Abstract

Coatings of transparent conductive oxides, especially indium tin oxide (ITO), are important in different fields. So far, application of these materials has been limited to substrates with high thermal stability. We describe an improved coating process for ITO based on plasma ion-assisted evaporation at a substrate temperature below 100°C, which is suitable for organic substrates. In characterizing the thin films, we used the classical Drude theory to calculate the resistivity from optical film properties and compared the data with linear four-point measurements. X-ray diffraction spectroscopy was used to determine the structural properties of the thin films.

© 2008 Optical Society of America

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  1. D. E. Carlson and C. R. Wronski, “Amorphous silicon solar cell,” Appl. Phys. Lett. 28, 671-673 (1976).
    [Crossref]
  2. R. Wang and C. C. Lee, “Design of antireflection coating using indium tin oxide (ITO) film prepared by ion assisted deposition (IAD),” in Proceedings of the 42nd Annual Technical Conference Proceedings (Society of Vacuum Coaters, 1999), pp. 246-249.
  3. D. Mergel, W. Stass, G. Ehl, and D. Barthel, “Oxygen incorporation in thin films of In2O3:Sn prepared by radio frequency sputtering,” J. Appl. Phys. 88, 2437-2442 (2000).
    [Crossref]
  4. H. Izumi, T. Ishihara, H. Yoshioka, and M. Motoyama, “Electrical properties of crystalline ITO films prepared at room temperature by pulsed laser deposition on plastic substrates,” Thin Solid Films 411, 32-35 (2002).
    [Crossref]
  5. F. Niino, H. Hirasawa, and K. Kondo, “Deposition of low-resistivity ITO on plastic substrates by DC arc-discharge ion plating,” Thin Solid Films 441, 28-31 (2002).
    [Crossref]
  6. D.-H. Kim, M.-R. Park, and G.-H. Lee, “Preparation of high quality ITO films on a plastic substrate using rf magnetron sputtering,” Surf. Coat. Technol. 201, 927-931 (2006).
    [Crossref]
  7. S. Laux, N. Kaiser, A. Zöller, R. Götzelmann, H. Lauth, and H. Bernitzki, “Room-temperature deposition of indium tin oxide thin films with plasma ion-assisted evaporation,” Thin Solid Films 335, 1-5 (1998).
    [Crossref]
  8. I. Hamberg and C. G. Granqvist, “Evaporated Sn-doped In2O3 films: basic optical properties and applications to energy-efficient windows,” J. Appl. Phys. 60, R123-R159 (1986).
    [Crossref]
  9. J. C. Manifacier, “Thin metallic oxides as transparent conductors,” Thin Solid Films 90, 297-308 (1982).
    [Crossref]
  10. G. Frank and H. Köstlin, “Electrical properties and defect model of tin-doped indium oxide layers,” Appl. Phys. A 27, 197-206 (1982).
    [Crossref]
  11. M. H. Brodsky, ed., Amorphous Semiconductors, Vol. 36 of Topics in Applied Physics (Springer-Verlag, 1979).
  12. O. Stenzel, The Physics of Thin Film Optical Spectra: an Introduction, Springer Series in Surface Sciences (Springer, 2005).
  13. I. Hamberg and C. G. Granqvist, “Optical properties of transparent and heat-reflecting indium tin oxide films: the role of ionized impurity scattering,” Appl. Phys. Lett. 44, 721-723 (1984).
    [Crossref]
  14. K. L. Chopra, S. Major, and D. K. Pandya, “Transparent conductors--a status review,” Thin Solid Films 102, 1-46 (1983).
    [Crossref]
  15. K. Füchsel, U. Schulz, and N. Kaiser, Low temperature deposition of indium tin oxide films by plasma ion-assisted evaporation, in Proceedings of Optical Interference Coatings (Optical Society of America, 2007), paper ThB3.
  16. D. Mergel and Z. Qiao, “Dielectric modelling of optical spectra of thin In2O3:Sn films,” J. Phys. D 35, 794-801 (2002).
    [Crossref]
  17. S. Pongratz and A. Zöller, “Plasma ion-assisted deposition: a promising technique for optical coatings,” J. Vac. Sci. Technol. A 10, 1897-1904 (1992).
    [Crossref]
  18. D. Mergel, “Thin films of ITO as transparent electrodes,” Vak. Forsch. Prax. 18(S1), 15-18 (2006).
    [Crossref]

2007 (1)

K. Füchsel, U. Schulz, and N. Kaiser, Low temperature deposition of indium tin oxide films by plasma ion-assisted evaporation, in Proceedings of Optical Interference Coatings (Optical Society of America, 2007), paper ThB3.

