Abstract

Understanding the temperature dependence of the optical properties of thin metal films is critical for designing practical devices for high temperature applications in a variety of research areas, including plasmonics and near-field radiative heat transfer. Even though the optical properties of bulk metals at elevated temperatures have been studied, the temperature-dependent data for thin metal films, with thicknesses ranging from few tens to few hundreds of nanometers, is largely missing. In this work we report on the optical constants of single- and polycrystalline gold thin films at elevated temperatures in the wavelength range from 370 to 2000 nm. Our results show that while the real part of the dielectric function changes marginally with increasing temperature, the imaginary part changes drastically. For 200-nm-thick single- and polycrystalline gold films the imaginary part of the dielectric function at 500 °C becomes nearly twice larger than that at room temperature. In contrast, in thinner films (50-nm and 30-nm) the imaginary part can show either increasing or decreasing behavior within the same temperature range and eventually at 500 °C it becomes nearly 3-4 times larger than that at room temperature. The increase in the imaginary part at elevated temperatures significantly reduces the surface plasmon polariton propagation length and the quality factor of the localized surface plasmon resonance for a spherical particle. We provide experiment-fitted models to describe the temperature-dependent gold dielectric function as a sum of one Drude and two critical point oscillators. These causal analytical models could enable accurate multiphysics modelling of gold-based nanophotonic and plasmonic elements in both frequency and time domains.

© 2016 Optical Society of America

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References

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  39. R. T. Beach and R. W. Christy, “Electron-electron scattering in intraband optical conductivity of Cu, Ag, and Au,” Phys. Rev. B 16(12), 5277–5284 (1977).
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    [Crossref]
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  43. K. Fussgaen, W. Martiens, and H. Bilz, “Uv absorption of Ag+ doped alkali halide crystals,” Phys. Status Solidi 12(1), 383–397 (1965).
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2016 (1)

A. Naldoni, F. Riboni, U. Guler, A. Boltasseva, V. M. Shalaev, and A. V. Kildishev, “Solar-Powered plasmon-enhanced heterogeneous catalysis,” Nanophotonics 5(1), 112–133 (2016).

2015 (4)

Q. Gu, J. S. T. Smalley, J. Shane, O. Bondarenko, and Y. Fainman, “Temperature effects in metal-clad semiconductor nanolasers,” Nanophotonics-Berlin 4(1), 26–43 (2015).

U. Guler, V. M. Shalaev, and A. Boltasseva, “Nanoparticle plasmonics: going practical with transition metal nitrides,” Mater. Today 18(4), 227–237 (2015).
[Crossref]

O. D. Miller, S. G. Johnson, and A. W. Rodriguez, “Shape-independent limits to near-field radiative heat transfer,” Phys. Rev. Lett. 115(20), 204302 (2015).
[Crossref] [PubMed]

S. V. Jayanti, J. H. Park, A. Dejneka, D. Chvostova, K. M. McPeak, X. S. Chen, S. H. Oh, and D. J. Norris, “Low-temperature enhancement of plasmonic performance in silver films,” Opt. Mater. Express 5(5), 1147–1155 (2015).
[Crossref]

2014 (4)

U. Guler, A. Boltasseva, and V. M. Shalaev, “Applied physics. Refractory plasmonics,” Science 344(6181), 263–264 (2014).
[Crossref] [PubMed]

N. Zhou, X. F. Xu, A. T. Hammack, B. C. Stipe, K. Z. Gao, W. Scholz, and E. C. Gage, “Plasmonic near-field transducer for heat-assisted magnetic recording,” Nanophotonics 3(3), 141–155 (2014).
[Crossref]

I. A. Fedorov, V. M. Parfenyev, S. S. Vergeles, G. T. Tartakovsky, and A. K. Sarychev, “Allowable Number of Plasmons in Nanoparticle,” JETP Lett. 100(8), 530–534 (2014).
[Crossref]

J. Trollmann and A. Pucci, “Infrared dielectric function of gold films in relation to their morphology,” J. Phys. Chem. C 118(27), 15011–15018 (2014).
[Crossref]

2013 (4)

K. O. Aruda, M. Tagliazucchi, C. M. Sweeney, D. C. Hannah, and E. A. Weiss, “The role of interfacial charge transfer-type interactions in the decay of plasmon excitations in metal nanoparticles,” Phys. Chem. Chem. Phys. 15(20), 7441–7449 (2013).
[Crossref] [PubMed]

S. T. Sundari, K. Srinivasu, S. Dash, and A. K. Tyagi, “Temperature evolution of optical constants and their tuning in silver,” Solid State Commun. 167, 36–39 (2013).
[Crossref]

S. T. Sundari, S. Chandra, and A. K. Tyagi, “Temperature dependent optical properties of silver from spectroscopic ellipsometry and density functional theory calculations,” J. Appl. Phys. 114(3), 033515 (2013).
[Crossref]

O. A. Yeshchenko, I. S. Bondarchuk, V. S. Gurin, I. M. Dmitruk, and A. V. Kotko, “Temperature dependence of the surface plasmon resonance in gold nanoparticles,” Surf. Sci. 608, 275–281 (2013).
[Crossref]

2012 (3)

U. Guler, G. V. Naik, A. Boltasseva, V. M. Shalaev, and A. V. Kildishev, ““Performance analysis of nitride alternative plasmonic materials for localized surface plasmon applications,” Appl. Phys,” B-Lasers O 107(2), 285–291 (2012).
[Crossref]

J. S. G. Bouillard, W. Dickson, D. P. O’Connor, G. A. Wurtz, and A. V. Zayats, “Low-Temperature Plasmonics of Metallic Nanostructures,” Nano Lett. 12(3), 1561–1565 (2012).
[Crossref] [PubMed]

J. H. Park, P. Nagpal, S. H. Oh, and D. J. Norris, “Improved dielectric functions in metallic films obtained via template stripping,” Appl. Phys. Lett. 100(8), 081105 (2012).
[Crossref]

2011 (1)

