Abstract

Scatterometry is an optical technique that has been studied and tested in recent years in semiconductor fabrication metrology for critical dimensions. Previous work presented an iterative linearized method to retrieve surface-relief profile parameters from reflectance measurements upon diffraction. With the iterative linear solution model in this work, rigorous models are developed to represent the random and deterministic or offset errors in scatterometric measurements. The propagation of different types of error from the measurement data to the profile parameter estimates is then presented. The improvement in solution accuracies is then demonstrated with theoretical and experimental data by adjusting for the offset errors. In a companion paper (in process) an improved optimization method is presented to account for unknown offset errors in the measurements based on the offset error model.

© 2007 Optical Society of America

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References

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  1. 2004 International Technology Roadmap for Semiconductors, "Metrology 2004 Update," ITRS Web Resources, http://www.itrs.net/Links/2005ITRS/Metrology2005.pdf.
  2. S. S. H. Naqvi, J. R. McNeil, R. H. Krukar, and K. P. Bishop, "Scatterometry and the simulation of diffraction-based metrology," Microlithogr. World 2, 5-16 (1993).
  3. J. Bischoff, J. W. Baumgart, J. J. Bauer, and H. Truckenbrodt, "Light-scattering-based micrometrology," in Proc. SPIE 2775, 251-262 (1996).
    [CrossRef]
  4. C. J. Raymond, M. Littau, A. Chuprin, and S. Ward, "Comparison of solutions to the scatterometry inverse problem," in Proc. SPIE 5375, 564-575 (2004).
    [CrossRef]
  5. E. M. Drege and D. M. Byrne, "Lithographic process monitoring using diffraction measurements," in Proc. SPIE 3998, 147-157 (2000).
    [CrossRef]
  6. E. M. Drege, J. A. Reed, and D. M. Byrne, "Linearized inversion of scatterometric data to obtain surface profile information," Opt. Eng. 41, 225-236 (2002).
    [CrossRef]
  7. E. M. Drege, R. M. Al-Assaad, and D. M. Byrne, "Mathematical analysis of inverse scatterometry," in Proc. SPIE 4689, 151-162 (2002).
    [CrossRef]
  8. R. M. Al-Assaad, E. M. Drege, and D. M. Byrne, "Profile parameter accuracy determined from scatterometric measurements," in Proc. SPIE 4692, 17-28 (2002).
    [CrossRef]
  9. M. G. Moharam and T. K. Gaylord, "Three-dimensional vector coupled-wave analysis of planar-grating diffraction," J. Opt. Soc. Am. 73, 1105-1112 (1983).
    [CrossRef]
  10. N. Chateau and J. Hugonin, "Algorithm for the rigorous coupled-wave analysis of grating diffraction," J. Opt. Soc. Am. A 11, 1321-1331 (1994).
    [CrossRef]
  11. C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice Hall, 1974).
  12. J. W. Goodman, Statistical Optics (Wiley, 2000).

2004 (1)

C. J. Raymond, M. Littau, A. Chuprin, and S. Ward, "Comparison of solutions to the scatterometry inverse problem," in Proc. SPIE 5375, 564-575 (2004).
[CrossRef]

2002 (3)

E. M. Drege, J. A. Reed, and D. M. Byrne, "Linearized inversion of scatterometric data to obtain surface profile information," Opt. Eng. 41, 225-236 (2002).
[CrossRef]

E. M. Drege, R. M. Al-Assaad, and D. M. Byrne, "Mathematical analysis of inverse scatterometry," in Proc. SPIE 4689, 151-162 (2002).
[CrossRef]

R. M. Al-Assaad, E. M. Drege, and D. M. Byrne, "Profile parameter accuracy determined from scatterometric measurements," in Proc. SPIE 4692, 17-28 (2002).
[CrossRef]

2000 (1)

E. M. Drege and D. M. Byrne, "Lithographic process monitoring using diffraction measurements," in Proc. SPIE 3998, 147-157 (2000).
[CrossRef]

1996 (1)

J. Bischoff, J. W. Baumgart, J. J. Bauer, and H. Truckenbrodt, "Light-scattering-based micrometrology," in Proc. SPIE 2775, 251-262 (1996).
[CrossRef]

1994 (1)

1993 (1)

S. S. H. Naqvi, J. R. McNeil, R. H. Krukar, and K. P. Bishop, "Scatterometry and the simulation of diffraction-based metrology," Microlithogr. World 2, 5-16 (1993).

