Reducing the energy bandwidth of electrons in a lattice below the long-range Coulomb interaction energy promotes correlation effects. Moiré superlattices—which are created by stacking van der Waals heterostructures with a controlled twist angle1,2,3—enable the engineering of electron band structure. Exotic quantum phases can emerge in an engineered moiré flat band. The recent discovery of correlated insulator states, superconductivity and the quantum anomalous Hall effect in the flat band of magic-angle twisted bilayer graphene4,5,6,7,8 has sparked the exploration of correlated electron states in other moiré systems9,10,11. The electronic properties of van der Waals moiré superlattices can further be tuned by adjusting the interlayer coupling6 or the band structure of constituent layers9. Here, using van der Waals heterostructures of twisted double bilayer graphene (TDBG), we demonstrate a flat electron band that is tunable by perpendicular electric fields in a range of twist angles. Similarly to magic-angle twisted bilayer graphene, TDBG shows energy gaps at the half- and quarter-filled flat bands, indicating the emergence of correlated insulator states. We find that the gaps of these insulator states increase with in-plane magnetic field, suggesting a ferromagnetic order. On doping the half-filled insulator, a sudden drop in resistivity is observed with decreasing temperature. This critical behaviour is confined to a small area in the density–electric-field plane, and is attributed to a phase transition from a normal metal to a spin-polarized correlated state. The discovery of spin-polarized correlated states in electric-field-tunable TDBG provides a new route to engineering interaction-driven quantum phases.
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The data that support the findings of this study are available from the corresponding author upon reasonable request.
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The major experimental work is supported by DOE (DE-SC0012260). P.K. acknowledges support from the DoD Vannevar Bush Faculty Fellowship N00014-18-1-2877. Z.H. is supported by ARO MURI (W911NF-14-1-0247). A.V., J.Y.L. and E.K. were supported by a Simons Investigator Fellowship. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, A3 Foresight by JSPS and the CREST (JPMJCR15F3), JST. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by the National Science Foundation Cooperative Agreement Number DMR-1157490* and the State of Florida. Nanofabrication was performed at the Center for Nanoscale Systems at Harvard, supported in part by an NSF NNIN award ECS-00335765. We thank S. Fang, S. Carr, Y. Xie, E. Kaxiras, B. I. Halperin, A. F. Young, J. Waissman and A. Zimmerman for helpful discussion.
The authors declare no competing interests.
Peer review information Nature thanks Hu-Jong Lee, Ming-Hao Liu and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
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Extended data figures and tables
a, Calculated band structure of TDBG at zero displacement field and optimal displacement field D0 for the isolated flat band. b, Calculated parameter space for isolated conduction band (x axis is onsite potential difference U = Vt − Vb between the top and bottom graphene layer, y axis is twist angle). Colour represents the bandwidth of the first conduction band c1 (meV). In the coloured parameter space, c1 is isolated from the second conduction band and the first valence bands. The two dotted lines represent cuts at θ = 1.26 and θ = 1.33°. c, Resistivity as a function of filling fraction and displacement field in the θ = 1.33° sample. A cross-like feature of high resistivity is formed along two lines from (n, D) = (1, −0.2) to (1, 0.6) and (−1, 0.2) to (1, −0.6), passing through the half-filled insulating states. d, Density of states at the Fermi energy calculated by the continuum model. The single-particle insulators (n/ns = 0, ±1) in experiment match well with the gaps shown in the calculation and the van Hove singularity captures the cross-like pattern in experiment.
a, b, Longitudinal resistivity (a) and Hall resistance (b) of the θ = 1.41° device around half-filling at T = 1.5 K and under perpendicular magnetic field B⊥ = 1 T. Data are symmetrized between positive and negative fields to eliminate mixing. The halo structure is apparent around the half-filled insulator and a three-quarter-filled insulating state resides on the border of the halo. The Hall resistance changes sign across the half-filled insulator inside the halo. c, Illustration of electron orders for different regimes. The left half (right half) of the cartoon represents the band of spin down (up) electrons. For half-filling, only one species of spin is filled. Inside the halo, one spin species is populated more than the other. Outside the halo, both spins are equally populated.
