys. Lett. 78, 2985–3983 (2001).
[CrossRef]

1998

B. K. Minhas, S. A. Coulombe, S. Sohail, H. Naqvi, and J. R. McNeil, “Ellipsometric scatterometry for metrology of sub-0.1 μm linewidth structures,” Appl. Opt. 37, 5112–5115 (1998).
[CrossRef]

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

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, 3323–3336 (1998).
[CrossRef]

1996

G. E. Jellison and F. A. Modine, “Parameterization of the optical functions of amorphous materials in the interband region,” Appl. Phys. Lett. 69, 371–373 (1996).
[CrossRef]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[CrossRef]

1995

1990

1977

D. T. Lee and C. K. Wong, “Worst-case analysis for region and partial region searches in multidimensional binary search trees and balanced quad trees,” Acta Inform. 9, 23–29 (1977).
[CrossRef]

1975

J. L. Bentley, “Multidimensional binary search trees used for associative searching,” Commun. ACM 18, 509–517 (1975).
[CrossRef]

Attota, R.

H. J. Patrick, R. Attota, B. M. Barnes, T. A. Germer, R. G. Dixson, M. T. Stocker, R. M. Silver, and M. R. Bishop, “Optical critical dimension measurement of silicon grating targets using focal plane scatterfield microscopy,” J. Micro/Nanolith. MEMS MOEMS 7, 013012 (2008).

Bao, J.

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

Bär, M.

H. Gross, R. Model, M. Bär, M. Wurm, B. Bodermann, and A. Rathsfeld, “Mathematical modelling of indirect measurements in scatterometry,” Measurement 39, 782–794 (2006).
[CrossRef]

Barnes, B. M.

N. F. Zhang, R. M. Silver, H. Zhou, and B. M. Barnes, “Improving optical measurement uncertainty with combined multitool metrology using a Bayesian approach,” Appl. Opt. 51, 6196–6206 (2012).
[CrossRef]

H. J. Patrick, R. Attota, B. M. Barnes, T. A. Germer, R. G. Dixson, M. T. Stocker, R. M. Silver, and M. R. Bishop, “Optical critical dimension measurement of silicon grating targets using focal plane scatterfield microscopy,” J. Micro/Nanolith. MEMS MOEMS 7, 013012 (2008).

Bentley, J. L.

J. L. Bentley, “Multidimensional binary search trees used for associative searching,” Commun. ACM 18, 509–517 (1975).
[CrossRef]

Bishop, M. R.

H. J. Patrick, R. Attota, B. M. Barnes, T. A. Germer, R. G. Dixson, M. T. Stocker, R. M. Silver, and M. R. Bishop, “Optical critical dimension measurement of silicon grating targets using focal plane scatterfield microscopy,” J. Micro/Nanolith. MEMS MOEMS 7, 013012 (2008).

Bodermann, B.

H. Gross, R. Model, M. Bär, M. Wurm, B. Bodermann, and A. Rathsfeld, “Mathematical modelling of indirect measurements in scatterometry,” Measurement 39, 782–794 (2006).
[CrossRef]

Bruce, J.

M. Littau, D. Forman, J. Bruce, C. J. Raymond, and S. G. Hummel, “Diffraction signature analysis methods for improving scatterometry precision,” Proc. SPIE 6152, 615236 (2006).
[CrossRef]

Bunday, B.

T. A. Germer, H. J. Patrick, R. M. Silver, and B. Bunday, “Developing an uncertainty analysis for optical scatterometry,” Proc. SPIE 7272, 72720T (2009).
[CrossRef]

Byrne, D.

E. Drége, J. Reed, and D. Byrne, “Linearized inversion of scatterometric data to obtain surface profile information,” Opt. Eng. 41, 225–236 (2002).
[CrossRef]

Chen, X. G.

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, 2638–2646 (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, 081504 (2012).
[CrossRef]

Chuprin, A.

C. J. Raymond, M. Littau, A. Chuprin, and S. Ward, “Comparison of solutions to the scatterometry inverse problems,” Proc. SPIE 5375, 564–575 (2004).
[CrossRef]

Coulombe, S. A.

De Martino, A.

Dixson, R. G.

H. J. Patrick, R. Attota, B. M. Barnes, T. A. Germer, R. G. Dixson, M. T. Stocker, R. M. Silver, and M. R. Bishop, “Optical critical dimension measurement of silicon grating targets using focal plane scatterfield microscopy,” J. Micro/Nanolith. MEMS MOEMS 7, 013012 (2008).

