Abstract

The broadband complex conductivities of transparent conducting oxides (TCO), namely aluminum-doped zinc oxide (AZO), gallium-doped zinc oxide (GZO) and tin-doped indium oxide (ITO), were investigated by terahertz time domain spectroscopy (THz-TDS) in the frequency range from 0.5 to 18 THz using air plasma techniques, supplemented by the photoconductive antenna (PCA) method. The complex conductivities were accurately calculated using a thin film extraction algorithm and analyzed in terms of the Drude conductivity model. All the measured TCOs have a scattering time below 15 fs. We find that a phonon response must be included in the description of the broadband properties of AZO and GZO for an accurate extraction of the scattering time in these materials, which is strongly influenced by the zinc oxide phonon resonance tail even in the low frequency part of the spectrum. The conductivity of AZO is found to be more thickness dependent than GZO and ITO, indicating high importance of the surface states for electron dynamics in AZO. Finally, we measure the transmittance of the TCO films from 10 to 200 THz with Fourier transform infrared spectroscopy (FTIR) measurements, thus closing the gap between THz-TDS measurements (0.5-18 THz) and ellipsometry measurements (200-1000 THz).

© 2015 Optical Society of America

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References

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    [Crossref]
  5. G. V. Naik, J. Kim, and A. Boltasseva, “Oxides and nitrides as alternative plasmonic materials in the optical range,” Opt. Mater. Express 1(6), 1090–1099 (2011).
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  29. J. Dai, J. Zhang, W. Zhang, and D. Grischkowsky, “Terahertz time-domain spectroscopy characterization of the far-infrared absorption and index of refraction of high-resistivity, float-zone silicon,” J. Opt. Soc. Am. B 21(7), 1379 (2004).
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2014 (2)

T. Wang, P. Klarskov, and P. U. Jepsen, “Ultrabroadband THz time-domain spectroscopy of a free-flowing water film,” IEEE Trans. Terahertz Sci. Technol. 2014, 1–7 (2014).

A. Gorodetsky, A. D. Koulouklidis, M. Massaouti, and S. Tzortzakis, “Physics of the conical broadband terahertz emission from two-color laser-induced plasma filaments,” Phys. Rev. A 89(3), 033838 (2014).
[Crossref]

2013 (3)

J. Kim, G. V. Naik, A. V. Gavrilenko, K. Dondapati, V. I. Gavrilenko, S. M. Prokes, O. J. Glembocki, V. M. Shalaev, and A. Boltasseva, “Optical properties of gallium-doped zinc oxide—A low-loss plasmonic material: first-principles theory and experiment,” Phys. Rev. X 3, 041037 (2013).

P. Klarskov, A. C. Strikwerda, K. Iwaszczuk, and P. U. Jepsen, “Experimental three-dimensional beam profiling and modeling of a terahertz beam generated from a two-color air plasma,” New J. Phys. 15(7), 075012 (2013).
[Crossref]

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
[Crossref] [PubMed]

2012 (5)

J. S. Kim, J.-H. Jeong, J. K. Park, Y. J. Baik, I. H. Kim, T.-Y. Seong, and W. M. Kim, “Optical analysis of doped ZnO thin films using nonparabolic conduction-band parameters,” J. Appl. Phys. 111(12), 123507 (2012).
[Crossref]

J. B. Khurgin and A. Boltasseva, “Reflecting upon the losses in plasmonics and metamaterials,” MRS Bull. 37(08), 768–779 (2012).
[Crossref]

G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. 109(23), 8834–8838 (2012).
[Crossref] [PubMed]

D. Zhang, Z. Lü, C. Meng, X. Du, Z. Zhou, Z. Zhao, and J. Yuan, “Synchronizing terahertz wave generation with attosecond bursts,” Phys. Rev. Lett. 109(24), 243002 (2012).
[Crossref] [PubMed]

M. Zalkovskij, C. Z. Bisgaard, A. Novitsky, R. Malureanu, D. Savastru, A. Popescu, P. U. Jepsen, and A. V. Lavrinenko, “Ultrabroadband terahertz spectroscopy of chalcogenide glasses,” Appl. Phys. Lett. 100(3), 031901 (2012).
[Crossref]

2011 (3)

P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging - Modern techniques and applications,” Laser Photonics Rev. 5(1), 124–166 (2011).
[Crossref]

A. Boltasseva and H. A. Atwater, “Materials science. Low-loss plasmonic metamaterials,” Science 331(6015), 290–291 (2011).
[Crossref] [PubMed]

G. V. Naik, J. Kim, and A. Boltasseva, “Oxides and nitrides as alternative plasmonic materials in the optical range,” Opt. Mater. Express 1(6), 1090–1099 (2011).
[Crossref]

2010 (2)

X. Li, Z. Hong, J. He, and Y. Chen, “Precisely optical material parameter determination by time domain waveform rebuilding with THz time-domain spectroscopy,” Opt. Commun. 283(23), 4701–4706 (2010).
[Crossref]

C.-W. Chen, Y.-C. Lin, C.-H. Chang, P. Yu, J.-M. Shieh, and C.-L. Pan, “Frequency-dependent complex conductivities and dielectric responses of indium tin oxide thin films from the visible to the far-Infrared,” IEEE J. Quantum Electron. 46(12), 1746–1754 (2010).
[Crossref]

