## Abstract

Twisted van der Waals heterostructures with flat electronic bands have recently emerged as a platform for realizing correlated and topological states with a high degree of control and tunability. In graphene-based moiré heterostructures, the correlated phase diagram and band topology depend on the number of graphene layers and the details of the external environment from the encapsulating crystals. Here, we report that the system of twisted monolayer–bilayer graphene (tMBG) hosts a variety of correlated metallic and insulating states, as well as topological magnetic states. Because of its low symmetry, the phase diagram of tMBG approximates that of twisted bilayer graphene when an applied perpendicular electric field points from the bilayer towards the monolayer graphene, or twisted double bilayer graphene when the field is reversed. In the former case, we observe correlated states that undergo an orbitally driven insulating transition above a critical perpendicular magnetic field. In the latter case, we observe the emergence of electrically tunable ferromagnetism at one-quarter filling of the conduction band, and an associated anomalous Hall effect. The direction of the magnetization can be switched by electrostatic doping at zero magnetic field. Our results establish tMBG as a tunable platform for investigating correlated and topological states.

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## Data availability

All data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. Source data are provided with this paper.

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## Acknowledgements

We thank A. Young, H. Polshyn, A. Millis, C.-Z. Chang, J.-H. Chu and E. Khalaf for helpful discussions. The research on correlated states in twisted monolayer–bilayer graphene was primarily supported as part of Programmable Quantum Materials, an Energy Frontier Research Center funded by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under award DE-SC0019443. The measurements and understanding of ferromagnetism at the University of Washington were partially supported by NSF MRSEC 1719797. M.Y. is an Army Research Office Young Investigator (W911NF-20-1-0211). X.X. acknowledges support from the Boeing Distinguished Professorship in Physics. X.X. and M.Y. acknowledge support from the State of Washington funded Clean Energy Institute. This work made use of a dilution refrigerator system provided by NSF DMR-1725221. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan (grant no. JPMXP0112101001), JSPS KAKENHI (grant no. JP20H00354) and CREST (JPMJCR15F3), JST.

## Author information

### Affiliations

### Contributions

S.C., M.H. and V.H. fabricated the devices and performed the measurements. Y.-H.Z. performed the calculations. Z.F. and D.H.C. assisted with measurements in the dilution refrigerator. K.W. and T.T. grew the BN crystals. S.C., M.H., X.X., C.R.D. and M.Y. analysed the data and wrote the manuscript with input from all authors.

### Corresponding authors

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Additional information

**Publisher’s note** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data

### Extended Data Fig. 1 Optical microscope images and two-terminal conductance of the tMBG devices.

(**a-d**), the twist angle of each device is denoted at the bottom left corner of each image. All scale bars are 5 *μ*m. (**e-f**) show the two-terminal conductance measured between neighbouring pairs of contacts as indicated by the colour-coded bars in (**a-b**). Device D1 exhibits a small twist angle gradient of approximately 0.03° primarily along the longitudinal direction of the Hall bar. Device D2 exhibits a larger twist angle inhomogeneity of approximately 0.1°, consistent with the apparent doubling of some of the transport features in Fig. 4a of the main text.

### Extended Data Fig. 2 Transport in tMBG at a variety of twist angles at *T* = 300 mK.

*ρ* as a function of *n* and *D* for (**a**) device D3 (*θ* = 1.44°), and (**b**) device D4 (*θ* = 1.55°). At these twist angles, the gap at the charge neutrality points only opens for *D* > 0 (pointing from monolayer to bilayer graphene). Features corresponding to single particle vHs can be seen in both conduction and valence bands. Although some features may arise owing to correlations, we do not observe any states with insulating behavior at integer *ν* within the bands. **c-d**, *ρ* and *R*_{H} for device D1 using different contacts from Fig. 1d-e of the main text. The twist angle, *θ* = 1.05°, is slightly smaller than for the region of the device probed in the main text.

### Extended Data Fig. 3 Energy scales and density of states of tMBG.

**a**, Single-particle density of states (DOS) calculated for *θ* = 1.08° as a function of *ν* and *D*, following the model described in the text. **b**, Calculated single particle gaps at *ν* = 0, + 4 (left axis) as a function of interlayer potential, *δ*. Bandwidth of the conduction band, *W* (right axis). The bandwidth does not vary widely over our this range of *δ*. **c**, *W* calculated as function of *θ* for *δ* = − 40 meV. *W* does not change substantially for 0.85^{∘} < *θ* < 1. 1^{∘}. **d**, Experimentally measured energy gaps of *ν* = 0, + 2, + 4 as a function of *D* in device D1 (*θ* = 1.08^{∘}). The single particle gaps at *ν* = 0, + 4 are qualitatively consistent with the predictions in **b**. A CI gap at *ν* = + 2 only emerges over a finite range of *D* > 0.

