Irradiated graphene as a tunable Floquet topological insulator

In the presence of a circularly polarized mid-infrared radiation graphene develops dynamical band gaps in its quasienergy band structure and becomes a Floquet insulator. Here, we analyze how topologically protected edge states arise inside these gaps in the presence of an edge. Our results show that the gap appearing at $\hslash \Omega /2$, where $\hslash \Omega$ is the photon energy, is bridged by two chiral edge states whose propagation direction is set by the direction of the polarization of the radiation field. Therefore, both the propagation direction and the energy window where the states appear can be controlled externally. We present both analytical and numerical calculations that fully characterize these states. This is complemented by simple topological arguments that account for them and by numerical calculations for the case of the semi-infinite sample, thereby eliminating finite-size effects.

Figure 1

(a) Scheme of a bulk graphene sheet being illuminated by a laser (perpendicular to the graphene plane). (b) Scheme showing how the Dirac cone is being modified by the laser. The opening of dynamical gaps at $\pm \hslash \Omega /2$ is evident. The bands shown in this scheme are weighted on the $m=0$ Floquet channel; that is, these bands are the ones contributing to the dc density of states.

Figure 2

Scheme of the Floquet bands with $m=-1,0,1$ as used for the calculation of the number of chiral edge states hosted within each gap. The column on the right indicates the number $\Delta N$ of chiral edges states at each crossing of the $m=0$ replica.

Figure 3

Quasienergy dispersion of the edge states, $\overline{\varepsilon }=\varepsilon /\hslash \Omega$, for the (a) $K$ and (b) $K^{\prime }$ Dirac cones and $\eta =0.25$. The vertical and horizontal dashed lines indicate the position of the Dirac cone and the center of the dynamical gap, respectively. The red solid line corresponds to the edge states on a given side of a wide $W\gg \xi$ ribbon, while the blue dashed line corresponds to the opposite side. Note that the velocity is positive for the former and negative for the latter.

Figure 4

Comparison of the tight-binding quasienergy dispersion (small dots) projected onto the $m=0$ subspace (weight given by the color scale), with the analytical expression (blue open circles) for $h\Omega =0.3{\gamma }_0$ and $z={10}^{-3}$. The tight-binding data were obtained by numerically solving the Floquet equation with $N_{\mathrm{FR}}=4$ (see main text) and $M=1000$ transverse sites.

Figure 5

Comparison of the squared modulus of a numerically obtained wave function (symbols) projected onto the $m=0$ subspace with the analytical expression (solid and dashed lines). The empty circles (empty squares) correspond to the numerical results for the $A$ sites ($B$ sites), while the analytical expressions are shown with solid and dashed lines, respectively. These results were computed for a laser with $h\Omega =0.3{\gamma }_0$ and $z=0.01$. The plotted wave functions correspond to the two branches with $\varepsilon \sim 0.15{\gamma }_0$ and $N_{\mathrm{F}}=4$.

Figure 6

Quasienergy spectrum for different ribbon terminations: (a) zigzag, (b) armchair, (c) zigzag with Klein edges, and (d) zigzag with cove edges. The color scale indicates the weight contributing to the average density of states. In the calculation the ribbons are illuminated with circularly polarized light with $\hslash \Omega =0.8{\gamma }_0$ and $z=0.04$; (a), (c), and (d) have ribbons with $W=31.95$ nm, while (b) has $W=31.48$ nm. All calculations include the Floquet replicas between $n=-2$ and $n=+2$. The laser-induced states bridging the bulk dynamical gap are evident, while those at the Dirac point are barely developed.

Figure 7

Color map of the averaged local density of states ${\rho }_{\mathrm{edge}}(\varepsilon ,k)$ projected onto several sites near the edge of the semi-infinite sheet, $r=50$, as a function of $k_y$ and $\varepsilon$. Here, $\hslash \Omega =3{\gamma }_0$, $z=0.5$, $a=\sqrt{3}{a}_{cc}$, and $N_{\mathrm{FR}}=5$. The appearance of edge states bridging the gaps is apparent. Notice that the two states at the dynamical gap have chirality opposite the one appearing inside the gap developed at the Dirac point.

Figure 8

Color map of ${\rho }_{\mathrm{edge}}(\varepsilon ,k)$ for a realistic set of radiation parameters. The color scale is set for visualization purposes. (a) Large energy span showing both the dynamical $\hslash \Omega /2$ gap (${\Delta }_1\sim \eta \hslash \Omega$) and the Dirac point gap with the corresponding edge states. (b) Close-up of the $\hslash \Omega /2$ gap; dots correspond to the analytical solution. (c) Further close-up near the middle of the first-order gap showing the emergence of a second-generation gap (${\Delta }_2\sim {\eta }^3\hslash \Omega$) and the corresponding edge states. Note that the energy (momentum) scale is reduced (enlarged). Inside this gap there are three additional states; one of them leads to an anticrossing with the first-order edge state.