Abstract

Easy sample preparation and measurement makes ellipsometry attractive for surface and film investigations; however, specialized numerical methods are normally required to relate measurements to unknown physical attributes of reflecting surfaces. Although solution techniques have been developed, the problem of data analysis is by no means solved. This paper presents an alternative method for computation of the thickness and optical properties of an absorbing film overlying a known substrate from ellipsometer data obtained at two light incidence angles. Tests for selected cases required no a priori knowledge of the film, in contrast to other methods.

© 1993 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, 1977).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  5. Y. Yoriume, “Method for numerical inversion of the ellipsometry equations for transparent films,” J. Opt. Soc. Am. 73, 888–891 (1983).
    [CrossRef]
  6. F. L. McCrackin, J. P. Colson, “Computational techniques for the use of the exact Drude equations in reflection problems,” in Ellipsometry in the Measurement of Surfaces and Films, E. Passaglia, R. R. Stromberg, J. Kruger, eds., Natl. Bur. Stand. Misc. Publ.256, 61–82 (1964).
  7. L. V. Semenenko, K. K. Svitashev, A. I. Semenenko, V. K. Sokolov, “Ellipsometry of absorbing films,” Opt. Spektrosk. 32, 1204–1210 (1972)[Opt. Spectrosc. (USSR) 32, 655–658 (1972)].
  8. D. Aspnes, “Spectroscopic ellipsometry of solids,” in Optical Properties of Solids: New Developments, B. O. Seraphin, ed. (Elsevier, New York, 1976). pp. 799–846.
  9. A. B. Winterbottom, Optical Studies of Metal Surfaces (I Kommisjon Hos F. Bruns, Trondheim, Norway, 1956).
  10. A. R. Reinberg, “Ellipsometer data analysis with a small programmable desk calculator,” Appl. Opt. 11, 1273–1274 (1972).
    [CrossRef] [PubMed]
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    [CrossRef]
  12. E. Elizalde, J. M. Frigerio, J. Rivory, “Determination of thickness and optical constants of thin films from photometric and ellipsometric measurements,” Appl. Opt. 25, 4557–4561 (1986).
    [CrossRef] [PubMed]
  13. J. Lekner, “Analytic inversion of ellipsometric data for an unsupported nonabsorbing uniform layer,” J. Opt. Soc. Am. A 7, 1875–1877 (1990).
    [CrossRef]
  14. J. H. Ho, C. L. Lee, T. F. Lei, T. S. Chao, “Ellipsometry measurement of the complex refractive index and thickness of polysilicon thin films,” J. Opt. Soc. Am. A 7, 196–205 (1990).
    [CrossRef]
  15. R. Zhu, C. Lin, Y. Wei, “Ellipsometric methods for the determination of the parameters of non-absorbing uniaxial anisotropic films,” Thin Solid Films 203, 213–219 (1991).
    [CrossRef]
  16. P. G. Snyder, M. C. Rost, G. H. Bu-Abbud, J. S. Woollam, “Variable angle of incidence spectroscopic ellipsometry: application to GaAs–AlxGa1–x As multiple heterostructures,” J. Apply. Phys. 60, 3293–3301 (1986).
    [CrossRef]
  17. J. Humliček, “Sensitivity extrema in multiple-angle ellipsometry,” J. Opt. Soc. Am. A 2, 713–722 (1985).
    [CrossRef]
  18. Y. Gaillyová, E. Schmidt, J. Humlicek, “Multiple-angle ellipsometry of Si–SiO2 polycrystalline Si system,” J. Opt. Soc. Am. A 2, 723–726 (1985).
    [CrossRef]
  19. F. K. Urban, “Ellipsometer measurement of thickness and optical properties of thin absorbing films,” Appl. Surf. Sci. 33/34, 934–941 (1988).
    [CrossRef]
  20. W. H. Press, B. P. Flannery, S. A. Tenkolsky, W. T. Vetterling, Numerical Recipes, the Art of Scientific Computing (fortran Version) (Cambridge U. Press, Cambridge, 1989), pp. 523–528.
  21. G. H. Bu-Abbud, N. M. Bashara, “Parameter correlation and precision in multiple-angle ellipsometry,” Appl. Opt. 20, 3020–3026 (1981).
    [CrossRef] [PubMed]