### Extended Data Fig. 4 Transport in an in-plane magnetic field in device D1.

**a**, *ρ* as a function of *ν* and *D* at *B*_{∣∣} = 14 T. **b**, *ρ*(*T*) for *ν* = 1, 2, 3 at both *B* = 0 (solid curves) and *B*_{∣∣} = 14 T (dashed curves). In the former, *ν* = 2 exhibits insulating behavior whereas *ν* = 1 and 3 are metallic. In the latter, all become more resistive at low temperature, and *ν* = 1 and 3 begin to undergo an insulating transition at low temperature. **c**, Device resistance, *R*, versus *T*^{−1} at *ν* = 2 and *D* = + 0.39 V/nm. (Inset) Energy gaps, *Δ*_{ν=2}, as function of *B*_{∣∣}, extracted from the thermal activation measurements. Gaps are extracted by fitting the data in the main panel to \(R\propto {e}^{\Delta /2{k}_{B}T}\), where *k*_{B} is the Boltzmann constant. The gap grows with larger *B*_{∣∣} suggestive of a spin-polarized ground state. However, thermal activation measurements may be complicated by additional orbital contributions owing to the multilayer structure of tMBG^{15}.

### Extended Data Fig. 5 High magnetic field transport in device D1.

**a**, *ρ* as a function of *n* and *D* for *D* > 0 at *B*_{⊥} = 3 T. In addition to the CI state at *ν* = 2, vertical blue stripes correspond to the formation of Landau levels within the symmetry broken halo region. **b**, *ρ* for *D* < 0 at *B*_{⊥} = 3 T. Field-assisted CI states emerge over a finite range of *D*, along with associated quantum oscillations. **c**, Energy gaps of *ν* = 2 and 3 at *D* = − 0.33 V/nm as a function of *B*_{⊥}, measured by thermal activation. The gap at *ν* = 3 closes at high field, which may be related to the spin and valley ordering of the state, and/or competition with Landau level formation. **d**, *ρ*(*T*) corresponding to *ν* = 1, 2, 3 at *B*_{⊥} = 0 and 3.5 T. All exhibit unusual metallic temperature dependence at *B* = 0, but are insulating (*ν* = 2, 3) or near a crossover point (*ν* = 1) at high field.

### Extended Data Fig. 6 Temperature dependence of the AHE resistivity and coercive field in device D2.

Measurements correspond to the data set shown in Fig. 4b of the main text, with *ν* = 0.94 and *D* = + 0.415 V/nm.

### Extended Data Fig. 7 Doping-induced switching of the magnetic order with *B*_{⊥} in device D2.

**a**, *ρ*_{yx} as *ν* is swept back and forth from -0.5 to +2.5 at *D* = + 0.415 V/nm, with *B*_{⊥}= + 4 mT, + 2mT, 0mT, − 2 mT, − 4 mT. Doping-induced switching of the magnetic order is only observed within a small range of *B*_{⊥} from ~ + 2 mT to − 2 mT. At larger *B*_{⊥}, the magnetic state is aligned with the applied field regardless of the sweep direction. The measurement is performed subsequent to thermally-cycling the device to *T* = 4 K — above the Curie temperature — and then back to base temperature. **b**, The difference between *ρ*_{yx} sweeping up and down as a function of *B*_{⊥}. The red region denotes the regime in which the magnetization can be switched with doping.

### Extended Data Fig. 8 Dependence of doping-induced switching on magnetic field initialization in device D2.

**a-d**, show hysteresis loops at *ν* = 0.94 and *D* = + 0.415 V/nm, acquired after initiating the state with different magnetic field and doping trainings. The schematics to the left of each *ρ*_{yx} panel denote the sequence of *B*_{⊥} and *ν* sweeps as a function of time. Black lines denote the initialization operations, performed prior to data acquisition. The initialized magnetic state is indicated by the starting point of the blue *ρ*_{yx} curve at *B* = 0. Subsequent to the initialization, *ν* is held fixed at 0.94 while *B*_{⊥} is swept back and forth. Notably, independent of the field training history, *ρ*_{yx} is initialized to the positive state when *ν* is swept down from 2.5, and to the negative state when *ν* is swept up from − 0.5. Subsequent magnetic field sweeping forms a similar hysteresis loop independent of the initialized state. This helps to rule out any effects of the magnetic field initialization in our observations of the doping-induced switching effect at *B* = 0.

