Abstract

We theoretically examine the effect of carrier-carrier scattering processes on the intraband radiation absorption and their contribution to the net dynamic conductivity in optically or electrically pumped graphene. We demonstrate that the radiation absorption assisted by the carrier-carrier scattering is comparable with Drude absorption due to impurity scattering and is even stronger in sufficiently clean samples. Since the intraband absorption of radiation effectively competes with its interband amplification, this can substantially affect the conditions of the negative dynamic conductivity in the pumped graphene and, hence, the interband terahertz and infrared lasing. We find the threshold values of the frequency and quasi-Fermi energy of nonequilibrium carriers corresponding to the onset of negative dynamic conductivity. The obtained results show that the effect of carrier-carrier scattering shifts the threshold frequency of the radiation amplification in pumped graphene to higher values. In particular, the negative dynamic conductivity is attainable at the frequencies above 6 THz in graphene on SiO2 substrates at room temperature. The threshold frequency can be decreased to markedly lower values in graphene structures with high-κ substrates due to screening of the carrier-carrier scattering, particularly at lower temperatures.

© 2014 Optical Society of America

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  4. A. Dubinov, V.Ya. Aleshkin, M. Ryzhii, T. Otsuji, and V. Ryzhii, “Terahertz laser with optically pumped graphene layers and Fabri-Perot resonator,” Appl. Phys. Express 2, 092301 (2009).
    [Crossref]
  5. V. Ryzhii, M. Ryzhii, V. Mitin, and T. Otsuji, “Toward the creation of terahertz graphene injection laser,” J. Appl. Phys. 110, 094503 (2011).
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  6. V. Ryzhii, A. Dubinov, T. Otsuji, V.Ya. Aleshkin, M. Ryzhii, and M. Shur, “Double-graphene-layer terahertz laser: concept, characteristics, and comparison,” Opt. Express 21(25), 31567–31577 (2013).
    [Crossref]
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    [Crossref]
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  9. D. Brida, A. Tomadin, C. Manzoni, Y.J. Kim, A. Lombardo, S. Milana, R.R. Nair, K.S. Novoselov, A.C. Ferrari, G. Cerullo, and M. Polini, “Ultrafast collinear scattering and carrier multiplication in graphene,” Nat. Commun. 4, 1987 (2013).
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  31. A. Dubinov, V. Aleshkin, V. Mitin, T. Otsuji, and V. Ryzhii, “Terahertz surface plasmons in optically pumped graphene structures,”J. Phys.: Condens. Matter 23, 145302 (2011).

2014 (3)

A. Tredicucci and M. S. Vitiello, “Device concepts for graphene-based terahertz photonics,” IEEE J. Sel. Top. Quantum Electron. 20, 8500109 (2014).
[Crossref]

P. Weis, J. L. Garcia-Pomar, and M. Rahm, “Towards loss compensated and lasing terahertz metamaterials based on optically pumped graphene,” Opt. Express 22(7), 8473–8489 (2014).
[Crossref] [PubMed]

M. Mittendorff, T. Winzer, E. Malic, A. Knorr, C. Berger, W.A. de Heer, H. Schneider, M. Helm, and S. Winnerl, “Anisotropy of excitation and relaxation of photogenerated charge carriers in graphene,” Nano Lett. 14(3), 1504–1507 (2014).
[Crossref] [PubMed]

2013 (4)

A. Satou, V. Ryzhii, Y. Kurita, and T. Otsuji, “Threshold of terahertz population inversion and negative dynamic conductivity in graphene under pulse photoexcitation,” J. Appl. Phys. 113, 143108 (2013).
[Crossref]

V. Ryzhii, A. Dubinov, T. Otsuji, V.Ya. Aleshkin, M. Ryzhii, and M. Shur, “Double-graphene-layer terahertz laser: concept, characteristics, and comparison,” Opt. Express 21(25), 31567–31577 (2013).
[Crossref]

D. Brida, A. Tomadin, C. Manzoni, Y.J. Kim, A. Lombardo, S. Milana, R.R. Nair, K.S. Novoselov, A.C. Ferrari, G. Cerullo, and M. Polini, “Ultrafast collinear scattering and carrier multiplication in graphene,” Nat. Commun. 4, 1987 (2013).
[Crossref] [PubMed]

T. Watanabe, T. Fukushima, Y. Yabe, S.A. Boubanga Tombet, A. Satou, A. Dubinov, V. Ya. Aleshkin, V. Mitin, V. Ryzhii, and T. Otsuji, “The gain enhancement effect of surface plasmon polaritons on terahertz stimulated emission in optically pumped monolayer graphene,” New J. Phys. 15, 075003 (2013).

