Abstract

We investigate a generalized Cauchy power-series expansion for the index of refraction of an n-type elemental semiconductor in the region of IR transparency. A plot of index versus photon energy squared should be very nearly linear if all absorptions lie above the transparent region. However, free carriers produce far-IR absorption, and the dispersive signature of this is a deviation from linearity in the mid- to far-IR. By retaining terms with negative exponents in the index expansion, we find a substantially improved fit to index measurements. Moreover, the free-carrier density may be determined from the coefficients in the regression fit. The method has been used to evaluate several extensive index measurements for doped silicon found in the literature.

© 2006 Optical Society of America

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  54. L. E. Howarth and J. F. Gilbert, 'Determination of free-electron effective mass of n-type silicon,' J. Appl. Phys. 34, 236-237, (1963).
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    [CrossRef]
  56. E. Barta, 'Optical constants of various heavily doped p- and n-type silicon crystals by Kramers-Kronig analysis,' Infrared Phys. 17, 319-329 (1977).
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2001 (2)

S. Nashima, O. Morikawa, K. Takata, and M. Hangyo, 'Temperature dependence of optical and electronic properties of moderately doped silicon at terahertz frequencies,' J. Appl. Phys. 90, 837-842 (2001).
[CrossRef]

D. Y. Smith, M. Inokuti, and W. Karstens, 'A generalized Cauchy dispersion formula and the refractivity of elemental semiconductors,' J. Phys.: Condens. Matter 13, 3883-3893 (2001).
[CrossRef]

2000 (1)

D. Y. Smith, M. Inokuti, and W. Karstens, 'Photoresponse of condensed matter over the entire range of excitation energies: analysis of silicon,' Phys. Essays 13, 465-472 (2000).
[CrossRef]

1993 (1)

K.-L. Barth and F. Keilmann, 'Far-infrared ellipsometer,' Rev. Sci. Instrum. 64, 870-875 (1993).
[CrossRef]

1990 (1)

1988 (2)

J. Humlícek and K. Vojtechovský, 'Infrared optical constants of n-type silicon,' Czech. J. Phys., Sect. B 38, 1033-1049 (1988).
[CrossRef]

T. Ohba and Shun-ichi Ikawa, 'Far-infrared absorption of silicon crystals,' J. Appl. Phys. 64, 4141-4143 (1988).
[CrossRef]

1985 (1)

J. Humlícek and K. Vojtechovský, 'Infrared optical constants of intrinsic silicon,' Phys. Status Solidi A 92, 249-255 (1985).
[CrossRef]

1983 (2)

D. E. Aspnes and A. A. Studna, 'Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,' Phys. Rev. B 27, 985-1009 (1983).
[CrossRef]

R. T. Kinasewitz and B. Senitzky, 'Investigation of the complex permittivity of n-type silicon at millimeter wavelengths,' J. Appl. Phys. 54, 3394-3398 (1983).
[CrossRef]

1980 (4)

D. F. Edwards and E. Ochoa, 'Infrared refractive index of silicon,' Appl. Opt. 19, 4130-4131 (1980).
[CrossRef] [PubMed]

H. Engstrom, 'Infrared reflectivity and transmissivity of boron-implanted, laser-annealed silicon,' J. Appl. Phys. 51, 5245-5249 (1980).
[CrossRef]

H. H. Li, 'Refractive index of silicon and germanium and its wavelength and temperature derivatives,' J. Phys. Chem. Ref. Data 9, 561-658 (1980).
[CrossRef]

E. Shiles, T. Sasaki, M. Inokuti, and D. Y. Smith, 'Self-consistency and sum-rule tests in the Kramers-Kronig analysis of optical data: applications to aluminum,' Phys. Rev. B 22, 1612-1628 (1980).
[CrossRef]

1978 (1)

D. Y. Smith and E. Shiles, 'Finite-energy f-sum rules for valence electrons,' Phys. Rev. B 17, 4689-4694 (1978). See Eq. (14), but note that on the second line following Eq. (18), omegab should be replaced by ϵb.
[CrossRef]

1977 (2)

E. Barta, 'Determination of effective mass values by a Kramers-Kronig analysis for variously doped silicon crystals,' Infrared Phys. 17, 111-119 (1977).
[CrossRef]

E. Barta, 'Optical constants of various heavily doped p- and n-type silicon crystals by Kramers-Kronig analysis,' Infrared Phys. 17, 319-329 (1977).
[CrossRef]

1976 (1)

1973 (1)

1972 (2)

M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, 'Superconvergence and sum rules for the optical constants,' Phys. Rev. B 6, 4502-4509 (1972).
[CrossRef]

L. M. Lambert, 'Free carrier reflectivity in optically homogeneous silicon,' Phys. Status Solidi A 11, 461-467 (1972).
[CrossRef]

1971 (2)

P. A. Schumann, Jr., W. A. Keenan, A. H. Tong, H. H. Gegenwarth, and C. P. Schneider, 'Silicon optical constants in the infrared,' J. Electrochem. Soc. 118, 145-148 (1971).
[CrossRef]

W. Primak, 'Refractive index of silicon,' Appl. Opt. 10, 759-763 (1971).
[CrossRef] [PubMed]

1969 (1)

K. Sato, 'Contribution of surface electrons to the infrared optical properties of silicon,' J. Phys. Soc. Jpn. 27, 89-95 (1969).
[CrossRef]

1967 (1)

1965 (1)

T. Staflin, 'Infra-red absorption due to hole transitions involving the split-off valence-band in silicon,' Phys. Lett. 19, 84-85 (1965).
[CrossRef]

1963 (2)

Yu. I. Ukhanov, 'Faraday effect in silicon in the infrared region of wavelengths,' Sov. Phys. Solid State 4, 2010-2013 (1963).

