Abstract

We have found that the thickness and refractive indices for the film–substrate problem in ellipsometry can frequently be calculated with a least-squares technique. It is shown that one algorithm can be used with a variety of data to solve problems that were formerly solved with step-by-step searches through parameter space. Moreover, the least-squares technique leads to a statistically meaningful estimate of the variances of the parameters. It is shown that the variances obtained from a correct error analysis can be at least one hundred times as large as those obtained from a naive analysis. It is also shown how the error analysis can provide information that is helpful in deciding if a given experiment is worthwhile.

© 1971 Optical Society of America

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References

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  1. D. G. Schueler, Surface Sci. 16, 104 (1969).
    [CrossRef]
  2. M. Berman, H. R. Kerchner, and S. Ergun, J. Opt. Soc. Am. 60, 646 (1970).
    [CrossRef]
  3. D. L. Greenaway and G. Harbeke, Phys. Rev. 178, 1340 (1969).
    [CrossRef]
  4. R. Deutsch, Estimation Theory (Prentice-Hall, Englewood Cliffs, N. J., 1965).
  5. F. L. McCrackin and J. P. Colson, in Ellipsometry in the Measurement of Surfaces and Thin Films, edited by E. Passaglia, R. R. Stromberg, and J. Kruger, Nat’l. Bur. Std. (U.S.) Publ. 265 (U. S. Govt. Printing Office, Washington, D. C., 1964), p. 61.
  6. W. G. Oldham, Surface Sci. 16, 97 (1969).
    [CrossRef]
  7. F. L. McCrackin, in A Fortran Program for Analysis of Ellipsometer Measurements, edited by E. Passaglia, R. R. Stromberg, and J. Kruger, Nat’l. Bur. Std. (U.S.) Publ. 479 (U. S. Govt. Printing Office, Washington, D. C., 1969).
  8. P. F. Rusch, W. H. Kochler, and J. J. Lagowski, Spectrochim. Acta 25A, 1527 (1969).
  9. K. Vedam, W. H. Knausenberger, and F. Lukes, J. Opt. Soc. Am. 59, 64 (1969).
    [CrossRef]
  10. F. Lukes, W. H. Knausenberger, and K. Vedam, Surface Sci. 16, 112 (1969).
    [CrossRef]
  11. J. Shewchun and E. C. Rowe, J. Appl. Phys. 41, 4128 (1970).
    [CrossRef]
  12. Reference 4, p. 52.
  13. P. Drude, Wied. Ann. Physik 30, 532, 865 (1899).
  14. D. W. Marquardt, J. Soc. Appl. Math. 11, 431 (1963). A version of this algorithm called CO-OP Library Code E2 WISC GAUSHAUS (Contributed by the University of Wisconsin, 1965) suitably modified for a CDC 6600 was used. Listings of our program are available on request.
    [CrossRef]
  15. Reference 4, p. 62.
  16. L. Ward and A. Nag, J. Phys. D 3, 462 (1969).
    [CrossRef]
  17. Reference 4, Chs. 4–6.
  18. W. H. Knausenberger and K. Vedam, Phys. Letters 29A, 428 (1969).
  19. R. W. Fane and W. E. J. Neal, J. Opt. Soc. Am. 60, 790 (1970).
    [CrossRef]

1970 (3)

1969 (8)

L. Ward and A. Nag, J. Phys. D 3, 462 (1969).
[CrossRef]

W. H. Knausenberger and K. Vedam, Phys. Letters 29A, 428 (1969).

D. G. Schueler, Surface Sci. 16, 104 (1969).
[CrossRef]

D. L. Greenaway and G. Harbeke, Phys. Rev. 178, 1340 (1969).
[CrossRef]

W. G. Oldham, Surface Sci. 16, 97 (1969).
[CrossRef]

P. F. Rusch, W. H. Kochler, and J. J. Lagowski, Spectrochim. Acta 25A, 1527 (1969).

K. Vedam, W. H. Knausenberger, and F. Lukes, J. Opt. Soc. Am. 59, 64 (1969).
[CrossRef]

F. Lukes, W. H. Knausenberger, and K. Vedam, Surface Sci. 16, 112 (1969).
[CrossRef]

1963 (1)

D. W. Marquardt, J. Soc. Appl. Math. 11, 431 (1963). A version of this algorithm called CO-OP Library Code E2 WISC GAUSHAUS (Contributed by the University of Wisconsin, 1965) suitably modified for a CDC 6600 was used. Listings of our program are available on request.
[CrossRef]

1899 (1)

P. Drude, Wied. Ann. Physik 30, 532, 865 (1899).

Berman, M.