2012 (6)

F.T. Vasko, V.V. Mitin, V. Ryzhii, and T. Otsuji, “Interplay of intra- and interband absorption in a disordered graphene,” Phys. Rev. B 86, 235424 (2012).
[Crossref]

D. Sun, C. Divin, M. Mihnev, T. Winzer, E. Malic, A. Knorr, J.E. Sipe, C. Berger, W.A. de Heer, and P.N. First, “Current relaxation due to hot carrier scattering in graphene,” New J. Phys. 14, 105012 (2012).
[Crossref]

L. Ren, Q. Zhang, J. Yao, Z. Sun, R. Kaneko, Z. Yan, S. Nanot, Z. Jin, I. Kawayama, M. Tonouchi, J. M. Tour, and J. Kono, “Terahertz and Infrared Spectroscopy of Gated Large-Area Graphene,” Nano Lett. 12, 3711–3715 (2012).
[Crossref] [PubMed]

S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics and terahertz emission in optically pumped graphene at room temperature,” Phys. Rev. B 85, 035443 (2012).
[Crossref]

D. Svintsov, V. Vyurkov, S. Yurchenko, V. Ryzhii, and T. Otsuji, “Hydrodynamic model for electron-hole plasma in graphene,” J. Appl. Phys. 111(8), 083715 (2012).
[Crossref]

V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, “Electron-electron interactions in graphene: current status and perspectives,” Rev. Mod. Phys. 84, 1067 (2012).
[Crossref]

2011 (5)

M. Schütt, P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin, “Coulomb interaction in graphene: Relaxation rates and transport,” Phys. Rev. B. 83, 155441 (2011).
[Crossref]

M. Martl, J. Darmo, C. Deutsch, M. Brandstetter, A. M. Andrews, P. Klang, G. Strasser, and K. Unterrainer, “Gain and losses in THz quantum cascade laser with metal-metal waveguide,” Opt. Express 19, 733 (2011).
[Crossref] [PubMed]

V. Ryzhii, M. Ryzhii, V. Mitin, and T. Otsuji, “Toward the creation of terahertz graphene injection laser,” J. Appl. Phys. 110, 094503 (2011).
[Crossref]

A. Dubinov, V. Aleshkin, V. Mitin, T. Otsuji, and V. Ryzhii, “Terahertz surface plasmons in optically pumped graphene structures,”J. Phys.: Condens. Matter 23, 145302 (2011).

V. Ryzhii, M. Ryzhii, V. Mitin, A. Satou, and T. Otsuji, “Effect of heating and cooling of photogenerated electron-hole plasma in optically pumped graphene on population inversion,” Jpn. J. Appl. Phys. 50, 094001 (2011).

2010 (1)

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
[Crossref]

2009 (1)

A. Dubinov, V.Ya. Aleshkin, M. Ryzhii, T. Otsuji, and V. Ryzhii, “Terahertz laser with optically pumped graphene layers and Fabri-Perot resonator,” Appl. Phys. Express 2, 092301 (2009).
[Crossref]

2008 (4)

K.F. Mak, M.Y. Sfeir, Y. Wu, C.H. Lui, J.A. Misewich, and T.F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008).
[Crossref] [PubMed]

J.M. Dawlaty, S. Shivaraman, M. Chandrashekhar, F. Rana, and M.G. Spencer, “Measurement of ultrafast carrier dynamics in epitaxial graphene,” Appl. Phys. Lett. 92, 042116 (2008).
[Crossref]

A.B. Kashuba, “Conductivity of defectless graphene,” Phys. Rev. B 78, 085415 (2008).
[Crossref]

L. Fritz, J. Schmalian, M. Müller, and S. Sachdev, “Quantum critical transport in clean graphene,” Phys. Rev. B 78, 085416 (2008).
[Crossref]

2007 (4)

L. A. Falkovsky and S. S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76, 153410 (2007).
[Crossref]

V. Ryzhii, M. Ryzhii, and T. Otsuji, “Negative dynamic conductivity of graphene with optical pumping,” J. Appl. Phys. 101, 083114 (2007).
[Crossref]

F. T. Vasko and V. Ryzhii, “Voltage and temperature dependencies of conductivity in gated graphene,” Phys. Rev. B 76, 233404 (2007).

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

1998 (1)

G. G. Zegrya and V. E. Perlin, “Intraband absorption of light in quantum wells induced by electron-electron collisions,” Semiconductors 32, 417–422 (1998).
[Crossref]

Aleshkin, V.

A. Dubinov, V. Aleshkin, V. Mitin, T. Otsuji, and V. Ryzhii, “Terahertz surface plasmons in optically pumped graphene structures,”J. Phys.: Condens. Matter 23, 145302 (2011).

Aleshkin, V. Ya.

T. Watanabe, T. Fukushima, Y. Yabe, S.A. Boubanga Tombet, A. Satou, A. Dubinov, V. Ya. Aleshkin, V. Mitin, V. Ryzhii, and T. Otsuji, “The gain enhancement effect of surface plasmon polaritons on terahertz stimulated emission in optically pumped monolayer graphene,” New J. Phys. 15, 075003 (2013).

Aleshkin, V.Ya.

V. Ryzhii, A. Dubinov, T. Otsuji, V.Ya. Aleshkin, M. Ryzhii, and M. Shur, “Double-graphene-layer terahertz laser: concept, characteristics, and comparison,” Opt. Express 21(25), 31567–31577 (2013).
[Crossref]

A. Dubinov, V.Ya. Aleshkin, M. Ryzhii, T. Otsuji, and V. Ryzhii, “Terahertz laser with optically pumped graphene layers and Fabri-Perot resonator,” Appl. Phys. Express 2, 092301 (2009).
[Crossref]

Andrews, A. M.

Berger, C.

