Abstract

A dispersion model describing two-phonon absorption is developed using several simplifications of the quasiparticle approach. The dielectric response is constructed from absorption bands corresponding to individual additive and subtractive combinations of phonon branches. The model also includes thermal effects, changes of the transition strength with temperature, originating in Bose-Einstein statistics, and the shift of phonon frequencies accompanying thermal expansion. The model is applied to the analysis of experimental data measured in the IR range on crystalline silicon. The modeled spectral dependencies of optical constants are capable of describing all features in the transmittance spectra 70–1000 cm−1 observable at 300 K for float-zone silicon. The phonon frequencies in the points of symmetry are obtained independently in good agreement with ab initio calculations. The model of thermal effects is verified using ellipsometric measurements 300–1000 cm−1 in the temperature range of 300–500 K. The agreement between the modeled and experimental data is good, except for the spectral range 750–850 cm−1, in which a better agreement at temperatures above 300 K would require including the three-phonon absorption. The analysis provides a reliable value of the thermal coefficient describing the phonon frequency shift and proves that changes of structure broadening with temperature are negligible within the temperature range of 300–500 K.

© 2014 Optical Society of America

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    [CrossRef]
  4. D. A. Neumayer and E. Cartier, “Materials characterization of ZrO2–SiO2 and HfO2–SiO2 binary oxides deposited by chemical solution deposition,” J. Appl. Phys.90, 1801–1808 (2001).
    [CrossRef]
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    [CrossRef]
  6. F. L. Martínez, M. Toledano-Luque, J. J. Gandía, J. Cárabe, W. Bohne, J. Röhrich, E. Strub, and I. Mártil, “Optical properties and structure of HfO2 thin films grown by high pressure reactive sputtering,” J. Phys. D Appl. Phys.40, 5256–5265 (2007).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2013 (1)

D. Franta, D. Nečas, and L. Zajíčková, “Application of Thomas–Reiche–Kuhn sum rule to construction of advanced dispersion models,” Thin Solid Films534, 432–441 (2013).
[CrossRef]

2011 (2)

O. Stenzel, S. Wilbrandt, S. Yulin, N. Kaiser, M. Held, A. Tünnermann, J. Biskupek, and U. Kaiser, “Plasma ion assisted deposition of hafnium dioxide using argon and xenon as process gases,” Opt. Mater. Express1, 278 (2011).
[CrossRef]

L. Zajíčková, D. Franta, D. Nečas, V. Buršíková, M. Muresan, V. Peřina, and C. Cobet, “Dielectric response and structure of amorphous hydrogenated carbon films with nitrogen admixture,” Thin Solid Films519, 4299–4308 (2011).
[CrossRef]

2009 (1)

2007 (1)

F. L. Martínez, M. Toledano-Luque, J. J. Gandía, J. Cárabe, W. Bohne, J. Röhrich, E. Strub, and I. Mártil, “Optical properties and structure of HfO2 thin films grown by high pressure reactive sputtering,” J. Phys. D Appl. Phys.40, 5256–5265 (2007).
[CrossRef]

2006 (1)

M. Modreanu, J. Sancho-Parramon, O. Durand, B. Servet, M. Stchakovsky, C. Eypert, C. Naudin, A. Knowles, F. Bridou, and M.-F. Ravet, “Investigation of thermal annealing effects on microstructural and optical properties of HfO2 thin films,” Appl. Surf. Sci.253, 328–334 (2006).
[CrossRef]

2002 (1)

A. Rogalski and K. Chrzanowski, “Infrared devices and techniques,” Opto-Electron. Rev.10, 111–136 (2002).

2001 (1)

D. A. Neumayer and E. Cartier, “Materials characterization of ZrO2–SiO2 and HfO2–SiO2 binary oxides deposited by chemical solution deposition,” J. Appl. Phys.90, 1801–1808 (2001).
[CrossRef]

1998 (1)

M. E. Thomas, S. K. Andersson, R. M. Sova, and R. I. Joseph, “Frequency and temperature dependence of the refractive index of sapphire,” Infrared Phys. Technol.39, 235–249 (1998).
[CrossRef]

1994 (1)

