Abstract

Tauc-Lorentz model is commonly used to describe the dielectric constant of amorphous semiconductors as a function of few parameters. However, this model is not fully analytic and presents other mathematical shortcomings. A modified self-consistent model based on the integration of [E’-(E + ia)]-1 functions using Tauc-Lorentz`s ε2 expression as a weight function is presented. This new model is analytic and meets all other mathematical requirements of optical constants. The main difference with TL model stands at photon energies close to or smaller than the bandgap energy. The new model has been satisfactorily tested on SiC optical constants. Additionally, an analytic extension of the new model has been also developed to include the Urbach tail. The complete model has been tested with Si3N4 optical constants, and it enables to extend the optical-constant characterization of materials down to zero energy.

© 2016 Optical Society of America

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References

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  1. A. R. Forouhi and I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B Condens. Matter 34(10), 7018–7026 (1986).
    [Crossref] [PubMed]
  2. D. Campi and C. Coriasso, “Prediction of optical properties of amorphous tetrahedrally bonded materials,” J. Appl. Phys. 64(8), 4128–4134 (1988).
    [Crossref]
  3. G. E. Jellison and F. A. Modine, “Parameterization of the optical functions of amorphous materials in the interband region,” Appl. Phys. Lett. 69(3), 371–373 (1996).
    [Crossref]
  4. G. D. Cody, Semiconductors and Semimetals, vol. 21 (Academic, 1984).
  5. D. Franta, D. Nečas, I. Ohlidal, and A. Giglia, “Dispersion model for optical thin films applicable in wide spectral range,” Proc. SPIE 9628, 96281U (2015).
    [Crossref]
  6. M. Foldyna, K. Postava, J. Bouchala, J. Pitora, and T. Yamaguchi, “Model dielectric function of amorphous materials including Urbach tail,” Proc. SPIE 5445, 301–305 (2003).
  7. A. S. Ferlauto, J. Koh, P. I. Rovira, C. R. Wronski, R. W. Collins, and G. Ganguly, “Modeling the dielectric functions of silicon-based films in the amorphous, nanocrystalline and microcrystalline regimes,” J. Non-Cryst. Solids 266–269, 269–273 (2000).
    [Crossref]
  8. J. I. Larruquert and L. V. Rodríguez-de Marcos, Instituto de Optica-CSIC, Serrano 144, Madrid 28006 Spain, are preparing a manuscript to be called “Procedure to convert optical-constant models into analytic.”
  9. J. Tauc, R. Grigorovici, and A. Vancu, “Optical properties and electronic structure of amorphous germanium,” Phys. Status Solidi 15(2), 627–637 (1966).
    [Crossref]
  10. B. von Blanckenhagen, D. Tonova, and J. Ullmann, “Application of the Tauc-Lorentz formulation to the interband absorption of optical coating materials,” Appl. Opt. 41(16), 3137–3141 (2002).
    [Crossref] [PubMed]
  11. R. Brendel and D. Bormann, “An infrared dielectric function model for amorphous solids,” J. Appl. Phys. 71(1), 1–6 (1992).
    [Crossref]
  12. J. I. Larruquert, A. P. Pérez-Marín, S. García-Cortés, L. Rodríguez-de Marcos, J. A. Aznárez, and J. A. Méndez, “Self-consistent optical constants of SiC thin films,” J. Opt. Soc. Am. A 28(11), 2340–2345 (2011).
    [Crossref] [PubMed]
  13. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover Publications Inc., 1965).
  14. V. Pegoraro and P. Slusallek, “On the evaluation of the complex-valued exponential integral,” J. Graphics GPU Game Tools 15(3), 183–198 (2011).
    [Crossref]
  15. H. R. Philipp, “Optical properties of silicon nitride,” J. Electrochem. Soc. 120(2), 295–300 (1973).
    [Crossref]
  16. H. R. Philipp, “Silicon nitride (Si3N4) (Noncrystalline),” in Handbook of Optical Constants of Solids D. Palik ed. (Academic Press, 1991).

