Abstract

The influence of edge roughness in angle-resolved scatterometry at periodically structured surfaces is investigated. A good description of the radiation interaction with structured surfaces is crucial for the understanding of optical imaging processes such as, e.g., in photolithography. We compared an analytical two-dimensional (2D) model and a numerical three-dimensional simulation with respect to the characterization of 2D diffraction of a line grating involving structure roughness. The results show a remarkably high agreement. The diffraction intensities of a rough structure can therefore be estimated using the numerical simulation result of an undisturbed structure and an analytically derived correction function. This work allows to improve scatterometric results for the case of practically relevant 2D structures.

© 2012 Optical Society of America

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  1. T. A. Germer, “Effect of line and trench profile variation on specular and diffuse reflectance from a periodic structure,” J. Opt. Soc. Am. A 24, 696–701 (2007).
    [CrossRef]
  2. T. A. Germer, “Modeling the effect of line profile variation on optical critical dimension metrology,” Proc. SPIE 6518, 65180Z (2007).
    [CrossRef]
  3. P. P. Naulleau, “Effect of mask-roughness on printed contact-size variation in extreme-ultraviolet lithography,” Appl. Opt. 44, 183–189 (2005).
    [CrossRef]
  4. H. Gross, A. Rathsfeld, F. Scholze, and M. Bär, “Profile reconstruction in extreme ultraviolet (EUV) scatterometry: modeling and uncertainty estimates,” Meas. Sci. Technol. 20, 105102 (2009).
    [CrossRef]
  5. P. Boher, J. Petit, T. Leroux, J. Foucher, Y. Desières, J. Hazart, and P. Chaton, “Optical Fourier transform scatterometry for LER and LWR metrology,” Proc. SPIE 5752, 192–203 (2005).
    [CrossRef]
  6. C. Wang, R. L. Jones, E. K. Lin, W. Wu, B. J. Rice, K. Choi, G. Thompson, S. J. Weigand, and D. T. Keane, “Characterization of correlated line edge roughness of nanoscale line gratings using small angle x-ray scattering,” J. Appl. Phys. 102, 024901 (2007).
    [CrossRef]
  7. F. Scholze and C. Laubis, “Use of EUV scatterometry for the characterization of line profiles and line roughness on photomasks,” Proc. SPIE 6792, 67920U (2008).
    [CrossRef]
  8. B. Bodermann, M. Wurm, A. Diener, F. Scholze, and H. Groß, “EUV and DUV scatterometry for CD and edge profile metrology on EUV masks,” Proc. SPIE 7470, 74700F (2009).
    [CrossRef]
  9. J. Pomplun, S. Burger, F. Schmidt, F. Scholze, C. Laubis, and U. Dersch, “Metrology of EUV masks by EUV-scatterometry and finite element analysis,” Proc. SPIE 7028, 70280P(2008).
    [CrossRef]
  10. S. Burger, L. Zschiedrich, J. Pomplun, and F. Schmidt, “Rigorous simulations of 3D patterns on extreme ultraviolet lithography masks,” Proc. SPIE 8083, 80831B (2011).
    [CrossRef]
  11. F. Scholze, B. Bodermann, H. Groß, A. Kato, and M. Wurm, “First steps towards traceability in scatterometry,” Proc. SPIE 7985, 79850G (2011).
    [CrossRef]
  12. A. Kato and F. Scholze, “The effect of line roughness on the diffraction intensities in angular resolved scatterometry,” Proc. SPIE 8083, 80830K (2011).
    [CrossRef]
  13. A. Kato and F. Scholze, “Effect of line roughness on the diffraction intensities in angular resolved scatterometry,” Appl. Opt. 49, 6102–6110 (2010).
    [CrossRef]
  14. H. Gross, M.-A. Henn, A. Rathsfeld, and M. Bär, “Stochastic modeling aspects for an improved solution of the inverse problem in scatterometry,” in Advanced Mathematical and Computional Tools in Metrology and Testing IX, Vol. 84 of Series on Advances in Mathematics for Applied Sciences (World Scientific, 2012), pp. 202–209.
  15. T. Schuster, S. Rafler, K. Frenner, and W. Osten, “Influence of line edge roughness (LER) on angular resolved and on spectroscopic scatterometry,” Proc. SPIE 7155, 71550W (2008).
    [CrossRef]
  16. B. C. Bergner, T. A. Germer, and T. J. Suleski, “Effective medium approximation for modeling optical reflectance from gratings with rough edges,” J. Opt. Soc. Am. A 27, 1083–1089 (2010).
    [CrossRef]
  17. T. Schuster, S. Rafler, V. Ferreras Paz, K. Frenner, and W. Osten, “Fieldstitching with Kirchhoff-boundaries as a model based description for line edge roughness (LER) in scatterometry,” Microelectron. Eng. 86, 1029–1032 (2009).
    [CrossRef]
  18. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Vol. 93 of Applied Mathematical Sciences (Springer, 1998).
  19. J. Pomplun, S. Burger, L. Zschiedrich, and F. Schmidt, “Adaptive finite element method for simulation of optical nano structures,” Phys. Status Solidi B 244, 3419–3434 (2007).
    [CrossRef]
  20. D. Lockau, L. Zschiedrich, S. Burger, F. Schmidt, F. Ruske, and B. Rech, “Rigorous optical simulation of light management in crystalline silicon thin film solar cells with rough interface textures,” Proc. SPIE 7933, 79330M (2011).
    [CrossRef]
  21. A. Schädle, L. Zschiedrich, S. Burger, R. Klose, and F. Schmidt, “Domain decomposition method for Maxwell’s equations: Scattering off periodic structures,” J. Comput. Phys. 226, 477–493 (2007).
    [CrossRef]
  22. S. Burger, L. Zschiedrich, J. Pomplun, F. Schmidt, A. Kato, C. Laubis, and F. Scholze, “Investigation of 3D patterns on EUV masks by means of scatterometry and comparison to numerical simulations,” Proc. SPIE 8166, 81661Q (2011).
    [CrossRef]