S. Wei and M. Y. Chou, “Phonon dispersions of silicon and germanium from first-principles calculations,” Phys. Rev. B50, 2221–2226 (1994).
[CrossRef]

1992 (1)

C. C. Kim, J. W. Garland, H. Abad, and P. M. Raccah, “Modeling the optical dielectric function of semiconductors: extension of the critical-poit parabolic-band approximation,” Phys. Rev. B45, 11749–11767 (1992).
[CrossRef]

1959 (1)

F. A. Johnson, “Lattice absorption bands in silicon,” Proc. Phys. Soc.73, 265 (1959).
[CrossRef]

1955 (1)

M. Lax and E. Burstein, “Infrared lattice absorption in ionic and homopolar crystals,” Phys. Rev.97, 39–52 (1955).
[CrossRef]

1953 (1)

L. Van Hove, “The occurence of singularities in the elastic frequency distribution of a crystal,” Phys. Rev.89, 1189–1193 (1953).
[CrossRef]

1918 (1)

J. Czochralski, “Ein neues Verfahren zur Messung des. Kristallisationsgeschwindigkeit der Metalle,” Z. Phys. Chem.92, 219–221 (1918).

Abad, H.

C. C. Kim, J. W. Garland, H. Abad, and P. M. Raccah, “Modeling the optical dielectric function of semiconductors: extension of the critical-poit parabolic-band approximation,” Phys. Rev. B45, 11749–11767 (1992).
[CrossRef]

Andersson, S. K.

M. E. Thomas, S. K. Andersson, R. M. Sova, and R. I. Joseph, “Frequency and temperature dependence of the refractive index of sapphire,” Infrared Phys. Technol.39, 235–249 (1998).
[CrossRef]

Balkanski, M.

M. Balkanski, “Photon-phonon interactions in solids,” in “Optical properties of solids,”, F. Abelès, ed. (North–Holland, 1972), pp. 529–651.

Biskupek, J.

Bohne, W.

F. L. Martínez, M. Toledano-Luque, J. J. Gandía, J. Cárabe, W. Bohne, J. Röhrich, E. Strub, and I. Mártil, “Optical properties and structure of HfO2 thin films grown by high pressure reactive sputtering,” J. Phys. D Appl. Phys.40, 5256–5265 (2007).
[CrossRef]

Bridou, F.

M. Modreanu, J. Sancho-Parramon, O. Durand, B. Servet, M. Stchakovsky, C. Eypert, C. Naudin, A. Knowles, F. Bridou, and M.-F. Ravet, “Investigation of thermal annealing effects on microstructural and optical properties of HfO2 thin films,” Appl. Surf. Sci.253, 328–334 (2006).
[CrossRef]

Buršíková, V.

L. Zajíčková, D. Franta, D. Nečas, V. Buršíková, M. Muresan, V. Peřina, and C. Cobet, “Dielectric response and structure of amorphous hydrogenated carbon films with nitrogen admixture,” Thin Solid Films519, 4299–4308 (2011).
[CrossRef]

Burstein, E.

M. Lax and E. Burstein, “Infrared lattice absorption in ionic and homopolar crystals,” Phys. Rev.97, 39–52 (1955).
[CrossRef]

Cárabe, J.

F. L. Martínez, M. Toledano-Luque, J. J. Gandía, J. Cárabe, W. Bohne, J. Röhrich, E. Strub, and I. Mártil, “Optical properties and structure of HfO2 thin films grown by high pressure reactive sputtering,” J. Phys. D Appl. Phys.40, 5256–5265 (2007).
[CrossRef]

Cardona, M.

P. Y. Yu and M. Cardona, Fundamentals of Semiconductors (Springer, 2001).

Cartier, E.

D. A. Neumayer and E. Cartier, “Materials characterization of ZrO2–SiO2 and HfO2–SiO2 binary oxides deposited by chemical solution deposition,” J. Appl. Phys.90, 1801–1808 (2001).
[CrossRef]

Choo, H.

Chou, M. Y.

S. Wei and M. Y. Chou, “Phonon dispersions of silicon and germanium from first-principles calculations,” Phys. Rev. B50, 2221–2226 (1994).
[CrossRef]

Chrzanowski, K.

