Abstract

We developed a multichannel three-polarizer spectroscopic ellipsometer based on a data acquisition algorithm for achieving optimized precision. This algorithm measures unnormalized Fourier coefficients accurately and precisely. Offset angles for optical elements were obtained as wavelength-independent values using regression calibration. Derived subsets of data reduction functions were used to calculate sample parameters. Correlation coefficients of Fourier coefficients were used to calculate errors in the sample parameters. Mean standard deviations of the sample parameters for each data reduction method were compared to identify the best method. This approach could be used to identify suitable precision optimization methods for other rotating-element ellipsometers.

© 2013 Optical Society of America

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References

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  1. D. E. Aspnes, “Expanding horizons: new developments in ellipsometry and polarimetry,” Thin Solid Films 455–456, 3–13 (2004).
    [CrossRef]
  2. M. Losurdo, “Applications of ellipsometry in nanoscale science: needs, status, achievements and future challenges,” Thin Solid Films 519, 2575–2583 (2011).
    [CrossRef]
  3. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).
  4. M. Schubert, Infrared Ellipsometry on Semiconductor Layer Structures: Phonons, Plasmons, and Polaritons (Springer, 2004).
  5. R. W. Collins, I. An, J. Lee, and J. A. Zapien, “Multichannel ellipsometry,” in Handbook of Ellipsometry, H. G. Tompkins and E. A. Irene, eds. (William Andrew, 2005).
  6. H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (Wiley, 2007).
  7. H. G. Tompkins, ed., Proceedings of the 5th International Conference on Spectroscopic Ellipsometry (Elsevier, 2011).
  8. I. An and R. W. Collins, “Waveform analysis with optical multichannel detectors: applications for rapid-scan spectroscopic ellipsometry,” Rev. Sci. Instrum. 62, 1904–1911 (1991).
    [CrossRef]
  9. N. V. Nguyen, B. S. Pudliner, I. An, and R. W. Collins, “Error correction for calibration and data reduction in rotating-polarizer ellipsometry: applications to a novel multichannel ellipsometer,” J. Opt. Soc. Am. A 8, 919–931 (1991).
    [CrossRef]
  10. D. E. Aspnes, “Optimizing precision of rotating-analyzer ellipsometers,” J. Opt. Soc. Am. 64, 639–646 (1974).
    [CrossRef]
  11. D. E. Aspnes, “Precision bounds to ellipsometric systems,” Appl. Opt. 14, 1131–1136 (1975).
    [CrossRef]
  12. R. W. Stobie, B. Rao, and M. J. Dignam, “Analysis of a novel ellipsometric technique with special advantages for infrared spectroscopy,” J. Opt. Soc. Am. 65, 25–28 (1975).
    [CrossRef]
  13. J. M. M. de Nijs and A. V. Silfhout, “Systematic and random errors in rotating-analyzer ellipsometry,” J. Opt. Soc. Am. A 5, 773–781 (1988).
    [CrossRef]
  14. R. Kleim, L. Kuntzler, and A. E. Ghemmaz, “Systematic errors in rotating-compensator ellipsometry,” J. Opt. Soc. Am. A 11, 2550–2559 (1994).
    [CrossRef]
  15. S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. E. Ghemmaz, N. Stein, and R. Kleim, “Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).
    [CrossRef]
  16. Z. Huang and J. Chu, “Optimizing precision of fixed-polarizer, rotating-polarizer, sample, and fixed-analyzer spectroscopic ellipsometry,” Appl. Opt. 39, 6390–6395 (2000).
    [CrossRef]
  17. A. E. Naciri, L. Broch, L. Johann, and R. Kleim, “Fixed polarizer, rotating-polarizer and fixed analyzer spectroscopic ellipsometer: accurate calibration method, effect of errors and testing,” Thin Solid Films 406, 103–112 (2002).
    [CrossRef]
  18. D. E. Aspnes, “Optimizing precision of rotating-analyzer and rotating-compensator ellipsometers,” J. Opt. Soc. Am. A 21, 403–410 (2004).
    [CrossRef]
  19. B. Johs and C. M. Herzinger, “Precision in ellipsometrically determined sample parameters: simulation and experiment,” Thin Solid Films 455–456, 66–71 (2004).
    [CrossRef]
  20. B. Johs and C. M. Herzinger, “Quantifying the accuracy of ellipsometer system,” Phys. Status Solidi C 5, 1031–1035 (2008).
    [CrossRef]
  21. L. Broch and L. Johann, “Optimizing precision of rotating compensator ellipsometry,” Phys. Status Solidi C 5, 1036–1040 (2008).
    [CrossRef]
  22. Y. J. Cho, W. Chegal, and H. M. Cho, “Fourier analysis for rotating-element ellipsometers,” Opt. Lett. 36, 118–120 (2011).
    [CrossRef]
  23. G. E. Jellison, “Spectroscopic ellipsometry data analysis: measured versus calculated quantities,” Thin Solid Films 313–314, 33–39 (1998).
    [CrossRef]
  24. B. Johs, “Regression calibration method for rotating element ellipsometers,” Thin Solid Films 234, 395–398 (1993).
    [CrossRef]
  25. B. N. Taylor and C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” Tech. Note NIST 1297 (National Institute of Standards and Technology, 1994), pp. 1–20.
  26. S. M. Ross, Introduction to Probability and Statistics for Engineers and Scientists (Elsevier, 2009).
  27. C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83, 3323–3336 (1998).
    [CrossRef]
  28. Y. J. Cho, W. Chegal, and H. M. Cho, Department of Industrial Metrology, Korea Research Institute of Standards and Science, 267 Gajeong-Ro, Yuseong-Gu, Daejeon 305-340, Korea, are preparing a manuscript to be called “Optimized precision of multichannel rotating-element spectroscopic ellipsometers.”

