Abstract

We investigate the impact of line-edge and line-width roughness (LER, LWR) on the measured diffraction intensities in angular resolved extreme ultraviolet (EUV) scatterometry for a periodic line-space structure designed for EUV lithography. LER and LWR with typical amplitudes of a few nanometers were previously neglected in the course of the profile reconstruction. The two-dimensional (2D) rigorous numerical simulations of the diffraction process for periodic structures are carried out with the finite element method providing a numerical solution of the 2D Helmholtz equation. To model roughness, multiple calculations are performed for domains with large periods, containing many pairs of line and space with stochastically chosen line and space widths. A systematic decrease of the mean efficiencies for higher diffraction orders along with increasing variances is observed and established for different degrees of roughness. In particular, we obtain simple analytical expressions for the bias in the mean efficiencies and the additional uncertainty contribution stemming from the presence of LER and/or LWR. As a consequence this bias can easily be included into the reconstruction model to provide accurate values for the evaluated profile parameters. We resolve the sensitivity of the reconstruction from this bias by using simulated data with LER/LWR perturbed efficiencies for multiple reconstructions. If the scattering efficiencies are bias-corrected, significant improvements are found in the reconstructed bottom and top widths toward the nominal values.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. M. Wurm, S. Bonifer, B. Bodermann, and J. Richter, “Deep ultraviolet scatterometer for dimensional characterization of nanostructures: system improvements and test measurements,” Meas. Sci. Technol. 22, 094024 (2011).
    [CrossRef]
  4. J. Perlich, F. Kamm, J. Rau, F. Scholze, and G. Ulm, “Characterization of extreme ultraviolet masks by extreme ultraviolet scatterometry,” J. Vac. Sci. Technol. B 22, 3059–3062 (2004).
    [CrossRef]
  5. F. Scholze and C. Laubis, “Use of EUV scatterometry for the characterization of line profiles and line roughness on photomasks,” in EMLC 2008, 24th European Mask and Lithography Conference (VDE Verlag, 2008), pp. 374–382.
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    [CrossRef]
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    [CrossRef]
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  26. T. A. Germer, “Effect of line and trench profile variation on specular and diffusive reflectance from periodic structure,” J. Opt. Soc. Am. A 24, 696–701 (2007).
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  27. T. Schuster, S. Rafler, V. F. Paz, F. 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]

2012

2011

M. Wurm, S. Bonifer, B. Bodermann, and J. Richter, “Deep ultraviolet scatterometer for dimensional characterization of nanostructures: system improvements and test measurements,” Meas. Sci. Technol. 22, 094024 (2011).
[CrossRef]

2010

2009

T. Schuster, S. Rafler, V. F. Paz, F. 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]

C. Raymond, M. Murnane, S. Prins, S. Sohail, H. Naqvi, J. McNeil, and J. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B 15, 361–368 (2009).
[CrossRef]

M.-A. Henn, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, and H. Gross, “On numerical reconstructions of lithographic masks in DUV scatterometry,” Proc. SPIE 7390, 73900Q (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]

H. Patrick, T. A. Germer, Y. Ding, H. Ro, L. Richter, and C. Soles, “In situ measurement of annealing-induced line shape evolution in nanoimprinted polymers using scatterometry,” Proc. SPIE 7271, 727128 (2009).
[CrossRef]

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

2008

H. Gross, A. Rathsfeld, F. Scholze, R. Model, and M. Bär, “Computational methods estimating uncertainties for profile reconstruction in scatterometry,” Proc. SPIE 6995, 6995OT (2008).

2007

2006

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]

2004

J. Perlich, F. Kamm, J. Rau, F. Scholze, and G. Ulm, “Characterization of extreme ultraviolet masks by extreme ultraviolet scatterometry,” J. Vac. Sci. Technol. B 22, 3059–3062 (2004).
[CrossRef]

2002

A. Tavrov, M. Totzeck, N. Kerwien, and H. Tiziani, “Rigorous coupled-wave analysis calculus of submicrometer interference pattern and resolving edge position versus signal-to-noise ratio,” Opt. Eng. 41, 1886–1892 (2002).
[CrossRef]

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139–156 (2002).
[CrossRef]

1998

O. Cessenat and B. Despres, “Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem,” SIAM J. Numer. Anal. 35, 255–299 (1998).
[CrossRef]

1997

1996

1995

1980

J. Chandezon, G. Raoult, and D. Maystre, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).
[CrossRef]

Al-Assaad, R.

