Nearly flat Chern bands in moiré superlattices

Topology and electron interactions are two central themes in modern condensed matter physics. Here, we propose graphene-based systems where both the band topology and interaction effects can be simply controlled with electric fields. We study a number of systems of twisted double layers with small twist angle where a moiré superlattice is formed. Each layer is chosen to be either AB-stacked bilayer graphene, ABC-stacked trilayer graphene, or hexagonal boron nitride. In these systems, a vertical applied electric field enables control of the bandwidth, and interestingly also the Chern number. We find that the Chern numbers of the bands associated with each of the two microscopic valleys can be $\pm 0,\pm 1,\pm 2,\pm 3$ depending on the specific system and vertical electrical field. We show that these graphene moiré superlattices are promising platforms to realize a number of fascinating many-body phenomena, including (fractional) quantum anomalous Hall effects. We also discuss conceptual similarities and implications for modeling twisted bilayer graphene systems.

Figure 1

Bandwidth of valence band with the applied voltage $U$ for the TG/$h$-BN system.

Figure 2

Possibility of gate-driven nearly flat Chern band for twisted graphene/$h$-BN systems. $U$ is the potential difference between the top and bottom layers, controlled by applied vertical electric field. Figures are generated for BG/$h$-BN. TG/$h$-BN shows quite similar behavior. $n=2$ for BG/$h$-BN and $n=3$ for TG/$h$-BN. Two sides with opposite $U$ have similar band dispersions with Chern number $\left|C\right|=0$ and $n$, respectively, which enables to switch between Hubbard model physics and quantum Hall physics within one sample. Twist angle $\theta =0$ and $a_M\approx 15$ nm. $K^{\prime }=(0,\frac{4\pi }{3a_M}),K^{″}=(\frac{2\pi }{\sqrt{3}{a}_M},\frac{2\pi }{3a_M})$, and $K=(0,-\frac{4\pi }{3a_M})$ are equivalent in the MBZ.

Figure 3

Bandwidth of conduction band with twist angle for the BG/BG system.

Figure 4

Chern number $\left|C\right|$ of BG/BG for conduction band. $U$ is in units of meV. There is a clear phase boundary $U_c\left(\theta \right)$, at which Chern number jumps from $\left|C\right|=2$ to 1 because the hybridization gap between conduction band and the band above closes at $\mathrm{\Gamma }$ point. $U_c\left({\theta }_M\right)=0$ at magic angle ${\theta }_M$ in our simple model. As explained in the main text, our method may underestimate $U_c\left(\theta \right)$ and $U_c\left(\theta \right)$ may remain large even at magic angle in realistic systems.

Figure 5

Bandwidth and band gap (in units of meV) with applied voltage difference $U$. (a) BG/$h$-BN; (b) TG/$h$-BN.

Figure 6

Integration of Berry curvature in region within MBZ with distance smaller than $r$ to $\mathrm{\Gamma }$ point. $r$ has been normalized by $|{\mathbf{K}}_M|=\frac{4\pi }{3a_M}$. The value at $r=1$ is the Chern number. (a) BG/$h$-BN; (b) TG/$h$-BN.

Figure 7

BG/BG system with both type-I and -II stackings. At magic angle $\theta \approx 1.{08}^{\circ }$, bandwidth reduces to almost zero. Meanwhile, the hybridization gap between conduction band and the band above also goes to zero. Roughly, $U_c\left(\theta \right)\sim {\mathrm{\Delta }}_{\mathrm{\Gamma }}\left(\theta \right)$. AB on BA stacking has very different $U_c\left(\theta \right)$ when $\theta >{\theta }_M$. $t_M=110$ meV in these calculations. (a) Bandwidth of conduction band with twist angle; (b) the gap ${\mathrm{\Delta }}_{\mathrm{\Gamma }}$ (in units of meV) between conduction band and the band above at $\mathrm{\Gamma }$ point.

Figure 8

Band structures for type-I BG/BG system. Twist angle is $\theta =1.{08}^{\circ }$. $K^{\prime }=(0,\frac{4\pi }{3a_M})$. $K^{″}=(\frac{2\pi }{\sqrt{3}{a}_M},\frac{2\pi }{3a_M})$ and $K=(0,-\frac{4\pi }{3a_M})$ are equivalent. At $U=\pm 1.8$ meV, mass term of one cone at $K$ or $K^{\prime }$ changes sign. (a) $U=0$ meV; (b) $U=10$ meV; (c) $U=-1.8$ meV; (d) $U=1.8$ meV.

Figure 9

Dispersion around $\mathrm{\Gamma }$ point for type-I BG/BG system. Twist angle is $\theta =1.{17}^{\circ }$. Gap closing is obvious at $U_c\left(\theta \right)=12$ meV. (a) $U=0$ meV; (b) $U=12$ meV; (c) $U=20$ meV.

Figure 10

Magic angle for type-I TG/TG and TG/BG systems. $t_M$ are hopping parameters for modeling moiré hopping term. Real system corresponds to $t_M=110$ meV. (a) TG/TG; (b) TG/BG.

Figure 11

Chern number $C$ for valley $+$ of twisted graphene on graphene systems. (a) Type-I BG/BG: conduction band; (b) type-II BG/BG: conduction band; (c) type-II BG/BG: valence band; (d) type-I TG/TG: conduction band; (e) type-I TG/BG: conduction band; (f) type-I TG/BG: valence band.