Abstract
The microscopic quantum theory of nonlinear interaction of strong coherent electromagnetic radiation with gapped bilayer graphene is used for consideration of the high harmonic generation at low-energy photon excitation Lifshitz transition. The Liouville–von Neumann equation for the density matrix is solved numerically at the nonadiabatic multiphoton excitation regime. By numerical solutions, we examine the rates of the second and third harmonics generation at the particle-hole annihilation at Lifshitz transitions in the linearly polarized coherent electromagnetic wave. The obtained results show that the gapped bilayer graphene can serve as an effective medium for the generation of even and odd high harmonics in the sub-THz domain of frequencies.
Similar content being viewed by others
REFERENCES
K. S. Novoselov, A. K. Geim, S. V. Morozov, et al., Science (Washington, DC, U. S.) 306, 666 (2004).
A. K. Geim, Science (Washington, DC, U. S.) 324, 1530 (2009).
A. V. Rozhkov, A. O. Sboychakov, A. L. Rakhmanov, and F. Nori, Phys. Rep. 648, 1 (2016).
A. H. Castro Neto, F. Guinea, N. M. R. Peres, et al., Rev. Mod. Phys. 81, 109 (2009).
T. Brabec and F. Krausz, Rev. Mod. Phys. 72, 545 (2000).
H. K. Avetissian, Relativistic Nonlinear Electrodynamics, The QED Vacuum and Matter in Super-Strong Radiation Fields (Springer, Berlin, 2016).
Sh. Ghimire, A. D. DiChiara, E. Sistrunk, et al., Nature (London, U.K.) 7, 138 (2011).
O. Schubert, M. Hohenleutner, F. Langer, et al., Nat. Photon. 8, 119 (2014).
G. Vampa, T. J. Hammond, N. Thirat, et al., Nature (London, U.K.) 522, 462 (2015).
G. Ndabashimiye, S. Ghimire, M. Wu, et al., Nature (London, U.K.) 534, 520 (2016).
Y. S. You, D. A. Reis, and S. Ghimire, Nat. Phys. 13, 345 (2017).
H. Liu, C. Guo, G. Vampa, et al., Nat. Phys. 14, 1006 (2018).
S. A. Mikhailov and K. Ziegler, J. Phys.: Condens. Matter 20, 384204 (2008).
S. V. Syzranov, Ya. I. Rodionov, K. I. Kugel, and F. Nori, Phys. Rev. B 88, 241112(R) (2013).
Ya. I. Rodionov, K. I. Kugel, and F. Nori, Phys. Rev. B 94, 195108 (2016).
