Abstract

A general mathematical formulation for solving the ellipsometric equation of a substrate covered by an arbitrary number of transparent or absorbing layers is presented. For single-wavelength and single-angle-of-incidence ellipsometry, from a single measurement (Δ, Ψ), the procedure allows the finding of any two-parameter combination, provided that one of the unknowns is a thickness.

© 1995 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, Amsterdam, 1977), Chap. 4, p. 269.
  2. A. R. Reinberg, “Ellipsometer data analysis with a small programmable desk calculator,” Appl. Opt. 11, 1273–1274 (1972).
    [Crossref] [PubMed]
  3. S. S. So, “Ellipsometric analyses for an absorbing surface film on an absorbing substrate with or without an intermediate surface layer,” Surf. Sci. 56, 97–108 (1976).
    [Crossref]
  4. 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]
  5. S. Bosch, “Double layer ellipsometry: an efficient numerical method for data analysis,” Surf. Sci. 289, 411–417 (1993).
    [Crossref]
  6. S. Bosch, F. Monzonís, “New algorithm for monochromatic ellipsometry of double absorbing layers,” Surf. Sci. 321, 156–160 (1994).
    [Crossref]
  7. A. Vasicek, Optics of Thin Films (North-Holland, New York, 1960).
  8. F. Abelès, “Recherches sur la propagation des ondes électromagnetiques sinusoidales dans les milieux stratifiés,” Ann. Phys. 5, 596–640 (1950).
  9. P. C. S. Hayfield, G. W. T. White, in Ellipsometry in the Measurement of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, J. Kruger eds., Natl. Bur. Std. Misc. Publ. 256 (U.S. Government Printing Office, Washington, D.C., 1964), p. 157.
  10. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Wetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), Chap. 9, p. 255.

1994 (1)

S. Bosch, F. Monzonís, “New algorithm for monochromatic ellipsometry of double absorbing layers,” Surf. Sci. 321, 156–160 (1994).
[Crossref]

1993 (1)

S. Bosch, “Double layer ellipsometry: an efficient numerical method for data analysis,” Surf. Sci. 289, 411–417 (1993).
[Crossref]

1988 (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]

1976 (1)

S. S. So, “Ellipsometric analyses for an absorbing surface film on an absorbing substrate with or without an intermediate surface layer,” Surf. Sci. 56, 97–108 (1976).
[Crossref]

1972 (1)

1950 (1)

F. Abelès, “Recherches sur la propagation des ondes électromagnetiques sinusoidales dans les milieux stratifiés,” Ann. Phys. 5, 596–640 (1950).

Abelès, F.

F. Abelès, “Recherches sur la propagation des ondes électromagnetiques sinusoidales dans les milieux stratifiés,” Ann. Phys. 5, 596–640 (1950).

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 4, p. 269.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 4, p. 269.

Bosch, S.

S. Bosch, F. Monzonís, “New algorithm for monochromatic ellipsometry of double absorbing layers,” Surf. Sci. 321, 156–160 (1994).
[Crossref]

S. Bosch, “Double layer ellipsometry: an efficient numerical method for data analysis,” Surf. Sci. 289, 411–417 (1993).
[Crossref]

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]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Wetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), Chap. 9, p. 255.

Hayfield, P. C. S.

P. C. S. Hayfield, G. W. T. White, in Ellipsometry in the Measurement of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, J. Kruger eds., Natl. Bur. Std. Misc. Publ. 256 (U.S. Government Printing Office, Washington, D.C., 1964), p. 157.

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]

Monzonís, F.

S. Bosch, F. Monzonís, “New algorithm for monochromatic ellipsometry of double absorbing layers,” Surf. Sci. 321, 156–160 (1994).
[Crossref]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Wetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), Chap. 9, p. 255.

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.

So, S. S.

S. S. So, “Ellipsometric analyses for an absorbing surface film on an absorbing substrate with or without an intermediate surface layer,” Surf. Sci. 56, 97–108 (1976).
[Crossref]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Wetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), Chap. 9, p. 255.

Vasicek, A.

A. Vasicek, Optics of Thin Films (North-Holland, New York, 1960).

Wetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Wetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), Chap. 9, p. 255.

White, G. W. T.

P. C. S. Hayfield, G. W. T. White, in Ellipsometry in the Measurement of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, J. Kruger eds., Natl. Bur. Std. Misc. Publ. 256 (U.S. Government Printing Office, Washington, D.C., 1964), p. 157.

Ann. Phys. (1)

F. Abelès, “Recherches sur la propagation des ondes électromagnetiques sinusoidales dans les milieux stratifiés,” Ann. Phys. 5, 596–640 (1950).

