Abstract

We describe a method for accurately determining critical point parameters from optical spectra in which digital filtering in real (energy) and reciprocal (Fourier-coefficient) space is treated on an equivalent basis. Experimental and theoretical line shapes are also filtered in parallel, thereby eliminating systematic errors that can arise in the standard approach in which only the data are processed. Real-space filtering is done using false data to isolate individual or groups of critical points in complicated spectra, to provide a more accurate representation of the data in reciprocal space, and to minimize the effects of end-point discontinuities and truncation errors on the Fourier coefficients calculated from these spectra. Reciprocal-space filtering is done by numerically differentiating the data to maximize the amplitudes of the Fourier coefficients carrying the critical point information, followed by truncating low- and high-order coefficients to minimize artifacts that are due to baseline effects and noise. The optimum order of differentiation (not necessarily integral) is determined from the coefficients themselves. We show that a least-squares regression (LSR) analysis of a restricted interval of equally weighted points in reciprocal space is equivalent to the LSR analysis of all data points equally weighted in real space, making LSR particularly useful for analyzing higher-derivative spectra, where the real-space line shapes rapidly approach zero outside the central structure. For a specific example discussed here, maximum accuracy is obtained if the data are analyzed in the form of a third derivative, as was previously concluded empirically from numerical processing in real space.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Ehrenreich and M. H. Cohen, Phys. Rev. 115, 786 (1959).
    [Crossref]
  2. H. R. Philipp and H. Ehrenreich, in Semiconductors and Semimetals 3: Optical Properties of III–V Compounds, R. K. Willardson and A. C. Beer, eds. (Academic, New York, 1967), p. 93.
    [Crossref]
  3. L. van Hove, Phys. Rev. 89, 1189 (1953).
    [Crossref]
  4. J. C. Phillips, Rev. Rev. 104, 1263 (1956).
    [Crossref]
  5. M. Cardona, Modulation Spectroscopy (Academic, New York, 1967).
  6. A. Savitzky and M. J. E. Golay, Anal. Chem. 36, 1627 (1964).
    [Crossref]
  7. J. Steinier, Y. Termonia, and J. Deltour, Anal. Chem. 44, 1906 (1972).
    [Crossref] [PubMed]
  8. R. J. Wonnacott and T. H. Wonnacott, Econometrics (Wiley, New York, 1970), Chap. 6.
  9. R. P. Vasquez, J. D. Klein, J. J. Barton, and F. J. Grunthaner, J. Electron. Spectrosc. 23, 63 (1981).
    [Crossref]
  10. D. E. Aspnes, Surf. Sci. (to be published).
  11. D. E. Aspnes and A. A. Studna, Phys. Rev. B 27, 985 (1983).
    [Crossref]
  12. J. R. Chelikowsky and M. L. Cohen, Phys. Rev. B 10, 5095 (1974).
    [Crossref]
  13. D. E. Aspnes and J. E. Rowe, Phys. Rev. Lett. 27, 188 (1971).
    [Crossref]
  14. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).
  15. D. E. Aspnes, Phys. Rev. Lett. 28, 168 (1972).
    [Crossref]
  16. S. M. Kelso, D. E. Aspnes, M. A. Pollack, and R. E. Nahory, Phys. Rev. B 26, 6669 (1982).
    [Crossref]
  17. S. Haykin and S. Kessler, in Nonlinear Methods of Spectral Analysis, Vol. 34 of Topics in Applied Physics, S. Haykin, ed. (Springer–Verlag, Berlin, 1979), p. 9.
    [Crossref]
  18. J. K. Kauppinen, D. J. Moffatt, H. H. Mantsch, and D. G. Cameron, Appl. Spectrosc. 35, 271 (1981).
    [Crossref]
  19. A. G. Ferrige and J. C. Lindon, J. Mag. Reson. 31, 337 (1978).

1983 (1)

D. E. Aspnes and A. A. Studna, Phys. Rev. B 27, 985 (1983).
[Crossref]

1982 (1)

S. M. Kelso, D. E. Aspnes, M. A. Pollack, and R. E. Nahory, Phys. Rev. B 26, 6669 (1982).
[Crossref]

1981 (2)

J. K. Kauppinen, D. J. Moffatt, H. H. Mantsch, and D. G. Cameron, Appl. Spectrosc. 35, 271 (1981).
[Crossref]

R. P. Vasquez, J. D. Klein, J. J. Barton, and F. J. Grunthaner, J. Electron. Spectrosc. 23, 63 (1981).
[Crossref]

1978 (1)

A. G. Ferrige and J. C. Lindon, J. Mag. Reson. 31, 337 (1978).

1974 (1)

