Gate-Tunable Topological Flat Bands in Trilayer Graphene Boron-Nitride Moiré Superlattices

We investigate the electronic structure of the flat bands induced by moiré superlattices and electric fields in nearly aligned $ABC$ trilayer graphene (TLG) boron-nitride (BN) interfaces where Coulomb effects can lead to correlated gapped phases. Our calculations indicate that valley-spin resolved isolated superlattice flat bands that carry a finite Chern number $C=3$ proportional to the layer number can appear near charge neutrality for appropriate perpendicular electric fields and twist angles. When the degeneracy of the bands is lifted by Coulomb interactions, these topological bands can lead to anomalous quantum Hall phases that embody orbital and spin magnetism. Narrow bandwidths of $\sim 10\mathrm{meV}$ achievable for a continuous range of twist angles $\theta \lesssim 0.6°$ with moderate interlayer potential differences of $\sim 50\mathrm{meV}$ make the TLG-BN systems a promising platform for the study of electric-field tunable Coulomb-interaction-driven spontaneous Hall phases.

Figure 1

(a) The band structure of ABC-TLG for zero twist angle near charge neutrality in the folded zones representation subject to $V_A^M=H_{\xi =1}^M$ moiré patterns and interlayer potential differences of $\mathrm{\Delta }=10\mathrm{meV}$ that give rise to flat Chern bands, with $C=3$ represented in red and the trivial $C=0$ band represented in blue. (b) The Berry curvatures for the valence and conduction band structures of panel (a), where we see Berry curvature hot spots near the trigonal warping band edges and mBZ boundaries that add up in the Chern band. In the trivial band, we see sharp peaks with opposite Berry curvatures, mainly at the mBZ boundaries, that cancel out.

Figure 2

(a) Chern numbers for the valence and conduction flat bands for aligned TLG on BN for the bands in Eqs. (1) and (2), bilayer on BN using $H_2^R$ from Ref. [20], and corresponding minimal $H_{\xi }^{\mathrm{R}}=0$ models, calculated using a total of 18 361 $k$ points in the mBZ. An abrupt transition happens when the sign of $\mathrm{\Delta }$ changes near zero field. (b) Schematic illustration of the Chern weights near $\tilde{\mathrm{\Gamma }}$ and the mBZ boundaries in the limit of $|\mathrm{\Delta }|\ll 1$ that add up into an integer $C^{e/h}=w_P^{e/h}+w_S^{e/h}$, where $w_P^e=-w_P^h$ and $w_S^e=w_S^h$, so that either the electron or hole band has a finite Chern number. (c) Bandwidth phase diagram for the valence and conduction bands as a function of interlayer potential difference $\mathrm{\Delta }$ and twist angle $\theta$ calculated from the difference between the maximum and minimum eigenvalue within a band. (d) Local stacking and energy-dependent LDOS and DOS manifesting the localization of flatband wave functions.

Figure 3

For the valence flat bands associated with $\mathrm{\Delta }>0$ and $\xi =1$ moiré potentials, we schematically represent how the occupation of valley $\nu =(K,K^{\prime })$ and spin $s=(\uparrow ,\downarrow )$ resolved CFBs contributes towards the generation of orbital and spin magnetism of different signs. Different band occupations can be pictured by shifting the band energies by a $g(\nu ,s)$ function. Four different configurations of valley spin are possible for (b) $1/4$ and (c) $3/4$ fillings, and two for each $1/2$ filling represented in (d)–(f). The total charge Hall conductivity ${\sigma }_H^{\text{tot}}=\sum _i{\sigma }_H^i$ of the bands is contributed by ${\sigma }_H^i=C_i{e}^2/h$ for each occupied valley-spin flavor $i$, where $C_i=N\nu \xi {\delta }_{\text{sgn}(\mathrm{\Delta })·\xi ,b}$, and the valley Hall conductivity is proportional to filling. A variety of spontaneous quantum anomalous, valley, and spin Hall effects should be expected when interaction-driven gaps open for $1/4$, $1/2$, and $3/4$ fillings of the CFB. In particular, $1/4$ and $3/4$ fillings are found to simultaneously have spin and orbital magnetism.