2006 (2)

D. Mergel, “Thin films of ITO as transparent electrodes,” Vak. Forsch. Prax. 18(S1), 15-18 (2006).
[Crossref]

D.-H. Kim, M.-R. Park, and G.-H. Lee, “Preparation of high quality ITO films on a plastic substrate using rf magnetron sputtering,” Surf. Coat. Technol. 201, 927-931 (2006).
[Crossref]

2005 (1)

O. Stenzel, The Physics of Thin Film Optical Spectra: an Introduction, Springer Series in Surface Sciences (Springer, 2005).

2002 (3)

D. Mergel and Z. Qiao, “Dielectric modelling of optical spectra of thin In2O3:Sn films,” J. Phys. D 35, 794-801 (2002).
[Crossref]

H. Izumi, T. Ishihara, H. Yoshioka, and M. Motoyama, “Electrical properties of crystalline ITO films prepared at room temperature by pulsed laser deposition on plastic substrates,” Thin Solid Films 411, 32-35 (2002).
[Crossref]

F. Niino, H. Hirasawa, and K. Kondo, “Deposition of low-resistivity ITO on plastic substrates by DC arc-discharge ion plating,” Thin Solid Films 441, 28-31 (2002).
[Crossref]

2000 (1)

D. Mergel, W. Stass, G. Ehl, and D. Barthel, “Oxygen incorporation in thin films of In2O3:Sn prepared by radio frequency sputtering,” J. Appl. Phys. 88, 2437-2442 (2000).
[Crossref]

1999 (1)

R. Wang and C. C. Lee, “Design of antireflection coating using indium tin oxide (ITO) film prepared by ion assisted deposition (IAD),” in Proceedings of the 42nd Annual Technical Conference Proceedings (Society of Vacuum Coaters, 1999), pp. 246-249.

1998 (1)

S. Laux, N. Kaiser, A. Zöller, R. Götzelmann, H. Lauth, and H. Bernitzki, “Room-temperature deposition of indium tin oxide thin films with plasma ion-assisted evaporation,” Thin Solid Films 335, 1-5 (1998).
[Crossref]

1992 (1)

S. Pongratz and A. Zöller, “Plasma ion-assisted deposition: a promising technique for optical coatings,” J. Vac. Sci. Technol. A 10, 1897-1904 (1992).
[Crossref]

1986 (1)

I. Hamberg and C. G. Granqvist, “Evaporated Sn-doped In2O3 films: basic optical properties and applications to energy-efficient windows,” J. Appl. Phys. 60, R123-R159 (1986).
[Crossref]

1984 (1)

I. Hamberg and C. G. Granqvist, “Optical properties of transparent and heat-reflecting indium tin oxide films: the role of ionized impurity scattering,” Appl. Phys. Lett. 44, 721-723 (1984).
[Crossref]

1983 (1)

K. L. Chopra, S. Major, and D. K. Pandya, “Transparent conductors--a status review,” Thin Solid Films 102, 1-46 (1983).
[Crossref]

1982 (2)

J. C. Manifacier, “Thin metallic oxides as transparent conductors,” Thin Solid Films 90, 297-308 (1982).
[Crossref]

G. Frank and H. Köstlin, “Electrical properties and defect model of tin-doped indium oxide layers,” Appl. Phys. A 27, 197-206 (1982).
[Crossref]

1979 (1)

M. H. Brodsky, ed., Amorphous Semiconductors, Vol. 36 of Topics in Applied Physics (Springer-Verlag, 1979).

1976 (1)

D. E. Carlson and C. R. Wronski, “Amorphous silicon solar cell,” Appl. Phys. Lett. 28, 671-673 (1976).
[Crossref]

Barthel, D.

D. Mergel, W. Stass, G. Ehl, and D. Barthel, “Oxygen incorporation in thin films of In2O3:Sn prepared by radio frequency sputtering,” J. Appl. Phys. 88, 2437-2442 (2000).
[Crossref]

Bernitzki, H.