Bär, M.

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

M.-A. Henn, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, and H. Gross, “On numerical reconstructions of lithographic masks in DUV scatterometry,” Proc. SPIE 7390, 73900Q (2009).
[CrossRef]

H. Gross, A. Rathsfeld, F. Scholze, R. Model, and M. Bär, “Computational methods estimating uncertainties for profile reconstruction in scatterometry,” Proc. SPIE 6995, 6995OT (2008).

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]

Bodermann, B.

M. Wurm, S. Bonifer, B. Bodermann, and J. Richter, “Deep ultraviolet scatterometer for dimensional characterization of nanostructures: system improvements and test measurements,” Meas. Sci. Technol. 22, 094024 (2011).
[CrossRef]

M.-A. Henn, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, and H. Gross, “On numerical reconstructions of lithographic masks in DUV scatterometry,” Proc. SPIE 7390, 73900Q (2009).
[CrossRef]

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]

Bonifer, S.

M. Wurm, S. Bonifer, B. Bodermann, and J. Richter, “Deep ultraviolet scatterometer for dimensional characterization of nanostructures: system improvements and test measurements,” Meas. Sci. Technol. 22, 094024 (2011).
[CrossRef]

Botten, L.

R. Petit and L. Botten, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).

Bunday, B.

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

Byrne, D.

Cessenat, O.

O. Cessenat and B. Despres, “Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem,” SIAM J. Numer. Anal. 35, 255–299 (1998).
[CrossRef]

Chandezon, J.

J. Chandezon, G. Raoult, and D. Maystre, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).
[CrossRef]

Ciarlet, P.

P. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, 1978).

Despres, B.

O. Cessenat and B. Despres, “Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem,” SIAM J. Numer. Anal. 35, 255–299 (1998).
[CrossRef]

Ding, Y.

H. Patrick, T. A. Germer, Y. Ding, H. Ro, L. Richter, and C. Soles, “In situ measurement of annealing-induced line shape evolution in nanoimprinted polymers using scatterometry,” Proc. SPIE 7271, 727128 (2009).
[CrossRef]

Elschner, J.

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139–156 (2002).
[CrossRef]

Elster, C.

Frenner, F.

T. Schuster, S. Rafler, V. F. Paz, F. 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]

Gaylord, T.

Germer, T.

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

Germer, T. A.

H. Patrick, T. A. Germer, Y. Ding, H. Ro, L. Richter, and C. Soles, “In situ measurement of annealing-induced line shape evolution in nanoimprinted polymers using scatterometry,” Proc. SPIE 7271, 727128 (2009).
[CrossRef]

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

Grann, E.

Gross, H.

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

M.-A. Henn, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, and H. Gross, “On numerical reconstructions of lithographic masks in DUV scatterometry,” Proc. SPIE 7390, 73900Q (2009).
[CrossRef]

H. Gross, A. Rathsfeld, F. Scholze, R. Model, and M. Bär, “Computational methods estimating uncertainties for profile reconstruction in scatterometry,” Proc. SPIE 6995, 6995OT (2008).

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]

Henn, M.-A.

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

M.-A. Henn, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, and H. Gross, “On numerical reconstructions of lithographic masks in DUV scatterometry,” Proc. SPIE 7390, 73900Q (2009).
[CrossRef]

Hinder, R.

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139–156 (2002).
[CrossRef]

Hosch, J.

C. Raymond, M. Murnane, S. Prins, S. Sohail, H. Naqvi, J. McNeil, and J. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B 15, 361–368 (2009).
[CrossRef]

Kamm, F.

J. Perlich, F. Kamm, J. Rau, F. Scholze, and G. Ulm, “Characterization of extreme ultraviolet masks by extreme ultraviolet scatterometry,” J. Vac. Sci. Technol. B 22, 3059–3062 (2004).
[CrossRef]

Kato, A.

Kerwien, N.

A. Tavrov, M. Totzeck, N. Kerwien, and H. Tiziani, “Rigorous coupled-wave analysis calculus of submicrometer interference pattern and resolving edge position versus signal-to-noise ratio,” Opt. Eng. 41, 1886–1892 (2002).
[CrossRef]

Lalanne, P.

Laubis, C.

F. Scholze and C. Laubis, “Use of EUV scatterometry for the characterization of line profiles and line roughness on photomasks,” in EMLC 2008, 24th European Mask and Lithography Conference (VDE Verlag, 2008), pp. 374–382.

Li, L.

Maystre, D.

