Abstract

The optical constants of ion-beam-sputtered SiC films have been measured by ellipsometry in the 190 to 950nm range. The set of data has been extended both toward shorter and longer wavelengths with data in the literature, along with inter- and extrapolations, in order to obtain a self-consistent set of data by means of Kramers–Krönig analysis. All data correspond to films that were deposited by sputtering on nonheated substrates, and hence they are expected to be amorphous. A bandgap of 1.9eV for the films was fitted from the obtained optical constants. A good global accuracy of the data was estimated through the use of various sum rules. The consistent dataset includes the visible to the extreme ultraviolet (EUV); this large spectrum of characterization will enable the design of multilayer coatings that combine a high reflectance in parts of the EUV with desired performance at a secondary range, such as the visible. To our knowledge, this paper provides the first compilation of the optical constants of amorphous SiC films.

© 2011 Optical Society of America

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    [CrossRef]

2011 (1)

J. A. Guerra, L. Montañez, O. Erlenbach, G. Galvez, F. De Zela, A. Winnacker, and R. Weingärtner, “Determination of the optical bandgap and disorder energies of thin amorphous SiC and AIN films produced by radio frequency magnetron sputtering,” J. Phys. 274, 012113 (2011).
[CrossRef]

2009 (3)

R. Soufli, S. L. Baker, J. C. Robinson, T. J. McCarville, M. J. Pivovaroff, S. P. Hau-Riege, and R. Bionta, “Morphology, microstructure, stress and damage properties of thin film coatings for the LCLS x-ray mirrors,” Proc. SPIE 7361, 73610U (2009).
[CrossRef]

M. Fernández-Perea, M. Vidal-Dasilva, J. I. Larruquert, J. A. Aznárez, J. A. Méndez, E. Gullikson, A. Aquila, and R. Soufli, “Optical constants of evaporation-deposited silicon monoxide films in the 7.1–800 eV photon energy range,” J. Appl. Phys. 105, 113505 (2009).
[CrossRef]

M. Fernández-Perea, J. A. Méndez, José A. Aznárez, and Juan I. Larruquert, “In situ reflectance and optical constants of ion-beam-sputtered SiC films in the 58.4 to 149.2 nm region,” Appl. Opt. 48, 4698–4672 (2009).
[CrossRef] [PubMed]

2007 (2)

U. Schühle, H. Uhlig, W. Curdt, T. Feigl, A. Theissen, and L. Teriaca, “Thin silicon carbide coating of the primary mirror of VUV imaging instruments of solar orbiter,” in The Second Solar Orbiter Workshop, E.Marsch, K.Tsinganos, R.Marsden, and L.Conroy, eds., ESA SP-641 (ESA Publications Division, 2007), paper P83.

M. Fernández-Perea, J. I. Larruquert, J. A. Aznárez, J. A. Méndez, M. Vidal, E. Gullikson, A. Aquila, R. Soufli, and J. L. G. Fierro, “Optical constants of electron-beam evaporated boron films in the 6.8–900 eV photon energy range,” J. Opt. Soc. Am. A 24, 3800–3807 (2007).
[CrossRef]

2006 (1)

2005 (3)

T. Ejima, A. Yamazaki, T. Banse, K. Saito, Y. Kondo, S. Ichimaru, and H. Takenaka, “Aging and thermal stability of Mg/SiC and Mg/Y2O3 reflection multilayers in the 25–35 nm region,” Appl. Opt. 44, 5446–5453 (2005).
[CrossRef] [PubMed]

O. Stenzel, The Physics of Thin Film Optical Spectra: An Introduction (Springer-Verlag, 2005), p. 214.

R. Soufli, D. L. Windt, J. C. Robinson, E. A. Spiller, F. J. Dollar, A. L. Aquila, E. M. Gullikson, B. Kjornrattanawanich, J. F. Seely, and L. Golub, “Development and testing of EUV multilayer coatings for the Atmospheric Imaging Assembly instrument aboard the Solar Dynamics Observatory,” Proc. SPIE 5901, 59010M(2005).
[CrossRef]

2002 (2)

K. B. Sundaram, Z. Alizadeh, and L. Chow, “The effects of oxidation on the optical properties of amorphous SiC films,” Mater. Sci. Eng. B 90, 47–49 (2002).
[CrossRef]

J. I. Larruquert and R. A. M. Keski-Kuha, “Sub-quarterwave multilayer coatings with high reflectance in the extreme ultraviolet,” Appl. Opt. 41, 5398–5404 (2002).
[CrossRef] [PubMed]

2000 (1)

1999 (1)

1997 (1)

P. Musumeci, R. Reitano, L. Calcagno, F. Roccaforte, A. Makhtari, and M. G. Grimaldi, “Relaxation and crystallization of amorphous silicon carbide probed by optical measurements,” Philos. Mag. B 76, 323–333 (1997).
[CrossRef]

1996 (1)

G. E. Jellison, Jr., and F. A. Modine, “Parameterization of the optical functions of amorphous materials in the interband region,” Appl. Phys. Lett. 69, 371–373 (1996).
[CrossRef]

1995 (1)

S. Heckens and J. A. Woollam, “In-situ ellipsometry on sputtered dielectric and magneto-optic thin films,” Thin Solid Films 270, 65–68 (1995).
[CrossRef]

1993 (1)

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50–30000 eV, Z=1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

1988 (2)

1985 (1)

W. J. Choyke and E. D. Palik, “Silicon carbide (SiC),” in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985), pp. 587–596.