M. Mittendorff, T. Winzer, E. Malic, A. Knorr, C. Berger, W.A. de Heer, H. Schneider, M. Helm, and S. Winnerl, “Anisotropy of excitation and relaxation of photogenerated charge carriers in graphene,” Nano Lett. 14(3), 1504–1507 (2014).
[Crossref] [PubMed]

D. Sun, C. Divin, M. Mihnev, T. Winzer, E. Malic, A. Knorr, J.E. Sipe, C. Berger, W.A. de Heer, and P.N. First, “Current relaxation due to hot carrier scattering in graphene,” New J. Phys. 14, 105012 (2012).
[Crossref]

Bonaccorso, F.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
[Crossref]

Boubanga Tombet, S.A.

T. Watanabe, T. Fukushima, Y. Yabe, S.A. Boubanga Tombet, A. Satou, A. Dubinov, V. Ya. Aleshkin, V. Mitin, V. Ryzhii, and T. Otsuji, “The gain enhancement effect of surface plasmon polaritons on terahertz stimulated emission in optically pumped monolayer graphene,” New J. Phys. 15, 075003 (2013).

Boubanga-Tombet, S.

S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics and terahertz emission in optically pumped graphene at room temperature,” Phys. Rev. B 85, 035443 (2012).
[Crossref]

Brandstetter, M.

Brida, D.

D. Brida, A. Tomadin, C. Manzoni, Y.J. Kim, A. Lombardo, S. Milana, R.R. Nair, K.S. Novoselov, A.C. Ferrari, G. Cerullo, and M. Polini, “Ultrafast collinear scattering and carrier multiplication in graphene,” Nat. Commun. 4, 1987 (2013).
[Crossref] [PubMed]

Castro Neto, A. H.

V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, “Electron-electron interactions in graphene: current status and perspectives,” Rev. Mod. Phys. 84, 1067 (2012).
[Crossref]

Cerullo, G.

D. Brida, A. Tomadin, C. Manzoni, Y.J. Kim, A. Lombardo, S. Milana, R.R. Nair, K.S. Novoselov, A.C. Ferrari, G. Cerullo, and M. Polini, “Ultrafast collinear scattering and carrier multiplication in graphene,” Nat. Commun. 4, 1987 (2013).
[Crossref] [PubMed]

Chan, S.

S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics and terahertz emission in optically pumped graphene at room temperature,” Phys. Rev. B 85, 035443 (2012).
[Crossref]

Chandrashekhar, M.

J.M. Dawlaty, S. Shivaraman, M. Chandrashekhar, F. Rana, and M.G. Spencer, “Measurement of ultrafast carrier dynamics in epitaxial graphene,” Appl. Phys. Lett. 92, 042116 (2008).
[Crossref]

Darmo, J.

Das Sarma, S.

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

Dawlaty, J.M.

J.M. Dawlaty, S. Shivaraman, M. Chandrashekhar, F. Rana, and M.G. Spencer, “Measurement of ultrafast carrier dynamics in epitaxial graphene,” Appl. Phys. Lett. 92, 042116 (2008).
[Crossref]

de Heer, W.A.

M. Mittendorff, T. Winzer, E. Malic, A. Knorr, C. Berger, W.A. de Heer, H. Schneider, M. Helm, and S. Winnerl, “Anisotropy of excitation and relaxation of photogenerated charge carriers in graphene,” Nano Lett. 14(3), 1504–1507 (2014).
[Crossref] [PubMed]

D. Sun, C. Divin, M. Mihnev, T. Winzer, E. Malic, A. Knorr, J.E. Sipe, C. Berger, W.A. de Heer, and P.N. First, “Current relaxation due to hot carrier scattering in graphene,” New J. Phys. 14, 105012 (2012).
[Crossref]

Deutsch, C.

Divin, C.

D. Sun, C. Divin, M. Mihnev, T. Winzer, E. Malic, A. Knorr, J.E. Sipe, C. Berger, W.A. de Heer, and P.N. First, “Current relaxation due to hot carrier scattering in graphene,” New J. Phys. 14, 105012 (2012).
[Crossref]

Dubinov, A.

T. Watanabe, T. Fukushima, Y. Yabe, S.A. Boubanga Tombet, A. Satou, A. Dubinov, V. Ya. Aleshkin, V. Mitin, V. Ryzhii, and T. Otsuji, “The gain enhancement effect of surface plasmon polaritons on terahertz stimulated emission in optically pumped monolayer graphene,” New J. Phys. 15, 075003 (2013).

V. Ryzhii, A. Dubinov, T. Otsuji, V.Ya. Aleshkin, M. Ryzhii, and M. Shur, “Double-graphene-layer terahertz laser: concept, characteristics, and comparison,” Opt. Express 21(25), 31567–31577 (2013).
[Crossref]

A. Dubinov, V. Aleshkin, V. Mitin, T. Otsuji, and V. Ryzhii, “Terahertz surface plasmons in optically pumped graphene structures,”J. Phys.: Condens. Matter 23, 145302 (2011).

A. Dubinov, V.Ya. Aleshkin, M. Ryzhii, T. Otsuji, and V. Ryzhii, “Terahertz laser with optically pumped graphene layers and Fabri-Perot resonator,” Appl. Phys. Express 2, 092301 (2009).
[Crossref]

Falkovsky, L. A.

L. A. Falkovsky and S. S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76, 153410 (2007).
[Crossref]

Ferrari, A. C.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
[Crossref]

Ferrari, A.C.