Colson, J. P.

F. L. McCrackin and J. P. Colson, in Ellipsometry in the Measurement of Surfaces and Thin Films, edited by E. Passaglia, R. R. Stromberg, and J. Kruger, Nat’l. Bur. Std. (U.S.) Publ. 265 (U. S. Govt. Printing Office, Washington, D. C., 1964), p. 61.

Deutsch, R.

R. Deutsch, Estimation Theory (Prentice-Hall, Englewood Cliffs, N. J., 1965).

Drude, P.

P. Drude, Wied. Ann. Physik 30, 532, 865 (1899).

Ergun, S.

Fane, R. W.

Greenaway, D. L.

D. L. Greenaway and G. Harbeke, Phys. Rev. 178, 1340 (1969).
[CrossRef]

Harbeke, G.

D. L. Greenaway and G. Harbeke, Phys. Rev. 178, 1340 (1969).
[CrossRef]

Kerchner, H. R.

Knausenberger, W. H.

K. Vedam, W. H. Knausenberger, and F. Lukes, J. Opt. Soc. Am. 59, 64 (1969).
[CrossRef]

F. Lukes, W. H. Knausenberger, and K. Vedam, Surface Sci. 16, 112 (1969).
[CrossRef]

W. H. Knausenberger and K. Vedam, Phys. Letters 29A, 428 (1969).

Kochler, W. H.

P. F. Rusch, W. H. Kochler, and J. J. Lagowski, Spectrochim. Acta 25A, 1527 (1969).

Lagowski, J. J.

P. F. Rusch, W. H. Kochler, and J. J. Lagowski, Spectrochim. Acta 25A, 1527 (1969).

Lukes, F.

K. Vedam, W. H. Knausenberger, and F. Lukes, J. Opt. Soc. Am. 59, 64 (1969).
[CrossRef]

F. Lukes, W. H. Knausenberger, and K. Vedam, Surface Sci. 16, 112 (1969).
[CrossRef]

Marquardt, D. W.

D. W. Marquardt, J. Soc. Appl. Math. 11, 431 (1963). A version of this algorithm called CO-OP Library Code E2 WISC GAUSHAUS (Contributed by the University of Wisconsin, 1965) suitably modified for a CDC 6600 was used. Listings of our program are available on request.
[CrossRef]

McCrackin, F. L.

F. L. McCrackin, in A Fortran Program for Analysis of Ellipsometer Measurements, edited by E. Passaglia, R. R. Stromberg, and J. Kruger, Nat’l. Bur. Std. (U.S.) Publ. 479 (U. S. Govt. Printing Office, Washington, D. C., 1969).

F. L. McCrackin and J. P. Colson, in Ellipsometry in the Measurement of Surfaces and Thin Films, edited by E. Passaglia, R. R. Stromberg, and J. Kruger, Nat’l. Bur. Std. (U.S.) Publ. 265 (U. S. Govt. Printing Office, Washington, D. C., 1964), p. 61.

Nag, A.

L. Ward and A. Nag, J. Phys. D 3, 462 (1969).
[CrossRef]

Neal, W. E. J.

Oldham, W. G.

W. G. Oldham, Surface Sci. 16, 97 (1969).
[CrossRef]

Rowe, E. C.

J. Shewchun and E. C. Rowe, J. Appl. Phys. 41, 4128 (1970).
[CrossRef]

Rusch, P. F.

P. F. Rusch, W. H. Kochler, and J. J. Lagowski, Spectrochim. Acta 25A, 1527 (1969).

Schueler, D. G.

D. G. Schueler, Surface Sci. 16, 104 (1969).
[CrossRef]

Shewchun, J.

J. Shewchun and E. C. Rowe, J. Appl. Phys. 41, 4128 (1970).
[CrossRef]

Vedam, K.

W. H. Knausenberger and K. Vedam, Phys. Letters 29A, 428 (1969).

K. Vedam, W. H. Knausenberger, and F. Lukes, J. Opt. Soc. Am. 59, 64 (1969).
[CrossRef]

F. Lukes, W. H. Knausenberger, and K. Vedam, Surface Sci. 16, 112 (1969).
[CrossRef]

Ward, L.

