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.

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  1. D. G. Schueler, Surface Sci. 16, 104 (1969).
  2. M. Berman, H. R. Kerchner, and S. Ergun, J. Opt. Soc. Am. 60, 646 (1970).
  3. D. L. Greenaway and G. Harbeke, Phys. Rev. 178, 1340 (1969).
  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).
  7. F. L. McCrackin, in A Fortran Program for Analysis of Ellipsometer Measurements, edited by E. Passaglia, R. R. Stromberg, and J. Kruger, Natrsquo;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).
  10. F. Lukes, W. H. Knausenberger, and K. Vedam, Surface Sci. 16, 112 (1969).
  11. J. Shewchun and E. C. Rowe, J. Appl. Phys. 41, 4128 (1970).
  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.
  15. Reference 4, p. 62.
  16. L. Ward and A. Nag, J. Phys. D 3, 462 (1969).
  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).

Berman, M.

M. Berman, H. R. Kerchner, and S. Ergun, J. Opt. Soc. Am. 60, 646 (1970).

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.

M. Berman, H. R. Kerchner, and S. Ergun, J. Opt. Soc. Am. 60, 646 (1970).

Fane, R. W.

R. W. Fane and W. E. J. Neal, J. Opt. Soc. Am. 60, 790 (1970).

Greenaway, D. L.

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

Harbeke, G.

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

Kerchner, H. R.

M. Berman, H. R. Kerchner, and S. Ergun, J. Opt. Soc. Am. 60, 646 (1970).

Knausenberger, W. H.

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

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

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).

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

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.

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, Natrsquo;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).

Neal, W. E. J.

R. W. Fane and W. E. J. Neal, J. Opt. Soc. Am. 60, 790 (1970).

Oldham, W. G.

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

Rowe, E. C.

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

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).

Shewchun, J.

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

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).

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

Ward, L.

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

Other (19)

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

M. Berman, H. R. Kerchner, and S. Ergun, J. Opt. Soc. Am. 60, 646 (1970).

D. L. Greenaway and G. Harbeke, Phys. Rev. 178, 1340 (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.

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

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

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).

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

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

Reference 4, p. 52.

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

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.

Reference 4, p. 62.

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

Reference 4, Chs. 4–6.

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

R. W. Fane and W. E. J. Neal, J. Opt. Soc. Am. 60, 790 (1970).

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