Abstract

A procedure is described for simultaneously fitting reflectance data obtained for various photon energies and angles of incidence using a simple physical model for (ω), the complex frequency-dependent dielectric constant. The mutual consistency of the model and of the experimental data is tested, within the accuracy of each measurement. As an example of the technique, reflectance data in the vacuum ultraviolet obtained by Toots, Fowler, and Marton for Be, Ge, Sb, and Bi have been satisfactorily fitted. The model parameters have been used to derive the optical constants n and k (in satisfactory agreement with conventional determinations) and can be readily related to other relevant experimental results or theoretical calculations of (ω) if these are available.

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  1. See, for example, R. P. Madden, in Physics of Thin Films, Vol. 1, G. Hass, Ed. (Academic Press Inc., New York, 1963), p. 123; O. S. Heavens, in Physics of Thin Films, Vol. 2, G. Hass and R. E. Thun, Eds. (Academic Press Inc., New York, 1964), p. 193.
  2. H. W. Verleur, J. Opt. Soc. Am. 58, 1356 (1968).
    [Crossref]
  3. C. J. Powell, J. Opt. Soc. Am. 60, 78 (1970).
    [Crossref]
  4. C. J. Powell, J. Opt. Soc. Am. 59, 738 (1969).
    [Crossref]
  5. W. R. Hunter, J. Opt. Soc. Am. 55, 1197 (1965); Appl. Opt. 6, 2140 (1967).
    [Crossref]
  6. L. Marton and J. Toots, Phys. Rev. 160, 602 (1967).
    [Crossref]
  7. J. Toots, H. A. Fowler, and L. Marton, Phys. Rev. 172, 670 (1968).
    [Crossref]
  8. J. Toots and L. Marton, J. Opt. Soc. Am. 59, 1305 (1969).
    [Crossref]
  9. M. H. Cohen, Phil. Mag. 3, 762 (1958).
    [Crossref]
  10. R . Tousey, J. Opt. Soc. Am. 29, 235 (1939).
    [Crossref]
  11. D . W. Marquardt, J. Soc. Indust. Appl. Math. 11, 431 (1963); T. Baumeister and D. W. Marquardt, Least-Squares Estimation of Nonlinear Parameters, Share General Purpose Library, Distribution No. 1428, December 1962. A revised version of the program is available with the same title as Share Library Distribution No. 309401, August 1966.
    [Crossref]
  12. The program used here11 supplies several estimates of the uncertainty of each parameter as briefly described in the Appendix to paper II3.
  13. J. G. Collins, Appl. Sci. Res. Sec. B, 7, 1 (1958).
    [Crossref]
  14. K. L. Kliewer and R. Fuchs, Phys. Rev. 172, 607 (1968).
    [Crossref]
  15. K. Rabinovitch, L. R. Canfield, and R. P. Madden, Appl. Opt. 4, 1005 (1965); D. C. Hinson, J. Opt. Soc. Am. 56, 408 (1966); R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, Phys. Rev. Letters 21, 1530 (1968).
    [Crossref]
  16. R. H. Ritchie, Surface Sci. 3, 497 (1965); P. Dobberstein, A. Hampe, and G. Sauerbrey, Phys. Letters 27A, 256 (1968); J. L. Stanford, H. E. Bennett, J. M. Bennett, E. J. Ashley, and E. T. Arakawa, Bull. Am. Phys. Soc. 13, 989 (1968); S. E. Schnatterly, 13, 989 (1968).
    [Crossref]
  17. See Figs. 5 and 6 of Ref. 8.
  18. The uncertainties of the parameters in the Tables are the one-parameter uncertainties defined in the Appendix to paper II3. These uncertainties indicate the limits on a single parameter (evaluated in this paper at a confidence level of 95%) if all other parameters are assumed to be correctly estimated.
  19. The standard errors of n and k (shown in the captions of Figs. 2, 4, 6, and 8) were computed in each case from the standard errors of the parameters of the fits to the reflectance data, the elements of the parameter correlation matrix, and the partial derivatives of n and k with respect to the parameters evaluated at representative energies3.
  20. C. J. Powell, Proc. Phys. Soc. (London) 76, 593 (1960).
    [Crossref]
  21. R. E. LaVilla and H. Mendlowitz, J. Appl. Phys. 40, 3297 (1969).
    [Crossref]
  22. B. Feuerbacher, M. Skibowski, R. P. Godwin, and T. Sasaki, J. Opt. Soc. Am. 58, 1434 (1968).
    [Crossref]
  23. J. Toots, private communication.
  24. J. A. Bearden and A. F. Burr, Rev. Mod. Phys. 39, 125 (1967).
    [Crossref]
  25. W. R. Hunter, D. W. Angel, and R. Tousey, Appl. Opt. 4, 891 (1965); C. J. Powell, Phys. Rev. 175, 972 (1968).
    [Crossref]
  26. C. J. Powell, Phys. Rev. Letters 15, 852 (1965).
    [Crossref]
  27. M. Cardona and D. L. Greenaway, Phys. Rev. 133, A1685 (1964).
    [Crossref]
  28. M. D. Hersey, J. Res. Natl. Bur. Std. (U. S.) 69B, 139 (1965); J. E. Monahan and A. Langsdorf, Ann. Phys. (N. Y.) 34, 238 (1965).
    [Crossref]

