Abstract

Silicon is widely used as the material of choice for semiconductor and insulator applications in nanoelectronics, micro-electro-mechanical systems, solar cells, and on-chip photonics. In stark contrast, in this paper, we explore silicon’s metallic properties and show that it can support propagating surface plasmons, collective charge oscillations, in the extreme ultraviolet (EUV) energy regime not possible with other plasmonic materials such as aluminum, silver, or gold. This is fundamentally different from conventional approaches, where doping semiconductors is considered necessary to observe plasmonic behavior. We experimentally map the photonic band structure of EUV surface and bulk plasmons in silicon using momentum-resolved electron energy loss spectroscopy. Our experimental observations are validated by macroscopic electrodynamic electron energy loss theory simulations as well as quantum density functional theory calculations. As an example of exploiting these EUV plasmons for applications, we propose a tunable and broadband thresholdless Cherenkov radiation source in the EUV using silicon plasmonic metamaterials. Our work can pave the way for the field of EUV plasmonics.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
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    [Crossref]

2018 (3)

A. H. Atabaki, S. Moazeni, F. Pavanello, H. Gevorgyan, J. Notaros, L. Alloatti, M. T. Wade, C. Sun, S. A. Kruger, H. Meng, K. A. Qubaisi, I. Wang, B. Zhang, A. Khilo, C. V. Baiocco, M. A. Popović, V. M. Stojanović, and R. J. Ram, “Integrating photonics with silicon nanoelectronics for the next generation of systems on a chip,” Nature 556, 349–354 (2018).
[Crossref]

E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, M. Faraji-Dana, and A. Faraon, “MEMS-tunable dielectric metasurface lens,” Nat. Commun. 9, 812 (2018).
[Crossref]

S. Jahani, S. Kim, J. Atkinson, J. C. Wirth, F. Kalhor, A. A. Noman, W. D. Newman, P. Shekhar, K. Han, V. Van, R. G. DeCorby, L. Chrostowski, M. Qi, and Z. Jacob, “Controlling evanescent waves using silicon photonic all-dielectric metamaterials for dense integration,” Nat. Commun. 9, 1893 (2018).
[Crossref]

2017 (3)

I. Staude and J. Schilling, “Metamaterial-inspired silicon nanophotonics,” Nat. Photonics 11, 274–284 (2017).
[Crossref]

P. Shekhar, M. Malac, V. Gaind, N. Dalili, A. Meldrum, and Z. Jacob, “Momentum-resolved electron energy loss spectroscopy for mapping the photonic density of states,” ACS Photon. 4, 1009–1014 (2017).
[Crossref]

F. Liu, L. Xiao, Y. Ye, M. Wang, K. Cui, X. Feng, W. Zhang, and Y. Huang, “Integrated Cherenkov radiation emitter eliminating the electron velocity threshold,” Nat. Photonics 11, 289–292 (2017).
[Crossref]

2016 (2)

I. Kaminer, Y. T. Katan, H. Buljan, Y. Shen, O. Ilic, J. J. López, L. J. Wong, J. D. Joannopoulos, and M. Soljačič, “Efficient plasmonic emission by the quantum Čerenkov effect from hot carriers in graphene,” Nat. Commun. 7, 11880 (2016).
[Crossref]

S. Jahani and Z. Jacob, “All-dielectric metamaterials,” Nat. Nanotechnol. 11, 23–36 (2016).
[Crossref]

2014 (3)

F. Priolo, T. Gregorkiewicz, M. Galli, and T. F. Krauss, “Silicon nanostructures for photonics and photovoltaics,” Nat. Nanotechnol. 9, 19–32 (2014).
[Crossref]

P. Shekhar and Z. Jacob, “Strong coupling in hyperbolic metamaterials,” Phys. Rev. B 90, 045313 (2014).
[Crossref]

V. Ginis, J. Danckaert, I. Veretennicoff, and P. Tassin, “Controlling Cherenkov radiation with transformation-optical metamaterials,” Phys. Rev. Lett. 113, 167402 (2014).
[Crossref]

2013 (1)