L. E. Howarth and J. F. Gilbert, 'Determination of free-electron effective mass of n-type silicon,' J. Appl. Phys. 34, 236-237, (1963).
[CrossRef]

1960 (1)

V. S. Vavilov, 'The absorption of free charge carriers by infrared radiation in silicon,' Sov. Phys. Solid State 2, 346-349 (1960).

1959 (1)

M. Cardona, W. Paul, and H. Brooks, 'Dielectric constant of germanium and silicon as a function of volume,' J. Phys. Chem. Solids 8, 204-206 (1959).
[CrossRef]

1957 (3)

C. D. Salzberg and J. J. Villa, 'Infrared refractive indices of silicon, germanium, and modified selenium glass,' J. Opt. Soc. Am. 47, 244-246 (1957).
[CrossRef]

W. G. Spitzer and H. Y. Fan, 'Infrared absorption in n-type silicon,' Phys. Rev. 108, 268-271 (1957).
[CrossRef]

W. G. Spitzer and H. Y. Fan, 'Determination of optical constants and carrier effective mass of semiconductors,' Phys. Rev. 106, 882-890 (1957).
[CrossRef]

1956 (2)

R. N. Dexter, H. J. Zeiger, and B. Lax, 'Cyclotron resonance experiments in silicon and germanium,' Phys. Rev. 104, 637-644 (1956).
[CrossRef]

H. Y. Fan, 'Infra-red absorption in semiconductors,' Rep. Prog. Phys. 19, 107-155 (1956).
[CrossRef]

1955 (1)

A. H. Kahn, 'Theory of the infrared absorption of carriers in germanium and silicon,' Phys. Rev. 97, 1647-1652 (1955).
[CrossRef]

1950 (1)

H. B. Briggs, 'Infrared absorption in silicon,' Phys. Rev. 77, 727-728 (1950).
[CrossRef]

1949 (1)

M. Becker and H. Y. Fan, 'Optical properties of semiconductors. III. Infra-red transmission of silicon,' Phys. Rev. 76, 1531-1532 (1949).
[CrossRef]

1933 (1)

C. Zener, 'Remarkable optical properties of the alkali metals,' Nature 132, 968 (1933).
[CrossRef]

1900 (1)

P. Drude, 'Electronic theory of metals I,' Ann. Phys. 1, 566-613 (1900).
[CrossRef]

1872 (1)

W. Sellmeier, 'Über die durch die Aetherschwingungen erregten Mitschwingungen der Körpertheilchen und deren Rückwirkung auf die ersteren, besonders zur Erklärung der Dispersion und ihrer Anomalien' (part 1), Ann. Phys. Chem. 145, 399-421 (1872);W. Sellmeier, 'Über die durch die Aetherschwingungen erregten Mitschwingungen der Köpertheilchen und deren Rückwirkung auf die ersteren, besonders zur Erklärung der Dispersion und ihrer Anomalien' (part 2), Ann. Phys. Chem. 145, 520-549 (1872); W. Sellmeier, 'Über die durch die Aetherschwingungen erregten Mitschwingungen der Köpertheilchen und deren Rückwirkung auf die ersteren, besonders zur Erklärung der Dispersion und ihrer Anomalien'(part 3), Ann. Phys. Chem. 147, 386-403 (1872); W. Sellmeier, 'Über die durch die Aetherschwingungen erregten Mitschwingungen der Köpertheilchen und deren Rückwirkung auf die ersteren, besonders zur Erklärung der Dispersion und ihrer Anomalien'(part 4), Ann. Phys. Chem. 147, 525-554 (1872).
[CrossRef]

1871 (1)

W. Sellmeier, 'Zur Erklärung der abnormen Farbenfolge in Spectrum einiger Substanzen,' Ann. Phys. Chem. 143, 272-282 (1871).
[CrossRef]

1852 (1)

A. Beer, 'Bestimmung der Absorption des rothen Lichts in farbigen Flüssigkeiten,' Ann. Phys. Chem. 86, 78-88 (1852).
[CrossRef]

1830 (1)

A. L. Cauchy, 'Sur la refraction et la réflexion de la lumière,' Bull. Sci. Math. 14, 6-10 (1830) (LC Control No. 06004562).

Altarelli, M.

M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, 'Superconvergence and sum rules for the optical constants,' Phys. Rev. B 6, 4502-4509 (1972).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes and A. A. Studna, 'Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,' Phys. Rev. B 27, 985-1009 (1983).
[CrossRef]

D. E. Aspnes, 'Optical functions,' in Properties of Silicon (Inspec, Institution of Electrical Engineers London, 1988), Chap. 2, pp. 61-79.

D. E. Aspnes, 'Optical properties,' in Properties of Crystalline Silicon, R.Hull, ed. (Inspec, Institution of Electrical Engineers London, 1999), Chap. 12, pp. 679-695.

Auslender, M.

M. Auslender and S. Hava, 'Doped n-type silicon,' in Handbook of Optical Constants of Solids, Vol. 3, E.D.Palik, ed. (Academic, 1997), pp. 155-186.

Barta, E.

E. Barta, 'Determination of effective mass values by a Kramers-Kronig analysis for variously doped silicon crystals,' Infrared Phys. 17, 111-119 (1977).
[CrossRef]

E. Barta, 'Optical constants of various heavily doped p- and n-type silicon crystals by Kramers-Kronig analysis,' Infrared Phys. 17, 319-329 (1977).
[CrossRef]

Barth, K.-L.

K.-L. Barth and F. Keilmann, 'Far-infrared ellipsometer,' Rev. Sci. Instrum. 64, 870-875 (1993).
[CrossRef]

Becker, M.