1970 (1)

C. J. Powell, J. Opt. Soc. Am. 60, 78 (1970).
[Crossref]

1969 (3)

C. J. Powell, J. Opt. Soc. Am. 59, 738 (1969).
[Crossref]

J. Toots and L. Marton, J. Opt. Soc. Am. 59, 1305 (1969).
[Crossref]

R. E. LaVilla and H. Mendlowitz, J. Appl. Phys. 40, 3297 (1969).
[Crossref]

1968 (4)

B. Feuerbacher, M. Skibowski, R. P. Godwin, and T. Sasaki, J. Opt. Soc. Am. 58, 1434 (1968).
[Crossref]

K. L. Kliewer and R. Fuchs, Phys. Rev. 172, 607 (1968).
[Crossref]

H. W. Verleur, J. Opt. Soc. Am. 58, 1356 (1968).
[Crossref]

J. Toots, H. A. Fowler, and L. Marton, Phys. Rev. 172, 670 (1968).
[Crossref]

1967 (2)

L. Marton and J. Toots, Phys. Rev. 160, 602 (1967).
[Crossref]

J. A. Bearden and A. F. Burr, Rev. Mod. Phys. 39, 125 (1967).
[Crossref]

1965 (6)

W. R. Hunter, D. W. Angel, and R. Tousey, Appl. Opt. 4, 891 (1965); C. J. Powell, Phys. Rev. 175, 972 (1968).
[Crossref]

C. J. Powell, Phys. Rev. Letters 15, 852 (1965).
[Crossref]

K. Rabinovitch, L. R. Canfield, and R. P. Madden, Appl. Opt. 4, 1005 (1965); D. C. Hinson, J. Opt. Soc. Am. 56, 408 (1966); R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, Phys. Rev. Letters 21, 1530 (1968).
[Crossref]

R. H. Ritchie, Surface Sci. 3, 497 (1965); P. Dobberstein, A. Hampe, and G. Sauerbrey, Phys. Letters 27A, 256 (1968); J. L. Stanford, H. E. Bennett, J. M. Bennett, E. J. Ashley, and E. T. Arakawa, Bull. Am. Phys. Soc. 13, 989 (1968); S. E. Schnatterly, 13, 989 (1968).
[Crossref]

W. R. Hunter, J. Opt. Soc. Am. 55, 1197 (1965); Appl. Opt. 6, 2140 (1967).
[Crossref]

M. D. Hersey, J. Res. Natl. Bur. Std. (U. S.) 69B, 139 (1965); J. E. Monahan and A. Langsdorf, Ann. Phys. (N. Y.) 34, 238 (1965).
[Crossref]

1964 (1)

M. Cardona and D. L. Greenaway, Phys. Rev. 133, A1685 (1964).
[Crossref]

1963 (1)

