Abstract

The errors introduced in Δ2 spectra by Kramers-Krönig analysis of modulated reflectivity data are investigated using an analytical model. It is found that the energy position of singularities is always reproduced with good accuracy even if the experimental spectrum of ΔR/R is cut barely above the last structure of interest. This procedure is instead completely insufficient when a quantitative line shape analysis is required. In such cases data up to very high energy are required for a meaningful analysis. Errors due to other sources, like baseline shifts or inaccurate static optical constants, are also investigated.

© 1975 Optical Society of America

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References

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  1. M. Cardona, in Solid State Physics,F. Seitz, D. Turnbull, H. Ehrenreich, Eds. (Academic, New York, 1972).
  2. R. K. Willardson, Albert C. Beer, Eds., Semiconductors and Semimetals (Academic, New York, 1972), Vol. 9.
    [Crossref]
  3. Proceedings of the First International Conference on Modulation Spectroscopy, Surf. Sci. 37, (1973).
  4. M. Garfinkel, J. J. Tiemann, W. E. Engeler, Phys. Rev. 148, 695 (1966).
    [Crossref]
  5. B. O. Seraphin, in Optical Properties of Solids,F. Abelès, Ed. (North Holland, Amsterdam, 1972).
  6. C. G. Olson, M. Piacentini, D. W. Lynch, Phys. Rev. Lett. 37, 644 (1974).
    [Crossref]
  7. J. H. Weaver, C. G. Olson, M. Piacentini, D. W. Lynch, Solid State Commun. (to be published).
  8. D. E. Aspnes, C. G. Olson, Phys. Rev. Lett. 33, 1605 (1974).
    [Crossref]
  9. W. J. Scouler, Phys. Rev. Lett. 18, 455 (1967).
    [Crossref]
  10. D. S. Szczepanek, R. Glosser, Univ. of Maryland Technical Rept. 73–123 (1973).
  11. J. H. Weaver, C. H. Culp, D. W. Lynch, R. Rosei (to be published).
  12. P. O. Nilsson, L. Munkby, Phys. Kondens. Mater. 10, 290 (1969).
  13. B. O. Seraphin, N. Bottka, Phys. Rev. 139A, 560 (1965).
    [Crossref]
  14. F. Stern, Solid State Physics,F. Seitz, D. Turnbull, H. Ehrenreich, Eds. (Academic, New York, 1963), Vol. 15.
    [Crossref]
  15. A. Balzarotti, M. Grandolfo, Phys. Rev. Lett. 20, 9 (1968).
    [Crossref]
  16. B. Batz, in Semiconductors and Semimetals, R. K. Willardson, A. C. Beer Eds. (Academic, New York, 1972), Vol. 9.
    [Crossref]
  17. R. Rosei, D. W. Lynch, Phys. Rev. B5, 3883 (1972).
  18. H. W. Verleur, J.Opt Soc. Am. 58, 1356 (1968).
    [Crossref]
  19. R. Rosei, E. Colavita (unpublished).
  20. Another important mechanism is due to the smearing of the Fermi distribution that cannot be simulated in our model.
  21. The K–K program was previously checked using the mathematical model itself. It was found that, by feeding the ΔR/R spectrum up to very high energy (∼100 eV), the Δ∊2 spectrum was reproduced with negligible errors.

1974 (2)

C. G. Olson, M. Piacentini, D. W. Lynch, Phys. Rev. Lett. 37, 644 (1974).
[Crossref]

D. E. Aspnes, C. G. Olson, Phys. Rev. Lett. 33, 1605 (1974).
[Crossref]

1973 (1)

Proceedings of the First International Conference on Modulation Spectroscopy, Surf. Sci. 37, (1973).

