Floquet chiral edge states in graphene

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Floquet chiral edge states in graphene

P. M. Perez-Piskunow, Gonzalo Usaj, C. A. Balseiro, and L. E. F. Foa Torres

Phys. Rev. B 89, 121401(R) – Published 5 March 2014

Abstract

We report on the emergence of laser-induced chiral edge states in graphene ribbons. Insights on the nature of these Floquet states is provided by an analytical solution which is complemented with numerical simulations of the transport properties. Guided by these results we show that graphene can be used for realizing nonequilibrium topological states with striking tunability: while the laser intensity can be used to control their velocity and decay length, changing the laser polarization switches their propagation direction.

Received 8 August 2013
Revised 20 February 2014

DOI:https://doi.org/10.1103/PhysRevB.89.121401

Authors & Affiliations

P. M. Perez-Piskunow¹, Gonzalo Usaj^2,3, C. A. Balseiro^2,3, and L. E. F. Foa Torres^1,*

¹Instituto de Física Enrique Gaviola (CONICET) and FaMAF, Universidad Nacional de Córdoba, Argentina
²Centro Atómico Bariloche and Instituto Balseiro, Comisión Nacional de Energía Atómica, 8400 Bariloche, Argentina
³Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina

^*Corresponding author: lfoa@famaf.unc.edu.ar

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Issue

Vol. 89, Iss. 12 — 15 March 2014

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Images

Figure 1
(a) Scheme of the setup: a laser of frequency $Ω$ illuminates a section of an eventually disordered graphene ribbon. The circular polarization is shown to induce chiral edge states (b)–(d). Typical evolution of the quasienergy dispersion as the system width $W$ is increased [(b) $W = 21.3$ nm, (c) $W = 42.6$ nm, and (d) $W = 426$ nm] obtained from a numerical solution of (1). The laser frequency in this plot is $ℏ Ω = 0.2 γ_{0}$ and $N_{\max} = 2$ . The color indicates the weight contributing to the average density of states. Note that there is also a minigap [24] at the Dirac point which was studied in [22, 23] but which is negligible for this parameter range. Enhancing this feature would require a much higher frequency and a laser power 6–7 orders of magnitude larger [22] than in this proposal.
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Figure 2
Quasienergy dispersion versus $k$ in the neighborhood of the $K$ [(a) inset] and $K^{'}$ [(b) inset] points. The weight on the $m = 0$ Floquet channel is shown in color scale from white (zero) to black (one). The FES bridging the gap at $ℏ Ω / 2$ which have a weight close to $0.5$ are highlighted with red and blue. The spatial distribution of the probability density for those states at each valley (for the $A$ sites) is shown in color scale in the respective main frames. Panel (c) shows a cut along the map shown in (a) for $ɛ_{α} = 0.15 γ_{0}$ , this time including type $A$ and $B$ sites (squares and circles) as well as the result from the analytical calculation (dashed and solid lines). The results correspond to circularly polarized light with $ℏ Ω = 0.3 γ_{0}$ and $η = 0.05$ and a zigzag graphene ribbon with $W = 213$ nm.
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Figure 3
Spatial distribution of the total transmission probabilities between point probes located sites $s_{1}$ on layer $L$ and $s_{2}$ on layer $R$ . These layers are located 63 nm from each other and are marked with dashed lines in the insets. $s_{1}$ is taken as the second atom counting from the left [panels (a), (c), and (e)] or from the right [panels (b), (d), and (f)] and are marked with a red dot on the insets. The $x$ coordinate of $s_{2}$ on layer $R$ is the horizontal axis in these plots ( $a_{c c}$ is the carbon-carbon distance). The transmission probabilities $T_{s_{1} \to s_{2}}$ and $T_{s_{2} \to s_{1}}$ are shown as blue and red points, respectively, and are normalized to their maximum value. Panels (a) and (b) are for a pristine sample (106 nm wide) irradiated with circular polarization ( $ℏ Ω = γ_{0}$ , $η = 3 / 16$ ) and with a Fermi energy close to the center of the dynamical gap ( $ɛ = 0.497 \times ℏ Ω$ ). The black dashed lines are exponentials with a decay rate of $1 / ξ$ , as indicated by the analytical solution. In panels (c) and (d) random vacancies at 0.1% are introduced between layers $L$ and $R$ , while in (e) and (f) a strip 2 nm wide and $15.7$ nm long was cut from the right side of the sample.
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