Abstract

The capability of coherence scanning interferometry has been extended recently to include the determination of the interfacial surface roughness between a thin film and a substrate when the surface perturbations are less than 10  nm in magnitude. The technique relies on introducing a first-order approximation to the helical complex field (HCF) function. This approximation of the HCF function enables a least-squares optimization to be carried out in every pixel of the scanned area to determine the heights of the substrate and/or the film layers in a multilayer stack. The method is fast but its implementation assumes that the noise variance in the frequency domain is statistically the same over the scanned area of the sample. This results in reconstructed surfaces that contain statistical fluctuations. In this paper we present an alternative least-squares optimization method, which takes into account the distribution of the noise variance-covariance in the frequency domain. The method is tested using results from a simulator and these show a significant improvement in the quality of the reconstructed surfaces.

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References

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  1. P. de Groot, “Coherence scanning interferometry,” in Optical Measurement of Surface Topography (Springer, 2011), Chap. 9, pp. 187–208.
  2. T. R. Thomas, “Other measurement topics,” in Rough Surfaces, 2nd ed. (Imperial College, 1999), Chap. 3, pp. 35–61.
  3. H. Yoshino, R. Smith, J. M. Walls, and D. Mansfield, “The development of thin film metrology by coherence scanning interferometry,” Proc. SPIE 9749, 97490P(2016).
  4. F. Blateyron, “The areal field parameters,” in Characterisation of Areal Surface Texture (Springer, 2013), pp. 15–43.
  5. B. S. Lee and T. C. Strand, “Profilometry with a coherence scanning microscope,” Appl. Opt. 29, 3784–3788 (1990).
    [Crossref]
  6. P. J. De Groot and X. C. de Lega, “Transparent film profiling and analysis by interference microscopy,” Proc. SPIE 7064, 70640I (2008).
  7. A. Bosseboeuf and S. Petitgrand, “Application of microscopic interferometry techniques in the MEMS field,” Proc. SPIE 5145, 1–16 (2003).
  8. S. Kim and G. Kim, “Method for measuring a thickness profile and a refractive index using white-light scanning interferometry and recording medium therefor,” U.S. patent6,545,763 (8 April 2003).
  9. P. J. de Groot, “Interferometry method for ellipsometry, reflectometry, and scatterometry measurements, including characterization of thin film structures,” U.S. patent7,403,289 (22 July 2008).
  10. Y. Ghim and S. Kim, “Spectrally resolved white-light interferometry for 3D inspection of a thin-film layer structure,” Appl. Opt. 48, 799–803 (2009).
    [Crossref]
  11. I. Abdulhalim, “Spectroscopic interference microscopy technique for measurement of layer parameters,” Meas. Sci. Technol. 12, 1996–2001 (2001).
    [Crossref]
  12. D. Mansfield, “Apparatus for and a method of determining characteristics of thin-layer structures using low-coherence interferometry,” WO patentPCT/GB2005/002,783 (19 January 2006).
  13. D. Mansfield, “The distorted helix: thin film extraction from scanning white light interferometry,” Proc. SPIE 6186, 61860O (2006).
  14. H. Yoshino, P. M. Kaminski, R. Smith, J. M. Walls, and D. Mansfield, “Refractive index determination by coherence scanning interferometry,” Appl. Opt. 55, 4253–4260 (2016).
    [Crossref]
  15. H. Yoshino, A. Abbas, P. M. Kaminski, R. Smith, J. M. Walls, and D. Mansfield, “Measurement of thin film interfacial surface roughness by coherence scanning interferometry,” J. Appl. Phys. 121, 105303 (2017).
    [Crossref]
  16. D. Mansfield, “Extraction of film interface surfaces from scanning white light interferometry,” Proc. SPIE 7101, 71010U (2008).
  17. T. Lewis and P. Odell, “A generalization of the Gauss-Markov theorem,” J. Am. Stat. Assoc. 61, 1063–1066 (1966).
    [Crossref]
  18. K. Sekihara, Introduction to Statistical Signal Processing (Kyouritsu Shuppan, 2011).
  19. D. I. Mansfield, “Apparatus for and a method of determining surface characteristics,” U.S. patent13/352,687 (12July2012).
  20. A. Albert, “The Gauss-Markov theorem for regression models with possibly singular covariances,” SIAM J. Appl. Math. 24, 182–187 (1973).
    [Crossref]