1972 (1)

R. Rosei, D. W. Lynch, Phys. Rev. B5, 3883 (1972).

1969 (1)

P. O. Nilsson, L. Munkby, Phys. Kondens. Mater. 10, 290 (1969).

1968 (2)

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

A. Balzarotti, M. Grandolfo, Phys. Rev. Lett. 20, 9 (1968).
[Crossref]

1967 (1)

W. J. Scouler, Phys. Rev. Lett. 18, 455 (1967).
[Crossref]

1966 (1)

M. Garfinkel, J. J. Tiemann, W. E. Engeler, Phys. Rev. 148, 695 (1966).
[Crossref]

1965 (1)

B. O. Seraphin, N. Bottka, Phys. Rev. 139A, 560 (1965).
[Crossref]

Aspnes, D. E.

D. E. Aspnes, C. G. Olson, Phys. Rev. Lett. 33, 1605 (1974).
[Crossref]

Balzarotti, A.

A. Balzarotti, M. Grandolfo, Phys. Rev. Lett. 20, 9 (1968).
[Crossref]

Batz, B.

B. Batz, in Semiconductors and Semimetals, R. K. Willardson, A. C. Beer Eds. (Academic, New York, 1972), Vol. 9.
[Crossref]

Bottka, N.

B. O. Seraphin, N. Bottka, Phys. Rev. 139A, 560 (1965).
[Crossref]

Cardona, M.

M. Cardona, in Solid State Physics,F. Seitz, D. Turnbull, H. Ehrenreich, Eds. (Academic, New York, 1972).

Colavita, E.

R. Rosei, E. Colavita (unpublished).

Culp, C. H.

J. H. Weaver, C. H. Culp, D. W. Lynch, R. Rosei (to be published).

Engeler, W. E.

M. Garfinkel, J. J. Tiemann, W. E. Engeler, Phys. Rev. 148, 695 (1966).
[Crossref]

Garfinkel, M.

M. Garfinkel, J. J. Tiemann, W. E. Engeler, Phys. Rev. 148, 695 (1966).
[Crossref]

Glosser, R.

D. S. Szczepanek, R. Glosser, Univ. of Maryland Technical Rept. 73–123 (1973).

Grandolfo, M.

A. Balzarotti, M. Grandolfo, Phys. Rev. Lett. 20, 9 (1968).
[Crossref]

Lynch, D. W.

C. G. Olson, M. Piacentini, D. W. Lynch, Phys. Rev. Lett. 37, 644 (1974).
[Crossref]

R. Rosei, D. W. Lynch, Phys. Rev. B5, 3883 (1972).

J. H. Weaver, C. H. Culp, D. W. Lynch, R. Rosei (to be published).

J. H. Weaver, C. G. Olson, M. Piacentini, D. W. Lynch, Solid State Commun. (to be published).

Munkby, L.

P. O. Nilsson, L. Munkby, Phys. Kondens. Mater. 10, 290 (1969).

Nilsson, P. O.

P. O. Nilsson, L. Munkby, Phys. Kondens. Mater. 10, 290 (1969).

Olson, C. G.

C. G. Olson, M. Piacentini, D. W. Lynch, Phys. Rev. Lett. 37, 644 (1974).
[Crossref]

D. E. Aspnes, C. G. Olson, Phys. Rev. Lett. 33, 1605 (1974).
[Crossref]

J. H. Weaver, C. G. Olson, M. Piacentini, D. W. Lynch, Solid State Commun. (to be published).

Piacentini, M.

C. G. Olson, M. Piacentini, D. W. Lynch, Phys. Rev. Lett. 37, 644 (1974).
[Crossref]

J. H. Weaver, C. G. Olson, M. Piacentini, D. W. Lynch, Solid State Commun. (to be published).

Rosei, R.

R. Rosei, D. W. Lynch, Phys. Rev. B5, 3883 (1972).

R. Rosei, E. Colavita (unpublished).

J. H. Weaver, C. H. Culp, D. W. Lynch, R. Rosei (to be published).

Scouler, W. J.

