Spin-orbit–driven ferromagnetism at half moiré filling in magic-angle twisted bilayer graphene

The van der Waals (vdW) moiré structures are an intriguing platform for exploring the interplay of correlation, topology and broken symmetry in 2-dimensional (2D) electronic systems. The rotational alignment between two sheets of vdW crystal gives rise to a flat moiré energy band where strong Coulomb correlation plays a dominating role in a rich landscape of emergent quantum phenomena (1–7). In a graphene moiré structure, breaking the C₂T symmetry was shown to stabilize spontaneous orbital ferromagnetism at quarter and three-quarter filling, which is manifested in robust anomalous Hall effect (AHE) with hysteretic switching transitions (8–11). Unlike one and three-quarter filling, a potential orbital ferromagnetic state at half-filled moiré band would feature a spin-unpolarized edge mode that is able to proximate superconducting pairing along a ferromagnet/superconducting interface (12). Such construction has been proposed to be key in realizing the Majorana mode. However, an orbital ferromagnet is predicted to be energetically unfavorable in twisted graphene structures, owing to the inter-valley Hund’s coupling (13–16).

As an essential ingredient in forming certain topological phases, spin-orbit coupling (SOC) provides an additional experimental knob to engineer the topological properties of moiré structures (17–19). It was recently proposed that SOC endows the moiré energy band with nonzero Berry curvature, making ferromagnetic order at half moiré filling a possibility without alignment with the hexagonal boron nitride (hBN) substrate (16, 20). Unlike bulk materials, where tuning the chemical composition is required to produce spin-orbit locking, an alternative route through the proximity effect exists in vdW structures. Close proximity between graphene and transition metal dichalcogenide crystals, such as tungsten diselenide (WSe₂), allows electron wavefunctions from both crystals to overlap and hybridize, endowing graphene with strong SOC (21–28). In this work, we use transport measurement to examine the effect of proximity-induced SOC on properties of the moiré band and its associated quantum phases.

The geometry of the twisted bilayer graphene (tBLG)/WSe₂ heterostructure is shown in Fig. 1A. An atomic interface is created by stacking a few-layer WSe₂ crystal on top of magic angle tBLG, which is further encapsulated with dual hBN and graphite crystals on top and bottom to achieve optimal sample quality (29). Transport measurement indicates excellent sample quality with low charge fluctuation δn ~ 0.08 (10¹² cm^–2) (see fig. S10). Longitudinal resistance R_xx measured from tBLG exhibits a series of well-defined resistance peaks emerging at partial filling of the moiré band, ν = –2, +1, +2, and +3, which are associated with the correlated insulator (CIs) states. The positions of these peaks are consistent with a twist angle of θ ≈ 0.98°. The tBLG and hBN substrate are maximally misaligned according to the optical image of the heterostructure (see fig. S2) (16), which is consistent with the fact that the sample appears gapless at the charge neutrality point (CNP) [see (16) for more discussions regarding the coupling between tBLG and hBN] (8, 9). Transverse resistance measurements reveal large Hall resistance R_xy at ν = +1 and +2; the Hall resistance exhibits hysteretic switching behavior as the field-effect induced doping in tBLG, n_tBLG, is swept back and forth (Fig. 1C). Hysteresis in magnetization reversal is also observed while sweeping an external magnetic field aligned perpendicular to the 2D interface, B_⊥ (Fig. 1, D and E). We note that the resistance peak at ν = +2 vanishes at large in-plane B-field (see fig. S9) (16), indicating a spin-unpolarized isospin configuration. As such, the ground state is likely valley-polarized and the ferromagnetic order is orbital. This is further illustrated by a schematic representation of the band structure (right panels of Fig. 1, D and E), where the two lowest conduction bands feature the same valley index with nonzero Chern number of C = –3 and +1.

A valley polarized state at ν = 2 is unfavorable in the absence of SOC because of the influence of inter-valley Hund’s coupling [see (16) for a more detailed discussion] (13–16). As a result, observing orbital ferromagnetism at the half-filled moiré band has remained elusive (8, 9, 30). To this end, the AHE at ν = +2 in our sample provides strong evidence that the moiré band structure is transformed by proximity-induced SOC, which is more dominant compared to inter-valley Hund’s coupling (20). Although the presence of SOC endows the moiré flatband with a nonzero Chern number, as shown in Fig. 1, D and E, the observed Hall resistance is much smaller than the expected value of the quantum anomalous Hall effect. We ascribe this behavior to the presence of bulk conduction channels parallel to the chiral edge conduction, which could result from the presence of magnetic domain walls (8) or sample disorder. A fully developed Chern insulator state could be realized in a sample with lower disorder or stronger SOC, hence a larger energy gap.