1992 (1)

1991 (1)

R. Zhu, C. Lin, Y. Wei, “Ellipsometric methods for the determination of the parameters of non-absorbing uniaxial anisotropic films,” Thin Solid Films 203, 213–219 (1991).
[CrossRef]

1990 (2)

1988 (2)

F. K. Urban, “Ellipsometer measurement of thickness and optical properties of thin absorbing films,” Appl. Surf. Sci. 33/34, 934–941 (1988).
[CrossRef]

T. Easwarakhanthan, C. Michel, S. Ravelet, “Numerical method for the ellipsometric determination of optical constants and thickness of thin films with microcomputers,” Surf. Sci. 197, 339–345 (1988).
[CrossRef]

1986 (2)

E. Elizalde, J. M. Frigerio, J. Rivory, “Determination of thickness and optical constants of thin films from photometric and ellipsometric measurements,” Appl. Opt. 25, 4557–4561 (1986).
[CrossRef] [PubMed]

P. G. Snyder, M. C. Rost, G. H. Bu-Abbud, J. S. Woollam, “Variable angle of incidence spectroscopic ellipsometry: application to GaAs–AlxGa1–x As multiple heterostructures,” J. Apply. Phys. 60, 3293–3301 (1986).
[CrossRef]

1985 (2)

1983 (1)

1981 (1)

1975 (1)

1972 (2)

L. V. Semenenko, K. K. Svitashev, A. I. Semenenko, V. K. Sokolov, “Ellipsometry of absorbing films,” Opt. Spektrosk. 32, 1204–1210 (1972)[Opt. Spectrosc. (USSR) 32, 655–658 (1972)].

A. R. Reinberg, “Ellipsometer data analysis with a small programmable desk calculator,” Appl. Opt. 11, 1273–1274 (1972).
[CrossRef] [PubMed]

1968 (1)

Aspnes, D.

D. Aspnes, “Spectroscopic ellipsometry of solids,” in Optical Properties of Solids: New Developments, B. O. Seraphin, ed. (Elsevier, New York, 1976). pp. 799–846.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).

Bashara, N. M.

Bu-Abbud, G. H.

P. G. Snyder, M. C. Rost, G. H. Bu-Abbud, J. S. Woollam, “Variable angle of incidence spectroscopic ellipsometry: application to GaAs–AlxGa1–x As multiple heterostructures,” J. Apply. Phys. 60, 3293–3301 (1986).
[CrossRef]

G. H. Bu-Abbud, N. M. Bashara, “Parameter correlation and precision in multiple-angle ellipsometry,” Appl. Opt. 20, 3020–3026 (1981).
[CrossRef] [PubMed]

Chao, T. S.

Colson, J. P.

F. L. McCrackin, J. P. Colson, “Computational techniques for the use of the exact Drude equations in reflection problems,” in Ellipsometry in the Measurement of Surfaces and Films, E. Passaglia, R. R. Stromberg, J. Kruger, eds., Natl. Bur. Stand. Misc. Publ.256, 61–82 (1964).

Easwarakhanthan, T.

T. Easwarakhanthan, C. Michel, S. Ravelet, “Numerical method for the ellipsometric determination of optical constants and thickness of thin films with microcomputers,” Surf. Sci. 197, 339–345 (1988).
[CrossRef]

Elizalde, E.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Tenkolsky, W. T. Vetterling, Numerical Recipes, the Art of Scientific Computing (fortran Version) (Cambridge U. Press, Cambridge, 1989), pp. 523–528.

Frigerio, J. M.

Gaillyová, Y.