2011 (2)

M. Losurdo, “Applications of ellipsometry in nanoscale science: needs, status, achievements and future challenges,” Thin Solid Films 519, 2575–2583 (2011).
[CrossRef]

Y. J. Cho, W. Chegal, and H. M. Cho, “Fourier analysis for rotating-element ellipsometers,” Opt. Lett. 36, 118–120 (2011).
[CrossRef]

2008 (2)

B. Johs and C. M. Herzinger, “Quantifying the accuracy of ellipsometer system,” Phys. Status Solidi C 5, 1031–1035 (2008).
[CrossRef]

L. Broch and L. Johann, “Optimizing precision of rotating compensator ellipsometry,” Phys. Status Solidi C 5, 1036–1040 (2008).
[CrossRef]

2004 (3)

D. E. Aspnes, “Expanding horizons: new developments in ellipsometry and polarimetry,” Thin Solid Films 455–456, 3–13 (2004).
[CrossRef]

B. Johs and C. M. Herzinger, “Precision in ellipsometrically determined sample parameters: simulation and experiment,” Thin Solid Films 455–456, 66–71 (2004).
[CrossRef]

D. E. Aspnes, “Optimizing precision of rotating-analyzer and rotating-compensator ellipsometers,” J. Opt. Soc. Am. A 21, 403–410 (2004).
[CrossRef]

2002 (1)

A. E. Naciri, L. Broch, L. Johann, and R. Kleim, “Fixed polarizer, rotating-polarizer and fixed analyzer spectroscopic ellipsometer: accurate calibration method, effect of errors and testing,” Thin Solid Films 406, 103–112 (2002).
[CrossRef]

2000 (1)

1998 (3)

C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83, 3323–3336 (1998).
[CrossRef]

G. E. Jellison, “Spectroscopic ellipsometry data analysis: measured versus calculated quantities,” Thin Solid Films 313–314, 33–39 (1998).
[CrossRef]

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. E. Ghemmaz, N. Stein, and R. Kleim, “Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).
[CrossRef]

1994 (1)

1993 (1)

B. Johs, “Regression calibration method for rotating element ellipsometers,” Thin Solid Films 234, 395–398 (1993).
[CrossRef]

1991 (2)

I. An and R. W. Collins, “Waveform analysis with optical multichannel detectors: applications for rapid-scan spectroscopic ellipsometry,” Rev. Sci. Instrum. 62, 1904–1911 (1991).
[CrossRef]

N. V. Nguyen, B. S. Pudliner, I. An, and R. W. Collins, “Error correction for calibration and data reduction in rotating-polarizer ellipsometry: applications to a novel multichannel ellipsometer,” J. Opt. Soc. Am. A 8, 919–931 (1991).
[CrossRef]

1988 (1)

1975 (2)

1974 (1)

An, I.