2008 (1)

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap,”,” Appl. Phys. Lett. 92(1), 011131 (2008).
[Crossref]

2007 (2)

M. Walther, D. G. Cooke, C. Sherstan, M. Hajar, M. R. Freeman, and F. A. Hegmann, “Terahertz conductivity of thin gold films at the metal-insulator percolation transition,” Phys. Rev. B 76(12), 125408 (2007).
[Crossref]

I. Pupeza, R. Wilk, and M. Koch, “Highly accurate optical material parameter determination with THz time-domain spectroscopy,” Opt. Express 15(7), 4335–4350 (2007).
[Crossref] [PubMed]

2006 (2)

X. Xie, J. Dai, and X.-C. Zhang, “Coherent control of THz wave generation in ambient air,” Phys. Rev. Lett. 96(7), 075005 (2006).
[Crossref] [PubMed]

J. Dai, X. Xie, and X.-C. Zhang, “Detection of broadband terahertz waves with a laser-induced plasma in gases,” Phys. Rev. Lett. 97(10), 103903 (2006).
[Crossref] [PubMed]

2004 (1)

1999 (1)

1998 (2)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9(1), 112–147 (1998).
[Crossref]

T.-I. Jeon and D. Grischkowsky, “Observation of a Cole–Davidson type complex conductivity in the limit of very low carrier densities in doped silicon,” Appl. Phys. Lett. 72(18), 2259 (1998).
[Crossref]

1997 (1)

T.-I. Jeon and D. Grischkowsky, “Nature of conduction in doped silicon,” Phys. Rev. Lett. 78(6), 1106–1109 (1997).
[Crossref]

1996 (1)

L. Duvillaret, F. Garet, and J.-L. Coutaz, “A reliable method for extraction of material parameters in terahertz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 2(3), 739–746 (1996).
[Crossref]

1994 (1)

S. Wei and M. Y. Chou, “Phonon dispersions of silicon and germanium from first-principles calculations,” Phys. Rev. B Condens. Matter 50(4), 2221–2226 (1994).
[Crossref] [PubMed]

1983 (1)

I. Hamberg and C. G. Granqvist, “Dielectric function of “undoped” In2O3,” Thin Solid Films 105(2), L83–L86 (1983).
[Crossref]

1973 (1)

C. S. Wang, J. M. Chen, R. Becker, and A. Zdetsis, “Second order raman spectrum and phonon density of states of silicon,” Phys. Lett. 44A(7), 517–518 (1973).
[Crossref]

1959 (1)

R. J. Collins and D. A. Kleinman, “Infrared reflectivity of zinc oxide,” J. Phys. Chem. 11, 190–194 (1959).

1956 (1)

M. Tinkham, “Energy gap interpretation of experiments on infrared transmission through superconducting films,” Phys. Rev. 104(3), 845–846 (1956).
[Crossref]

Atwater, H. A.

A. Boltasseva and H. A. Atwater, “Materials science. Low-loss plasmonic metamaterials,” Science 331(6015), 290–291 (2011).
[Crossref] [PubMed]

Baik, Y. J.

J. S. Kim, J.-H. Jeong, J. K. Park, Y. J. Baik, I. H. Kim, T.-Y. Seong, and W. M. Kim, “Optical analysis of doped ZnO thin films using nonparabolic conduction-band parameters,” J. Appl. Phys. 111(12), 123507 (2012).
[Crossref]

Becker, R.

C. S. Wang, J. M. Chen, R. Becker, and A. Zdetsis, “Second order raman spectrum and phonon density of states of silicon,” Phys. Lett. 44A(7), 517–518 (1973).
[Crossref]

Bisgaard, C. Z.

M. Zalkovskij, C. Z. Bisgaard, A. Novitsky, R. Malureanu, D. Savastru, A. Popescu, P. U. Jepsen, and A. V. Lavrinenko, “Ultrabroadband terahertz spectroscopy of chalcogenide glasses,” Appl. Phys. Lett. 100(3), 031901 (2012).
[Crossref]

Boltasseva, A.

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
[Crossref] [PubMed]

J. Kim, G. V. Naik, A. V. Gavrilenko, K. Dondapati, V. I. Gavrilenko, S. M. Prokes, O. J. Glembocki, V. M. Shalaev, and A. Boltasseva, “Optical properties of gallium-doped zinc oxide—A low-loss plasmonic material: first-principles theory and experiment,” Phys. Rev. X 3, 041037 (2013).

J. B. Khurgin and A. Boltasseva, “Reflecting upon the losses in plasmonics and metamaterials,” MRS Bull. 37(08), 768–779 (2012).
[Crossref]

G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. 109(23), 8834–8838 (2012).
[Crossref] [PubMed]

A. Boltasseva and H. A. Atwater, “Materials science. Low-loss plasmonic metamaterials,” Science 331(6015), 290–291 (2011).
[Crossref] [PubMed]

G. V. Naik, J. Kim, and A. Boltasseva, “Oxides and nitrides as alternative plasmonic materials in the optical range,” Opt. Mater. Express 1(6), 1090–1099 (2011).
[Crossref]

Chang, C.-H.