D . W. Marquardt, J. Soc. Indust. Appl. Math. 11, 431 (1963); T. Baumeister and D. W. Marquardt, Least-Squares Estimation of Nonlinear Parameters, Share General Purpose Library, Distribution No. 1428, December 1962. A revised version of the program is available with the same title as Share Library Distribution No. 309401, August 1966.
[Crossref]

1960 (1)

C. J. Powell, Proc. Phys. Soc. (London) 76, 593 (1960).
[Crossref]

1958 (2)

J. G. Collins, Appl. Sci. Res. Sec. B, 7, 1 (1958).
[Crossref]

M. H. Cohen, Phil. Mag. 3, 762 (1958).
[Crossref]

1939 (1)

R . Tousey, J. Opt. Soc. Am. 29, 235 (1939).
[Crossref]

Angel, D. W.

W. R. Hunter, D. W. Angel, and R. Tousey, Appl. Opt. 4, 891 (1965); C. J. Powell, Phys. Rev. 175, 972 (1968).
[Crossref]

Bearden, J. A.

J. A. Bearden and A. F. Burr, Rev. Mod. Phys. 39, 125 (1967).
[Crossref]

Burr, A. F.

J. A. Bearden and A. F. Burr, Rev. Mod. Phys. 39, 125 (1967).
[Crossref]

Canfield, L. R.

K. Rabinovitch, L. R. Canfield, and R. P. Madden, Appl. Opt. 4, 1005 (1965); D. C. Hinson, J. Opt. Soc. Am. 56, 408 (1966); R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, Phys. Rev. Letters 21, 1530 (1968).
[Crossref]

Cardona, M.

M. Cardona and D. L. Greenaway, Phys. Rev. 133, A1685 (1964).
[Crossref]

Cohen, M. H.

M. H. Cohen, Phil. Mag. 3, 762 (1958).
[Crossref]

Collins, J. G.

J. G. Collins, Appl. Sci. Res. Sec. B, 7, 1 (1958).
[Crossref]

Feuerbacher, B.

B. Feuerbacher, M. Skibowski, R. P. Godwin, and T. Sasaki, J. Opt. Soc. Am. 58, 1434 (1968).
[Crossref]

Fowler, H. A.

J. Toots, H. A. Fowler, and L. Marton, Phys. Rev. 172, 670 (1968).
[Crossref]

Fuchs, R.

K. L. Kliewer and R. Fuchs, Phys. Rev. 172, 607 (1968).
[Crossref]

Godwin, R. P.

B. Feuerbacher, M. Skibowski, R. P. Godwin, and T. Sasaki, J. Opt. Soc. Am. 58, 1434 (1968).
[Crossref]

Greenaway, D. L.

M. Cardona and D. L. Greenaway, Phys. Rev. 133, A1685 (1964).
[Crossref]

Hersey, M. D.

M. D. Hersey, J. Res. Natl. Bur. Std. (U. S.) 69B, 139 (1965); J. E. Monahan and A. Langsdorf, Ann. Phys. (N. Y.) 34, 238 (1965).
[Crossref]

Hunter, W. R.

W. R. Hunter, D. W. Angel, and R. Tousey, Appl. Opt. 4, 891 (1965); C. J. Powell, Phys. Rev. 175, 972 (1968).
[Crossref]

W. R. Hunter, J. Opt. Soc. Am. 55, 1197 (1965); Appl. Opt. 6, 2140 (1967).
[Crossref]

Kliewer, K. L.

K. L. Kliewer and R. Fuchs, Phys. Rev. 172, 607 (1968).
[Crossref]

LaVilla, R. E.

R. E. LaVilla and H. Mendlowitz, J. Appl. Phys. 40, 3297 (1969).
[Crossref]

Madden, R. P.

K. Rabinovitch, L. R. Canfield, and R. P. Madden, Appl. Opt. 4, 1005 (1965); D. C. Hinson, J. Opt. Soc. Am. 56, 408 (1966); R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, Phys. Rev. Letters 21, 1530 (1968).
[Crossref]

See, for example, R. P. Madden, in Physics of Thin Films, Vol. 1, G. Hass, Ed. (Academic Press Inc., New York, 1963), p. 123; O. S. Heavens, in Physics of Thin Films, Vol. 2, G. Hass and R. E. Thun, Eds. (Academic Press Inc., New York, 1964), p. 193.