A. Rogalski and K. Chrzanowski, “Infrared devices and techniques,” Opto-Electron. Rev.10, 111–136 (2002).

Clerjaud, B.

B. Pajot and B. Clerjaud, Optical Absorption of Impurities and Defects in Semiconducting Crystals, Springer Series in Solid-State Sciences 169 (Springer–Verlag, 2013).
[CrossRef]

Cobet, C.

L. Zajíčková, D. Franta, D. Nečas, V. Buršíková, M. Muresan, V. Peřina, and C. Cobet, “Dielectric response and structure of amorphous hydrogenated carbon films with nitrogen admixture,” Thin Solid Films519, 4299–4308 (2011).
[CrossRef]

Czochralski, J.

J. Czochralski, “Ein neues Verfahren zur Messung des. Kristallisationsgeschwindigkeit der Metalle,” Z. Phys. Chem.92, 219–221 (1918).

Deen, C. P.

Durand, O.

M. Modreanu, J. Sancho-Parramon, O. Durand, B. Servet, M. Stchakovsky, C. Eypert, C. Naudin, A. Knowles, F. Bridou, and M.-F. Ravet, “Investigation of thermal annealing effects on microstructural and optical properties of HfO2 thin films,” Appl. Surf. Sci.253, 328–334 (2006).
[CrossRef]

Eypert, C.

M. Modreanu, J. Sancho-Parramon, O. Durand, B. Servet, M. Stchakovsky, C. Eypert, C. Naudin, A. Knowles, F. Bridou, and M.-F. Ravet, “Investigation of thermal annealing effects on microstructural and optical properties of HfO2 thin films,” Appl. Surf. Sci.253, 328–334 (2006).
[CrossRef]

Franta, D.

D. Franta, D. Nečas, and L. Zajíčková, “Application of Thomas–Reiche–Kuhn sum rule to construction of advanced dispersion models,” Thin Solid Films534, 432–441 (2013).
[CrossRef]

L. Zajíčková, D. Franta, D. Nečas, V. Buršíková, M. Muresan, V. Peřina, and C. Cobet, “Dielectric response and structure of amorphous hydrogenated carbon films with nitrogen admixture,” Thin Solid Films519, 4299–4308 (2011).
[CrossRef]

D. Franta, D. Nečas, L. Zajíčková, and I. Ohlídal, “Utilization of the sum rule for construction of advanced dispersion model of crystalline silicon containing interstitial oxygen,” Thin Solid Films (2014).
[CrossRef]

D. Franta, D. Nečas, L. Zajíčková, and I. Ohlídal, “Broadening of dielectric response and sum rule conservation,” Thin Solid Films (2014).
[CrossRef]

Gandía, J. J.

F. L. Martínez, M. Toledano-Luque, J. J. Gandía, J. Cárabe, W. Bohne, J. Röhrich, E. Strub, and I. Mártil, “Optical properties and structure of HfO2 thin films grown by high pressure reactive sputtering,” J. Phys. D Appl. Phys.40, 5256–5265 (2007).
[CrossRef]

Garland, J. W.

C. C. Kim, J. W. Garland, H. Abad, and P. M. Raccah, “Modeling the optical dielectric function of semiconductors: extension of the critical-poit parabolic-band approximation,” Phys. Rev. B45, 11749–11767 (1992).
[CrossRef]

Held, M.

Jaffe, D. T.

Johnson, F. A.

F. A. Johnson, “Lattice absorption bands in silicon,” Proc. Phys. Soc.73, 265 (1959).
[CrossRef]

Joseph, R. I.

M. E. Thomas, S. K. Andersson, R. M. Sova, and R. I. Joseph, “Frequency and temperature dependence of the refractive index of sapphire,” Infrared Phys. Technol.39, 235–249 (1998).
[CrossRef]

Kaiser, N.

Kaiser, U.

Kim, C. C.

C. C. Kim, J. W. Garland, H. Abad, and P. M. Raccah, “Modeling the optical dielectric function of semiconductors: extension of the critical-poit parabolic-band approximation,” Phys. Rev. B45, 11749–11767 (1992).
[CrossRef]

Knowles, A.