I. An and R. W. Collins, “Waveform analysis with optical multichannel detectors: applications for rapid-scan spectroscopic ellipsometry,” Rev. Sci. Instrum. 62, 1904–1911 (1991).
[CrossRef]

N. V. Nguyen, B. S. Pudliner, I. An, and R. W. Collins, “Error correction for calibration and data reduction in rotating-polarizer ellipsometry: applications to a novel multichannel ellipsometer,” J. Opt. Soc. Am. A 8, 919–931 (1991).
[CrossRef]

R. W. Collins, I. An, J. Lee, and J. A. Zapien, “Multichannel ellipsometry,” in Handbook of Ellipsometry, H. G. Tompkins and E. A. Irene, eds. (William Andrew, 2005).

Aspnes, D. E.

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

Bertucci, S.

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. E. Ghemmaz, N. Stein, and R. Kleim, “Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).
[CrossRef]

Broch, L.

L. Broch and L. Johann, “Optimizing precision of rotating compensator ellipsometry,” Phys. Status Solidi C 5, 1036–1040 (2008).
[CrossRef]

A. E. Naciri, L. Broch, L. Johann, and R. Kleim, “Fixed polarizer, rotating-polarizer and fixed analyzer spectroscopic ellipsometer: accurate calibration method, effect of errors and testing,” Thin Solid Films 406, 103–112 (2002).
[CrossRef]

Chegal, W.

Y. J. Cho, W. Chegal, and H. M. Cho, “Fourier analysis for rotating-element ellipsometers,” Opt. Lett. 36, 118–120 (2011).
[CrossRef]

Y. J. Cho, W. Chegal, and H. M. Cho, Department of Industrial Metrology, Korea Research Institute of Standards and Science, 267 Gajeong-Ro, Yuseong-Gu, Daejeon 305-340, Korea, are preparing a manuscript to be called “Optimized precision of multichannel rotating-element spectroscopic ellipsometers.”

Cho, H. M.

Y. J. Cho, W. Chegal, and H. M. Cho, “Fourier analysis for rotating-element ellipsometers,” Opt. Lett. 36, 118–120 (2011).
[CrossRef]

Y. J. Cho, W. Chegal, and H. M. Cho, Department of Industrial Metrology, Korea Research Institute of Standards and Science, 267 Gajeong-Ro, Yuseong-Gu, Daejeon 305-340, Korea, are preparing a manuscript to be called “Optimized precision of multichannel rotating-element spectroscopic ellipsometers.”

Cho, Y. J.

Y. J. Cho, W. Chegal, and H. M. Cho, “Fourier analysis for rotating-element ellipsometers,” Opt. Lett. 36, 118–120 (2011).
[CrossRef]

Y. J. Cho, W. Chegal, and H. M. Cho, Department of Industrial Metrology, Korea Research Institute of Standards and Science, 267 Gajeong-Ro, Yuseong-Gu, Daejeon 305-340, Korea, are preparing a manuscript to be called “Optimized precision of multichannel rotating-element spectroscopic ellipsometers.”

Chu, J.

Collins, R. W.

I. An and R. W. Collins, “Waveform analysis with optical multichannel detectors: applications for rapid-scan spectroscopic ellipsometry,” Rev. Sci. Instrum. 62, 1904–1911 (1991).
[CrossRef]

N. V. Nguyen, B. S. Pudliner, I. An, and R. W. Collins, “Error correction for calibration and data reduction in rotating-polarizer ellipsometry: applications to a novel multichannel ellipsometer,” J. Opt. Soc. Am. A 8, 919–931 (1991).
[CrossRef]

R. W. Collins, I. An, J. Lee, and J. A. Zapien, “Multichannel ellipsometry,” in Handbook of Ellipsometry, H. G. Tompkins and E. A. Irene, eds. (William Andrew, 2005).

de Nijs, J. M. M.

Dignam, M. J.

Fujiwara, H.

H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (Wiley, 2007).

Ghemmaz, A. E.

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. E. Ghemmaz, N. Stein, and R. Kleim, “Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).
[CrossRef]

R. Kleim, L. Kuntzler, and A. E. Ghemmaz, “Systematic errors in rotating-compensator ellipsometry,” J. Opt. Soc. Am. A 11, 2550–2559 (1994).
[CrossRef]

Herzinger, C. M.

B. Johs and C. M. Herzinger, “Quantifying the accuracy of ellipsometer system,” Phys. Status Solidi C 5, 1031–1035 (2008).
[CrossRef]

B. Johs and C. M. Herzinger, “Precision in ellipsometrically determined sample parameters: simulation and experiment,” Thin Solid Films 455–456, 66–71 (2004).
[CrossRef]

C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83, 3323–3336 (1998).
[CrossRef]

Huang, Z.

Jellison, G. E.