Marquardt, D . W.

D . W. Marquardt, J. Soc. Indust. Appl. Math. 11, 431 (1963); T. Baumeister and D. W. Marquardt, Least-Squares Estimation of Nonlinear Parameters, Share General Purpose Library, Distribution No. 1428, December 1962. A revised version of the program is available with the same title as Share Library Distribution No. 309401, August 1966.
[Crossref]

Marton, L.

J. Toots and L. Marton, J. Opt. Soc. Am. 59, 1305 (1969).
[Crossref]

J. Toots, H. A. Fowler, and L. Marton, Phys. Rev. 172, 670 (1968).
[Crossref]

L. Marton and J. Toots, Phys. Rev. 160, 602 (1967).
[Crossref]

Mendlowitz, H.

R. E. LaVilla and H. Mendlowitz, J. Appl. Phys. 40, 3297 (1969).
[Crossref]

Powell, C. J.

C. J. Powell, J. Opt. Soc. Am. 60, 78 (1970).
[Crossref]

C. J. Powell, J. Opt. Soc. Am. 59, 738 (1969).
[Crossref]

C. J. Powell, Phys. Rev. Letters 15, 852 (1965).
[Crossref]

C. J. Powell, Proc. Phys. Soc. (London) 76, 593 (1960).
[Crossref]

Rabinovitch, K.

K. Rabinovitch, L. R. Canfield, and R. P. Madden, Appl. Opt. 4, 1005 (1965); D. C. Hinson, J. Opt. Soc. Am. 56, 408 (1966); R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, Phys. Rev. Letters 21, 1530 (1968).
[Crossref]

Ritchie, R. H.

R. H. Ritchie, Surface Sci. 3, 497 (1965); P. Dobberstein, A. Hampe, and G. Sauerbrey, Phys. Letters 27A, 256 (1968); J. L. Stanford, H. E. Bennett, J. M. Bennett, E. J. Ashley, and E. T. Arakawa, Bull. Am. Phys. Soc. 13, 989 (1968); S. E. Schnatterly, 13, 989 (1968).
[Crossref]

Sasaki, T.

B. Feuerbacher, M. Skibowski, R. P. Godwin, and T. Sasaki, J. Opt. Soc. Am. 58, 1434 (1968).
[Crossref]

Skibowski, M.

B. Feuerbacher, M. Skibowski, R. P. Godwin, and T. Sasaki, J. Opt. Soc. Am. 58, 1434 (1968).
[Crossref]

Toots, J.

J. Toots and L. Marton, J. Opt. Soc. Am. 59, 1305 (1969).
[Crossref]

J. Toots, H. A. Fowler, and L. Marton, Phys. Rev. 172, 670 (1968).
[Crossref]

L. Marton and J. Toots, Phys. Rev. 160, 602 (1967).
[Crossref]

J. Toots, private communication.

Tousey, R .

R . Tousey, J. Opt. Soc. Am. 29, 235 (1939).
[Crossref]

Tousey, R.

W. R. Hunter, D. W. Angel, and R. Tousey, Appl. Opt. 4, 891 (1965); C. J. Powell, Phys. Rev. 175, 972 (1968).
[Crossref]

Verleur, H. W.

H. W. Verleur, J. Opt. Soc. Am. 58, 1356 (1968).
[Crossref]

Other (28)

See, for example, R. P. Madden, in Physics of Thin Films, Vol. 1, G. Hass, Ed. (Academic Press Inc., New York, 1963), p. 123; O. S. Heavens, in Physics of Thin Films, Vol. 2, G. Hass and R. E. Thun, Eds. (Academic Press Inc., New York, 1964), p. 193.