S. Laux, N. Kaiser, A. Zöller, R. Götzelmann, H. Lauth, and H. Bernitzki, “Room-temperature deposition of indium tin oxide thin films with plasma ion-assisted evaporation,” Thin Solid Films 335, 1-5 (1998).
[Crossref]

Brodsky, M. H.

M. H. Brodsky, ed., Amorphous Semiconductors, Vol. 36 of Topics in Applied Physics (Springer-Verlag, 1979).

Carlson, D. E.

D. E. Carlson and C. R. Wronski, “Amorphous silicon solar cell,” Appl. Phys. Lett. 28, 671-673 (1976).
[Crossref]

Chopra, K. L.

K. L. Chopra, S. Major, and D. K. Pandya, “Transparent conductors--a status review,” Thin Solid Films 102, 1-46 (1983).
[Crossref]

Ehl, G.

D. Mergel, W. Stass, G. Ehl, and D. Barthel, “Oxygen incorporation in thin films of In2O3:Sn prepared by radio frequency sputtering,” J. Appl. Phys. 88, 2437-2442 (2000).
[Crossref]

Frank, G.

G. Frank and H. Köstlin, “Electrical properties and defect model of tin-doped indium oxide layers,” Appl. Phys. A 27, 197-206 (1982).
[Crossref]

Füchsel, K.

K. Füchsel, U. Schulz, and N. Kaiser, Low temperature deposition of indium tin oxide films by plasma ion-assisted evaporation, in Proceedings of Optical Interference Coatings (Optical Society of America, 2007), paper ThB3.

Götzelmann, R.

S. Laux, N. Kaiser, A. Zöller, R. Götzelmann, H. Lauth, and H. Bernitzki, “Room-temperature deposition of indium tin oxide thin films with plasma ion-assisted evaporation,” Thin Solid Films 335, 1-5 (1998).
[Crossref]

Granqvist, C. G.

I. Hamberg and C. G. Granqvist, “Evaporated Sn-doped In2O3 films: basic optical properties and applications to energy-efficient windows,” J. Appl. Phys. 60, R123-R159 (1986).
[Crossref]

I. Hamberg and C. G. Granqvist, “Optical properties of transparent and heat-reflecting indium tin oxide films: the role of ionized impurity scattering,” Appl. Phys. Lett. 44, 721-723 (1984).
[Crossref]

Hamberg, I.

I. Hamberg and C. G. Granqvist, “Evaporated Sn-doped In2O3 films: basic optical properties and applications to energy-efficient windows,” J. Appl. Phys. 60, R123-R159 (1986).
[Crossref]

I. Hamberg and C. G. Granqvist, “Optical properties of transparent and heat-reflecting indium tin oxide films: the role of ionized impurity scattering,” Appl. Phys. Lett. 44, 721-723 (1984).
[Crossref]

Hirasawa, H.

F. Niino, H. Hirasawa, and K. Kondo, “Deposition of low-resistivity ITO on plastic substrates by DC arc-discharge ion plating,” Thin Solid Films 441, 28-31 (2002).
[Crossref]

Ishihara, T.

H. Izumi, T. Ishihara, H. Yoshioka, and M. Motoyama, “Electrical properties of crystalline ITO films prepared at room temperature by pulsed laser deposition on plastic substrates,” Thin Solid Films 411, 32-35 (2002).
[Crossref]

Izumi, H.

H. Izumi, T. Ishihara, H. Yoshioka, and M. Motoyama, “Electrical properties of crystalline ITO films prepared at room temperature by pulsed laser deposition on plastic substrates,” Thin Solid Films 411, 32-35 (2002).
[Crossref]

Kaiser, N.

K. Füchsel, U. Schulz, and N. Kaiser, Low temperature deposition of indium tin oxide films by plasma ion-assisted evaporation, in Proceedings of Optical Interference Coatings (Optical Society of America, 2007), paper ThB3.

S. Laux, N. Kaiser, A. Zöller, R. Götzelmann, H. Lauth, and H. Bernitzki, “Room-temperature deposition of indium tin oxide thin films with plasma ion-assisted evaporation,” Thin Solid Films 335, 1-5 (1998).
[Crossref]

Kim, D.-H.

D.-H. Kim, M.-R. Park, and G.-H. Lee, “Preparation of high quality ITO films on a plastic substrate using rf magnetron sputtering,” Surf. Coat. Technol. 201, 927-931 (2006).
[Crossref]

Kondo, K.