Ghezzo, M.

Ho, J. H.

Humlicek, J.

Lee, C. L.

Lei, T. F.

Lekner, J.

Lin, C.

R. Zhu, C. Lin, Y. Wei, “Ellipsometric methods for the determination of the parameters of non-absorbing uniaxial anisotropic films,” Thin Solid Films 203, 213–219 (1991).
[CrossRef]

McCrackin, F. L.

F. L. McCrackin, J. P. Colson, “Computational techniques for the use of the exact Drude equations in reflection problems,” in Ellipsometry in the Measurement of Surfaces and Films, E. Passaglia, R. R. Stromberg, J. Kruger, eds., Natl. Bur. Stand. Misc. Publ.256, 61–82 (1964).

Michel, C.

T. Easwarakhanthan, C. Michel, S. Ravelet, “Numerical method for the ellipsometric determination of optical constants and thickness of thin films with microcomputers,” Surf. Sci. 197, 339–345 (1988).
[CrossRef]

Namioka, T.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Tenkolsky, W. T. Vetterling, Numerical Recipes, the Art of Scientific Computing (fortran Version) (Cambridge U. Press, Cambridge, 1989), pp. 523–528.

Ravelet, S.

T. Easwarakhanthan, C. Michel, S. Ravelet, “Numerical method for the ellipsometric determination of optical constants and thickness of thin films with microcomputers,” Surf. Sci. 197, 339–345 (1988).
[CrossRef]

Reinberg, A. R.

Rivory, J.

Rost, M. C.

P. G. Snyder, M. C. Rost, G. H. Bu-Abbud, J. S. Woollam, “Variable angle of incidence spectroscopic ellipsometry: application to GaAs–AlxGa1–x As multiple heterostructures,” J. Apply. Phys. 60, 3293–3301 (1986).
[CrossRef]

Schmidt, E.

Semenenko, A. I.

L. V. Semenenko, K. K. Svitashev, A. I. Semenenko, V. K. Sokolov, “Ellipsometry of absorbing films,” Opt. Spektrosk. 32, 1204–1210 (1972)[Opt. Spectrosc. (USSR) 32, 655–658 (1972)].

Semenenko, L. V.

L. V. Semenenko, K. K. Svitashev, A. I. Semenenko, V. K. Sokolov, “Ellipsometry of absorbing films,” Opt. Spektrosk. 32, 1204–1210 (1972)[Opt. Spectrosc. (USSR) 32, 655–658 (1972)].

Snyder, P. G.

P. G. Snyder, M. C. Rost, G. H. Bu-Abbud, J. S. Woollam, “Variable angle of incidence spectroscopic ellipsometry: application to GaAs–AlxGa1–x As multiple heterostructures,” J. Apply. Phys. 60, 3293–3301 (1986).
[CrossRef]

Sokolov, V. K.

L. V. Semenenko, K. K. Svitashev, A. I. Semenenko, V. K. Sokolov, “Ellipsometry of absorbing films,” Opt. Spektrosk. 32, 1204–1210 (1972)[Opt. Spectrosc. (USSR) 32, 655–658 (1972)].

Svitashev, K. K.

L. V. Semenenko, K. K. Svitashev, A. I. Semenenko, V. K. Sokolov, “Ellipsometry of absorbing films,” Opt. Spektrosk. 32, 1204–1210 (1972)[Opt. Spectrosc. (USSR) 32, 655–658 (1972)].

Takahashi, H.

Tenkolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Tenkolsky, W. T. Vetterling, Numerical Recipes, the Art of Scientific Computing (fortran Version) (Cambridge U. Press, Cambridge, 1989), pp. 523–528.

Urban, F. K.

F. K. Urban, “Ellipsometer measurement of thickness and optical properties of thin absorbing films,” Appl. Surf. Sci. 33/34, 934–941 (1988).
[CrossRef]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Tenkolsky, W. T. Vetterling, Numerical Recipes, the Art of Scientific Computing (fortran Version) (Cambridge U. Press, Cambridge, 1989), pp. 523–528.