J. R. Chelikowsky and M. L. Cohen, Phys. Rev. B 10, 5095 (1974).
[Crossref]

1972 (2)

J. Steinier, Y. Termonia, and J. Deltour, Anal. Chem. 44, 1906 (1972).
[Crossref] [PubMed]

D. E. Aspnes, Phys. Rev. Lett. 28, 168 (1972).
[Crossref]

1971 (1)

D. E. Aspnes and J. E. Rowe, Phys. Rev. Lett. 27, 188 (1971).
[Crossref]

1964 (1)

A. Savitzky and M. J. E. Golay, Anal. Chem. 36, 1627 (1964).
[Crossref]

1959 (1)

H. Ehrenreich and M. H. Cohen, Phys. Rev. 115, 786 (1959).
[Crossref]

1956 (1)

J. C. Phillips, Rev. Rev. 104, 1263 (1956).
[Crossref]

1953 (1)

L. van Hove, Phys. Rev. 89, 1189 (1953).
[Crossref]

Aspnes, D. E.

D. E. Aspnes and A. A. Studna, Phys. Rev. B 27, 985 (1983).
[Crossref]

S. M. Kelso, D. E. Aspnes, M. A. Pollack, and R. E. Nahory, Phys. Rev. B 26, 6669 (1982).
[Crossref]

D. E. Aspnes, Phys. Rev. Lett. 28, 168 (1972).
[Crossref]

D. E. Aspnes and J. E. Rowe, Phys. Rev. Lett. 27, 188 (1971).
[Crossref]

D. E. Aspnes, Surf. Sci. (to be published).

Barton, J. J.

R. P. Vasquez, J. D. Klein, J. J. Barton, and F. J. Grunthaner, J. Electron. Spectrosc. 23, 63 (1981).
[Crossref]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

Cameron, D. G.

Cardona, M.

M. Cardona, Modulation Spectroscopy (Academic, New York, 1967).

Chelikowsky, J. R.

J. R. Chelikowsky and M. L. Cohen, Phys. Rev. B 10, 5095 (1974).
[Crossref]

Cohen, M. H.

H. Ehrenreich and M. H. Cohen, Phys. Rev. 115, 786 (1959).
[Crossref]

Cohen, M. L.

J. R. Chelikowsky and M. L. Cohen, Phys. Rev. B 10, 5095 (1974).
[Crossref]

Deltour, J.

J. Steinier, Y. Termonia, and J. Deltour, Anal. Chem. 44, 1906 (1972).
[Crossref] [PubMed]

Ehrenreich, H.

H. Ehrenreich and M. H. Cohen, Phys. Rev. 115, 786 (1959).
[Crossref]

H. R. Philipp and H. Ehrenreich, in Semiconductors and Semimetals 3: Optical Properties of III–V Compounds, R. K. Willardson and A. C. Beer, eds. (Academic, New York, 1967), p. 93.
[Crossref]

Ferrige, A. G.

A. G. Ferrige and J. C. Lindon, J. Mag. Reson. 31, 337 (1978).

Golay, M. J. E.

A. Savitzky and M. J. E. Golay, Anal. Chem. 36, 1627 (1964).
[Crossref]

Grunthaner, F. J.

R. P. Vasquez, J. D. Klein, J. J. Barton, and F. J. Grunthaner, J. Electron. Spectrosc. 23, 63 (1981).
[Crossref]

Haykin, S.

S. Haykin and S. Kessler, in Nonlinear Methods of Spectral Analysis, Vol. 34 of Topics in Applied Physics, S. Haykin, ed. (Springer–Verlag, Berlin, 1979), p. 9.
[Crossref]

Kauppinen, J. K.

Kelso, S. M.

S. M. Kelso, D. E. Aspnes, M. A. Pollack, and R. E. Nahory, Phys. Rev. B 26, 6669 (1982).
[Crossref]

Kessler, S.

S. Haykin and S. Kessler, in Nonlinear Methods of Spectral Analysis, Vol. 34 of Topics in Applied Physics, S. Haykin, ed. (Springer–Verlag, Berlin, 1979), p. 9.
[Crossref]

Klein, J. D.

R. P. Vasquez, J. D. Klein, J. J. Barton, and F. J. Grunthaner, J. Electron. Spectrosc. 23, 63 (1981).
[Crossref]

Lindon, J. C.

A. G. Ferrige and J. C. Lindon, J. Mag. Reson. 31, 337 (1978).

Mantsch, H. H.

Moffatt, D. J.

Nahory, R. E.

S. M. Kelso, D. E. Aspnes, M. A. Pollack, and R. E. Nahory, Phys. Rev. B 26, 6669 (1982).
[Crossref]

Philipp, H. R.

