Outline High-frequency (optical) conductivity of graphene; Optical - PowerPoint PPT Presentation

Д.А. Свинцов 1,2 , В.И. Рыжий 3 , T. Otsuji 3 1. Физико - технологический институт РАН 2. Московский физико - технический институт 3. Research institute of electrical communication, Tohoku university

Outline  High-frequency (optical) conductivity of graphene;  Optical conductivity under population inversion  Direct interband transitions  Carrier-carrier scattering and intraband absorption  Indirect interband transitions

Graphene-based THz electronics? THz lasing in optically or electrically pumped graphene? GOOD IDEA! But what about • Intraband Drude absorption (strong at low frequencies)? F. Bonaccorso, Z. Sun, T. Hasan , and A.C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, (2010). A. Tredicucci and M. S. Vitiello , “Device concepts for graphene - based terahertz photonics,” IEEE J. Sel. Top. Quant. 20 (2014). 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 (2009). V. Ryzhii, M. Ryzhii, V. Mitin, and T. Otsuji , “Toward the creation of terahertz graphene injection laser,” J. Appl. Phys. 110 (2011).

Optical conductivity of graphene   4 1   4 e T e      F T / Re l n 1      2 4  2    1 4 e            / 2 / 2  f f  V C 4 L. A. Falkovsky and A. A. Varlamov, Eur. Phys. J. B 56, 281

Optical conductivity of graphene   4 1   8 e T e      F T / Re l n 1      2 4  2    1      4 / 2 e  V. Ryzhii, M. Ryzhii, and T. Otsuji, F tanh   J. Appl. Phys. 101 , 083114 (2007).   4 2 T

Drude conductivity of clean graphene under population inversion Electron-hole, electron-electron and hole-hole scattering govern the intraband conductivity Velocity-momentum decoupling  0 / p v v p p    p p p p 1 2 3 4 Does not necessarily lead to    v v v v 1 2 3 4 1 e        , 0 V A v v v v  1 2 3 4 fi c F. T. Vasko and V. Ryzhii, Phys. Rev. B 77 , 195433 (2008). D. Sun et. al. New J. Phys. 14 , 105012 (2012). D. Svintsov et. al., Optics Express 22 , 19873 (2014).

Drude conductivity of clean graphene under population inversion Fermi golden rule + occupation numbers of initial and final states: 2    3  2 2   1 e e T          / T    Re 1 / , /  e I T T    ee ee F 3   4  v  0 0 2    3  2 2   2 e e T          / T    Re 1 / , /  e I T T    eh eh F 3   4   v 0 0 d d Q k k d                 2 2 2 1 2 ( ) cos / 2 cos / 2 [ / ] I n k k T k k         , 1 2 1 2 1 2 ee 2 2 ee ( ) Q Q       ( ) ( ) 1 ( ) 1 ( ) f k f k f k f k     1 2 1 2 d d Q k k d                    2 2 2 2 2 1 2 ( ) cos / 2 cos / 2 sin / 2 sin / 2 I n         eh , eh 1 2 1 2 2 2 Q            [ / ] ( ) ( ) 1 ( ) 1 ( ) k k T k k f k f k f k f k         1 2 1 2 1 2 1 2

Interband amplification vs. carrier- carrier absorption Real parts of net dynamic conductivity Re( σ intra + σ inter ) normalized by σ q =e 2 /4 ђ at different quasi-Fermi energies ε F in graphene structures with different background dielectric constants κ0 (T = 300 K). • Negative conductivity is possible in suspended graphene above ~6 THz; • In high- k background above ~3 THz; • Negative dynamic conductivity threshold slowly moves to lower values as k 0 increases.