H. K. Avetissian, G. F. Mkrtchian, K. G. Batrakov, et al., Phys. Rev. B 88, 165411 (2013).
P. Bowlan, E. Martinez-Moreno, K. Reimann, et al., Phys. Rev. B 89, 041408 (2014).
I. Al-Naib, J. E. Sipe, and M. M. Dignam, New J. Phys. 17, 113018 (2015).
L. A. Chizhova, F. Libisch, and J. Burgdorfer, Phys. Rev. B 94, 075412 (2016).
H. K. Avetissian and G. F. Mkrtchian, Phys. Rev. B 94, 045419 (2016).
H. K. Avetissian, A. G. Ghazaryan, G. F. Mkrtchian, et al., J. Nanophoton. 11, 016004 (2017).
H. K. Avetissian, B. R Avchyan, G. F. Mkrtchian, et al., J. Nanophoton. 14, 026018 (2020).
L. A. Chizhova, F. Libisch, and J. Burgdorfer, Phys. Rev. B 95, 085436 (2017).
D. Dimitrovski, L. B. Madsen, and T. G. Pedersen, Phys. Rev. B 95, 035405 (2017).
N. Yoshikawa, T. Tamaya, and K. Tanaka, Science (Washington, DC, U. S.) 356, 736 (2017).
A. Golub, R. Egger, C. Muller et al., Phys. Rev. Lett. 124, 110403 (2020).
H. K. Avetissian and G. F. Mkrtchian, Phys. Rev. B 97, 115454 (2018).
A. K. Avetissian, A. G. Ghazaryan, and Kh. V. Sedrakian, J. Nanophoton. 13, 036010 (2019).
A. G. Ghazaryan and Kh. V. Sedrakian, J. Nanophoton. 13, 046004 (2019).
A. G. Ghazaryan and Kh. V. Sedrakian, J. Nanophoton. 13, 046008 (2019).
A. K. Avetissian, A. G. Ghazaryan, K. V. Sedrakian, et al., J. Nanophoton. 11, 036004 (2017).
A. K. Avetissian, A. G. Ghazaryan, K. V. Sedrakian, et al., J. Nanophoton. 12, 016006 (2018).
H. K. Avetissian, A. K. Avetissian, A. G. Ghazaryan, et al., J. Nanophoton. 14, 026004 (2020).
Yu. Bludov, N. Peres, and M. Vasilevskiy, Phys. Rev. B 101, 075415 (2020).