C.-W. Chen, Y.-C. Lin, C.-H. Chang, P. Yu, J.-M. Shieh, and C.-L. Pan, “Frequency-dependent complex conductivities and dielectric responses of indium tin oxide thin films from the visible to the far-Infrared,” IEEE J. Quantum Electron. 46(12), 1746–1754 (2010).
[Crossref]

Chen, C.-W.

C.-W. Chen, Y.-C. Lin, C.-H. Chang, P. Yu, J.-M. Shieh, and C.-L. Pan, “Frequency-dependent complex conductivities and dielectric responses of indium tin oxide thin films from the visible to the far-Infrared,” IEEE J. Quantum Electron. 46(12), 1746–1754 (2010).
[Crossref]

Chen, J. M.

C. S. Wang, J. M. Chen, R. Becker, and A. Zdetsis, “Second order raman spectrum and phonon density of states of silicon,” Phys. Lett. 44A(7), 517–518 (1973).
[Crossref]

Chen, Y.

X. Li, Z. Hong, J. He, and Y. Chen, “Precisely optical material parameter determination by time domain waveform rebuilding with THz time-domain spectroscopy,” Opt. Commun. 283(23), 4701–4706 (2010).
[Crossref]

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap,”,” Appl. Phys. Lett. 92(1), 011131 (2008).
[Crossref]

Chou, M. Y.

S. Wei and M. Y. Chou, “Phonon dispersions of silicon and germanium from first-principles calculations,” Phys. Rev. B Condens. Matter 50(4), 2221–2226 (1994).
[Crossref] [PubMed]

Collins, R. J.

R. J. Collins and D. A. Kleinman, “Infrared reflectivity of zinc oxide,” J. Phys. Chem. 11, 190–194 (1959).

Cooke, D. G.

P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging - Modern techniques and applications,” Laser Photonics Rev. 5(1), 124–166 (2011).
[Crossref]

M. Walther, D. G. Cooke, C. Sherstan, M. Hajar, M. R. Freeman, and F. A. Hegmann, “Terahertz conductivity of thin gold films at the metal-insulator percolation transition,” Phys. Rev. B 76(12), 125408 (2007).
[Crossref]

Coutaz, J.-L.

L. Duvillaret, F. Garet, and J.-L. Coutaz, “Highly precise determination of optical constants and sample thickness in terahertz time-domain spectroscopy,” Appl. Opt. 38(2), 409–415 (1999).
[Crossref] [PubMed]

L. Duvillaret, F. Garet, and J.-L. Coutaz, “A reliable method for extraction of material parameters in terahertz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 2(3), 739–746 (1996).
[Crossref]

Dai, J.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap,”,” Appl. Phys. Lett. 92(1), 011131 (2008).
[Crossref]

X. Xie, J. Dai, and X.-C. Zhang, “Coherent control of THz wave generation in ambient air,” Phys. Rev. Lett. 96(7), 075005 (2006).
[Crossref] [PubMed]

J. Dai, X. Xie, and X.-C. Zhang, “Detection of broadband terahertz waves with a laser-induced plasma in gases,” Phys. Rev. Lett. 97(10), 103903 (2006).
[Crossref] [PubMed]

J. Dai, J. Zhang, W. Zhang, and D. Grischkowsky, “Terahertz time-domain spectroscopy characterization of the far-infrared absorption and index of refraction of high-resistivity, float-zone silicon,” J. Opt. Soc. Am. B 21(7), 1379 (2004).
[Crossref]

Dondapati, K.

J. Kim, G. V. Naik, A. V. Gavrilenko, K. Dondapati, V. I. Gavrilenko, S. M. Prokes, O. J. Glembocki, V. M. Shalaev, and A. Boltasseva, “Optical properties of gallium-doped zinc oxide—A low-loss plasmonic material: first-principles theory and experiment,” Phys. Rev. X 3, 041037 (2013).

Du, X.

D. Zhang, Z. Lü, C. Meng, X. Du, Z. Zhou, Z. Zhao, and J. Yuan, “Synchronizing terahertz wave generation with attosecond bursts,” Phys. Rev. Lett. 109(24), 243002 (2012).
[Crossref] [PubMed]

Duvillaret, L.

L. Duvillaret, F. Garet, and J.-L. Coutaz, “Highly precise determination of optical constants and sample thickness in terahertz time-domain spectroscopy,” Appl. Opt. 38(2), 409–415 (1999).
[Crossref] [PubMed]

L. Duvillaret, F. Garet, and J.-L. Coutaz, “A reliable method for extraction of material parameters in terahertz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 2(3), 739–746 (1996).
[Crossref]

Fletcher, C.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap,”,” Appl. Phys. Lett. 92(1), 011131 (2008).
[Crossref]

Freeman, M. R.

M. Walther, D. G. Cooke, C. Sherstan, M. Hajar, M. R. Freeman, and F. A. Hegmann, “Terahertz conductivity of thin gold films at the metal-insulator percolation transition,” Phys. Rev. B 76(12), 125408 (2007).
[Crossref]

Garet, F.

L. Duvillaret, F. Garet, and J.-L. Coutaz, “Highly precise determination of optical constants and sample thickness in terahertz time-domain spectroscopy,” Appl. Opt. 38(2), 409–415 (1999).
[Crossref] [PubMed]

L. Duvillaret, F. Garet, and J.-L. Coutaz, “A reliable method for extraction of material parameters in terahertz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 2(3), 739–746 (1996).
[Crossref]

Gavrilenko, A. V.