1983 (1)

R. Dutta, P. K. Banerjee, and S. S. Mitra, “Amorphous silicon-carbon-fluorine alloy films,” Phys. Rev. B 27, 5032–5038 (1983).
[CrossRef]

1982 (1)

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, and B. K. Fujikawa, “Low-energy x-ray interaction coefficients: photoabsorption, scattering, and reflection, E=100–2000 eV, Z=1–94,” At. Data Nucl. Data Tables 27, 1–144 (1982).
[CrossRef]

1981 (1)

M. M. Kelly, J. B. West, and D. E. Lloyd, “Reflectance of silicon carbide in the vacuum ultraviolet,” J. Phys. D 14, 401–404(1981).
[CrossRef]

1980 (1)

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 aluminium,” Phys. Rev. B 22, 1612–1628 (1980).
[CrossRef]

1979 (1)

H. Matsunami, H. Masahiro, and T. Tanaka, “Structures and physical properties of sputtered amorphous SiC films,” J. Electron. Mater. 8, 249–260 (1979).
[CrossRef]

1977 (1)

1974 (3)

M. L. Theye, “Optical properties of a-Ge, a-Si and a-III-V compounds,” in Amorphous and Liquid Semiconductors, Volume  1, J.Stuke and W.Brenig, eds. (Taylor & Francis, 1974). It contains part of the proceedings of the International Conference on Amorphous and Liquid Semiconductors held at Garmisch-Partenkirchen, Germany, in 1973.

M. Altarelli and D. Y. Smith, “Superconvergence and sum rules for the optical constants: physical meaning, comparison with experiment, and generalization,” Phys. Rev. B 9, 1290–1298(1974).
[CrossRef]

E. A. Fagen, “Optical and electrical properties of amorphous silicon carbide films,” in Amorphous and Liquid Semiconductors, J.Stuke and W.Brenig, eds. (Taylor & Francis, 1974), Vol.  1. It contains part of the proceedings of the International Conference on Amorphous and Liquid Semiconductors held at Garmisch-Partenkirchen, Germany, in 1973.

1972 (2)

D. T. Pierce and W. E. Spicer, “Electronic structure of amorphous Si from photoemission and optical studies,” Phys. Rev. B 5, 3017–3029 (1972).
[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]

1968 (1)

J. Tauc, “Optical properties and electronic structure of amorphous Ge and Si,” Mat. Res. Bull. 3, 37–46 (1968).
[CrossRef]

1966 (1)

J. Tauc, R. Grigorovici, and A. Vancu, “Optical properties and electronic structure of amorphous germanium,” Phys. Status Solidi 15, 627–637 (1966).
[CrossRef]

1948 (1)

S. Tolansky, Multiple-Beam Interferometry of Surfaces and Films (Oxford Univ. Press, 1948).

Alizadeh, Z.

K. B. Sundaram, Z. Alizadeh, and L. Chow, “The effects of oxidation on the optical properties of amorphous SiC films,” Mater. Sci. Eng. B 90, 47–49 (2002).
[CrossRef]

Altarelli, M.

M. Altarelli and D. Y. Smith, “Superconvergence and sum rules for the optical constants: physical meaning, comparison with experiment, and generalization,” Phys. Rev. B 9, 1290–1298(1974).
[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]

Aquila, A.

M. Fernández-Perea, M. Vidal-Dasilva, J. I. Larruquert, J. A. Aznárez, J. A. Méndez, E. Gullikson, A. Aquila, and R. Soufli, “Optical constants of evaporation-deposited silicon monoxide films in the 7.1–800 eV photon energy range,” J. Appl. Phys. 105, 113505 (2009).
[CrossRef]

M. Fernández-Perea, J. I. Larruquert, J. A. Aznárez, J. A. Méndez, M. Vidal, E. Gullikson, A. Aquila, R. Soufli, and J. L. G. Fierro, “Optical constants of electron-beam evaporated boron films in the 6.8–900 eV photon energy range,” J. Opt. Soc. Am. A 24, 3800–3807 (2007).
[CrossRef]

Aquila, A. L.

R. Soufli, D. L. Windt, J. C. Robinson, E. A. Spiller, F. J. Dollar, A. L. Aquila, E. M. Gullikson, B. Kjornrattanawanich, J. F. Seely, and L. Golub, “Development and testing of EUV multilayer coatings for the Atmospheric Imaging Assembly instrument aboard the Solar Dynamics Observatory,” Proc. SPIE 5901, 59010M(2005).
[CrossRef]

Aznárez, J. A.

M. Fernández-Perea, M. Vidal-Dasilva, J. I. Larruquert, J. A. Aznárez, J. A. Méndez, E. Gullikson, A. Aquila, and R. Soufli, “Optical constants of evaporation-deposited silicon monoxide films in the 7.1–800 eV photon energy range,” J. Appl. Phys. 105, 113505 (2009).
[CrossRef]

M. Fernández-Perea, J. I. Larruquert, J. A. Aznárez, J. A. Méndez, M. Vidal, E. Gullikson, A. Aquila, R. Soufli, and J. L. G. Fierro, “Optical constants of electron-beam evaporated boron films in the 6.8–900 eV photon energy range,” J. Opt. Soc. Am. A 24, 3800–3807 (2007).
[CrossRef]

Aznárez, José A.

Baker, S. L.

R. Soufli, S. L. Baker, J. C. Robinson, T. J. McCarville, M. J. Pivovaroff, S. P. Hau-Riege, and R. Bionta, “Morphology, microstructure, stress and damage properties of thin film coatings for the LCLS x-ray mirrors,” Proc. SPIE 7361, 73610U (2009).
[CrossRef]

Banerjee, P. K.

R. Dutta, P. K. Banerjee, and S. S. Mitra, “Amorphous silicon-carbon-fluorine alloy films,” Phys. Rev. B 27, 5032–5038 (1983).
[CrossRef]

Banse, T.

Bionta, R.

R. Soufli, S. L. Baker, J. C. Robinson, T. J. McCarville, M. J. Pivovaroff, S. P. Hau-Riege, and R. Bionta, “Morphology, microstructure, stress and damage properties of thin film coatings for the LCLS x-ray mirrors,” Proc. SPIE 7361, 73610U (2009).
[CrossRef]

Brandt, G. B.