2015 (1)

D. Franta, D. Nečas, I. Ohlidal, and A. Giglia, “Dispersion model for optical thin films applicable in wide spectral range,” Proc. SPIE 9628, 96281U (2015).
[Crossref]

2011 (2)

2003 (1)

M. Foldyna, K. Postava, J. Bouchala, J. Pitora, and T. Yamaguchi, “Model dielectric function of amorphous materials including Urbach tail,” Proc. SPIE 5445, 301–305 (2003).

2002 (1)

2000 (1)

A. S. Ferlauto, J. Koh, P. I. Rovira, C. R. Wronski, R. W. Collins, and G. Ganguly, “Modeling the dielectric functions of silicon-based films in the amorphous, nanocrystalline and microcrystalline regimes,” J. Non-Cryst. Solids 266–269, 269–273 (2000).
[Crossref]

1996 (1)

G. E. Jellison and F. A. Modine, “Parameterization of the optical functions of amorphous materials in the interband region,” Appl. Phys. Lett. 69(3), 371–373 (1996).
[Crossref]

1992 (1)

R. Brendel and D. Bormann, “An infrared dielectric function model for amorphous solids,” J. Appl. Phys. 71(1), 1–6 (1992).
[Crossref]

1988 (1)

D. Campi and C. Coriasso, “Prediction of optical properties of amorphous tetrahedrally bonded materials,” J. Appl. Phys. 64(8), 4128–4134 (1988).
[Crossref]

1986 (1)

A. R. Forouhi and I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B Condens. Matter 34(10), 7018–7026 (1986).
[Crossref] [PubMed]

1973 (1)

H. R. Philipp, “Optical properties of silicon nitride,” J. Electrochem. Soc. 120(2), 295–300 (1973).
[Crossref]

1966 (1)

J. Tauc, R. Grigorovici, and A. Vancu, “Optical properties and electronic structure of amorphous germanium,” Phys. Status Solidi 15(2), 627–637 (1966).
[Crossref]

Aznárez, J. A.

Bloomer, I.

A. R. Forouhi and I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B Condens. Matter 34(10), 7018–7026 (1986).
[Crossref] [PubMed]

Bormann, D.

R. Brendel and D. Bormann, “An infrared dielectric function model for amorphous solids,” J. Appl. Phys. 71(1), 1–6 (1992).
[Crossref]

Bouchala, J.

M. Foldyna, K. Postava, J. Bouchala, J. Pitora, and T. Yamaguchi, “Model dielectric function of amorphous materials including Urbach tail,” Proc. SPIE 5445, 301–305 (2003).

Brendel, R.

R. Brendel and D. Bormann, “An infrared dielectric function model for amorphous solids,” J. Appl. Phys. 71(1), 1–6 (1992).
[Crossref]

Campi, D.

D. Campi and C. Coriasso, “Prediction of optical properties of amorphous tetrahedrally bonded materials,” J. Appl. Phys. 64(8), 4128–4134 (1988).
[Crossref]

Collins, R. W.

A. S. Ferlauto, J. Koh, P. I. Rovira, C. R. Wronski, R. W. Collins, and G. Ganguly, “Modeling the dielectric functions of silicon-based films in the amorphous, nanocrystalline and microcrystalline regimes,” J. Non-Cryst. Solids 266–269, 269–273 (2000).
[Crossref]

Coriasso, C.

D. Campi and C. Coriasso, “Prediction of optical properties of amorphous tetrahedrally bonded materials,” J. Appl. Phys. 64(8), 4128–4134 (1988).
[Crossref]

Ferlauto, A. S.

A. S. Ferlauto, J. Koh, P. I. Rovira, C. R. Wronski, R. W. Collins, and G. Ganguly, “Modeling the dielectric functions of silicon-based films in the amorphous, nanocrystalline and microcrystalline regimes,” J. Non-Cryst. Solids 266–269, 269–273 (2000).
[Crossref]

Foldyna, M.

M. Foldyna, K. Postava, J. Bouchala, J. Pitora, and T. Yamaguchi, “Model dielectric function of amorphous materials including Urbach tail,” Proc. SPIE 5445, 301–305 (2003).

Forouhi, A. R.

A. R. Forouhi and I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B Condens. Matter 34(10), 7018–7026 (1986).
[Crossref] [PubMed]

Franta, D.