A. Ono, M. Kikawada, R. Akimoto, W. Inami, and Y. Kawata, “Fluorescence enhancement with deep-ultraviolet surface plasmon excitation,” Opt. Express 21, 17447–17453 (2013).
[Crossref]

2012 (3)

J. A. Scholl, A. L. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature 483, 421–427 (2012).
[Crossref]

D. E. Fernandes, S. I. Maslovski, and M. G. Silveirinha, “Cherenkov emission in a nanowire material,” Phys. Rev. B 85, 155107 (2012).
[Crossref]

M. W. Knight, L. Liu, Y. Wang, L. Brown, S. Mukherjee, N. S. King, H. O. Everitt, P. Nordlander, and N. J. Halas, “Aluminum plasmonic nanoantennas,” Nano Lett. 12, 6000–6004 (2012).
[Crossref]

2011 (1)

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6, 630–634 (2011).
[Crossref]

2010 (4)

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

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4, 83–91 (2010).
[Crossref]

F. J. Garcia de Abajo, “Optical excitations in electron microscopy,” Rev. Mod. Phys. 82, 209–275 (2010).
[Crossref]

J.-K. So, J.-H. Won, M. A. Sattorov, S.-H. Bak, K.-H. Jang, G.-S. Park, D. S. Kim, and F. J. Garcia-Vidal, “Cerenkov radiation in metallic metamaterials,” Appl. Phys. Lett. 97, 151107 (2010).
[Crossref]

2009 (1)

G. Adamo, K. F. MacDonald, Y. H. Fu, C.-M. Wang, D. P. Tsai, F. J. García de Abajo, and N. I. Zheludev, “Light well: a tunable free-electron light source on a chip,” Phys. Rev. Lett. 103, 113901 (2009).
[Crossref]

2008 (3)

R. Erni and N. D. Browning, “The impact of surface and retardation losses on valence electron energy-loss spectroscopy,” Ultramicroscopy 108, 84–99 (2008).
[Crossref]

F. J. Garcia de Abajo and M. Kociak, “Probing the photonic local density of states with electron energy loss spectroscopy,” Phys. Rev. Lett. 100, 106804 (2008).
[Crossref]

A. Yurtsever, M. Couillard, and D. A. Muller, “Formation of guided Cherenkov radiation in silicon-based nanocomposites,” Phys. Rev. Lett. 100, 217402 (2008).
[Crossref]

2007 (1)

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
[Crossref]

2005 (1)

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref]

2004 (1)

I. Gryczynski, J. Malicka, Z. Gryczynski, K. Nowaczyk, and J. R. Lakowicz, “Ultraviolet surface plasmon-coupled emission using thin aluminum films,” Anal. Chem. 76, 4076–4081 (2004).
[Crossref]

2003 (1)

W. Knulst, M. J. van der Wiel, O. J. Luiten, and J. Verhoeven, “High-brightness, narrowband, and compact soft x-ray Cherenkov sources in the water window,” Appl. Phys. Lett. 83, 4050–4052 (2003).
[Crossref]

2002 (1)

F. J. García de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B 65, 115418 (2002).
[Crossref]

2001 (1)

W. Knulst, O. J. Luiten, M. J. van der Wiel, and J. Verhoeven, “Observation of narrow-band Si L-edge Čerenkov radiation generated by 5  MeV electrons,” Appl. Phys. Lett. 79, 2999–3001 (2001).
[Crossref]

1996 (1)

G. Kresse and J. Furthmüller, “Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set,” Phys. Rev. B 54, 11169–11186 (1996).
[Crossref]

1987 (1)

P. Lautenschlager, M. Garriga, L. Vina, and M. Cardona, “Temperature dependence of the dielectric function and interband critical points in silicon,” Phys. Rev. B 36, 4821–4830 (1987).
[Crossref]

1979 (1)

C. H. Chen and J. Silcox, “Calculations of the electron-energy-loss probability in thin uniaxial crystals at oblique incidence,” Phys. Rev. B 20, 3605–3614 (1979).
[Crossref]

1975 (1)

C. H. Chen, J. Silcox, and R. Vincent, “Electron-energy losses in silicon: bulk and surface plasmons and Čerenkov radiation,” Phys. Rev. B 12, 64–71 (1975).
[Crossref]

1963 (1)

H. R. Philipp and H. Ehrenreich, “Optical properties of semiconductors,” Phys. Rev. 129, 1550–1560 (1963).
[Crossref]

1962 (1)

T. Sasaki and K. Ishiguro, “Optical constants of silicon in the extreme ultraviolet region,” Phys. Rev. 127, 1091–1092 (1962).
[Crossref]

Adamo, G.