M. Becker and H. Y. Fan, 'Optical properties of semiconductors. III. Infra-red transmission of silicon,' Phys. Rev. 76, 1531-1532 (1949).
[CrossRef]

H. Y Fan and M. Becker, 'Infra-red optical properties of silicon and germanium,' in Semiconducting Materials, H.K.Henisch, ed. (Butterworth, 1951), pp. 132-147.

Beer, A.

A. Beer, 'Bestimmung der Absorption des rothen Lichts in farbigen Flüssigkeiten,' Ann. Phys. Chem. 86, 78-88 (1852).
[CrossRef]

A. Beer, Grundriss des Photometrischen Calcüles (Vieweg, Braunschweig, 1854).

Briggs, H. B.

H. B. Briggs, 'Infrared absorption in silicon,' Phys. Rev. 77, 727-728 (1950).
[CrossRef]

Brooks, H.

M. Cardona, W. Paul, and H. Brooks, 'Dielectric constant of germanium and silicon as a function of volume,' J. Phys. Chem. Solids 8, 204-206 (1959).
[CrossRef]

Cardona, M.

M. Cardona, W. Paul, and H. Brooks, 'Dielectric constant of germanium and silicon as a function of volume,' J. Phys. Chem. Solids 8, 204-206 (1959).
[CrossRef]

Cauchy, A. L.

A. L. Cauchy, 'Sur la refraction et la réflexion de la lumière,' Bull. Sci. Math. 14, 6-10 (1830) (LC Control No. 06004562).

A. L. Cauchy, Mémoire sur la dispersion de la lumière (Calve, Prague, 1836).

Chelikowsky, J. R.

M. L. Cohen and J. R. Chelikowsky, Electronic Structure and Optical Properties of Semiconductors, 2nd ed., Springer Series on Solid-State Science (Springer, 1989).
[CrossRef]

Cohen, M. L.

M. L. Cohen and J. R. Chelikowsky, Electronic Structure and Optical Properties of Semiconductors, 2nd ed., Springer Series on Solid-State Science (Springer, 1989).
[CrossRef]

Dexter, D. L.

M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, 'Superconvergence and sum rules for the optical constants,' Phys. Rev. B 6, 4502-4509 (1972).
[CrossRef]

Dexter, R. N.

R. N. Dexter, H. J. Zeiger, and B. Lax, 'Cyclotron resonance experiments in silicon and germanium,' Phys. Rev. 104, 637-644 (1956).
[CrossRef]

Drude, P.

P. Drude, 'Electronic theory of metals I,' Ann. Phys. 1, 566-613 (1900).
[CrossRef]

Edwards, D. F.

D. F. Edwards and E. Ochoa, 'Infrared refractive index of silicon,' Appl. Opt. 19, 4130-4131 (1980).
[CrossRef] [PubMed]

D. F. Edwards, 'Silicon (Si),' in Handbook of Optical Constants of Solids, Vol. 1, E.D.Palik, ed. (Academic, 1985), pp. 547-569.

Einspruch, N. G.

F. Seitz and N. G. Einspruch, Electronic Genie: The Tangled History of Silicon (U. Illinois Press, 1998).

Engstrom, H.

H. Engstrom, 'Infrared reflectivity and transmissivity of boron-implanted, laser-annealed silicon,' J. Appl. Phys. 51, 5245-5249 (1980).
[CrossRef]

Fan, H. Y

H. Y Fan and M. Becker, 'Infra-red optical properties of silicon and germanium,' in Semiconducting Materials, H.K.Henisch, ed. (Butterworth, 1951), pp. 132-147.

Fan, H. Y.

W. G. Spitzer and H. Y. Fan, 'Infrared absorption in n-type silicon,' Phys. Rev. 108, 268-271 (1957).
[CrossRef]

W. G. Spitzer and H. Y. Fan, 'Determination of optical constants and carrier effective mass of semiconductors,' Phys. Rev. 106, 882-890 (1957).
[CrossRef]

H. Y. Fan, 'Infra-red absorption in semiconductors,' Rep. Prog. Phys. 19, 107-155 (1956).
[CrossRef]

M. Becker and H. Y. Fan, 'Optical properties of semiconductors. III. Infra-red transmission of silicon,' Phys. Rev. 76, 1531-1532 (1949).
[CrossRef]

Fattinger, Ch.

Gegenwarth, H. H.

P. A. Schumann, Jr., W. A. Keenan, A. H. Tong, H. H. Gegenwarth, and C. P. Schneider, 'Silicon optical constants in the infrared,' J. Electrochem. Soc. 118, 145-148 (1971).
[CrossRef]

P. A Schumann, Jr., W. A. Keenan, A. H. Tong, H. H. Gegenwarth, and C. P. Schneider, 'Optical constants of silicon in the wavelength range 2.5 to 40 µm,' IBM Tech. Rep. TR 22.1008 (IBM, 1970).

Gilbert, J. F.

L. E. Howarth and J. F. Gilbert, 'Determination of free-electron effective mass of n-type silicon,' J. Appl. Phys. 34, 236-237, (1963).
[CrossRef]

Greenaway, D. L.

D. L. Greenaway and G. Harbeke, Optical Properties and Band Structure of Semiconductors (Pergamon, 1968).

Grischkowsky, D.

Hangyo, M.

S. Nashima, O. Morikawa, K. Takata, and M. Hangyo, 'Temperature dependence of optical and electronic properties of moderately doped silicon at terahertz frequencies,' J. Appl. Phys. 90, 837-842 (2001).
[CrossRef]

Harbeke, G.

D. L. Greenaway and G. Harbeke, Optical Properties and Band Structure of Semiconductors (Pergamon, 1968).

Hava, S.

M. Auslender and S. Hava, 'Doped n-type silicon,' in Handbook of Optical Constants of Solids, Vol. 3, E.D.Palik, ed. (Academic, 1997), pp. 155-186.