M. Modreanu, J. Sancho-Parramon, O. Durand, B. Servet, M. Stchakovsky, C. Eypert, C. Naudin, A. Knowles, F. Bridou, and M.-F. Ravet, “Investigation of thermal annealing effects on microstructural and optical properties of HfO2 thin films,” Appl. Surf. Sci.253, 328–334 (2006).
[CrossRef]

Lax, M.

M. Lax and E. Burstein, “Infrared lattice absorption in ionic and homopolar crystals,” Phys. Rev.97, 39–52 (1955).
[CrossRef]

Ling, H.

Mar, D. J.

Marsh, J. P.

Mártil, I.

F. L. Martínez, M. Toledano-Luque, J. J. Gandía, J. Cárabe, W. Bohne, J. Röhrich, E. Strub, and I. Mártil, “Optical properties and structure of HfO2 thin films grown by high pressure reactive sputtering,” J. Phys. D Appl. Phys.40, 5256–5265 (2007).
[CrossRef]

Martínez, F. L.

F. L. Martínez, M. Toledano-Luque, J. J. Gandía, J. Cárabe, W. Bohne, J. Röhrich, E. Strub, and I. Mártil, “Optical properties and structure of HfO2 thin films grown by high pressure reactive sputtering,” J. Phys. D Appl. Phys.40, 5256–5265 (2007).
[CrossRef]

Modreanu, M.

M. Modreanu, J. Sancho-Parramon, O. Durand, B. Servet, M. Stchakovsky, C. Eypert, C. Naudin, A. Knowles, F. Bridou, and M.-F. Ravet, “Investigation of thermal annealing effects on microstructural and optical properties of HfO2 thin films,” Appl. Surf. Sci.253, 328–334 (2006).
[CrossRef]

Muresan, M.

L. Zajíčková, D. Franta, D. Nečas, V. Buršíková, M. Muresan, V. Peřina, and C. Cobet, “Dielectric response and structure of amorphous hydrogenated carbon films with nitrogen admixture,” Thin Solid Films519, 4299–4308 (2011).
[CrossRef]

Naudin, C.

M. Modreanu, J. Sancho-Parramon, O. Durand, B. Servet, M. Stchakovsky, C. Eypert, C. Naudin, A. Knowles, F. Bridou, and M.-F. Ravet, “Investigation of thermal annealing effects on microstructural and optical properties of HfO2 thin films,” Appl. Surf. Sci.253, 328–334 (2006).
[CrossRef]

Necas, D.

D. Franta, D. Nečas, and L. Zajíčková, “Application of Thomas–Reiche–Kuhn sum rule to construction of advanced dispersion models,” Thin Solid Films534, 432–441 (2013).
[CrossRef]

L. Zajíčková, D. Franta, D. Nečas, V. Buršíková, M. Muresan, V. Peřina, and C. Cobet, “Dielectric response and structure of amorphous hydrogenated carbon films with nitrogen admixture,” Thin Solid Films519, 4299–4308 (2011).
[CrossRef]

D. Franta, D. Nečas, L. Zajíčková, and I. Ohlídal, “Utilization of the sum rule for construction of advanced dispersion model of crystalline silicon containing interstitial oxygen,” Thin Solid Films (2014).
[CrossRef]

D. Franta, D. Nečas, L. Zajíčková, and I. Ohlídal, “Broadening of dielectric response and sum rule conservation,” Thin Solid Films (2014).
[CrossRef]

Neumayer, D. A.

D. A. Neumayer and E. Cartier, “Materials characterization of ZrO2–SiO2 and HfO2–SiO2 binary oxides deposited by chemical solution deposition,” J. Appl. Phys.90, 1801–1808 (2001).
[CrossRef]

Ohlídal, I.

D. Franta, D. Nečas, L. Zajíčková, and I. Ohlídal, “Broadening of dielectric response and sum rule conservation,” Thin Solid Films (2014).
[CrossRef]

D. Franta, D. Nečas, L. Zajíčková, and I. Ohlídal, “Utilization of the sum rule for construction of advanced dispersion model of crystalline silicon containing interstitial oxygen,” Thin Solid Films (2014).
[CrossRef]

Pajot, B.