G. Adamo, K. F. MacDonald, Y. H. Fu, C.-M. Wang, D. P. Tsai, F. J. García de Abajo, and N. I. Zheludev, “Light well: a tunable free-electron light source on a chip,” Phys. Rev. Lett. 103, 113901 (2009).
[Crossref]

Akimoto, R.

A. Ono, M. Kikawada, R. Akimoto, W. Inami, and Y. Kawata, “Fluorescence enhancement with deep-ultraviolet surface plasmon excitation,” Opt. Express 21, 17447–17453 (2013).
[Crossref]

Alekseyev, L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
[Crossref]

Alloatti, L.

A. H. Atabaki, S. Moazeni, F. Pavanello, H. Gevorgyan, J. Notaros, L. Alloatti, M. T. Wade, C. Sun, S. A. Kruger, H. Meng, K. A. Qubaisi, I. Wang, B. Zhang, A. Khilo, C. V. Baiocco, M. A. Popović, V. M. Stojanović, and R. J. Ram, “Integrating photonics with silicon nanoelectronics for the next generation of systems on a chip,” Nature 556, 349–354 (2018).
[Crossref]

Arbabi, A.

E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, M. Faraji-Dana, and A. Faraon, “MEMS-tunable dielectric metasurface lens,” Nat. Commun. 9, 812 (2018).
[Crossref]

Arbabi, E.

E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, M. Faraji-Dana, and A. Faraon, “MEMS-tunable dielectric metasurface lens,” Nat. Commun. 9, 812 (2018).
[Crossref]

Atabaki, A. H.

A. H. Atabaki, S. Moazeni, F. Pavanello, H. Gevorgyan, J. Notaros, L. Alloatti, M. T. Wade, C. Sun, S. A. Kruger, H. Meng, K. A. Qubaisi, I. Wang, B. Zhang, A. Khilo, C. V. Baiocco, M. A. Popović, V. M. Stojanović, and R. J. Ram, “Integrating photonics with silicon nanoelectronics for the next generation of systems on a chip,” Nature 556, 349–354 (2018).
[Crossref]

Atkinson, J.

S. Jahani, S. Kim, J. Atkinson, J. C. Wirth, F. Kalhor, A. A. Noman, W. D. Newman, P. Shekhar, K. Han, V. Van, R. G. DeCorby, L. Chrostowski, M. Qi, and Z. Jacob, “Controlling evanescent waves using silicon photonic all-dielectric metamaterials for dense integration,” Nat. Commun. 9, 1893 (2018).
[Crossref]

Baiocco, C. V.

A. H. Atabaki, S. Moazeni, F. Pavanello, H. Gevorgyan, J. Notaros, L. Alloatti, M. T. Wade, C. Sun, S. A. Kruger, H. Meng, K. A. Qubaisi, I. Wang, B. Zhang, A. Khilo, C. V. Baiocco, M. A. Popović, V. M. Stojanović, and R. J. Ram, “Integrating photonics with silicon nanoelectronics for the next generation of systems on a chip,” Nature 556, 349–354 (2018).
[Crossref]

Bak, S.-H.

J.-K. So, J.-H. Won, M. A. Sattorov, S.-H. Bak, K.-H. Jang, G.-S. Park, D. S. Kim, and F. J. Garcia-Vidal, “Cerenkov radiation in metallic metamaterials,” Appl. Phys. Lett. 97, 151107 (2010).
[Crossref]

Bechtel, H. A.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6, 630–634 (2011).
[Crossref]

Boltasseva, A.