Howarth, L. E.

L. E. Howarth and J. F. Gilbert, 'Determination of free-electron effective mass of n-type silicon,' J. Appl. Phys. 34, 236-237, (1963).
[CrossRef]

Humlícek, J.

J. Humlícek and K. Vojtechovský, 'Infrared optical constants of n-type silicon,' Czech. J. Phys., Sect. B 38, 1033-1049 (1988).
[CrossRef]

J. Humlícek and K. Vojtechovský, 'Infrared optical constants of intrinsic silicon,' Phys. Status Solidi A 92, 249-255 (1985).
[CrossRef]

Ikawa, Shun-ichi

T. Ohba and Shun-ichi Ikawa, 'Far-infrared absorption of silicon crystals,' J. Appl. Phys. 64, 4141-4143 (1988).
[CrossRef]

Inokuti, M.

D. Y. Smith, M. Inokuti, and W. Karstens, 'A generalized Cauchy dispersion formula and the refractivity of elemental semiconductors,' J. Phys.: Condens. Matter 13, 3883-3893 (2001).
[CrossRef]

D. Y. Smith, M. Inokuti, and W. Karstens, 'Photoresponse of condensed matter over the entire range of excitation energies: analysis of silicon,' Phys. Essays 13, 465-472 (2000).
[CrossRef]

E. Shiles, T. Sasaki, M. Inokuti, and D. Y. Smith, 'Self-consistency and sum-rule tests in the Kramers-Kronig analysis of optical data: applications to aluminum,' Phys. Rev. B 22, 1612-1628 (1980).
[CrossRef]

Kahn, A. H.

A. H. Kahn, 'Theory of the infrared absorption of carriers in germanium and silicon,' Phys. Rev. 97, 1647-1652 (1955).
[CrossRef]

Karstens, W.

D. Y. Smith, M. Inokuti, and W. Karstens, 'A generalized Cauchy dispersion formula and the refractivity of elemental semiconductors,' J. Phys.: Condens. Matter 13, 3883-3893 (2001).
[CrossRef]

D. Y. Smith, M. Inokuti, and W. Karstens, 'Photoresponse of condensed matter over the entire range of excitation energies: analysis of silicon,' Phys. Essays 13, 465-472 (2000).
[CrossRef]

Keenan, W. A.

P. A. Schumann, Jr., W. A. Keenan, A. H. Tong, H. H. Gegenwarth, and C. P. Schneider, 'Silicon optical constants in the infrared,' J. Electrochem. Soc. 118, 145-148 (1971).
[CrossRef]

P. A Schumann, Jr., W. A. Keenan, A. H. Tong, H. H. Gegenwarth, and C. P. Schneider, 'Optical constants of silicon in the wavelength range 2.5 to 40 µm,' IBM Tech. Rep. TR 22.1008 (IBM, 1970).

Keiding, S.

Keilmann, F.

K.-L. Barth and F. Keilmann, 'Far-infrared ellipsometer,' Rev. Sci. Instrum. 64, 870-875 (1993).
[CrossRef]

Kinasewitz, R. T.

R. T. Kinasewitz and B. Senitzky, 'Investigation of the complex permittivity of n-type silicon at millimeter wavelengths,' J. Appl. Phys. 54, 3394-3398 (1983).
[CrossRef]

Kingslake, R.

R. Kingslake, Lens Design Fundamentals (Academic, 1978).

Lambert, L. M.

L. M. Lambert, 'Free carrier reflectivity in optically homogeneous silicon,' Phys. Status Solidi A 11, 461-467 (1972).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1972).

Lax, B.

R. N. Dexter, H. J. Zeiger, and B. Lax, 'Cyclotron resonance experiments in silicon and germanium,' Phys. Rev. 104, 637-644 (1956).
[CrossRef]

Li, H. H.

H. H. Li, 'Refractive index of silicon and germanium and its wavelength and temperature derivatives,' J. Phys. Chem. Ref. Data 9, 561-658 (1980).
[CrossRef]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1972).

Loewenstein, E. V.

Lorentz, H. A.

H. A. Lorentz, The Theory of Electrons (Teubner, Leipzig, 1909; Stechert, New York, 1923).

Morgan, R. L.

Morikawa, O.

S. Nashima, O. Morikawa, K. Takata, and M. Hangyo, 'Temperature dependence of optical and electronic properties of moderately doped silicon at terahertz frequencies,' J. Appl. Phys. 90, 837-842 (2001).
[CrossRef]

Moss, T. S.

T. S. Moss, Optical Properties of Semiconductors (Butterworth, 1959).

Nashima, S.

S. Nashima, O. Morikawa, K. Takata, and M. Hangyo, 'Temperature dependence of optical and electronic properties of moderately doped silicon at terahertz frequencies,' J. Appl. Phys. 90, 837-842 (2001).
[CrossRef]

Nussenzveig, H. M.

M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, 'Superconvergence and sum rules for the optical constants,' Phys. Rev. B 6, 4502-4509 (1972).
[CrossRef]

Ochoa, E.

Ohba, T.

T. Ohba and Shun-ichi Ikawa, 'Far-infrared absorption of silicon crystals,' J. Appl. Phys. 64, 4141-4143 (1988).
[CrossRef]

Paul, W.

M. Cardona, W. Paul, and H. Brooks, 'Dielectric constant of germanium and silicon as a function of volume,' J. Phys. Chem. Solids 8, 204-206 (1959).
[CrossRef]

Potter, R. F.

R. F. Potter, 'Germanium (Ge),' in Handbook of Optical Constants of Solids, Vol. 1, E.D.Palik, ed. (Academic, 1985), pp. 465-478.