D. Brida, A. Tomadin, C. Manzoni, Y.J. Kim, A. Lombardo, S. Milana, R.R. Nair, K.S. Novoselov, A.C. Ferrari, G. Cerullo, and M. Polini, “Ultrafast collinear scattering and carrier multiplication in graphene,” Nat. Commun. 4, 1987 (2013).
[Crossref] [PubMed]

First, P.N.

D. Sun, C. Divin, M. Mihnev, T. Winzer, E. Malic, A. Knorr, J.E. Sipe, C. Berger, W.A. de Heer, and P.N. First, “Current relaxation due to hot carrier scattering in graphene,” New J. Phys. 14, 105012 (2012).
[Crossref]

Fritz, L.

L. Fritz, J. Schmalian, M. Müller, and S. Sachdev, “Quantum critical transport in clean graphene,” Phys. Rev. B 78, 085416 (2008).
[Crossref]

Fukushima, T.

T. Watanabe, T. Fukushima, Y. Yabe, S.A. Boubanga Tombet, A. Satou, A. Dubinov, V. Ya. Aleshkin, V. Mitin, V. Ryzhii, and T. Otsuji, “The gain enhancement effect of surface plasmon polaritons on terahertz stimulated emission in optically pumped monolayer graphene,” New J. Phys. 15, 075003 (2013).

Garcia-Pomar, J. L.

Gornyi, I. V.

M. Schütt, P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin, “Coulomb interaction in graphene: Relaxation rates and transport,” Phys. Rev. B. 83, 155441 (2011).
[Crossref]

Guinea, F.

V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, “Electron-electron interactions in graphene: current status and perspectives,” Rev. Mod. Phys. 84, 1067 (2012).
[Crossref]

Hasan, T.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
[Crossref]

Heinz, T.F.

K.F. Mak, M.Y. Sfeir, Y. Wu, C.H. Lui, J.A. Misewich, and T.F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008).
[Crossref] [PubMed]

Helm, M.

M. Mittendorff, T. Winzer, E. Malic, A. Knorr, C. Berger, W.A. de Heer, H. Schneider, M. Helm, and S. Winnerl, “Anisotropy of excitation and relaxation of photogenerated charge carriers in graphene,” Nano Lett. 14(3), 1504–1507 (2014).
[Crossref] [PubMed]

Hwang, E. H.

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

Jin, Z.

L. Ren, Q. Zhang, J. Yao, Z. Sun, R. Kaneko, Z. Yan, S. Nanot, Z. Jin, I. Kawayama, M. Tonouchi, J. M. Tour, and J. Kono, “Terahertz and Infrared Spectroscopy of Gated Large-Area Graphene,” Nano Lett. 12, 3711–3715 (2012).
[Crossref] [PubMed]

Kaneko, R.

L. Ren, Q. Zhang, J. Yao, Z. Sun, R. Kaneko, Z. Yan, S. Nanot, Z. Jin, I. Kawayama, M. Tonouchi, J. M. Tour, and J. Kono, “Terahertz and Infrared Spectroscopy of Gated Large-Area Graphene,” Nano Lett. 12, 3711–3715 (2012).
[Crossref] [PubMed]

Kashuba, A.B.

A.B. Kashuba, “Conductivity of defectless graphene,” Phys. Rev. B 78, 085415 (2008).
[Crossref]

Kawayama, I.

L. Ren, Q. Zhang, J. Yao, Z. Sun, R. Kaneko, Z. Yan, S. Nanot, Z. Jin, I. Kawayama, M. Tonouchi, J. M. Tour, and J. Kono, “Terahertz and Infrared Spectroscopy of Gated Large-Area Graphene,” Nano Lett. 12, 3711–3715 (2012).
[Crossref] [PubMed]

Kim, Y.J.

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V. Ryzhii, A. Dubinov, T. Otsuji, V.Ya. Aleshkin, M. Ryzhii, and M. Shur, “Double-graphene-layer terahertz laser: concept, characteristics, and comparison,” Opt. Express 21(25), 31567–31577 (2013).
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A. Dubinov, V.Ya. Aleshkin, M. Ryzhii, T. Otsuji, and V. Ryzhii, “Terahertz laser with optically pumped graphene layers and Fabri-Perot resonator,” Appl. Phys. Express 2, 092301 (2009).
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V. Ryzhii, A. Dubinov, T. Otsuji, V.Ya. Aleshkin, M. Ryzhii, and M. Shur, “Double-graphene-layer terahertz laser: concept, characteristics, and comparison,” Opt. Express 21(25), 31567–31577 (2013).
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V. Ryzhii, M. Ryzhii, V. Mitin, and T. Otsuji, “Toward the creation of terahertz graphene injection laser,” J. Appl. Phys. 110, 094503 (2011).
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A. Dubinov, V.Ya. Aleshkin, M. Ryzhii, T. Otsuji, and V. Ryzhii, “Terahertz laser with optically pumped graphene layers and Fabri-Perot resonator,” Appl. Phys. Express 2, 092301 (2009).
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A. Satou, V. Ryzhii, Y. Kurita, and T. Otsuji, “Threshold of terahertz population inversion and negative dynamic conductivity in graphene under pulse photoexcitation,” J. Appl. Phys. 113, 143108 (2013).
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T. Watanabe, T. Fukushima, Y. Yabe, S.A. Boubanga Tombet, A. Satou, A. Dubinov, V. Ya. Aleshkin, V. Mitin, V. Ryzhii, and T. Otsuji, “The gain enhancement effect of surface plasmon polaritons on terahertz stimulated emission in optically pumped monolayer graphene,” New J. Phys. 15, 075003 (2013).