J. Kim, G. V. Naik, A. V. Gavrilenko, K. Dondapati, V. I. Gavrilenko, S. M. Prokes, O. J. Glembocki, V. M. Shalaev, and A. Boltasseva, “Optical properties of gallium-doped zinc oxide—A low-loss plasmonic material: first-principles theory and experiment,” Phys. Rev. X 3, 041037 (2013).

Gavrilenko, V. I.

J. Kim, G. V. Naik, A. V. Gavrilenko, K. Dondapati, V. I. Gavrilenko, S. M. Prokes, O. J. Glembocki, V. M. Shalaev, and A. Boltasseva, “Optical properties of gallium-doped zinc oxide—A low-loss plasmonic material: first-principles theory and experiment,” Phys. Rev. X 3, 041037 (2013).

Glembocki, O. J.

J. Kim, G. V. Naik, A. V. Gavrilenko, K. Dondapati, V. I. Gavrilenko, S. M. Prokes, O. J. Glembocki, V. M. Shalaev, and A. Boltasseva, “Optical properties of gallium-doped zinc oxide—A low-loss plasmonic material: first-principles theory and experiment,” Phys. Rev. X 3, 041037 (2013).

Gorodetsky, A.

A. Gorodetsky, A. D. Koulouklidis, M. Massaouti, and S. Tzortzakis, “Physics of the conical broadband terahertz emission from two-color laser-induced plasma filaments,” Phys. Rev. A 89(3), 033838 (2014).
[Crossref]

Granqvist, C. G.

I. Hamberg and C. G. Granqvist, “Dielectric function of “undoped” In2O3,” Thin Solid Films 105(2), L83–L86 (1983).
[Crossref]

Grischkowsky, D.

J. Dai, J. Zhang, W. Zhang, and D. Grischkowsky, “Terahertz time-domain spectroscopy characterization of the far-infrared absorption and index of refraction of high-resistivity, float-zone silicon,” J. Opt. Soc. Am. B 21(7), 1379 (2004).
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T.-I. Jeon and D. Grischkowsky, “Observation of a Cole–Davidson type complex conductivity in the limit of very low carrier densities in doped silicon,” Appl. Phys. Lett. 72(18), 2259 (1998).
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T.-I. Jeon and D. Grischkowsky, “Nature of conduction in doped silicon,” Phys. Rev. Lett. 78(6), 1106–1109 (1997).
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Hajar, M.

M. Walther, D. G. Cooke, C. Sherstan, M. Hajar, M. R. Freeman, and F. A. Hegmann, “Terahertz conductivity of thin gold films at the metal-insulator percolation transition,” Phys. Rev. B 76(12), 125408 (2007).
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Hamberg, I.

I. Hamberg and C. G. Granqvist, “Dielectric function of “undoped” In2O3,” Thin Solid Films 105(2), L83–L86 (1983).
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He, J.

X. Li, Z. Hong, J. He, and Y. Chen, “Precisely optical material parameter determination by time domain waveform rebuilding with THz time-domain spectroscopy,” Opt. Commun. 283(23), 4701–4706 (2010).
[Crossref]

Hegmann, F. A.

M. Walther, D. G. Cooke, C. Sherstan, M. Hajar, M. R. Freeman, and F. A. Hegmann, “Terahertz conductivity of thin gold films at the metal-insulator percolation transition,” Phys. Rev. B 76(12), 125408 (2007).
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Hong, Z.

X. Li, Z. Hong, J. He, and Y. Chen, “Precisely optical material parameter determination by time domain waveform rebuilding with THz time-domain spectroscopy,” Opt. Commun. 283(23), 4701–4706 (2010).
[Crossref]

Iwaszczuk, K.

P. Klarskov, A. C. Strikwerda, K. Iwaszczuk, and P. U. Jepsen, “Experimental three-dimensional beam profiling and modeling of a terahertz beam generated from a two-color air plasma,” New J. Phys. 15(7), 075012 (2013).
[Crossref]

Jeon, T.-I.

T.-I. Jeon and D. Grischkowsky, “Observation of a Cole–Davidson type complex conductivity in the limit of very low carrier densities in doped silicon,” Appl. Phys. Lett. 72(18), 2259 (1998).
[Crossref]

T.-I. Jeon and D. Grischkowsky, “Nature of conduction in doped silicon,” Phys. Rev. Lett. 78(6), 1106–1109 (1997).
[Crossref]

Jeong, J.-H.

J. S. Kim, J.-H. Jeong, J. K. Park, Y. J. Baik, I. H. Kim, T.-Y. Seong, and W. M. Kim, “Optical analysis of doped ZnO thin films using nonparabolic conduction-band parameters,” J. Appl. Phys. 111(12), 123507 (2012).
[Crossref]

Jepsen, P. U.

T. Wang, P. Klarskov, and P. U. Jepsen, “Ultrabroadband THz time-domain spectroscopy of a free-flowing water film,” IEEE Trans. Terahertz Sci. Technol. 2014, 1–7 (2014).