L. J. Prokopeva, J. D. Borneman, and A. V. Kildishev, “Optical Dispersion Models for Time-Domain Modeling of Metal-Dielectric Nanostructures,” Ieee T. Magn. 47(5), 1150–1153 (2011).
[Crossref]

2010 (4)

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[Crossref] [PubMed]

M. L. Brongersma and V. M. Shalaev, “Applied physics. The case for plasmonics,” Science 328(5977), 440–441 (2010).
[Crossref] [PubMed]

K. P. Chen, V. P. Drachev, J. D. Borneman, A. V. Kildishev, and V. M. Shalaev, “Drude relaxation rate in grained gold nanoantennas,” Nano Lett. 10(3), 916–922 (2010).
[Crossref] [PubMed]

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev. 4(6), 795–808 (2010).
[Crossref]

2008 (1)

T. Brandt, M. Hovel, B. Gompf, and M. Dressel, “Temperature- and frequency-dependent optical properties of ultrathin Au films,” Phys. Rev. B 78(20), 205409 (2008).
[Crossref]

2007 (2)

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold (vol 125, pg 164705, 2006),” J. Chem. Phys. 127(18), 189901 (2007).
[Crossref]

A. Vial, “Implementation of the critical points model in the recursive convolution method for modelling dispersive media with the finite-difference time domain method,” J. Opt. A, Pure Appl. Opt. 9(7), 745–748 (2007).
[Crossref]

2006 (2)

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
[Crossref] [PubMed]

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006).
[Crossref] [PubMed]

2005 (1)

S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98(1), 011101 (2005).
[Crossref]

1997 (1)

Y. K. Sun, X. A. Zhang, and C. P. Grigoropoulos, “Spectral optical functions of silicon in the range of 1.13-4.96 eV at elevated temperatures,”Int. J. Heat Mass Tran. 40(7), 1591–1600 (1997).
[Crossref]

1984 (1)

L. Viña, S. Logothetidis, and M. Cardona, “Temperature-dependence of the dielectric function of germanium,” Phys. Rev. B 30(4), 1979–1991 (1984).
[Crossref]

1981 (1)

G. R. Parkins, W. E. Lawrence, and R. W. Christy, “Intraband optical conductivity sigma(omega,tau) of Cu, Ag, and Au - contribution from electron-electron scattering,” Phys. Rev. B 23(12), 6408–6416 (1981).
[Crossref]

1977 (1)

R. T. Beach and R. W. Christy, “Electron-electron scattering in intraband optical conductivity of Cu, Ag, and Au,” Phys. Rev. B 16(12), 5277–5284 (1977).
[Crossref]

1976 (2)

W. E. Lawrence, “Electron-electron scattering in low-temperature resistivity of noble-metals,” Phys. Rev. B 13(12), 5316–5319 (1976).
[Crossref]

P. Winsemius, F. F. Vankampen, H. P. Lengkeek, and C. G. Vanwent, “Temperature-Dependence of Optical-Properties of Au, Ag and Cu,” J. Phys. F Met. Phys. 6(8), 1583–1606 (1976).
[Crossref]

1973 (1)

W. E. Lawrence and J. W. Wilkins, “Electron-electron scattering in transport coefficients of simple metals,” Phys. Rev. B 7(6), 2317–2332 (1973).
[Crossref]

1971 (1)

N. E. Christensen and B. O. Seraphin, “Relativistic band calculation and optical properties of gold,” Phys. Rev. B-Solid St. 4(10), 3321–3344 (1971).

1969 (1)

C. Y. Young, “Frequency and Temperature Dependence of Optical Effective Mass of Conduction Electrons in Simple Metals,” J. Phys. Chem. Solids 30(12), 2765–2769 (1969).
[Crossref]

1965 (1)

K. Fussgaen, W. Martiens, and H. Bilz, “Uv absorption of Ag+ doped alkali halide crystals,” Phys. Status Solidi 12(1), 383–397 (1965).
[Crossref]

1964 (1)

T. Holstein, “Theory of transport phenomena in an electron-phonon gas,” Ann. Phys. 29(3), 410–535 (1964).
[Crossref]

1958 (1)

R. N. Gurzhi, “On the Theory of the Infrared Absorptivity of Metals,” Sov. Phys. Jetp-Ussr 6(3), 506–512 (1958).

1957 (1)

A. D. Liehr and C. J. Ballhausen, “Intensities in inorganic complexes,” Phys. Rev. 106(6), 1161–1163 (1957).
[Crossref]

Aruda, K. O.

K. O. Aruda, M. Tagliazucchi, C. M. Sweeney, D. C. Hannah, and E. A. Weiss, “The role of interfacial charge transfer-type interactions in the decay of plasmon excitations in metal nanoparticles,” Phys. Chem. Chem. Phys. 15(20), 7441–7449 (2013).
[Crossref] [PubMed]

Atwater, H. A.

S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98(1), 011101 (2005).
[Crossref]

Ballhausen, C. J.

A. D. Liehr and C. J. Ballhausen, “Intensities in inorganic complexes,” Phys. Rev. 106(6), 1161–1163 (1957).
[Crossref]

Barnard, E. S.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[Crossref] [PubMed]

Beach, R. T.

R. T. Beach and R. W. Christy, “Electron-electron scattering in intraband optical conductivity of Cu, Ag, and Au,” Phys. Rev. B 16(12), 5277–5284 (1977).
[Crossref]

Bilz, H.

K. Fussgaen, W. Martiens, and H. Bilz, “Uv absorption of Ag+ doped alkali halide crystals,” Phys. Status Solidi 12(1), 383–397 (1965).
[Crossref]

Boltasseva, A.

A. Naldoni, F. Riboni, U. Guler, A. Boltasseva, V. M. Shalaev, and A. V. Kildishev, “Solar-Powered plasmon-enhanced heterogeneous catalysis,” Nanophotonics 5(1), 112–133 (2016).