V. Ryzhii, A. Dubinov, T. Otsuji, V.Ya. Aleshkin, M. Ryzhii, and M. Shur, “Double-graphene-layer terahertz laser: concept, characteristics, and comparison,” Opt. Express 21(25), 31567–31577 (2013).
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[Crossref]

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[Crossref]

D. Svintsov, V. Vyurkov, S. Yurchenko, V. Ryzhii, and T. Otsuji, “Hydrodynamic model for electron-hole plasma in graphene,” J. Appl. Phys. 111(8), 083715 (2012).
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V. Ryzhii, M. Ryzhii, V. Mitin, and T. Otsuji, “Toward the creation of terahertz graphene injection laser,” J. Appl. Phys. 110, 094503 (2011).
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A. Dubinov, V. Aleshkin, V. Mitin, T. Otsuji, and V. Ryzhii, “Terahertz surface plasmons in optically pumped graphene structures,”J. Phys.: Condens. Matter 23, 145302 (2011).

A. Dubinov, V.Ya. Aleshkin, M. Ryzhii, T. Otsuji, and V. Ryzhii, “Terahertz laser with optically pumped graphene layers and Fabri-Perot resonator,” Appl. Phys. Express 2, 092301 (2009).
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V. Ryzhii, M. Ryzhii, and T. Otsuji, “Negative dynamic conductivity of graphene with optical pumping,” J. Appl. Phys. 101, 083114 (2007).
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T. Watanabe, T. Fukushima, Y. Yabe, S.A. Boubanga Tombet, A. Satou, A. Dubinov, V. Ya. Aleshkin, V. Mitin, V. Ryzhii, and T. Otsuji, “The gain enhancement effect of surface plasmon polaritons on terahertz stimulated emission in optically pumped monolayer graphene,” New J. Phys. 15, 075003 (2013).

S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics and terahertz emission in optically pumped graphene at room temperature,” Phys. Rev. B 85, 035443 (2012).
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J.M. Dawlaty, S. Shivaraman, M. Chandrashekhar, F. Rana, and M.G. Spencer, “Measurement of ultrafast carrier dynamics in epitaxial graphene,” Appl. Phys. Lett. 92, 042116 (2008).
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D. Sun, C. Divin, M. Mihnev, T. Winzer, E. Malic, A. Knorr, J.E. Sipe, C. Berger, W.A. de Heer, and P.N. First, “Current relaxation due to hot carrier scattering in graphene,” New J. Phys. 14, 105012 (2012).
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J.M. Dawlaty, S. Shivaraman, M. Chandrashekhar, F. Rana, and M.G. Spencer, “Measurement of ultrafast carrier dynamics in epitaxial graphene,” Appl. Phys. Lett. 92, 042116 (2008).
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D. Brida, A. Tomadin, C. Manzoni, Y.J. Kim, A. Lombardo, S. Milana, R.R. Nair, K.S. Novoselov, A.C. Ferrari, G. Cerullo, and M. Polini, “Ultrafast collinear scattering and carrier multiplication in graphene,” Nat. Commun. 4, 1987 (2013).
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L. Ren, Q. Zhang, J. Yao, Z. Sun, R. Kaneko, Z. Yan, S. Nanot, Z. Jin, I. Kawayama, M. Tonouchi, J. M. Tour, and J. Kono, “Terahertz and Infrared Spectroscopy of Gated Large-Area Graphene,” Nano Lett. 12, 3711–3715 (2012).
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L. Ren, Q. Zhang, J. Yao, Z. Sun, R. Kaneko, Z. Yan, S. Nanot, Z. Jin, I. Kawayama, M. Tonouchi, J. M. Tour, and J. Kono, “Terahertz and Infrared Spectroscopy of Gated Large-Area Graphene,” Nano Lett. 12, 3711–3715 (2012).
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F.T. Vasko, V.V. Mitin, V. Ryzhii, and T. Otsuji, “Interplay of intra- and interband absorption in a disordered graphene,” Phys. Rev. B 86, 235424 (2012).
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S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics and terahertz emission in optically pumped graphene at room temperature,” Phys. Rev. B 85, 035443 (2012).
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Winnerl, S.

M. Mittendorff, T. Winzer, E. Malic, A. Knorr, C. Berger, W.A. de Heer, H. Schneider, M. Helm, and S. Winnerl, “Anisotropy of excitation and relaxation of photogenerated charge carriers in graphene,” Nano Lett. 14(3), 1504–1507 (2014).
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M. Mittendorff, T. Winzer, E. Malic, A. Knorr, C. Berger, W.A. de Heer, H. Schneider, M. Helm, and S. Winnerl, “Anisotropy of excitation and relaxation of photogenerated charge carriers in graphene,” Nano Lett. 14(3), 1504–1507 (2014).
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K.F. Mak, M.Y. Sfeir, Y. Wu, C.H. Lui, J.A. Misewich, and T.F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008).
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L. Ren, Q. Zhang, J. Yao, Z. Sun, R. Kaneko, Z. Yan, S. Nanot, Z. Jin, I. Kawayama, M. Tonouchi, J. M. Tour, and J. Kono, “Terahertz and Infrared Spectroscopy of Gated Large-Area Graphene,” Nano Lett. 12, 3711–3715 (2012).
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Yurchenko, S.

