Floquet bound states around defects and adatoms in graphene

Recent studies have focused on laser-induced gaps in graphene which have been shown to have a topological origin, thereby hosting robust states at the sample edges. While the focus has remained mainly on these topological chiral edge states, the Floquet bound states around defects lack a detailed study. In this paper we present such a study covering large defects of different shape and also vacancy-like defects and adatoms at the dynamical gap at $\hslash \mathrm{\Omega }/2$ $(\hslash \mathrm{\Omega }$ being the photon energy). Our results, based on analytical calculations as well as numerics for full tight-binding models, show that the bound states are chiral and appear in a number which grows with the defect size. Furthermore, while the bound states exist regardless of the type of the defect's edge termination (zigzag, armchair, mixed), the spectrum is strongly dependent on it. In the case of top adatoms, the bound state quasienergies depend on the adatoms energy. The appearance of such bound states might open the door to the presence of topological effects on the bulk transport properties of dirty graphene.

Figure 1

Scheme of irradiated graphene with different defects on the graphene lattice: holes, adatoms, or regions with a staggering sublattice potential. The arrows indicate the chirality of the probability currents associated with the Floquet bound states around the defects.

Figure 2

Energy levels for $\eta =0.15$ and $l=0,\pm 1,\pm 2$, ... as a function of the size of the defect. Solid and dashed lines correspond to the different Dirac cones. In both cases, the thicker lines correspond to $l=0$ and energy levels with $l>0$ $\left(l<0\right)$ emerge from the top (bottom) of the dynamical gap.

Figure 3

Energy levels for the case of IMBC. Parameters as in Fig. 2. The solid (dashed) line corresponds to $K\left(K^{\prime }\right)$ cone. There is no solution with $l=0$.

Figure 4

Probability (top) and current (bottom) densities as a function of the radial coordinate $\xi$ for two different boundary conditions studied in Sec. 3. Notice the log scale on the horizontal axis. In all cases, the defect boundary is located at ${\xi }_0=1$ $\left(R=30a_{cc}\right)$. Probabilities and current densities with $\left|l\right|>3$ are several orders of magnitude smaller than those shown in the figure and were omitted for clarity. In the case of the IMBC, curves with $l$ and $-l$ are coincident $(l>0$ are shown). The spatial range shown in the figure corresponds to the distance from the center of the defect to the end of the samples of graphene sheets used in Sec. 5 for the numerical tight-binding calculations $(500\times \sqrt{3}{a}_{cc})$.

Figure 5

Dispersion relation of a nanoribbon of $\sim {10}^6$ atoms width with zigzag (top row) and infinite-mass (down row) boundary conditions obtained numerically by decimation procedures and a tight-binding model. All parameters such as $\hslash \mathrm{\Omega },\eta$, and mass $\delta$ are the same as used in Sec. 5.

Figure 6

Total current as a function of the quantum number $l$ for different defect sizes $\left({\xi }_0=1,5,10,20\right)$. Open (closed) symbols correspond to Floquet states lying in the $K$ $\left(K^{\prime }\right)$ cone (lines are only guides for the eye). A curve with a larger span on $l$ corresponds to a larger ${\xi }_0$. The dotted black line in the top panel represents the $\eta /(1+{\eta }^2)$ value expected for the current of the Floquet edge state in a semi-infinite irradiated graphene sheet with zigzag termination. The inset shows how this limit is reached for $l=0$ when the size of the defect increases.

Figure 7

Color map of the Floquet local density of states (FLDOS) projected on the $m=0$ replica and onto sites located around the boundary of the defects for different sizes of the defects, ${\overline{\xi }}_0=k_0\overline{R}$ (see main text). Top and bottom panels show the case of triangular and hexagonal holes, respectively, with zigzag (left) and armchair (right) edge termination. The appearance of Floquet bound states inside the bulk dynamical gap is apparent from the figure. The dashed lines in the zigzag triangular case correspond to the analytical solution found in Sec. 3a for a “zigzag circle.”

Figure 8

Same as Fig. 7 but for the case of where a staggered potential is included inside the defect region (IMBC). Dashed lines on the bottom panels correspond to the analytical solutions calculated for the IMBC hexagon as explained in the appendix. Notice that, unlike the hole defects, the FLDOS for the IMBC does not depend on the termination of the defects—except for the zigzag triangular defect (see main text).

Figure 9

Same as Fig. 7 but for a circular defect: (i) hole (top); (ii) staggered potential (bottom).

Figure 10

Local retarded Green's functions $\left(G_{nm}\right)$ for the irradiated pristine graphene corresponding to the Floquet subspaces $m,n=0,1$. These calculations were obtained numerically by decimation procedures, projecting onto only one C atom and using the same parameters $\eta$ and $\hslash \mathrm{\Omega }$ as earlier. The black arrows show the zeros of $G_{00}$ and $G_{11}$, i.e., the quasienergies of the bound states for the vacancy (see main text).

Figure 11

Floquet local density of states around the center of the dynamical gap as a function of hybridization strength, projected on the three first neighbors of the carbon atom at which the impurity is adsorbed. Left and right panels correspond to projections onto $m=0$ and $m=1$ Floquet subspaces, respectively. The bound states inside the dynamical gap were obtained separately for adatoms with single-orbital energies ${\epsilon }_0=0,{\epsilon }_1=\hslash \mathrm{\Omega }/2$, and ${\epsilon }_2=\hslash \mathrm{\Omega }$. At the right of each panel we show the vacancy limit where both the adatom and the C atom below it are removed (red and yellow atoms in the inset).

Figure 12

Unit vectors $\widehat{\mathbf{\nu }}$ and $\widehat{\mathit{n}}$ determine the boundary conditions. Each one acts on distinct degrees of freedom: $\widehat{\mathbf{\nu }}$ acts on the isospin (valleys) and $\widehat{\mathit{n}}$ acts on the pseudospin (sublattices). Unit vector $\widehat{\mathit{n}}$ depends on the number of sides $N$ of the regular polygon via the normal unit vector ${\widehat{\mathit{n}}}_B$ for ABC and IMBC and the alternating nature of the sublattice terminations for ZZBC. This dependence also implies a dependence with the polar angle $\phi$. However, for a triangular defect there is only one type of sublattice termination (all atoms belong to the same sublattice, the $A$ sublattice in the right scheme) and the matrix $M$ in Eq. (A6) becomes $\phi$-independent.

Figure 13

Relative contributions, measured by ${\overline{a}}_{mN,N}=a_{mN,N}/a_{0,N}$, of higher orders in the $N\phi$-dependence to the zero-order expansion in the Eq. (C2) for triangular and hexagonal defects.

Figure 14

Same as Fig. 7 for the three possible distinct hexagonal hole defects. The FLDOS in the top panel is the same as shown in Fig. 7 for this kind of defect. The FLDOS for remaining configuration is shown in the bottom panel. It depends on the microscopic details beyond the zigzag or armchair terminations.