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

Bozhevolnyi, S. I.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4, 83–91 (2010).
[Crossref]

Brown, L.

M. W. Knight, L. Liu, Y. Wang, L. Brown, S. Mukherjee, N. S. King, H. O. Everitt, P. Nordlander, and N. J. Halas, “Aluminum plasmonic nanoantennas,” Nano Lett. 12, 6000–6004 (2012).
[Crossref]

Browning, N. D.

R. Erni and N. D. Browning, “The impact of surface and retardation losses on valence electron energy-loss spectroscopy,” Ultramicroscopy 108, 84–99 (2008).
[Crossref]

Buljan, H.

I. Kaminer, Y. T. Katan, H. Buljan, Y. Shen, O. Ilic, J. J. López, L. J. Wong, J. D. Joannopoulos, and M. Soljačič, “Efficient plasmonic emission by the quantum Čerenkov effect from hot carriers in graphene,” Nat. Commun. 7, 11880 (2016).
[Crossref]

Cardona, M.

P. Lautenschlager, M. Garriga, L. Vina, and M. Cardona, “Temperature dependence of the dielectric function and interband critical points in silicon,” Phys. Rev. B 36, 4821–4830 (1987).
[Crossref]

Chen, C. H.

C. H. Chen and J. Silcox, “Calculations of the electron-energy-loss probability in thin uniaxial crystals at oblique incidence,” Phys. Rev. B 20, 3605–3614 (1979).
[Crossref]

C. H. Chen, J. Silcox, and R. Vincent, “Electron-energy losses in silicon: bulk and surface plasmons and Čerenkov radiation,” Phys. Rev. B 12, 64–71 (1975).
[Crossref]

Chrostowski, L.

S. Jahani, S. Kim, J. Atkinson, J. C. Wirth, F. Kalhor, A. A. Noman, W. D. Newman, P. Shekhar, K. Han, V. Van, R. G. DeCorby, L. Chrostowski, M. Qi, and Z. Jacob, “Controlling evanescent waves using silicon photonic all-dielectric metamaterials for dense integration,” Nat. Commun. 9, 1893 (2018).
[Crossref]

Couillard, M.

A. Yurtsever, M. Couillard, and D. A. Muller, “Formation of guided Cherenkov radiation in silicon-based nanocomposites,” Phys. Rev. Lett. 100, 217402 (2008).
[Crossref]

Cui, K.

F. Liu, L. Xiao, Y. Ye, M. Wang, K. Cui, X. Feng, W. Zhang, and Y. Huang, “Integrated Cherenkov radiation emitter eliminating the electron velocity threshold,” Nat. Photonics 11, 289–292 (2017).
[Crossref]

Dalili, N.

P. Shekhar, M. Malac, V. Gaind, N. Dalili, A. Meldrum, and Z. Jacob, “Momentum-resolved electron energy loss spectroscopy for mapping the photonic density of states,” ACS Photon. 4, 1009–1014 (2017).
[Crossref]

Danckaert, J.

V. Ginis, J. Danckaert, I. Veretennicoff, and P. Tassin, “Controlling Cherenkov radiation with transformation-optical metamaterials,” Phys. Rev. Lett. 113, 167402 (2014).
[Crossref]

DeCorby, R. G.

S. Jahani, S. Kim, J. Atkinson, J. C. Wirth, F. Kalhor, A. A. Noman, W. D. Newman, P. Shekhar, K. Han, V. Van, R. G. DeCorby, L. Chrostowski, M. Qi, and Z. Jacob, “Controlling evanescent waves using silicon photonic all-dielectric metamaterials for dense integration,” Nat. Commun. 9, 1893 (2018).
[Crossref]

Dionne, J. A.