D. Svintsov, V. Vyurkov, S. Yurchenko, V. Ryzhii, and T. Otsuji, “Hydrodynamic model for electron-hole plasma in graphene,” J. Appl. Phys. 111(8), 083715 (2012).
[Crossref]

Zegrya, G. G.

G. G. Zegrya and V. E. Perlin, “Intraband absorption of light in quantum wells induced by electron-electron collisions,” Semiconductors 32, 417–422 (1998).
[Crossref]

Zhang, Q.

L. Ren, Q. Zhang, J. Yao, Z. Sun, R. Kaneko, Z. Yan, S. Nanot, Z. Jin, I. Kawayama, M. Tonouchi, J. M. Tour, and J. Kono, “Terahertz and Infrared Spectroscopy of Gated Large-Area Graphene,” Nano Lett. 12, 3711–3715 (2012).
[Crossref] [PubMed]

Ziman, J. M.

J. M. Ziman, Electrons and Phonons (Oxford University Press, 1960).

Appl. Phys. Express (1)

A. Dubinov, V.Ya. Aleshkin, M. Ryzhii, T. Otsuji, and V. Ryzhii, “Terahertz laser with optically pumped graphene layers and Fabri-Perot resonator,” Appl. Phys. Express 2, 092301 (2009).
[Crossref]

Appl. Phys. Lett. (1)

J.M. Dawlaty, S. Shivaraman, M. Chandrashekhar, F. Rana, and M.G. Spencer, “Measurement of ultrafast carrier dynamics in epitaxial graphene,” Appl. Phys. Lett. 92, 042116 (2008).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

A. Tredicucci and M. S. Vitiello, “Device concepts for graphene-based terahertz photonics,” IEEE J. Sel. Top. Quantum Electron. 20, 8500109 (2014).
[Crossref]

J. Appl. Phys. (4)

V. Ryzhii, M. Ryzhii, V. Mitin, and T. Otsuji, “Toward the creation of terahertz graphene injection laser,” J. Appl. Phys. 110, 094503 (2011).
[Crossref]

V. Ryzhii, M. Ryzhii, and T. Otsuji, “Negative dynamic conductivity of graphene with optical pumping,” J. Appl. Phys. 101, 083114 (2007).
[Crossref]

D. Svintsov, V. Vyurkov, S. Yurchenko, V. Ryzhii, and T. Otsuji, “Hydrodynamic model for electron-hole plasma in graphene,” J. Appl. Phys. 111(8), 083715 (2012).
[Crossref]

A. Satou, V. Ryzhii, Y. Kurita, and T. Otsuji, “Threshold of terahertz population inversion and negative dynamic conductivity in graphene under pulse photoexcitation,” J. Appl. Phys. 113, 143108 (2013).
[Crossref]

J. Phys.: Condens. Matter (1)

A. Dubinov, V. Aleshkin, V. Mitin, T. Otsuji, and V. Ryzhii, “Terahertz surface plasmons in optically pumped graphene structures,”J. Phys.: Condens. Matter 23, 145302 (2011).

Jpn. J. Appl. Phys. (1)

V. Ryzhii, M. Ryzhii, V. Mitin, A. Satou, and T. Otsuji, “Effect of heating and cooling of photogenerated electron-hole plasma in optically pumped graphene on population inversion,” Jpn. J. Appl. Phys. 50, 094001 (2011).

Nano Lett. (2)

L. Ren, Q. Zhang, J. Yao, Z. Sun, R. Kaneko, Z. Yan, S. Nanot, Z. Jin, I. Kawayama, M. Tonouchi, J. M. Tour, and J. Kono, “Terahertz and Infrared Spectroscopy of Gated Large-Area Graphene,” Nano Lett. 12, 3711–3715 (2012).
[Crossref] [PubMed]

M. Mittendorff, T. Winzer, E. Malic, A. Knorr, C. Berger, W.A. de Heer, H. Schneider, M. Helm, and S. Winnerl, “Anisotropy of excitation and relaxation of photogenerated charge carriers in graphene,” Nano Lett. 14(3), 1504–1507 (2014).
[Crossref] [PubMed]

Nat. Commun. (1)

D. Brida, A. Tomadin, C. Manzoni, Y.J. Kim, A. Lombardo, S. Milana, R.R. Nair, K.S. Novoselov, A.C. Ferrari, G. Cerullo, and M. Polini, “Ultrafast collinear scattering and carrier multiplication in graphene,” Nat. Commun. 4, 1987 (2013).
[Crossref] [PubMed]

Nat. Photonics (1)

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
[Crossref]

New J. Phys. (2)

T. Watanabe, T. Fukushima, Y. Yabe, S.A. Boubanga Tombet, A. Satou, A. Dubinov, V. Ya. Aleshkin, V. Mitin, V. Ryzhii, and T. Otsuji, “The gain enhancement effect of surface plasmon polaritons on terahertz stimulated emission in optically pumped monolayer graphene,” New J. Phys. 15, 075003 (2013).

