Abstract

Terahertz plasmon emission is the key to getting terahertz radiation, which has resulted in numerous studies on it. In this paper, we present the results of a theoretical investigation of terahertz plasmon emission by drifting electrons in a grated graphene system driven by an electric field by applying the Boltzmann’s equilibrium equation method. The results show that plasmon frequencies from terahertz to infrared are generated by drifting electrons through the interaction between plasmons and electrons. Obvious increase of the plasmon emission strength with the driving electric field can be seen when the electric field is more than a certain strength (e.g. 1.0 kV/cm). The effects of electron density and the grating period on the emission strength of plasmons were also investigated. It was found that terahertz plasmons can be obtained by applying a grating with appropriate period. The plasmon frequencies can be tuned using either the driving electric field or the electron density controlled by the gate voltage or the grating parameters. This work may help to gain insight into graphene plasmonics and be pertinent to the application of graphene-based structures as electrically tunable terahertz plasmonic devices.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  1. J. N. Chen, M. Badioli, P. A. Gonzáez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
    [Crossref]
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    [Crossref]
  3. L. Ju, B. S. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. G. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
    [Crossref]
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    [Crossref]
  5. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
    [Crossref]
  6. D. Li, Y. Wang, M. Nakajima, Y. Wei, and S. Miyamoto, “Harmonics radiation of graphene surface plasmon polaritons in terahertz regime,” Phys. Lett. A 380(25-26), 2181–2184 (2016).
    [Crossref]
  7. H. L. Zhao, G. J. Ren, F. liu, H. P. Xin, Y. B. Bai, and J. Q. Yao, “Tunable terahertz source via liquid crystal grating coated with electron beam excited graphene: a theoretical analysis,” Opt. Commun. 390, 137–139 (2017).
    [Crossref]
  8. A. V. Muraviev, S. L. Rumyantsev, G. Liu, A. A. Balandin, W. Knap, and M. S. Shur, “Plasmonic and bolometric terahertz detection by graphene field-effect transistor,” Appl. Phys. Lett. 103(18), 181114 (2013).
    [Crossref]
  9. W. Gao, G. Shi, Z. Jin, S. J Q. Zhang, R. Vajtai, P. M. Ajayan, J. Kono, and Q. Xu, “Excitation and active control of propagating surface plasmon polaritons in graphene,” Nano Lett. 13(8), 3698–3702 (2013).
    [Crossref]
  10. Y. V. Bludov, A. Ferreira, N. Peres, and M. Vasilevskiy, “A primer on surface plasmon-polaritons in graphene,” Int. J. Mod. Phys. B 27(10), 1341001 (2013).
    [Crossref]
  11. C. M. Aryal and B. Y. K. Hu, “Plasma wave instabilities in nonequilibrium graphene,” Phys. Rev. B 94(11), 115401 (2016).
    [Crossref]
  12. O. V. Polischuka, D. V. Fateeva, and V. V. Popov, “Electrical tunability of terahertz amplification in a periodic plasmon graphene structure with charge-carrier injection,” Semiconductors 52(12), 1534–1539 (2018).
    [Crossref]
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    [Crossref]
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    [Crossref]
  15. C. X. Zhao and F. M. p. W. Xu, “Cerenkov emission of terahertz acoustic-phonons from graphene,” Appl. Phys. Lett. 102(22), 222101 (2013).
    [Crossref]
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    [Crossref]
  17. L. L. Li and W. Xu, “Surface plasmon polaritons in a topological insulator embedded in an optical cavity,” Appl. Phys. Lett. 104(11), 111603 (2014).
    [Crossref]
  18. H. M. Dong, L. L. Li, W. Y. Wang, S. H. Zhang, C. X. Zhao, and W. Xu, “Terahertz plasmon and infrared coupled plasmon-phonon modes in graphene,” Phys. E 44(9), 1889–1893 (2012).
    [Crossref]
  19. W. Xu and C. Zhang, “Nonlinear transport in steady-state terahertz-driven two-dimensional electron gases,” Phys. Rev. B 55(8), 5259–5265 (1997).
    [Crossref]
  20. H. M. Dong, W. Xu, and F. M. Peeters, “High-field transport properties of graphene,” J. Appl. Phys. 110(6), 063704 (2011).
    [Crossref]
  21. E. H. Hwang and S. D. Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75(20), 205418 (2007).
    [Crossref]
  22. F. Caruso and F. Giustino, “Theory of electron-plasmon coupling in semiconductors,” Phys. Rev. B 94(11), 115208 (2016).
    [Crossref]
  23. C. X. Zhao, W. Xu, and F. M. Peeters, “Plasmon and coupled plasmon-phonon modes in graphene in the presence of a driving electric field,” Phys. Rev. B 89(19), 195447 (2014).
    [Crossref]
  24. Y. D. Huang, H. Qin, B. S. Zhang, J. B. Wu, G. C. Zhou, and B. B. Jin, “Excitation of terahertz plasmon-polariton in a grating coupled two-dimensional electron gas with a fabry-pérot cavity,” Appl. Phys. Lett. 102(25), 253106 (2013).
    [Crossref]
  25. S. D. Mo and W. Y. Ching, “Electronic and optical properties of $\theta -$θ−Al$_2$2O$_3$3 and comparison to $\alpha -$α−Al$_2$2O$_3$3,” Phys. Rev. B 57(24), 15219–15228 (1998).
    [Crossref]
  26. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
    [Crossref]