Calcagno, L.

P. Musumeci, R. Reitano, L. Calcagno, F. Roccaforte, A. Makhtari, and M. G. Grimaldi, “Relaxation and crystallization of amorphous silicon carbide probed by optical measurements,” Philos. Mag. B 76, 323–333 (1997).
[CrossRef]

Chow, L.

K. B. Sundaram, Z. Alizadeh, and L. Chow, “The effects of oxidation on the optical properties of amorphous SiC films,” Mater. Sci. Eng. B 90, 47–49 (2002).
[CrossRef]

Choyke, W. J.

W. J. Choyke and E. D. Palik, “Silicon carbide (SiC),” in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985), pp. 587–596.

W. J. Choyke, W. D. Partlow, E. P. Supertzi, F. J. Venskytis, and G. B. Brandt, “Silicon-carbide diffraction grating for the vacuum ultraviolet: feasibility,” Appl. Opt. 16, 2013–2014 (1977).
[CrossRef] [PubMed]

Curdt, W.

U. Schühle, H. Uhlig, W. Curdt, T. Feigl, A. Theissen, and L. Teriaca, “Thin silicon carbide coating of the primary mirror of VUV imaging instruments of solar orbiter,” in The Second Solar Orbiter Workshop, E.Marsch, K.Tsinganos, R.Marsden, and L.Conroy, eds., ESA SP-641 (ESA Publications Division, 2007), paper P83.

Davis, J. C.

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50–30000 eV, Z=1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

De Zela, F.

J. A. Guerra, L. Montañez, O. Erlenbach, G. Galvez, F. De Zela, A. Winnacker, and R. Weingärtner, “Determination of the optical bandgap and disorder energies of thin amorphous SiC and AIN films produced by radio frequency magnetron sputtering,” J. Phys. 274, 012113 (2011).
[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]

Dollar, F. J.

R. Soufli, D. L. Windt, J. C. Robinson, E. A. Spiller, F. J. Dollar, A. L. Aquila, E. M. Gullikson, B. Kjornrattanawanich, J. F. Seely, and L. Golub, “Development and testing of EUV multilayer coatings for the Atmospheric Imaging Assembly instrument aboard the Solar Dynamics Observatory,” Proc. SPIE 5901, 59010M(2005).
[CrossRef]

Dutta, R.

R. Dutta, P. K. Banerjee, and S. S. Mitra, “Amorphous silicon-carbon-fluorine alloy films,” Phys. Rev. B 27, 5032–5038 (1983).
[CrossRef]

Ejima, T.

Erlenbach, O.

J. A. Guerra, L. Montañez, O. Erlenbach, G. Galvez, F. De Zela, A. Winnacker, and R. Weingärtner, “Determination of the optical bandgap and disorder energies of thin amorphous SiC and AIN films produced by radio frequency magnetron sputtering,” J. Phys. 274, 012113 (2011).
[CrossRef]

Fagen, E. A.

E. A. Fagen, “Optical and electrical properties of amorphous silicon carbide films,” in Amorphous and Liquid Semiconductors, J.Stuke and W.Brenig, eds. (Taylor & Francis, 1974), Vol.  1. It contains part of the proceedings of the International Conference on Amorphous and Liquid Semiconductors held at Garmisch-Partenkirchen, Germany, in 1973.

Feigl, T.

U. Schühle, H. Uhlig, W. Curdt, T. Feigl, A. Theissen, and L. Teriaca, “Thin silicon carbide coating of the primary mirror of VUV imaging instruments of solar orbiter,” in The Second Solar Orbiter Workshop, E.Marsch, K.Tsinganos, R.Marsden, and L.Conroy, eds., ESA SP-641 (ESA Publications Division, 2007), paper P83.

Fernández-Perea, M.

Fierro, J. L. G.

Fujikawa, B. K.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, and B. K. Fujikawa, “Low-energy x-ray interaction coefficients: photoabsorption, scattering, and reflection, E=100–2000 eV, Z=1–94,” At. Data Nucl. Data Tables 27, 1–144 (1982).
[CrossRef]

Galvez, G.

J. A. Guerra, L. Montañez, O. Erlenbach, G. Galvez, F. De Zela, A. Winnacker, and R. Weingärtner, “Determination of the optical bandgap and disorder energies of thin amorphous SiC and AIN films produced by radio frequency magnetron sputtering,” J. Phys. 274, 012113 (2011).
[CrossRef]

Golub, L.

R. Soufli, D. L. Windt, J. C. Robinson, E. A. Spiller, F. J. Dollar, A. L. Aquila, E. M. Gullikson, B. Kjornrattanawanich, J. F. Seely, and L. Golub, “Development and testing of EUV multilayer coatings for the Atmospheric Imaging Assembly instrument aboard the Solar Dynamics Observatory,” Proc. SPIE 5901, 59010M(2005).
[CrossRef]

Grigorovici, R.

J. Tauc, R. Grigorovici, and A. Vancu, “Optical properties and electronic structure of amorphous germanium,” Phys. Status Solidi 15, 627–637 (1966).
[CrossRef]

Grimaldi, M. G.

P. Musumeci, R. Reitano, L. Calcagno, F. Roccaforte, A. Makhtari, and M. G. Grimaldi, “Relaxation and crystallization of amorphous silicon carbide probed by optical measurements,” Philos. Mag. B 76, 323–333 (1997).
[CrossRef]

Guerra, J. A.

J. A. Guerra, L. Montañez, O. Erlenbach, G. Galvez, F. De Zela, A. Winnacker, and R. Weingärtner, “Determination of the optical bandgap and disorder energies of thin amorphous SiC and AIN films produced by radio frequency magnetron sputtering,” J. Phys. 274, 012113 (2011).
[CrossRef]

Gullikson, E.