a, b, Longitudinal resistivity at B = 0 (a) and Hall resistance at B⊥ = 0.5 T (b) in the θ = 1.26° sample at T = 1.5 K. The Hall resistance here is symmetrized with both directions of the magnetic field. c, d, Fan diagram of longitudinal resistivity (c) and Hall resistance (d) at T = 1.5 K at a constant displacement field. The black line marks the expected position for ν = 4 Chern insulator state originating from half-filling. e, Longitudinal resistivity and Hall resistance along the black line shown in c and d.
a, Resistivity in 1.26° device plotted against filling factor and displacement field. b, Resistivity as a function of displacement field and temperature along the constant density line shown in a. c, Resistivity as a function of filling and temperature along the tilted line in a. The dome of the low resistance state can be seen next to the half-filled insulator. d, Resistivity on a log scale as a function of filling and perpendicular magnetic field. e, The power α in V ∝ Iα as a function of temperature from fitting the top left inset of Fig. 3c. α = 3 is defined as the BKT transition temperature. f, Differential resistance as a function of current and displacement field along the constant density line shown in a. g, I–V curves outside the halo.
a, Resistivity as a function of in-plane magnetic field across the half-filled insulator and superconducting-like state in the 1.26° device. b, Resistivity as a function of temperature and in-plane magnetic field at optimal doping and displacement field. TBKT denotes the BKT temperature extracted from nonlinear IV measurements. T50% marks the temperature where resistance is half of the normal resistance. c, Line traces at different in-plane magnetic fields. d, Illustration of pairing in spin-polarized superconductor. The blue (red) surface represents the spin down (up) electron band. The two bands are filled differently due to the parent ferromagnetic metallic state. The hexagon represents the Brillouin zone of graphene lattice. Pairing thus happens between Fermi surfaces of the same spin and opposite valleys.
R(T) curve measured in the superconducting regime of the 1.26° device in two measurement configurations. The voltage probes are kept the same between the two configurations while the source and the drain contacts are switched.
Extended Data Fig. 7 Landau fan diagram as a function of filling fraction and perpendicular magnetic field.
a, The 1.33° device. The numbers next to the guiding lines indicate Landau-level filling factors. b, The 1.26° device. Horizontal lines highlight the Hofstadter’s butterfly features that occur when a simple fraction of the flux quantum ϕ0/N (N is an integer) penetrates through a moiré unit cell.
a–c, Temperature-dependent SdH oscillations in the θ = 1.33° device at a few representative density points: n = −1.3 × 1012 cm−2 (a), 1.45 × 1012 cm−2 (b) and 2.65 × 1012 cm−2 (c). d, Extracted oscillation amplitudes as a function of T/B for the density configuration shown in a–c and corresponding fitting curves. e, Temperature-dependent SdH oscillations in the θ = 1.26° device at n = 0.61 × 1012 cm−2, which is above half-filling and inside the halo. f, Extracted oscillation amplitudes as a function of T/B in the θ = 1.26° device.
a, c–g, Device structure, optical image and four-terminal resistivity map of each device: 1.26° (a), 1.32° (c), 1.41° (d), 1.48° (e), 1.53° (f) and 2.00° (g). For the 1.26° device, the active device is the four-terminal Van der Pauw sample. The structure of each device is depicted by the cross-section illustration on the left of the optical image. b, Two-terminal resistance measured in the 1.26° device in the same gate voltage range presented in a. Dashed square marks the active area studied. h, Structure and optical image of the 1.33° device.
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Liu, X., Hao, Z., Khalaf, E. et al. Tunable spin-polarized correlated states in twisted double bilayer graphene. Nature 583, 221–225 (2020). https://doi.org/10.1038/s41586-020-2458-7
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