P. Klarskov, A. C. Strikwerda, K. Iwaszczuk, and P. U. Jepsen, “Experimental three-dimensional beam profiling and modeling of a terahertz beam generated from a two-color air plasma,” New J. Phys. 15(7), 075012 (2013).
[Crossref]

M. Zalkovskij, C. Z. Bisgaard, A. Novitsky, R. Malureanu, D. Savastru, A. Popescu, P. U. Jepsen, and A. V. Lavrinenko, “Ultrabroadband terahertz spectroscopy of chalcogenide glasses,” Appl. Phys. Lett. 100(3), 031901 (2012).
[Crossref]

P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging - Modern techniques and applications,” Laser Photonics Rev. 5(1), 124–166 (2011).
[Crossref]

Johnson, K.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap,”,” Appl. Phys. Lett. 92(1), 011131 (2008).
[Crossref]

Karpowicz, N.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap,”,” Appl. Phys. Lett. 92(1), 011131 (2008).
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J. B. Khurgin and A. Boltasseva, “Reflecting upon the losses in plasmonics and metamaterials,” MRS Bull. 37(08), 768–779 (2012).
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G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. 109(23), 8834–8838 (2012).
[Crossref] [PubMed]

Kim, I. H.

J. S. Kim, J.-H. Jeong, J. K. Park, Y. J. Baik, I. H. Kim, T.-Y. Seong, and W. M. Kim, “Optical analysis of doped ZnO thin films using nonparabolic conduction-band parameters,” J. Appl. Phys. 111(12), 123507 (2012).
[Crossref]

Kim, J.

J. Kim, G. V. Naik, A. V. Gavrilenko, K. Dondapati, V. I. Gavrilenko, S. M. Prokes, O. J. Glembocki, V. M. Shalaev, and A. Boltasseva, “Optical properties of gallium-doped zinc oxide—A low-loss plasmonic material: first-principles theory and experiment,” Phys. Rev. X 3, 041037 (2013).

G. V. Naik, J. Kim, and A. Boltasseva, “Oxides and nitrides as alternative plasmonic materials in the optical range,” Opt. Mater. Express 1(6), 1090–1099 (2011).
[Crossref]

Kim, J. S.

J. S. Kim, J.-H. Jeong, J. K. Park, Y. J. Baik, I. H. Kim, T.-Y. Seong, and W. M. Kim, “Optical analysis of doped ZnO thin films using nonparabolic conduction-band parameters,” J. Appl. Phys. 111(12), 123507 (2012).
[Crossref]

Kim, W. M.

J. S. Kim, J.-H. Jeong, J. K. Park, Y. J. Baik, I. H. Kim, T.-Y. Seong, and W. M. Kim, “Optical analysis of doped ZnO thin films using nonparabolic conduction-band parameters,” J. Appl. Phys. 111(12), 123507 (2012).
[Crossref]

Klarskov, P.

T. Wang, P. Klarskov, and P. U. Jepsen, “Ultrabroadband THz time-domain spectroscopy of a free-flowing water film,” IEEE Trans. Terahertz Sci. Technol. 2014, 1–7 (2014).

P. Klarskov, A. C. Strikwerda, K. Iwaszczuk, and P. U. Jepsen, “Experimental three-dimensional beam profiling and modeling of a terahertz beam generated from a two-color air plasma,” New J. Phys. 15(7), 075012 (2013).
[Crossref]

Kleinman, D. A.

R. J. Collins and D. A. Kleinman, “Infrared reflectivity of zinc oxide,” J. Phys. Chem. 11, 190–194 (1959).

Koch, M.

P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging - Modern techniques and applications,” Laser Photonics Rev. 5(1), 124–166 (2011).
[Crossref]

I. Pupeza, R. Wilk, and M. Koch, “Highly accurate optical material parameter determination with THz time-domain spectroscopy,” Opt. Express 15(7), 4335–4350 (2007).
[Crossref] [PubMed]

Koulouklidis, A. D.

A. Gorodetsky, A. D. Koulouklidis, M. Massaouti, and S. Tzortzakis, “Physics of the conical broadband terahertz emission from two-color laser-induced plasma filaments,” Phys. Rev. A 89(3), 033838 (2014).
[Crossref]

Lagarias, J. C.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9(1), 112–147 (1998).
[Crossref]

Lavrinenko, A. V.

M. Zalkovskij, C. Z. Bisgaard, A. Novitsky, R. Malureanu, D. Savastru, A. Popescu, P. U. Jepsen, and A. V. Lavrinenko, “Ultrabroadband terahertz spectroscopy of chalcogenide glasses,” Appl. Phys. Lett. 100(3), 031901 (2012).
[Crossref]

Lesimple, A.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap,”,” Appl. Phys. Lett. 92(1), 011131 (2008).
[Crossref]

Li, X.

X. Li, Z. Hong, J. He, and Y. Chen, “Precisely optical material parameter determination by time domain waveform rebuilding with THz time-domain spectroscopy,” Opt. Commun. 283(23), 4701–4706 (2010).
[Crossref]

Lin, Y.-C.

C.-W. Chen, Y.-C. Lin, C.-H. Chang, P. Yu, J.-M. Shieh, and C.-L. Pan, “Frequency-dependent complex conductivities and dielectric responses of indium tin oxide thin films from the visible to the far-Infrared,” IEEE J. Quantum Electron. 46(12), 1746–1754 (2010).
[Crossref]

Liu, J.

G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. 109(23), 8834–8838 (2012).
[Crossref] [PubMed]

Lu, X.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap,”,” Appl. Phys. Lett. 92(1), 011131 (2008).
[Crossref]

Lü, Z.