Primak, W.

Randall, C. M.

Rawcliffe, R. D.

Salzberg, C. D.

Sasaki, T.

E. Shiles, T. Sasaki, M. Inokuti, and D. Y. Smith, 'Self-consistency and sum-rule tests in the Kramers-Kronig analysis of optical data: applications to aluminum,' Phys. Rev. B 22, 1612-1628 (1980).
[CrossRef]

Sato, K.

K. Sato, 'Contribution of surface electrons to the infrared optical properties of silicon,' J. Phys. Soc. Jpn. 27, 89-95 (1969).
[CrossRef]

Schneider, C. P.

P. A. Schumann, Jr., W. A. Keenan, A. H. Tong, H. H. Gegenwarth, and C. P. Schneider, 'Silicon optical constants in the infrared,' J. Electrochem. Soc. 118, 145-148 (1971).
[CrossRef]

P. A Schumann, Jr., W. A. Keenan, A. H. Tong, H. H. Gegenwarth, and C. P. Schneider, 'Optical constants of silicon in the wavelength range 2.5 to 40 µm,' IBM Tech. Rep. TR 22.1008 (IBM, 1970).

Schumann, P. A

P. A Schumann, Jr., W. A. Keenan, A. H. Tong, H. H. Gegenwarth, and C. P. Schneider, 'Optical constants of silicon in the wavelength range 2.5 to 40 µm,' IBM Tech. Rep. TR 22.1008 (IBM, 1970).

Schumann, P. A.

P. A. Schumann, Jr., W. A. Keenan, A. H. Tong, H. H. Gegenwarth, and C. P. Schneider, 'Silicon optical constants in the infrared,' J. Electrochem. Soc. 118, 145-148 (1971).
[CrossRef]

Seitz, F.

F. Seitz and N. G. Einspruch, Electronic Genie: The Tangled History of Silicon (U. Illinois Press, 1998).

Sellmeier, W.

W. Sellmeier, 'Über die durch die Aetherschwingungen erregten Mitschwingungen der Körpertheilchen und deren Rückwirkung auf die ersteren, besonders zur Erklärung der Dispersion und ihrer Anomalien' (part 1), Ann. Phys. Chem. 145, 399-421 (1872);W. Sellmeier, 'Über die durch die Aetherschwingungen erregten Mitschwingungen der Köpertheilchen und deren Rückwirkung auf die ersteren, besonders zur Erklärung der Dispersion und ihrer Anomalien' (part 2), Ann. Phys. Chem. 145, 520-549 (1872); W. Sellmeier, 'Über die durch die Aetherschwingungen erregten Mitschwingungen der Köpertheilchen und deren Rückwirkung auf die ersteren, besonders zur Erklärung der Dispersion und ihrer Anomalien'(part 3), Ann. Phys. Chem. 147, 386-403 (1872); W. Sellmeier, 'Über die durch die Aetherschwingungen erregten Mitschwingungen der Köpertheilchen und deren Rückwirkung auf die ersteren, besonders zur Erklärung der Dispersion und ihrer Anomalien'(part 4), Ann. Phys. Chem. 147, 525-554 (1872).
[CrossRef]

W. Sellmeier, 'Zur Erklärung der abnormen Farbenfolge in Spectrum einiger Substanzen,' Ann. Phys. Chem. 143, 272-282 (1871).
[CrossRef]

Senitzky, B.

R. T. Kinasewitz and B. Senitzky, 'Investigation of the complex permittivity of n-type silicon at millimeter wavelengths,' J. Appl. Phys. 54, 3394-3398 (1983).
[CrossRef]

Shiles, E.

E. Shiles, T. Sasaki, M. Inokuti, and D. Y. Smith, 'Self-consistency and sum-rule tests in the Kramers-Kronig analysis of optical data: applications to aluminum,' Phys. Rev. B 22, 1612-1628 (1980).
[CrossRef]

D. Y. Smith and E. Shiles, 'Finite-energy f-sum rules for valence electrons,' Phys. Rev. B 17, 4689-4694 (1978). See Eq. (14), but note that on the second line following Eq. (18), omegab should be replaced by ϵb.
[CrossRef]

Shimura, F.

F. Shimura, Oxygen in Silicon (Academic, 1994).

Smith, D. R.

Smith, D. Y.

D. Y. Smith, M. Inokuti, and W. Karstens, 'A generalized Cauchy dispersion formula and the refractivity of elemental semiconductors,' J. Phys.: Condens. Matter 13, 3883-3893 (2001).
[CrossRef]

D. Y. Smith, M. Inokuti, and W. Karstens, 'Photoresponse of condensed matter over the entire range of excitation energies: analysis of silicon,' Phys. Essays 13, 465-472 (2000).
[CrossRef]

E. Shiles, T. Sasaki, M. Inokuti, and D. Y. Smith, 'Self-consistency and sum-rule tests in the Kramers-Kronig analysis of optical data: applications to aluminum,' Phys. Rev. B 22, 1612-1628 (1980).
[CrossRef]

D. Y. Smith and E. Shiles, 'Finite-energy f-sum rules for valence electrons,' Phys. Rev. B 17, 4689-4694 (1978). See Eq. (14), but note that on the second line following Eq. (18), omegab should be replaced by ϵb.
[CrossRef]

D. Y. Smith, 'Comments on the dispersion relations for the complex refractive index of circularly and elliptically polarized light,' J. Opt. Soc. Am. 66, 454-460 (1976).
[CrossRef]

M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, 'Superconvergence and sum rules for the optical constants,' Phys. Rev. B 6, 4502-4509 (1972).
[CrossRef]

Spitzer, W. G.