2011 (5)

S. Burger, L. Zschiedrich, J. Pomplun, and F. Schmidt, “Rigorous simulations of 3D patterns on extreme ultraviolet lithography masks,” Proc. SPIE 8083, 80831B (2011).
[CrossRef]

F. Scholze, B. Bodermann, H. Groß, A. Kato, and M. Wurm, “First steps towards traceability in scatterometry,” Proc. SPIE 7985, 79850G (2011).
[CrossRef]

A. Kato and F. Scholze, “The effect of line roughness on the diffraction intensities in angular resolved scatterometry,” Proc. SPIE 8083, 80830K (2011).
[CrossRef]

D. Lockau, L. Zschiedrich, S. Burger, F. Schmidt, F. Ruske, and B. Rech, “Rigorous optical simulation of light management in crystalline silicon thin film solar cells with rough interface textures,” Proc. SPIE 7933, 79330M (2011).
[CrossRef]

S. Burger, L. Zschiedrich, J. Pomplun, F. Schmidt, A. Kato, C. Laubis, and F. Scholze, “Investigation of 3D patterns on EUV masks by means of scatterometry and comparison to numerical simulations,” Proc. SPIE 8166, 81661Q (2011).
[CrossRef]

2010 (2)

2009 (3)

T. Schuster, S. Rafler, V. Ferreras Paz, K. Frenner, and W. Osten, “Fieldstitching with Kirchhoff-boundaries as a model based description for line edge roughness (LER) in scatterometry,” Microelectron. Eng. 86, 1029–1032 (2009).
[CrossRef]

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

B. Bodermann, M. Wurm, A. Diener, F. Scholze, and H. Groß, “EUV and DUV scatterometry for CD and edge profile metrology on EUV masks,” Proc. SPIE 7470, 74700F (2009).
[CrossRef]

2008 (3)

J. Pomplun, S. Burger, F. Schmidt, F. Scholze, C. Laubis, and U. Dersch, “Metrology of EUV masks by EUV-scatterometry and finite element analysis,” Proc. SPIE 7028, 70280P(2008).
[CrossRef]

F. Scholze and C. Laubis, “Use of EUV scatterometry for the characterization of line profiles and line roughness on photomasks,” Proc. SPIE 6792, 67920U (2008).
[CrossRef]

T. Schuster, S. Rafler, K. Frenner, and W. Osten, “Influence of line edge roughness (LER) on angular resolved and on spectroscopic scatterometry,” Proc. SPIE 7155, 71550W (2008).
[CrossRef]

2007 (5)

J. Pomplun, S. Burger, L. Zschiedrich, and F. Schmidt, “Adaptive finite element method for simulation of optical nano structures,” Phys. Status Solidi B 244, 3419–3434 (2007).
[CrossRef]

A. Schädle, L. Zschiedrich, S. Burger, R. Klose, and F. Schmidt, “Domain decomposition method for Maxwell’s equations: Scattering off periodic structures,” J. Comput. Phys. 226, 477–493 (2007).
[CrossRef]

C. Wang, R. L. Jones, E. K. Lin, W. Wu, B. J. Rice, K. Choi, G. Thompson, S. J. Weigand, and D. T. Keane, “Characterization of correlated line edge roughness of nanoscale line gratings using small angle x-ray scattering,” J. Appl. Phys. 102, 024901 (2007).
[CrossRef]

T. A. Germer, “Effect of line and trench profile variation on specular and diffuse reflectance from a periodic structure,” J. Opt. Soc. Am. A 24, 696–701 (2007).
[CrossRef]

T. A. Germer, “Modeling the effect of line profile variation on optical critical dimension metrology,” Proc. SPIE 6518, 65180Z (2007).
[CrossRef]

2005 (2)

P. P. Naulleau, “Effect of mask-roughness on printed contact-size variation in extreme-ultraviolet lithography,” Appl. Opt. 44, 183–189 (2005).
[CrossRef]

P. Boher, J. Petit, T. Leroux, J. Foucher, Y. Desières, J. Hazart, and P. Chaton, “Optical Fourier transform scatterometry for LER and LWR metrology,” Proc. SPIE 5752, 192–203 (2005).
[CrossRef]

Bär, M.

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

H. Gross, M.-A. Henn, A. Rathsfeld, and M. Bär, “Stochastic modeling aspects for an improved solution of the inverse problem in scatterometry,” in Advanced Mathematical and Computional Tools in Metrology and Testing IX, Vol. 84 of Series on Advances in Mathematics for Applied Sciences (World Scientific, 2012), pp. 202–209.

Bergner, B. C.

Bodermann, B.