G. E. Jellison, “Spectroscopic ellipsometry data analysis: measured versus calculated quantities,” Thin Solid Films 313–314, 33–39 (1998).
[CrossRef]

Johann, L.

L. Broch and L. Johann, “Optimizing precision of rotating compensator ellipsometry,” Phys. Status Solidi C 5, 1036–1040 (2008).
[CrossRef]

A. E. Naciri, L. Broch, L. Johann, and R. Kleim, “Fixed polarizer, rotating-polarizer and fixed analyzer spectroscopic ellipsometer: accurate calibration method, effect of errors and testing,” Thin Solid Films 406, 103–112 (2002).
[CrossRef]

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. E. Ghemmaz, N. Stein, and R. Kleim, “Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).
[CrossRef]

Johs, B.

B. Johs and C. M. Herzinger, “Quantifying the accuracy of ellipsometer system,” Phys. Status Solidi C 5, 1031–1035 (2008).
[CrossRef]

B. Johs and C. M. Herzinger, “Precision in ellipsometrically determined sample parameters: simulation and experiment,” Thin Solid Films 455–456, 66–71 (2004).
[CrossRef]

C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83, 3323–3336 (1998).
[CrossRef]

B. Johs, “Regression calibration method for rotating element ellipsometers,” Thin Solid Films 234, 395–398 (1993).
[CrossRef]

Kleim, R.

A. E. Naciri, L. Broch, L. Johann, and R. Kleim, “Fixed polarizer, rotating-polarizer and fixed analyzer spectroscopic ellipsometer: accurate calibration method, effect of errors and testing,” Thin Solid Films 406, 103–112 (2002).
[CrossRef]

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. E. Ghemmaz, N. Stein, and R. Kleim, “Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).
[CrossRef]

R. Kleim, L. Kuntzler, and A. E. Ghemmaz, “Systematic errors in rotating-compensator ellipsometry,” J. Opt. Soc. Am. A 11, 2550–2559 (1994).
[CrossRef]

Kuntzler, L.

Kuyatt, C. E.

B. N. Taylor and C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” Tech. Note NIST 1297 (National Institute of Standards and Technology, 1994), pp. 1–20.

Lee, J.

R. W. Collins, I. An, J. Lee, and J. A. Zapien, “Multichannel ellipsometry,” in Handbook of Ellipsometry, H. G. Tompkins and E. A. Irene, eds. (William Andrew, 2005).

Losurdo, M.

M. Losurdo, “Applications of ellipsometry in nanoscale science: needs, status, achievements and future challenges,” Thin Solid Films 519, 2575–2583 (2011).
[CrossRef]

McGahan, W. A.

C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83, 3323–3336 (1998).
[CrossRef]

Naciri, A. E.

A. E. Naciri, L. Broch, L. Johann, and R. Kleim, “Fixed polarizer, rotating-polarizer and fixed analyzer spectroscopic ellipsometer: accurate calibration method, effect of errors and testing,” Thin Solid Films 406, 103–112 (2002).
[CrossRef]

Nguyen, N. V.

Nicolas, N.

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. E. Ghemmaz, N. Stein, and R. Kleim, “Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).
[CrossRef]

Paulson, W.

C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83, 3323–3336 (1998).
[CrossRef]

Pawlowski, A.

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. E. Ghemmaz, N. Stein, and R. Kleim, “Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).
[CrossRef]

Pudliner, B. S.

Rao, B.

Ross, S. M.

S. M. Ross, Introduction to Probability and Statistics for Engineers and Scientists (Elsevier, 2009).

Schubert, M.

M. Schubert, Infrared Ellipsometry on Semiconductor Layer Structures: Phonons, Plasmons, and Polaritons (Springer, 2004).

Silfhout, A. V.

Stein, N.

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. E. Ghemmaz, N. Stein, and R. Kleim, “Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).
[CrossRef]

Stobie, R. W.

Taylor, B. N.

B. N. Taylor and C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” Tech. Note NIST 1297 (National Institute of Standards and Technology, 1994), pp. 1–20.

Woollam, J. A.

C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83, 3323–3336 (1998).
[CrossRef]

Zapien, J. A.

R. W. Collins, I. An, J. Lee, and J. A. Zapien, “Multichannel ellipsometry,” in Handbook of Ellipsometry, H. G. Tompkins and E. A. Irene, eds. (William Andrew, 2005).

