Electrically Tunable Flat Bands and Magnetism in Twisted Bilayer Graphene

Twisted graphene bilayers provide a versatile platform to engineer metamaterials with novel emergent properties by exploiting the resulting geometric moiré superlattice. Such superlattices are known to host bulk valley currents at tiny angles ($\alpha \approx 0.3°$) and flat bands at magic angles ($\alpha \approx 1°$). We show that tuning the twist angle to ${\alpha }^{*}\approx 0.8°$ generates flat bands away from charge neutrality with a triangular superlattice periodicity. When doped with $\pm 6$ electrons per moiré cell, these bands are half-filled and electronic interactions produce a symmetry-broken ground state (Stoner instability) with spin-polarized regions that order ferromagnetically. Application of an interlayer electric field breaks inversion symmetry and introduces valley-dependent dispersion that quenches the magnetic order. With these results, we propose a solid-state platform that realizes electrically tunable strong correlations.

Figure 1

Tunability of the single-particle bilayer graphene properties as a function of the twist angle. (a) Stacking two honeycomb lattices (light/dark blue) with small twist angle $\alpha$ leads to a hexagonal moiré superlattice. The long-range structure has a moiré unit cell (opaque overlay) with varying local stacking order $AA/AB/BA$, where $A$ and $B$ correspond to the two atomic sites in each of the stacked graphene unit cells. (b) Density of states (DOS) as a function of twist angle $\alpha$, showing peaks ascribed to emergent flat bands associated with three main pseudo-Landau levels (PLLs). Grey dotted lines denote the position of Fermi levels at half-filled $\pm 1$-PLL bands: The two 0-PLL bands near charge neutrality hold 4 electrons each; similarly, each of the $\pm 1$-PLL bands holds 4 electrons and thus is half-filled at a doping level of $\pm 6$ electrons per moiré cell. At the flat-band angle ${\alpha }^{*}=0.8°$, the bandwidth of the $\pm 1$ PLLs becomes minimal. (c) Local density of states (LDOS) of states in the $-1$ PLL, showing the emergence of an effective triangular lattice of states localized around the $AA$ regions. Dashed lines mark effective hopping amplitudes in an effective triangular superlattice Hamiltonian, see text. (d)–(f) Band structures for $\alpha <{\alpha }^{*}$, $\alpha ={\alpha }^{*}$ and $\alpha >{\alpha }^{*}$ illustrating how the $\pm 1$-PLL bands evolve to generate flat bands at ${\alpha }^{*}$ (marked in blue). (f) The 0 PLL flattens at the magic angle [15, 16]. Note that there is a gap to the $\pm 1$ PLLs (small in this plotted scale).

Figure 2

Effect of local mean-field interactions [Eq. (2) with $U=2t$] in twisted bilayer graphene at physical twist angle ${\alpha }^{*}=0.8°$. (a) At $-6$ electron doping, the half-filled flat band [cf. Fig. 1] exhibits an exchange splitting between the up and down spins. (b) The resulting moiré structure of local magnetic moments. It forms a superlattice of finite collinear moments (circled “$\downarrow$”) in real space. The red (blue) color indicates a positive (negative) local expectation value $⟨S_z⟩=M_z$ of the spin operator that is associated with a net ferromagnetic moment of each $AA$ region and a weak local antiferromagnetic texture therein. For numerical calculation, we rescaled the parameters to $\alpha \simeq 2.86°$, $t_{\perp }=0.46t$ [43].

Figure 3

Effects of interlayer bias $V$ on the flat bands of bilayer graphene at twist angle ${\alpha }^{*}=0.8°$, (a)–(c) without and (d) with interactions. (a) The sharp peaks in the density of states (DOS) broaden and vanish with increasing interlayer bias $V$; dotted lines mark Fermi energies for dopings $\pm 6$. (b) At $-6$ electron doping, the half-filled flat bands disperse upon introduction of a small interlayer voltage bias [$V/t=0.07$, dashed line in panel (a)] to form two independent valley-polarized bands (green and purple) [cf. Fig. 1]. (c) The local valley-Berry curvature in position space $\partial {\mathrm{\Omega }}_v(\mathit{r})/\partial \omega$ [Eq. (4)] [integrated over the bands shown in (b)] can be understood in terms of an emergent valley magnetic field in the $AB/BA$ regions. This field destroys the fine-tuned interference, inducing a finite bandwidth in the originally flat bands. (d) The total interaction-induced [$U=2t$] mean-field magnetization $M_z$ at the twist angle ${\alpha }^{*}$ decreases with interlayer bias $V$. The bias-induced dispersion quenches the interactions and the formation of magnetic order. For numerical calculation, we rescaled the physical parameters to $\alpha \simeq 2.86°$, $t_{\perp }=0.46t$ [43].