B. Pajot and B. Clerjaud, Optical Absorption of Impurities and Defects in Semiconducting Crystals, Springer Series in Solid-State Sciences 169 (Springer–Verlag, 2013).
[CrossRef]

Perina, V.

L. Zajíčková, D. Franta, D. Nečas, V. Buršíková, M. Muresan, V. Peřina, and C. Cobet, “Dielectric response and structure of amorphous hydrogenated carbon films with nitrogen admixture,” Thin Solid Films519, 4299–4308 (2011).
[CrossRef]

Raccah, P. M.

C. C. Kim, J. W. Garland, H. Abad, and P. M. Raccah, “Modeling the optical dielectric function of semiconductors: extension of the critical-poit parabolic-band approximation,” Phys. Rev. B45, 11749–11767 (1992).
[CrossRef]

Ravet, M.-F.

M. Modreanu, J. Sancho-Parramon, O. Durand, B. Servet, M. Stchakovsky, C. Eypert, C. Naudin, A. Knowles, F. Bridou, and M.-F. Ravet, “Investigation of thermal annealing effects on microstructural and optical properties of HfO2 thin films,” Appl. Surf. Sci.253, 328–334 (2006).
[CrossRef]

Rogalski, A.

A. Rogalski and K. Chrzanowski, “Infrared devices and techniques,” Opto-Electron. Rev.10, 111–136 (2002).

Röhrich, J.

F. L. Martínez, M. Toledano-Luque, J. J. Gandía, J. Cárabe, W. Bohne, J. Röhrich, E. Strub, and I. Mártil, “Optical properties and structure of HfO2 thin films grown by high pressure reactive sputtering,” J. Phys. D Appl. Phys.40, 5256–5265 (2007).
[CrossRef]

Sancho-Parramon, J.

M. Modreanu, J. Sancho-Parramon, O. Durand, B. Servet, M. Stchakovsky, C. Eypert, C. Naudin, A. Knowles, F. Bridou, and M.-F. Ravet, “Investigation of thermal annealing effects on microstructural and optical properties of HfO2 thin films,” Appl. Surf. Sci.253, 328–334 (2006).
[CrossRef]

Servet, B.

M. Modreanu, J. Sancho-Parramon, O. Durand, B. Servet, M. Stchakovsky, C. Eypert, C. Naudin, A. Knowles, F. Bridou, and M.-F. Ravet, “Investigation of thermal annealing effects on microstructural and optical properties of HfO2 thin films,” Appl. Surf. Sci.253, 328–334 (2006).
[CrossRef]

Sova, R. M.

M. E. Thomas, S. K. Andersson, R. M. Sova, and R. I. Joseph, “Frequency and temperature dependence of the refractive index of sapphire,” Infrared Phys. Technol.39, 235–249 (1998).
[CrossRef]

Stchakovsky, M.

M. Modreanu, J. Sancho-Parramon, O. Durand, B. Servet, M. Stchakovsky, C. Eypert, C. Naudin, A. Knowles, F. Bridou, and M.-F. Ravet, “Investigation of thermal annealing effects on microstructural and optical properties of HfO2 thin films,” Appl. Surf. Sci.253, 328–334 (2006).
[CrossRef]

Stenzel, O.

Strub, E.

F. L. Martínez, M. Toledano-Luque, J. J. Gandía, J. Cárabe, W. Bohne, J. Röhrich, E. Strub, and I. Mártil, “Optical properties and structure of HfO2 thin films grown by high pressure reactive sputtering,” J. Phys. D Appl. Phys.40, 5256–5265 (2007).
[CrossRef]

Thomas, M. E.

M. E. Thomas, S. K. Andersson, R. M. Sova, and R. I. Joseph, “Frequency and temperature dependence of the refractive index of sapphire,” Infrared Phys. Technol.39, 235–249 (1998).
[CrossRef]

Toledano-Luque, M.

F. L. Martínez, M. Toledano-Luque, J. J. Gandía, J. Cárabe, W. Bohne, J. Röhrich, E. Strub, and I. Mártil, “Optical properties and structure of HfO2 thin films grown by high pressure reactive sputtering,” J. Phys. D Appl. Phys.40, 5256–5265 (2007).
[CrossRef]

Tünnermann, A.