M. Fernández-Perea, M. Vidal-Dasilva, J. I. Larruquert, J. A. Aznárez, J. A. Méndez, E. Gullikson, A. Aquila, and R. Soufli, “Optical constants of evaporation-deposited silicon monoxide films in the 7.1–800 eV photon energy range,” J. Appl. Phys. 105, 113505 (2009).
[CrossRef]

M. Fernández-Perea, J. I. Larruquert, J. A. Aznárez, J. A. Méndez, M. Vidal, E. Gullikson, A. Aquila, R. Soufli, and J. L. G. Fierro, “Optical constants of electron-beam evaporated boron films in the 6.8–900 eV photon energy range,” J. Opt. Soc. Am. A 24, 3800–3807 (2007).
[CrossRef]

Gullikson, E. M.

R. Soufli, D. L. Windt, J. C. Robinson, E. A. Spiller, F. J. Dollar, A. L. Aquila, E. M. Gullikson, B. Kjornrattanawanich, J. F. Seely, and L. Golub, “Development and testing of EUV multilayer coatings for the Atmospheric Imaging Assembly instrument aboard the Solar Dynamics Observatory,” Proc. SPIE 5901, 59010M(2005).
[CrossRef]

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50–30000 eV, Z=1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

Gum, J. S.

Hau-Riege, S. P.

R. Soufli, S. L. Baker, J. C. Robinson, T. J. McCarville, M. J. Pivovaroff, S. P. Hau-Riege, and R. Bionta, “Morphology, microstructure, stress and damage properties of thin film coatings for the LCLS x-ray mirrors,” Proc. SPIE 7361, 73610U (2009).
[CrossRef]

Heckens, S.

S. Heckens and J. A. Woollam, “In-situ ellipsometry on sputtered dielectric and magneto-optic thin films,” Thin Solid Films 270, 65–68 (1995).
[CrossRef]

Henke, B. L.

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50–30000 eV, Z=1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, and B. K. Fujikawa, “Low-energy x-ray interaction coefficients: photoabsorption, scattering, and reflection, E=100–2000 eV, Z=1–94,” At. Data Nucl. Data Tables 27, 1–144 (1982).
[CrossRef]

Herzig, H.

Ichimaru, S.

Inokuti, M.

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 aluminium,” Phys. Rev. B 22, 1612–1628 (1980).
[CrossRef]

Jellison, G. E.

G. E. Jellison, Jr., and F. A. Modine, “Parameterization of the optical functions of amorphous materials in the interband region,” Appl. Phys. Lett. 69, 371–373 (1996).
[CrossRef]

Kelly, M. M.

M. M. Kelly, J. B. West, and D. E. Lloyd, “Reflectance of silicon carbide in the vacuum ultraviolet,” J. Phys. D 14, 401–404(1981).
[CrossRef]

Keski-Kuha, R. A. M.

Kjornrattanawanich, B.

B. Kjornrattanawanich, D. L. Windt, J. F. Seely, and Yu. A. Uspenskii, “SiC/Tb and Si/Tb multilayer coatings for extreme ultraviolet solar imaging,” Appl. Opt. 45, 1765–1772 (2006).
[CrossRef] [PubMed]

R. Soufli, D. L. Windt, J. C. Robinson, E. A. Spiller, F. J. Dollar, A. L. Aquila, E. M. Gullikson, B. Kjornrattanawanich, J. F. Seely, and L. Golub, “Development and testing of EUV multilayer coatings for the Atmospheric Imaging Assembly instrument aboard the Solar Dynamics Observatory,” Proc. SPIE 5901, 59010M(2005).
[CrossRef]

Kondo, Y.

Kortright, J. B.

Larruquert, J. I.

Larruquert, Juan I.

Lee, P.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, and B. K. Fujikawa, “Low-energy x-ray interaction coefficients: photoabsorption, scattering, and reflection, E=100–2000 eV, Z=1–94,” At. Data Nucl. Data Tables 27, 1–144 (1982).
[CrossRef]

Lloyd, D. E.

M. M. Kelly, J. B. West, and D. E. Lloyd, “Reflectance of silicon carbide in the vacuum ultraviolet,” J. Phys. D 14, 401–404(1981).
[CrossRef]

Makhtari, A.

P. Musumeci, R. Reitano, L. Calcagno, F. Roccaforte, A. Makhtari, and M. G. Grimaldi, “Relaxation and crystallization of amorphous silicon carbide probed by optical measurements,” Philos. Mag. B 76, 323–333 (1997).
[CrossRef]

Masahiro, H.

H. Matsunami, H. Masahiro, and T. Tanaka, “Structures and physical properties of sputtered amorphous SiC films,” J. Electron. Mater. 8, 249–260 (1979).
[CrossRef]

Matsunami, H.

H. Matsunami, H. Masahiro, and T. Tanaka, “Structures and physical properties of sputtered amorphous SiC films,” J. Electron. Mater. 8, 249–260 (1979).
[CrossRef]

McCarville, T. J.

R. Soufli, S. L. Baker, J. C. Robinson, T. J. McCarville, M. J. Pivovaroff, S. P. Hau-Riege, and R. Bionta, “Morphology, microstructure, stress and damage properties of thin film coatings for the LCLS x-ray mirrors,” Proc. SPIE 7361, 73610U (2009).
[CrossRef]

Méndez, J. A.

Mitra, S. S.

R. Dutta, P. K. Banerjee, and S. S. Mitra, “Amorphous silicon-carbon-fluorine alloy films,” Phys. Rev. B 27, 5032–5038 (1983).
[CrossRef]

Modine, F. A.

G. E. Jellison, Jr., and F. A. Modine, “Parameterization of the optical functions of amorphous materials in the interband region,” Appl. Phys. Lett. 69, 371–373 (1996).
[CrossRef]

Montañez, L.