Appl. Opt. (2)

J. Appl. Phys. (1)

C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83, 3323–3336 (1998).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Opt. Lett. (1)

Phys. Status Solidi C (2)

B. Johs and C. M. Herzinger, “Quantifying the accuracy of ellipsometer system,” Phys. Status Solidi C 5, 1031–1035 (2008).
[CrossRef]

L. Broch and L. Johann, “Optimizing precision of rotating compensator ellipsometry,” Phys. Status Solidi C 5, 1036–1040 (2008).
[CrossRef]

Rev. Sci. Instrum. (1)

I. An and R. W. Collins, “Waveform analysis with optical multichannel detectors: applications for rapid-scan spectroscopic ellipsometry,” Rev. Sci. Instrum. 62, 1904–1911 (1991).
[CrossRef]

Thin Solid Films (7)

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. E. Ghemmaz, N. Stein, and R. Kleim, “Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).
[CrossRef]

B. Johs and C. M. Herzinger, “Precision in ellipsometrically determined sample parameters: simulation and experiment,” Thin Solid Films 455–456, 66–71 (2004).
[CrossRef]

G. E. Jellison, “Spectroscopic ellipsometry data analysis: measured versus calculated quantities,” Thin Solid Films 313–314, 33–39 (1998).
[CrossRef]

B. Johs, “Regression calibration method for rotating element ellipsometers,” Thin Solid Films 234, 395–398 (1993).
[CrossRef]

D. E. Aspnes, “Expanding horizons: new developments in ellipsometry and polarimetry,” Thin Solid Films 455–456, 3–13 (2004).
[CrossRef]

M. Losurdo, “Applications of ellipsometry in nanoscale science: needs, status, achievements and future challenges,” Thin Solid Films 519, 2575–2583 (2011).
[CrossRef]

A. E. Naciri, L. Broch, L. Johann, and R. Kleim, “Fixed polarizer, rotating-polarizer and fixed analyzer spectroscopic ellipsometer: accurate calibration method, effect of errors and testing,” Thin Solid Films 406, 103–112 (2002).
[CrossRef]

Other (8)

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

M. Schubert, Infrared Ellipsometry on Semiconductor Layer Structures: Phonons, Plasmons, and Polaritons (Springer, 2004).

R. W. Collins, I. An, J. Lee, and J. A. Zapien, “Multichannel ellipsometry,” in Handbook of Ellipsometry, H. G. Tompkins and E. A. Irene, eds. (William Andrew, 2005).

H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (Wiley, 2007).

H. G. Tompkins, ed., Proceedings of the 5th International Conference on Spectroscopic Ellipsometry (Elsevier, 2011).

B. N. Taylor and C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” Tech. Note NIST 1297 (National Institute of Standards and Technology, 1994), pp. 1–20.

S. M. Ross, Introduction to Probability and Statistics for Engineers and Scientists (Elsevier, 2009).

Y. J. Cho, W. Chegal, and H. M. Cho, Department of Industrial Metrology, Korea Research Institute of Standards and Science, 267 Gajeong-Ro, Yuseong-Gu, Daejeon 305-340, Korea, are preparing a manuscript to be called “Optimized precision of multichannel rotating-element spectroscopic ellipsometers.”

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Figures (12)

Fig. 1.
Fig. 1.

Schematic diagram of a three-polarizer ellipsometer design where P , P r , and A represent the azimuth angles of the transmission axes of the polarizing prisms in the fixed polarizer, rotating polarizer, and fixed analyzer, respectively, with respect to the plane of incidence, and ϕ is the angle of incidence.

Fig. 2.
Fig. 2.

Plot of the experimental and fit data of the normalized Fourier coefficients versus the index angle of the fixed analyzer, as varied by the stepping motor, for the nominally 30 nm-thick thermal SiO 2 film grown on c-Si wafers.

Fig. 3.
Fig. 3.

Offset angles of the transmission axes of the fixed polarizer, fixed analyzer, and rotating polarizer with respect to the plane of incidence. The fit results were obtained by using the regression calibration [24] for each wavelength as shown in Fig. 2.

Fig. 4.
Fig. 4.

Using the RTSE, the mean values of the experimental spectra of the Fourier coefficients were obtained from the nominally 30 nm-thick thermal SiO 2 film grown on c-Si wafers where the spectra had been averaged over 60 mechanical turns.

Fig. 5.
Fig. 5.

Spectra of the sample standard deviations of the Fourier coefficients in Fig. 4.

Fig. 6.
Fig. 6.

Experimental spectra of the correlation coefficients between the Fourier coefficients, which were obtained using the RTSE under the same measurement conditions as in Fig. 4.

Fig. 7.
Fig. 7.