To better understand the influence of proximity-induced SOC, we note that the introduction of SOC adds an extra term to the Hamiltonian:Here τ, σ and s denote the Pauli matrix for valley, sublattice and spin at each momentum k, whereas λ_I and λ_R represent the Ising and Rashba SOC coefficients, respectively (See eqs. S3 to S7 for more detailed discussions) (16). The Ising SOC locks the valley moment τ_z with the spin moment s_z, whereas the Rashba term λ_R locks the in-plane spin s_x, s_y with the sublattice σ_x, σ_y (the locking depends on the valley τ_z). For tBLG without SOC, there is a C₂ symmetry defined as C₂:τ_xσ_x and time reversal symmetry defined as T:iτ_xs_yK, where K is the complex conjugation. Both the Ising and Rashba SOC break the C₂ symmetry, while preserving the time reversal. Therefore, C₂T is broken in the presence of proximity-induced SOC.

The combination of Rashba SOC and time reversal symmetry gives rise to a valley-contrasting spin texture within the mini Brillouin Zone (MBZ) for each band: for any momentum k, spin for the two valleys points in opposite directions, s_K(k) = –s_K´(–k). The average value of in-plane spin moment of each band is obtained by integrating over the MBZ for valley K (K´),where N is the system size. We note that a broken C₃ rotation symmetry could lead to nonzero $〈S_{\parallel }^K〉$ and $〈S_{\parallel }^{K^{\prime }}〉$. In this scenario, time reversal symmetry is preserved by the valley-contrasting spin texture $〈S_K^{\parallel }〉=-〈S_{K^{\prime }}^{\parallel }〉$. On the other hand, an orbital ferromagnetic state emerges because valley-polarized Chern bands are occupied, spontaneously breaking time reversal symmetry. Most remarkably, the combination of SOC, valley polarization and a broken C₃ rotation symmetry allows an in-plane magnetic field to couple to the orbital magnetic order through the in-plane Zeeman energy.

This control of the out-of-plane magnetic order using an in-plane magnetic field B_|| is demonstrated in Fig. 2, B and C: as B_|| is swept back and forth, R_xy exhibits hysteretic switching behavior at both ν = +1 and +2. Using Hall resistance measurements, we show that the out-of-plane component of the B-field is negligible compared to the out-of-plane coercive field, confirming that the magnetic order is indeed controlled by the in-plane component of the B-field, B_|| (see fig. S4). At the same time, direct coupling between the valley order and B_|| is shown to be absent in tBLG samples without SOC (31), which rules out the orbital effect as a possible origin for the observed B_||-dependence. Taken together, we conclude that B_⊥ and B_|| couple to the magnetic order through different mechanisms: B_⊥ directly controls the magnetic ground state through valley Zeeman coupling, $E_Z^v=-{\gamma }_v{B}_{\perp }{\tau }_z$, whereas the influence of B_|| arises from the combination of SOC and spin Zeeman coupling, $E_{\text{Z}}^s=-{\gamma }_s{\tau }_z{B}_{\parallel }\cdot \langle S_{\parallel }\rangle$. Here τ_z corresponds to the valley polarization, γ_v and γ_s are valley and spin gyromagnetic ratio, respectively.

We note a few intriguing properties of the B-induced hysteresis behavior in Figs. 1 and 2: (i) a large B_|| stabilizes opposite magnetic orders at ν = +1 and +2, which is evidenced by R_xy with opposite signs (Fig. 2, B and C). This indicates that 〈S_||〉 points in opposite directions for ν = +1 and +2. On the other hand, an out-of-plane B-field stabilizes the same magnetic order at different fillings (Fig. 1, D and E), providing further confirmation that ν = +1 and +2 feature the same valley index and the ground state at half-filling is valley polarized; (ii) although the out-of-plane coercive fields are similar, the in-plane coercive field at ν = 2 is much bigger compared to ν = 1. This suggests that the average in-plane spin moment 〈S_||〉 is much smaller at ν = 2, which is consistent with a predominantly spin unpolarized ground state (fig. S9) (16). It is worth pointing out that the observed behavior at ν = +2 does not rule out alternative isospin configurations. Our results provide important constraints for future theoretical work to examine such possibilities. The effective control of in-plane B-field on the magnetic order not only provides further validation that the orbital ferromagnetic order is stabilized by proximity-induced SOC, it also reveals that a broken C₃ rotational symmetry gives rise to a nonzero 〈S_||〉. A spontaneously broken C₃ symmetry naturally derives from the combination of strong SOC and nematic charge order (32–34). Alternatively, a preferred in-plane direction for spin could result from small amount of uniaxial strain in the moiré lattice (Fig. 2D), which is shown to be common for tBLG samples (35, 36).

The proximity-induced SOC arises from wavefunction overlap across the interface (22). The role of wavefunction overlap was recently demonstrated experimentally, as SOC strength was shown to depend on interlayer separation, which is tunable with hydrostatic pressure (37). In the same vein, we show that SOC strength can be controlled with a perpendicular electric field D: under a positive (negative) D, charge carriers are polarized toward (away from) the WSe₂ crystal, resulting in increased (decreased) wavefunction overlap and stronger (weaker) SOC (Fig. 3A) (22, 37). Figure 3 demonstrates the effect of D by plotting the evolution of B-induced hysteresis loops: with increasing D, hysteresis loops exhibit larger coercive fields ($B_{\uparrow }^{\perp }$ and $B_{\downarrow }^{\perp }$) for both ν = +1 and +2 (Fig. 3, B and C). Transport measurements near the CNP show that the width of the disorder regime remains the same over a wide range of D-field (see fig. S10B), suggesting that changes in the coercive field are not caused by the influence of disorder across two graphene layers. Because the value of the coercive field reflects the robustness of magnetic order, the D-dependence shown in Fig. 3C provides another indication that the orbital ferromagnetism is stabilized by proximity-induced SOC.