1983 (1)

Al-Assaad, R. M.

E. M. Drege, R. M. Al-Assaad, and D. M. Byrne, "Mathematical analysis of inverse scatterometry," in Proc. SPIE 4689, 151-162 (2002).
[CrossRef]

R. M. Al-Assaad, E. M. Drege, and D. M. Byrne, "Profile parameter accuracy determined from scatterometric measurements," in Proc. SPIE 4692, 17-28 (2002).
[CrossRef]

Bauer, J. J.

J. Bischoff, J. W. Baumgart, J. J. Bauer, and H. Truckenbrodt, "Light-scattering-based micrometrology," in Proc. SPIE 2775, 251-262 (1996).
[CrossRef]

Baumgart, J. W.

J. Bischoff, J. W. Baumgart, J. J. Bauer, and H. Truckenbrodt, "Light-scattering-based micrometrology," in Proc. SPIE 2775, 251-262 (1996).
[CrossRef]

Bischoff, J.

J. Bischoff, J. W. Baumgart, J. J. Bauer, and H. Truckenbrodt, "Light-scattering-based micrometrology," in Proc. SPIE 2775, 251-262 (1996).
[CrossRef]

Bishop, K. P.

S. S. H. Naqvi, J. R. McNeil, R. H. Krukar, and K. P. Bishop, "Scatterometry and the simulation of diffraction-based metrology," Microlithogr. World 2, 5-16 (1993).

Byrne, D. M.

E. M. Drege, J. A. Reed, and D. M. Byrne, "Linearized inversion of scatterometric data to obtain surface profile information," Opt. Eng. 41, 225-236 (2002).
[CrossRef]

E. M. Drege, R. M. Al-Assaad, and D. M. Byrne, "Mathematical analysis of inverse scatterometry," in Proc. SPIE 4689, 151-162 (2002).
[CrossRef]

R. M. Al-Assaad, E. M. Drege, and D. M. Byrne, "Profile parameter accuracy determined from scatterometric measurements," in Proc. SPIE 4692, 17-28 (2002).
[CrossRef]

E. M. Drege and D. M. Byrne, "Lithographic process monitoring using diffraction measurements," in Proc. SPIE 3998, 147-157 (2000).
[CrossRef]

Chateau, N.

Chuprin, A.

C. J. Raymond, M. Littau, A. Chuprin, and S. Ward, "Comparison of solutions to the scatterometry inverse problem," in Proc. SPIE 5375, 564-575 (2004).
[CrossRef]

Drege, E. M.

E. M. Drege, J. A. Reed, and D. M. Byrne, "Linearized inversion of scatterometric data to obtain surface profile information," Opt. Eng. 41, 225-236 (2002).
[CrossRef]

E. M. Drege, R. M. Al-Assaad, and D. M. Byrne, "Mathematical analysis of inverse scatterometry," in Proc. SPIE 4689, 151-162 (2002).
[CrossRef]

R. M. Al-Assaad, E. M. Drege, and D. M. Byrne, "Profile parameter accuracy determined from scatterometric measurements," in Proc. SPIE 4692, 17-28 (2002).
[CrossRef]

E. M. Drege and D. M. Byrne, "Lithographic process monitoring using diffraction measurements," in Proc. SPIE 3998, 147-157 (2000).
[CrossRef]

Gaylord, T. K.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 2000).

Hanson, R. J.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice Hall, 1974).

Hugonin, J.

Krukar, R. H.

S. S. H. Naqvi, J. R. McNeil, R. H. Krukar, and K. P. Bishop, "Scatterometry and the simulation of diffraction-based metrology," Microlithogr. World 2, 5-16 (1993).

Lawson, C. L.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice Hall, 1974).

Littau, M.

C. J. Raymond, M. Littau, A. Chuprin, and S. Ward, "Comparison of solutions to the scatterometry inverse problem," in Proc. SPIE 5375, 564-575 (2004).
[CrossRef]

McNeil, J. R.

S. S. H. Naqvi, J. R. McNeil, R. H. Krukar, and K. P. Bishop, "Scatterometry and the simulation of diffraction-based metrology," Microlithogr. World 2, 5-16 (1993).

Moharam, M. G.

Naqvi, S. S. H.