2018 (1)

O. V. Polischuka, D. V. Fateeva, and V. V. Popov, “Electrical tunability of terahertz amplification in a periodic plasmon graphene structure with charge-carrier injection,” Semiconductors 52(12), 1534–1539 (2018).
[Crossref]

2017 (2)

T. Zhao, M. Hu, R. B. Zhong, S. Gong, C. Zhang, and S. G. Liu, “Cherenkov terahertz radiation from graphene surface plasmon polaritons excited by an electron beam,” Appl. Phys. Lett. 110(23), 231102 (2017).
[Crossref]

H. L. Zhao, G. J. Ren, F. liu, H. P. Xin, Y. B. Bai, and J. Q. Yao, “Tunable terahertz source via liquid crystal grating coated with electron beam excited graphene: a theoretical analysis,” Opt. Commun. 390, 137–139 (2017).
[Crossref]

2016 (5)

I. Kaminer, Y. T. Katan, H. Buljan, Y. C. Shen, O. llic, J. J. López, L. J. Wong, J. D. Joannopoulos, and M. Soljačić, “Efficient plasmonic emission by the quantum cerenkov effect from hot carriers in graphene,” Nat. Commun. 7(1), ncomms11880 (2016).
[Crossref]

S. S. Kubakaddi, “Cerenkov emission of acoustic phonons electrically generated from three-dimensional dirac semimetals,” J. Appl. Phys. 119(19), 195701 (2016).
[Crossref]

D. Li, Y. Wang, M. Nakajima, Y. Wei, and S. Miyamoto, “Harmonics radiation of graphene surface plasmon polaritons in terahertz regime,” Phys. Lett. A 380(25-26), 2181–2184 (2016).
[Crossref]

F. Caruso and F. Giustino, “Theory of electron-plasmon coupling in semiconductors,” Phys. Rev. B 94(11), 115208 (2016).
[Crossref]

C. M. Aryal and B. Y. K. Hu, “Plasma wave instabilities in nonequilibrium graphene,” Phys. Rev. B 94(11), 115401 (2016).
[Crossref]

2014 (2)

C. X. Zhao, W. Xu, and F. M. Peeters, “Plasmon and coupled plasmon-phonon modes in graphene in the presence of a driving electric field,” Phys. Rev. B 89(19), 195447 (2014).
[Crossref]

L. L. Li and W. Xu, “Surface plasmon polaritons in a topological insulator embedded in an optical cavity,” Appl. Phys. Lett. 104(11), 111603 (2014).
[Crossref]

2013 (5)

C. X. Zhao and F. M. p. W. Xu, “Cerenkov emission of terahertz acoustic-phonons from graphene,” Appl. Phys. Lett. 102(22), 222101 (2013).
[Crossref]