F. Scholze, B. Bodermann, H. Groß, A. Kato, and M. Wurm, “First steps towards traceability in scatterometry,” Proc. SPIE 7985, 79850G (2011).
[CrossRef]

B. Bodermann, M. Wurm, A. Diener, F. Scholze, and H. Groß, “EUV and DUV scatterometry for CD and edge profile metrology on EUV masks,” Proc. SPIE 7470, 74700F (2009).
[CrossRef]

Boher, P.

P. Boher, J. Petit, T. Leroux, J. Foucher, Y. Desières, J. Hazart, and P. Chaton, “Optical Fourier transform scatterometry for LER and LWR metrology,” Proc. SPIE 5752, 192–203 (2005).
[CrossRef]

Burger, S.

S. Burger, L. Zschiedrich, J. Pomplun, and F. Schmidt, “Rigorous simulations of 3D patterns on extreme ultraviolet lithography masks,” Proc. SPIE 8083, 80831B (2011).
[CrossRef]

D. Lockau, L. Zschiedrich, S. Burger, F. Schmidt, F. Ruske, and B. Rech, “Rigorous optical simulation of light management in crystalline silicon thin film solar cells with rough interface textures,” Proc. SPIE 7933, 79330M (2011).
[CrossRef]

S. Burger, L. Zschiedrich, J. Pomplun, F. Schmidt, A. Kato, C. Laubis, and F. Scholze, “Investigation of 3D patterns on EUV masks by means of scatterometry and comparison to numerical simulations,” Proc. SPIE 8166, 81661Q (2011).
[CrossRef]

J. Pomplun, S. Burger, F. Schmidt, F. Scholze, C. Laubis, and U. Dersch, “Metrology of EUV masks by EUV-scatterometry and finite element analysis,” Proc. SPIE 7028, 70280P(2008).
[CrossRef]

J. Pomplun, S. Burger, L. Zschiedrich, and F. Schmidt, “Adaptive finite element method for simulation of optical nano structures,” Phys. Status Solidi B 244, 3419–3434 (2007).
[CrossRef]

A. Schädle, L. Zschiedrich, S. Burger, R. Klose, and F. Schmidt, “Domain decomposition method for Maxwell’s equations: Scattering off periodic structures,” J. Comput. Phys. 226, 477–493 (2007).
[CrossRef]

Chaton, P.

P. Boher, J. Petit, T. Leroux, J. Foucher, Y. Desières, J. Hazart, and P. Chaton, “Optical Fourier transform scatterometry for LER and LWR metrology,” Proc. SPIE 5752, 192–203 (2005).
[CrossRef]

Choi, K.

C. Wang, R. L. Jones, E. K. Lin, W. Wu, B. J. Rice, K. Choi, G. Thompson, S. J. Weigand, and D. T. Keane, “Characterization of correlated line edge roughness of nanoscale line gratings using small angle x-ray scattering,” J. Appl. Phys. 102, 024901 (2007).
[CrossRef]

Colton, D.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Vol. 93 of Applied Mathematical Sciences (Springer, 1998).

Dersch, U.

J. Pomplun, S. Burger, F. Schmidt, F. Scholze, C. Laubis, and U. Dersch, “Metrology of EUV masks by EUV-scatterometry and finite element analysis,” Proc. SPIE 7028, 70280P(2008).
[CrossRef]

Desières, Y.

P. Boher, J. Petit, T. Leroux, J. Foucher, Y. Desières, J. Hazart, and P. Chaton, “Optical Fourier transform scatterometry for LER and LWR metrology,” Proc. SPIE 5752, 192–203 (2005).
[CrossRef]

Diener, A.

B. Bodermann, M. Wurm, A. Diener, F. Scholze, and H. Groß, “EUV and DUV scatterometry for CD and edge profile metrology on EUV masks,” Proc. SPIE 7470, 74700F (2009).
[CrossRef]

Foucher, J.

P. Boher, J. Petit, T. Leroux, J. Foucher, Y. Desières, J. Hazart, and P. Chaton, “Optical Fourier transform scatterometry for LER and LWR metrology,” Proc. SPIE 5752, 192–203 (2005).
[CrossRef]

Frenner, K.

T. Schuster, S. Rafler, V. Ferreras Paz, K. Frenner, and W. Osten, “Fieldstitching with Kirchhoff-boundaries as a model based description for line edge roughness (LER) in scatterometry,” Microelectron. Eng. 86, 1029–1032 (2009).
[CrossRef]

T. Schuster, S. Rafler, K. Frenner, and W. Osten, “Influence of line edge roughness (LER) on angular resolved and on spectroscopic scatterometry,” Proc. SPIE 7155, 71550W (2008).
[CrossRef]

Germer, T. A.

Groß, H.

F. Scholze, B. Bodermann, H. Groß, A. Kato, and M. Wurm, “First steps towards traceability in scatterometry,” Proc. SPIE 7985, 79850G (2011).
[CrossRef]

Gross, H.

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

Groß, H.

B. Bodermann, M. Wurm, A. Diener, F. Scholze, and H. Groß, “EUV and DUV scatterometry for CD and edge profile metrology on EUV masks,” Proc. SPIE 7470, 74700F (2009).
[CrossRef]

Gross, H.