F. Niino, H. Hirasawa, and K. Kondo, “Deposition of low-resistivity ITO on plastic substrates by DC arc-discharge ion plating,” Thin Solid Films 441, 28-31 (2002).
[Crossref]

Köstlin, H.

G. Frank and H. Köstlin, “Electrical properties and defect model of tin-doped indium oxide layers,” Appl. Phys. A 27, 197-206 (1982).
[Crossref]

Lauth, H.

S. Laux, N. Kaiser, A. Zöller, R. Götzelmann, H. Lauth, and H. Bernitzki, “Room-temperature deposition of indium tin oxide thin films with plasma ion-assisted evaporation,” Thin Solid Films 335, 1-5 (1998).
[Crossref]

Laux, S.

S. Laux, N. Kaiser, A. Zöller, R. Götzelmann, H. Lauth, and H. Bernitzki, “Room-temperature deposition of indium tin oxide thin films with plasma ion-assisted evaporation,” Thin Solid Films 335, 1-5 (1998).
[Crossref]

Lee, C. C.

R. Wang and C. C. Lee, “Design of antireflection coating using indium tin oxide (ITO) film prepared by ion assisted deposition (IAD),” in Proceedings of the 42nd Annual Technical Conference Proceedings (Society of Vacuum Coaters, 1999), pp. 246-249.

Lee, G.-H.

D.-H. Kim, M.-R. Park, and G.-H. Lee, “Preparation of high quality ITO films on a plastic substrate using rf magnetron sputtering,” Surf. Coat. Technol. 201, 927-931 (2006).
[Crossref]

Major, S.

K. L. Chopra, S. Major, and D. K. Pandya, “Transparent conductors--a status review,” Thin Solid Films 102, 1-46 (1983).
[Crossref]

Manifacier, J. C.

J. C. Manifacier, “Thin metallic oxides as transparent conductors,” Thin Solid Films 90, 297-308 (1982).
[Crossref]

Mergel, D.

D. Mergel, “Thin films of ITO as transparent electrodes,” Vak. Forsch. Prax. 18(S1), 15-18 (2006).
[Crossref]

D. Mergel and Z. Qiao, “Dielectric modelling of optical spectra of thin In2O3:Sn films,” J. Phys. D 35, 794-801 (2002).
[Crossref]

D. Mergel, W. Stass, G. Ehl, and D. Barthel, “Oxygen incorporation in thin films of In2O3:Sn prepared by radio frequency sputtering,” J. Appl. Phys. 88, 2437-2442 (2000).
[Crossref]

Motoyama, M.

H. Izumi, T. Ishihara, H. Yoshioka, and M. Motoyama, “Electrical properties of crystalline ITO films prepared at room temperature by pulsed laser deposition on plastic substrates,” Thin Solid Films 411, 32-35 (2002).
[Crossref]

Niino, F.

F. Niino, H. Hirasawa, and K. Kondo, “Deposition of low-resistivity ITO on plastic substrates by DC arc-discharge ion plating,” Thin Solid Films 441, 28-31 (2002).
[Crossref]

Pandya, D. K.

K. L. Chopra, S. Major, and D. K. Pandya, “Transparent conductors--a status review,” Thin Solid Films 102, 1-46 (1983).
[Crossref]

Park, M.-R.

D.-H. Kim, M.-R. Park, and G.-H. Lee, “Preparation of high quality ITO films on a plastic substrate using rf magnetron sputtering,” Surf. Coat. Technol. 201, 927-931 (2006).
[Crossref]

Pongratz, S.

S. Pongratz and A. Zöller, “Plasma ion-assisted deposition: a promising technique for optical coatings,” J. Vac. Sci. Technol. A 10, 1897-1904 (1992).
[Crossref]

Qiao, Z.

D. Mergel and Z. Qiao, “Dielectric modelling of optical spectra of thin In2O3:Sn films,” J. Phys. D 35, 794-801 (2002).
[Crossref]

Schulz, U.

K. Füchsel, U. Schulz, and N. Kaiser, Low temperature deposition of indium tin oxide films by plasma ion-assisted evaporation, in Proceedings of Optical Interference Coatings (Optical Society of America, 2007), paper ThB3.

Stass, W.

D. Mergel, W. Stass, G. Ehl, and D. Barthel, “Oxygen incorporation in thin films of In2O3:Sn prepared by radio frequency sputtering,” J. Appl. Phys. 88, 2437-2442 (2000).
[Crossref]

Stenzel, O.

