Graphene superlattices in strong circularly polarized fields: Chirality, Berry phase, and attosecond dynamics

Hamed Koochaki Kelardeh, Vadym Apalkov, and Mark I. Stockman

Phys. Rev. B 96, 075409 – Published 7 August 2017

Abstract

We propose and theoretically explore states of graphene superlattices with relaxed $P$ and $T$ symmetries created by strong circularly polarized ultrashort pulses. The conduction-band electron distribution in the reciprocal space forms an interferogram with discontinuities related to topological (Berry) fluxes at the Dirac points. This can be studied using time- and angle-resolved photoemission spectroscopy (TR-ARPES). Our findings hold promise for control and observation of ultrafast electron dynamics in topological solids and may be applied to petahertz-scale information processing.

Received 28 March 2017

DOI:https://doi.org/10.1103/PhysRevB.96.075409

Physics Subject Headings (PhySH)

Geometric & topological phases Population dynamics Ultrafast phenomena Valleytronics

Graphene

Nonlinear DynamicsAtomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Hamed Koochaki Kelardeh^*, Vadym Apalkov^†, and Mark I. Stockman^‡

Center for Nano-Optics (CeNO) and Department of Physics and Astronomy, Georgia State University, Atlanta, Georgia 30303, USA

^*hkoochakikelardeh1@student.gsu.edu
^†vapalkov@gsu.edu
^‡mstockman@gsu.edu

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Issue

Vol. 96, Iss. 7 — 15 August 2017

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Images

Figure 1
Real- and reciprocal-space structure of the system and electron trajectories for a single-oscillation circularly polarized ultrashort pulse. (a) Electron dispersion of a graphene monolayer obtained within the tight-binding approximation. Energies of the highest valence band ( $π$ band) and the lowest conduction band ( $π^{*}$ band) in the reciprocal space are displayed as functions of wave vector $k = {k_{x}, k_{y}}$ . The two distinct sets of Dirac points are labeled $K$ and $K^{'}$ . (b) Schematic of the proposed structure. A graphene monolayer is positioned over a superlattice formed by nanowires with period $L$ in the $y$ direction. Inset: Electric field waveform $F (t) = {F_{x} (t), F_{y} (t)}$ as a function of time $t$ . (c) An illustration of an electron trajectory (dashed red line) in the reciprocal space, which starts and ends at a $k_{0}$ point outside the separatrix and passes close to a $K$ point without circling around it. The separatrix (solid blue line) separates the $k_{0}$ points for those trajectoris that circle around the $K$ point and those that do not. (d) The same as in panel (c) but for the $k_{0}$ point inside the separatrix. (e) Schematic of the reciprocal space trajectories and transitions caused by the Bragg reflections for the $k_{0}$ point outside of the separatrix, corresponding to the case of panel (c). The red line shows an electron trajectory where the solid and dashed segments correspond to the VB and CB, respectively, as indicated. The thin dash-dot green and blue lines are the Bragg-shifted replica of the original trajectory. See other details in the text. (f) The same as in panel (e) but for the $k_{0}$ point inside the separatrix, corresponding to the case of panel (d).
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Figure 2
Conduction band population $N_{c} (k, t)$ in the reciprocal space in the extended Brillouin zone picture plotted (color coded) as a function of crystal momentum $k$ at time $t$ at the end of the optical pulse. The pulse has one optical oscillation with a circular polarization, duration of 5 fs, and amplitude of $F_{0} = 0.5 V / Å$ , as shown in the inset. Magnified distributions around the two nonequivalent Dirac points, $K$ and $K^{'}$ , are shown in the right panels. The separatrix is indicated by the dashed green line superimposed on the population distribution. The three discontinuities are clearly seen at the separatrix and its replicas Bragg-shifted by $\pm Q$ .
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Figure 3
Similar to Fig. 2 but with a two-cycle pulse with opposite circularities (clockwise then counterclockwise) for the two periods. The field amplitude ratio for the first and second periods is $α = 0.75$ . The expanded images of CB population near the $K$ and $K^{'}$ points are shown on the separate panels to the right. The two separatrices corresponding to the two optical-oscillation periods are indicated by the dashed green lines.
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Figure 4
Residual CB population plotted as a function of $q_{y}$ with a fixed value of $q_{x}$ . Panel (a) corresponds to a one-cycle pulse (Fig. 2) with $q_{x} = 0.1 Å^{- 1}$ (black) and $q_{x} = 0.2 Å^{- 1}$ (red). The three jumps stemming from the nontrivial geometric (Berry) phase are evident. Panel (b) corresponds to a two-cycle pulse (Fig. 3) with the second cycle of the opposite rotation and an amplitude ratio of $α = 0.75$ . For this plot, $q_{x} = 0.1 Å^{- 1}$ . There are six jumps visible originating from the nontrivial Berry phase.
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