H. Gross, M.-A. Henn, A. Rathsfeld, and M. Bär, “Stochastic modeling aspects for an improved solution of the inverse problem in scatterometry,” in Advanced Mathematical and Computional Tools in Metrology and Testing IX, Vol. 84 of Series on Advances in Mathematics for Applied Sciences (World Scientific, 2012), pp. 202–209.

Hazart, J.

P. Boher, J. Petit, T. Leroux, J. Foucher, Y. Desières, J. Hazart, and P. Chaton, “Optical Fourier transform scatterometry for LER and LWR metrology,” Proc. SPIE 5752, 192–203 (2005).
[CrossRef]

Henn, M.-A.

H. Gross, M.-A. Henn, A. Rathsfeld, and M. Bär, “Stochastic modeling aspects for an improved solution of the inverse problem in scatterometry,” in Advanced Mathematical and Computional Tools in Metrology and Testing IX, Vol. 84 of Series on Advances in Mathematics for Applied Sciences (World Scientific, 2012), pp. 202–209.

Jones, R. L.

C. Wang, R. L. Jones, E. K. Lin, W. Wu, B. J. Rice, K. Choi, G. Thompson, S. J. Weigand, and D. T. Keane, “Characterization of correlated line edge roughness of nanoscale line gratings using small angle x-ray scattering,” J. Appl. Phys. 102, 024901 (2007).
[CrossRef]

Kato, A.

F. Scholze, B. Bodermann, H. Groß, A. Kato, and M. Wurm, “First steps towards traceability in scatterometry,” Proc. SPIE 7985, 79850G (2011).
[CrossRef]

A. Kato and F. Scholze, “The effect of line roughness on the diffraction intensities in angular resolved scatterometry,” Proc. SPIE 8083, 80830K (2011).
[CrossRef]

S. Burger, L. Zschiedrich, J. Pomplun, F. Schmidt, A. Kato, C. Laubis, and F. Scholze, “Investigation of 3D patterns on EUV masks by means of scatterometry and comparison to numerical simulations,” Proc. SPIE 8166, 81661Q (2011).
[CrossRef]

A. Kato and F. Scholze, “Effect of line roughness on the diffraction intensities in angular resolved scatterometry,” Appl. Opt. 49, 6102–6110 (2010).
[CrossRef]

Keane, D. T.

C. Wang, R. L. Jones, E. K. Lin, W. Wu, B. J. Rice, K. Choi, G. Thompson, S. J. Weigand, and D. T. Keane, “Characterization of correlated line edge roughness of nanoscale line gratings using small angle x-ray scattering,” J. Appl. Phys. 102, 024901 (2007).
[CrossRef]

Klose, R.

A. Schädle, L. Zschiedrich, S. Burger, R. Klose, and F. Schmidt, “Domain decomposition method for Maxwell’s equations: Scattering off periodic structures,” J. Comput. Phys. 226, 477–493 (2007).
[CrossRef]

Kress, R.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Vol. 93 of Applied Mathematical Sciences (Springer, 1998).

Laubis, C.

S. Burger, L. Zschiedrich, J. Pomplun, F. Schmidt, A. Kato, C. Laubis, and F. Scholze, “Investigation of 3D patterns on EUV masks by means of scatterometry and comparison to numerical simulations,” Proc. SPIE 8166, 81661Q (2011).
[CrossRef]

F. Scholze and C. Laubis, “Use of EUV scatterometry for the characterization of line profiles and line roughness on photomasks,” Proc. SPIE 6792, 67920U (2008).
[CrossRef]

J. Pomplun, S. Burger, F. Schmidt, F. Scholze, C. Laubis, and U. Dersch, “Metrology of EUV masks by EUV-scatterometry and finite element analysis,” Proc. SPIE 7028, 70280P(2008).
[CrossRef]

Leroux, T.

P. Boher, J. Petit, T. Leroux, J. Foucher, Y. Desières, J. Hazart, and P. Chaton, “Optical Fourier transform scatterometry for LER and LWR metrology,” Proc. SPIE 5752, 192–203 (2005).
[CrossRef]

Lin, E. K.

C. Wang, R. L. Jones, E. K. Lin, W. Wu, B. J. Rice, K. Choi, G. Thompson, S. J. Weigand, and D. T. Keane, “Characterization of correlated line edge roughness of nanoscale line gratings using small angle x-ray scattering,” J. Appl. Phys. 102, 024901 (2007).
[CrossRef]

Lockau, D.

D. Lockau, L. Zschiedrich, S. Burger, F. Schmidt, F. Ruske, and B. Rech, “Rigorous optical simulation of light management in crystalline silicon thin film solar cells with rough interface textures,” Proc. SPIE 7933, 79330M (2011).
[CrossRef]

Naulleau, P. P.

Osten, W.

T. Schuster, S. Rafler, V. Ferreras Paz, K. Frenner, and W. Osten, “Fieldstitching with Kirchhoff-boundaries as a model based description for line edge roughness (LER) in scatterometry,” Microelectron. Eng. 86, 1029–1032 (2009).
[CrossRef]

T. Schuster, S. Rafler, K. Frenner, and W. Osten, “Influence of line edge roughness (LER) on angular resolved and on spectroscopic scatterometry,” Proc. SPIE 7155, 71550W (2008).
[CrossRef]

Paz, V. Ferreras

T. Schuster, S. Rafler, V. Ferreras Paz, K. Frenner, and W. Osten, “Fieldstitching with Kirchhoff-boundaries as a model based description for line edge roughness (LER) in scatterometry,” Microelectron. Eng. 86, 1029–1032 (2009).
[CrossRef]

Petit, J.