O. Stenzel, The Physics of Thin Film Optical Spectra: an Introduction, Springer Series in Surface Sciences (Springer, 2005).

Wang, R.

R. Wang and C. C. Lee, “Design of antireflection coating using indium tin oxide (ITO) film prepared by ion assisted deposition (IAD),” in Proceedings of the 42nd Annual Technical Conference Proceedings (Society of Vacuum Coaters, 1999), pp. 246-249.

Wronski, C. R.

D. E. Carlson and C. R. Wronski, “Amorphous silicon solar cell,” Appl. Phys. Lett. 28, 671-673 (1976).
[Crossref]

Yoshioka, H.

H. Izumi, T. Ishihara, H. Yoshioka, and M. Motoyama, “Electrical properties of crystalline ITO films prepared at room temperature by pulsed laser deposition on plastic substrates,” Thin Solid Films 411, 32-35 (2002).
[Crossref]

Zöller, A.

S. Laux, N. Kaiser, A. Zöller, R. Götzelmann, H. Lauth, and H. Bernitzki, “Room-temperature deposition of indium tin oxide thin films with plasma ion-assisted evaporation,” Thin Solid Films 335, 1-5 (1998).
[Crossref]

S. Pongratz and A. Zöller, “Plasma ion-assisted deposition: a promising technique for optical coatings,” J. Vac. Sci. Technol. A 10, 1897-1904 (1992).
[Crossref]

Appl. Phys. A (1)

G. Frank and H. Köstlin, “Electrical properties and defect model of tin-doped indium oxide layers,” Appl. Phys. A 27, 197-206 (1982).
[Crossref]

Appl. Phys. Lett. (2)

I. Hamberg and C. G. Granqvist, “Optical properties of transparent and heat-reflecting indium tin oxide films: the role of ionized impurity scattering,” Appl. Phys. Lett. 44, 721-723 (1984).
[Crossref]

D. E. Carlson and C. R. Wronski, “Amorphous silicon solar cell,” Appl. Phys. Lett. 28, 671-673 (1976).
[Crossref]

J. Appl. Phys. (2)

D. Mergel, W. Stass, G. Ehl, and D. Barthel, “Oxygen incorporation in thin films of In2O3:Sn prepared by radio frequency sputtering,” J. Appl. Phys. 88, 2437-2442 (2000).
[Crossref]

I. Hamberg and C. G. Granqvist, “Evaporated Sn-doped In2O3 films: basic optical properties and applications to energy-efficient windows,” J. Appl. Phys. 60, R123-R159 (1986).
[Crossref]

J. Phys. D (1)

D. Mergel and Z. Qiao, “Dielectric modelling of optical spectra of thin In2O3:Sn films,” J. Phys. D 35, 794-801 (2002).
[Crossref]

J. Vac. Sci. Technol. A (1)

S. Pongratz and A. Zöller, “Plasma ion-assisted deposition: a promising technique for optical coatings,” J. Vac. Sci. Technol. A 10, 1897-1904 (1992).
[Crossref]

Surf. Coat. Technol. (1)

D.-H. Kim, M.-R. Park, and G.-H. Lee, “Preparation of high quality ITO films on a plastic substrate using rf magnetron sputtering,” Surf. Coat. Technol. 201, 927-931 (2006).
[Crossref]

Thin Solid Films (5)

S. Laux, N. Kaiser, A. Zöller, R. Götzelmann, H. Lauth, and H. Bernitzki, “Room-temperature deposition of indium tin oxide thin films with plasma ion-assisted evaporation,” Thin Solid Films 335, 1-5 (1998).
[Crossref]

J. C. Manifacier, “Thin metallic oxides as transparent conductors,” Thin Solid Films 90, 297-308 (1982).
[Crossref]

H. Izumi, T. Ishihara, H. Yoshioka, and M. Motoyama, “Electrical properties of crystalline ITO films prepared at room temperature by pulsed laser deposition on plastic substrates,” Thin Solid Films 411, 32-35 (2002).
[Crossref]

F. Niino, H. Hirasawa, and K. Kondo, “Deposition of low-resistivity ITO on plastic substrates by DC arc-discharge ion plating,” Thin Solid Films 441, 28-31 (2002).
[Crossref]

K. L. Chopra, S. Major, and D. K. Pandya, “Transparent conductors--a status review,” Thin Solid Films 102, 1-46 (1983).
[Crossref]

Vak. Forsch. Prax. (1)

D. Mergel, “Thin films of ITO as transparent electrodes,” Vak. Forsch. Prax. 18(S1), 15-18 (2006).
[Crossref]

Other (4)

K. Füchsel, U. Schulz, and N. Kaiser, Low temperature deposition of indium tin oxide films by plasma ion-assisted evaporation, in Proceedings of Optical Interference Coatings (Optical Society of America, 2007), paper ThB3.

