Abstract

The relationship between infrared refractive index and near-infrared, visible, and ultraviolet absorption spectra is examined. The long-wavelength limit and dispersion are determined as simple functions of composition. The computed results are compared with infrared ellipsometric measurements.

© 1992 Optical Society of America

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References

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  1. R. F. Potter, “Germanium,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, New York, 1985), pp. 465–478.
  2. D. F. Edwards, “Silicon,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, New York, 1985), pp. 547–569. ibid., pp. 547–569.
  3. J. J. Miceli, D. P. Naughton, “Model for gradient formation in polycrystalline germanium-silicon alloy GRIN crystals via Czochralski crystal growing,” Appl. Opt. 27, 500–504 (1988).
    [CrossRef] [PubMed]
  4. D. P. Naughton, J. J. Miceli, D. T. Moore, “Measurement of the refractive-index profile in polycrystalline germanium-silicon alloy GRIN crystals,” Appl. Opt. 27, 505–507 (1988).
    [CrossRef] [PubMed]
  5. J. Humlíček, M. Garriga, M. I. Alonso, M. Cardona, “Optical spectra of SxGe1−x alloys,” J. Appl. Phys. 65, 2827–2832 (1989).
    [CrossRef]
  6. A. Röseler, “Spectroscopic infrared-ellipsometry with the Fourier-transform-spectrometer,” preprint 85-4 (Zentralinstitut Optik und Spektroskopie, Berlin, Germany, 1985); A. Röseler, “Spectroscopic infrared ellipsometry by means of FTS,” Microchim. Acta 11, 79–83 (1988).
  7. R. Braunstein, A. R. Moore, F. Herman, “Optical absorption in germanium-silicon alloys,” Phys. Rev. 109, 695–710 (1958).
    [CrossRef]
  8. R. Braunstein, “Lattice vibration spectra of germanium-silicon alloys,” Phys. Rev. 130, 879–887 (1963).
    [CrossRef]
  9. A. E. Cosand, W. G. Spitzer, “Infrared absorption of lattice modes and the silicon local mode in GexSi1−x alloys,” J. Appl. Phys. 42, 5241–5249 (1971).
    [CrossRef]
  10. J. Humlíček, K. Vojtčchovský, “Infrared optical constants of intrinsic silicon,” Phys. Status Solidi A 92, 249–255 (1985).
    [CrossRef]

1989

J. Humlíček, M. Garriga, M. I. Alonso, M. Cardona, “Optical spectra of SxGe1−x alloys,” J. Appl. Phys. 65, 2827–2832 (1989).
[CrossRef]

1988

1985

J. Humlíček, K. Vojtčchovský, “Infrared optical constants of intrinsic silicon,” Phys. Status Solidi A 92, 249–255 (1985).
[CrossRef]

1971

A. E. Cosand, W. G. Spitzer, “Infrared absorption of lattice modes and the silicon local mode in GexSi1−x alloys,” J. Appl. Phys. 42, 5241–5249 (1971).
[CrossRef]

1963

R. Braunstein, “Lattice vibration spectra of germanium-silicon alloys,” Phys. Rev. 130, 879–887 (1963).
[CrossRef]

1958

R. Braunstein, A. R. Moore, F. Herman, “Optical absorption in germanium-silicon alloys,” Phys. Rev. 109, 695–710 (1958).
[CrossRef]

Alonso, M. I.

J. Humlíček, M. Garriga, M. I. Alonso, M. Cardona, “Optical spectra of SxGe1−x alloys,” J. Appl. Phys. 65, 2827–2832 (1989).
[CrossRef]

Braunstein, R.

R. Braunstein, “Lattice vibration spectra of germanium-silicon alloys,” Phys. Rev. 130, 879–887 (1963).
[CrossRef]

R. Braunstein, A. R. Moore, F. Herman, “Optical absorption in germanium-silicon alloys,” Phys. Rev. 109, 695–710 (1958).
[CrossRef]

Cardona, M.

J. Humlíček, M. Garriga, M. I. Alonso, M. Cardona, “Optical spectra of SxGe1−x alloys,” J. Appl. Phys. 65, 2827–2832 (1989).
[CrossRef]

Cosand, A. E.

A. E. Cosand, W. G. Spitzer, “Infrared absorption of lattice modes and the silicon local mode in GexSi1−x alloys,” J. Appl. Phys. 42, 5241–5249 (1971).
[CrossRef]

Edwards, D. F.

D. F. Edwards, “Silicon,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, New York, 1985), pp. 547–569. ibid., pp. 547–569.

Garriga, M.

J. Humlíček, M. Garriga, M. I. Alonso, M. Cardona, “Optical spectra of SxGe1−x alloys,” J. Appl. Phys. 65, 2827–2832 (1989).
[CrossRef]

Herman, F.

R. Braunstein, A. R. Moore, F. Herman, “Optical absorption in germanium-silicon alloys,” Phys. Rev. 109, 695–710 (1958).
[CrossRef]

Humlícek, J.

J. Humlíček, M. Garriga, M. I. Alonso, M. Cardona, “Optical spectra of SxGe1−x alloys,” J. Appl. Phys. 65, 2827–2832 (1989).
[CrossRef]

J. Humlíček, K. Vojtčchovský, “Infrared optical constants of intrinsic silicon,” Phys. Status Solidi A 92, 249–255 (1985).
[CrossRef]

Miceli, J. J.