Van Hove, L.

L. Van Hove, “The occurence of singularities in the elastic frequency distribution of a crystal,” Phys. Rev.89, 1189–1193 (1953).
[CrossRef]

Wei, S.

S. Wei and M. Y. Chou, “Phonon dispersions of silicon and germanium from first-principles calculations,” Phys. Rev. B50, 2221–2226 (1994).
[CrossRef]

Wilbrandt, S.

Yu, P. Y.

P. Y. Yu and M. Cardona, Fundamentals of Semiconductors (Springer, 2001).

Yulin, S.

Zajícková, L.

D. Franta, D. Nečas, and L. Zajíčková, “Application of Thomas–Reiche–Kuhn sum rule to construction of advanced dispersion models,” Thin Solid Films534, 432–441 (2013).
[CrossRef]

L. Zajíčková, D. Franta, D. Nečas, V. Buršíková, M. Muresan, V. Peřina, and C. Cobet, “Dielectric response and structure of amorphous hydrogenated carbon films with nitrogen admixture,” Thin Solid Films519, 4299–4308 (2011).
[CrossRef]

D. Franta, D. Nečas, L. Zajíčková, and I. Ohlídal, “Utilization of the sum rule for construction of advanced dispersion model of crystalline silicon containing interstitial oxygen,” Thin Solid Films (2014).
[CrossRef]

D. Franta, D. Nečas, L. Zajíčková, and I. Ohlídal, “Broadening of dielectric response and sum rule conservation,” Thin Solid Films (2014).
[CrossRef]

Appl. Opt. (1)

Appl. Surf. Sci. (1)

M. Modreanu, J. Sancho-Parramon, O. Durand, B. Servet, M. Stchakovsky, C. Eypert, C. Naudin, A. Knowles, F. Bridou, and M.-F. Ravet, “Investigation of thermal annealing effects on microstructural and optical properties of HfO2 thin films,” Appl. Surf. Sci.253, 328–334 (2006).
[CrossRef]

Infrared Phys. Technol. (1)

M. E. Thomas, S. K. Andersson, R. M. Sova, and R. I. Joseph, “Frequency and temperature dependence of the refractive index of sapphire,” Infrared Phys. Technol.39, 235–249 (1998).
[CrossRef]

J. Appl. Phys. (1)

D. A. Neumayer and E. Cartier, “Materials characterization of ZrO2–SiO2 and HfO2–SiO2 binary oxides deposited by chemical solution deposition,” J. Appl. Phys.90, 1801–1808 (2001).
[CrossRef]

J. Phys. D Appl. Phys. (1)

F. L. Martínez, M. Toledano-Luque, J. J. Gandía, J. Cárabe, W. Bohne, J. Röhrich, E. Strub, and I. Mártil, “Optical properties and structure of HfO2 thin films grown by high pressure reactive sputtering,” J. Phys. D Appl. Phys.40, 5256–5265 (2007).
[CrossRef]

Opt. Mater. Express (1)

Opto-Electron. Rev. (1)

A. Rogalski and K. Chrzanowski, “Infrared devices and techniques,” Opto-Electron. Rev.10, 111–136 (2002).

Phys. Rev. (2)

M. Lax and E. Burstein, “Infrared lattice absorption in ionic and homopolar crystals,” Phys. Rev.97, 39–52 (1955).
[CrossRef]

L. Van Hove, “The occurence of singularities in the elastic frequency distribution of a crystal,” Phys. Rev.89, 1189–1193 (1953).
[CrossRef]

Phys. Rev. B (2)

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[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Phonon dispersion curves of c-Si calculated from the first principles calculation taken from [17]. The points denote phonon frequencies determined using the dispersion model presented here.

Fig. 2
Fig. 2

A schematic diagram of functions Li(E) modeling 3D and 2D Van Hove singularities in critical points (Arabic digits denote the contribution i).

Fig. 3
Fig. 3

A schematic diagram of function H(E) modifying the resulting shape of the transition strength function of the individual absorption bands FA±B(E, T).