J. Chandezon, G. Raoult, and D. Maystre, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).
[CrossRef]

McNeil, J.

C. Raymond, M. Murnane, S. Prins, S. Sohail, H. Naqvi, J. McNeil, and J. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B 15, 361–368 (2009).
[CrossRef]

Model, R.

M.-A. Henn, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, and H. Gross, “On numerical reconstructions of lithographic masks in DUV scatterometry,” Proc. SPIE 7390, 73900Q (2009).
[CrossRef]

H. Gross, A. Rathsfeld, F. Scholze, R. Model, and M. Bär, “Computational methods estimating uncertainties for profile reconstruction in scatterometry,” Proc. SPIE 6995, 6995OT (2008).

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]

Moharam, M.

Morris, G.

Murnane, M.

C. Raymond, M. Murnane, S. Prins, S. Sohail, H. Naqvi, J. McNeil, and J. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B 15, 361–368 (2009).
[CrossRef]

Naqvi, H.

C. Raymond, M. Murnane, S. Prins, S. Sohail, H. Naqvi, J. McNeil, and J. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B 15, 361–368 (2009).
[CrossRef]

Osten, W.

T. Schuster, S. Rafler, V. F. Paz, F. 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]

Patrick, H.

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

H. Patrick, T. A. Germer, Y. Ding, H. Ro, L. Richter, and C. Soles, “In situ measurement of annealing-induced line shape evolution in nanoimprinted polymers using scatterometry,” Proc. SPIE 7271, 727128 (2009).
[CrossRef]

Paz, V. F.

T. Schuster, S. Rafler, V. F. Paz, F. 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]

Perlich, J.

J. Perlich, F. Kamm, J. Rau, F. Scholze, and G. Ulm, “Characterization of extreme ultraviolet masks by extreme ultraviolet scatterometry,” J. Vac. Sci. Technol. B 22, 3059–3062 (2004).
[CrossRef]

Petit, R.

R. Petit and L. Botten, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).

Pommet, D.

Prins, S.

C. Raymond, M. Murnane, S. Prins, S. Sohail, H. Naqvi, J. McNeil, and J. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B 15, 361–368 (2009).
[CrossRef]

Rafler, S.

T. Schuster, S. Rafler, V. F. Paz, F. 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]

Raoult, G.

J. Chandezon, G. Raoult, and D. Maystre, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).
[CrossRef]

Rathsfeld, A.

M.-A. Henn, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, and H. Gross, “On numerical reconstructions of lithographic masks in DUV scatterometry,” Proc. SPIE 7390, 73900Q (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]

H. Gross, A. Rathsfeld, F. Scholze, R. Model, and M. Bär, “Computational methods estimating uncertainties for profile reconstruction in scatterometry,” Proc. SPIE 6995, 6995OT (2008).

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]

Rau, J.

J. Perlich, F. Kamm, J. Rau, F. Scholze, and G. Ulm, “Characterization of extreme ultraviolet masks by extreme ultraviolet scatterometry,” J. Vac. Sci. Technol. B 22, 3059–3062 (2004).
[CrossRef]

Raymond, C.

C. Raymond, M. Murnane, S. Prins, S. Sohail, H. Naqvi, J. McNeil, and J. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B 15, 361–368 (2009).
[CrossRef]

Richter, J.

M. Wurm, S. Bonifer, B. Bodermann, and J. Richter, “Deep ultraviolet scatterometer for dimensional characterization of nanostructures: system improvements and test measurements,” Meas. Sci. Technol. 22, 094024 (2011).
[CrossRef]

Richter, L.

H. Patrick, T. A. Germer, Y. Ding, H. Ro, L. Richter, and C. Soles, “In situ measurement of annealing-induced line shape evolution in nanoimprinted polymers using scatterometry,” Proc. SPIE 7271, 727128 (2009).
[CrossRef]

Ro, H.

H. Patrick, T. A. Germer, Y. Ding, H. Ro, L. Richter, and C. Soles, “In situ measurement of annealing-induced line shape evolution in nanoimprinted polymers using scatterometry,” Proc. SPIE 7271, 727128 (2009).
[CrossRef]

Schmidt, G.

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139–156 (2002).
[CrossRef]

Scholze, F.

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

A. Kato and F. Scholze, “Effect of line roughness on the diffraction intensities in angular resolved scatterometry,” Appl. Opt. 49, 6102–6111 (2010).
[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]

H. Gross, A. Rathsfeld, F. Scholze, R. Model, and M. Bär, “Computational methods estimating uncertainties for profile reconstruction in scatterometry,” Proc. SPIE 6995, 6995OT (2008).