J. A. Guerra, L. Montañez, O. Erlenbach, G. Galvez, F. De Zela, A. Winnacker, and R. Weingärtner, “Determination of the optical bandgap and disorder energies of thin amorphous SiC and AIN films produced by radio frequency magnetron sputtering,” J. Phys. 274, 012113 (2011).
[CrossRef]

Musumeci, P.

P. Musumeci, R. Reitano, L. Calcagno, F. Roccaforte, A. Makhtari, and M. G. Grimaldi, “Relaxation and crystallization of amorphous silicon carbide probed by optical measurements,” Philos. Mag. B 76, 323–333 (1997).
[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]

Osantowski, J. F.

Palik, E. D.

W. J. Choyke and E. D. Palik, “Silicon carbide (SiC),” in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985), pp. 587–596.

Partlow, W. D.

Pierce, D. T.

D. T. Pierce and W. E. Spicer, “Electronic structure of amorphous Si from photoemission and optical studies,” Phys. Rev. B 5, 3017–3029 (1972).
[CrossRef]

Pivovaroff, M. J.

R. Soufli, S. L. Baker, J. C. Robinson, T. J. McCarville, M. J. Pivovaroff, S. P. Hau-Riege, and R. Bionta, “Morphology, microstructure, stress and damage properties of thin film coatings for the LCLS x-ray mirrors,” Proc. SPIE 7361, 73610U (2009).
[CrossRef]

Reitano, R.

P. Musumeci, R. Reitano, L. Calcagno, F. Roccaforte, A. Makhtari, and M. G. Grimaldi, “Relaxation and crystallization of amorphous silicon carbide probed by optical measurements,” Philos. Mag. B 76, 323–333 (1997).
[CrossRef]

Robinson, J. C.

R. Soufli, S. L. Baker, J. C. Robinson, T. J. McCarville, M. J. Pivovaroff, S. P. Hau-Riege, and R. Bionta, “Morphology, microstructure, stress and damage properties of thin film coatings for the LCLS x-ray mirrors,” Proc. SPIE 7361, 73610U (2009).
[CrossRef]

R. Soufli, D. L. Windt, J. C. Robinson, E. A. Spiller, F. J. Dollar, A. L. Aquila, E. M. Gullikson, B. Kjornrattanawanich, J. F. Seely, and L. Golub, “Development and testing of EUV multilayer coatings for the Atmospheric Imaging Assembly instrument aboard the Solar Dynamics Observatory,” Proc. SPIE 5901, 59010M(2005).
[CrossRef]

Roccaforte, F.

P. Musumeci, R. Reitano, L. Calcagno, F. Roccaforte, A. Makhtari, and M. G. Grimaldi, “Relaxation and crystallization of amorphous silicon carbide probed by optical measurements,” Philos. Mag. B 76, 323–333 (1997).
[CrossRef]

Saito, K.

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 aluminium,” Phys. Rev. B 22, 1612–1628 (1980).
[CrossRef]

Schühle, U.

U. Schühle, H. Uhlig, W. Curdt, T. Feigl, A. Theissen, and L. Teriaca, “Thin silicon carbide coating of the primary mirror of VUV imaging instruments of solar orbiter,” in The Second Solar Orbiter Workshop, E.Marsch, K.Tsinganos, R.Marsden, and L.Conroy, eds., ESA SP-641 (ESA Publications Division, 2007), paper P83.

Seely, J. F.

B. Kjornrattanawanich, D. L. Windt, J. F. Seely, and Yu. A. Uspenskii, “SiC/Tb and Si/Tb multilayer coatings for extreme ultraviolet solar imaging,” Appl. Opt. 45, 1765–1772 (2006).
[CrossRef] [PubMed]

R. Soufli, D. L. Windt, J. C. Robinson, E. A. Spiller, F. J. Dollar, A. L. Aquila, E. M. Gullikson, B. Kjornrattanawanich, J. F. Seely, and L. Golub, “Development and testing of EUV multilayer coatings for the Atmospheric Imaging Assembly instrument aboard the Solar Dynamics Observatory,” Proc. SPIE 5901, 59010M(2005).
[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 aluminium,” Phys. Rev. B 22, 1612–1628 (1980).
[CrossRef]

Shimabukuro, R. L.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, and B. K. Fujikawa, “Low-energy x-ray interaction coefficients: photoabsorption, scattering, and reflection, E=100–2000 eV, Z=1–94,” At. Data Nucl. Data Tables 27, 1–144 (1982).
[CrossRef]

Smith, D. Y.

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 aluminium,” Phys. Rev. B 22, 1612–1628 (1980).
[CrossRef]

M. Altarelli and D. Y. Smith, “Superconvergence and sum rules for the optical constants: physical meaning, comparison with experiment, and generalization,” Phys. Rev. B 9, 1290–1298(1974).
[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]

Soufli, R.

M. Fernández-Perea, M. Vidal-Dasilva, J. I. Larruquert, J. A. Aznárez, J. A. Méndez, E. Gullikson, A. Aquila, and R. Soufli, “Optical constants of evaporation-deposited silicon monoxide films in the 7.1–800 eV photon energy range,” J. Appl. Phys. 105, 113505 (2009).
[CrossRef]

R. Soufli, S. L. Baker, J. C. Robinson, T. J. McCarville, M. J. Pivovaroff, S. P. Hau-Riege, and R. Bionta, “Morphology, microstructure, stress and damage properties of thin film coatings for the LCLS x-ray mirrors,” Proc. SPIE 7361, 73610U (2009).
[CrossRef]

M. Fernández-Perea, J. I. Larruquert, J. A. Aznárez, J. A. Méndez, M. Vidal, E. Gullikson, A. Aquila, R. Soufli, and J. L. G. Fierro, “Optical constants of electron-beam evaporated boron films in the 6.8–900 eV photon energy range,” J. Opt. Soc. Am. A 24, 3800–3807 (2007).
[CrossRef]

R. Soufli, D. L. Windt, J. C. Robinson, E. A. Spiller, F. J. Dollar, A. L. Aquila, E. M. Gullikson, B. Kjornrattanawanich, J. F. Seely, and L. Golub, “Development and testing of EUV multilayer coatings for the Atmospheric Imaging Assembly instrument aboard the Solar Dynamics Observatory,” Proc. SPIE 5901, 59010M(2005).
[CrossRef]

Spicer, W. E.

