Topological edge states on time-periodically strained armchair graphene nanoribbons

We report the emergence of electronic edge states in time-periodically driven strained armchair terminated graphene nanoribbons. This is done by considering a short-pulse spatial-periodic strain field. Then, the tight-binding Hamiltonian of the system is mapped into a one-dimensional ladder. The time periodicity is considered within the Floquet formalism. Thus the quasienergy spectrum is found numerically by diagonalizing the evolution operator. For some particular cases, the quasienergy spectrum is found analytically. We found that the system is able to support gapless and gapped phases. Very different edge states emerge for both the gapless and the gapful phases. In the case of the gapped phase, edge states emerge at the gap centered at zero quasienergy, although the Chern number is zero due to the chiral symmetry of the system. For the gapless phase, besides edge states at zero quasienergy, cosinelike edge states which merge and coexist with the bulk band are observed. To confirm the topological nature of these edge states, we analytically obtained the effective Hamiltonian and its spectrum for a particular case, finding that the edge states are topologically weak. Finally, we found analytically the evolution of band edges and their crossings as a function of the driven period. Topological modes arise at such crossings.

Figure 1

Schematic representation of the periodically delta driving layout for an armchair graphene nanoribbon. (a) For $t\ne T$, the strain field is turned off. (b) The strain field is turned on for $t=mT$. Here $T$ is the period of the delta driving. For uniaxial strain along the $y$ direction (in particular we consider a sinusoidal strain with wavelength $\sigma$), the strained armchair graphene nanoribbon can be mapped onto a one-dimensional effective system, which is represented by linear ladders on the left of the figure. Solid dots represent the position of carbon atoms. The shaded areas correspond to some unitary cells of pristine graphene, while the red dotted lines correspond to the limits of the unit cell in the $x$ direction. $j$ labels the rows in graphene, while A and B are used to denote each of the bipartite lattices [59]. Carbons on each sublattice are indicated by green and blue.

Figure 2

Gapless quasienergy band structure obtained from the numerical diagonalization of the matrix representation given by Eq. (9). The parameters used are $\sigma =1/\sqrt{3}$, $\lambda =0.2$, $\tau =\pi$, $\varphi =\pi \sigma$, and $N=324$, for (a) cyclic boundary conditions and (b) fixed boundary conditions. In panel (a), note that no edge states appear since cyclic boundary conditions were used. In panel (b), two kinds of edge states emerge. One appears around zero quasienergy. The others are cosinelike edge states that merge and coexist with the bulk bands. The colors represent the inverse participation ratio. For red color the states are highly localized, whereas for blue color they are totally delocalized.

Figure 3

Gapless quasienergy band structure obtained from the numerical diagonalization of the matrix representation of Eq. (9). The parameters used are $\sigma =1/\sqrt{3}$, $\lambda =0.2$, $\tau =\pi$, $N=324$, and using fixed boundary conditions for (a) $\varphi =0$ and (b) $\varphi =\pi \sigma /2$. Note that for panels (a) and (b) the edge states deeply penetrate into the bulk bands. Also, for panel (b), edge states around zero quasienergy are decoupled. The same color code as in Fig. 2 was used to display localization of each mode.

Figure 4

Gapful quasienergy band structure obtained from the numerical diagonalization of the matrix representation of Eq. (9) for $\sigma =1/2\sqrt{3}$, $\lambda =0.2$, $\tau =\pi$, $\varphi =\pi \sigma$, and $N=324$ using fixed boundary conditions. (a) An amplification around zero quasienergy to highlight the zero quasienergy edge modes. Observe that edge states also emerge at other gaps. The same color code as in Fig. 2 was used to display localization of each mode.

Figure 5

Quasienergy band spectrum for $k_x=0$ as a function of $\tau$ obtained by numerical diagonalization of Eq. (9) using $N=324$, $\sigma =1/\sqrt{3}$, $\lambda =0.2$, $\varphi =\pi \sigma$, and fixed boundary conditions compared with the band edges obtained from the analytical calculation given by Eqs. (30) and (32). The quasienergy band edge $\tau {\omega }^{+}$ ($\tau {\omega }^{-}$) is denoted by solid black (dotted gray) lines. Note that such band edges cross each other for the first time at $\tau ={\tau }_c^{\pm }$ at quasienergies different from zero or $\pm \pi$. It is interesting that for $\tau ={\tau }_c^{+}$ edge states emerge, which is not the case for ${\tau }_c^{-}$. This is related with the nature of the crossings. Also, observe that the edge states are not flat bands but they depend linearly on $\tau$. See Sec. 5.

Figure 6

The winding of the vector ${\widehat{\mathbf{h}}}_{\text{eff}}\left(k_y\right)$ for $k_x=0$, $\lambda =0.2$, $\sigma =1/\sqrt{3}$, and $\varphi =\pi \sigma$ using $\tau =\pi$ (a) and $\tau =2\pi$ (b). The winding number for panel (a) is 2, whereas for panel (b) it is 4. Observe that by increasing the driving period $\tau$ the winding number is also increased.