W. G. Spitzer and H. Y. Fan, 'Determination of optical constants and carrier effective mass of semiconductors,' Phys. Rev. 106, 882-890 (1957).
[CrossRef]

W. G. Spitzer and H. Y. Fan, 'Infrared absorption in n-type silicon,' Phys. Rev. 108, 268-271 (1957).
[CrossRef]

Staflin, T.

T. Staflin, 'Infra-red absorption due to hole transitions involving the split-off valence-band in silicon,' Phys. Lett. 19, 84-85 (1965).
[CrossRef]

Stern, F.

F. Stern, 'Elementary theory of the optical properties of solids,' in Solid State Physics, Vol. 15, F.Seitz and D.Turnbull, eds. (Academic, 1963), pp. 299-408.
[CrossRef]

Studna, A. A.

D. E. Aspnes and A. A. Studna, 'Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,' Phys. Rev. B 27, 985-1009 (1983).
[CrossRef]

Takata, K.

S. Nashima, O. Morikawa, K. Takata, and M. Hangyo, 'Temperature dependence of optical and electronic properties of moderately doped silicon at terahertz frequencies,' J. Appl. Phys. 90, 837-842 (2001).
[CrossRef]

Tong, A. H.

P. A. Schumann, Jr., W. A. Keenan, A. H. Tong, H. H. Gegenwarth, and C. P. Schneider, 'Silicon optical constants in the infrared,' J. Electrochem. Soc. 118, 145-148 (1971).
[CrossRef]

P. A Schumann, Jr., W. A. Keenan, A. H. Tong, H. H. Gegenwarth, and C. P. Schneider, 'Optical constants of silicon in the wavelength range 2.5 to 40 µm,' IBM Tech. Rep. TR 22.1008 (IBM, 1970).

Ukhanov, Yu. I.

Yu. I. Ukhanov, 'Faraday effect in silicon in the infrared region of wavelengths,' Sov. Phys. Solid State 4, 2010-2013 (1963).

van Exter, M.

Vavilov, V. S.

V. S. Vavilov, 'The absorption of free charge carriers by infrared radiation in silicon,' Sov. Phys. Solid State 2, 346-349 (1960).

Villa, J. J.

Vojtechovský, K.

J. Humlícek and K. Vojtechovský, 'Infrared optical constants of n-type silicon,' Czech. J. Phys., Sect. B 38, 1033-1049 (1988).
[CrossRef]

J. Humlícek and K. Vojtechovský, 'Infrared optical constants of intrinsic silicon,' Phys. Status Solidi A 92, 249-255 (1985).
[CrossRef]

Zeiger, H. J.

R. N. Dexter, H. J. Zeiger, and B. Lax, 'Cyclotron resonance experiments in silicon and germanium,' Phys. Rev. 104, 637-644 (1956).
[CrossRef]

Zener, C.

C. Zener, 'Remarkable optical properties of the alkali metals,' Nature 132, 968 (1933).
[CrossRef]

Ann. Phys. (1)

P. Drude, 'Electronic theory of metals I,' Ann. Phys. 1, 566-613 (1900).
[CrossRef]

Ann. Phys. Chem. (3)

A. Beer, 'Bestimmung der Absorption des rothen Lichts in farbigen Flüssigkeiten,' Ann. Phys. Chem. 86, 78-88 (1852).
[CrossRef]

W. Sellmeier, 'Zur Erklärung der abnormen Farbenfolge in Spectrum einiger Substanzen,' Ann. Phys. Chem. 143, 272-282 (1871).
[CrossRef]

W. Sellmeier, 'Über die durch die Aetherschwingungen erregten Mitschwingungen der Körpertheilchen und deren Rückwirkung auf die ersteren, besonders zur Erklärung der Dispersion und ihrer Anomalien' (part 1), Ann. Phys. Chem. 145, 399-421 (1872);W. Sellmeier, 'Über die durch die Aetherschwingungen erregten Mitschwingungen der Köpertheilchen und deren Rückwirkung auf die ersteren, besonders zur Erklärung der Dispersion und ihrer Anomalien' (part 2), Ann. Phys. Chem. 145, 520-549 (1872); W. Sellmeier, 'Über die durch die Aetherschwingungen erregten Mitschwingungen der Köpertheilchen und deren Rückwirkung auf die ersteren, besonders zur Erklärung der Dispersion und ihrer Anomalien'(part 3), Ann. Phys. Chem. 147, 386-403 (1872); W. Sellmeier, 'Über die durch die Aetherschwingungen erregten Mitschwingungen der Köpertheilchen und deren Rückwirkung auf die ersteren, besonders zur Erklärung der Dispersion und ihrer Anomalien'(part 4), Ann. Phys. Chem. 147, 525-554 (1872).
[CrossRef]

Appl. Opt. (4)

Bull. Sci. Math. (1)

A. L. Cauchy, 'Sur la refraction et la réflexion de la lumière,' Bull. Sci. Math. 14, 6-10 (1830) (LC Control No. 06004562).