2017 (1)

H. Yoshino, A. Abbas, P. M. Kaminski, R. Smith, J. M. Walls, and D. Mansfield, “Measurement of thin film interfacial surface roughness by coherence scanning interferometry,” J. Appl. Phys. 121, 105303 (2017).
[Crossref]

2016 (2)

H. Yoshino, R. Smith, J. M. Walls, and D. Mansfield, “The development of thin film metrology by coherence scanning interferometry,” Proc. SPIE 9749, 97490P(2016).

H. Yoshino, P. M. Kaminski, R. Smith, J. M. Walls, and D. Mansfield, “Refractive index determination by coherence scanning interferometry,” Appl. Opt. 55, 4253–4260 (2016).
[Crossref]

2009 (1)

2008 (2)

P. J. De Groot and X. C. de Lega, “Transparent film profiling and analysis by interference microscopy,” Proc. SPIE 7064, 70640I (2008).

D. Mansfield, “Extraction of film interface surfaces from scanning white light interferometry,” Proc. SPIE 7101, 71010U (2008).

2006 (1)

D. Mansfield, “The distorted helix: thin film extraction from scanning white light interferometry,” Proc. SPIE 6186, 61860O (2006).

2003 (1)

A. Bosseboeuf and S. Petitgrand, “Application of microscopic interferometry techniques in the MEMS field,” Proc. SPIE 5145, 1–16 (2003).

2001 (1)

I. Abdulhalim, “Spectroscopic interference microscopy technique for measurement of layer parameters,” Meas. Sci. Technol. 12, 1996–2001 (2001).
[Crossref]

1990 (1)

1973 (1)

A. Albert, “The Gauss-Markov theorem for regression models with possibly singular covariances,” SIAM J. Appl. Math. 24, 182–187 (1973).
[Crossref]

1966 (1)

T. Lewis and P. Odell, “A generalization of the Gauss-Markov theorem,” J. Am. Stat. Assoc. 61, 1063–1066 (1966).
[Crossref]

Abbas, A.

H. Yoshino, A. Abbas, P. M. Kaminski, R. Smith, J. M. Walls, and D. Mansfield, “Measurement of thin film interfacial surface roughness by coherence scanning interferometry,” J. Appl. Phys. 121, 105303 (2017).
[Crossref]

Abdulhalim, I.

I. Abdulhalim, “Spectroscopic interference microscopy technique for measurement of layer parameters,” Meas. Sci. Technol. 12, 1996–2001 (2001).
[Crossref]

Albert, A.

A. Albert, “The Gauss-Markov theorem for regression models with possibly singular covariances,” SIAM J. Appl. Math. 24, 182–187 (1973).
[Crossref]

Blateyron, F.

F. Blateyron, “The areal field parameters,” in Characterisation of Areal Surface Texture (Springer, 2013), pp. 15–43.

Bosseboeuf, A.

A. Bosseboeuf and S. Petitgrand, “Application of microscopic interferometry techniques in the MEMS field,” Proc. SPIE 5145, 1–16 (2003).

de Groot, P.

P. de Groot, “Coherence scanning interferometry,” in Optical Measurement of Surface Topography (Springer, 2011), Chap. 9, pp. 187–208.

De Groot, P. J.

P. J. De Groot and X. C. de Lega, “Transparent film profiling and analysis by interference microscopy,” Proc. SPIE 7064, 70640I (2008).

P. J. de Groot, “Interferometry method for ellipsometry, reflectometry, and scatterometry measurements, including characterization of thin film structures,” U.S. patent7,403,289 (22 July 2008).

de Lega, X. C.

P. J. De Groot and X. C. de Lega, “Transparent film profiling and analysis by interference microscopy,” Proc. SPIE 7064, 70640I (2008).