W. J. Scouler, Phys. Rev. Lett. 18, 455 (1967).
[Crossref]

Seraphin, B. O.

B. O. Seraphin, N. Bottka, Phys. Rev. 139A, 560 (1965).
[Crossref]

B. O. Seraphin, in Optical Properties of Solids,F. Abelès, Ed. (North Holland, Amsterdam, 1972).

Stern, F.

F. Stern, Solid State Physics,F. Seitz, D. Turnbull, H. Ehrenreich, Eds. (Academic, New York, 1963), Vol. 15.
[Crossref]

Szczepanek, D. S.

D. S. Szczepanek, R. Glosser, Univ. of Maryland Technical Rept. 73–123 (1973).

Tiemann, J. J.

M. Garfinkel, J. J. Tiemann, W. E. Engeler, Phys. Rev. 148, 695 (1966).
[Crossref]

Verleur, H. W.

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

Weaver, J. H.

J. H. Weaver, C. G. Olson, M. Piacentini, D. W. Lynch, Solid State Commun. (to be published).

J. H. Weaver, C. H. Culp, D. W. Lynch, R. Rosei (to be published).

J.Opt Soc. Am. (1)

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

Phys. Kondens. Mater. (1)

P. O. Nilsson, L. Munkby, Phys. Kondens. Mater. 10, 290 (1969).

Phys. Rev. (3)

B. O. Seraphin, N. Bottka, Phys. Rev. 139A, 560 (1965).
[Crossref]

R. Rosei, D. W. Lynch, Phys. Rev. B5, 3883 (1972).

M. Garfinkel, J. J. Tiemann, W. E. Engeler, Phys. Rev. 148, 695 (1966).
[Crossref]

Phys. Rev. Lett. (4)

D. E. Aspnes, C. G. Olson, Phys. Rev. Lett. 33, 1605 (1974).
[Crossref]

W. J. Scouler, Phys. Rev. Lett. 18, 455 (1967).
[Crossref]

A. Balzarotti, M. Grandolfo, Phys. Rev. Lett. 20, 9 (1968).
[Crossref]

C. G. Olson, M. Piacentini, D. W. Lynch, Phys. Rev. Lett. 37, 644 (1974).
[Crossref]

Proceedings of the First International Conference on Modulation Spectroscopy (1)

Proceedings of the First International Conference on Modulation Spectroscopy, Surf. Sci. 37, (1973).

Other (11)

B. Batz, in Semiconductors and Semimetals, R. K. Willardson, A. C. Beer Eds. (Academic, New York, 1972), Vol. 9.
[Crossref]

J. H. Weaver, C. G. Olson, M. Piacentini, D. W. Lynch, Solid State Commun. (to be published).

F. Stern, Solid State Physics,F. Seitz, D. Turnbull, H. Ehrenreich, Eds. (Academic, New York, 1963), Vol. 15.
[Crossref]

D. S. Szczepanek, R. Glosser, Univ. of Maryland Technical Rept. 73–123 (1973).

J. H. Weaver, C. H. Culp, D. W. Lynch, R. Rosei (to be published).

B. O. Seraphin, in Optical Properties of Solids,F. Abelès, Ed. (North Holland, Amsterdam, 1972).

M. Cardona, in Solid State Physics,F. Seitz, D. Turnbull, H. Ehrenreich, Eds. (Academic, New York, 1972).

R. K. Willardson, Albert C. Beer, Eds., Semiconductors and Semimetals (Academic, New York, 1972), Vol. 9.
[Crossref]

R. Rosei, E. Colavita (unpublished).

Another important mechanism is due to the smearing of the Fermi distribution that cannot be simulated in our model.

The K–K program was previously checked using the mathematical model itself. It was found that, by feeding the ΔR/R spectrum up to very high energy (∼100 eV), the Δ∊2 spectrum was reproduced with negligible errors.