Fig. 4
Fig. 4

Spectral dependencies of dielectric function for three different temperatures of TO+TA two-phonon absorption band. The position of critical points are plotted for 300 K.

Fig. 5
Fig. 5

Two-phonon absorption spectra: (a) transmittances of 580 μm and 14 mm thick float-zone silicon samples; (b) full line – modeled dielectric function of c-Si; dotted line –individual bands of two-phonon absorption.

Fig. 6
Fig. 6

Associated ellipsometric parameter IcIII = cos2Ψ illustrating the temperature dependence of two-phonon absorption of float-zone silicon for 70 °.

Tables (5)

Tables Icon

Table 1 Mathematical expression of functions Li(E) modeling 3D and 2D Van Hove singularities in critical points.

Tables Icon

Table 2 Phonon frequencies in points of symmetry determined using the optical characterization of c-Si based on the transmittance spectra at 300 K. The values are in THz units and dagger means fixed value.

Tables Icon

Table 3 The identification of the critical points of the two-phonon absorption bands in mid-infrared region (above 450 cm−1).

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Table 4 The identification of the critical points of the two-phonon absorption bands in far-infrared region (below 450 cm−1).

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Table 5 The parameters of the thermal independent part of the dispersion model. The daggers mean fixed values.

Equations (33)