D. Sun, C. Divin, M. Mihnev, T. Winzer, E. Malic, A. Knorr, J.E. Sipe, C. Berger, W.A. de Heer, and P.N. First, “Current relaxation due to hot carrier scattering in graphene,” New J. Phys. 14, 105012 (2012).
[Crossref]

Opt. Express (3)

Phys. Rev. B (7)

S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics and terahertz emission in optically pumped graphene at room temperature,” Phys. Rev. B 85, 035443 (2012).
[Crossref]

A.B. Kashuba, “Conductivity of defectless graphene,” Phys. Rev. B 78, 085415 (2008).
[Crossref]

L. Fritz, J. Schmalian, M. Müller, and S. Sachdev, “Quantum critical transport in clean graphene,” Phys. Rev. B 78, 085416 (2008).
[Crossref]

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

L. A. Falkovsky and S. S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76, 153410 (2007).
[Crossref]

F.T. Vasko, V.V. Mitin, V. Ryzhii, and T. Otsuji, “Interplay of intra- and interband absorption in a disordered graphene,” Phys. Rev. B 86, 235424 (2012).
[Crossref]

F. T. Vasko and V. Ryzhii, “Voltage and temperature dependencies of conductivity in gated graphene,” Phys. Rev. B 76, 233404 (2007).

Phys. Rev. B. (1)

M. Schütt, P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin, “Coulomb interaction in graphene: Relaxation rates and transport,” Phys. Rev. B. 83, 155441 (2011).
[Crossref]

Phys. Rev. Lett. (1)

K.F. Mak, M.Y. Sfeir, Y. Wu, C.H. Lui, J.A. Misewich, and T.F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, “Electron-electron interactions in graphene: current status and perspectives,” Rev. Mod. Phys. 84, 1067 (2012).
[Crossref]

Semiconductors (1)

G. G. Zegrya and V. E. Perlin, “Intraband absorption of light in quantum wells induced by electron-electron collisions,” Semiconductors 32, 417–422 (1998).
[Crossref]

Other (2)

L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965).

J. M. Ziman, Electrons and Phonons (Oxford University Press, 1960).

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Figures (7)

Fig. 1
Fig. 1 Schematic views of band diagrams of (A) n-type graphene in equilibrium and (B) pumped graphene. Wavy arrows indicate photon absorption and emission processes. Diagram of photon absorption by an electron (e) associated with (C) electron-electron (electron-hole) and (D) impurity (i) scattering.
Fig. 2
Fig. 2 Real parts of the intraband (upper panels) and interband (lower panels) contributions, Reσintra and Reσinter, to dynamic conductivity normalized by σq at different quasi-Fermi energies εF in graphene structures with different background dielectric constants κ0 (T = 300 K).
Fig. 3
Fig. 3 Real parts of net dynamic conductivity Re(σintra +σinter) normalized by σq at different quasi-Fermi energies εF in graphene structures with different background dielectric constants κ0 (T = 300 K).
Fig. 4
Fig. 4 Color map of real part of net dynamic conductivity Re(σintra + σinter)/σq vs frequency and quasi-Fermi energy for κ0 = 5: (A) at T = 300 K and (B) at T = 200 K. The area Re(σintra +σinter)/σq < 0.75 is filled in solid color
Fig. 5
Fig. 5 (A) Thresholds of negative dynamic conductivity at different values of background dielectric constant κ0. Solid lines correspond to Re(σintra +σinter)= 0. (B) Threshold frequencies ω0/2π vs background dielectric constant at different temperatures T (εF = 3kBT)
Fig. 6
Fig. 6 Real parts of intraband dynamic conductivity Reσintra (solid lines) and interband conductivity −Reσinter (dashed line) as functions of quasi-Fermi energy εF at fixed frequency ω/(2π)= 6 THz and different values of background dielectric constant κ0.
Fig. 7
Fig. 7 Separate contributions of e-h, e-e, and h-h scattering processes to real part of interband conductivity: (A) for pumped graphene and (B) for electron-doped graphene in thermodynamic equilibrium (κ0 = 1, ω/2π = 6 THz, and T = 300 K).

Equations (27)

