High-harmonic generation from monolayer and bilayer graphene

M. S. Mrudul and Gopal Dixit

Phys. Rev. B 103, 094308 – Published 24 March 2021

Abstract

High-harmonic generation (HHG) in solids is an emerging method to probe ultrafast electron dynamics in solids at attosecond time scale. In this work, we study HHG from monolayer and bilayer graphene. Bilayer graphenes with AA and AB stacking are considered in this work. It is found that the monolayer and bilayer graphenes exhibit significantly different harmonic spectra. The difference in the spectra is attributed to the interlayer coupling between the two layers. Also, the intraband and interband contributions to the total harmonic spectrum play a significant role. Moreover, interesting polarization and ellipticity dependence are noticed in total harmonic spectrum for monolayer and bilayer graphene.

Received 1 December 2020
Accepted 17 March 2021

DOI:https://doi.org/10.1103/PhysRevB.103.094308

Physics Subject Headings (PhySH)

High-order harmonic generation

Graphene Monolayer films

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

M. S. Mrudul and Gopal Dixit^*

Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

^*gdixit@phy.iitb.ac.in

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Vol. 103, Iss. 9 — 1 March 2021

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Images

Figure 1
Monolayer and bilayer graphene (AB-stacked). (a) and (b) The top view of the monolayer and bilayer graphenes, respectively. The carbon atoms are arranged in a honeycomb lattice with two inequivalent carbon atoms (A and B). In bilayer graphene, the B atoms of the top layer are placed on top of A atoms of the bottom layer. (c) The band structure of the monolayer (green) and bilayer graphene (violet).
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Figure 2
(a) High-harmonic spectrum of the monolayer (green) and bilayer (violet) graphenes with AB stacking. A linearly polarized pulse along the $Γ$ -K direction is considered here. The high-harmonic intensity is normalized to the total number of electrons in monolayer and bilayer graphenes. The interband and intraband contributions to the high-harmonic spectrum for (b) monolayer and (c) bilayer graphenes. The total harmonic spectrum is also plotted for the reference. The relative harmonic yield (integrated) for different orders from interband and intraband contributions is plotted in the insets of (b) and (c), respectively. (d) The normalized optical joint density of states (JDOS) of monolayer (green) and bilayer (violet) graphenes.
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Figure 3
(a) High-harmonic spectrum of bilayer graphene with AB stacking as a function of interlayer coupling ( $t_{⊥}$ ), where $t_{⊥}$ = 0 corresponds to the HHG from monolayer graphene. The red dashed line corresponds to the actual value of $t_{⊥}$ used in the calculations. (b) HHG from bilayer graphene (in AB stacking) with $t_{⊥}$ ( $B_{1} − A_{2}$ ) coupling only (violet) and with both $t_{⊥}$ and $t_{3}$ ( $A_{1} − B_{2}$ ) coupling (orange). (c) HHG from bilayer graphene with AA stacking (red) and monolayer graphene (green). In (b) and (c), the band structures near the K-point are shown in the inset. (d) and (e) show the non-zero momentum matrix elements in AB and AA stacked bilayer graphene, respectively.
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Figure 4
Polarization dependence of the normalized harmonic yield for (a) monolayer and (b) bilayer graphenes. Here, $θ$ is an angle between laser polarization and the $x$ axis along $Γ$ -K in the BZ. An illustration of the semiclassical real-space model with nearest neighbors of (e) A-type and (d) B-type carbon atoms. The first, second, third, and fourth nearest neighbors are shown using brown, orange, green, and violet colors, respectively.
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Figure 5
High-harmonic spectrum for (a) monolayer and (b) bilayer graphenes for different ellipticities of the driving laser pulse. Here, $ε$ = 0 corresponds to a linearly polarized pulse and $ε$ = 1 corresponds to a circularly polarized pulse.
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Figure 6
Ellipticity dependence of the integrated harmonic yield for $3 rd$ (H3), $5 th$ (H5), and $7 th$ (H7) harmonics of the monolayer (top panel) and bilayer graphenes (bottom panel). The integrated harmonic yield and its $x$ and $y$ components are shown by brown, red, and orange color lines, respectively. The green line shows the ellipticity dependence on the averaged ellipticity of the emitted harmonics. An elliptically polarized pulse with an intensity of $1 \times 10^{11} W / {cm}^{2}$ is used for HHG.
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