M. H. Brodsky, ed., Amorphous Semiconductors, Vol. 36 of Topics in Applied Physics (Springer-Verlag, 1979).

O. Stenzel, The Physics of Thin Film Optical Spectra: an Introduction, Springer Series in Surface Sciences (Springer, 2005).

R. Wang and C. C. Lee, “Design of antireflection coating using indium tin oxide (ITO) film prepared by ion assisted deposition (IAD),” in Proceedings of the 42nd Annual Technical Conference Proceedings (Society of Vacuum Coaters, 1999), pp. 246-249.

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Figures (10)

Fig. 1
Fig. 1

Tauc plot for determination of the optical bandgap, the so-called Tauc gap.

Fig. 2
Fig. 2

Comparison of real and imaginary parts of the dielectric function between SWA and the classical Drude model for N = 6 × 10 26 m 3 and μ e = 20 × 10 4 m 2 V 1 s 1 .

Fig. 3
Fig. 3

Schematic diagram of the APS 904 deposition plant (Leybold Optics).

Fig. 4
Fig. 4

Influence of oxygen flow for a constant bias voltage (80 V) at a deposition rate of approximately 0.3 n m / s , substrate temperature of 90   ° C , and a constant argon flow of 15 sccm (sccm denotes cubic centimeters per minute at STP). Right, calculated electron density and electron mobility; left, calculated and measured resistivity and Tauc gap.

Fig. 5
Fig. 5

Influence of argon flow for a constant bias voltage (80 V) at a deposition rate of approximately 0.3 n m / s , substrate temperature of 90   ° C , and a constant oxygen flow of 15 sccm. Right, calculated electron density and electron mobility; left, calculated and measured resistivity and Tauc gap.

Fig. 6
Fig. 6

Influence of argon flow for a constant bias voltage (70 V) at a deposition rate of approximately 0.3 n m / s , substrate temperature of 90   ° C , and a constant oxygen flow of 15 sccm. Right, calculated electron density and electron mobility; left, calculated and measured resistivity and Tauc gap.

Fig. 7
Fig. 7

l n α versus energy for different bias voltages.

Fig. 8
Fig. 8

Influence of substrate temperature for a constant bias voltage (80 V) at a deposition rate of approximately 0.38 nm∕s and constant oxygen and argon flows of 15 and 18 sccm. Right, calculated electron density and electron mobility; left, calculated and measured resistivity and Tauc gap.

Fig. 9
Fig. 9

XRD spectra for films deposited at different substrate temperatures for constant oxygen and argon flows of 15 and 18 SCCM and a bias voltage of 80 V.

Fig. 10
Fig. 10

ITO film ( 230 n m ) with a resistivity of 4.0 × 10 6 Ω m deposited at 0.3 n m / s with a gas flow composition of O 2 / A r = 15 : 15. Left, reflection and transmission; right, extinction coefficient and refractive index computed by reverse engineering with a harmonic oscillator model.

Equations (10)

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

α ( ω ) ω ( h ω 2 π E 0 ) ,
ρ = ( e N μ e ) 1 ,
ϵ = ( n i k ) 2 = ϵ λ 2 λ P 2 ( 1 + i λ / ( 2 π c τ ) ) ,
λ p 2 = ( 2 π c ) 2 ϵ 0 m * N e 2 ,
τ = μ e m * e
ϵ r = n 2 k 2 = ϵ λ 2 λ p 2 = ϵ γ λ 2 ,
ϵ i = 2 n k = 1 λ p 2 2 π c τ λ 3 = κ λ 3 .
ρ c a l = ( 2 π c ϵ 0 γ 2 κ ) 1 .
N = ( 2 π c ) 2 ϵ 0 m * e 2 γ ,
μ e = 1 2 π c e m * γ κ .

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