Comparison between the experimental and theoretically calculated spectra of the standard deviations for the ellipsometric sample parameters, N 1 and C 1 , of Eqs. (15)–(17). The theoretical spectra in (a) and (b) assume zero correlation coefficients, while those in (c) and (d) incorporate the measured correlation coefficients from Fig. 6.

Fig. 8.
Fig. 8.

Circular and triangular symbols represent the experimental spectra of the ellipsometric sample parameters, N 1 and C 1 , for the nominally 30 nm-thick thermal SiO 2 film grown on c-Si wafers using the RTSE. The solid lines in (a) denote the best-fit one-parameter simulation, yielding an oxide thickness of 33.085 nm ± 0.007 nm . δ N 1 and δ C 1 in (b) denote the differences between the fit and measured spectra in (a).

Fig. 9.
Fig. 9.

Circular and triangular symbols represent the experimental spectra of the ellipsometric sample parameters, N 1 and C 1 , for a native oxide film grown on c-Si wafers using the RTSE. The solid lines in (a) denote the best-fit one-parameter simulation, yielding an oxide thickness of 1.76 nm ± 0.03 nm . δ N 1 and δ C 1 in (b) denote the differences between the fit and measured spectra in (a).

Fig. 10.
Fig. 10.

Circular and triangular symbols represent the experimental spectra of the ellipsometric sample parameters, N 1 and C 1 , for a nominally 100 nm-thick thermal SiO 2 film grown on c-Si wafers using the RTSE. The solid lines in (a) denote the best-fit one-parameter simulation, yielding an oxide thickness of 102.28 nm ± 0.02 nm . δ N 1 and δ C 1 in (b) denote the differences between the fit and measured spectra in (a).

Fig. 11.
Fig. 11.

Circular and triangular symbols represent the experimental spectra of the ellipsometric sample parameters, N 1 and C 1 , for a nominally 500 nm-thick thermal SiO 2 film grown on c-Si wafers using the RTSE. The solid lines in (a) denote the best-fit one-parameter simulation, yielding an oxide thickness of 489.37 nm ± 0.03 nm . δ N 1 and δ C 1 in (b) denote the differences between the fit and measured spectra in (a).

Fig. 12.
Fig. 12.

Experimental spectra of the ellipsometric sample parameters of N 1 and C 1 for the native oxide sample obtained by using the RTSE in a better measurement condition than that used in Fig. 9. The solid lines in (a) denote the best-fit one-parameter simulation, yielding an oxide thickness of 1.725 nm ± 0.009 nm . δ N 1 and δ C 1 in (b) denote the differences between the fit and measured spectra in (a).

Tables (1)

Tables Icon

Table 1. Comparison between the Mean Values of the Standard Deviations (SDs) of the Ellipsometric Sample Parameters Measured by Using Each Subset from among the Ten Subsets of the Data Reduction Functions of Eqs. (15)–(44)a

Equations (50)