H. W. Verleur, J. Opt. Soc. Am. 58, 1356 (1968).
[Crossref]

C. J. Powell, J. Opt. Soc. Am. 60, 78 (1970).
[Crossref]

C. J. Powell, J. Opt. Soc. Am. 59, 738 (1969).
[Crossref]

W. R. Hunter, J. Opt. Soc. Am. 55, 1197 (1965); Appl. Opt. 6, 2140 (1967).
[Crossref]

L. Marton and J. Toots, Phys. Rev. 160, 602 (1967).
[Crossref]

J. Toots, H. A. Fowler, and L. Marton, Phys. Rev. 172, 670 (1968).
[Crossref]

J. Toots and L. Marton, J. Opt. Soc. Am. 59, 1305 (1969).
[Crossref]

M. H. Cohen, Phil. Mag. 3, 762 (1958).
[Crossref]

R . Tousey, J. Opt. Soc. Am. 29, 235 (1939).
[Crossref]

D . W. Marquardt, J. Soc. Indust. Appl. Math. 11, 431 (1963); T. Baumeister and D. W. Marquardt, Least-Squares Estimation of Nonlinear Parameters, Share General Purpose Library, Distribution No. 1428, December 1962. A revised version of the program is available with the same title as Share Library Distribution No. 309401, August 1966.
[Crossref]

The program used here11 supplies several estimates of the uncertainty of each parameter as briefly described in the Appendix to paper II3.

J. G. Collins, Appl. Sci. Res. Sec. B, 7, 1 (1958).
[Crossref]

K. L. Kliewer and R. Fuchs, Phys. Rev. 172, 607 (1968).
[Crossref]

K. Rabinovitch, L. R. Canfield, and R. P. Madden, Appl. Opt. 4, 1005 (1965); D. C. Hinson, J. Opt. Soc. Am. 56, 408 (1966); R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, Phys. Rev. Letters 21, 1530 (1968).
[Crossref]

R. H. Ritchie, Surface Sci. 3, 497 (1965); P. Dobberstein, A. Hampe, and G. Sauerbrey, Phys. Letters 27A, 256 (1968); J. L. Stanford, H. E. Bennett, J. M. Bennett, E. J. Ashley, and E. T. Arakawa, Bull. Am. Phys. Soc. 13, 989 (1968); S. E. Schnatterly, 13, 989 (1968).
[Crossref]

See Figs. 5 and 6 of Ref. 8.

The uncertainties of the parameters in the Tables are the one-parameter uncertainties defined in the Appendix to paper II3. These uncertainties indicate the limits on a single parameter (evaluated in this paper at a confidence level of 95%) if all other parameters are assumed to be correctly estimated.

The standard errors of n and k (shown in the captions of Figs. 2, 4, 6, and 8) were computed in each case from the standard errors of the parameters of the fits to the reflectance data, the elements of the parameter correlation matrix, and the partial derivatives of n and k with respect to the parameters evaluated at representative energies3.

C. J. Powell, Proc. Phys. Soc. (London) 76, 593 (1960).
[Crossref]

R. E. LaVilla and H. Mendlowitz, J. Appl. Phys. 40, 3297 (1969).
[Crossref]

B. Feuerbacher, M. Skibowski, R. P. Godwin, and T. Sasaki, J. Opt. Soc. Am. 58, 1434 (1968).
[Crossref]

J. Toots, private communication.

J. A. Bearden and A. F. Burr, Rev. Mod. Phys. 39, 125 (1967).
[Crossref]

W. R. Hunter, D. W. Angel, and R. Tousey, Appl. Opt. 4, 891 (1965); C. J. Powell, Phys. Rev. 175, 972 (1968).
[Crossref]

C. J. Powell, Phys. Rev. Letters 15, 852 (1965).
[Crossref]

M. Cardona and D. L. Greenaway, Phys. Rev. 133, A1685 (1964).
[Crossref]

M. D. Hersey, J. Res. Natl. Bur. Std. (U. S.) 69B, 139 (1965); J. E. Monahan and A. Langsdorf, Ann. Phys. (N. Y.) 34, 238 (1965).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1

The solid lines are a fit using a Drude model for (ω) to the Toots et al.7 reflectance data for Be (solid circles) as a function of photon energy for the angles of incidence indicated. The dashed lines are a similar fit to the data for angles of incidence of 20° and 80° only. The parameters describing each fit are shown in Table I.