Drége, E.

E. Drége, J. Reed, and D. Byrne, “Linearized inversion of scatterometric data to obtain surface profile information,” Opt. Eng. 41, 225–236 (2002).
[CrossRef]

Drévillon, B.

Forman, D.

M. Littau, D. Forman, J. Bruce, C. J. Raymond, and S. G. Hummel, “Diffraction signature analysis methods for improving scatterometry precision,” Proc. SPIE 6152, 615236 (2006).
[CrossRef]

Gaylord, T. K.

Germer, T. A.

T. A. Germer, H. J. Patrick, R. M. Silver, and B. Bunday, “Developing an uncertainty analysis for optical scatterometry,” Proc. SPIE 7272, 72720T (2009).
[CrossRef]

H. J. Patrick, R. Attota, B. M. Barnes, T. A. Germer, R. G. Dixson, M. T. Stocker, R. M. Silver, and M. R. Bishop, “Optical critical dimension measurement of silicon grating targets using focal plane scatterfield microscopy,” J. Micro/Nanolith. MEMS MOEMS 7, 013012 (2008).

Gionis, A.

A. Gionis, P. Indyk, and P. Motwani, “Similarity search in high dimensions via hashing,” Proc. VLDB, 518–529 (1999).

Grann, E. B.

Gross, H.

H. Gross, R. Model, M. Bär, M. Wurm, B. Bodermann, and A. Rathsfeld, “Mathematical modelling of indirect measurements in scatterometry,” Measurement 39, 782–794 (2006).
[CrossRef]

Hatit, S. B.

Henry, D.

P. Thony, D. Herisson, D. Henry, E. Severgnini, and M. Vasconi, “Review of CD measurement and scatterometry,” AIP Conf. Proc. 683, 381–388 (2003).
[CrossRef]

Herisson, D.

P. Thony, D. Herisson, D. Henry, E. Severgnini, and M. Vasconi, “Review of CD measurement and scatterometry,” AIP Conf. Proc. 683, 381–388 (2003).
[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, 3323–3336 (1998).
[CrossRef]

Hu, J.

J. Li, J. J. Hwu, Y. Liu, S. Rabello, Z. Liu, and J. Hu, “Mueller matrix measurement of asymmetric gratings,” J. Micro/Nanolith. MEMS MOEMS 9, 041305 (2010).

Huang, H. T.

H. T. Huang, W. Kong, and F. L. Terry, “Normal-incidence spectroscopic ellipsometry for critical dimension monitoring,” Appl. Phys. Lett. 78, 2985–3983 (2001).
[CrossRef]

Hummel, S. G.

M. Littau, D. Forman, J. Bruce, C. J. Raymond, and S. G. Hummel, “Diffraction signature analysis methods for improving scatterometry precision,” Proc. SPIE 6152, 615236 (2006).
[CrossRef]

Hwu, J. J.

J. Li, J. J. Hwu, Y. Liu, S. Rabello, Z. Liu, and J. Hu, “Mueller matrix measurement of asymmetric gratings,” J. Micro/Nanolith. MEMS MOEMS 9, 041305 (2010).

Ichikawa, H.

Indyk, P.

A. Gionis, P. Indyk, and P. Motwani, “Similarity search in high dimensions via hashing,” Proc. VLDB, 518–529 (1999).

Jakatdar, N.

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

Jellison, G. E.

G. E. Jellison and F. A. Modine, “Parameterization of the optical functions of amorphous materials in the interband region,” Appl. Phys. Lett. 69, 371–373 (1996).
[CrossRef]

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, 3323–3336 (1998).
[CrossRef]

Kashiba, M.

Ko, C. H.

Kong, W.

H. T. Huang, W. Kong, and F. L. Terry, “Normal-incidence spectroscopic ellipsometry for critical dimension monitoring,” Appl. Phys. Lett. 78, 2985–3983 (2001).
[CrossRef]

Ku, Y. S.

Lee, D. T.

D. T. Lee and C. K. Wong, “Worst-case analysis for region and partial region searches in multidimensional binary search trees and balanced quad trees,” Acta Inform. 9, 23–29 (1977).
[CrossRef]

Li, J.

J. Li, J. J. Hwu, Y. Liu, S. Rabello, Z. Liu, and J. Hu, “Mueller matrix measurement of asymmetric gratings,” J. Micro/Nanolith. MEMS MOEMS 9, 041305 (2010).

Li, L.

Littau, M.