Appl. Opt. (1)

Surf. Sci. (4)

S. S. So, “Ellipsometric analyses for an absorbing surface film on an absorbing substrate with or without an intermediate surface layer,” Surf. Sci. 56, 97–108 (1976).
[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]

S. Bosch, “Double layer ellipsometry: an efficient numerical method for data analysis,” Surf. Sci. 289, 411–417 (1993).
[Crossref]

S. Bosch, F. Monzonís, “New algorithm for monochromatic ellipsometry of double absorbing layers,” Surf. Sci. 321, 156–160 (1994).
[Crossref]

Other (4)

A. Vasicek, Optics of Thin Films (North-Holland, New York, 1960).

P. C. S. Hayfield, G. W. T. White, in Ellipsometry in the Measurement of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, J. Kruger eds., Natl. Bur. Std. Misc. Publ. 256 (U.S. Government Printing Office, Washington, D.C., 1964), p. 157.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Wetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), Chap. 9, p. 255.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 4, p. 269.

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Figures (4)

Fig. 1
Fig. 1

Sketch of the ideal sample considered for illustration.

Fig. 2
Fig. 2

Imaginary parts of the thicknesses computed from the roots obtained for values 3.50 < n2 < 4.50 (one branch) for orders 0, 1, …10.

Fig. 3
Fig. 3

Bands obtained in considering δΔ = ±0.04 and δΨ = ±0.02 around the exact values Δ = 145.446, Ψ = 14.094, which correspond to the ideal sample (only for orders 4–8).

Fig. 4
Fig. 4

Imaginary parts of the thicknesses computed from the roots obtained for values 0.00 < k2 < 0.10 (one branch) for orders 0, 1, …10.

Equations (18)

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ρ = tan ( Ψ ) exp ( i Δ ) .
X exp ( - 2 i β ) ,             β = 2 π ( d / λ ) N cos ( ϕ ) ,
S = I 0 , 1 L 1 I 1 , 2 L 2 L n I n , n + 1 , L j = [ exp ( i β j ) 0 0 exp ( - i β j ) ] , β j = 2 π λ N j d j cos ( ϕ j ) , I k , l = 1 t k , l [ 1 r k , l r k , l 1 ] ,
S = [ S 11 S 12 S 21 S 22 ] ,
R p = S 21 p S 11 p ,             R s = S 21 s / S 11 s , ρ ρ r = R p / R s .
S = M L j N , M = I 0 , 1 L 1 L j - 1 I j - 1 , j , N = I j , j + 1 L j + 1 L n I n , n + 1 .
S = [ M 11 M 12 M 21 M 22 ] [ X 0 0 X * ] [ N 11 N 12 N 21 N 22 ] ,
ρ = S 21 p S 11 s S 11 p S 21 s = ( M 21 p N 11 p X + M 22 p N 21 p X * ) ( M 11 s N 11 s X + M 12 s N 21 s X * ) ( M 11 P N 11 p X + M 12 p N 21 p X * ) ( M 21 s N 11 s X + M 22 s N 21 s X * ) ,
( M 21 p N 11 p M 11 s N 11 s - ρ M 21 s N 11 s M 11 p N 11 p ) X 2 + ( M 22 p N 21 p M 12 s N 21 s - ρ M 22 s N 21 s M 12 p N 21 p ) ( X * ) 2 + [ M 21 p N 11 p M 12 s N 21 s + M 22 p N 21 p M 11 s N 11 s - ρ × ( M 21 s N 11 s M 12 p N 21 p + M 22 s N 21 s M 11 p N 11 p ) ] X X * = 0.
a = M 21 p N 11 p M 11 s N 11 s - ρ M 21 s N 11 s M 11 p N 11 p , b = M 22 p N 21 p M 12 s N 21 s - ρ M 22 s N 21 s M 12 p N 21 p , c = M 21 p N 11 p M 12 s N 21 s + M 22 p N 21 p M 11 s N 11 s - ρ ( M 21 s N 11 s M 12 p N 21 p + M 22 s N 21 s M 11 p N 11 p ) ,
a X 2 + b ( X * ) 2 + c X X * = 0
a ( X / X * ) 2 + b ( X / X * ) + c = 0.
d j = - i λ log Z 4 π n j cos ( ϕ j )
d j = - i λ log Z 4 π N j cos ( ϕ j )
d M = d 0 + M λ 2 N j cos ( ϕ j ) ,             M = integer ,
d 0 = i λ ln ( r ) - λ θ 4 π N j cos ( ϕ j ) .
λ / [ 2 n j cos ( ϕ j ) ]
Im ( d j ) < 0 < Im ( d j )             or             Im ( d j ) > 0 > Im ( d j ) ,

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