D. T. Pierce and W. E. Spicer, “Electronic structure of amorphous Si from photoemission and optical studies,” Phys. Rev. B 5, 3017–3029 (1972).
[CrossRef]

Spiller, E. A.

R. Soufli, D. L. Windt, J. C. Robinson, E. A. Spiller, F. J. Dollar, A. L. Aquila, E. M. Gullikson, B. Kjornrattanawanich, J. F. Seely, and L. Golub, “Development and testing of EUV multilayer coatings for the Atmospheric Imaging Assembly instrument aboard the Solar Dynamics Observatory,” Proc. SPIE 5901, 59010M(2005).
[CrossRef]

Stenzel, O.

O. Stenzel, The Physics of Thin Film Optical Spectra: An Introduction (Springer-Verlag, 2005), p. 214.

Sundaram, K. B.

K. B. Sundaram, Z. Alizadeh, and L. Chow, “The effects of oxidation on the optical properties of amorphous SiC films,” Mater. Sci. Eng. B 90, 47–49 (2002).
[CrossRef]

Supertzi, E. P.

Takenaka, H.

Tanaka, T.

H. Matsunami, H. Masahiro, and T. Tanaka, “Structures and physical properties of sputtered amorphous SiC films,” J. Electron. Mater. 8, 249–260 (1979).
[CrossRef]

Tanaka, T. J.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, and B. K. Fujikawa, “Low-energy x-ray interaction coefficients: photoabsorption, scattering, and reflection, E=100–2000 eV, Z=1–94,” At. Data Nucl. Data Tables 27, 1–144 (1982).
[CrossRef]

Tauc, J.

J. Tauc, “Optical properties and electronic structure of amorphous Ge and Si,” Mat. Res. Bull. 3, 37–46 (1968).
[CrossRef]

J. Tauc, R. Grigorovici, and A. Vancu, “Optical properties and electronic structure of amorphous germanium,” Phys. Status Solidi 15, 627–637 (1966).
[CrossRef]

Teriaca, L.

U. Schühle, H. Uhlig, W. Curdt, T. Feigl, A. Theissen, and L. Teriaca, “Thin silicon carbide coating of the primary mirror of VUV imaging instruments of solar orbiter,” in The Second Solar Orbiter Workshop, E.Marsch, K.Tsinganos, R.Marsden, and L.Conroy, eds., ESA SP-641 (ESA Publications Division, 2007), paper P83.

Theissen, A.

U. Schühle, H. Uhlig, W. Curdt, T. Feigl, A. Theissen, and L. Teriaca, “Thin silicon carbide coating of the primary mirror of VUV imaging instruments of solar orbiter,” in The Second Solar Orbiter Workshop, E.Marsch, K.Tsinganos, R.Marsden, and L.Conroy, eds., ESA SP-641 (ESA Publications Division, 2007), paper P83.

Theye, M. L.

M. L. Theye, “Optical properties of a-Ge, a-Si and a-III-V compounds,” in Amorphous and Liquid Semiconductors, Volume  1, J.Stuke and W.Brenig, eds. (Taylor & Francis, 1974). It contains part of the proceedings of the International Conference on Amorphous and Liquid Semiconductors held at Garmisch-Partenkirchen, Germany, in 1973.

Toft, A. R.

Tolansky, S.

S. Tolansky, Multiple-Beam Interferometry of Surfaces and Films (Oxford Univ. Press, 1948).

Uhlig, H.

U. Schühle, H. Uhlig, W. Curdt, T. Feigl, A. Theissen, and L. Teriaca, “Thin silicon carbide coating of the primary mirror of VUV imaging instruments of solar orbiter,” in The Second Solar Orbiter Workshop, E.Marsch, K.Tsinganos, R.Marsden, and L.Conroy, eds., ESA SP-641 (ESA Publications Division, 2007), paper P83.

Uspenskii, Yu. A.

Vancu, A.

J. Tauc, R. Grigorovici, and A. Vancu, “Optical properties and electronic structure of amorphous germanium,” Phys. Status Solidi 15, 627–637 (1966).
[CrossRef]

Venskytis, F. J.

Vidal, M.

Vidal-Dasilva, M.

M. Fernández-Perea, M. Vidal-Dasilva, J. I. Larruquert, J. A. Aznárez, J. A. Méndez, E. Gullikson, A. Aquila, and R. Soufli, “Optical constants of evaporation-deposited silicon monoxide films in the 7.1–800 eV photon energy range,” J. Appl. Phys. 105, 113505 (2009).
[CrossRef]

Weingärtner, R.

J. A. Guerra, L. Montañez, O. Erlenbach, G. Galvez, F. De Zela, A. Winnacker, and R. Weingärtner, “Determination of the optical bandgap and disorder energies of thin amorphous SiC and AIN films produced by radio frequency magnetron sputtering,” J. Phys. 274, 012113 (2011).
[CrossRef]

West, J. B.

M. M. Kelly, J. B. West, and D. E. Lloyd, “Reflectance of silicon carbide in the vacuum ultraviolet,” J. Phys. D 14, 401–404(1981).
[CrossRef]

Windt, D. L.