J. Perlich, F. Kamm, J. Rau, F. Scholze, and G. Ulm, “Characterization of extreme ultraviolet masks by extreme ultraviolet scatterometry,” J. Vac. Sci. Technol. B 22, 3059–3062 (2004).
[CrossRef]

F. Scholze and C. Laubis, “Use of EUV scatterometry for the characterization of line profiles and line roughness on photomasks,” in EMLC 2008, 24th European Mask and Lithography Conference (VDE Verlag, 2008), pp. 374–382.

Schuster, T.

T. Schuster, S. Rafler, V. F. Paz, F. 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]

Silver, R.

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

Sohail, S.

C. Raymond, M. Murnane, S. Prins, S. Sohail, H. Naqvi, J. McNeil, and J. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B 15, 361–368 (2009).
[CrossRef]

Soles, C.

H. Patrick, T. A. Germer, Y. Ding, H. Ro, L. Richter, and C. Soles, “In situ measurement of annealing-induced line shape evolution in nanoimprinted polymers using scatterometry,” Proc. SPIE 7271, 727128 (2009).
[CrossRef]

Tarantola, A.

A. Tarantola, Inverse Problem Theory (Elsevier, 1987).

Tavrov, A.

A. Tavrov, M. Totzeck, N. Kerwien, and H. Tiziani, “Rigorous coupled-wave analysis calculus of submicrometer interference pattern and resolving edge position versus signal-to-noise ratio,” Opt. Eng. 41, 1886–1892 (2002).
[CrossRef]

Tiziani, H.

A. Tavrov, M. Totzeck, N. Kerwien, and H. Tiziani, “Rigorous coupled-wave analysis calculus of submicrometer interference pattern and resolving edge position versus signal-to-noise ratio,” Opt. Eng. 41, 1886–1892 (2002).
[CrossRef]

Totzeck, M.

A. Tavrov, M. Totzeck, N. Kerwien, and H. Tiziani, “Rigorous coupled-wave analysis calculus of submicrometer interference pattern and resolving edge position versus signal-to-noise ratio,” Opt. Eng. 41, 1886–1892 (2002).
[CrossRef]

Ulm, G.

J. Perlich, F. Kamm, J. Rau, F. Scholze, and G. Ulm, “Characterization of extreme ultraviolet masks by extreme ultraviolet scatterometry,” J. Vac. Sci. Technol. B 22, 3059–3062 (2004).
[CrossRef]

Wurm, M.

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

M. Wurm, S. Bonifer, B. Bodermann, and J. Richter, “Deep ultraviolet scatterometer for dimensional characterization of nanostructures: system improvements and test measurements,” Meas. Sci. Technol. 22, 094024 (2011).
[CrossRef]

M.-A. Henn, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, and H. Gross, “On numerical reconstructions of lithographic masks in DUV scatterometry,” Proc. SPIE 7390, 73900Q (2009).
[CrossRef]

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]

M. Wurm, “Über die dimensionelle Charakterisierung von Gitterstrukturen auf Fotomasken mit einem neuartigen DUV-Scatterometer,” Ph.D. thesis (Friedrich-Schiller-Universität Jena, 2008).

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[CrossRef]

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[CrossRef]

Meas. Sci. Technol.

M. Wurm, S. Bonifer, B. Bodermann, and J. Richter, “Deep ultraviolet scatterometer for dimensional characterization of nanostructures: system improvements and test measurements,” Meas. Sci. Technol. 22, 094024 (2011).
[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]

Measurement

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]

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T. Schuster, S. Rafler, V. F. Paz, F. 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).
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[CrossRef]

H. Gross, A. Rathsfeld, F. Scholze, R. Model, and M. Bär, “Computational methods estimating uncertainties for profile reconstruction in scatterometry,” Proc. SPIE 6995, 6995OT (2008).

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Figures (8)

Fig. 1.
Fig. 1.

Super cell containing many profile lines used for roughness modeling by randomly changed center positions for LER and randomly varied line widths for LWR (left); cross section of a single line whose profile parameters such as the horizontal coordinates of the corners p 2 , p 7 and the height p 6 have to be reconstructed (right).

Fig. 2.
Fig. 2.