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I ex ( t ) = I 0 + n = 1 N ho [ A 2 n cos ( 2 n ω t ) + B 2 n sin ( 2 n ω t ) ] ,
S j = ( j 1 ) T / J + T d ( j 1 ) T / J + T d + T i I ex ( t ) d t , ( j = 1 , , J ) .
H 2 n c + i H 2 n s = 2 M J j = 1 M J S j exp [ i 4 n π ( j 1 ) J ] .
I 0 = H 0 c 2 T i ,
A 2 n = C 2 n c H 2 n c C 2 n s H 2 n s ,
B 2 n = C 2 n c H 2 n s + C 2 n s H 2 n c ,
C 2 n c = 2 n π T sin ( 2 n π T i / T ) cos [ 2 n π ( T i + 2 T d T ) ] ,
C 2 n s = 2 n π T sin ( 2 n π T i / T ) sin [ 2 n π ( T i + 2 T d T ) ] .
I th ( P r ) = I 0 + A 2 cos ( 2 P r ) + B 2 sin ( 2 P r ) + A 4 cos ( 4 P r ) + B 4 sin ( 4 P r ) = I 0 [ 1 + α 2 cos ( 2 P r ) + β 2 sin ( 2 P r ) + α 4 cos ( 4 P r ) + β 4 sin ( 4 P r ) ] ,
I 0 = γ { 2 + cos ( 2 A ) cos ( 2 P ) N [ 2 cos ( 2 A ) + cos ( 2 P ) ] + C sin ( 2 A ) sin ( 2 P ) } ,
A 2 = 2 γ { cos ( 2 P ) + cos ( 2 A ) N [ 1 + cos ( 2 A ) cos ( 2 P ) ] } ,
B 2 = 2 γ [ sin ( 2 P ) N cos ( 2 A ) sin ( 2 P ) + C sin ( 2 A ) ] ,
A 4 = γ [ cos ( 2 A ) cos ( 2 P ) N cos ( 2 P ) C sin ( 2 A ) sin ( 2 P ) ] ,
B 4 = γ [ cos ( 2 A ) sin ( 2 P ) N sin ( 2 P ) + C sin ( 2 A ) cos ( 2 P ) ] .
N 1 = { 4 I 0 cos ( 2 P ) A 2 [ 3 + cos ( 4 P ) ] + 2 cos ( 2 A ) [ 2 I 0 A 2 cos ( 2 P ) B 2 sin ( 2 P ) ] B 2 sin ( 4 P ) } / D 1 ,
C 1 = sin ( 2 A ) { A 2 sin ( 4 P ) + B 2 [ 3 cos ( 4 P ) ] 4 I 0 sin ( 2 P ) } / D 1 ,
D 1 = 4 I 0 2 A 2 cos ( 2 P ) 2 B 2 sin ( 2 P ) + cos ( 2 A ) { 4 I 0 cos ( 2 P ) A 2 [ 3 + cos ( 4 P ) ] B 2 sin ( 4 P ) } ,
N 2 = 2 sin ( 2 P ) { ( I 0 + A 4 ) cos ( 2 P ) + cos ( 2 A ) [ I 0 + A 4 A 2 cos ( 2 P ) ] A 2 } / D 2 ,
C 2 = sin ( 2 A ) [ 2 A 2 cos ( 2 P ) ( I 0 A 4 ) cos ( 4 P ) I 0 3 A 4 ] / D 2 ,
D 2 = 2 sin ( 2 P ) { I 0 + A 4 A 2 cos ( 2 P ) cos ( 2 A ) [ A 2 ( I 0 + A 4 ) cos ( 2 P ) ] } ,
N 3 = { 2 cos ( 2 P ) [ A 2 I 0 cos ( 2 P ) + B 4 sin ( 2 P ) ] cos ( 2 A ) [ 2 I 0 cos ( 2 P ) A 2 cos ( 4 P ) 2 B 4 sin ( 2 P ) ] } / D 3 ,
C 3 = sin ( 2 A ) [ 2 A 2 sin ( 2 P ) 3 B 4 + B 4 cos ( 4 P ) I 0 sin ( 4 P ) ] / D 3 ,
D 3 = A 2 cos ( 4 P ) 2 I 0 cos ( 2 P ) + 2 B 4 sin ( 2 P ) + 2 cos ( 2 A ) cos ( 2 P ) [ A 2 I 0 cos ( 2 P ) + B 4 sin ( 2 P ) ] ,
N 4 = { 2 A 4 + cos ( 2 A ) cos ( 2 P ) [ A 4 I 0 + B 2 sin ( 2 P ) ] + sin ( 2 P ) [ B 2 ( A 4 + I 0 ) sin ( 2 P ) ] } / D 4 ,
C 4 = cos ( 2 P ) sin ( 2 A ) [ ( I 0 A 4 ) sin ( 2 P ) B 2 ] / D 4 ,
D 4 = cos ( 2 P ) [ A 4 I 0 + B 2 sin ( 2 P ) ] + cos ( 2 A ) { 2 A 4 + sin ( 2 P ) [ B 2 ( A 4 + I 0 ) sin ( 2 P ) ] } ,