G. L. Breton, A. Rubio, and N. Tancogne-Dejean, Phys. Rev. B 98, 165308 (2018).
H. Liu, Y. Li, Y. S. You et al., Nat. Phys. 13, 262 (2017).
G. F. Mkrtchian, A. Knorr, and M. Selig, Phys. Rev. B 100, 125401 (2020).
H. K. Avetissian, G. F. Mkrtchian, and K. Z. Hatsagortsyan, Phys. Rev. Res. 2, 023072 (2020).
H. K. Avetissian, A. K. Avetissian, B. R. Avchyan, et al., J. Phys.: Condens. Matter 30, 185302 (2018).
H. K. Avetissian, A. K. Avetissian, B. R. Avchyan, et al., Phys. Rev. B 100, 035434 (2019).
T. G. Pedersen, Phys. Rev. B 95, 235419 (2017).
H. K. Avetissian and G. F. Mkrtchian, Phys. Rev. B 99, 085432 (2019).
S. Almalki, A. M. Parks, G. Bart, et al., Phys. Rev. B 98, 144307 (2018).
B. Cheng, N. Kanda, T. N. Ikeda, et al., Rev. Lett. 124, 117402 (2020).
T. Cao, Z. Li, and S. G. Louie, Phys. Rev. Lett. 114, 236602 (2015).
L. Seixas, A. S. Rodin, A. Carvalho, et al., Phys. Rev. Lett. 116, 206803 (2016).
H. Sevinzli, Nano Lett. 17, 2589 (2017).
J. Faist, F. Capasso, D. L. Sivco, et al., Science (Washington, DC, U. S.) 264, 553 (1994).
D. S. L. Abergel and T. Chakraborty, Appl. Phys. Lett. 95, 062107 (2009).
E. Suarez Morell and L. E. F. Foa Torres, Phys. Rev. B 86, 125449 (2012).
J. J. Dean and H. M. van Driel, Phys. Rev. B 82, 125411 (2010).
S. Wu, L. Mao, A. M. Jones et al., Nano Lett. 12, 2032 (2012).
Y. S. Ang, S. Sultan, and C. Zhang, Appl. Phys. Lett. 97, 243110 (2010).
N. Kumar, J. Kumar, C. Gerstenkornet, et al., Phys. Rev. B 87, 121406 (2013).
E. V. Castro, K. S. Novoselov, S. V. Morozov, et al., Phys. Rev. Lett. 99, 216802 (2007).
J. B. Oostinga, H. B. Heersche, X. Liu, et al., Nat. Mater. 7, 151 (2008).
Y. B. Zhang, T.-T. Tang, C. Girit, et al., Nature (London, U.K.) 459, 820 (2009).
F. Guinea, A. H. C. Neto, and N. M. R. Peres, Phys. Rev. B 73, 245426 (2006).
E. McCann and V. I. Falko, Phys. Rev. Lett. 96, 086805 (2006).
M. Koshino and T. Ando, Phys. Rev. B 73, 245403 (2006).
A. Varleta, M. Mucha-Kruczynski, D. Bischoff, et al., Synth. Met. 210, 19 (2015).
M. Aoki and H. Amawashi, Solid State Commun. 142, 123 (2007).
L. A. Falkovsky, J. Exp. Theor. Phys. 110, 319 (2010).
K. Tang, R. Qin, J. Zhou, et al., J. Phys. Chem. C 115, 9458 (2011).
D. Xiao, M. C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
L. Vicarelli, M. S Vitiello, D. Coquillat, et al., Nat. Mater. 11, 865 (2012).
K. Wang, M. M. Elahi, L. Wang et al., Proc. Natl. Acad. Sci. U. S. A. 116, 201816119 (2019).
I. M. Lifshitz, Sov. Phys. JETP 11, 1130 (1960).
J. L. Manes, F. Guinea, and M. A. H. Vozmediano, Phys. Rev. B 75, 155424 (2007).
G. P. Mikitik and Yu. V. Sharlai, Phys. Rev. B 77, 113407 (2008).
E. McCann, Phys. Rev. B 74, 161403 (2006).
H. Min, B. Sahu, S. Banerjee, and A. H. MacDonald, Phys. Rev. B 75, 155115 (2007).
E. McCann, D. Abergel, and V. Falko, Solid State Commun. 143, 110 (2007).
M. Mucha-Kruczynski, E. McCann, and V. I. Falko, Solid State Commun. 149, 1111 (2009).
D. Suszalski, G. Rut, and A. Rycerz, Phys. Rev. B 97, 125403 (2018).
M. Mucha-Kruczynski, I. L. Aleiner, and V. I. Falko, Phys. Rev. B 84, 041404 (2011).
M. Mucha-Kruczynski, I. L. Aleiner, and V. I. Falko, Solid State Commun. 151, 1088 (2011).
A. Varlet, D. Bischo, P. Simonet, et al., Phys. Rev. Lett. 113, 116602 (2014).
A. Varlet, M. Mucha-Kruczynski, D. Bischo, et al., Synth. Met. 210, 19 (2015).
L. V. Keldysh, Sov. Phys. JETP 7, 788 (1958).
L. V. Keldysh, Sov. Phys. JETP 20, 1307 (1965).
I. F. Akyildiz, J. M. Jornet, and C. Han, Phys. Commun. 12, 16 (2014).
H. Vettikalladi, W. T. Sethi, A. F. Bin Abas, et al., Int. J. Anten. Propagat. 2019, 9573647 (2019).
M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, et al., Phys. Rev. A 49, 2117 (1994).
C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms. Introduction to Quantum Electrodynamics (Wiley, New York, 1989).
E. H. Hwang and S. Das Sarma, Phys. Rev. B 77, 115449 (2008).
J. K. Viljas and T. T. Heikkila, Phys. Rev. B 81, 245404 (2010).
Author information
Authors and Affiliations
Corresponding author
APPENDIX
APPENDIX
The Liouville–von Neumann equation for a single-particle density matrix can be presented by the form
where \({{\hat {a}}_{{{\mathbf{p}},\alpha }}}\)(t) obeys the Heisenberg equation
Due to the homogeneity of the problem we only need the p-diagonal elements of the density matrix. So, taking into account Eqs. (11)–(26), the evolutionary-equation will be
nondiagonal Γ—the damping rate. In Eq. (27) the nondiagonal elements are interband polarization ρ1, –1(p, t) = P(p, t) and its complex conjugate ρ–1, 1(p, t) = P*(p, t), and the diagonal elements represent particle distribution functions for conduction Nc(p, t) = ρ1,1(p, t) and valence \({{N}_{{v}}}\)(p, t) = ρ–1,1(p, t) bands. We will solve the set of differential equations for these functions:
The total mean dipole moments are
The components of the transition dipole moments are calculated via Eq. (14) by spinor wave functions (6) (see also in [30, 33]):
The components of the velocity operator given by the relation \({{{\mathbf{\hat {v}}}}_{\zeta }}\) = ∂\(\hat {H}\)/∂\({\mathbf{\hat {p}}}\) for the effective 2 × 2 Hamiltonian (3), can be presented by the expressions:
The intraband velocity V'(p) at HHG in bilayer AB-stacked graphene is given by the formula:
The author is deeply grateful to prof. H.K. Avetissian for permanent discussions and valuable recommendations. This work was supported by the Science Committee of Ministry of Education, Science, Culture and Sport of RA.
Rights and permissions
About this article
Cite this article
Ghazaryan, A.G. Higher Harmonic Generation of Coherent Sub-THz Radiation at Lifshitz Transition in Gapped Bilayer Graphene. J. Exp. Theor. Phys. 132, 843–851 (2021). https://doi.org/10.1134/S1063776121050034
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1063776121050034