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F. Liu, L. Xiao, Y. Ye, M. Wang, K. Cui, X. Feng, W. Zhang, and Y. Huang, “Integrated Cherenkov radiation emitter eliminating the electron velocity threshold,” Nat. Photonics 11, 289–292 (2017).
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ACS Photon. (1)

P. Shekhar, M. Malac, V. Gaind, N. Dalili, A. Meldrum, and Z. Jacob, “Momentum-resolved electron energy loss spectroscopy for mapping the photonic density of states,” ACS Photon. 4, 1009–1014 (2017).
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Anal. Chem. (1)

I. Gryczynski, J. Malicka, Z. Gryczynski, K. Nowaczyk, and J. R. Lakowicz, “Ultraviolet surface plasmon-coupled emission using thin aluminum films,” Anal. Chem. 76, 4076–4081 (2004).
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Appl. Phys. Lett. (3)

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P. West, S. Ishii, G. Naik, N. Emani, V. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photon. Rev. 4, 795–808 (2010).
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Nano Lett. (2)

M. W. Knight, L. Liu, Y. Wang, L. Brown, S. Mukherjee, N. S. King, H. O. Everitt, P. Nordlander, and N. J. Halas, “Aluminum plasmonic nanoantennas,” Nano Lett. 12, 6000–6004 (2012).
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Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
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E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, M. Faraji-Dana, and A. Faraon, “MEMS-tunable dielectric metasurface lens,” Nat. Commun. 9, 812 (2018).
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I. Kaminer, Y. T. Katan, H. Buljan, Y. Shen, O. Ilic, J. J. López, L. J. Wong, J. D. Joannopoulos, and M. Soljačič, “Efficient plasmonic emission by the quantum Čerenkov effect from hot carriers in graphene,” Nat. Commun. 7, 11880 (2016).
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Nat. Mater. (1)

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
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Nat. Nanotechnol. (3)

S. Jahani and Z. Jacob, “All-dielectric metamaterials,” Nat. Nanotechnol. 11, 23–36 (2016).
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L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6, 630–634 (2011).
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F. Priolo, T. Gregorkiewicz, M. Galli, and T. F. Krauss, “Silicon nanostructures for photonics and photovoltaics,” Nat. Nanotechnol. 9, 19–32 (2014).
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Nat. Photonics (3)

I. Staude and J. Schilling, “Metamaterial-inspired silicon nanophotonics,” Nat. Photonics 11, 274–284 (2017).
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F. Liu, L. Xiao, Y. Ye, M. Wang, K. Cui, X. Feng, W. Zhang, and Y. Huang, “Integrated Cherenkov radiation emitter eliminating the electron velocity threshold,” Nat. Photonics 11, 289–292 (2017).
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D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4, 83–91 (2010).
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Nature (2)

A. H. Atabaki, S. Moazeni, F. Pavanello, H. Gevorgyan, J. Notaros, L. Alloatti, M. T. Wade, C. Sun, S. A. Kruger, H. Meng, K. A. Qubaisi, I. Wang, B. Zhang, A. Khilo, C. V. Baiocco, M. A. Popović, V. M. Stojanović, and R. J. Ram, “Integrating photonics with silicon nanoelectronics for the next generation of systems on a chip,” Nature 556, 349–354 (2018).
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J. A. Scholl, A. L. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature 483, 421–427 (2012).
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Opt. Express (1)

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Supplementary Material (1)

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» Supplement 1       Suppemental document

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Figures (5)