D. Franta, D. Nečas, I. Ohlidal, and A. Giglia, “Dispersion model for optical thin films applicable in wide spectral range,” Proc. SPIE 9628, 96281U (2015).
[Crossref]

Ganguly, G.

A. S. Ferlauto, J. Koh, P. I. Rovira, C. R. Wronski, R. W. Collins, and G. Ganguly, “Modeling the dielectric functions of silicon-based films in the amorphous, nanocrystalline and microcrystalline regimes,” J. Non-Cryst. Solids 266–269, 269–273 (2000).
[Crossref]

García-Cortés, S.

Giglia, A.

D. Franta, D. Nečas, I. Ohlidal, and A. Giglia, “Dispersion model for optical thin films applicable in wide spectral range,” Proc. SPIE 9628, 96281U (2015).
[Crossref]

Grigorovici, R.

J. Tauc, R. Grigorovici, and A. Vancu, “Optical properties and electronic structure of amorphous germanium,” Phys. Status Solidi 15(2), 627–637 (1966).
[Crossref]

Jellison, G. E.

G. E. Jellison and F. A. Modine, “Parameterization of the optical functions of amorphous materials in the interband region,” Appl. Phys. Lett. 69(3), 371–373 (1996).
[Crossref]

Koh, J.

A. S. Ferlauto, J. Koh, P. I. Rovira, C. R. Wronski, R. W. Collins, and G. Ganguly, “Modeling the dielectric functions of silicon-based films in the amorphous, nanocrystalline and microcrystalline regimes,” J. Non-Cryst. Solids 266–269, 269–273 (2000).
[Crossref]

Larruquert, J. I.

Méndez, J. A.

Modine, F. A.

G. E. Jellison and F. A. Modine, “Parameterization of the optical functions of amorphous materials in the interband region,” Appl. Phys. Lett. 69(3), 371–373 (1996).
[Crossref]

Necas, D.

D. Franta, D. Nečas, I. Ohlidal, and A. Giglia, “Dispersion model for optical thin films applicable in wide spectral range,” Proc. SPIE 9628, 96281U (2015).
[Crossref]

Ohlidal, I.

D. Franta, D. Nečas, I. Ohlidal, and A. Giglia, “Dispersion model for optical thin films applicable in wide spectral range,” Proc. SPIE 9628, 96281U (2015).
[Crossref]

Pegoraro, V.

V. Pegoraro and P. Slusallek, “On the evaluation of the complex-valued exponential integral,” J. Graphics GPU Game Tools 15(3), 183–198 (2011).
[Crossref]

Pérez-Marín, A. P.

Philipp, H. R.

H. R. Philipp, “Optical properties of silicon nitride,” J. Electrochem. Soc. 120(2), 295–300 (1973).
[Crossref]

Pitora, J.

M. Foldyna, K. Postava, J. Bouchala, J. Pitora, and T. Yamaguchi, “Model dielectric function of amorphous materials including Urbach tail,” Proc. SPIE 5445, 301–305 (2003).

Postava, K.

M. Foldyna, K. Postava, J. Bouchala, J. Pitora, and T. Yamaguchi, “Model dielectric function of amorphous materials including Urbach tail,” Proc. SPIE 5445, 301–305 (2003).

Rodríguez-de Marcos, L.

Rovira, P. I.

A. S. Ferlauto, J. Koh, P. I. Rovira, C. R. Wronski, R. W. Collins, and G. Ganguly, “Modeling the dielectric functions of silicon-based films in the amorphous, nanocrystalline and microcrystalline regimes,” J. Non-Cryst. Solids 266–269, 269–273 (2000).
[Crossref]

Slusallek, P.

V. Pegoraro and P. Slusallek, “On the evaluation of the complex-valued exponential integral,” J. Graphics GPU Game Tools 15(3), 183–198 (2011).
[Crossref]

Tauc, J.

J. Tauc, R. Grigorovici, and A. Vancu, “Optical properties and electronic structure of amorphous germanium,” Phys. Status Solidi 15(2), 627–637 (1966).
[Crossref]

Tonova, D.

Ullmann, J.

Vancu, A.