H. R. Philipp and H. Ehrenreich, in Semiconductors and Semimetals 3: Optical Properties of III–V Compounds, R. K. Willardson and A. C. Beer, eds. (Academic, New York, 1967), p. 93.
[Crossref]

Phillips, J. C.

J. C. Phillips, Rev. Rev. 104, 1263 (1956).
[Crossref]

Pollack, M. A.

S. M. Kelso, D. E. Aspnes, M. A. Pollack, and R. E. Nahory, Phys. Rev. B 26, 6669 (1982).
[Crossref]

Rowe, J. E.

D. E. Aspnes and J. E. Rowe, Phys. Rev. Lett. 27, 188 (1971).
[Crossref]

Savitzky, A.

A. Savitzky and M. J. E. Golay, Anal. Chem. 36, 1627 (1964).
[Crossref]

Steinier, J.

J. Steinier, Y. Termonia, and J. Deltour, Anal. Chem. 44, 1906 (1972).
[Crossref] [PubMed]

Studna, A. A.

D. E. Aspnes and A. A. Studna, Phys. Rev. B 27, 985 (1983).
[Crossref]

Termonia, Y.

J. Steinier, Y. Termonia, and J. Deltour, Anal. Chem. 44, 1906 (1972).
[Crossref] [PubMed]

van Hove, L.

L. van Hove, Phys. Rev. 89, 1189 (1953).
[Crossref]

Vasquez, R. P.

R. P. Vasquez, J. D. Klein, J. J. Barton, and F. J. Grunthaner, J. Electron. Spectrosc. 23, 63 (1981).
[Crossref]

Wonnacott, R. J.

R. J. Wonnacott and T. H. Wonnacott, Econometrics (Wiley, New York, 1970), Chap. 6.

Wonnacott, T. H.

R. J. Wonnacott and T. H. Wonnacott, Econometrics (Wiley, New York, 1970), Chap. 6.

Anal. Chem. (2)

A. Savitzky and M. J. E. Golay, Anal. Chem. 36, 1627 (1964).
[Crossref]

J. Steinier, Y. Termonia, and J. Deltour, Anal. Chem. 44, 1906 (1972).
[Crossref] [PubMed]

Appl. Spectrosc. (1)

J. Electron. Spectrosc. (1)

R. P. Vasquez, J. D. Klein, J. J. Barton, and F. J. Grunthaner, J. Electron. Spectrosc. 23, 63 (1981).
[Crossref]

J. Mag. Reson. (1)

A. G. Ferrige and J. C. Lindon, J. Mag. Reson. 31, 337 (1978).

Phys. Rev. (2)

H. Ehrenreich and M. H. Cohen, Phys. Rev. 115, 786 (1959).
[Crossref]

L. van Hove, Phys. Rev. 89, 1189 (1953).
[Crossref]

Phys. Rev. B (3)

S. M. Kelso, D. E. Aspnes, M. A. Pollack, and R. E. Nahory, Phys. Rev. B 26, 6669 (1982).
[Crossref]

D. E. Aspnes and A. A. Studna, Phys. Rev. B 27, 985 (1983).
[Crossref]

J. R. Chelikowsky and M. L. Cohen, Phys. Rev. B 10, 5095 (1974).
[Crossref]

Phys. Rev. Lett. (2)

D. E. Aspnes and J. E. Rowe, Phys. Rev. Lett. 27, 188 (1971).
[Crossref]

D. E. Aspnes, Phys. Rev. Lett. 28, 168 (1972).
[Crossref]

Rev. Rev. (1)

J. C. Phillips, Rev. Rev. 104, 1263 (1956).
[Crossref]

Other (6)

M. Cardona, Modulation Spectroscopy (Academic, New York, 1967).

H. R. Philipp and H. Ehrenreich, in Semiconductors and Semimetals 3: Optical Properties of III–V Compounds, R. K. Willardson and A. C. Beer, eds. (Academic, New York, 1967), p. 93.
[Crossref]

D. E. Aspnes, Surf. Sci. (to be published).

R. J. Wonnacott and T. H. Wonnacott, Econometrics (Wiley, New York, 1970), Chap. 6.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

S. Haykin and S. Kessler, in Nonlinear Methods of Spectral Analysis, Vol. 34 of Topics in Applied Physics, S. Haykin, ed. (Springer–Verlag, Berlin, 1979), p. 9.
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Top: representative energy dependence of the imaginary part of the dielectric function of a typical semiconductor in the visible–near-ultraviolet spectral range. Bottom: schematic diagram of the major critical point singularities giving rise to the structure in the optical spectrum. The individual structures are labeled according to standard conventions. Greater distances below the real axis indicate larger broadening, and larger diameters indicate greater oscillator strength for the transition. Simple poles that contribute only to the background are indicated by the shading.