N 5 = { B 4 [ 3 + cos ( 4 P ) ] + I 0 sin ( 4 P ) cos ( 2 A ) [ 2 I 0 sin ( 2 P ) 2 B 4 cos ( 2 P ) + B 2 cos ( 4 P ) ] 2 B 2 cos ( 2 P ) } / D 5 ,
C 5 = 2 sin ( 2 A ) sin ( 2 P ) [ I 0 sin ( 2 P ) B 2 B 4 cos ( 2 P ) ] / D 5 ,
D 5 = cos ( 2 A ) [ I 0 sin ( 4 P ) 2 B 2 cos ( 2 P ) + 3 B 4 + B 4 cos ( 4 P ) ] + 2 B 4 cos ( 2 P ) B 2 cos ( 4 P ) 2 I 0 sin ( 2 P ) ,
N 6 = { cos ( 2 A ) [ I 0 A 4 cos ( 4 P ) B 4 sin ( 4 P ) ] 2 [ A 4 cos ( 2 P ) + B 4 sin ( 2 P ) ] } / D 6 ,
C 6 = 2 sin ( 2 A ) [ B 4 cos ( 2 P ) A 4 sin ( 2 P ) ] / D 6 ,
D 6 = I 0 A 4 cos ( 4 P ) B 4 sin ( 4 P ) 2 cos ( 2 A ) [ A 4 cos ( 2 P ) + B 4 sin ( 2 P ) ] ,
N 7 = { A 2 [ 1 cos ( 4 P ) ] 4 A 4 cos ( 2 P ) B 2 sin ( 4 P ) 2 cos ( 2 A ) [ 2 A 4 A 2 cos ( 2 P ) + B 2 sin ( 2 P ) ] } / D 7 ,
C 7 = sin ( 2 A ) [ B 2 + B 2 cos ( 4 P ) + 4 A 4 sin ( 2 P ) A 2 sin ( 4 P ) ] / D 7 ,
D 7 = 2 A 2 cos ( 2 P ) 2 [ 2 A 4 + B 2 sin ( 2 P ) ] cos ( 2 A ) { 4 A 4 cos ( 2 P ) + B 2 sin ( 4 P ) A 2 [ 1 cos ( 4 P ) ] } ,
N 8 = { A 2 sin ( 4 P ) + 4 B 4 cos ( 2 P ) B 2 [ 1 + cos ( 4 P ) ] + cos ( 2 A ) [ 4 B 4 2 B 2 cos ( 2 P ) 2 A 2 sin ( 2 P ) ] } / D 8 ,
C 8 = 2 sin ( 2 A ) sin ( 2 P ) [ A 2 sin ( 2 P ) 2 B 4 B 2 cos ( 2 P ) ] / D 8 ,
D 8 = 2 cos ( 2 P ) cos ( 2 A ) [ 2 B 4 + A 2 sin ( 2 P ) B 2 cos ( 2 P ) ] + 4 B 4 2 A 2 sin ( 2 P ) 2 B 2 cos ( 2 P ) ,
N 9 = { cos ( 2 A ) [ A 2 2 A 4 cos ( 2 P ) 2 B 4 sin ( 2 P ) ] B 4 sin ( 4 P ) A 4 [ 1 + cos ( 4 P ) ] } / D 9 ,
C 9 = 2 cos ( 2 P ) sin ( 2 A ) [ B 4 cos ( 2 P ) A 4 sin ( 2 P ) ] / D 9 ,
D 9 = A 2 2 B 4 sin ( 2 P ) 2 cos ( 2 P ) { A 4 + cos ( 2 A ) [ A 4 cos ( 2 P ) + B 4 sin ( 2 P ) ] } ,
N 10 = { B 4 [ cos ( 4 P ) 1 ] + cos ( 2 A ) [ B 2 2 B 4 cos ( 2 P ) + 2 A 4 sin ( 2 P ) ] A 4 sin ( 4 P ) } / D 10 ,
C 10 = 2 sin ( 2 A ) sin ( 2 P ) [ B 4 cos ( 2 P ) A 4 sin ( 2 P ) ] / D 10 ,
D 10 = B 2 2 cos ( 2 P ) [ B 4 + A 4 cos ( 2 A ) sin ( 2 P ) ] + 2 sin ( 2 P ) [ A 4 B 4 cos ( 2 A ) sin ( 2 P ) ] .
I 0 = I 0 ,
A 2 n = A 2 n cos ( 2 n P r 0 ) + B 2 n sin ( 2 n P r 0 ) ,
B 2 n = A 2 n sin ( 2 n P r 0 ) + B 2 n cos ( 2 n P r 0 ) ,
σ 2 ( Q ) = c x 1 2 σ 2 ( x 1 ) + c x 2 2 σ 2 ( x 2 ) + c x 3 2 σ 2 ( x 3 ) + 2 c x 1 c x 2 σ ( x 1 ) σ ( x 2 ) r ( x 1 , x 2 ) + 2 c x 1 c x 3 σ ( x 1 ) σ ( x 3 ) r ( x 1 , x 3 ) + 2 c x 2 c x 3 σ ( x 2 ) σ ( x 3 ) r ( x 2 , x 3 ) ,
r ( x i , x j ) = σ ( x i , x j ) σ ( x i ) σ ( x j ) .
σ ( x i , x j ) = 1 M 1 k = 1 M ( x i , k x i ) ( x j , k x j ) .

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