U. Guler, V. M. Shalaev, and A. Boltasseva, “Nanoparticle plasmonics: going practical with transition metal nitrides,” Mater. Today 18(4), 227–237 (2015).
[Crossref]

U. Guler, A. Boltasseva, and V. M. Shalaev, “Applied physics. Refractory plasmonics,” Science 344(6181), 263–264 (2014).
[Crossref] [PubMed]

U. Guler, G. V. Naik, A. Boltasseva, V. M. Shalaev, and A. V. Kildishev, ““Performance analysis of nitride alternative plasmonic materials for localized surface plasmon applications,” Appl. Phys,” B-Lasers O 107(2), 285–291 (2012).
[Crossref]

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev. 4(6), 795–808 (2010).
[Crossref]

Bondarchuk, I. S.

O. A. Yeshchenko, I. S. Bondarchuk, V. S. Gurin, I. M. Dmitruk, and A. V. Kotko, “Temperature dependence of the surface plasmon resonance in gold nanoparticles,” Surf. Sci. 608, 275–281 (2013).
[Crossref]

Bondarenko, O.

Q. Gu, J. S. T. Smalley, J. Shane, O. Bondarenko, and Y. Fainman, “Temperature effects in metal-clad semiconductor nanolasers,” Nanophotonics-Berlin 4(1), 26–43 (2015).

Borneman, J. D.

L. J. Prokopeva, J. D. Borneman, and A. V. Kildishev, “Optical Dispersion Models for Time-Domain Modeling of Metal-Dielectric Nanostructures,” Ieee T. Magn. 47(5), 1150–1153 (2011).
[Crossref]

K. P. Chen, V. P. Drachev, J. D. Borneman, A. V. Kildishev, and V. M. Shalaev, “Drude relaxation rate in grained gold nanoantennas,” Nano Lett. 10(3), 916–922 (2010).
[Crossref] [PubMed]

Bouillard, J. S. G.

J. S. G. Bouillard, W. Dickson, D. P. O’Connor, G. A. Wurtz, and A. V. Zayats, “Low-Temperature Plasmonics of Metallic Nanostructures,” Nano Lett. 12(3), 1561–1565 (2012).
[Crossref] [PubMed]

Brandt, T.

T. Brandt, M. Hovel, B. Gompf, and M. Dressel, “Temperature- and frequency-dependent optical properties of ultrathin Au films,” Phys. Rev. B 78(20), 205409 (2008).
[Crossref]

Brongersma, M. L.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[Crossref] [PubMed]

M. L. Brongersma and V. M. Shalaev, “Applied physics. The case for plasmonics,” Science 328(5977), 440–441 (2010).
[Crossref] [PubMed]

Cai, W.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[Crossref] [PubMed]

Cardona, M.

L. Viña, S. Logothetidis, and M. Cardona, “Temperature-dependence of the dielectric function of germanium,” Phys. Rev. B 30(4), 1979–1991 (1984).
[Crossref]

Chandra, S.

S. T. Sundari, S. Chandra, and A. K. Tyagi, “Temperature dependent optical properties of silver from spectroscopic ellipsometry and density functional theory calculations,” J. Appl. Phys. 114(3), 033515 (2013).
[Crossref]

Chen, K. P.

K. P. Chen, V. P. Drachev, J. D. Borneman, A. V. Kildishev, and V. M. Shalaev, “Drude relaxation rate in grained gold nanoantennas,” Nano Lett. 10(3), 916–922 (2010).
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Chen, X. S.

Christensen, N. E.

N. E. Christensen and B. O. Seraphin, “Relativistic band calculation and optical properties of gold,” Phys. Rev. B-Solid St. 4(10), 3321–3344 (1971).

Christy, R. W.

G. R. Parkins, W. E. Lawrence, and R. W. Christy, “Intraband optical conductivity sigma(omega,tau) of Cu, Ag, and Au - contribution from electron-electron scattering,” Phys. Rev. B 23(12), 6408–6416 (1981).
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Dash, S.

S. T. Sundari, K. Srinivasu, S. Dash, and A. K. Tyagi, “Temperature evolution of optical constants and their tuning in silver,” Solid State Commun. 167, 36–39 (2013).
[Crossref]

Dejneka, A.

Dickson, W.

J. S. G. Bouillard, W. Dickson, D. P. O’Connor, G. A. Wurtz, and A. V. Zayats, “Low-Temperature Plasmonics of Metallic Nanostructures,” Nano Lett. 12(3), 1561–1565 (2012).
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Dmitruk, I. M.

O. A. Yeshchenko, I. S. Bondarchuk, V. S. Gurin, I. M. Dmitruk, and A. V. Kotko, “Temperature dependence of the surface plasmon resonance in gold nanoparticles,” Surf. Sci. 608, 275–281 (2013).
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Drachev, V. P.

K. P. Chen, V. P. Drachev, J. D. Borneman, A. V. Kildishev, and V. M. Shalaev, “Drude relaxation rate in grained gold nanoantennas,” Nano Lett. 10(3), 916–922 (2010).
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T. Brandt, M. Hovel, B. Gompf, and M. Dressel, “Temperature- and frequency-dependent optical properties of ultrathin Au films,” Phys. Rev. B 78(20), 205409 (2008).
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P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev. 4(6), 795–808 (2010).
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P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold (vol 125, pg 164705, 2006),” J. Chem. Phys. 127(18), 189901 (2007).
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P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
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Fainman, Y.

Q. Gu, J. S. T. Smalley, J. Shane, O. Bondarenko, and Y. Fainman, “Temperature effects in metal-clad semiconductor nanolasers,” Nanophotonics-Berlin 4(1), 26–43 (2015).

Fedorov, I. A.

I. A. Fedorov, V. M. Parfenyev, S. S. Vergeles, G. T. Tartakovsky, and A. K. Sarychev, “Allowable Number of Plasmons in Nanoparticle,” JETP Lett. 100(8), 530–534 (2014).
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K. Fussgaen, W. Martiens, and H. Bilz, “Uv absorption of Ag+ doped alkali halide crystals,” Phys. Status Solidi 12(1), 383–397 (1965).
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Gage, E. C.