Moore, A. R.

R. Braunstein, A. R. Moore, F. Herman, “Optical absorption in germanium-silicon alloys,” Phys. Rev. 109, 695–710 (1958).
[CrossRef]

Moore, D. T.

Naughton, D. P.

Potter, R. F.

R. F. Potter, “Germanium,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, New York, 1985), pp. 465–478.

Röseler, A.

A. Röseler, “Spectroscopic infrared-ellipsometry with the Fourier-transform-spectrometer,” preprint 85-4 (Zentralinstitut Optik und Spektroskopie, Berlin, Germany, 1985); A. Röseler, “Spectroscopic infrared ellipsometry by means of FTS,” Microchim. Acta 11, 79–83 (1988).

Spitzer, W. G.

A. E. Cosand, W. G. Spitzer, “Infrared absorption of lattice modes and the silicon local mode in GexSi1−x alloys,” J. Appl. Phys. 42, 5241–5249 (1971).
[CrossRef]

Vojtcchovský, K.

J. Humlíček, K. Vojtčchovský, “Infrared optical constants of intrinsic silicon,” Phys. Status Solidi A 92, 249–255 (1985).
[CrossRef]

Appl. Opt.

J. Appl. Phys.

J. Humlíček, M. Garriga, M. I. Alonso, M. Cardona, “Optical spectra of SxGe1−x alloys,” J. Appl. Phys. 65, 2827–2832 (1989).
[CrossRef]

A. E. Cosand, W. G. Spitzer, “Infrared absorption of lattice modes and the silicon local mode in GexSi1−x alloys,” J. Appl. Phys. 42, 5241–5249 (1971).
[CrossRef]

Phys. Rev.

R. Braunstein, A. R. Moore, F. Herman, “Optical absorption in germanium-silicon alloys,” Phys. Rev. 109, 695–710 (1958).
[CrossRef]

R. Braunstein, “Lattice vibration spectra of germanium-silicon alloys,” Phys. Rev. 130, 879–887 (1963).
[CrossRef]

Phys. Status Solidi A

J. Humlíček, K. Vojtčchovský, “Infrared optical constants of intrinsic silicon,” Phys. Status Solidi A 92, 249–255 (1985).
[CrossRef]

Other

A. Röseler, “Spectroscopic infrared-ellipsometry with the Fourier-transform-spectrometer,” preprint 85-4 (Zentralinstitut Optik und Spektroskopie, Berlin, Germany, 1985); A. Röseler, “Spectroscopic infrared ellipsometry by means of FTS,” Microchim. Acta 11, 79–83 (1988).

R. F. Potter, “Germanium,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, New York, 1985), pp. 465–478.

D. F. Edwards, “Silicon,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, New York, 1985), pp. 547–569. ibid., pp. 547–569.

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Figures (5)

Fig. 1
Fig. 1

Extinction coefficients measured ellipsometrically in Ref. 5 for the compositions listed in Table I. The value of x increases from left to right.

Fig. 2
Fig. 2

Refractive index measured ellipsometrically in Ref. 5 for the compositions listed in Table I (for photon energies above 1.66 eV) and computed from the Kramers–Kronig relations (in the 0- to 1.4-eV range). The value of x increases from bottom to top.

Fig. 3
Fig. 3

Long-wavelength limit of the refractive index computed from the Kramers–Konig relations (crosses) and its quadratic approximation of Eq. (9) (solid line). Filled circles with error bars show the results of Fourier-transform ellipsometric measurements. The dotted curve is the refractive index obtained in Ref. 3.

Fig. 4
Fig. 4

Dispersion coefficient computed from the Kramers-Kronig relations (crosses) and its quadratic approximation of Eq. (10) (solid curve).

Fig. 5
Fig. 5

Extinction coefficient for the x = 0.26 sample of SixGe1−x alloy (solid line). The broken line is the refractive index computed from the Kramers–Kronig relations.

Tables (1)

Tables Icon

Table I Refractive Index and Dispersion Coefficients for SixGe1−x Alloys

Equations (11)

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n ( E 0 ) = 1 + 2 π 0 E k ( E ) E 2 E 0 2 d E ;
n ( E 0 ) = n m ( E 0 ) + n extr ( E 0 ) ,
n m ( E 0 ) = 1 + 2 π E min E max E k ( E ) E 2 E 0 2 d E , n extr ( E 0 ) = 2 π E max E k ( E ) E 2 E 0 2 d E .
k ( E ) = k ( E max ) ( E max E ) r for E > E max ,
r ( x ) = 1 . 61 + 0 . 20 x .
k ( 5 . 66 eV ) = 2 . 76 + 0 . 35 x .
n extr = 2 π k ( E max ) j = 0 1 r + 2 j ( E 0 E max ) 2 j .
n extr ( E 0 ) = 1 . 76 + 0 . 22 x 1 . 61 + 0 . 20 x + 1 . 76 + 0 . 22 x 3 . 61 + 0 . 20 x ( E 0 E max ) 2 , E 0 E max .
n ( E 0 ) = n 0 + n 1 E 0 2 = n 0 + n ν 1 2 .
n 0 ( x ) = 4 . 01 0 . 81 x + 0 . 22 x 2 ,
n 1 ( x ) = 0 . 216 0 . 211 x + 0 . 089 x 2 , n ( x ) 1 = 0 . 141 0 . 137 n + 0 . 058 x 2 ,

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