P. Boher, J. Petit, T. Leroux, J. Foucher, Y. Desières, J. Hazart, and P. Chaton, “Optical Fourier transform scatterometry for LER and LWR metrology,” Proc. SPIE 5752, 192–203 (2005).
[CrossRef]

Pomplun, J.

S. Burger, L. Zschiedrich, J. Pomplun, and F. Schmidt, “Rigorous simulations of 3D patterns on extreme ultraviolet lithography masks,” Proc. SPIE 8083, 80831B (2011).
[CrossRef]

S. Burger, L. Zschiedrich, J. Pomplun, F. Schmidt, A. Kato, C. Laubis, and F. Scholze, “Investigation of 3D patterns on EUV masks by means of scatterometry and comparison to numerical simulations,” Proc. SPIE 8166, 81661Q (2011).
[CrossRef]

J. Pomplun, S. Burger, F. Schmidt, F. Scholze, C. Laubis, and U. Dersch, “Metrology of EUV masks by EUV-scatterometry and finite element analysis,” Proc. SPIE 7028, 70280P(2008).
[CrossRef]

J. Pomplun, S. Burger, L. Zschiedrich, and F. Schmidt, “Adaptive finite element method for simulation of optical nano structures,” Phys. Status Solidi B 244, 3419–3434 (2007).
[CrossRef]

Rafler, S.

T. Schuster, S. Rafler, V. Ferreras Paz, K. Frenner, and W. Osten, “Fieldstitching with Kirchhoff-boundaries as a model based description for line edge roughness (LER) in scatterometry,” Microelectron. Eng. 86, 1029–1032 (2009).
[CrossRef]

T. Schuster, S. Rafler, K. Frenner, and W. Osten, “Influence of line edge roughness (LER) on angular resolved and on spectroscopic scatterometry,” Proc. SPIE 7155, 71550W (2008).
[CrossRef]

Rathsfeld, A.

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

H. Gross, M.-A. Henn, A. Rathsfeld, and M. Bär, “Stochastic modeling aspects for an improved solution of the inverse problem in scatterometry,” in Advanced Mathematical and Computional Tools in Metrology and Testing IX, Vol. 84 of Series on Advances in Mathematics for Applied Sciences (World Scientific, 2012), pp. 202–209.

Rech, B.

D. Lockau, L. Zschiedrich, S. Burger, F. Schmidt, F. Ruske, and B. Rech, “Rigorous optical simulation of light management in crystalline silicon thin film solar cells with rough interface textures,” Proc. SPIE 7933, 79330M (2011).
[CrossRef]

Rice, B. J.

C. Wang, R. L. Jones, E. K. Lin, W. Wu, B. J. Rice, K. Choi, G. Thompson, S. J. Weigand, and D. T. Keane, “Characterization of correlated line edge roughness of nanoscale line gratings using small angle x-ray scattering,” J. Appl. Phys. 102, 024901 (2007).
[CrossRef]

Ruske, F.

D. Lockau, L. Zschiedrich, S. Burger, F. Schmidt, F. Ruske, and B. Rech, “Rigorous optical simulation of light management in crystalline silicon thin film solar cells with rough interface textures,” Proc. SPIE 7933, 79330M (2011).
[CrossRef]

Schädle, A.

A. Schädle, L. Zschiedrich, S. Burger, R. Klose, and F. Schmidt, “Domain decomposition method for Maxwell’s equations: Scattering off periodic structures,” J. Comput. Phys. 226, 477–493 (2007).
[CrossRef]

Schmidt, F.

S. Burger, L. Zschiedrich, J. Pomplun, F. Schmidt, A. Kato, C. Laubis, and F. Scholze, “Investigation of 3D patterns on EUV masks by means of scatterometry and comparison to numerical simulations,” Proc. SPIE 8166, 81661Q (2011).
[CrossRef]

S. Burger, L. Zschiedrich, J. Pomplun, and F. Schmidt, “Rigorous simulations of 3D patterns on extreme ultraviolet lithography masks,” Proc. SPIE 8083, 80831B (2011).
[CrossRef]

D. Lockau, L. Zschiedrich, S. Burger, F. Schmidt, F. Ruske, and B. Rech, “Rigorous optical simulation of light management in crystalline silicon thin film solar cells with rough interface textures,” Proc. SPIE 7933, 79330M (2011).
[CrossRef]

J. Pomplun, S. Burger, F. Schmidt, F. Scholze, C. Laubis, and U. Dersch, “Metrology of EUV masks by EUV-scatterometry and finite element analysis,” Proc. SPIE 7028, 70280P(2008).
[CrossRef]

J. Pomplun, S. Burger, L. Zschiedrich, and F. Schmidt, “Adaptive finite element method for simulation of optical nano structures,” Phys. Status Solidi B 244, 3419–3434 (2007).
[CrossRef]

A. Schädle, L. Zschiedrich, S. Burger, R. Klose, and F. Schmidt, “Domain decomposition method for Maxwell’s equations: Scattering off periodic structures,” J. Comput. Phys. 226, 477–493 (2007).
[CrossRef]

Scholze, F.