S. S. H. Naqvi, J. R. McNeil, R. H. Krukar, and K. P. Bishop, "Scatterometry and the simulation of diffraction-based metrology," Microlithogr. World 2, 5-16 (1993).

Raymond, C. J.

C. J. Raymond, M. Littau, A. Chuprin, and S. Ward, "Comparison of solutions to the scatterometry inverse problem," in Proc. SPIE 5375, 564-575 (2004).
[CrossRef]

Reed, J. A.

E. M. Drege, J. A. Reed, and D. M. Byrne, "Linearized inversion of scatterometric data to obtain surface profile information," Opt. Eng. 41, 225-236 (2002).
[CrossRef]

Truckenbrodt, H.

J. Bischoff, J. W. Baumgart, J. J. Bauer, and H. Truckenbrodt, "Light-scattering-based micrometrology," in Proc. SPIE 2775, 251-262 (1996).
[CrossRef]

Ward, S.

C. J. Raymond, M. Littau, A. Chuprin, and S. Ward, "Comparison of solutions to the scatterometry inverse problem," in Proc. SPIE 5375, 564-575 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Microlithogr. World (1)

S. S. H. Naqvi, J. R. McNeil, R. H. Krukar, and K. P. Bishop, "Scatterometry and the simulation of diffraction-based metrology," Microlithogr. World 2, 5-16 (1993).

Opt. Eng. (1)

E. M. Drege, J. A. Reed, and D. M. Byrne, "Linearized inversion of scatterometric data to obtain surface profile information," Opt. Eng. 41, 225-236 (2002).
[CrossRef]

Proc. SPIE (5)

E. M. Drege, R. M. Al-Assaad, and D. M. Byrne, "Mathematical analysis of inverse scatterometry," in Proc. SPIE 4689, 151-162 (2002).
[CrossRef]

R. M. Al-Assaad, E. M. Drege, and D. M. Byrne, "Profile parameter accuracy determined from scatterometric measurements," in Proc. SPIE 4692, 17-28 (2002).
[CrossRef]

J. Bischoff, J. W. Baumgart, J. J. Bauer, and H. Truckenbrodt, "Light-scattering-based micrometrology," in Proc. SPIE 2775, 251-262 (1996).
[CrossRef]

C. J. Raymond, M. Littau, A. Chuprin, and S. Ward, "Comparison of solutions to the scatterometry inverse problem," in Proc. SPIE 5375, 564-575 (2004).
[CrossRef]

E. M. Drege and D. M. Byrne, "Lithographic process monitoring using diffraction measurements," in Proc. SPIE 3998, 147-157 (2000).
[CrossRef]

Other (3)

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice Hall, 1974).

J. W. Goodman, Statistical Optics (Wiley, 2000).

2004 International Technology Roadmap for Semiconductors, "Metrology 2004 Update," ITRS Web Resources, http://www.itrs.net/Links/2005ITRS/Metrology2005.pdf.

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Figures (9)

Fig. 1
Fig. 1

General 2 θ configuration.

Fig. 2
Fig. 2

Cross section of a grating on a substrate.

Fig. 3
Fig. 3

Propagation of solution by the iterative linear inverse method along the sse curve for the depth parameter from a starting or design value based on design ( 500 nm ) to the actual value ( 510 nm ) in the absence of noise in measurements. The angular reflectance data sets used to generate the sse curve were modeled with 90 data points from 0° to 89° with 632.8 nm wavelength and TE polarization. The actual top width, period, and slope angle values were set fixed for all data sets at 510 nm , 1 μ m , and 0°, respectively.

Fig. 4
Fig. 4

Propagation of solution by the iterative linear inverse method along the sse surface for the depth and the top width parameters from starting values for both based on design ( 500 nm ) to the actual values ( 510 nm ) in the absence of noise in measurements. The angular reflectance data sets used to generate the sse surface were modeled with 90 data points from 0° to 89° with 632.8 nm wavelength and TE polarization. The actual period and slope angle values were set fixed for all data sets at 1 μ m and 0°, respectively.

Fig. 5
Fig. 5

Propagation of the slope angle solution by the iterative linear inverse method along the sse curve for the slope angle parameter from a guessing value 0° to the actual value 5° at 1% noise level in measurements (top curve). The difference between the actual and the adjusted parameter estimate values is shown to be within a ± 2 σ Δ p uncertainty range.