L. Ward and A. Nag, J. Phys. D 3, 462 (1969).
[CrossRef]

J. Appl. Phys. (1)

J. Shewchun and E. C. Rowe, J. Appl. Phys. 41, 4128 (1970).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Phys. D (1)

L. Ward and A. Nag, J. Phys. D 3, 462 (1969).
[CrossRef]

J. Soc. Appl. Math. (1)

D. W. Marquardt, J. Soc. Appl. Math. 11, 431 (1963). A version of this algorithm called CO-OP Library Code E2 WISC GAUSHAUS (Contributed by the University of Wisconsin, 1965) suitably modified for a CDC 6600 was used. Listings of our program are available on request.
[CrossRef]

Phys. Letters (1)

W. H. Knausenberger and K. Vedam, Phys. Letters 29A, 428 (1969).

Phys. Rev. (1)

D. L. Greenaway and G. Harbeke, Phys. Rev. 178, 1340 (1969).
[CrossRef]

Spectrochim. Acta (1)

P. F. Rusch, W. H. Kochler, and J. J. Lagowski, Spectrochim. Acta 25A, 1527 (1969).

Surface Sci. (3)

D. G. Schueler, Surface Sci. 16, 104 (1969).
[CrossRef]

W. G. Oldham, Surface Sci. 16, 97 (1969).
[CrossRef]

F. Lukes, W. H. Knausenberger, and K. Vedam, Surface Sci. 16, 112 (1969).
[CrossRef]

Wied. Ann. Physik (1)

P. Drude, Wied. Ann. Physik 30, 532, 865 (1899).

Other (6)

Reference 4, p. 52.

Reference 4, p. 62.

Reference 4, Chs. 4–6.

F. L. McCrackin, in A Fortran Program for Analysis of Ellipsometer Measurements, edited by E. Passaglia, R. R. Stromberg, and J. Kruger, Nat’l. Bur. Std. (U.S.) Publ. 479 (U. S. Govt. Printing Office, Washington, D. C., 1969).

R. Deutsch, Estimation Theory (Prentice-Hall, Englewood Cliffs, N. J., 1965).

F. L. McCrackin and J. P. Colson, in Ellipsometry in the Measurement of Surfaces and Thin Films, edited by E. Passaglia, R. R. Stromberg, and J. Kruger, Nat’l. Bur. Std. (U.S.) Publ. 265 (U. S. Govt. Printing Office, Washington, D. C., 1964), p. 61.

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Figures (3)

Fig. 1
Fig. 1

Hypothetical-error contour for a two-parameter fit with β1(1)/(β1 − intercept) = 2.7.

Fig. 2
Fig. 2

Variation of σd and σnf (for d = 26.0 nm, nf = 1.80, ñs = 1.80 − 18.0j) with choice of angles of incidence. The abscissa labels give the first and last angle in a series of five, e.g., 20 40 stands or 20°, 25°, 30°, 35°, 40°.

Fig. 3
Fig. 3

Variations of σd, σnf, and σns with d, the film thickness, shown for ñf = 1.64 − 0.0i, ñs = 5.64 − 1.81i, δΔ = 0.2, δψ = 0.1, and δRef = 0.02.

Tables (2)

Tables Icon

Table I The sum of errors QQ0 is shown for the deviation vectors calculated from Eqs. (11) and (12). Also shown are the corresponding values for test vectors in which all deviations but one were set to zero.

Tables Icon

Table II The results of calculations for a four-parameter fit to multiple-angle data for d = 30.0 nm, nf = 1.65, and ñs = 1.00 − 6.00i are shown. In case B, the estimate of nf = 1.65 with σnf = 0.1 was included in the least-squares sum. Results are shown as converged or final value ± standard deviation.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

Q ( d , ñ f , ñ s ) = d i ( y i - f i ) 2 δ i 2 ,
f i ( b ) f i ( B ) + A i μ β μ ,
A i μ = f i ( B ) / b μ ,
β = b - B .
Q ( b ) Q 0 + q ( β ) ,
Q 0 = Q ( B ) ,
q ( β ) = β μ C μ ν β ν .
C μ ν = d i A μ i A ν i δ i 2 .
Γ = C - 1 .
σ μ = ( Γ μ μ ) 1 2 .
β μ = ± ( C μ μ ) - 1 2 = ± [ ( Γ - 1 ) μ μ ] - 1 2 .
β ν ( μ ) = Γ ν μ ( Γ μ μ ) - 1 2 .