Flat bands with fragile topology through superlattice engineering on single-layer graphene

Magic-angle twisted bilayer graphene has received a lot of interest due to its flat bands with potentially nontrivial topology that lead to intricate correlated phases. A spectrum with flat bands, however, does not require a twist between multiple sheets of two-dimensional materials, but can be realized with an appropriate periodic potential. Here, we propose the imposition of a tailored potential onto a single graphene layer through local perturbations that could be created via lithography or adatom manipulation, which also results in an energy spectrum featuring flat bands. First-principle calculations for an appropriate adatom decoration of graphene indeed show the presence of flat bands and a symmetry-indicator analysis further reveals the bands' topological nature. This nontrivial topology manifests itself in corner-localized states with a filling anomaly as we show using a tight-binding calculation. Our proposal of a single decorated graphene sheet provides a new versatile route to study correlated phases in topologically nontrivial, flat band structures.

Figure 1

System setup and electronic structure. (a) The unit cell consisting of a graphene monolayer with adatoms placed in the center of the unit cell as indicated by the red circle. The lattice vectors (${\mathit{v}}_{\mathit{1}},{\mathit{v}}_{\mathit{2}}$) are related to the lattice vectors of pure graphene (${\mathit{a}}_{\mathit{1}}$, ${\mathit{a}}_{\mathit{2}}$) by Eq. (1) with $(n,m)=(-1,2)$. (b) The Brillouin zone (BZ) of monolayer graphene shown in red and the reduced BZ of our setup depicted in black. The $K$ and $K^{\prime }$ points of graphene map to the $\mathrm{\Gamma }$ point in the reduced BZ of the superstructure. (c) For comparison, one of the two BZs of TBG (in black) tilted with respect to the two BZs of graphene twisted by a small angle (in blue and red). (d) The DFT spectrum of graphene with tungsten adatoms. Irreducible representations of the bands at high-symmetry points are indicated in the figure (see Appendix pp2 for details). The right side shows the density of states projected onto the adatom and carbon $p_z$ orbitals, respectively.

Figure 2

Tight-binding calculations for open geometries. (a) The geometry of $C_6$-symmetric flake with the number of the unit cells $N_{uc}=61$. (b) Band structure of the tight-binding model (in blue) with flat bands highlighted in yellow and $C_2$-symmetry eigenvalues at $\mathrm{\Gamma }$ and M points indicating the topological nature of the bands. The spectrum for the flake (in red) shows in-gap states around the energy indicated by the dashed green line. (c) The unit cell of the ribbon geometry. (d) The spectrum for the ribbon geometry (in blue) shows a small gap around the energy indicated in green. In the flake geometry, there are six additional states that lie in this gap (in red).

Figure 3

Corner-localized in-gap states. (a) The spatial distribution of the absolute values squared of the eigenvectors corresponding to the in-gap states for $C_6$-symmetric graphene flake shown in Fig. 2 with $N_{\text{uc}}=331$. (b) Line profile of the local density of states at the bound state, integrated along the blue and red lines given in (a). Dashed red line indicates exponential fit. (c) Finite-size scaling of the in-gap states showing their degeneracy in the thermodynamic limit. An arrow indicates the states with spatial distribution given in (a).

Figure 4

The values of the hopping parameters between four different $d$ orbitals of adatom shown in the center of each hexagon and the nearest $p_z$ orbitals of graphene.

Figure 5

DFT band structures. The DFT band structure with indicated irreducible representations at $\mathrm{\Gamma }$ and M momenta and highlighted flat bands featuring fragile topology for the following adatoms: Mo, Nb, Ru, Cr, Os, Re, Ir, Ta, and Rh.