N. Zhou, X. F. Xu, A. T. Hammack, B. C. Stipe, K. Z. Gao, W. Scholz, and E. C. Gage, “Plasmonic near-field transducer for heat-assisted magnetic recording,” Nanophotonics 3(3), 141–155 (2014).
[Crossref]

Gao, K. Z.

N. Zhou, X. F. Xu, A. T. Hammack, B. C. Stipe, K. Z. Gao, W. Scholz, and E. C. Gage, “Plasmonic near-field transducer for heat-assisted magnetic recording,” Nanophotonics 3(3), 141–155 (2014).
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Gompf, B.

T. Brandt, M. Hovel, B. Gompf, and M. Dressel, “Temperature- and frequency-dependent optical properties of ultrathin Au films,” Phys. Rev. B 78(20), 205409 (2008).
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Grigoropoulos, C. P.

Y. K. Sun, X. A. Zhang, and C. P. Grigoropoulos, “Spectral optical functions of silicon in the range of 1.13-4.96 eV at elevated temperatures,”Int. J. Heat Mass Tran. 40(7), 1591–1600 (1997).
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Gu, Q.

Q. Gu, J. S. T. Smalley, J. Shane, O. Bondarenko, and Y. Fainman, “Temperature effects in metal-clad semiconductor nanolasers,” Nanophotonics-Berlin 4(1), 26–43 (2015).

Guler, U.

A. Naldoni, F. Riboni, U. Guler, A. Boltasseva, V. M. Shalaev, and A. V. Kildishev, “Solar-Powered plasmon-enhanced heterogeneous catalysis,” Nanophotonics 5(1), 112–133 (2016).

U. Guler, V. M. Shalaev, and A. Boltasseva, “Nanoparticle plasmonics: going practical with transition metal nitrides,” Mater. Today 18(4), 227–237 (2015).
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U. Guler, A. Boltasseva, and V. M. Shalaev, “Applied physics. Refractory plasmonics,” Science 344(6181), 263–264 (2014).
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U. Guler, G. V. Naik, A. Boltasseva, V. M. Shalaev, and A. V. Kildishev, ““Performance analysis of nitride alternative plasmonic materials for localized surface plasmon applications,” Appl. Phys,” B-Lasers O 107(2), 285–291 (2012).
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Gurin, V. S.

O. A. Yeshchenko, I. S. Bondarchuk, V. S. Gurin, I. M. Dmitruk, and A. V. Kotko, “Temperature dependence of the surface plasmon resonance in gold nanoparticles,” Surf. Sci. 608, 275–281 (2013).
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R. N. Gurzhi, “On the Theory of the Infrared Absorptivity of Metals,” Sov. Phys. Jetp-Ussr 6(3), 506–512 (1958).

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N. Zhou, X. F. Xu, A. T. Hammack, B. C. Stipe, K. Z. Gao, W. Scholz, and E. C. Gage, “Plasmonic near-field transducer for heat-assisted magnetic recording,” Nanophotonics 3(3), 141–155 (2014).
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Hannah, D. C.

K. O. Aruda, M. Tagliazucchi, C. M. Sweeney, D. C. Hannah, and E. A. Weiss, “The role of interfacial charge transfer-type interactions in the decay of plasmon excitations in metal nanoparticles,” Phys. Chem. Chem. Phys. 15(20), 7441–7449 (2013).
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T. Holstein, “Theory of transport phenomena in an electron-phonon gas,” Ann. Phys. 29(3), 410–535 (1964).
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Hovel, M.

T. Brandt, M. Hovel, B. Gompf, and M. Dressel, “Temperature- and frequency-dependent optical properties of ultrathin Au films,” Phys. Rev. B 78(20), 205409 (2008).
[Crossref]

Ishii, S.

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev. 4(6), 795–808 (2010).
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Jayanti, S. V.

Johnson, S. G.

O. D. Miller, S. G. Johnson, and A. W. Rodriguez, “Shape-independent limits to near-field radiative heat transfer,” Phys. Rev. Lett. 115(20), 204302 (2015).
[Crossref] [PubMed]

Jun, Y. C.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
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Kildishev, A. V.

A. Naldoni, F. Riboni, U. Guler, A. Boltasseva, V. M. Shalaev, and A. V. Kildishev, “Solar-Powered plasmon-enhanced heterogeneous catalysis,” Nanophotonics 5(1), 112–133 (2016).

U. Guler, G. V. Naik, A. Boltasseva, V. M. Shalaev, and A. V. Kildishev, ““Performance analysis of nitride alternative plasmonic materials for localized surface plasmon applications,” Appl. Phys,” B-Lasers O 107(2), 285–291 (2012).
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L. J. Prokopeva, J. D. Borneman, and A. V. Kildishev, “Optical Dispersion Models for Time-Domain Modeling of Metal-Dielectric Nanostructures,” Ieee T. Magn. 47(5), 1150–1153 (2011).
[Crossref]

K. P. Chen, V. P. Drachev, J. D. Borneman, A. V. Kildishev, and V. M. Shalaev, “Drude relaxation rate in grained gold nanoantennas,” Nano Lett. 10(3), 916–922 (2010).
[Crossref] [PubMed]

Kotko, A. V.

O. A. Yeshchenko, I. S. Bondarchuk, V. S. Gurin, I. M. Dmitruk, and A. V. Kotko, “Temperature dependence of the surface plasmon resonance in gold nanoparticles,” Surf. Sci. 608, 275–281 (2013).
[Crossref]

Lawrence, W. E.