B. Kjornrattanawanich, D. L. Windt, J. F. Seely, and Yu. A. Uspenskii, “SiC/Tb and Si/Tb multilayer coatings for extreme ultraviolet solar imaging,” Appl. Opt. 45, 1765–1772 (2006).
[CrossRef] [PubMed]

R. Soufli, D. L. Windt, J. C. Robinson, E. A. Spiller, F. J. Dollar, A. L. Aquila, E. M. Gullikson, B. Kjornrattanawanich, J. F. Seely, and L. Golub, “Development and testing of EUV multilayer coatings for the Atmospheric Imaging Assembly instrument aboard the Solar Dynamics Observatory,” Proc. SPIE 5901, 59010M(2005).
[CrossRef]

J. B. Kortright and D. L. Windt, “Amorphous silicon carbide coatings for EUV optics,” Appl. Opt. 27, 2841–2846 (1988).
[CrossRef] [PubMed]

Winnacker, A.

J. A. Guerra, L. Montañez, O. Erlenbach, G. Galvez, F. De Zela, A. Winnacker, and R. Weingärtner, “Determination of the optical bandgap and disorder energies of thin amorphous SiC and AIN films produced by radio frequency magnetron sputtering,” J. Phys. 274, 012113 (2011).
[CrossRef]

Woollam, J. A.

S. Heckens and J. A. Woollam, “In-situ ellipsometry on sputtered dielectric and magneto-optic thin films,” Thin Solid Films 270, 65–68 (1995).
[CrossRef]

Yamazaki, A.

Appl. Opt. (9)

W. J. Choyke, W. D. Partlow, E. P. Supertzi, F. J. Venskytis, and G. B. Brandt, “Silicon-carbide diffraction grating for the vacuum ultraviolet: feasibility,” Appl. Opt. 16, 2013–2014 (1977).
[CrossRef] [PubMed]

J. B. Kortright and D. L. Windt, “Amorphous silicon carbide coatings for EUV optics,” Appl. Opt. 27, 2841–2846 (1988).
[CrossRef] [PubMed]

J. I. Larruquert and R. A. M. Keski-Kuha, “Multilayer coatings with high reflectance in the EUV spectral region from 50 to 121.6 nm,” Appl. Opt. 38, 1231–1236 (1999).
[CrossRef]

J. I. Larruquert and R. A. M. Keski-Kuha, “Reflectance measurements and optical constants in the extreme ultraviolet for thin films of ion-beam-deposited SiC, Mo, Mg2Si, and InSb and of evaporated Cr,” Appl. Opt. 39, 2772–2781 (2000).
[CrossRef]

J. I. Larruquert and R. A. M. Keski-Kuha, “Sub-quarterwave multilayer coatings with high reflectance in the extreme ultraviolet,” Appl. Opt. 41, 5398–5404 (2002).
[CrossRef] [PubMed]

T. Ejima, A. Yamazaki, T. Banse, K. Saito, Y. Kondo, S. Ichimaru, and H. Takenaka, “Aging and thermal stability of Mg/SiC and Mg/Y2O3 reflection multilayers in the 25–35 nm region,” Appl. Opt. 44, 5446–5453 (2005).
[CrossRef] [PubMed]

B. Kjornrattanawanich, D. L. Windt, J. F. Seely, and Yu. A. Uspenskii, “SiC/Tb and Si/Tb multilayer coatings for extreme ultraviolet solar imaging,” Appl. Opt. 45, 1765–1772 (2006).
[CrossRef] [PubMed]

R. A. M. Keski-Kuha, J. F. Osantowski, H. Herzig, J. S. Gum, and A. R. Toft, “Normal incidence reflectance of ion beam deposited SiC films in the EUV,” Appl. Opt. 27, 2815–2816 (1988).
[CrossRef] [PubMed]

M. Fernández-Perea, J. A. Méndez, José A. Aznárez, and Juan I. Larruquert, “In situ reflectance and optical constants of ion-beam-sputtered SiC films in the 58.4 to 149.2 nm region,” Appl. Opt. 48, 4698–4672 (2009).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

G. E. Jellison, Jr., and F. A. Modine, “Parameterization of the optical functions of amorphous materials in the interband region,” Appl. Phys. Lett. 69, 371–373 (1996).
[CrossRef]

At. Data Nucl. Data Tables (2)

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, and B. K. Fujikawa, “Low-energy x-ray interaction coefficients: photoabsorption, scattering, and reflection, E=100–2000 eV, Z=1–94,” At. Data Nucl. Data Tables 27, 1–144 (1982).
[CrossRef]

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50–30000 eV, Z=1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

J. Appl. Phys. (1)

M. Fernández-Perea, M. Vidal-Dasilva, J. I. Larruquert, J. A. Aznárez, J. A. Méndez, E. Gullikson, A. Aquila, and R. Soufli, “Optical constants of evaporation-deposited silicon monoxide films in the 7.1–800 eV photon energy range,” J. Appl. Phys. 105, 113505 (2009).
[CrossRef]

J. Electron. Mater. (1)

H. Matsunami, H. Masahiro, and T. Tanaka, “Structures and physical properties of sputtered amorphous SiC films,” J. Electron. Mater. 8, 249–260 (1979).
[CrossRef]

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

J. Phys. (1)

J. A. Guerra, L. Montañez, O. Erlenbach, G. Galvez, F. De Zela, A. Winnacker, and R. Weingärtner, “Determination of the optical bandgap and disorder energies of thin amorphous SiC and AIN films produced by radio frequency magnetron sputtering,” J. Phys. 274, 012113 (2011).
[CrossRef]

J. Phys. D (1)

M. M. Kelly, J. B. West, and D. E. Lloyd, “Reflectance of silicon carbide in the vacuum ultraviolet,” J. Phys. D 14, 401–404(1981).
[CrossRef]

Mat. Res. Bull. (1)

J. Tauc, “Optical properties and electronic structure of amorphous Ge and Si,” Mat. Res. Bull. 3, 37–46 (1968).
[CrossRef]