Czech. J. Phys., Sect. B (1)

J. Humlícek and K. Vojtechovský, 'Infrared optical constants of n-type silicon,' Czech. J. Phys., Sect. B 38, 1033-1049 (1988).
[CrossRef]

Infrared Phys. (2)

E. Barta, 'Determination of effective mass values by a Kramers-Kronig analysis for variously doped silicon crystals,' Infrared Phys. 17, 111-119 (1977).
[CrossRef]

E. Barta, 'Optical constants of various heavily doped p- and n-type silicon crystals by Kramers-Kronig analysis,' Infrared Phys. 17, 319-329 (1977).
[CrossRef]

J. Appl. Phys. (5)

L. E. Howarth and J. F. Gilbert, 'Determination of free-electron effective mass of n-type silicon,' J. Appl. Phys. 34, 236-237, (1963).
[CrossRef]

T. Ohba and Shun-ichi Ikawa, 'Far-infrared absorption of silicon crystals,' J. Appl. Phys. 64, 4141-4143 (1988).
[CrossRef]

H. Engstrom, 'Infrared reflectivity and transmissivity of boron-implanted, laser-annealed silicon,' J. Appl. Phys. 51, 5245-5249 (1980).
[CrossRef]

R. T. Kinasewitz and B. Senitzky, 'Investigation of the complex permittivity of n-type silicon at millimeter wavelengths,' J. Appl. Phys. 54, 3394-3398 (1983).
[CrossRef]

S. Nashima, O. Morikawa, K. Takata, and M. Hangyo, 'Temperature dependence of optical and electronic properties of moderately doped silicon at terahertz frequencies,' J. Appl. Phys. 90, 837-842 (2001).
[CrossRef]

J. Electrochem. Soc. (1)

P. A. Schumann, Jr., W. A. Keenan, A. H. Tong, H. H. Gegenwarth, and C. P. Schneider, 'Silicon optical constants in the infrared,' J. Electrochem. Soc. 118, 145-148 (1971).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. B (1)

J. Phys. Chem. Ref. Data (1)

H. H. Li, 'Refractive index of silicon and germanium and its wavelength and temperature derivatives,' J. Phys. Chem. Ref. Data 9, 561-658 (1980).
[CrossRef]

J. Phys. Chem. Solids (1)

M. Cardona, W. Paul, and H. Brooks, 'Dielectric constant of germanium and silicon as a function of volume,' J. Phys. Chem. Solids 8, 204-206 (1959).
[CrossRef]

J. Phys. Soc. Jpn. (1)

K. Sato, 'Contribution of surface electrons to the infrared optical properties of silicon,' J. Phys. Soc. Jpn. 27, 89-95 (1969).
[CrossRef]

J. Phys.: Condens. Matter (1)

D. Y. Smith, M. Inokuti, and W. Karstens, 'A generalized Cauchy dispersion formula and the refractivity of elemental semiconductors,' J. Phys.: Condens. Matter 13, 3883-3893 (2001).
[CrossRef]

Nature (1)

C. Zener, 'Remarkable optical properties of the alkali metals,' Nature 132, 968 (1933).
[CrossRef]

Phys. Essays (1)

D. Y. Smith, M. Inokuti, and W. Karstens, 'Photoresponse of condensed matter over the entire range of excitation energies: analysis of silicon,' Phys. Essays 13, 465-472 (2000).
[CrossRef]

Phys. Lett. (1)

T. Staflin, 'Infra-red absorption due to hole transitions involving the split-off valence-band in silicon,' Phys. Lett. 19, 84-85 (1965).
[CrossRef]

Phys. Rev. (6)

W. G. Spitzer and H. Y. Fan, 'Infrared absorption in n-type silicon,' Phys. Rev. 108, 268-271 (1957).
[CrossRef]

M. Becker and H. Y. Fan, 'Optical properties of semiconductors. III. Infra-red transmission of silicon,' Phys. Rev. 76, 1531-1532 (1949).
[CrossRef]

H. B. Briggs, 'Infrared absorption in silicon,' Phys. Rev. 77, 727-728 (1950).
[CrossRef]

A. H. Kahn, 'Theory of the infrared absorption of carriers in germanium and silicon,' Phys. Rev. 97, 1647-1652 (1955).
[CrossRef]

R. N. Dexter, H. J. Zeiger, and B. Lax, 'Cyclotron resonance experiments in silicon and germanium,' Phys. Rev. 104, 637-644 (1956).
[CrossRef]

W. G. Spitzer and H. Y. Fan, 'Determination of optical constants and carrier effective mass of semiconductors,' Phys. Rev. 106, 882-890 (1957).
[CrossRef]

Phys. Rev. B (4)

D. Y. Smith and E. Shiles, 'Finite-energy f-sum rules for valence electrons,' Phys. Rev. B 17, 4689-4694 (1978). See Eq. (14), but note that on the second line following Eq. (18), omegab should be replaced by ϵb.
[CrossRef]

M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, 'Superconvergence and sum rules for the optical constants,' Phys. Rev. B 6, 4502-4509 (1972).
[CrossRef]

E. Shiles, T. Sasaki, M. Inokuti, and D. Y. Smith, 'Self-consistency and sum-rule tests in the Kramers-Kronig analysis of optical data: applications to aluminum,' Phys. Rev. B 22, 1612-1628 (1980).
[CrossRef]

D. E. Aspnes and A. A. Studna, 'Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,' Phys. Rev. B 27, 985-1009 (1983).
[CrossRef]

Phys. Status Solidi A (2)

L. M. Lambert, 'Free carrier reflectivity in optically homogeneous silicon,' Phys. Status Solidi A 11, 461-467 (1972).
[CrossRef]

J. Humlícek and K. Vojtechovský, 'Infrared optical constants of intrinsic silicon,' Phys. Status Solidi A 92, 249-255 (1985).
[CrossRef]

Rep. Prog. Phys. (1)

H. Y. Fan, 'Infra-red absorption in semiconductors,' Rep. Prog. Phys. 19, 107-155 (1956).
[CrossRef]

Rev. Sci. Instrum. (1)

K.-L. Barth and F. Keilmann, 'Far-infrared ellipsometer,' Rev. Sci. Instrum. 64, 870-875 (1993).
[CrossRef]

Sov. Phys. Solid State (2)

V. S. Vavilov, 'The absorption of free charge carriers by infrared radiation in silicon,' Sov. Phys. Solid State 2, 346-349 (1960).

Yu. I. Ukhanov, 'Faraday effect in silicon in the infrared region of wavelengths,' Sov. Phys. Solid State 4, 2010-2013 (1963).