Ghim, Y.

Kaminski, P. M.

H. Yoshino, A. Abbas, P. M. Kaminski, R. Smith, J. M. Walls, and D. Mansfield, “Measurement of thin film interfacial surface roughness by coherence scanning interferometry,” J. Appl. Phys. 121, 105303 (2017).
[Crossref]

H. Yoshino, P. M. Kaminski, R. Smith, J. M. Walls, and D. Mansfield, “Refractive index determination by coherence scanning interferometry,” Appl. Opt. 55, 4253–4260 (2016).
[Crossref]

Kim, G.

S. Kim and G. Kim, “Method for measuring a thickness profile and a refractive index using white-light scanning interferometry and recording medium therefor,” U.S. patent6,545,763 (8 April 2003).

Kim, S.

Y. Ghim and S. Kim, “Spectrally resolved white-light interferometry for 3D inspection of a thin-film layer structure,” Appl. Opt. 48, 799–803 (2009).
[Crossref]

S. Kim and G. Kim, “Method for measuring a thickness profile and a refractive index using white-light scanning interferometry and recording medium therefor,” U.S. patent6,545,763 (8 April 2003).

Lee, B. S.

Lewis, T.

T. Lewis and P. Odell, “A generalization of the Gauss-Markov theorem,” J. Am. Stat. Assoc. 61, 1063–1066 (1966).
[Crossref]

Mansfield, D.

H. Yoshino, A. Abbas, P. M. Kaminski, R. Smith, J. M. Walls, and D. Mansfield, “Measurement of thin film interfacial surface roughness by coherence scanning interferometry,” J. Appl. Phys. 121, 105303 (2017).
[Crossref]

H. Yoshino, P. M. Kaminski, R. Smith, J. M. Walls, and D. Mansfield, “Refractive index determination by coherence scanning interferometry,” Appl. Opt. 55, 4253–4260 (2016).
[Crossref]

H. Yoshino, R. Smith, J. M. Walls, and D. Mansfield, “The development of thin film metrology by coherence scanning interferometry,” Proc. SPIE 9749, 97490P(2016).

D. Mansfield, “Extraction of film interface surfaces from scanning white light interferometry,” Proc. SPIE 7101, 71010U (2008).

D. Mansfield, “The distorted helix: thin film extraction from scanning white light interferometry,” Proc. SPIE 6186, 61860O (2006).

D. Mansfield, “Apparatus for and a method of determining characteristics of thin-layer structures using low-coherence interferometry,” WO patentPCT/GB2005/002,783 (19 January 2006).

Mansfield, D. I.

D. I. Mansfield, “Apparatus for and a method of determining surface characteristics,” U.S. patent13/352,687 (12July2012).

Odell, P.

T. Lewis and P. Odell, “A generalization of the Gauss-Markov theorem,” J. Am. Stat. Assoc. 61, 1063–1066 (1966).
[Crossref]

Petitgrand, S.

A. Bosseboeuf and S. Petitgrand, “Application of microscopic interferometry techniques in the MEMS field,” Proc. SPIE 5145, 1–16 (2003).

Sekihara, K.

K. Sekihara, Introduction to Statistical Signal Processing (Kyouritsu Shuppan, 2011).

Smith, R.

H. Yoshino, A. Abbas, P. M. Kaminski, R. Smith, J. M. Walls, and D. Mansfield, “Measurement of thin film interfacial surface roughness by coherence scanning interferometry,” J. Appl. Phys. 121, 105303 (2017).
[Crossref]

H. Yoshino, P. M. Kaminski, R. Smith, J. M. Walls, and D. Mansfield, “Refractive index determination by coherence scanning interferometry,” Appl. Opt. 55, 4253–4260 (2016).
[Crossref]

H. Yoshino, R. Smith, J. M. Walls, and D. Mansfield, “The development of thin film metrology by coherence scanning interferometry,” Proc. SPIE 9749, 97490P(2016).

Strand, T. C.

Thomas, T. R.

T. R. Thomas, “Other measurement topics,” in Rough Surfaces, 2nd ed. (Imperial College, 1999), Chap. 3, pp. 35–61.

Walls, J. M.