### Extended Data Fig. 9 Quantum oscillations in device D3 (*θ* = 1.44°).

**a**, *ρ*_{xx} and **b**, *R*_{xy} as a function of *n* and *B*_{⊥} at *D* = 0. **c**, Comparable *R*_{xy} map at *D* = − 0.2 V/nm. **d**, Schematic illustration of the prominent quantum oscillations observed in **a**. Quantum oscillations exhibit four-fold degeneracy at each value of *n*. For *n* < 0, we observe a sequence of quantum oscillations with filling factors *ν*_{LL} = − 2, − 6, − 10. . . . For *n* > 0, the sequence shifts from + 2, + 6, + 10. . . to + 4, + 8, + 12, . . . and back again as *n* is raised. These sequences can be understood by considering the monolayer- and bilayer-like corners of the moiré Brillouin zone as approximately uncoupled owing to the larger twist angle, in which the dominant quantum oscillations are the sum of the contributions from each band. **e**, Dashed blue lines denote the Landau levels of the monolayer-like bands as a function of energy, following \({E}_{LL,m}=\sqrt{2e\hslash {v}_{F}^{2}NB}\), where *N* is the Landau level index and *v*_{F} is the Fermi velocity. Dashed red lines denote the same for the bilayer-like bands, with \({E}_{LL,b}=\frac{e\hslash B}{{m}^{* }}\sqrt{N(N-1)}\), where *m*^{*} is the effective mass. The total filling factor *ν*_{LL} within each gap is given by the sum of the Landau level indexes for each band (solid bars). The experimentally observed sequence of quantum oscillations is well reproduced taking *v*_{F} = 1.44 × 10^{5} m/s, *m*^{*} = 0.14*m*_{0}, and including a charge neutrality band gap in the monolayer spectrum of *Δ*_{M} = 5 meV and an offset between the monolayer and bilayer charge neutrality points of *δ* = 14.5 meV (indicative of band overlap). **f**, Similar energy diagram for *D* = − 0.2 V/nm. To account for the observation of *ν*_{LL} = − 2 at high magnetic field, a band gap for the bilayer spectrum of *Δ*_{B} = 8 meV is included, and we take *δ* = 19.5 meV.

### Extended Data Fig. 10 Quantum oscillations at *D* = + 0.5 V/nm in device D1 (*θ* = 1.08°).

Landau fan diagram up to full filling of the moiré unit cell with *D* = + 0.5 V/nm at *T* = 300 mK. Separate sequences of quantum oscillations emerge from *ν* = − 4 with a dominant sequence of *ν*_{LL} = + 4, + 8, + 12, . . . , from *ν* = 0 with dominant sequences of *ν*_{LL} = − 2, − 6, − 10, . . . and *ν*_{LL} = + 6, + 12, + 18, . . . , and from *ν* = + 2 with a dominant state at *ν*_{LL} = + 3. Quantum oscillations emerging from *ν* = + 2 with larger filling factor do not follow a simple sequence, which may be a consequence of structural disorder in the sample. These states are illustrated schematically in the bottom panel with dashed lines. Dotted lines show additional states which emerge at higher field. Gray shaded regions denote band insulators and the orange shaded region denotes the CI state. The apparent 6-fold degeneracy of electron-like quantum oscillations emerging from *ν* = 0 may indicate the presence of multiple Fermi surface pockets within the moiré Brillouin zone.

## Supplementary information

### Supplementary Information

Supplementary Figs. 1–6.

## Source data

### Source Data Fig. 1

Data for Fig. 1b–e..

### Source Data Fig. 2

Data for line plots in Fig. 2 all panels

### Source Data Fig. 3

Data for Fig. 3a–d.

### Source Data Fig. 4

Data for Fig. 4 all panels.

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Chen, S., He, M., Zhang, Y. *et al.* Electrically tunable correlated and topological states in twisted monolayer–bilayer graphene.
*Nat. Phys.* (2020). https://doi.org/10.1038/s41567-020-01062-6

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