G. R. Parkins, W. E. Lawrence, and R. W. Christy, “Intraband optical conductivity sigma(omega,tau) of Cu, Ag, and Au - contribution from electron-electron scattering,” Phys. Rev. B 23(12), 6408–6416 (1981).
[Crossref]

W. E. Lawrence, “Electron-electron scattering in low-temperature resistivity of noble-metals,” Phys. Rev. B 13(12), 5316–5319 (1976).
[Crossref]

W. E. Lawrence and J. W. Wilkins, “Electron-electron scattering in transport coefficients of simple metals,” Phys. Rev. B 7(6), 2317–2332 (1973).
[Crossref]

Le Ru, E. C.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold (vol 125, pg 164705, 2006),” J. Chem. Phys. 127(18), 189901 (2007).
[Crossref]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
[Crossref] [PubMed]

Lengkeek, H. P.

P. Winsemius, F. F. Vankampen, H. P. Lengkeek, and C. G. Vanwent, “Temperature-Dependence of Optical-Properties of Au, Ag and Cu,” J. Phys. F Met. Phys. 6(8), 1583–1606 (1976).
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A. D. Liehr and C. J. Ballhausen, “Intensities in inorganic complexes,” Phys. Rev. 106(6), 1161–1163 (1957).
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Logothetidis, S.

L. Viña, S. Logothetidis, and M. Cardona, “Temperature-dependence of the dielectric function of germanium,” Phys. Rev. B 30(4), 1979–1991 (1984).
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Maier, S. A.

S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98(1), 011101 (2005).
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Martiens, W.

K. Fussgaen, W. Martiens, and H. Bilz, “Uv absorption of Ag+ doped alkali halide crystals,” Phys. Status Solidi 12(1), 383–397 (1965).
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McPeak, K. M.

Meyer, M.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold (vol 125, pg 164705, 2006),” J. Chem. Phys. 127(18), 189901 (2007).
[Crossref]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
[Crossref] [PubMed]

Miller, O. D.

O. D. Miller, S. G. Johnson, and A. W. Rodriguez, “Shape-independent limits to near-field radiative heat transfer,” Phys. Rev. Lett. 115(20), 204302 (2015).
[Crossref] [PubMed]

Nagpal, P.

J. H. Park, P. Nagpal, S. H. Oh, and D. J. Norris, “Improved dielectric functions in metallic films obtained via template stripping,” Appl. Phys. Lett. 100(8), 081105 (2012).
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Naik, G. V.

U. Guler, G. V. Naik, A. Boltasseva, V. M. Shalaev, and A. V. Kildishev, ““Performance analysis of nitride alternative plasmonic materials for localized surface plasmon applications,” Appl. Phys,” B-Lasers O 107(2), 285–291 (2012).
[Crossref]

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev. 4(6), 795–808 (2010).
[Crossref]

Naldoni, A.

A. Naldoni, F. Riboni, U. Guler, A. Boltasseva, V. M. Shalaev, and A. V. Kildishev, “Solar-Powered plasmon-enhanced heterogeneous catalysis,” Nanophotonics 5(1), 112–133 (2016).

Norris, D. J.

S. V. Jayanti, J. H. Park, A. Dejneka, D. Chvostova, K. M. McPeak, X. S. Chen, S. H. Oh, and D. J. Norris, “Low-temperature enhancement of plasmonic performance in silver films,” Opt. Mater. Express 5(5), 1147–1155 (2015).
[Crossref]

J. H. Park, P. Nagpal, S. H. Oh, and D. J. Norris, “Improved dielectric functions in metallic films obtained via template stripping,” Appl. Phys. Lett. 100(8), 081105 (2012).
[Crossref]

O’Connor, D. P.

J. S. G. Bouillard, W. Dickson, D. P. O’Connor, G. A. Wurtz, and A. V. Zayats, “Low-Temperature Plasmonics of Metallic Nanostructures,” Nano Lett. 12(3), 1561–1565 (2012).
[Crossref] [PubMed]

Oh, S. H.

S. V. Jayanti, J. H. Park, A. Dejneka, D. Chvostova, K. M. McPeak, X. S. Chen, S. H. Oh, and D. J. Norris, “Low-temperature enhancement of plasmonic performance in silver films,” Opt. Mater. Express 5(5), 1147–1155 (2015).
[Crossref]

J. H. Park, P. Nagpal, S. H. Oh, and D. J. Norris, “Improved dielectric functions in metallic films obtained via template stripping,” Appl. Phys. Lett. 100(8), 081105 (2012).
[Crossref]

Ozbay, E.

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006).
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Parfenyev, V. M.

I. A. Fedorov, V. M. Parfenyev, S. S. Vergeles, G. T. Tartakovsky, and A. K. Sarychev, “Allowable Number of Plasmons in Nanoparticle,” JETP Lett. 100(8), 530–534 (2014).
[Crossref]

Park, J. H.

S. V. Jayanti, J. H. Park, A. Dejneka, D. Chvostova, K. M. McPeak, X. S. Chen, S. H. Oh, and D. J. Norris, “Low-temperature enhancement of plasmonic performance in silver films,” Opt. Mater. Express 5(5), 1147–1155 (2015).
[Crossref]

J. H. Park, P. Nagpal, S. H. Oh, and D. J. Norris, “Improved dielectric functions in metallic films obtained via template stripping,” Appl. Phys. Lett. 100(8), 081105 (2012).
[Crossref]

Parkins, G. R.

G. R. Parkins, W. E. Lawrence, and R. W. Christy, “Intraband optical conductivity sigma(omega,tau) of Cu, Ag, and Au - contribution from electron-electron scattering,” Phys. Rev. B 23(12), 6408–6416 (1981).
[Crossref]

Prokopeva, L. J.

L. J. Prokopeva, J. D. Borneman, and A. V. Kildishev, “Optical Dispersion Models for Time-Domain Modeling of Metal-Dielectric Nanostructures,” Ieee T. Magn. 47(5), 1150–1153 (2011).
[Crossref]

Pucci, A.

J. Trollmann and A. Pucci, “Infrared dielectric function of gold films in relation to their morphology,” J. Phys. Chem. C 118(27), 15011–15018 (2014).
[Crossref]

Riboni, F.