Fig. 1.
Fig. 1. Plasmonics across the EM spectrum. (a) Measured SP resonance for various materials across the EM spectrum from the terahertz ( 10 2    eV / 124    μm ) to the EUV (11.5 eV/107 nm); doped semiconductors are limited to the mid-infrared region, whereas transparent conducting oxides have plasmon resonances in the near-infrared. Alternative plasmonic media and conventional materials (Ag, Au) work well in the visible range. Aluminum is the material of choice for UV applications. Plasmonic behavior in the EUV has remained largely ignored. Here, we explore silicon for its EUV plasmonic properties at more than double the energy of aluminum. (b) Experimental (from Palik [18]) and theoretical (DFT) calculations with the GW approximation) of the permittivity of silicon showing its metallic character in the EUV ( ε < 0 in the 10–16 eV (124–77 nm) regime); (c) electronic band structure of silicon calculated with DFT+GW approximations. Arrows indicate indirect interband transitions that are very weak in the EUV. This results in a sea of unbound electrons in the valence band that leads to silicon’s metallic character [20].
Fig. 2.
Fig. 2. EUV plasmons and CR in silicon measured with k -EELS. (a) Schematic showing the key components of the k -EELS technique for measuring the momentum-resolved photonic band structure of silicon. The k -EELS experiment was performed with a Hitachi HF-3300 TEM with a GIF Tridiem in k -EELS mode at 300 keV incident energy with parallel illumination resulting in a quantitative energy-momentum dispersion map of the excitations in the sample (details in Supplement 1). The photonic band structure of (b) 200 nm; (c) 100 nm; and (d) 60 nm thick silicon films measured with k -EELS (error bars show 95% confidence interval). All three films show evidence of the BP at ( 16    eV ) and the SP at ( 4 11.5    eV ) in the EUV as well as CR in the visible in the ( 2 4    eV ) region mapped to large scattering angles (large momentum with k > 5 * k 0 ). A good agreement to the macroscopic electrodynamic energy loss function (red line) is seen for all three thicknesses. (e), (f), and (g) show the electron scattering probability for the three excitations as measured by k -EELS integrated over the indicated scattering angles for the 200, 100, and 60 nm silicon films, respectively. Insets in (f) and (g) show scanning electron microscope images of the free-standing silicon films prepared via FIB milling and mounted to the TEM grid.
Fig. 3.
Fig. 3. k -EELS scattering intensity scaling with momentum ( k ) The experimental (blue circles) SP and the BP scattering peak intensity ratio is plotted as a function of k x (scattering angle) for the (a) 200 nm and (b) 60 nm silicon films. In macroscopic electrodynamic electron energy loss theory, surface contributions (such as the SP) and bulk contributions (such as the BP) scale with momentum as k 3 and k 2 , respectively [24,26]. As a result, the k -space scaling of the ratio of the surface to bulk intensity goes as k 1 . This is evident from the red line in the figure. We can thus unambiguously separate the SPP and BP contributions using k -EELS.
Fig. 4.
Fig. 4. TCR in the EUV. (a) Schematic of TCR ( v z c ) in the EUV excited in a HMM composed of a 100 nm thick Si / SiO 2 multilayer stack in the effective medium limit. k c is the TCR wave vector and θ c is the TCR cone angle. (b) Uniaxial effective medium permittivity at 0.35 metallic fill fraction for the Si / SiO 2 multilayer stack highlighting the regions of type I and type II behavior where TCR can be observed. (c) Type II ( ε x < 0 , ε z > 0 ) HMM isofrequency typology that supports TCR. In the ideal limit, the strongest TCR resonance occurs as v z 0 , where θ c lies along the asymptotes of the hyperbola in k -space (defined by angle θ r ) (details in Supplement 1). (d) Normalized E z fields in the x–y plane of CR in the dielectric and hyperbolic regimes of the Si / SiO 2 multilayer stack at different electron velocities in a lossless structure. Opposite trends are observed where the field strengths increase for the hyperbolic regime, while they are suppressed in the dielectric regime as the electron velocity decreases. The type I regime is seen to support TCR in the EUV ( 11 15.5    eV ).
Fig. 5.
Fig. 5. TCR dispersion from the DUV to the EUV. Simulation of the momentum-resolved electron scattering probability (calculated via the energy loss function) for the Si / SiO 2 structure described in Fig. 4 for different incident electron velocities. The simulations are performed in the effective medium (homogenized) low-loss limit. The dashed black line shows the analytical CR dispersion in a uniaxial metamaterial described in Eq. (S1) of Supplement 1. TCR is clearly observed in the type I and type II HMM regimes, where the scattering probability increases with decreasing electron velocity. Note that the type I region has an upper threshold when v z c / ε x and is suppressed at v z = 0.9 c . The type II region has no velocity threshold. Additionally, the TCR dispersion extends to larger wave vectors as the velocity decreases, as it is approaching the resonance condition described in Fig. 4(b), where infinitely large wave vectors are supported in the structure and v phase = ω / k 0 .

Equations (3)

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tan ( θ c ) = ( v z c ) 2 ε z ε z ε x ,
v z c / ε x Type    I ,
0 v z c Type    II .

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