D. Zhang, Z. Lü, C. Meng, X. Du, Z. Zhou, Z. Zhao, and J. Yuan, “Synchronizing terahertz wave generation with attosecond bursts,” Phys. Rev. Lett. 109(24), 243002 (2012).
[Crossref] [PubMed]

Malureanu, R.

M. Zalkovskij, C. Z. Bisgaard, A. Novitsky, R. Malureanu, D. Savastru, A. Popescu, P. U. Jepsen, and A. V. Lavrinenko, “Ultrabroadband terahertz spectroscopy of chalcogenide glasses,” Appl. Phys. Lett. 100(3), 031901 (2012).
[Crossref]

Mamer, O.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap,”,” Appl. Phys. Lett. 92(1), 011131 (2008).
[Crossref]

Massaouti, M.

A. Gorodetsky, A. D. Koulouklidis, M. Massaouti, and S. Tzortzakis, “Physics of the conical broadband terahertz emission from two-color laser-induced plasma filaments,” Phys. Rev. A 89(3), 033838 (2014).
[Crossref]

Meng, C.

D. Zhang, Z. Lü, C. Meng, X. Du, Z. Zhou, Z. Zhao, and J. Yuan, “Synchronizing terahertz wave generation with attosecond bursts,” Phys. Rev. Lett. 109(24), 243002 (2012).
[Crossref] [PubMed]

Naik, G. V.

J. Kim, G. V. Naik, A. V. Gavrilenko, K. Dondapati, V. I. Gavrilenko, S. M. Prokes, O. J. Glembocki, V. M. Shalaev, and A. Boltasseva, “Optical properties of gallium-doped zinc oxide—A low-loss plasmonic material: first-principles theory and experiment,” Phys. Rev. X 3, 041037 (2013).

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
[Crossref] [PubMed]

G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. 109(23), 8834–8838 (2012).
[Crossref] [PubMed]

G. V. Naik, J. Kim, and A. Boltasseva, “Oxides and nitrides as alternative plasmonic materials in the optical range,” Opt. Mater. Express 1(6), 1090–1099 (2011).
[Crossref]

Novitsky, A.

M. Zalkovskij, C. Z. Bisgaard, A. Novitsky, R. Malureanu, D. Savastru, A. Popescu, P. U. Jepsen, and A. V. Lavrinenko, “Ultrabroadband terahertz spectroscopy of chalcogenide glasses,” Appl. Phys. Lett. 100(3), 031901 (2012).
[Crossref]

Pan, C.-L.

C.-W. Chen, Y.-C. Lin, C.-H. Chang, P. Yu, J.-M. Shieh, and C.-L. Pan, “Frequency-dependent complex conductivities and dielectric responses of indium tin oxide thin films from the visible to the far-Infrared,” IEEE J. Quantum Electron. 46(12), 1746–1754 (2010).
[Crossref]

Park, J. K.

J. S. Kim, J.-H. Jeong, J. K. Park, Y. J. Baik, I. H. Kim, T.-Y. Seong, and W. M. Kim, “Optical analysis of doped ZnO thin films using nonparabolic conduction-band parameters,” J. Appl. Phys. 111(12), 123507 (2012).
[Crossref]

Popescu, A.

M. Zalkovskij, C. Z. Bisgaard, A. Novitsky, R. Malureanu, D. Savastru, A. Popescu, P. U. Jepsen, and A. V. Lavrinenko, “Ultrabroadband terahertz spectroscopy of chalcogenide glasses,” Appl. Phys. Lett. 100(3), 031901 (2012).
[Crossref]

Price-Gallagher, M.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap,”,” Appl. Phys. Lett. 92(1), 011131 (2008).
[Crossref]

Prokes, S. M.

J. Kim, G. V. Naik, A. V. Gavrilenko, K. Dondapati, V. I. Gavrilenko, S. M. Prokes, O. J. Glembocki, V. M. Shalaev, and A. Boltasseva, “Optical properties of gallium-doped zinc oxide—A low-loss plasmonic material: first-principles theory and experiment,” Phys. Rev. X 3, 041037 (2013).

Pupeza, I.

Reeds, J. A.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9(1), 112–147 (1998).
[Crossref]

Savastru, D.

M. Zalkovskij, C. Z. Bisgaard, A. Novitsky, R. Malureanu, D. Savastru, A. Popescu, P. U. Jepsen, and A. V. Lavrinenko, “Ultrabroadband terahertz spectroscopy of chalcogenide glasses,” Appl. Phys. Lett. 100(3), 031901 (2012).
[Crossref]

Seong, T.-Y.

J. S. Kim, J.-H. Jeong, J. K. Park, Y. J. Baik, I. H. Kim, T.-Y. Seong, and W. M. Kim, “Optical analysis of doped ZnO thin films using nonparabolic conduction-band parameters,” J. Appl. Phys. 111(12), 123507 (2012).
[Crossref]

Shalaev, V. M.

J. Kim, G. V. Naik, A. V. Gavrilenko, K. Dondapati, V. I. Gavrilenko, S. M. Prokes, O. J. Glembocki, V. M. Shalaev, and A. Boltasseva, “Optical properties of gallium-doped zinc oxide—A low-loss plasmonic material: first-principles theory and experiment,” Phys. Rev. X 3, 041037 (2013).