Dependence on background dielectric constant Threshold frequencies ( solutions of Re( σ intra +σ inter ) = 0 ) vs. background dielectric constant at different temperatures T ( ε F = 75 meV) 2 2   1 e e T Thomas-Fermi screening leads to weak     F T / , 8 ln 1 V q e    dependence on k 0 q TF q q v v 0 T F 0 0 0

Raise interband amplification above 2.3% D. Svintsov, V. Ryzhii, T. Otsuji “Negative dynamic Drude conductivity in pumped graphene” arXiv:1408.7023

Conductivity due to indirect inter- and intraband transitions   ( , ) ( , ) v v A v v A e e ˆ ˆ           p p 0 p p 0 cc cv | | | | V p c V p c V p c V p v     S S pp pp 2 2 c c    2 g  2 ˆ 2                         D Re ( ) ( ) | | f f p V p v v               intra q p p p p S p p 3   , , p p Energy conservation requires q=| p - p’ |> w /v 0 !    2 g  2 ˆ 2                       D Re ( ) ( ) | | . f f p c V p v v v           inter q v p c p p p S p p 3  , p p Energy conservation requires q=| p - p’ |< w /v 0 ! 2 ˆ    | | Need scattering potentials with p q V p S q   either singular at or quickly decreasing as ! q  0

Conductivity due to indirect inter- and intraband transitions  Scattering by Gaussian correlated disorder          2 2 2 ( ) ( ) exp | | / c V r V r V r r l       1 cos 2 ˆ           2 ( /2) pp 2 2 ql | | p p V l V e c S c 2 Intraband Drude absorption is suppressed due to requirement q=| p - p’ |> w /v 0 F. T. Vasko and V. Ryzhii, Phys. Rev. B 76 , 233404 (2007). G. M. Rutter, J.N. Crain, N.P. Guisinger, T. Li, P.N. First, J.A. Stroscio, Science 317, 219 (2007)

Conductivity due to indirect inter- and intraband transitions Calculated frequency dependencies of the interband, Real part of the net Drude conductivity (normalized intraband and net Drude conductivity (normalized by s q ) in by s q ) vs. frequency at different correlation lengths l c . pumped graphene with quasi-Fermi energy e F = 50 meV. The distribution of impurities is Gaussian. The dashed line indicates the region w < n , where our calculations are not rigorous.

Net dynamic conductivity of graphene with population inversion

Conclusions  Graphene with population inversion exhibits negative dynamic conductivity (optical gain) in THz and IR range;  Carrier-carrier scattering sets the threshold of negative dynamic conductivity in clean graphene;  Indirect interband transitions can improve the optical gain above 2.3%.

Conductivity of graphene            2 2 d p v f f 2 ie   x p k /2 p k + /2                    2 1 i      ,   p k /2 p k + /2 p k + / 2 p k /2           2 d p v v f f  2 Intraband term 2 ie   12 x 21 x p k /2 p k + /2       2 2 2                1 i       p k + /2 p k /2 p k + /2 p k /2 Interband term 2 e                Uniform field R e / 2 / 2  f f   0 k V C k=0 4 L. A. Falkovsky and A. A. Varlamov, Eur. Phys. J. B 56, 281

Optical conductivity of graphene: experiment  Optical / IR range  Far IR and THz range   2 4 Re    e 4 Re     inter   2.3% k intr a e k inter    int ra c c 2 2 c e K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, Phys. Rev. Lett. 101 , 196405 (2008). L. Ren, Q. Zhang et.al., Nano Lett. 12 , 3711 (2012).

Population inversion in graphene  Optical pumping + fast relaxation + slow recombination S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, Phys. Rev. B 85 , 035443 (2012). T. Li, L. Luo, M. Hupalo, J. Zhang, M. C. Tringides, J. Schmalian, and J.Wang, Phys. Rev. Lett. 108 , 167401 (2012).

Negative conductivity and optical gain under population inversion S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, Phys. Rev. B 85 , 035443 (2012). T. Li, L. Luo, M. Hupalo, J. Zhang, M. C. Tringides, J. Schmalian, and J.Wang, Phys. Rev. Lett. 108 , 167401 (2012).