A. Naldoni, F. Riboni, U. Guler, A. Boltasseva, V. M. Shalaev, and A. V. Kildishev, “Solar-Powered plasmon-enhanced heterogeneous catalysis,” Nanophotonics 5(1), 112–133 (2016).

Rodriguez, A. W.

O. D. Miller, S. G. Johnson, and A. W. Rodriguez, “Shape-independent limits to near-field radiative heat transfer,” Phys. Rev. Lett. 115(20), 204302 (2015).
[Crossref] [PubMed]

Sarychev, A. K.

I. A. Fedorov, V. M. Parfenyev, S. S. Vergeles, G. T. Tartakovsky, and A. K. Sarychev, “Allowable Number of Plasmons in Nanoparticle,” JETP Lett. 100(8), 530–534 (2014).
[Crossref]

Scholz, W.

N. Zhou, X. F. Xu, A. T. Hammack, B. C. Stipe, K. Z. Gao, W. Scholz, and E. C. Gage, “Plasmonic near-field transducer for heat-assisted magnetic recording,” Nanophotonics 3(3), 141–155 (2014).
[Crossref]

Schuller, J. A.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[Crossref] [PubMed]

Seraphin, B. O.

N. E. Christensen and B. O. Seraphin, “Relativistic band calculation and optical properties of gold,” Phys. Rev. B-Solid St. 4(10), 3321–3344 (1971).

Shalaev, V. M.

A. Naldoni, F. Riboni, U. Guler, A. Boltasseva, V. M. Shalaev, and A. V. Kildishev, “Solar-Powered plasmon-enhanced heterogeneous catalysis,” Nanophotonics 5(1), 112–133 (2016).

U. Guler, V. M. Shalaev, and A. Boltasseva, “Nanoparticle plasmonics: going practical with transition metal nitrides,” Mater. Today 18(4), 227–237 (2015).
[Crossref]

U. Guler, A. Boltasseva, and V. M. Shalaev, “Applied physics. Refractory plasmonics,” Science 344(6181), 263–264 (2014).
[Crossref] [PubMed]

U. Guler, G. V. Naik, A. Boltasseva, V. M. Shalaev, and A. V. Kildishev, ““Performance analysis of nitride alternative plasmonic materials for localized surface plasmon applications,” Appl. Phys,” B-Lasers O 107(2), 285–291 (2012).
[Crossref]

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev. 4(6), 795–808 (2010).
[Crossref]

K. P. Chen, V. P. Drachev, J. D. Borneman, A. V. Kildishev, and V. M. Shalaev, “Drude relaxation rate in grained gold nanoantennas,” Nano Lett. 10(3), 916–922 (2010).
[Crossref] [PubMed]

M. L. Brongersma and V. M. Shalaev, “Applied physics. The case for plasmonics,” Science 328(5977), 440–441 (2010).
[Crossref] [PubMed]

Shane, J.

Q. Gu, J. S. T. Smalley, J. Shane, O. Bondarenko, and Y. Fainman, “Temperature effects in metal-clad semiconductor nanolasers,” Nanophotonics-Berlin 4(1), 26–43 (2015).

Smalley, J. S. T.

Q. Gu, J. S. T. Smalley, J. Shane, O. Bondarenko, and Y. Fainman, “Temperature effects in metal-clad semiconductor nanolasers,” Nanophotonics-Berlin 4(1), 26–43 (2015).

Srinivasu, K.

S. T. Sundari, K. Srinivasu, S. Dash, and A. K. Tyagi, “Temperature evolution of optical constants and their tuning in silver,” Solid State Commun. 167, 36–39 (2013).
[Crossref]

Stipe, B. C.

N. Zhou, X. F. Xu, A. T. Hammack, B. C. Stipe, K. Z. Gao, W. Scholz, and E. C. Gage, “Plasmonic near-field transducer for heat-assisted magnetic recording,” Nanophotonics 3(3), 141–155 (2014).
[Crossref]

Sun, Y. K.

Y. K. Sun, X. A. Zhang, and C. P. Grigoropoulos, “Spectral optical functions of silicon in the range of 1.13-4.96 eV at elevated temperatures,”Int. J. Heat Mass Tran. 40(7), 1591–1600 (1997).
[Crossref]

Sundari, S. T.

S. T. Sundari, S. Chandra, and A. K. Tyagi, “Temperature dependent optical properties of silver from spectroscopic ellipsometry and density functional theory calculations,” J. Appl. Phys. 114(3), 033515 (2013).
[Crossref]

S. T. Sundari, K. Srinivasu, S. Dash, and A. K. Tyagi, “Temperature evolution of optical constants and their tuning in silver,” Solid State Commun. 167, 36–39 (2013).
[Crossref]

Sweeney, C. M.

K. O. Aruda, M. Tagliazucchi, C. M. Sweeney, D. C. Hannah, and E. A. Weiss, “The role of interfacial charge transfer-type interactions in the decay of plasmon excitations in metal nanoparticles,” Phys. Chem. Chem. Phys. 15(20), 7441–7449 (2013).
[Crossref] [PubMed]

Tagliazucchi, M.

K. O. Aruda, M. Tagliazucchi, C. M. Sweeney, D. C. Hannah, and E. A. Weiss, “The role of interfacial charge transfer-type interactions in the decay of plasmon excitations in metal nanoparticles,” Phys. Chem. Chem. Phys. 15(20), 7441–7449 (2013).
[Crossref] [PubMed]

Tartakovsky, G. T.

I. A. Fedorov, V. M. Parfenyev, S. S. Vergeles, G. T. Tartakovsky, and A. K. Sarychev, “Allowable Number of Plasmons in Nanoparticle,” JETP Lett. 100(8), 530–534 (2014).
[Crossref]

Trollmann, J.

J. Trollmann and A. Pucci, “Infrared dielectric function of gold films in relation to their morphology,” J. Phys. Chem. C 118(27), 15011–15018 (2014).
[Crossref]

Tyagi, A. K.