M. Littau, D. Forman, J. Bruce, C. J. Raymond, and S. G. Hummel, “Diffraction signature analysis methods for improving scatterometry precision,” Proc. SPIE 6152, 615236 (2006).
[CrossRef]

C. J. Raymond, M. Littau, A. Chuprin, and S. Ward, “Comparison of solutions to the scatterometry inverse problems,” Proc. SPIE 5375, 564–575 (2004).
[CrossRef]

Liu, S. Y.

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, 2638–2646 (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, 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, 2327–2335 (2009).
[CrossRef]

Liu, Y.

J. Li, J. J. Hwu, Y. Liu, S. Rabello, Z. Liu, and J. Hu, “Mueller matrix measurement of asymmetric gratings,” J. Micro/Nanolith. MEMS MOEMS 9, 041305 (2010).

Liu, Z.

J. Li, J. J. Hwu, Y. Liu, S. Rabello, Z. Liu, and J. Hu, “Mueller matrix measurement of asymmetric gratings,” J. Micro/Nanolith. MEMS MOEMS 9, 041305 (2010).

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, 081504 (2012).
[CrossRef]

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, 3323–3336 (1998).
[CrossRef]

McNeil, J. R.

Minhas, B. K.

Model, R.

H. Gross, R. Model, M. Bär, M. Wurm, B. Bodermann, and A. Rathsfeld, “Mathematical modelling of indirect measurements in scatterometry,” Measurement 39, 782–794 (2006).
[CrossRef]

Modine, F. A.

G. E. Jellison and F. A. Modine, “Parameterization of the optical functions of amorphous materials in the interband region,” Appl. Phys. Lett. 69, 371–373 (1996).
[CrossRef]

Moharam, M. G.

Motwani, P.

A. Gionis, P. Indyk, and P. Motwani, “Similarity search in high dimensions via hashing,” Proc. VLDB, 518–529 (1999).

Nakata, Y.

Naqvi, H.

Niu, X.

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

Novikova, T.

Patrick, H. J.

T. A. Germer, H. J. Patrick, R. M. Silver, and B. Bunday, “Developing an uncertainty analysis for optical scatterometry,” Proc. SPIE 7272, 72720T (2009).
[CrossRef]

H. J. Patrick, R. Attota, B. M. Barnes, T. A. Germer, R. G. Dixson, M. T. Stocker, R. M. Silver, and M. R. Bishop, “Optical critical dimension measurement of silicon grating targets using focal plane scatterfield microscopy,” J. Micro/Nanolith. MEMS MOEMS 7, 013012 (2008).

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, 3323–3336 (1998).
[CrossRef]

Pommet, D. A.

Pomplun, J.

J. Pomplun and F. Schmidt, “Accelerated a posterior error estimation for the reduced basis method with application to 3D electromagnetic scattering problems,” SIAM J. Sci. Comput. 32, 498–520 (2010).
[CrossRef]

Rabello, S.

J. Li, J. J. Hwu, Y. Liu, S. Rabello, Z. Liu, and J. Hu, “Mueller matrix measurement of asymmetric gratings,” J. Micro/Nanolith. MEMS MOEMS 9, 041305 (2010).

Rathsfeld, A.

H. Gross, R. Model, M. Bär, M. Wurm, B. Bodermann, and A. Rathsfeld, “Mathematical modelling of indirect measurements in scatterometry,” Measurement 39, 782–794 (2006).
[CrossRef]

Raymond, C.

C. Raymond, “Overview of scatterometry applications in high volume silicon manufacturing,” AIP Conf. Proc. 788, 394–402 (2005).
[CrossRef]

Raymond, C. J.

M. Littau, D. Forman, J. Bruce, C. J. Raymond, and S. G. Hummel, “Diffraction signature analysis methods for improving scatterometry precision,” Proc. SPIE 6152, 615236 (2006).
[CrossRef]

C. J. Raymond, M. Littau, A. Chuprin, and S. Ward, “Comparison of solutions to the scatterometry inverse problems,” Proc. SPIE 5375, 564–575 (2004).
[CrossRef]

C. J. Raymond, “Scatterometry for semiconductor metrology,” in Handbook of Silicon Semiconductor Metrology, A. C. Diebold, ed. (Academic, 2001), Chap. 18, pp. 477–514.

Reed, J.

E. Drége, J. Reed, and D. Byrne, “Linearized inversion of scatterometric data to obtain surface profile information,” Opt. Eng. 41, 225–236 (2002).
[CrossRef]

Schmidt, F.