A. V. Muraviev, S. L. Rumyantsev, G. Liu, A. A. Balandin, W. Knap, and M. S. Shur, “Plasmonic and bolometric terahertz detection by graphene field-effect transistor,” Appl. Phys. Lett. 103(18), 181114 (2013).
[Crossref]

W. Gao, G. Shi, Z. Jin, S. J Q. Zhang, R. Vajtai, P. M. Ajayan, J. Kono, and Q. Xu, “Excitation and active control of propagating surface plasmon polaritons in graphene,” Nano Lett. 13(8), 3698–3702 (2013).
[Crossref]

Y. V. Bludov, A. Ferreira, N. Peres, and M. Vasilevskiy, “A primer on surface plasmon-polaritons in graphene,” Int. J. Mod. Phys. B 27(10), 1341001 (2013).
[Crossref]

Y. D. Huang, H. Qin, B. S. Zhang, J. B. Wu, G. C. Zhou, and B. B. Jin, “Excitation of terahertz plasmon-polariton in a grating coupled two-dimensional electron gas with a fabry-pérot cavity,” Appl. Phys. Lett. 102(25), 253106 (2013).
[Crossref]

2012 (3)

J. N. Chen, M. Badioli, P. A. Gonzáez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[Crossref]

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[Crossref]

H. M. Dong, L. L. Li, W. Y. Wang, S. H. Zhang, C. X. Zhao, and W. Xu, “Terahertz plasmon and infrared coupled plasmon-phonon modes in graphene,” Phys. E 44(9), 1889–1893 (2012).
[Crossref]

2011 (3)

L. Ju, B. S. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. G. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[Crossref]

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[Crossref]

H. M. Dong, W. Xu, and F. M. Peeters, “High-field transport properties of graphene,” J. Appl. Phys. 110(6), 063704 (2011).
[Crossref]

2009 (1)

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80(24), 245435 (2009).
[Crossref]

2007 (2)

E. H. Hwang and S. D. Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75(20), 205418 (2007).
[Crossref]

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[Crossref]

1998 (1)

S. D. Mo and W. Y. Ching, “Electronic and optical properties of $\theta -$θ−Al$_2$2O$_3$3 and comparison to $\alpha -$α−Al$_2$2O$_3$3,” Phys. Rev. B 57(24), 15219–15228 (1998).
[Crossref]

1997 (1)

W. Xu and C. Zhang, “Nonlinear transport in steady-state terahertz-driven two-dimensional electron gases,” Phys. Rev. B 55(8), 5259–5265 (1997).
[Crossref]

Ajayan, P. M.

W. Gao, G. Shi, Z. Jin, S. J Q. Zhang, R. Vajtai, P. M. Ajayan, J. Kono, and Q. Xu, “Excitation and active control of propagating surface plasmon polaritons in graphene,” Nano Lett. 13(8), 3698–3702 (2013).
[Crossref]

Andreev, G. O.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[Crossref]

Aryal, C. M.

C. M. Aryal and B. Y. K. Hu, “Plasma wave instabilities in nonequilibrium graphene,” Phys. Rev. B 94(11), 115401 (2016).
[Crossref]

Badioli, M.

J. N. Chen, M. Badioli, P. A. Gonzáez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[Crossref]

Bai, Y. B.

H. L. Zhao, G. J. Ren, F. liu, H. P. Xin, Y. B. Bai, and J. Q. Yao, “Tunable terahertz source via liquid crystal grating coated with electron beam excited graphene: a theoretical analysis,” Opt. Commun. 390, 137–139 (2017).
[Crossref]

Balandin, A. A.

A. V. Muraviev, S. L. Rumyantsev, G. Liu, A. A. Balandin, W. Knap, and M. S. Shur, “Plasmonic and bolometric terahertz detection by graphene field-effect transistor,” Appl. Phys. Lett. 103(18), 181114 (2013).
[Crossref]

Bao, W.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[Crossref]

Basov, D. N.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[Crossref]

Bechtel, H. A.

L. Ju, B. S. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. G. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[Crossref]

Bludov, Y. V.

Y. V. Bludov, A. Ferreira, N. Peres, and M. Vasilevskiy, “A primer on surface plasmon-polaritons in graphene,” Int. J. Mod. Phys. B 27(10), 1341001 (2013).
[Crossref]

Buljan, H.