Wei, Y.

R. Zhu, C. Lin, Y. Wei, “Ellipsometric methods for the determination of the parameters of non-absorbing uniaxial anisotropic films,” Thin Solid Films 203, 213–219 (1991).
[CrossRef]

Winterbottom, A. B.

A. B. Winterbottom, Optical Studies of Metal Surfaces (I Kommisjon Hos F. Bruns, Trondheim, Norway, 1956).

Woollam, J. S.

P. G. Snyder, M. C. Rost, G. H. Bu-Abbud, J. S. Woollam, “Variable angle of incidence spectroscopic ellipsometry: application to GaAs–AlxGa1–x As multiple heterostructures,” J. Apply. Phys. 60, 3293–3301 (1986).
[CrossRef]

Yamaguchi, T.

Yamamoto, M.

Yoriume, Y.

Zhu, R.

R. Zhu, C. Lin, Y. Wei, “Ellipsometric methods for the determination of the parameters of non-absorbing uniaxial anisotropic films,” Thin Solid Films 203, 213–219 (1991).
[CrossRef]

Appl. Opt. (5)

Appl. Surf. Sci. (1)

F. K. Urban, “Ellipsometer measurement of thickness and optical properties of thin absorbing films,” Appl. Surf. Sci. 33/34, 934–941 (1988).
[CrossRef]

J. Apply. Phys. (1)

P. G. Snyder, M. C. Rost, G. H. Bu-Abbud, J. S. Woollam, “Variable angle of incidence spectroscopic ellipsometry: application to GaAs–AlxGa1–x As multiple heterostructures,” J. Apply. Phys. 60, 3293–3301 (1986).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Spektrosk. (1)

L. V. Semenenko, K. K. Svitashev, A. I. Semenenko, V. K. Sokolov, “Ellipsometry of absorbing films,” Opt. Spektrosk. 32, 1204–1210 (1972)[Opt. Spectrosc. (USSR) 32, 655–658 (1972)].

Surf. Sci. (1)

T. Easwarakhanthan, C. Michel, S. Ravelet, “Numerical method for the ellipsometric determination of optical constants and thickness of thin films with microcomputers,” Surf. Sci. 197, 339–345 (1988).
[CrossRef]

Thin Solid Films (1)

R. Zhu, C. Lin, Y. Wei, “Ellipsometric methods for the determination of the parameters of non-absorbing uniaxial anisotropic films,” Thin Solid Films 203, 213–219 (1991).
[CrossRef]

Other (5)

D. Aspnes, “Spectroscopic ellipsometry of solids,” in Optical Properties of Solids: New Developments, B. O. Seraphin, ed. (Elsevier, New York, 1976). pp. 799–846.

A. B. Winterbottom, Optical Studies of Metal Surfaces (I Kommisjon Hos F. Bruns, Trondheim, Norway, 1956).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).

F. L. McCrackin, J. P. Colson, “Computational techniques for the use of the exact Drude equations in reflection problems,” in Ellipsometry in the Measurement of Surfaces and Films, E. Passaglia, R. R. Stromberg, J. Kruger, eds., Natl. Bur. Stand. Misc. Publ.256, 61–82 (1964).

W. H. Press, B. P. Flannery, S. A. Tenkolsky, W. T. Vetterling, Numerical Recipes, the Art of Scientific Computing (fortran Version) (Cambridge U. Press, Cambridge, 1989), pp. 523–528.

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Figures (5)

Fig. 1
Fig. 1

Schematic drawing of light incident upon a film-covered surface. The light incidence angle is α0. The single layer film of thickness d is medium 1 and the substrate is medium 2. The complex indices of refraction of the surrounding medium, film, and substrate are n ̂ 0, n ̂ 1, and n ̂ 2, respectively.