J. Pomplun and F. Schmidt, “Accelerated a posterior error estimation for the reduced basis method with application to 3D electromagnetic scattering problems,” SIAM J. Sci. Comput. 32, 498–520 (2010).
[CrossRef]

Severgnini, E.

P. Thony, D. Herisson, D. Henry, E. Severgnini, and M. Vasconi, “Review of CD measurement and scatterometry,” AIP Conf. Proc. 683, 381–388 (2003).
[CrossRef]

Shi, T. L.

Shyu, D. M.

Silver, R. M.

N. F. Zhang, R. M. Silver, H. Zhou, and B. M. Barnes, “Improving optical measurement uncertainty with combined multitool metrology using a Bayesian approach,” Appl. Opt. 51, 6196–6206 (2012).
[CrossRef]

T. A. Germer, H. J. Patrick, R. M. Silver, and B. Bunday, “Developing an uncertainty analysis for optical scatterometry,” Proc. SPIE 7272, 72720T (2009).
[CrossRef]

H. J. Patrick, R. Attota, B. M. Barnes, T. A. Germer, R. G. Dixson, M. T. Stocker, R. M. Silver, and M. R. Bishop, “Optical critical dimension measurement of silicon grating targets using focal plane scatterfield microscopy,” J. Micro/Nanolith. MEMS MOEMS 7, 013012 (2008).

Smith, N.

Sohail, S.

Spanos, C. J.

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

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P. Thony, D. Herisson, D. Henry, E. Severgnini, and M. Vasconi, “Review of CD measurement and scatterometry,” AIP Conf. Proc. 683, 381–388 (2003).
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Figures (7)

Fig. 1.
Fig. 1.

Geometrical illustration of the corrected solution xc to the inverse problem in optical scatterometry.

Fig. 2.
Fig. 2.

Scheme of the dual-rotating compensator ellipsometer.

Fig. 3.
Fig. 3.

Schematic diagram of the silicon wafer and the photoresist grating structure.

Fig. 4.
Fig. 4.

Absolute errors of the structural parameters TCD, Hgt1, and SWA obtained with Library 1# by the conventional library search method, the interpolation-based library search method, and the correction-based library search method. The mean absolute errors of the extracted structural parameters for the 35 dies of the investigated silicon wafer are shown in the bottom right corner.

Fig. 5.
Fig. 5.

(a) Measured Stokes vector elements for No. 16 die of the grating sample shown in Fig. 3 and the fitted Stokes vector elements calculated from the structural parameters presented in Table 2 extracted by the conventional library search method, the interpolation-based library search method, the correction-based library search method, and the ANN-LM combined method. (b) The absolute fitting errors between the calculated and measured Stokes vector elements.

Fig. 6.
Fig. 6.

Time cost of the conventional library search method, the interpolation-based library search method, and the correction-based library search method in the parameter extraction with Library 1# for the 35 dies of the investigated silicon wafer.

Fig. 7.
Fig. 7.

Absolute errors of the structural parameters TCD, Hgt1, and SWA obtained by the conventional library search method with Library 1#, the correction-based library search method as well as the interpolation-based library search method with Library 2#. The mean absolute errors of the extracted structural parameters for the 35 dies of the investigated silicon wafer are shown in the bottom right corner.

Tables (2)

Tables Icon

Table 1. Details of the Signature Libraries Used in the Experiments

Tables Icon

Table 2. Comparison of the Structural Parameters Extracted by the Conventional Library Search Method, the Interpolation-Based Library Search Method, the Correction-Based Library Search Method, and the ANN-LM Combined Method

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

χ2=i=1Nwi[yifi(x)]2,
χ2=[yf(x)]TW[yf(x)],
x^=argminxΩ{[yf(x)]TW[yf(x)]},
χmin2=[yf(xe)]TW[yf(xe)].
Δx=xexs.
f(x)=f(xs)+J·(xxs),
[J]ij=fi(x)xj|x=xs.
f(xe)=f(xs)+J·(xexs)=f(xs)+JΔx.
χmin2=[yf(xs)JΔx]TW[yf(xs)JΔx].
J˜Δx=W1/2[yf(xs)],
Δx=J˜+W1/2[yf(xs)],
xc=xs+J˜+W1/2[yf(xs)],
fi(x)xj|x=xs=fi(xs+δjej)fi(xsδjej)2δj+O(δj2),
εl=|x^(l)x0(l)|,ε¯=1Ll=1Lεl,

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