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Figures (8)

Fig. 1
Fig. 1

The real and imaginary parts of the dielectric constant calculated from model I.

Fig. 2
Fig. 2

The modulated reflectance ΔR/R calculated from model I. The arrows show the energies where the spectrum has been interrupted and the broken lines the extrapolations used in each case.

Fig. 3
Fig. 3

Imaginary part of the modulated dielectric constant: (a) —— true Δ2 generated from model I; (b) ----- Δ2 spectrum obtained from the K-K analysis truncating the model at 30 eV; (c) ····· Δ2 spectrum obtained from the K-K analysis truncating the model at 7.5 eV; (d) -·-·- Δ2 spectrum obtained from the K-K analysis truncating the model at 5.9 eV; (e) -·-·· Δ2 spectrum obtained from the K-K analysis truncating the model at 4.3 eV.

Fig. 4
Fig. 4

Imaginary part of the modulated dielectric constant: ——true Δ2 generated from model I;------ Δ2 spectrum obtained from the K-K analysis adding a baseline shift to the modulated reflectivity ΔR/R.

Fig. 5
Fig. 5

Imaginary part of the modulated dielectric constant: —— true Δ2 generated from model I;- - - - - - Δ2 spectrum obtained from the K-K analysis increasing the index of refraction n(ℏω) by a factor 1.25.

Fig. 6
Fig. 6

Imaginary part of the modulated dielectric constant: —— true Δ2 generated from model I;----- Δ2 spectrum obtained from the K-K analysis increasing the index of extinction k(ℏω) by a factor 1.25.

Fig. 7
Fig. 7

Imaginary part of the dielectric constant 2 (model II): —— contribution of the test oscillator;----- contribution of the ghost oscillator in its successive energy positions (see text).

Fig. 8
Fig. 8

Imaginary part of the modulated dielectric constant Δ2 (model II): (a′) ——generated from the model; (b′) ----- obtained from the K-K analysis when the ghost oscillator is placed at 12 eV; (c′) obtained from the K-K analysis when the ghost oscillator is placed at 9 eV; (d′) --- obtained from the K-K analysis when the ghost oscillator is placed at 6 eV; (e′) ----- obtained from the K-K analysis when the ghost oscillator is placed at 4.5 eV.

Tables (2)

Tables Icon

Table I Best Fit Parameters of the Two Test Oscillators Used in Model IIa

Tables Icon

Table II Parameters of the Lorentzian Oscillators Used to Reproduce the Optical Spectra in Analytical Model I

Equations (7)

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θ ( ω 0 ) = ω 0 π P 0 ln R ( ω ) d ω ω 2 ω 0 2 ,
Δ θ = ω 0 π P 0 Δ R R d ω ω 2 ω 0 2 .
Δ 1 = 1 2 [ n ( 1 1 ) k 2 ] Δ R R + [ k ( 1 1 ) + n 2 ] Δ θ , Δ 2 = 1 2 [ k ( 1 1 ) + n 2 ] Δ R R [ n ( 1 1 ) k 2 ] Δ θ .
1 = 0 β 1 Γ 1 ω 2 + Γ 1 2 + i = 2 N β i Γ i ( ω 0 i 2 ω 2 ) ( ω 0 i 2 ω 2 ) 2 + ω 2 Γ i 2 ,
2 = β 1 Γ 1 2 ω ( ω 2 + Γ 1 2 ) + i = 2 N β i Γ i 2 ω ( ω 0 i 2 ω 2 ) 2 + ω 2 Γ i 2 .
Δ 1 i = 1 i ( β i + Δ β i , Γ i + Δ Γ i , ω 0 i + Δ ω 0 i ) + 1 i ( β i , Γ i , ω 0 i ) ,
Δ 2 i = 2 i ( β i + Δ β i , Γ i + Δ Γ i , ω 0 i + Δ ω 0 i ) + 2 i ( β i , Γ i , ω 0 i ) .

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