Fig. 2
Fig. 2

Solution curves of n1 versus k1 and d versus k1 for a film on silicon (n2 = 3.85, k2 = 0.02) by using 632.8-nm light. For the film, d = 50 nm and n ̂ 1 = 2 i 2. The solid curve shows solution curves for an incidence angle of 70° (ψ = 25.92° and Δ = 102.80°) while the dashed curve shows solution curves for a different incident angle, greatly exaggerated for illustrative purposes.

Fig. 3
Fig. 3

Flow diagram of an algorithm that solves the ellipsometry equations for the optical index of refraction ( n ̂ 1 = n 1 i k 1) and thickness (d) of unknown absorbing films overlying known substrates. The algorithm requires two ellipsometer measurements at different incidence angles, the light angle of incidence, light wavelength in vacuum, substrate complex index, and complex index of the surrounding medium as input data.

Fig. 4
Fig. 4

Solution curves of d versus k1 for two different angles of incidence in which the difference is greatly exaggerated for clarity. The approximate location of the intersection (solution) may be found by first locating the approximate end points by starting from k1 = 0 and increasing until a solution is found at k1 = 1 (low n1 and d) and then decreasing from k1 = 5 until a solution is found at k1 = 3 (high n1 and d). The continuity of the curves is then tested by attempting solutions at a number of points in the range between low and high solutions. The solution is between the points at which the difference changes sign.

Fig. 5
Fig. 5

Segments of solution curves between k1 = 2.0 and k1 = 2.25. The curvature has been exaggerated to show the solution process more clearly. The end points (circles) are connected by straight lines and the k1 of their intersection computed. New n1 and d solutions are computed at the k1 intersection (squares). The process is repeated by connecting the new solutions by straight lines to the rightmost solutions and finding a new k1 intersection. The intersection of the straight lines rapidly approaches the intersection of the curves.

Tables (2)

Tables Icon

Table 1 Computed ψ and Δ Valuesa

Tables Icon

Table 2 Solutions for Film Parameters n1, k1, and da

Equations (7)

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R ̂ p = r ̂ 1 p + r ̂ 2 p exp ( i δ ̂ ) 1 + r ̂ 1 p r ̂ 2 p exp ( i δ ̂ ) ,
R ̂ s = r ̂ 1 s + r ̂ 2 s exp ( i δ ̂ ) 1 + r ̂ 1 s r ̂ 2 s exp ( i δ ̂ ) .
δ ̂ = 4 π d n ̂ 1 cos ( α ̂ 1 ) / λ ,
tan ( Ψ ) exp ( i Δ ) = R ̂ p / R ̂ s .
Ψ i c ( B , θ i ) = Ψ i c ( B 0 , θ i ) + [ Ψ i c ( B 0 , θ i ) ] · δ B , Δ i c ( B , θ i ) = Δ i c ( B 0 , θ i ) + [ Δ i c ( B 0 , θ i ) ] · δ B ,
d error = ( d d 1 + d d 2 ) / [ ( d k ) 2 ( d k ) 1 ] , k d error = [ ( d k ) 2 d d 1 + ( d k ) 1 d d 2 ] / [ ( d k ) 2 ( d k ) 1 ] , n error = ( d n 1 + d n 2 ) / [ ( n k ) 2 ( n k ) 1 ] , k n error = [ ( n k ) 2 d n 1 + ( n k ) 1 d n 2 ] / [ ( n k ) 2 ( n k ) 1 ] ,
n k = ( Ψ k Δ d Δ k Ψ d ) / ( Δ n Ψ d Ψ n Δ d ) , d k = ( Δ n Ψ k Ψ n Δ k ) / ( Ψ n Δ d Δ n Ψ d ) , d d = ( δ Δ Ψ n δ Ψ Δ n ) / ( Ψ n Δ d Δ n Ψ d ) , d n = ( δ Δ Δ d δ Ψ Ψ d ) / ( Δ n Ψ d Ψ n Δ d ) ,

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