S. Burger, L. Zschiedrich, J. Pomplun, F. Schmidt, A. Kato, C. Laubis, and F. Scholze, “Investigation of 3D patterns on EUV masks by means of scatterometry and comparison to numerical simulations,” Proc. SPIE 8166, 81661Q (2011).
[CrossRef]

F. Scholze, B. Bodermann, H. Groß, A. Kato, and M. Wurm, “First steps towards traceability in scatterometry,” Proc. SPIE 7985, 79850G (2011).
[CrossRef]

A. Kato and F. Scholze, “The effect of line roughness on the diffraction intensities in angular resolved scatterometry,” Proc. SPIE 8083, 80830K (2011).
[CrossRef]

A. Kato and F. Scholze, “Effect of line roughness on the diffraction intensities in angular resolved scatterometry,” Appl. Opt. 49, 6102–6110 (2010).
[CrossRef]

B. Bodermann, M. Wurm, A. Diener, F. Scholze, and H. Groß, “EUV and DUV scatterometry for CD and edge profile metrology on EUV masks,” Proc. SPIE 7470, 74700F (2009).
[CrossRef]

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

F. Scholze and C. Laubis, “Use of EUV scatterometry for the characterization of line profiles and line roughness on photomasks,” Proc. SPIE 6792, 67920U (2008).
[CrossRef]

J. Pomplun, S. Burger, F. Schmidt, F. Scholze, C. Laubis, and U. Dersch, “Metrology of EUV masks by EUV-scatterometry and finite element analysis,” Proc. SPIE 7028, 70280P(2008).
[CrossRef]

Schuster, T.

T. Schuster, S. Rafler, V. Ferreras Paz, K. Frenner, and W. Osten, “Fieldstitching with Kirchhoff-boundaries as a model based description for line edge roughness (LER) in scatterometry,” Microelectron. Eng. 86, 1029–1032 (2009).
[CrossRef]

T. Schuster, S. Rafler, K. Frenner, and W. Osten, “Influence of line edge roughness (LER) on angular resolved and on spectroscopic scatterometry,” Proc. SPIE 7155, 71550W (2008).
[CrossRef]

Suleski, T. J.

Thompson, G.

C. Wang, R. L. Jones, E. K. Lin, W. Wu, B. J. Rice, K. Choi, G. Thompson, S. J. Weigand, and D. T. Keane, “Characterization of correlated line edge roughness of nanoscale line gratings using small angle x-ray scattering,” J. Appl. Phys. 102, 024901 (2007).
[CrossRef]

Wang, C.

C. Wang, R. L. Jones, E. K. Lin, W. Wu, B. J. Rice, K. Choi, G. Thompson, S. J. Weigand, and D. T. Keane, “Characterization of correlated line edge roughness of nanoscale line gratings using small angle x-ray scattering,” J. Appl. Phys. 102, 024901 (2007).
[CrossRef]

Weigand, S. J.

C. Wang, R. L. Jones, E. K. Lin, W. Wu, B. J. Rice, K. Choi, G. Thompson, S. J. Weigand, and D. T. Keane, “Characterization of correlated line edge roughness of nanoscale line gratings using small angle x-ray scattering,” J. Appl. Phys. 102, 024901 (2007).
[CrossRef]

Wu, W.

C. Wang, R. L. Jones, E. K. Lin, W. Wu, B. J. Rice, K. Choi, G. Thompson, S. J. Weigand, and D. T. Keane, “Characterization of correlated line edge roughness of nanoscale line gratings using small angle x-ray scattering,” J. Appl. Phys. 102, 024901 (2007).
[CrossRef]

Wurm, M.

F. Scholze, B. Bodermann, H. Groß, A. Kato, and M. Wurm, “First steps towards traceability in scatterometry,” Proc. SPIE 7985, 79850G (2011).
[CrossRef]

B. Bodermann, M. Wurm, A. Diener, F. Scholze, and H. Groß, “EUV and DUV scatterometry for CD and edge profile metrology on EUV masks,” Proc. SPIE 7470, 74700F (2009).
[CrossRef]

Zschiedrich, L.

S. Burger, L. Zschiedrich, J. Pomplun, and F. Schmidt, “Rigorous simulations of 3D patterns on extreme ultraviolet lithography masks,” Proc. SPIE 8083, 80831B (2011).
[CrossRef]

D. Lockau, L. Zschiedrich, S. Burger, F. Schmidt, F. Ruske, and B. Rech, “Rigorous optical simulation of light management in crystalline silicon thin film solar cells with rough interface textures,” Proc. SPIE 7933, 79330M (2011).
[CrossRef]

S. Burger, L. Zschiedrich, J. Pomplun, F. Schmidt, A. Kato, C. Laubis, and F. Scholze, “Investigation of 3D patterns on EUV masks by means of scatterometry and comparison to numerical simulations,” Proc. SPIE 8166, 81661Q (2011).
[CrossRef]

A. Schädle, L. Zschiedrich, S. Burger, R. Klose, and F. Schmidt, “Domain decomposition method for Maxwell’s equations: Scattering off periodic structures,” J. Comput. Phys. 226, 477–493 (2007).
[CrossRef]

J. Pomplun, S. Burger, L. Zschiedrich, and F. Schmidt, “Adaptive finite element method for simulation of optical nano structures,” Phys. Status Solidi B 244, 3419–3434 (2007).
[CrossRef]