I. Kaminer, Y. T. Katan, H. Buljan, Y. C. Shen, O. llic, J. J. López, L. J. Wong, J. D. Joannopoulos, and M. Soljačić, “Efficient plasmonic emission by the quantum cerenkov effect from hot carriers in graphene,” Nat. Commun. 7(1), ncomms11880 (2016).
[Crossref]

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80(24), 245435 (2009).
[Crossref]

Camara, N.

J. N. Chen, M. Badioli, P. A. Gonzáez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[Crossref]

Caruso, F.

F. Caruso and F. Giustino, “Theory of electron-plasmon coupling in semiconductors,” Phys. Rev. B 94(11), 115208 (2016).
[Crossref]

Centeno, A.

J. N. Chen, M. Badioli, P. A. Gonzáez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[Crossref]

Chen, J. N.

J. N. Chen, M. Badioli, P. A. Gonzáez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[Crossref]

Ching, W. Y.

S. D. Mo and W. Y. Ching, “Electronic and optical properties of $\theta -$θ−Al$_2$2O$_3$3 and comparison to $\alpha -$α−Al$_2$2O$_3$3,” Phys. Rev. B 57(24), 15219–15228 (1998).
[Crossref]

de Abajo, F. J. G.

J. N. Chen, M. Badioli, P. A. Gonzáez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[Crossref]

Dominguez, G.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[Crossref]

Dong, H. M.

H. M. Dong, L. L. Li, W. Y. Wang, S. H. Zhang, C. X. Zhao, and W. Xu, “Terahertz plasmon and infrared coupled plasmon-phonon modes in graphene,” Phys. E 44(9), 1889–1893 (2012).
[Crossref]

H. M. Dong, W. Xu, and F. M. Peeters, “High-field transport properties of graphene,” J. Appl. Phys. 110(6), 063704 (2011).
[Crossref]

Elorza, A. Z.

J. N. Chen, M. Badioli, P. A. Gonzáez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[Crossref]

Engheta, N.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[Crossref]

Fateeva, D. V.

O. V. Polischuka, D. V. Fateeva, and V. V. Popov, “Electrical tunability of terahertz amplification in a periodic plasmon graphene structure with charge-carrier injection,” Semiconductors 52(12), 1534–1539 (2018).
[Crossref]

Fei, Z.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[Crossref]

Ferreira, A.

Y. V. Bludov, A. Ferreira, N. Peres, and M. Vasilevskiy, “A primer on surface plasmon-polaritons in graphene,” Int. J. Mod. Phys. B 27(10), 1341001 (2013).
[Crossref]

Fogler, M. M.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[Crossref]

Gao, W.

W. Gao, G. Shi, Z. Jin, S. J Q. Zhang, R. Vajtai, P. M. Ajayan, J. Kono, and Q. Xu, “Excitation and active control of propagating surface plasmon polaritons in graphene,” Nano Lett. 13(8), 3698–3702 (2013).
[Crossref]

Geim, A. K.

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[Crossref]

Geng, B. S.

L. Ju, B. S. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. G. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[Crossref]

Girit, C.

L. Ju, B. S. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. G. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[Crossref]

Giustino, F.

F. Caruso and F. Giustino, “Theory of electron-plasmon coupling in semiconductors,” Phys. Rev. B 94(11), 115208 (2016).
[Crossref]

Godignon, P.

J. N. Chen, M. Badioli, P. A. Gonzáez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[Crossref]

Gong, S.

T. Zhao, M. Hu, R. B. Zhong, S. Gong, C. Zhang, and S. G. Liu, “Cherenkov terahertz radiation from graphene surface plasmon polaritons excited by an electron beam,” Appl. Phys. Lett. 110(23), 231102 (2017).
[Crossref]

Gonzáez, P. A.

J. N. Chen, M. Badioli, P. A. Gonzáez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[Crossref]

Hao, Z.

