Abstract

Optical properties of bulk materials and thin films have long been determined by spectral ellipsometers (SE’s). Optical properties are determined by finding the global minimum of a merit function χ2. Even in the simplest cases χ2 is dependent on at least five parameters. The global minimization of χ2 benefits from careful selection of the SE instrument state such that χ2 is optimally smooth in some sense. Minimization methods that assume analyticity, such as the popular Levenberg–Marquardt algorithm, encounter problems as the number of nondifferentiable points in χ2 increases. The purpose of the paper is to examine the distribution of local minimum and discontinuities in χ2 as a function of incident angle and wavelength-range selection. With proper attention to the selection of the incident angle and wavelength range the robustness of the Levenberg–Marquardt algorithm may be extended in fixed-angle SE’s.

© 1998 Optical Society of America

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References

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  1. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1996).
  2. D. P. Arndt, R. M. A. Azzam, J. M. Bennett, J. P. Borgogno, C. K. Carniglia, W. E. Case, J. A. Dobrowolski, U. J. Gibson, T. T. Hart, F. C. Ho, V. A. Hodgkin, W. P. Klapp, H. A. Macleod, E. Pelletier, M. K. Purvis, D. M. Quinn, D. H. Strome, R. Swenson, P. A. Temple, T. F. Thonn, “Multiple determination of the optical constants of thin-film coating materials,” Appl. Opt. 23, 3571–3596 (1984).
    [CrossRef] [PubMed]
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    [CrossRef]
  4. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1991).
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    [CrossRef] [PubMed]
  6. C. M. Herzinger, H. Yao, P. G. Snyder, “Determination of AlAs optical constants by variable angle spectroscopic ellipsometry and a multisample analysis,” J. Appl. Phys. 77, 4677–4687 (1995).
    [CrossRef]
  7. A Verity Instruments Series-10 Spectral Ellipsometer was used in obtaining data. The wavelength increment is determined by the spectrograph optics and the grating dispersion, which varies from 9.5 nm at the short-wavelength end to 9.1 nm at the long-wavelength end.
  8. E. D. Palik, ed., Handbook of Optical Constants of Solids, Volume 1 and 2 (Academic, New York, 1991).
  9. K. Price, R. Storn, “Differential evolution—a simple and efficient adaptive scheme for global optimization over continuous spaces,” (International Computer Science Institute, University of California at Berkeley, Berkeley, Calif., 1995) (available through ftp.icsi.berkeley.edu).
  10. R. Storn, K. Price, “Minimizing the real functions of the ICEC’96 contest by differential evolution,” in Proceedings of the 1996 IEEE Conference on Evolutionary Computation (ICEC ’96) (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 842–844.

1995

C. M. Herzinger, H. Yao, P. G. Snyder, “Determination of AlAs optical constants by variable angle spectroscopic ellipsometry and a multisample analysis,” J. Appl. Phys. 77, 4677–4687 (1995).
[CrossRef]

1994

1991

1984

Arndt, D. P.

Azzam, R. M. A.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1996).

Bennett, J. M.

Borgogno, J. P.

Boyanov, M. I.

Carniglia, C. K.

Case, W. E.

Dobrowolski, J. A.

Drolet, J. P.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1991).

Gibson, U. J.

Hart, T. T.

Herzinger, C. M.

C. M. Herzinger, H. Yao, P. G. Snyder, “Determination of AlAs optical constants by variable angle spectroscopic ellipsometry and a multisample analysis,” J. Appl. Phys. 77, 4677–4687 (1995).
[CrossRef]

Ho, F. C.

Hodgkin, V. A.

Jellison, G. E.

Klapp, W. P.

Leblanc, R. M.

Macleod, H. A.

Pelletier, E.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1991).

Price, K.

R. Storn, K. Price, “Minimizing the real functions of the ICEC’96 contest by differential evolution,” in Proceedings of the 1996 IEEE Conference on Evolutionary Computation (ICEC ’96) (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 842–844.

K. Price, R. Storn, “Differential evolution—a simple and efficient adaptive scheme for global optimization over continuous spaces,” (International Computer Science Institute, University of California at Berkeley, Berkeley, Calif., 1995) (available through ftp.icsi.berkeley.edu).

Purvis, M. K.

Quinn, D. M.

Russev, S. C.

Snyder, P. G.