H. Yoshino, A. Abbas, P. M. Kaminski, R. Smith, J. M. Walls, and D. Mansfield, “Measurement of thin film interfacial surface roughness by coherence scanning interferometry,” J. Appl. Phys. 121, 105303 (2017).
[Crossref]

H. Yoshino, P. M. Kaminski, R. Smith, J. M. Walls, and D. Mansfield, “Refractive index determination by coherence scanning interferometry,” Appl. Opt. 55, 4253–4260 (2016).
[Crossref]

H. Yoshino, R. Smith, J. M. Walls, and D. Mansfield, “The development of thin film metrology by coherence scanning interferometry,” Proc. SPIE 9749, 97490P(2016).

Yoshino, H.

H. Yoshino, A. Abbas, P. M. Kaminski, R. Smith, J. M. Walls, and D. Mansfield, “Measurement of thin film interfacial surface roughness by coherence scanning interferometry,” J. Appl. Phys. 121, 105303 (2017).
[Crossref]

H. Yoshino, P. M. Kaminski, R. Smith, J. M. Walls, and D. Mansfield, “Refractive index determination by coherence scanning interferometry,” Appl. Opt. 55, 4253–4260 (2016).
[Crossref]

H. Yoshino, R. Smith, J. M. Walls, and D. Mansfield, “The development of thin film metrology by coherence scanning interferometry,” Proc. SPIE 9749, 97490P(2016).

Appl. Opt. (3)

J. Am. Stat. Assoc. (1)

T. Lewis and P. Odell, “A generalization of the Gauss-Markov theorem,” J. Am. Stat. Assoc. 61, 1063–1066 (1966).
[Crossref]

J. Appl. Phys. (1)

H. Yoshino, A. Abbas, P. M. Kaminski, R. Smith, J. M. Walls, and D. Mansfield, “Measurement of thin film interfacial surface roughness by coherence scanning interferometry,” J. Appl. Phys. 121, 105303 (2017).
[Crossref]

Meas. Sci. Technol. (1)

I. Abdulhalim, “Spectroscopic interference microscopy technique for measurement of layer parameters,” Meas. Sci. Technol. 12, 1996–2001 (2001).
[Crossref]

Proc. SPIE (5)

D. Mansfield, “Extraction of film interface surfaces from scanning white light interferometry,” Proc. SPIE 7101, 71010U (2008).

D. Mansfield, “The distorted helix: thin film extraction from scanning white light interferometry,” Proc. SPIE 6186, 61860O (2006).

P. J. De Groot and X. C. de Lega, “Transparent film profiling and analysis by interference microscopy,” Proc. SPIE 7064, 70640I (2008).

A. Bosseboeuf and S. Petitgrand, “Application of microscopic interferometry techniques in the MEMS field,” Proc. SPIE 5145, 1–16 (2003).

H. Yoshino, R. Smith, J. M. Walls, and D. Mansfield, “The development of thin film metrology by coherence scanning interferometry,” Proc. SPIE 9749, 97490P(2016).

SIAM J. Appl. Math. (1)

A. Albert, “The Gauss-Markov theorem for regression models with possibly singular covariances,” SIAM J. Appl. Math. 24, 182–187 (1973).
[Crossref]

Other (8)

K. Sekihara, Introduction to Statistical Signal Processing (Kyouritsu Shuppan, 2011).

D. I. Mansfield, “Apparatus for and a method of determining surface characteristics,” U.S. patent13/352,687 (12July2012).

D. Mansfield, “Apparatus for and a method of determining characteristics of thin-layer structures using low-coherence interferometry,” WO patentPCT/GB2005/002,783 (19 January 2006).

F. Blateyron, “The areal field parameters,” in Characterisation of Areal Surface Texture (Springer, 2013), pp. 15–43.

P. de Groot, “Coherence scanning interferometry,” in Optical Measurement of Surface Topography (Springer, 2011), Chap. 9, pp. 187–208.