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F ( E , T ) = π ε 0 V ( e h ¯ m e ) 2 i , f i f exp ( Ω E i k B T ) | f | p ^ x | i | 2 E f E i [ δ ( E f E i E ) + δ ( E i E f E ) ] ,
p ^ x = p ^ x e Z Si m e m Si p ^ x Si ,
F 2 ph ( E , T ) = π ε 0 V | E | ( e h ¯ m e ) 2 A , B k BZ i exp ( Ω E i k B T ) × [ n A , k = 0 n B , k = 0 | i , n A , k + 1 , n B , k + 1 | p ^ x | i , n A , k , n B , k | 2 c A , B , k + δ [ E A + B ( k ) | E | ] n A , k = 1 n B , k = 1 | i , n A , k 1 , n B , k 1 | p ^ x | i , n A , k , n B , k | 2 c A , B , k + δ [ E A + B ( k ) | E | ] + n A , k = 0 n B , k = 1 | i , n A , k + 1 , n B , k 1 | p ^ x | i , n A , k , n B , k | 2 c A , B , k δ [ E A B ( k ) | E | ] n A , k = 1 n B , k = 0 | i , n A , k 1 , n B , k + 1 | p ^ x | i , n A , k , n B , k | 2 c A , B , k δ [ E A B ( k ) | E | ] ] ,
c A , B , k ± = exp ( n A , k E A ( k ) + n B , k E B ( k ) k B T ) , E A ± B ( k ) = E A ( k ) ± E B ( k ) .
exp ( Ω k B T ) = i n A , k = 0 n B , ± k = 0 exp ( E i k B T ) c A , B , k ± .
| i , n A , k + 1 , n B , k + 1 | p ^ x | i , n A , k , n B , k | 2 ( n A , k + 1 ) ( n B , k + 1 ) | i , 1 , 1 | p ^ x | i , 0 , 0 | 2
| i , n A , k + 1 , n B , k 1 | p ^ x | i , n A , k , n B , k | 2 ( n A , k + 1 ) ( n B , k ) | i , 1 , 0 | p ^ x | i , 0 , 1 | 2 .
F 2 ph ( E , T ) = A , B [ F A + B ( E , T ) + F A B ( E , T ) ] ,
F A ± B ( E , T ) = k BZ f A ± B ( k , T ) P A ± B ( k ) δ [ E A ± B ( k ) | E | ] .
f A + B ( k , T ) = n A ( k , T ) + n B ( k , T ) + 1 , f A B ( k , T ) = n B ( k , T ) n A ( k , T ) ,
n p ( k , T ) = f BE [ E p ( k ) ] = 1 exp [ E p ( k ) / k B T ] 1 .
P A + B ( k ) = π ε 0 V | E | ( e h ¯ m e ) 2 | i , 1 , 1 | p ^ x | i , 0 , 0 | 2 ,
P A B ( k ) = π ε 0 V | E | ( e h ¯ m e ) 2 | i , 1 , 0 | p ^ x | i , 0 , 1 | 2 .
F A ± B ( E , T ) = f A ± B ( E , T ) P A ± B ( E ) ,
E t , A ± B = h [ ν A ( M t ) ± ν B ( M t ) ] [ 1 + ν T ( T 300 ) ] ν iso ,
ν LA ( Γ ) = ν TA 1 ( Γ ) = ν TA 2 ( Γ ) ν A ( Γ ) = 0 , ν LO ( Γ ) = ν TO 1 ( Γ ) = ν TO 2 ( Γ ) ν O ( Γ ) , ν TA 1 ( X ) = ν TA 2 ( X ) ν TA ( X ) , ν LA ( X ) = ν LO ( X ) ν L ( X ) , ν TO 1 ( X ) = ν TO 2 ( X ) ν TO ( X ) , ν TA 1 ( L ) = ν TA 2 ( L ) ν TA ( L ) , ν TO 1 ( L ) = ν TO 2 ( L ) ν TO ( L ) , ν TA 1 ( W ) = ν TA 2 ( W ) ν TA ( W ) , ν LA ( W ) = ν LO ( W ) ν L ( W ) , ν TO 1 ( W ) = ν TO 2 ( W ) ν TO ( W ) .
X I ( E ) = E E 0 E 1 E 0 Π E 0 , E 1 ( E ) , Y I ( E ) = E 1 E E 1 E 0 Π E 0 , E 1 ( E ) , X II ( E ) = E E 1 E 2 E 1 Π E 1 , E 2 ( E ) , Y II ( E ) = E 2 E E 2 E 1 Π E 1 , E 2 ( E ) , X III ( E ) = E E 2 E 3 E 2 Π E 2 , E 3 ( E ) , Y III ( E ) = E 3 E E 3 E 2 Π E 2 , E 3 ( E ) ,
Π E min , E max ( E ) = { 1 E min < E < E max 0 otherwise .
P A ± B ( E ) = H ( E ) i = 0 9 A i L i ( E ) ,
H ( E ) = κ 0 Y I ( E ) + κ 1 X I ( E ) ( κ 1 1 ) X I ( E ) + 1 + κ 2 Y II ( E ) ( κ 2 1 ) Y II ( E ) + 1 + κ 3 X II ( E ) ( κ 3 1 ) X II ( E ) + 1 + κ 4 X III ( E ) ( κ 4 1 ) Y III ( E ) + 1 + κ 5 X III ( E )
H ( E ) = κ 0 , H ( E 1 ) = 1 , H ( E 2 ) = 1 , H ( E 3 ) = κ 5 , H ( E ) = Π E 0 , E 3 ( E ) for κ j = 1 .
f A + B ( E , T ) = 1 + f BE [ E A ( E ) ] + f BE [ E B ( E ) ]
f A B ( E , T ) = f BE [ E B ( E ) ] f BE [ E A ( E ) ]
E A ( E ) = h { ν A ( M 0 ) Y I ( E ) + ν A ( M 1 ) [ X I ( E ) + Y II ( E ) ] + ν A ( M 2 ) [ X II ( E ) + Y III ( E ) ] + ν A ( M 3 ) X III ( E ) } [ 1 + ν T ( T 300 ) ] ν iso .
F A ± B 0 ( E ) = 1 𝒞 N f A ± B ( E , T ) P A ± B ( E ) ,
𝒞 N = E 0 E 3 f A ± B ( E , 300 ) P A ± B ( E ) d E .
β ^ ( x ) = 2 π B D ( x 2 B ) + i 1 2 π B exp ( x 2 2 B 2 ) ,
β = 2 2 ln 2 h c B .
ε ^ A ± B 0 ( E ) = β ^ * F A ± B 0 E ,
ε ^ TO + TA 0 ( E ) = j = 1 3 C TO + TA ( j ) ε ^ TO + TA ( j ) 0 ( E ) j = 1 3 C TO + TA ( j )
ε ^ A ± B ( E ) = α A ± B N Si ε ^ A ± B 0 .
e 2 ph * = Z Si e A ± B α A ± B .
ν T = 3.4 × 10 5 ( K 1 ) .

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