Spontaneous Valley Spirals in Magnetically Encapsulated Twisted Bilayer Graphene

Van der Waals heterostructures provide a rich platform for emergent physics due to their tunable hybridization of layers, orbitals, and spin. Here, we find that twisted bilayer graphene stacked between antialigned ferromagnetic insulators can feature flat electronic bands due to the interplay between twist, exchange proximity, and spin–orbit coupling. These flat bands are nearly degenerate in valley only and are effectively described by a triangular superlattice model. At half filling, we find that interactions induce spontaneous valley correlations that favor spiral order and derive a low-energy valley-Heisenberg model with symmetric and antisymmetric exchange couplings. We also show how electric interlayer bias broadens the bands and tunes these couplings. Our results put forward magnetic van der Waals heterostructures as a platform to explore valley-correlated states.

Figure 1

Structure and single-particle electronic properties of twisted bilayer graphene (TBG) encapsulated between ferromagnetic insulators, see (a) where arrows denote the magnetization of each FI. (b) Moiré pattern arising from stacking two graphene layers with relative twist angle $\alpha$. The pattern has a length scale ${\ell }_m$ with $AA$, $AB$, and $BA$ regions. It generates a hexagonal mini-Brillouin zone with high-symmetry points. (c)–(e) Band structures at twist angle $\alpha \simeq 2°$, interlayer coupling $t_{\perp }=0.12t$, and no interlayer bias ($V=0$): for (c) the isolated TBG, including (d) local exchange fields with $m=t_{\perp }/3$, and including (e) both exchange fields with $m=t_{\perp }/3$ and Rashba SOC ${\lambda }_R=t_{\perp }/3$. The band colors indicate the expectation value of the valley-spin operator $⟨v^z{s}^z⟩$, showing fixed spin and valley in (d) and finite spin mixing at fixed valley in (e). The light blue box marks the flatband below charge neutrality.

Figure 2

Effective triangular lattice model for the moiré orbitals of the flatband. (a) Integrated local density of states $n(E,\mathit{x})$ squared of the flatband marked in Fig. 1, with $AA$ regions highlighted. (b) A sketch of the lattice model ${\mathcal{H}}_0$, see Eq. (2), where cyan circles represent the $AA$ regions, the black lines denote hoppings ${\gamma }_1$, ${\gamma }_2$, respectively, and the red and blue triangles represent the staggered flux patterns associated with corresponding hopping phases ${\varphi }_1$ and ${\varphi }_2$. (c, top) Closeup on the flatband in Figs. 1 and 1, bottom) comparison with the band of the phenomenological model [cf. Eq. (2) with ${\gamma }_1/t_{\perp }=0.03$, ${\gamma }_2/t_{\perp }=0.09$, ${\varphi }_1=0$, ${\varphi }_2=-0.4$]. The band color indicates the valley index $v=⟨v^z⟩$ and illustrates the valley degeneracy along $\gamma \text{−}\kappa \text{−}{\kappa }^{\prime }\text{−}\gamma$ (green on top of magenta). (d) (In-plane) valley spiral appearing in the mean-field ground state of ${\mathcal{H}}_0+{\mathcal{H}}_U$, cf. Eqs. (2) and (3). Arrows illustrate the valley polarization $⟨{\mathit{v}}_I⟩$ of the respective moiré Wannier orbital (inset).

Figure 3

Effect of interlayer bias $V$ on single-particle properties and effective valley–valley exchange interactions in the valley-Heisenberg model ${\mathcal{H}}_v$ (4). (a) Local valley (Berry) flux $\mathrm{\Phi }(\mathit{r},E)$ near half filling (at energy $E\simeq -0.1t_{\perp }$), averaged over the microscopic scale of model (1) including both layers, see Eq. (5). The staggered flux is largest in $AB/BA$ regions and vanishes in the $AA$ regions. (b) Flatband as obtained from the microscopic model (1) (top) compared with the phenomenological model (2) (bottom) at finite interlayer bias $V=0.33t_{\perp }$, and with ${\gamma }_1={\gamma }_2=0.07t_{\perp }$ and ${\varphi }_1=-{\varphi }_2=0.7$. Note the difference with the $V=0$ result in Fig. 2. (c)–(e) Isotropic ($J_n$), anisotropic (${\mathrm{\Delta }}_n$), and antisymmetric ($D_n$) valley exchange couplings, cf. Eq. (4), for first and second neighbors ($n=1$, 2) as interlayer bias $V$ increases. The interlayer bias can enhance the first-neighbor hopping to the point where ${\gamma }_1={\gamma }_2$ and ${\varphi }_1=-{\varphi }_2$, resulting in exchange couplings of the same magnitude for first and second neighbors (here at $V\simeq 0.33t_{\perp }$). (d) The numerical mean-field (MF) result (open circles) superimposed on the analytical result (solid and dashed lines) [58].

Figure 4

Schematic setup for an experiment signaling the presence of a valley spiral state. (a) Standard four-terminal device, with valley Hall effect (VHE) driven by a charge current and inverse VHE driven by a valley current and producing a finite voltage $V_H$ [74, 75]. (b) FI-encapsulated TBG (FI-TBG) at half filling of the flatband acts as a filter blocking the valley current and suppresses the voltage $V_H$ of the inverse VHE.