T. R. Thomas, “Other measurement topics,” in Rough Surfaces, 2nd ed. (Imperial College, 1999), Chap. 3, pp. 35–61.

S. Kim and G. Kim, “Method for measuring a thickness profile and a refractive index using white-light scanning interferometry and recording medium therefor,” U.S. patent6,545,763 (8 April 2003).

P. J. de Groot, “Interferometry method for ellipsometry, reflectometry, and scatterometry measurements, including characterization of thin film structures,” U.S. patent7,403,289 (22 July 2008).

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Figures (13)

Fig. 1.
Fig. 1. Determined HCF function of a 520 nm SiO 2 thin film on a Si substrate: (a) The global determined HCF function HCF d ¯ , obtained from the full 21 × 21 matrix of four pixels; (b) the HCF function HCF px d determined from four pixels at the edge of the measurement area; (c) the locally determined HCF function HCF px d at the center of the measurement area.
Fig. 2.
Fig. 2. Noise variance-covariance matrix Σ o from a silicon reference sample with M ref = 21 × 21    pixels (actual CSI measurement): (a) real part, (b) imaginary part (color available online).
Fig. 3.
Fig. 3. Noise variance in the frequency domain (actual CSI measurement). The diagonal element of the (a) real and (b) imaginary parts of the noise variance-covariance matrix illustrated in Fig. 2.
Fig. 4.
Fig. 4. Schematic drawing of the model. The number of the pixels in the measurement area is M together with the pixels having the feature is M f . Note that the global film thickness d ^ d .
Fig. 5.
Fig. 5. Noise variance in the frequency domain: The diagonal element of the (a) real and (b) imaginary parts of the noise variance-covariance matrix given to the simulations. Note that the signal-to-noise (S/N) ratio is 2000.
Fig. 6.
Fig. 6. Comparisons between the three computational methods on the sample: SiO 2 (thicknes s = 514    nm ) on a Si substrate; the feature height is 5 nm: (a) the ISR method (with noise), (b) the ISR-NC method (with noise), (c) the ISR-NF method (noise free). The S/N ratio is set at 10 2 to correspond to Sim 3-1 in Table 1 (color available online).
Fig. 7.
Fig. 7. Surface roughness ( S q ) as a function of film thickness: Red circles, top ISR; red squares, Sub ISR; blue circles, top ISR-NC; blue squares, sub ISR-NC.
Fig. 8.
Fig. 8. Feature height sensitivity as a function of feature height: black circles, ISR-NF; red triangles, ISR; blue squares, ISR-NC.
Fig. 9.
Fig. 9. Signal-to-noise ratio sensitivity to the determined feature height: black circles, ISR-NF; red triangles, ISR; blue squares, ISR-NC.
Fig. 10.
Fig. 10. Surface roughness ( S q ) as a function of the S/N ratio: red circles, top ISR; red squares, sub ISR; blue circles, top ISR-NC; blue squares, sub ISR-NC.
Fig. 11.
Fig. 11. Erroneous determined feature height (originally set as 5 nm): black circles, ISR-NF; red triangles, ISR; blue squares, ISR-NC.
Fig. 12.
Fig. 12. HCF functions generated at the feature pixel (simulation 2-2 with 20 nm feature height): (a) true HCF function (without noise) denoted by “Org” and its first-order approximation by “aprx”; (b) spectral difference between the true HCF function HCF px d and the HCF functions produced by each method HCF px s (noise occurs in both ISR and ISR-NC, and NF stands for ISR-NF) (color available online).
Fig. 13.
Fig. 13. HCF functions generated at the feature pixel (simulation 5-1 with Ta 2 O 5 film): (a) the true HCF function (without noise) denoted by “Org” and its first-order approximation by “aprx”; (b) the spectral difference between the true HCF function HCF px d and the HCF functions produced by each method HCF px s (noise exists for ISR and ISR-NC, and NF stands for ISR-NF); (c) the spectral difference between the real and imaginary parts of the true amplitude reflection coefficient and its first-order approximation. Note that the dotted lines (black and pink) represent the maximum deviations of the real ( Re [ r r aprx ] ), imaginary ( Im [ r r aprx ] ), and the reflectivity R , respectively (color available online).