Fig. 6
Fig. 6

Representing random noise level in measurements and modeling its distribution.

Fig. 7
Fig. 7

Simulated angular reflectance measurements with two different noise distributions and the reflectance curves for the obtained parameter estimates.

Fig. 8
Fig. 8

SEM profile image and the resolved profile from scatterometry for a 1 μ m grating composed of developed PMMA on a silicon wafer.

Fig. 9
Fig. 9

Experimental data and generated reflectance values from solution versus incident angle for sample 1 and sample 2 gratings: (a) biased noisy measurements, (b) unbiased noisy measurements.

Tables (2)

Tables Icon

Table 1 Estimated Parameter Values and Errors for Two Different Noise Levels

Tables Icon

Table 2 Estimated Parameter Values and Errors from Biased and Unbiased Measurement Sets for Two Fabricated Grating Samples

Equations (35)

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sin θ inc + sin θ m = m λ a .
R ( u i , p ) R RCWT ( u i , p ) .
R RCWT ( u i , p ( a ) ) R RCWT ( u i , p ( 0 ) ) + j = 1 l [ R ( u i , p ) p j ] p j ( 0 ) Δ p j .
R ( u i , p ( a ) ) R RCWT ( u i , p ( 0 ) ) = Δ R ( u i ) j = 1 N [ R RCWT ( u i , p ) p j ] p j ( 0 ) Δ p j .
Δ R = M Δ p ,
M i j = [ R RCWT ( u i , p ) p j ] p j ( 0 ) .
Δ p = ( M T M ) 1 M T Δ R .
r i = Δ R exp i j = 1 l M i j Δ p j ,
i = 1 N r i 2 = i = 1 N ( Δ R i j = 1 l M i j Δ p j ) 2 ,
Δ p q ( i = 1 N r i 2 ) = Δ p q { i = 1 N ( Δ R i j = 1 l M i j Δ p j ) 2 } = 0 .
i = 1 N [ 2 ( Δ R i j = 1 l M i j Δ p j ) M i q ] = 0 ,
i = 1 N M i q Δ R i = j = 1 l Δ p j i = 1 N M i j M i q .
M T Δ R = ( M T M ) Δ p Δ p = ( M T M ) 1 M T Δ R .
C i j = E [ ( x i x ¯ i ) ( x j x ¯ j ) ] .
C Δ p = E [ ( Δ p Δ p ¯ ) ( Δ p Δ p ¯ ) T ] = E [ { ( M T M ) 1 M T Δ R ( M T M ) 1 M T Δ R ¯ } { ( M T M ) 1 M T Δ R ( M T M ) 1 M T Δ R ¯ } T ] .
C Δ p = ( M T M ) 1 M T C Δ R M ( M T M ) 1 .
C Δ p = σ noise 2 ( M T M ) 1 ,
σ Δ p j 2 = σ noise 2 ( M T M ) j j 1 ,
m Δ p j 2 = σ Δ p j 2 σ noise 2 = ( M T M ) j j 1 .
μ Δ p = ( M T M ) 1 M T μ Δ R .
p act ( p est k 1 + Δ p k ) = p act p est k = δ Δ p k .
lim k δ Δ p k = 0 lim k ( p act ( p est k 1 + Δ p k ) ) = lim k ( p act p est k ) = p act p est = 0 .
p act ( p est k 1 + Δ p k ) = p act p est k = ϵ Δ p k μ Δ p k + δ Δ p k ,
lim k δ Δ p k = 0 lim k ( p act p est k ) = p act p est = ϵ Δ p μ Δ p .
K σ Δ p ϵ Δ p K σ Δ p .
K σ Δ p p act p est + μ Δ p K σ Δ p .
p act j + μ Δ p j K σ Δ p j p est j p act j + μ Δ p j + K Δ p j ,
p est j ¯ p est j = ϵ Δ p j .
p est j ¯ = E [ p est j ] = p act j + μ Δ p j .
p adj j = p est j μ Δ p j .
p adj j ¯ = E [ p est j μ Δ p j ] = p act j .
p ( ϵ Δ R i ) = 1 2 π σ noise 2 exp ( ϵ Δ R i 2 2 σ noise 2 ) .
R rms = 1 ( θ 2 θ 1 ) θ 1 θ 2 R RCWT 2 ( θ , p ) d θ .
σ noise = f R rms .
R noise = R sim + ϵ Δ R + μ Δ R .

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