S. T. Sundari, K. Srinivasu, S. Dash, and A. K. Tyagi, “Temperature evolution of optical constants and their tuning in silver,” Solid State Commun. 167, 36–39 (2013).
[Crossref]

S. T. Sundari, S. Chandra, and A. K. Tyagi, “Temperature dependent optical properties of silver from spectroscopic ellipsometry and density functional theory calculations,” J. Appl. Phys. 114(3), 033515 (2013).
[Crossref]

Vankampen, F. F.

P. Winsemius, F. F. Vankampen, H. P. Lengkeek, and C. G. Vanwent, “Temperature-Dependence of Optical-Properties of Au, Ag and Cu,” J. Phys. F Met. Phys. 6(8), 1583–1606 (1976).
[Crossref]

Vanwent, C. G.

P. Winsemius, F. F. Vankampen, H. P. Lengkeek, and C. G. Vanwent, “Temperature-Dependence of Optical-Properties of Au, Ag and Cu,” J. Phys. F Met. Phys. 6(8), 1583–1606 (1976).
[Crossref]

Vergeles, S. S.

I. A. Fedorov, V. M. Parfenyev, S. S. Vergeles, G. T. Tartakovsky, and A. K. Sarychev, “Allowable Number of Plasmons in Nanoparticle,” JETP Lett. 100(8), 530–534 (2014).
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Vial, A.

A. Vial, “Implementation of the critical points model in the recursive convolution method for modelling dispersive media with the finite-difference time domain method,” J. Opt. A, Pure Appl. Opt. 9(7), 745–748 (2007).
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Viña, L.

L. Viña, S. Logothetidis, and M. Cardona, “Temperature-dependence of the dielectric function of germanium,” Phys. Rev. B 30(4), 1979–1991 (1984).
[Crossref]

Weiss, E. A.

K. O. Aruda, M. Tagliazucchi, C. M. Sweeney, D. C. Hannah, and E. A. Weiss, “The role of interfacial charge transfer-type interactions in the decay of plasmon excitations in metal nanoparticles,” Phys. Chem. Chem. Phys. 15(20), 7441–7449 (2013).
[Crossref] [PubMed]

West, P. R.

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev. 4(6), 795–808 (2010).
[Crossref]

White, J. S.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[Crossref] [PubMed]

Wilkins, J. W.

W. E. Lawrence and J. W. Wilkins, “Electron-electron scattering in transport coefficients of simple metals,” Phys. Rev. B 7(6), 2317–2332 (1973).
[Crossref]

Winsemius, P.

P. Winsemius, F. F. Vankampen, H. P. Lengkeek, and C. G. Vanwent, “Temperature-Dependence of Optical-Properties of Au, Ag and Cu,” J. Phys. F Met. Phys. 6(8), 1583–1606 (1976).
[Crossref]

Wurtz, G. A.

J. S. G. Bouillard, W. Dickson, D. P. O’Connor, G. A. Wurtz, and A. V. Zayats, “Low-Temperature Plasmonics of Metallic Nanostructures,” Nano Lett. 12(3), 1561–1565 (2012).
[Crossref] [PubMed]

Xu, X. F.

N. Zhou, X. F. Xu, A. T. Hammack, B. C. Stipe, K. Z. Gao, W. Scholz, and E. C. Gage, “Plasmonic near-field transducer for heat-assisted magnetic recording,” Nanophotonics 3(3), 141–155 (2014).
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Yeshchenko, O. A.

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Figures (15)

Fig. 1
Fig. 1

Temperature dependent dielectric function of 200 nm thick PC films. (a) and (b) show the real and imaginary parts of the dielectric function, respectively for the first cycle. Likewise (c), (d) and (e), (f) are those of second and third cycle. Different colors represent the dielectric functions at different temperatures (legend in (a) shows the color coding). The imaginary part increases monotonically with increasing temperature whereas the real part decreases with increasing temperature up to 200 °C and increases when the temperature is increased further. The same trend is observed for all three cycles. Insets show the real and imaginary parts for a selected wavelength range.

Fig. 2
Fig. 2

Temperature dependent dielectric function of 200 nm thick SC films. (a) and (b) show the real and imaginary parts of the dielectric function, respectively for the first cycle. Likewise (c), (d) are those of the second cycle. Different colors represent the dielectric functions at different temperatures (legend in (a) shows the color coding). Similar to the PC films the imaginary part increases monotonically with increasing temperature whereas the real part decreases with increasing temperature and saturates at 500 °C. The same trend is observed for both cycles. Insets show the real and imaginary parts for a selected wavelength range.

Fig. 3
Fig. 3

Temperature dependent dielectric function of 50 nm thick poly crystalline films. (a), (b) and (c) show the real part and (d), (e) and (f) show the imaginary part of the dielectric function for different temperature regions. Different colors correspond to dielectric functions at different temperatures (shown in the legend of each figure). As the temperature is increased from room temperature the imaginary part (d) increases up to 200 °C. But for the temperature range from 200 °C- 350 °C the imaginary part (e) reduces, unlike the thicker films. Increasing the temperature further increases the imaginary part drastically reducing the film quality significantly as shown in (f). The real part also displays increasing and deceasing behavior with temperature, depending on the temperature range.

Fig. 4
Fig. 4

Room temperature measurements on the 50 nm and 30 nm thick gold film. The black and red curves represent the room temperature dielectric function on the same sample before and after heating, respectively. Both the real part (a,c) and the imaginary part (b,d) increase after heating the sample.

Fig. 5
Fig. 5

Temperature dependent dielectric function of 30 nm thick poly crystalline films. (a), (b) and (c) show the real part and (d), (e) and (f) show the imaginary part of the dielectric function for different temperature regions. Different colors correspond to dielectric functions at different temperatures (shown in the legend of each figure). Initially, the imaginary part (d) increases as the temperature is increased from room temperature to 200 °C. Similar to the 50 nm thick samples, the imaginary part (e) reduces when the temperature is increased to 250 °C. When the temperature is increased over 300 °C the imaginary part (f) increases and becomes extremely large. The real part also displays increasing and decreasing behavior with temperature, depending on the temperature range.