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V ^ F ( t ) = e 2 c ( A 0 , v ^ 1 + v ^ 2 ) ( e i ω t + e i ω t ) ,
V C ( q ) = 2 π e 2 h ¯ q κ 0 κ ( q ) u p 1 ( e ) u p 3 ( e ) u p 2 ( e ) u p 4 ( e ) ,
κ ( q ) = 1 + q TF / q ,
f | V ^ | i = m f | V ^ F ω | m m | V ^ C | i ( ε p 1 + ε p 2 ) ( ε p 1 m + ε p 2 m ) + f | V ^ C | m m | V ^ F ω | i ( ε p 1 + ε p 2 + h ¯ ω ) ( ε p 1 m + ε p 2 m ) ,
ε p 1 + ε p 2 + h ¯ ω = ε p 3 + ε p 4 ,
f | V | i = e 2 c V C ( q ) h ¯ ω ( A 0 , v p 1 + v p 2 v p 3 v p 4 ) cos ( θ 13 / 2 ) cos ( θ 24 / 2 ) .
P = h ¯ ω 2 2 π g 2 h ¯ p 1 , p 2 , q | f | V | i | 2 δ ( ε p 1 + ε p 2 + h ¯ ω ε p 3 ε p 4 ) × f e ( p 1 ) f e ( p 2 ) [ 1 f e ( p 3 ) ] [ 1 f e ( p 4 ) ] [ 1 exp ( h ¯ ω / k B T ) ] .
f e ( p ) = [ 1 + exp ( ε p μ e k B T ) ] 1 .
Re σ e e = σ q α c 2 π 3 ( k B T h ¯ ω ) 3 [ 1 exp ( h ¯ ω k B T ) ] I e e , ω ,
I e e , ω = d Q d k 1 d k 2 Q 2 κ 2 ( Q ) ( Δ n e e ) 2 cos 2 ( θ 1 ± / 2 ) cos 2 ( θ 2 ± / 2 ) × δ [ k 1 + + k 2 + h ¯ ω / ( k B T ) k 1 k 2 + ] F ( k 1 + ) F ( k 2 ) [ 1 F ( k 1 ) ] [ 1 F ( k 2 + ) ] .
Re σ e h = σ q 2 α c 2 π 3 ( k B T h ¯ ω ) 3 [ 1 exp ( h ¯ ω k B T ) ] I e h , ω ,
I e h , ω = d Q d k 1 d k 2 Q 2 κ 2 ( Δ n e e ) 2 [ cos 2 ( θ 1 ± / 2 ) cos 2 ( θ 2 ± / 2 ) + sin 2 ( θ 1 ± / 2 ) sin 2 ( θ 2 ± / 2 ) ] × δ [ k 1 + + k 2 + h ¯ ω / ( k B T ) k 1 k 2 + ] F ( k 1 + ) F ( k 2 ) [ 1 F ( k 1 ) ] [ 1 F ( k 2 + ) ] .
Re σ e e = σ q 2 α c 2 π 3 ( k B T h ¯ ω ) 3 [ 1 exp ( h ¯ ω k B T ) ] ( I e e , ω + I e h , ω ) .
Re σ inter = σ q tanh ( h ¯ ω / 2 ε F 2 k B T ) .
q TF 8 α c k B T v F ln [ 1 + exp ( ε F k B T ) ] .
Re σ intra = 2 e 2 π h ¯ k B T h ¯ ln [ 1 + exp ( ε F k B T ) ] v c c ω 2 + v c c 2 ,
v c c = α c 2 4 π 2 k B T h ¯ I e e , 0 + I e h , 0 ln [ 1 + exp ( ε F k B T ) ] ,
( τ p 1 ) i = 2 π α c 2 p v F h ¯ h ¯ 2 n i q TF 2 0 π sin 2 θ d θ [ 1 + ( 2 p / q TF ) sin ( θ / 2 ) ] 2 .
C e e ( ε 1 , ε 2 , δ 3 , Q ) = d k 1 d k 2 ( Δ n e e ) 2 cos 2 ( θ 1 ± / 2 ) cos 2 ( θ 2 ± / 2 ) δ ( ε 1 k 1 + ) δ ( ε 2 k 2 ) × δ [ δ ε ( k 2 + k 2 ω / 2 ) ] δ [ δ ε ( k 1 + k 1 + ω / 2 ) ] .
I e e , ω = w / 2 Q d Q ( Q + Q TF ) 2 δ ε min δ ε max d δ ε ε 1 min d ε 1 ε 2 min d ε 2 C e e ( ε 1 , ε 2 , δ ε , Q ) F ( ε 1 ) F ( ε 2 ) × [ 1 F ( ε 2 + δ ε + w / 2 ) ] [ 1 F ( ε 1 δ ε + w / 2 ) ] .
k i = Q 2 { cosh u i cos v i , sinh u i sin v i } ,
4 Q 4 ( cos v 2 δ ε + w / 2 Q ) δ ( cos v 1 δ ε w / 2 Q ) δ ( cos u 1 w / 2 δ ε + 2 ε 1 Q ) × δ ( cosh u 2 w / 2 + δ ε + 2 ε 2 Q ) .
ε 1 > ( Q + δ ε w / 2 ) 2 ; ε 2 > ( Q + δ ε w / 2 ) 2 , Q > w / 2 ; Q + w / 2 < δ ε < Q w / 2 .
C ( ε 1 , ε 2 , δ ε , Q ) = 4 ( Δ n ) 2 Q 4 cos 2 ( θ 1 ± / 2 ) cos 2 ( θ 2 ± / 2 ) [ sin v 1 sin v 2 sinh u 1 sinh u 2 ] k 1 + k 1 k 2 + k 2 .
cos 2 ( θ 1 ± / 2 ) cos 2 ( θ 2 ± / 2 ) k 1 + k 1 k 2 + k 2 = 1 16 [ ( 2 ε 1 δ ε + w / 2 ) 2 Q 2 ] [ ( 2 ε 2 + δ ε + w / 2 ) 2 Q 2 ] .
C e e ( ε 1 , ε 2 , δ ε , Q ) = ( Δ n ) e e 2 4 sinh u 1 sinh u 2 sin v 1 sin v 2 = ( Δ n ) e e 2 4 [ ( 2 ε 1 δ ε + w / 2 ) 2 Q 2 ] [ ( 2 ε 2 + δ ε + w / 2 ) 2 Q 2 ] [ Q 2 ( δ ε + w / 2 ) 2 ] [ Q 2 ( δ ε w / 2 ) 2 ] .
( Δ n ) e e 2 = 2 ( Q 2 δ ε 2 ) ε 1 ε 2 ( ε 1 δ ε ) ( δ ε + ε 2 ) × [ δ ε 2 + 4 δ ε ( ε 2 ε 1 ) + 2 ( ε 1 ε 2 ) 2 δ ε 2 ( δ ε 2 ε 1 ) ( δ ε + 2 ε 2 ) Q 2 ] .

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