Simulated diffraction patterns for randomly perturbed line-space structures at λ = 13.389 nm . Diamond symbols depict the mean efficiencies of all samples; 24 lines per FEM domain, two different random perturbations. (a) LER with σ x = 5.6 nm ( σ edge = 5.6 nm ) and (b) LWR with σ C D = 5.6 nm ( σ edge = 2.8 nm ).

Fig. 3.
Fig. 3.

Normalized deviations from the efficiencies of the unperturbed reference line structure, depicted as circles; 24 lines per FEM domain; diamond symbols represent the mean over all samples; dashed lines indicate the mean ± standard deviation ; solid lines depict the exponential approximation according to Eq. (5); two different random perturbations: (a) LER with σ x = 5.6 nm ( σ edge = 5.6 nm ) and (b) LWR with σ C D = 5.6 nm ( σ edge = 2.8 nm ).

Fig. 4.
Fig. 4.

Normalized deviations from the efficiencies of the unperturbed reference line structure, depicted as circles; 24 lines per FEM domain; diamond symbols represent the mean over all samples; dashed lines indicate the mean ± standard deviation ; solid lines depict the exponential approximation according to Eq. (5); two different random LEWR perturbations: (a) with σ x = σ C D = 5.6 nm ( σ edge = 6.26 nm ) and (b) with σ x = σ C D = 2.8 nm ( σ edge = 3.13 nm ).

Fig. 5.
Fig. 5.

Normalized deviations from the efficiencies of the unperturbed reference line structure, depicted as circles; 48 lines per FEM domain; diamond symbols represent the mean; dashed lines indicate the mean ± standard deviation and solid lines depict the exponential approximations according to Eq. (5): (a) LER perturbations with σ x = 5.6 nm ( σ edge = 5.6 nm ), (b) LWR perturbations with σ C D = 5.6 nm ( σ edge = 2.8 nm ), and (c) LEWR perturbations with σ x = σ C D = 5.6 nm ( σ edge = 6.26 nm ).

Fig. 6.
Fig. 6.

Variation coefficients of the efficiencies for 2% LER, LWR, and LEWR perturbations obtained with 24 lines per super-cell [left column: (a), (c), and (e)] and with 48 lines per super cell [right column: (b), (d), and (f)]; solid lines depict curves applying Eq. (6) with σ r the values of Table 2 characterizing the damping of the mean efficiencies.

Fig. 7.
Fig. 7.

Impact of normally distributed random perturbations of the center positions and widths (1% LEWR: σ x = σ C D = 2.8 nm ; 100 samples) on the reconstructed parameters, right lower and upper corner p 2 , p 7 and height p 6 of the TaN absorber layer (a), (b), on width CDm of the absorber layer at middle height (c), and on the side-wall angle SWA of the absorber line (d).

Fig. 8.
Fig. 8.

Impact of normally distributed random perturbations of the center positions and widths (1% LEWR: σ x = σ C D = 2.8 nm ; 100 samples) on the reconstructed parameters, right lower and upper corner p 2 , p 7 and height p 6 of the TaN absorber layer (a),(b), on width CDm of the absorber layer at middle height (c), and on the side-wall angle SWA of the absorber line (d); outliers depicted as plus signs; bias-corrected datasets of efficiencies are used.

Tables (2)

Tables Icon

Table 1. Complex Indices of Refractiona

Tables Icon

Table 2. Results σ r Applying the Proposed Exponential Approximation of Eq. (5) for all Calculated Examples and Comparison with the Expected Values σ edge = σ x 2 + σ C D 2 / 4 for σ x = σ C D σ edge = 1.118 · σ x (all σ values are given in nanometers)

Equations (8)

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Δ u ( x , y ) + k 2 u ( x , y ) = 0 .
χ 2 ( p ) = f ( p ) y 2 = j = 1 m ω j [ f j ( p ) y j ] 2 .
σ j 2 = ( a · f j ( p ) ) 2 + b 2 .
y j = f j ( p ) + ϵ j .
f j , ref ( p ) f j , pert ( p ) ¯ f j , ref ( p ) 1 exp ( σ r 2 k j 2 ) = 1 exp ( ( α σ edge ) 2 k j 2 ) , k j 2 π n j d .
Σ ( N , n j ) Σ ( N , k j , σ r ) 24 N ( 1 exp ( σ r 2 k j 2 / 3 ) ) .
y j = exp ( σ r 2 k j 2 ) · f j ( p ) + ϵ j ,
σ j 2 = ( a · exp ( σ r 2 k j 2 ) · f j ( p ) ) 2 + b 2 .

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