Fig. 6
Fig. 6

Plasma frequency and Drude broadening of 200-nm-thick poly- (a, b) and singlecrystalline (c, d) films. Depending on the temperature range the plasma frequency either increases or decreases in polycrystalline film while it increases monotonically in singlecrystalline film. On the other hand, Drude broadening increases monotonically with increasing temperature for both samples. The red curve is the fit obtained using Eq. (10).

Fig. 7
Fig. 7

Oscillator strengths (a,b), Oscillator dampings (c,d) and Oscillator energies (e,f) of 200-nm-thick polycrystalline films. The red curve shows the fit obtained using the empirical expressions discussed in the Theory section.

Fig. 8
Fig. 8

Oscillator strengths (a,b), Oscillator dampings (c,d) and Oscillator energies (e,f) of 200-nm-thick singlecrystalline films. The red curve shows the fit obtained using the empirical expressions discussed in the Theory section.

Fig. 9
Fig. 9

Schematic of the experimental setup without (a) and with (b) pinhole in the reflected beam path. Introducing the pinhole (b) significantly suppresses the intensity of background thermal radiation reaching the detector while still allowing most of the reflected beam to pass through.

Fig. 10
Fig. 10

Room temperature dielectric function of the 200-nm-thick poly-crystalline (a,b) and single-crystalline (c,d) films after each cycle. After the first heating cycle the imaginary part reduces (green curves in (a) and (b)) thus improving the film quality. But when the film is subjected to subsequent heat cycles the imaginary part start to increase, gradually degrading the film quality (blue and red curves in (b)). For the case of single crystalline films, the imaginary part increases after each cycle (green and blue curves in (d)). In both the samples the real part only changes marginally with repeated heating.

Fig. 11
Fig. 11

Temperature dependence of the imaginary part of the dielectric function at 1900 nm wavelength for 50-nm-thick (a) and 30-nm-thick films (b). Depending on the temperature range the imaginary part either increase or decreases.

Fig. 12
Fig. 12

AFM images of 50-nm-thick poly-crystalline films. The mean roughness (Ra), which represents the average of the deviations from the center plane, after the heat treatment (b) increased significantly compared to the same samples before heating (a).

Fig. 13
Fig. 13

AFM images of 30-nm-thick poly-crystalline films. Similar to the 50-nm-thick films the mean roughness (Ra) increased after the heat treatment (b) compared to the same samples before heating (a).

Fig. 14
Fig. 14

Optical images of the 50-nm- and 30-nm-thick films. Images before (a) and after heating (b) confirm that the 50-nm-thick film has degraded significantly. Several cracks can be seen in the film after heating (b). Similar behavior is seen in 30-nm-thick films (c, d).

Fig. 15
Fig. 15

Computed values of temperature dependent SPP propagation lengths and QLSPR ( ε 1 ε 2 ) using the optical constants of 200-nm-thick poly-crystalline (a,b) and single crystalline (c,d) gold films. Legends in Figures (a) and (c) show the color coding. In both cases, the propagation lengths and QLSPR reduce by nearly a factor of two compared to the room temperature results when the temperature is raised to 500 °C. These results were computed using the third cycle and second cycle optical constants for the polycrystalline and single crystalline gold films, respectively.

Tables (9)

Tables Icon

Table 1 Comparison of SPP propagation lengths and Q LSPR at 820 nm at Room temperature and 500 °C for 200 nm thick films.

Tables Icon

Table 2 Comparison of SPP propagation lengths and Q LSPR at 820 nm at Room temperature and 450 °C for 50 nm and 30 nm thick films (# represents computed values of propagation lengths and Q LSPR at 500 °C).

Tables Icon

Table 3 200-nm-thick poly-crystalline gold film

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Table 6 200-nm-thick single-crystalline gold film

Tables Icon

Table 8 50-nm-thick gold film

Tables Icon

Table 9 30-nm-thick gold film

Equations (18)

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

ε ^ (ω)= ε ω p 2 ω 2 +i Γ D ω + j=1 2 C j Ω j ( e i ϕ j Ω j ωi γ j + e i ϕ j Ω j +ω+i γ j )
ε ^ (ω) α 0 +(iω) α 1 +...+ (iω) p α p +...+ (iω) m α m β 0 +(iω) β 1 ...+ (iω) q β q +...+...+ (iω) n
ε ^ (ω)= ε σ iω ε 0 + j I 1 a 0,j b 0,j iω + j I 2 a 0,j iω a 1,j b 0,j iω b 1,j ω 2
ω p 2 = N e 2 m * ε 0
N= N 0 1+γ(T T 0 ) ,
Γ D = Γ ee + Γ eϕ ,
1 τ D = 1 τ ee + 1 τ eϕ .
1 τ ee = 1 12 π 3 ΓΔ( 1 E F )[ ( K B T) 2 + ( ω 2π ) 2 ],
1 τ eϕ = 1 τ o [ 2 5 +4 ( T θ ) 5 0 θ T z 4 e z 1 dz ].
1 τ eϕ = 1 τ 0 [ 2 5 + T θ ]
ε ^ (ω,T)= ε ω p (T) 2 ω 2 +i Γ D (T)ω .
d ω p dT = ω 2 ( Γ D 2 ω 2 1 ) ε 1 T +2 Γ D ω ε 2 T 2 ω p
d Γ D dT = ω 3 1+ Γ D 2 / ω 2 ω p 2 ( Γ D ω ε 1 T + ε 2 T )
d ω p dT ω 2 2 ω p ε 1 T
d Γ D dT ω 3 ω p 2 ε 2 T
C(T)= C 0 coth( θ 2T )+α
γ(T)= γ 0 coth( θ 2T )+ γ 1
ε(ω)= ε ω p 2 ω 2 +i Γ D ω + j=1 2 C j Ω j ( e i ϕ j Ω j ωi γ j + e i ϕ j Ω j +ω+i γ j )

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