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H. L. Zhao, G. J. Ren, F. liu, H. P. Xin, Y. B. Bai, and J. Q. Yao, “Tunable terahertz source via liquid crystal grating coated with electron beam excited graphene: a theoretical analysis,” Opt. Commun. 390, 137–139 (2017).
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T. Zhao, M. Hu, R. B. Zhong, S. Gong, C. Zhang, and S. G. Liu, “Cherenkov terahertz radiation from graphene surface plasmon polaritons excited by an electron beam,” Appl. Phys. Lett. 110(23), 231102 (2017).
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T. Zhao, M. Hu, R. B. Zhong, S. Gong, C. Zhang, and S. G. Liu, “Cherenkov terahertz radiation from graphene surface plasmon polaritons excited by an electron beam,” Appl. Phys. Lett. 110(23), 231102 (2017).
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Figures (6)

Fig. 1.
Fig. 1. Black solid line is the plasmon spectrum in graphene obtained by random phase approximation (RPA) in (a)-(d). The grey areas below the red lines (depicted by $\gamma q$) are the single particle excitation (SPE) regions for different grating periods.
Fig. 2.
Fig. 2. Device model that combines a gold-grated graphene with a dielectric substrate. From top to bottom is the gold grating, Al$_2$O$_3$ dielectric medium, graphene, and SiO$_2$/Si. S and D are, respectively, the source and drain electrode.
Fig. 3.
Fig. 3. (a) Plasmon emission distribution in the direction of $\theta =0^\circ$ for different driving electric fields at a fixed grating period and electron density. The drifting electron velocity v$_x$ and temperature T$_e$ for the driving electric fields of 15 kV/cm, 10 kV/cm, 5.0 kV/cm, 1.0 kV/cm, and 0.2 kV/cm are, respectively, 1.76$\times$10$^7$ cm/s, 1.49$\times$10$^7$ cm/s, 1.06$\times$10$^7$ cm/s, 3.35$\times$10$^6$ cm/s, and 0.72$\times$10$^6$ cm/s, at 712.01 K, 602.70K, 474.78 K, 324.25 K, and 301.26 K. (b)-(d) Angular and frequency dependence of plasmon emission for different electric fields $F_x=15,\ 10$ and $5.0$ kV/cm at a fixed electron density $n_e=1.0\times 10^{12}$ cm$^{-2}$.
Fig. 4.
Fig. 4. Plasmon emission distribution for different driving electric fields at a fixed grating period and electron density. The drifting electron velocity v$_x$ and temperature T$_e$ for the different driving electric fields are the same as in Fig. 3.
Fig. 5.
Fig. 5. Plasmon emission distribution for different driving electric fields at high frequency for the grating period $L=100$ nm. The inset is similar to Fig. 1(b) except that $q$ is extended to a larger value.
Fig. 6.
Fig. 6. Plasmon emission spectrum for different electron densities with a fixed electric field and grating parameters (i.e., $L=20$ nm and $w=10$ nm). The electron temperature (423.72 K, 457.89 K, 474.78 K, 514.38 K, and 556.12 K) is dependent on the electron density with decreasing electron density.

Equations (10)

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e F x f ( k ) k x = g s g v k' [ F ( k , k ) F ( k , k ) ] ,
W ( k , k ) = j 2 π | U j ( q ) | 2 δ [ E ( k ) E ( k ) ± ω j ] ,
q = q .
| U p e ( k , q ) | 2 = V ( q ) [ ε ( q , ω ) ω ] ω p 1 1 | q | 2 ( < Ψ k | e i q r | Ψ k > ) 2 ,
ε ( q , ω ) = 1 V ( q ) k Π ( k , q , ω ) ,
Π ( k , q , ω ) = g s g v F [ E ( k + q ) ] F [ E ( k ) ] ω + E ( k + q ) E ( k ) + i δ ,
| U ( k , q ) | 2 = 3 ω p 3 π q 2 E F ( 1 + cos φ ) 2 4
e F x = 16 n e k' , k ( k x k x ) F ( k , k ) ,
P t = 16 k',{\textbf k} ω p F ( k , k ) .
P ± ( ω p , θ ) = 3 ω p 3 ( N q + 1 / 2 1 / 2 ) 2 ε 0 E F π 2 d k d ϕ f ( k ) × [ 1 f ( k ) ] k ( 1 + cos φ ) 2 δ [ E ( k ) E ( k' ) ± ω p ]

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