Appl. Opt. (2)

J. Appl. Phys. (1)

C. Wang, R. L. Jones, E. K. Lin, W. Wu, B. J. Rice, K. Choi, G. Thompson, S. J. Weigand, and D. T. Keane, “Characterization of correlated line edge roughness of nanoscale line gratings using small angle x-ray scattering,” J. Appl. Phys. 102, 024901 (2007).
[CrossRef]

J. Comput. Phys. (1)

A. Schädle, L. Zschiedrich, S. Burger, R. Klose, and F. Schmidt, “Domain decomposition method for Maxwell’s equations: Scattering off periodic structures,” J. Comput. Phys. 226, 477–493 (2007).
[CrossRef]

J. Opt. Soc. Am. A (2)

Meas. Sci. Technol. (1)

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

Microelectron. Eng. (1)

T. Schuster, S. Rafler, V. Ferreras Paz, K. Frenner, and W. Osten, “Fieldstitching with Kirchhoff-boundaries as a model based description for line edge roughness (LER) in scatterometry,” Microelectron. Eng. 86, 1029–1032 (2009).
[CrossRef]

Phys. Status Solidi B (1)

J. Pomplun, S. Burger, L. Zschiedrich, and F. Schmidt, “Adaptive finite element method for simulation of optical nano structures,” Phys. Status Solidi B 244, 3419–3434 (2007).
[CrossRef]

Proc. SPIE (11)

D. Lockau, L. Zschiedrich, S. Burger, F. Schmidt, F. Ruske, and B. Rech, “Rigorous optical simulation of light management in crystalline silicon thin film solar cells with rough interface textures,” Proc. SPIE 7933, 79330M (2011).
[CrossRef]

S. Burger, L. Zschiedrich, J. Pomplun, F. Schmidt, A. Kato, C. Laubis, and F. Scholze, “Investigation of 3D patterns on EUV masks by means of scatterometry and comparison to numerical simulations,” Proc. SPIE 8166, 81661Q (2011).
[CrossRef]

P. Boher, J. Petit, T. Leroux, J. Foucher, Y. Desières, J. Hazart, and P. Chaton, “Optical Fourier transform scatterometry for LER and LWR metrology,” Proc. SPIE 5752, 192–203 (2005).
[CrossRef]

T. A. Germer, “Modeling the effect of line profile variation on optical critical dimension metrology,” Proc. SPIE 6518, 65180Z (2007).
[CrossRef]

F. Scholze and C. Laubis, “Use of EUV scatterometry for the characterization of line profiles and line roughness on photomasks,” Proc. SPIE 6792, 67920U (2008).
[CrossRef]

B. Bodermann, M. Wurm, A. Diener, F. Scholze, and H. Groß, “EUV and DUV scatterometry for CD and edge profile metrology on EUV masks,” Proc. SPIE 7470, 74700F (2009).
[CrossRef]

J. Pomplun, S. Burger, F. Schmidt, F. Scholze, C. Laubis, and U. Dersch, “Metrology of EUV masks by EUV-scatterometry and finite element analysis,” Proc. SPIE 7028, 70280P(2008).
[CrossRef]

S. Burger, L. Zschiedrich, J. Pomplun, and F. Schmidt, “Rigorous simulations of 3D patterns on extreme ultraviolet lithography masks,” Proc. SPIE 8083, 80831B (2011).
[CrossRef]

F. Scholze, B. Bodermann, H. Groß, A. Kato, and M. Wurm, “First steps towards traceability in scatterometry,” Proc. SPIE 7985, 79850G (2011).
[CrossRef]

A. Kato and F. Scholze, “The effect of line roughness on the diffraction intensities in angular resolved scatterometry,” Proc. SPIE 8083, 80830K (2011).
[CrossRef]

T. Schuster, S. Rafler, K. Frenner, and W. Osten, “Influence of line edge roughness (LER) on angular resolved and on spectroscopic scatterometry,” Proc. SPIE 7155, 71550W (2008).
[CrossRef]

Other (2)

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Vol. 93 of Applied Mathematical Sciences (Springer, 1998).

H. Gross, M.-A. Henn, A. Rathsfeld, and M. Bär, “Stochastic modeling aspects for an improved solution of the inverse problem in scatterometry,” in Advanced Mathematical and Computional Tools in Metrology and Testing IX, Vol. 84 of Series on Advances in Mathematics for Applied Sciences (World Scientific, 2012), pp. 202–209.

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Figures (6)

Fig. 1.
Fig. 1.

2D binary grating without roughness (a). Deterministic “roughness” models: LER (b) and LWR (c). The framed boxes indicate the unit cells of the numerical simulation; cf. Section 3.

Fig. 2.
Fig. 2.

(a) Visualization of parts of a FEM mesh discretizing a EUV line mask with sinusoidal LWR, amplitude a=10nm, and period dr=600nm (red, dark gray, absorber; light gray, buffer). Mesh elements discretizing the surrounding vacuum and the multilayer mirror below the structure are not shown. Mesh generated using the automatic mesh generator JCMgeo. (b) Pseudocolor visualization of the electromagnetic field intensity distribution I(r) on a logarithmic scale [colors blue-yellow-red (dark gray-white-light gray) correspond to log(I)=3+1] in a x-z cross section through the upper part of the computational domain containing the absorber structure. (c) Same, in a x-y cross section through the center of the absorber structure.