Tables (2)

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Table 1. Simulation Conditions Together with the Number of Pixels Allocated for the “Global” and “Featured” Areas

Tables Icon

Table 2. Corresponding Effective Quarter-Wavelength Optical Thickness Values at the Wavelength of 600 nm

Equations (14)

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HCF d ( ν ; d ) = r ¯ ref ( ν ) · F [ I ] S B + F [ I ref ] S B + , HCF s ( ν , d ) = r ¯ ( ν , d ) · exp ( j 4 π ν Δ z HCF cos θ ¯ ) ,
HCF px d = r ¯ ref ( ν ) · F [ I px + ε px ] S B + F [ I ref ] S B + , HCF px s ( ν ; d ^ + Δ d ) HCF d ¯ + j 4 π ν cos θ ¯ · HCF d ¯ { Δ d sub + l = 1 L G l ( ν ; d ^ ) Δ d l } ,
G l ( ν ; d ^ ) = 1 + 1 4 π ν cos θ ¯ χ ( d ^ ) d l , arg ( r ¯ ) = χ , HCF d ¯ = r ¯ ref ( ν ) · F [ E [ I px + ε px ] ] S B + F [ I ref ] S B + .
HCF px d = [ HCF px d ( ν 1 ) , HCF px d ( ν 2 ) , , HCF px d ( ν m ) ] , HCF px s HCF d ¯ + Diag [ HCF d ¯ ] G Δ d .
HCF px d HCF d ¯ + Diag [ HCF d ¯ ] G Δ d + ϵ o ,
G = j 4 π cos θ ¯ [ ν 1 ν 1 · G 1 ( ν 1 ) ν 1 · G L ( ν 1 ) ν 2 ν 2 · G 1 ( ν 2 ) ν m ν m · G 1 ( ν m ) ν m · G L ( ν m ) ] , Diag [ HCF d ¯ ] = [ HCF d ¯ ( ν 1 ) 0 0 0 HCF d ¯ ( ν 2 ) 0 0 0 HCF d ¯ ( ν m ) ] , HCF d ¯ = [ HCF d ¯ ( ν 1 ) , HCF d ¯ ( ν 2 ) , , HCF d ¯ ( ν m ) ] , ϵ o = [ ϵ o 1 , ϵ o 2 , , ϵ o m ] N ( ϵ o | 0 , σ o 2 I ) .
Δ ^ d = ( G G ) 1 G T u ,
u = { Diag [ HCF d ¯ ] } 1 [ HCF px d HCF d ¯ ] .
u = G Δ d + ϵ , where    ϵ = { Diag [ HCF d ¯ ] } 1 ϵ o ,
p ( u ) = 1 ( 2 π ) m 2 | Σ | 1 2 exp [ 1 2 ( u G Δ d ) Σ 1 ( u G Δ d ) ] , L ( Δ d ) = log p ( u ) = 1 2 ( u G Δ d ) Σ 1 ( u G Δ d ) + C ,
minimize Δ d J px = ( u G Δ d ) Σ 1 ( u G Δ d ) , subject    to    ϵ N ( ϵ | 0 , Σ ) .
Δ ^ d = ( G Σ 1 G ) 1 G Σ 1 u .
Σ o = E [ ( HCF ref d HCF ref d ¯ ) ( HCF ref d HCF ref d ¯ ) ] 1 M ref i = 1 M ref ( HCF ref d , i HCF ref d ¯ ) ( HCF ref d , i HCF ref d ¯ ) , Σ Diag [ HCF d ¯ ] 1 Σ o { Diag [ HCF d ¯ ] 1 } ,
HCF ref d ¯ ( ν ) = r ¯ ref ( ν ) · F [ I ref ] S B + F [ I ref ] S B + = r ¯ ref ( ν ) , HCF ref d , i ( ν ) = r ¯ ref ( ν ) · F [ I ref i ] S B + F [ I ref ] S B + + ϵ i , I ref 1 M ref i = 1 M ref I ref i .

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