C. M. Herzinger, H. Yao, P. G. Snyder, “Determination of AlAs optical constants by variable angle spectroscopic ellipsometry and a multisample analysis,” J. Appl. Phys. 77, 4677–4687 (1995).
[CrossRef]

Storn, R.

R. Storn, K. Price, “Minimizing the real functions of the ICEC’96 contest by differential evolution,” in Proceedings of the 1996 IEEE Conference on Evolutionary Computation (ICEC ’96) (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 842–844.

K. Price, R. Storn, “Differential evolution—a simple and efficient adaptive scheme for global optimization over continuous spaces,” (International Computer Science Institute, University of California at Berkeley, Berkeley, Calif., 1995) (available through ftp.icsi.berkeley.edu).

Strome, D. H.

Swenson, R.

Temple, P. A.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1991).

Thonn, T. F.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1991).

Yao, H.

C. M. Herzinger, H. Yao, P. G. Snyder, “Determination of AlAs optical constants by variable angle spectroscopic ellipsometry and a multisample analysis,” J. Appl. Phys. 77, 4677–4687 (1995).
[CrossRef]

Appl. Opt.

J. Appl. Phys.

C. M. Herzinger, H. Yao, P. G. Snyder, “Determination of AlAs optical constants by variable angle spectroscopic ellipsometry and a multisample analysis,” J. Appl. Phys. 77, 4677–4687 (1995).
[CrossRef]

J. Opt. Soc. Am. A

Other

A Verity Instruments Series-10 Spectral Ellipsometer was used in obtaining data. The wavelength increment is determined by the spectrograph optics and the grating dispersion, which varies from 9.5 nm at the short-wavelength end to 9.1 nm at the long-wavelength end.

E. D. Palik, ed., Handbook of Optical Constants of Solids, Volume 1 and 2 (Academic, New York, 1991).

K. Price, R. Storn, “Differential evolution—a simple and efficient adaptive scheme for global optimization over continuous spaces,” (International Computer Science Institute, University of California at Berkeley, Berkeley, Calif., 1995) (available through ftp.icsi.berkeley.edu).

R. Storn, K. Price, “Minimizing the real functions of the ICEC’96 contest by differential evolution,” in Proceedings of the 1996 IEEE Conference on Evolutionary Computation (ICEC ’96) (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 842–844.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1996).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1991).

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Figures (6)

Fig. 1
Fig. 1

χ2 as a function of depth and angle for the SiO2-on-Si system. Depth is in micrometers (0.0–1.0 µm) and angle is in degrees (35°–85°). The actual depth of the model film is 560 nm, and the wavelength range is λ(min)=343.4 nm to λ(max)=1047 nm in steps of 9.4 nm.

Fig. 2
Fig. 2

Three cuts of the χ2 function defined in Fig. 1 at three fixed angles, 45° (shallowest modulation), 65°, and 75°. At 75° χ2 is highly discontinuous.

Fig. 3
Fig. 3

Complex reflectivity (ρ) parameterized by wavelength for angles around the critical angle (where ρ). At the critical angle, rs is identically 0. For angles less than the critical angle, ρ spirals around the -1 attractor, and for angles larger than the critical angle, ρ spirals around the +1 attractor. Each loop around +1 results in a phase jump of 2π for the normal tangent branch cut along the positive real axis.

Fig. 4
Fig. 4

χ2 for the SiO2Si system close to the global minimum for angles of incidence where all wavelengths are below the critical angle (65°), close to the critical angle (72° and 75°), and greater than the critical angle (80°).

Fig. 5
Fig. 5

Plot of ρ for the system defined in Fig. 1 in the complex plane for incidence angle of 75°. Wavelength increases from 340 nm (starting on the left) to 1050 nm. Several specific wavelengths (in micrometers) are marked on the plot.

Fig. 6
Fig. 6

Contour plot of critical-angle–wavelength regions for the SiO2-on-Si system. The contours are labeled with maximum wavelength (in micrometers). Thus, for a film thickness of 0.7 µm and an incident angle of 70°, wavelengths below 0.9 µm will generally result in real (ρ)<0.

Equations (3)

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ρ=rprs=tan(ψ)exp(iΔ),
ρ=ρ(nsup, nsub, ksub, θ, λ),
χ2(d, θ)=1Ni=1Nψ[λ(i)]-ψ¯iσi2+Δ[λ(i)]-Δ¯iτi2,

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