Fig. 3.
Fig. 3.

(a) and (b) Pseudocolor visualization of the result of FEM simulations for the structure from Fig. 2 for LER (left) and LWR (right) on a logarithmic scale [colors blue-yellow-red (dark gray-white-light gray) correspond to log(I)=51]. Numerical values are shown in (c) and (d) for the zeroth (red circles), first (blue crosses), and second (green triangles) orders in y of (a) and (b), respectively.

Fig. 4.
Fig. 4.

Black curves: Bessel squared functions according to Eqs. (7), (14), and (16). Data points: ratios of the diffraction intensities obtained from FEM simulations with respect to the FEM simulation of the undisturbed structure, denoted by I0: LER (red circles) and LWR (blue crosses). The damping ratio for the zeroth order in y is shown in (a) for LER and LWR at a=10nm and in (b) for LER at a=5nm, both with dr=600nm. (c) First order in y for LER at amplitude a=10nm and dr=300nm (closed circles) or 600 nm (open circles), respectively. (d) Ratio of third and first order in y for LWR at a=10nm and dr=600 nm.

Fig. 5.
Fig. 5.

The black polygonal chain connects the diffraction intensities of the undisturbed grating (triangles). The red points represent the sum of the LER-disturbed intensities over the orders along the y-direction. LER with a=10nm, dr=600nm.

Fig. 6.
Fig. 6.

FEM results for ky=±2π/dr at kx=0 in the case of LWR. Shown are the ratios of the orders +1 (red circles) and 1 (blue crosses) in y to the zeroth order I˜g(0,0) for six amplitudes a at c=100nm. The dashed line shows the slope 1/const obtained from the correction factor derived in Eq. (20) to account for the difference between I˜g(0,0) and Ig(0,0) as obtained by FEM and the Fraunhofer approximation, respectively.

Equations (21)

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r(x,y)=j=nnδ(xjd)*rect(xc)for allx[Nd2,Nd2],yR.
E(kx,ky)F{r}(kx,ky),
Ir(kx,ky)=|E(kx,ky)|2(csinkxc2kxc2sinNkxd2sinkxd22πδ(ky))2.
f(x,y)=j=nnδ(x(jdacos(2πy/dr)))*rect(xc)for allx[Nd2,Nd2],yR,
If(kx,ky)|F{f}(kx,ky)|2=|csinkxc2kxc2sinkxdN2sinkxd2m=imJm(akx)2πδ(kym2π/dr)|2.
If(kx,ky)|csinkxc2kxc2sinkxdN2sinkxd2imJm(akx)2π|2.
If(kx,m2π/dr)=Jm2(akx)×Ir(kx,0),
observablemIf(kx,m2π/dr)=Ir(kx,0)observablemJm2(akx)Ir(kx,0)mZJm2(akx)=Ir(kx,0).
g(x,y)=j=nnδ(xjd)*rect(xc2acos(2πy/dr))for allx[Nd2,Nd2],yR.
F{g}(kx,ky)=j=nnδ(xjd)*rect(xc2acos(2πy/dr))exp(ikxxikyy)dxdy=kx0j=nnexp(ikxjd)2kxsinkx(c2acos(2πy/dr))2exp(ikyy)dy=j=nnF{exp(ikxjd)2kxsinkx(c2acos(2πy/dr))2}(ky)=csinkxdN2sinkxd2[sinkxc2kxc2J0(kxa)2πδ(ky)+sinkxc2kxc22n=1(1)nJ2n(kxa)π[δ(ky+2n2π/dr)+δ(ky2n2π/dr)]+coskxc2kxc22n=1(1)nJ2n1(kxa)π[δ(ky+(2n1)2π/dr)+δ(ky(2n1)2π/dr)]].
F{g}(kx,ky)2πcsinkxdN2sinkxd2sinkxc2kxc2(1)|m|J2|m|(kxa).
F{g}(kx,ky)2πcsinkxdN2sinkxd2coskxc2kxc2(1)mJ2m1(kxa)form>0,
F{g}(kx,ky)2πcsinkxdN2sinkxd2coskxc2kxc2(1)|m|+1J2|m|+1(kxa)form0.
Ig(kx,2m2π/dr)=J2m2(akx)×Ir(kx,0),
If,g(kx,0)Ir(kx,0)=J02(akx)exp((akx)22),
Ig(kx,(2m1)2π/dr)Ig(kx,(2p1)2π/dr)=J2m12(akx)J2p12(akx),
F{f}(kx,ky)=j=nnF{exp(ikxjd)(c2acos(2πy/dr))}(ky)
F{f}(0,ky)=2πN(cδ(ky)a[δ(ky+2π/dr)+δ(ky2π/dr)]).
Ig(0,±2π/dr)Ig(0,0)=(ac)2
const=2I˜r(0,0)I˜r(2π/d,0)+I˜r(2π/d,0)×[2Ir(0,0)Ir(2pi/d,0)+Ir(2pi/d,0)]1=2I˜r(0,0)I˜r(2π/d,0)+I˜r(2π/d,0)×(sin(πc/d)πc/d)2=0.468.
(ac)2=const×I˜g(0,±2π/dr)I˜g(0,0),

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