Flat bands with Berry curvature in multilayer graphene

We demonstrate that flat bands with local Berry curvature arise naturally in chiral ($\mathit{ABC}$) multilayer graphene placed on a boron nitride (BN) substrate. The degree of flatness can be tuned by varying the number of graphene layers $N$. For $N=7$ the bands become nearly flat, with a small bandwidth $\sim 3.6$ meV. The two nearly flat bands coming from the $K$ and $K^{\prime }$ valleys cross along lines in the reduced zone. Weak intervalley tunneling turns the band crossing into an avoided crossing, producing two nearly flat bands with global Chern number zero, but with local Berry curvature. The flatness of the bands suggests that many body effects will dominate the physics, while the local Berry curvature of the bands endows the system with a nontrivial quantum geometry. The quantum geometry effects manifest themselves through the quantum distance (Fubini-Study) metric, rather than the more conventional Chern number. Multilayer graphene on BN thus provides a platform for investigating the effect of interactions in a system with a nontrivial quantum distance metric, without the complication of nonzero Chern numbers. We note in passing that flat bands with nonzero Chern number can also be realized by making use of magnetic adatoms, and explicitly breaking time reversal symmetry.

Figure 1

Low energy band structure for $N$ layer $\mathit{ABC}$ stacked graphene in the presence of a vertical electric field. The band structure is plotted in the vicinity of the $\mathbf{K}$ point, assuming that the potential difference between the top and bottom layers is $\Delta =0.167t_0\approx 50$ meV.

Figure 2

Bandwidth $\Lambda$ of the lowest conduction band for $N$ layer chiral graphene. For $N>5$, the bandwidth comes mainly from umklapp scattering at the zone boundary.

Figure 3

Dispersions of the three lowest conduction bands for $N=7$ along the high symmetry directions.

Figure 4

Contour plot of Berry curvature in the lowest flat conduction band for $N=7$. The red/yellow regions come mainly from the $K$ valley and have positive curvature, while the blue regions come mainly from the $K^{\prime }$ valley and have negative curvature. The Berry curvature integrated over the band is zero.