J. Tauc, R. Grigorovici, and A. Vancu, “Optical properties and electronic structure of amorphous germanium,” Phys. Status Solidi 15(2), 627–637 (1966).
[Crossref]

von Blanckenhagen, B.

Wronski, C. R.

A. S. Ferlauto, J. Koh, P. I. Rovira, C. R. Wronski, R. W. Collins, and G. Ganguly, “Modeling the dielectric functions of silicon-based films in the amorphous, nanocrystalline and microcrystalline regimes,” J. Non-Cryst. Solids 266–269, 269–273 (2000).
[Crossref]

Yamaguchi, T.

M. Foldyna, K. Postava, J. Bouchala, J. Pitora, and T. Yamaguchi, “Model dielectric function of amorphous materials including Urbach tail,” Proc. SPIE 5445, 301–305 (2003).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

G. E. Jellison and F. A. Modine, “Parameterization of the optical functions of amorphous materials in the interband region,” Appl. Phys. Lett. 69(3), 371–373 (1996).
[Crossref]

J. Appl. Phys. (2)

D. Campi and C. Coriasso, “Prediction of optical properties of amorphous tetrahedrally bonded materials,” J. Appl. Phys. 64(8), 4128–4134 (1988).
[Crossref]

R. Brendel and D. Bormann, “An infrared dielectric function model for amorphous solids,” J. Appl. Phys. 71(1), 1–6 (1992).
[Crossref]

J. Electrochem. Soc. (1)

H. R. Philipp, “Optical properties of silicon nitride,” J. Electrochem. Soc. 120(2), 295–300 (1973).
[Crossref]

J. Graphics GPU Game Tools (1)

V. Pegoraro and P. Slusallek, “On the evaluation of the complex-valued exponential integral,” J. Graphics GPU Game Tools 15(3), 183–198 (2011).
[Crossref]

J. Non-Cryst. Solids (1)

A. S. Ferlauto, J. Koh, P. I. Rovira, C. R. Wronski, R. W. Collins, and G. Ganguly, “Modeling the dielectric functions of silicon-based films in the amorphous, nanocrystalline and microcrystalline regimes,” J. Non-Cryst. Solids 266–269, 269–273 (2000).
[Crossref]

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

Phys. Rev. B Condens. Matter (1)

A. R. Forouhi and I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B Condens. Matter 34(10), 7018–7026 (1986).
[Crossref] [PubMed]

Phys. Status Solidi (1)

J. Tauc, R. Grigorovici, and A. Vancu, “Optical properties and electronic structure of amorphous germanium,” Phys. Status Solidi 15(2), 627–637 (1966).
[Crossref]

Proc. SPIE (2)

D. Franta, D. Nečas, I. Ohlidal, and A. Giglia, “Dispersion model for optical thin films applicable in wide spectral range,” Proc. SPIE 9628, 96281U (2015).
[Crossref]

M. Foldyna, K. Postava, J. Bouchala, J. Pitora, and T. Yamaguchi, “Model dielectric function of amorphous materials including Urbach tail,” Proc. SPIE 5445, 301–305 (2003).

Other (4)

J. I. Larruquert and L. V. Rodríguez-de Marcos, Instituto de Optica-CSIC, Serrano 144, Madrid 28006 Spain, are preparing a manuscript to be called “Procedure to convert optical-constant models into analytic.”

G. D. Cody, Semiconductors and Semimetals, vol. 21 (Academic, 1984).

H. R. Philipp, “Silicon nitride (Si3N4) (Noncrystalline),” in Handbook of Optical Constants of Solids D. Palik ed. (Academic Press, 1991).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover Publications Inc., 1965).

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Figures (3)

Fig. 1
Fig. 1 Linear-axis (a) and log-axis (b) plot of the dielectric constant of thin films of SiC versus the logarithm of photon energy. Experimental data from Larruquert et al. [12] are compared with fittings with TL and TL-an models with parameters plotted in Table 1
Fig. 2
Fig. 2 Log-log plot of the dielectric constant of thin films of SiC versus photon energy extended to include the reststrahlen band. Experimental data from Larruquert et al. [12] are compared with fittings with TL and TL-an model using parameters plotted in Table 1, except for the modification of a to 0.06 eV. The models were added a Lorentz oscillator to account for the reststrahlen band
Fig. 3
Fig. 3 Linear-axis (a) and log-axis (b) plot of the dielectric constant of Si3N4 versus energy. Experimental data of Philipp [15] are compared with a fitting of ε2 with TLU-an model and with ε1 obtained in the latter fitting using parameters plotted in Table 2.