Mater. Sci. Eng. B (1)

K. B. Sundaram, Z. Alizadeh, and L. Chow, “The effects of oxidation on the optical properties of amorphous SiC films,” Mater. Sci. Eng. B 90, 47–49 (2002).
[CrossRef]

Philos. Mag. B (1)

P. Musumeci, R. Reitano, L. Calcagno, F. Roccaforte, A. Makhtari, and M. G. Grimaldi, “Relaxation and crystallization of amorphous silicon carbide probed by optical measurements,” Philos. Mag. B 76, 323–333 (1997).
[CrossRef]

Phys. Rev. B (5)

R. Dutta, P. K. Banerjee, and S. S. Mitra, “Amorphous silicon-carbon-fluorine alloy films,” Phys. Rev. B 27, 5032–5038 (1983).
[CrossRef]

D. T. Pierce and W. E. Spicer, “Electronic structure of amorphous Si from photoemission and optical studies,” Phys. Rev. B 5, 3017–3029 (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 aluminium,” Phys. Rev. B 22, 1612–1628 (1980).
[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]

M. Altarelli and D. Y. Smith, “Superconvergence and sum rules for the optical constants: physical meaning, comparison with experiment, and generalization,” Phys. Rev. B 9, 1290–1298(1974).
[CrossRef]

Phys. Status Solidi (1)

J. Tauc, R. Grigorovici, and A. Vancu, “Optical properties and electronic structure of amorphous germanium,” Phys. Status Solidi 15, 627–637 (1966).
[CrossRef]

Proc. SPIE (2)

R. Soufli, S. L. Baker, J. C. Robinson, T. J. McCarville, M. J. Pivovaroff, S. P. Hau-Riege, and R. Bionta, “Morphology, microstructure, stress and damage properties of thin film coatings for the LCLS x-ray mirrors,” Proc. SPIE 7361, 73610U (2009).
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R. Soufli, D. L. Windt, J. C. Robinson, E. A. Spiller, F. J. Dollar, A. L. Aquila, E. M. Gullikson, B. Kjornrattanawanich, J. F. Seely, and L. Golub, “Development and testing of EUV multilayer coatings for the Atmospheric Imaging Assembly instrument aboard the Solar Dynamics Observatory,” Proc. SPIE 5901, 59010M(2005).
[CrossRef]

Thin Solid Films (1)

S. Heckens and J. A. Woollam, “In-situ ellipsometry on sputtered dielectric and magneto-optic thin films,” Thin Solid Films 270, 65–68 (1995).
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Other (9)

S. Tolansky, Multiple-Beam Interferometry of Surfaces and Films (Oxford Univ. Press, 1948).

E. A. Fagen, “Optical and electrical properties of amorphous silicon carbide films,” in Amorphous and Liquid Semiconductors, J.Stuke and W.Brenig, eds. (Taylor & Francis, 1974), Vol.  1. It contains part of the proceedings of the International Conference on Amorphous and Liquid Semiconductors held at Garmisch-Partenkirchen, Germany, in 1973.

http://henke.lbl.gov/optical_constants/.

The data are available on request at the following e-mail address: larruquert@io.cfmac.csic.es.

U. Schühle, H. Uhlig, W. Curdt, T. Feigl, A. Theissen, and L. Teriaca, “Thin silicon carbide coating of the primary mirror of VUV imaging instruments of solar orbiter,” in The Second Solar Orbiter Workshop, E.Marsch, K.Tsinganos, R.Marsden, and L.Conroy, eds., ESA SP-641 (ESA Publications Division, 2007), paper P83.

W. J. Choyke and E. D. Palik, “Silicon carbide (SiC),” in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985), pp. 587–596.

Downloaded from the following web of Physical Reference Data, Physics Laboratory at NIST: http://physics.nist.gov/PhysRefData/FFast/html/form.html.

M. L. Theye, “Optical properties of a-Ge, a-Si and a-III-V compounds,” in Amorphous and Liquid Semiconductors, Volume  1, J.Stuke and W.Brenig, eds. (Taylor & Francis, 1974). It contains part of the proceedings of the International Conference on Amorphous and Liquid Semiconductors held at Garmisch-Partenkirchen, Germany, in 1973.

O. Stenzel, The Physics of Thin Film Optical Spectra: An Introduction (Springer-Verlag, 2005), p. 214.

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Figures (5)

Fig. 1
Fig. 1

Ellipsometry parameters tan ψ and cos Δ , both experimental and fitted, as a function of wavelength.

Fig. 2
Fig. 2

Optical constants n and k obtained from ellipsometry measurements as a function of wavelength. The curve thickness stands for the error-bar length.

Fig. 3
Fig. 3

(a) Obtained optical constants n and (b) the logarithm of k versus energy (a) and its logarithm (b) compared with the literature data of Fagen [14], Matsunami et al. [15], Dutta et al. [16], Heckens and Woollam [17], Sundaram et al. [18], Guerra et al. [19], and Musumeci et al. [20].

Fig. 4
Fig. 4

(a) Self-consistent set of SiC optical constants n and (b) the logarithm of k versus the logarithm of photon energy in a spectral range including from the reststrahlen band to the Si L 2 , 3 edge.

Fig. 5
Fig. 5

Tauc plot for amorphous SiC; the bandgap is obtained as the abscissa intercept of the linear extrapolation of α E .

Equations (6)

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n ( E ) 1 = 2 π P 0 E k ( E ) E 2 E 2 d E ,
k ( E ) = 2 E π P 0 [ n ( E ) 1 ] E 2 E 2 d E ,
α E ( E E G ) ,
n eff ( E ) = 4 ε 0 m π N mol e 2 2 0 E E k ( E ) d E ,
0 [ n ( E ) 1 ] d E = 0 ,
ζ = 0 [ n ( E ) 1 ] d E 0 | n ( E ) 1 | d E .

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