Other (18)

F. Seitz and N. G. Einspruch, Electronic Genie: The Tangled History of Silicon (U. Illinois Press, 1998).

H. Y Fan and M. Becker, 'Infra-red optical properties of silicon and germanium,' in Semiconducting Materials, H.K.Henisch, ed. (Butterworth, 1951), pp. 132-147.

M. L. Cohen and J. R. Chelikowsky, Electronic Structure and Optical Properties of Semiconductors, 2nd ed., Springer Series on Solid-State Science (Springer, 1989).
[CrossRef]

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1972).

F. Stern, 'Elementary theory of the optical properties of solids,' in Solid State Physics, Vol. 15, F.Seitz and D.Turnbull, eds. (Academic, 1963), pp. 299-408.
[CrossRef]

P. A Schumann, Jr., W. A. Keenan, A. H. Tong, H. H. Gegenwarth, and C. P. Schneider, 'Optical constants of silicon in the wavelength range 2.5 to 40 µm,' IBM Tech. Rep. TR 22.1008 (IBM, 1970).

M. Auslender and S. Hava, 'Doped n-type silicon,' in Handbook of Optical Constants of Solids, Vol. 3, E.D.Palik, ed. (Academic, 1997), pp. 155-186.

T. S. Moss, Optical Properties of Semiconductors (Butterworth, 1959).

D. L. Greenaway and G. Harbeke, Optical Properties and Band Structure of Semiconductors (Pergamon, 1968).

D. F. Edwards, 'Silicon (Si),' in Handbook of Optical Constants of Solids, Vol. 1, E.D.Palik, ed. (Academic, 1985), pp. 547-569.

D. E. Aspnes, 'Optical functions,' in Properties of Silicon (Inspec, Institution of Electrical Engineers London, 1988), Chap. 2, pp. 61-79.

D. E. Aspnes, 'Optical properties,' in Properties of Crystalline Silicon, R.Hull, ed. (Inspec, Institution of Electrical Engineers London, 1999), Chap. 12, pp. 679-695.

R. F. Potter, 'Germanium (Ge),' in Handbook of Optical Constants of Solids, Vol. 1, E.D.Palik, ed. (Academic, 1985), pp. 465-478.

R. Kingslake, Lens Design Fundamentals (Academic, 1978).

A. L. Cauchy, Mémoire sur la dispersion de la lumière (Calve, Prague, 1836).

H. A. Lorentz, The Theory of Electrons (Teubner, Leipzig, 1909; Stechert, New York, 1923).

A. Beer, Grundriss des Photometrischen Calcüles (Vieweg, Braunschweig, 1854).

F. Shimura, Oxygen in Silicon (Academic, 1994).

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Figures (6)

Fig. 1
Fig. 1

Rapid convergence of Eq. (3), the series expansion of the Kramers–Kronig relation for n ( ω ) , below the onset of band-to-band transitions in the elemental semiconductors, is illustrated by the observed refractive index of high-purity silicon and germanium. Silicon data were taken from the compilations of Li 36 and Edwards;[37] germanium data are from Potter.[47]

Fig. 2
Fig. 2

Refractive index for nominally pure silicon. The deviation from linearity at lower energy is due to the presence of free carriers. Data are from Loewenstein,[49] and Randall and Rawcliffe.[50]

Fig. 3
Fig. 3

Generalized Cauchy fits to the index data of Schumann et al.[27] The dashed line gives the low-frequency limit of the index associated with interband transitions.

Fig. 4
Fig. 4

Expanded generalized Cauchy fits to the index data of Schumann et al.[27] for N = 1.02 × 10 18 cm 3 . The difference between the observed index and the expected Cauchy equation, Eq. (8), is attributed to an oscillator-like dispersion associated with impurity absorption near 0.1 eV.

Fig. 5
Fig. 5

Generalized Cauchy fits to the index data of Auslender and Hava.[28] The dashed line gives the low-frequency limit of the index associated with interband transitions.

Fig. 6
Fig. 6

Generalized Cauchy fit to the index data of Auslender and Hava[28] for N = 1.0 × 10 16 cm 3 . Note the two points that are clearly transposed.

Tables (1)

Tables Icon

Table 1 Comparison of Free-carrier Densities as Determined by Resistivity Measurements N e x p and by the Generalized Cauchy Fit N c a l c a

Equations (11)

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

n ( ω ) 1 = 2 π P 0 ω κ ( ω ) ω 2 ω 2 d ω ,
n ( ω ) 1 = 2 π 0 ω l ω κ ( ω ) ω 2 ω 2 d ω + Δ ( ω ) + 2 π ω h ω κ ( ω ) ω 2 ω 2 d ω .
2 π ω h ω κ ( ω ) ω 2 ω 2 d ω = n 0 + n 2 ω 2 + n 4 ω 4 + , ω < ω h ,
n j = δ j , 0 + 2 π ω h κ ( ω ) ω j + 1 d ω , j = 0 , 2 , 4 , 6 , .
2 π 0 ω l ω κ ( ω ) ω 2 ω 2 d ω = n 2 ω 2 + n 4 ω 4 + , ω > ω l ,
n j = 2 π 0 ω l ω j 1 κ ( ω ) d ω , j = 2 , 4 , 6 , .
n ( ω ) = n 0 + n 2 ω 2 + n 4 ( ω 2 ) 2 + .
n ( ω ) = n 6 ω 6 + n 4 ω 4 + n 2 ω 2 + n 0 + n 2 ω 2 ,
0 E l E κ f c ( E ) d E = π 2 Ω p 2 4 n 0 = π 2 n 2 ,
Ω p 2 = 4 π N e 2 m * ,
N = n 0 n 2 2 π 2 e 2 m * .

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