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
[Crossref] [PubMed]

G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. 109(23), 8834–8838 (2012).
[Crossref] [PubMed]

Sherstan, C.

M. Walther, D. G. Cooke, C. Sherstan, M. Hajar, M. R. Freeman, and F. A. Hegmann, “Terahertz conductivity of thin gold films at the metal-insulator percolation transition,” Phys. Rev. B 76(12), 125408 (2007).
[Crossref]

Shieh, J.-M.

C.-W. Chen, Y.-C. Lin, C.-H. Chang, P. Yu, J.-M. Shieh, and C.-L. Pan, “Frequency-dependent complex conductivities and dielectric responses of indium tin oxide thin films from the visible to the far-Infrared,” IEEE J. Quantum Electron. 46(12), 1746–1754 (2010).
[Crossref]

Strikwerda, A. C.

P. Klarskov, A. C. Strikwerda, K. Iwaszczuk, and P. U. Jepsen, “Experimental three-dimensional beam profiling and modeling of a terahertz beam generated from a two-color air plasma,” New J. Phys. 15(7), 075012 (2013).
[Crossref]

Tinkham, M.

M. Tinkham, “Energy gap interpretation of experiments on infrared transmission through superconducting films,” Phys. Rev. 104(3), 845–846 (1956).
[Crossref]

Tzortzakis, S.

A. Gorodetsky, A. D. Koulouklidis, M. Massaouti, and S. Tzortzakis, “Physics of the conical broadband terahertz emission from two-color laser-induced plasma filaments,” Phys. Rev. A 89(3), 033838 (2014).
[Crossref]

Walther, M.

M. Walther, D. G. Cooke, C. Sherstan, M. Hajar, M. R. Freeman, and F. A. Hegmann, “Terahertz conductivity of thin gold films at the metal-insulator percolation transition,” Phys. Rev. B 76(12), 125408 (2007).
[Crossref]

Wang, C. S.

C. S. Wang, J. M. Chen, R. Becker, and A. Zdetsis, “Second order raman spectrum and phonon density of states of silicon,” Phys. Lett. 44A(7), 517–518 (1973).
[Crossref]

Wang, T.

T. Wang, P. Klarskov, and P. U. Jepsen, “Ultrabroadband THz time-domain spectroscopy of a free-flowing water film,” IEEE Trans. Terahertz Sci. Technol. 2014, 1–7 (2014).

Wei, S.

S. Wei and M. Y. Chou, “Phonon dispersions of silicon and germanium from first-principles calculations,” Phys. Rev. B Condens. Matter 50(4), 2221–2226 (1994).
[Crossref] [PubMed]

Wilk, R.

Wright, M. H.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9(1), 112–147 (1998).
[Crossref]

Wright, P. E.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9(1), 112–147 (1998).
[Crossref]

Xie, X.

J. Dai, X. Xie, and X.-C. Zhang, “Detection of broadband terahertz waves with a laser-induced plasma in gases,” Phys. Rev. Lett. 97(10), 103903 (2006).
[Crossref] [PubMed]

X. Xie, J. Dai, and X.-C. Zhang, “Coherent control of THz wave generation in ambient air,” Phys. Rev. Lett. 96(7), 075005 (2006).
[Crossref] [PubMed]

Yamaguchi, M.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap,”,” Appl. Phys. Lett. 92(1), 011131 (2008).
[Crossref]

Yu, P.

C.-W. Chen, Y.-C. Lin, C.-H. Chang, P. Yu, J.-M. Shieh, and C.-L. Pan, “Frequency-dependent complex conductivities and dielectric responses of indium tin oxide thin films from the visible to the far-Infrared,” IEEE J. Quantum Electron. 46(12), 1746–1754 (2010).
[Crossref]

Yuan, J.

D. Zhang, Z. Lü, C. Meng, X. Du, Z. Zhou, Z. Zhao, and J. Yuan, “Synchronizing terahertz wave generation with attosecond bursts,” Phys. Rev. Lett. 109(24), 243002 (2012).
[Crossref] [PubMed]

Zalkovskij, M.

M. Zalkovskij, C. Z. Bisgaard, A. Novitsky, R. Malureanu, D. Savastru, A. Popescu, P. U. Jepsen, and A. V. Lavrinenko, “Ultrabroadband terahertz spectroscopy of chalcogenide glasses,” Appl. Phys. Lett. 100(3), 031901 (2012).
[Crossref]

Zdetsis, A.

C. S. Wang, J. M. Chen, R. Becker, and A. Zdetsis, “Second order raman spectrum and phonon density of states of silicon,” Phys. Lett. 44A(7), 517–518 (1973).
[Crossref]

Zhang, C.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap,”,” Appl. Phys. Lett. 92(1), 011131 (2008).
[Crossref]

Zhang, D.

D. Zhang, Z. Lü, C. Meng, X. Du, Z. Zhou, Z. Zhao, and J. Yuan, “Synchronizing terahertz wave generation with attosecond bursts,” Phys. Rev. Lett. 109(24), 243002 (2012).
[Crossref] [PubMed]

Zhang, J.

Zhang, L.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap,”,” Appl. Phys. Lett. 92(1), 011131 (2008).
[Crossref]

Zhang, W.