Fig. 2
Fig. 2

Upper solid line: 1 spectrum of crystalline silicon from 2.2 to 4.2 eV. Lower solid line: spectrum of E21 with false data appended at energies indicated by solid arrows and then subtracted as described in the text. Dashed line: like lower solid line but with false data appended at the dashed arrows. The actual critical point energy is indicated by the vertical line on each spectrum.

Fig. 3
Fig. 3

Top solid line: normalized relative amplitudes of the Fourier coefficients of the solid-line E21 spectrum of Fig. 2 and its first four integral-order derivatives. Top dashed line: relative amplitudes of the Fourier coefficients of the dashed-line E21 spectrum of Fig. 2 to the same scale as the corresponding solid-line curve. Bottom: phases calculated assuming a Fourier origin at 3.38 eV. Phases for all derivatives are the same, except for additive constants −κπ/2, where κ is the order of differentiation. Theoretical phase variations corresponding to hypothetical critical point energies at 3.37, 3.38, and 3.39 eV are also shown.

Fig. 4
Fig. 4

Reciprocal-space LSR fits of theoretical line shapes to the first- and third-deviative data of Fig. 3. The zero of the phase scale has been shifted by π for the first derivative. Fitting ranges are indicated by the vertical lines.

Fig. 5
Fig. 5

Reconstructed real-space derivative line shapes for the experimental and best-reciprocal-space-fit theoretical line shapes from Fig. 4. Top: fully reconstructed first derivatives with no reciprocal-space filtering. Middle: first derivative reconstructed over the index range 6–36 used in reciprocal-space LSR. Bottom: like middle but for third derivative. The reconstructed experimental and theoretical line shapes are identical for the middle and lower curves on the scale shown. The energies at which false data were appended are indicated by the vertical lines.

Fig. 6
Fig. 6

Expanded view of several reconstructed real-space line shapes for the third derivative in Fig. 4. Solid curve: data reconstructed over the index range 6–36 used in reciprocal-space LSR. Dotted–dashed curve: like the solid curve but for theoretical line shape. Dashed curve: fully reconstructed theoretical line shape with no reciprocal-space filtering. The fully reconstructed experimental line shape is not shown, but its noise amplitude is indicated by the vertical bar at the right.

Equations (24)

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

x κ δ 2 = x κ j = m 1 m 2 [ F j - f ( x , E j ) ] 2 ,
= - 2 j = m 1 m 2 [ F j - f ( x , E j ) ] x κ f ( x , E j ) ,
= 0 ,             κ = 1 , 2 , , τ ,
E j = E 1 + ( j - 1 ) ( E F - E 1 ) / ( M - 1 )
F j = n = 0 N ( A n cos n θ j + B n sin n θ j ) ,
f ( x , E ) = n = 0 N [ a n ( x ) cos n θ + b n ( x ) sin n θ ] ,
θ j = 2 π M ( j - 1 ) ,
θ = ( E - E 1 ) / E s ,
E s = M ( E F - E 1 ) / [ 2 π ( M - 1 ) ] ,
0 = j = m 1 m 2 n = 0 N n = 0 N [ ( A n - a n ) cos n θ j + ( B n - b n ) sin n θ j ] * ( a n x κ cos n θ j + b n x κ sin n θ j ) ,             κ = 1 , 2 , , τ .
0 = ( A 0 - a 0 ) a 0 x κ + n = 1 N [ ( A n - a n ) a n x κ + ( B n - b n ) b n x κ ] ,             κ = 1 , 2 , , τ .
δ x a = 1.7 { δ 2 [ M - 1 ] α α N p - p - 1 } 1 / 2 ,
[ C ] α β = [ M - 1 ] α β / ( [ M - 1 ] α α [ M - 1 ] β β ) 1 / 2 ,
[ M ] α β = j = m 1 m 2 f ( x , E j ) x α f ( x , E j ) x β ,
δ f 2 = n = n 1 n 2 [ ( A n - a n ) 2 + ( B n - b n ) 2 ] ,
[ M f ] α β = n = n 1 n 2 [ a n ( x ) x α a n ( x ) x β + b n ( x ) x α b n ( x ) x β ] .
δ 2 = M 2 δ f 2 ,
[ M ] α β = M 2 [ M f ] α β .
N p = min [ ( m 2 - m 1 + 1 ) , ( n 2 - n 1 + 1 ) ]
A 0 = 1 M j = 1 M F j ,
A M / 2 = 1 M j = 1 M ( - 1 ) j + 1 F j ,
B 0 = B M / 2 = 0 ,
A n = 2 M j = 1 M F j cos n θ j ,
B n = 2 M j = 1 M F j sin n θ j ,