Most interestingly, the effect of varying D at ν = +2 is drastically different in the presence of an in-plane magnetic field B_||: changing D induces a magnetization reversal, which is evidenced by hysteresis loops with opposite signs in R_xy (Fig. 3D and fig. S12A) (16). The D-induced reversal is also manifested in the sign change of coercive field $B_{\uparrow }^{\parallel }$ and $B_{\downarrow }^{\parallel }$ at D ~ –100 mV/nm (Fig. 3E). A possible explanation for this unique D-dependence is obtained by examining the average in-plane spin moment for valley K, $〈S_K^{\parallel }〉$. Calculations using the 4-band model show that $〈S_K^{\parallel }〉$ changes sign with varying D at ν = 2 (Fig. 3F), indicating that an in-plane B-field favors opposite valley polarizations at different D. By comparison, Fig. 3F shows that no sign change in $〈S_K^{\parallel }〉$ is expected at ν = 1, which is consistent with our experimental observation (Fig. 3E). Because carrier density remains the same when changing D, the D-controlled magnetization reversal represents electric field control of the magnetic order, which is made possible by proximity induced SOC.

Next, we turn our attention to the effect of SOC on the isospin polarization and superconductivity. In the absence of SOC, the ground state at ν = –1 was shown to be isospin unpolarized at B = 0 (38, 39). The application of a large in-plane magnetic field lifts the isospin degeneracy, stabilizing an isospin ferromagnetic state (IF₃) near ν = –1, which is separated from the unpolarized state (IU) by a resistance peak in R_xx and a step in the Hall density (38, 39). Figure 4, A and B, shows that this phase boundary between IF₃ and IU extends to B_|| = 0 in the presence of proximity-induced SOC. Taken together, the influence of SOC on the iso-spin degeneracy is comparable to a large in-plane magnetic field (see fig. S7) (16). Notably, we show that the superconducting phase is unstable against proximity-induced SOC in our sample. As shown in Fig. 4C, no zero-resistance state is observed over the full density range of the moiré band. In addition, R_xx – T traces exhibit no clear downturn with decreasing T at moiré filling ν = –2 – δ, where a robust superconducting phase usually emerges in magic-angle tBLG (Fig. 4D). Because strong Ising and Rashba SOC break C₂T symmetry, the absence of superconductivity, combined with the emergence of AHE are potentially consistent with a recent theoretical proposal that C₂T symmetry is essential for stabilizing the superconducting phase in tBLG (40). Our results are distinct from another experimental report showing robust superconductivity stabilized by SOC in tBLG away from the magic angle (28). Apart from the difference in twist angle range, these distinct observations could result from a few other factors that will be discussed in the following. The D-dependence shown in Fig. 3 suggests that a dual-gated geometry, which allows independent control on D and n_tBLG, is key to investigating the influence of proximity-induced SOC in WSe₂/tBLG samples. When carrier density is controlled with only the bottom gate electrode (28), doping tBLG with electron also gives rise to a D-field that pulls electrons away from the WSe₂ crystal (see fig. S1) (16). This results in weaker SOC strength, which could contribute to the observation of superconductivity in singly-gated tBLG/WSe₂ samples (28). In addition, it is proposed that the strength of proximity-induced SOC is sensitive to the rotational alignment between graphene and WSe₂: strong SOC is expected when graphene is rotationally misaligned with WSe₂ by 10–20°, whereas perfect alignment produces weak proximity-induced SOC (41). If confirmed, this rotational degree of freedom could provide an additional experimental knob to engineer moiré band structure (42). Our sample features a twist angle of ~16° (fig. S1F) between tBLG and WSe₂, falling in the range that is predicted to induce the strongest SOC strength. The effect of rotational misalignment between tBLG and WSe₂ is investigated in two additional samples near the magic angle: AHE and hysteresis loops are observed at hν = +2 in sample A1 where tBLG and WSe₂ are misaligned at ~10° (fig. S14). On the other hand, tBLG and WSe₂ are perfectly aligned in sample A2, where AHE is absent (fig. S15) (16). The superconducting phase is absent or significantly suppressed in all samples. These observations provide experimental support for the notion that the rotational misalignment between tBLG/WSe₂, and thus the SOC strength, play a key role in determining the stability of the ferromagnetic and superconducting states. Although transport measurement alone cannot definitively confirm the influence of SOC on the superconducting phase near the magic angle, our results could motivate future efforts, both theoretical and experimental, to investigate the influence of SOC on moiré structures as a function of graphene twist angle and graphene/WSe₂ misalignment (42).