Double flat bands in kagome twisted bilayers

We have studied how a generic bilayer kagome lattice behaves upon layer rotation. We employed a tight-binding model with one orbital per site and found: (i) for low rotational angles and at low energies, the same flat bands structure, such as in twisted bilayer graphene; although, for a larger value of the magic angle. Moreover, (ii) at high energies due to the superstructure symmetry regions, we found the characteristic three-band dispersion of the kagome lattice. In the latter, its bandwidth decreases for lower angles confining them within a few meV. Therefore, we found, in the twisted kagome lattice, the coexistence of two sets of flat bands in different energies and lying in different spatial regions of the bilayer system.

Figure 1

Different stackings of kagome bilayer (a1) $C_6$, (a2) $C_3$, and (a3) $C_2$ symmetry; (b1)–(b3) the band structures for the corresponding stacking (red lines) $C_6,C_3$, and $C_2$, respectively, the dashed lines are the monolayer kagome band structure; (c) identification of symmetry regions of a $3.{89}^{\circ }$ twisted kagome bilayer lattice; the blue line in the left panel shows the unit cell.

Figure 2

Twisted kagome band structure close to Dirac bands for (a) $21.{78}^{\circ }$, (b) $6.{01}^{\circ }$, (c) $3.{14}^{\circ }$, and (d) $2.{28}^{\circ }$. The bands close to the Dirac dispersion are highlighted in red, whereas blue dashed lines are the monolayer dispersion.

Figure 3

Projected band structure in the symmetric regions for twisted angles of (a1) $21.{78}^{\circ }$, (a2) $6.{01}^{\circ }$, (a3) $3.{14}^{\circ }$, and (a4) $2.{28}^{\circ }$. Real-space distribution of the Dirac states for twisted angles of (b1) $21.{78}^{\circ }$, (b2) $6.{01}^{\circ }$, (b3) $3.{14}^{\circ }$, and (b4) $2.{28}^{\circ }$.

Figure 4

Twisted kagome symmetric site projected band structures close to flat bands [shaded regions in Fig. 1–1] for (a1) $21.{78}^{\circ }$, (a2) $13.{17}^{\circ }$, (a3) $9.{43}^{\circ }$, and (a4) $7.{34}^{\circ }$. Real-space distribution of the states shown in (a) for (b1) $21.{78}^{\circ }$, (b2) $13.{17}^{\circ }$, (b3) $9.{43}^{\circ }$, and (b4) $7.{34}^{\circ }$.

Figure 5

Bandwidth evolution as a function of the $1/\alpha$ parameter for different angles.