Fig. 2
Fig. 2

Plots of n and k for Be as a function of photon energy, based on the parameters (Table I) derived from the fits to the reflectance data shown in Fig. 1. The solid lines are based on the fit using all of the data shown in Fig. 1, whereas the dashed lines represent the values derived from the data for angles of incidence of 20° and 80°. For each fit, the computed standard error of n was between 0.0021 and 0.0056, and the computed standard error of k was between 0.005 and 0.008 for energies between 10 and 19 eV and decreased to 0.0013 at 26 eV.19 The solid and open circles are the n and k values derived by Toots et al.7 from their reflectance data at 20° and 80°.

Fig. 3
Fig. 3

The solid lines are a fit using a Drude model for (ω) to the Marton and Toots6 reflectance data for Ge (solid and open circles) as a function of photon energy for the angles of incidence indicated. The dashed lines represent a fit to the same data with one interband transition in the expression for (b)(ω) [Eq. (3)]. The parameters describing each fit are listed in Table II.

Fig. 4
Fig. 4

Plots of n and k for Ge as a function of photon energy, based on the parameters (Table II) derived from the fits to the reflectance data in Fig. 3. The solid lines are based on the two-parameter fit [Drude model for (ω)], whereas the dashed lines are based on the five-parameter fit [one interband transition contributing to (ω)]. In the former fit, the computed standard error of n decreased from 0.0075 to 0.0023 for photon energies between 8 and 25 eV, whereas the computed standard error of k decreased from 0.009 to 0.001 over the same range; in the five-parameter fit, the standard errors of n and k were approximately twice those for the same quantities derived from the two-parameter fit.19 The solid and open circles are the n and k values derived by Marton and Toots.6

Fig. 5
Fig. 5

The solid lines are a fit to the Toots and Marton8 reflectance data for Sb (solid and open circles) as a function of photon energy using a model (ω) with one interband transition. The parameters that describe the fit are listed in Table III.

Fig. 6
Fig. 6

Plots of n and k (solid lines) for Sb as a function of photon energy, based on the parameters shown in Table III for the fit in Fig. 5. The computed standard error of n was between 0.003 and 0.005, whereas the computed standard error of k decreased from 0.011 at 10.4 eV to 0.0016 at 23 eV and then increased to 0.0032 at 26 eV.19 The solid and open circles are the n and k values derived from locally smoothed values of reflectance by Toots and Marton.8

Fig. 7
Fig. 7

The solid lines are a fit to the Toots and Marton8 reflectance data for Bi (solid and open circles) as a function of photon energy using a model (ω) with three interband transitions. The parameters that describe the fit are listed in Table IV.

Fig. 8
Fig. 8

Plots of n and k (solid lines) for Bi as a function of photon energy, based on the parameters in Table IV (from the fit of Fig. 7). The computed standard error of n decreased from 0.015 at 11.6 eV to 0.003 at 23 eV and then increased to 0.07 at 26 eV. The computed standard error of k decreased from 0.03 at 11.6 eV to 0.004 at 23 eV and then increased to 0.09 at 26 eV.19 Because of the systematic deviations in Fig. 7 (which are comparable with the rms deviation), the limits on the computed n and k at a specified confidence level should be increased, possibly by a factor of 2. The solid and open circles are the n and k values derived from locally smoothed reflectance data by Toots and Marton.8

Tables (4)

Tables Icon

Table I Parameters for the fits to the Be data shown in Fig. 1. The plasma energy ħωp = 18.4 eV.

Tables Icon

Table II Parameters for the fits to the Ge data in Fig. 3. The plasma energy ħωp = 15.8 eV.

Tables Icon

Table III Parameters for the fit to the Sb data in Fig. 5. The plasma energy ħωp = 15.0 eV.

Tables Icon

Table IV Parameters for the fit to the Bi data shown in Fig. 7. The plasma energy ħωp = 13.94 eV.

Equations (3)

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

( ω ) = ( f ) ( ω ) + ( b ) ( ω ) ,
( f ) ( ω ) = 1 - ( f 0 ω p 2 ) / ( ω 2 + i g 0 ω ) ,
( b ) ( ω ) = - j = 1 r f j ω p 2 ω 2 - ω j 2 + i g j ω .