Zhang, X.-C.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap,”,” Appl. Phys. Lett. 92(1), 011131 (2008).
[Crossref]

X. Xie, J. Dai, and X.-C. Zhang, “Coherent control of THz wave generation in ambient air,” Phys. Rev. Lett. 96(7), 075005 (2006).
[Crossref] [PubMed]

J. Dai, X. Xie, and X.-C. Zhang, “Detection of broadband terahertz waves with a laser-induced plasma in gases,” Phys. Rev. Lett. 97(10), 103903 (2006).
[Crossref] [PubMed]

Zhao, H.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap,”,” Appl. Phys. Lett. 92(1), 011131 (2008).
[Crossref]

Zhao, Z.

D. Zhang, Z. Lü, C. Meng, X. Du, Z. Zhou, Z. Zhao, and J. Yuan, “Synchronizing terahertz wave generation with attosecond bursts,” Phys. Rev. Lett. 109(24), 243002 (2012).
[Crossref] [PubMed]

Zhou, Z.

D. Zhang, Z. Lü, C. Meng, X. Du, Z. Zhou, Z. Zhao, and J. Yuan, “Synchronizing terahertz wave generation with attosecond bursts,” Phys. Rev. Lett. 109(24), 243002 (2012).
[Crossref] [PubMed]

Adv. Mater. (1)

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (3)

T.-I. Jeon and D. Grischkowsky, “Observation of a Cole–Davidson type complex conductivity in the limit of very low carrier densities in doped silicon,” Appl. Phys. Lett. 72(18), 2259 (1998).
[Crossref]

M. Zalkovskij, C. Z. Bisgaard, A. Novitsky, R. Malureanu, D. Savastru, A. Popescu, P. U. Jepsen, and A. V. Lavrinenko, “Ultrabroadband terahertz spectroscopy of chalcogenide glasses,” Appl. Phys. Lett. 100(3), 031901 (2012).
[Crossref]

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap,”,” Appl. Phys. Lett. 92(1), 011131 (2008).
[Crossref]

IEEE J. Quantum Electron. (1)

C.-W. Chen, Y.-C. Lin, C.-H. Chang, P. Yu, J.-M. Shieh, and C.-L. Pan, “Frequency-dependent complex conductivities and dielectric responses of indium tin oxide thin films from the visible to the far-Infrared,” IEEE J. Quantum Electron. 46(12), 1746–1754 (2010).
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IEEE J. Sel. Top. Quantum Electron. (1)

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Figures (5)

Fig. 1
Fig. 1 (a) Time traces of terahertz pulses generated by photoconductive antenna transmitted through 525 µm thick HR Si (black line) scaled down by a factor of 4 for comparision, 525 µm thick HR Si with 215 nm thick GZO (red line), 260 nm thick AZO (green line) and 210 nm thick ITO (blue line). Inset: sketch of the sample and illumination geometry. (b) Time trace of terahertz pulse generated by air plasma. Inset of (b): (c) the amplitude spectra of the PCA and air plasma THz pulses.
Fig. 2
Fig. 2 Optical constants of TCOs: 110 nm thick ITO (black), 140 nm thick AZO (red) and 122 nm thick GZO (blue), with real (solid lines) and imaginary (dashed lines) parts of the complex index of refraction.
Fig. 3
Fig. 3 The extracted real (filled symbols) and imaginary (open symbols) parts of the complex conductivity of the TCOs. Drude-Lorentz fits (red and blue curves) up to 18 THz for (a) GZO and (b) AZO. (c) Drude fits to 18 THz for ITO. The error bars for 215 nm thick GZO is the standard deviation calculated from 3 measurements, which is also representative for AZO and ITO measurements. The data between 0.5 and 2 THz were obtained with PCA measurements.
Fig. 4
Fig. 4 Comparison between Drude fit (blue lines) and Drude-Lorentz fit (green lines) for the extracted complex conductivity of 215 nm thick GZO. The real and imaginary parts are plotted in dots and open circles, respectively.
Fig. 5
Fig. 5 Transmittance of GZO, AZO and ITO from 0.5 THz to 1000 THz measured with THz-TDS, FTIR spectroscopy and ellipsometry.

Tables (2)

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Table 1 Fit parameters extracted from Drude fits to 8 THz

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Table 2 Fit parameters extracted from Drude-Lorentz fits to 18 THz in Fig. 3

Equations (7)

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Δ(ω)= ( A calc (ω) A exp (ω) ) 2 +| θ calc (ω) θ exp (ω) |
σ ' init (ω)= n sub +1 Z 0 d [ cos( θ exp (ω)) A exp (ω) 1 ]
σ' ' init (ω)= n sub +1 Z 0 d sin( θ exp (ω)) A exp (ω) ,
σ'(ω)= σ dc 1+ (ωτ) 2 ,
σ''(ω)= σ dc ωτ 1+ (ωτ) 2 .
σ'(ω)= σ dc 1+ (ωτ) 2 + ε 0 ω pl 2 ω 2 γ ( ω 0 2 ω 2 ) 2 + (ωγ) 2 ,
σ''(ω)= σ dc ωτ 1+ (ωτ) 2 ε 0 ω pl 2 ω( ω 0 2 ω 2 ) ( ω 0 2 ω 2 ) 2 + (ωγ) 2 ,

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