Tables (2)

Tables Icon

Table 1 Fitting parameters for the TL and the TL-an models applied to SiC

Tables Icon

Table 2 Optimum parameters for TLU-an models applied to Si3N4

Equations (23)

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k FB = A ( E E g ) 2 E 2 BE+C
n FB = n + B 0 E+ C 0 E 2 BE+C
ε 2,TL ( E;A, E 0 , E g ,C )=Θ( E E g ) ( E E g ) 2 A E 0 C ( E 2 E 0 2 ) 2 + C 2 E 2 1 E
ε 1,TL ( E;A, E 0 , E g ,C, ε 1 ( ) )= ε 1 ( )+ 2 π E g ε 2,TL ( E ) E E 2 E 2 d E
ε ˜ an ( E )1= 1 π ε 2,mod ( E ) E Eia d E
ε ˜ an ( E )1= i π ε 1,mod ( E )1 E Eia d E
ε ˜ TLan ( E;A, E 0 , E g ,C,a )=1+ 1 π ε 2,TL ( E ) E Eia d E
ε ˜ TLan ( E;A, E 0 , E g ,C,a )=1+ A E 0 C π [ F( b,d, d * )+F( d, d * ,b )+F( d * ,b,d ) ]
F( α,β,γ )= ( E g +α ) 2 Log( E g +α ) ( E g α ) 2 Log( E g α ) α( α 2 β 2 )( α 2 γ 2 )
bb( E )=E+ia
d= E 0 2 ( C 2 ) 2 i C 2
α( E )= α 0 exp( E E 0 )
ε 2,TLU ( E;A, E 0 , E g ,C, E c )= A u E exp( E E u ) E< E c
ε 2,TLU ( E;A, E 0 , E g ,C, E c )= ( E E g ) 2 A E 0 C ( E 2 E 0 2 ) 2 + C 2 E 2 1 E E³ E c
ε 2,U ( E;A, E 0 , E g ,C, E c )=E A u exp( E E u ) at E< E c
ε 2,TL ( E;A, E 0 , E g ,C, E c )= ( E E g ) 2 A E 0 C ( E 2 E 0 2 ) 2 + C 2 E 2 1 E at E³ E c
E u = [ E c 4 +( C 2 2 E 0 2 ) E c 2 + E 0 4 ]( E c E g ) E c 4 E c 5 +6 E g E c 4 +( 4 E 0 2 2 C 2 ) E c 3 +4 E g ( C 2 2 E 0 2 ) E c 2 +2 E g E 0 4
A u = A E 0 C ( E c E g ) 2 exp( E c E u ) [ E c 4 +( C 2 2 E 0 2 ) E c 2 + E 0 4 ] E c 2
ε ˜ TLUan ( E )= ε ˜ Uan ( E;A, E 0 , E g ,C,a, E c )+ ε ˜ TLan ( E;A, E 0 , E g ,C,a, E c )
ε ˜ Uan ( E;A, E 0 , E g ,C,a, E c )=1+ 1 π ε 2,U ( E ) E Eia d E = A u π { bexp( b E u )[ Ei( E c b E u )Ei( b E u ) ]+bexp( b E u )[ Ei( b E u )Ei( E c +b E u ) ] +2exp( E c E u ) E u 2 E u }
ε ˜ TLan ( E;A, E 0 , E g ,C,a, E c )=1+ A E 0 C π [ F ¯ ( b,d, d * )+ F ¯ ( d, d * ,b )+ F ¯ ( d * ,b,d ) ]
F ¯ ( α,β,γ )= ( E g +α ) 2 Log( E c +α ) ( E g α ) 2 Log( E c α ) α( α 2 β 2 )( α 2 γ 2 )
Ei( z )= z e t t dt | arg( z ) |<π

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