Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

# Tunable spin-polarized correlated states in twisted double bilayer graphene

## Abstract

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.

## Access options

from\$8.99

All prices are NET prices.

## Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## References

1. 1.

Dean, C. R. et al. Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices. Nature 497, 598–602 (2013).

2. 2.

Ponomarenko, L. A. et al. Cloning of Dirac fermions in graphene superlattices. Nature 497, 594–597 (2013).

3. 3.

Hunt, B. et al. Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure. Science 340, 1427–1430 (2013).

4. 4.

Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).

5. 5.

Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).

6. 6.

Yankowitz, M. et al. Tuning superconductivity in twisted bilayer graphene. Science 363, 1059–1064 (2019).

7. 7.

Sharpe, A. L. et al. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene. Science 365, 605–608 (2019).

8. 8.

Serlin, M. et al. Intrinsic quantized anomalous Hall effect in a moiré heterostructure. Science 367, 900–903 (2020).

9. 9.

Chen, G. et al. Evidence of a gate-tunable Mott insulator in a trilayer graphene moiré superlattice. Nat. Phys. 15, 237–241 (2019).

10. 10.

Wang, L. et al. Magic continuum in twisted bilayer WSe2. Preprint at http://arxiv.org/abs/1910.12147 (2019).

11. 11.

Regan, E. C. et al. Mott and generalized Wigner crystal states in WSe2/WS2 moiré superlattices. Nature 579, 359–363 (2020).

12. 12.

Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proc. Natl Acad. Sci. USA 108, 12233–12237 (2011).

13. 13.

Keimer, B., Kivelson, S. A., Norman, M. R., Uchida, S. & Zaanen, J. From quantum matter to high-temperature superconductivity in copper oxides. Nature 518, 179–186 (2015).

14. 14.

Koshino, M. et al. Maximally localized Wannier orbitals and the extended Hubbard model for twisted bilayer graphene. Phys. Rev. X 8, 031087 (2018).

15. 15.

Kang, J. & Vafek, O. Symmetry, maximally localized Wannier states, and a low-energy model for twisted bilayer graphene narrow bands. Phys. Rev. X 8, 031088 (2018).

16. 16.

Po, H. C., Zou, L., Vishwanath, A. & Senthil, T. Origin of Mott insulating behavior and superconductivity in twisted bilayer graphene. Phys. Rev. X 8, 031089 (2018).

17. 17.

Stepanov, P. et al. The interplay of insulating and superconducting orders in magic-angle graphene bilayers. Preprint at http://arxiv.org/abs/1911.09198 (2019).

18. 18.

Saito, Y., Ge, J., Watanabe, K., Taniguchi, T. & Young, A. F. Decoupling superconductivity and correlated insulators in twisted bilayer graphene. Preprint at http://arxiv.org/abs/1911.13302 (2019).

19. 19.

Arora, H. S. et al. Superconductivity without insulating states in twisted bilayer graphene stabilized by monolayer WSe2. Preprint at http://arxiv.org/abs/2002.03003 (2020).

20. 20.

Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K. The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009).

21. 21.

McCann, E. & Koshino, M. The electronic properties of bilayer graphene. Rep. Prog. Phys. 76, 056503 (2013).

22. 22.

Zhang, Y. et al. Direct observation of a widely tunable bandgap in bilayer graphene. Nature 459, 820–823 (2009).

23. 23.

Lee, J. Y. et al. Theory of correlated insulating behaviour and spin-triplet superconductivity in twisted double bilayer graphene. Nat. Commun. 10, 5333 (2019).

24. 24.

Zhang, Y.-H., Mao, D., Cao, Y., Jarillo-Herrero, P. & Senthil, T. Nearly flat Chern bands in moiré superlattices. Phys. Rev. B 99, 075127 (2019).

25. 25.

Koshino, M. Band structure and topological properties of twisted double bilayer graphene. Phys. Rev. B 99, 235406 (2019).

26. 26.

Chebrolu, N. R., Chittari, B. L. & Jung, J. Flat bands in twisted double bilayer graphene. Phys. Rev. B 99, 235417 (2019).

27. 27.

Choi, Y. W. & Choi, H. J. Intrinsic band gap and electrically tunable flat bands in twisted double bilayer graphene. Phys. Rev. B 100, 201402 (2019).

28. 28.

Cao, Y. et al. Superlattice-induced insulating states and valley-protected orbits in twisted bilayer graphene. Phys. Rev. Lett. 117, 116804 (2016).

29. 29.

Kim, K. et al. van der Waals heterostructures with high accuracy rotational alignment. Nano Lett. 16, 1989–1995 (2016).

30. 30.

Li, J. et al. Effective mass in bilayer graphene at low carrier densities: the role of potential disorder and electron–electron interaction. Phys. Rev. B 94, 161406 (2016).

31. 31.

He, M. et al. Tunable correlation-driven symmetry breaking in twisted double bilayer graphene. Preprint at http://arxiv.org/abs/2002.08904 (2020).

32. 32.

Shen, C. et al. Correlated states in twisted double bilayer graphene. Nat. Phys. 16, 520–525 (2020).

33. 33.

Cao, Y. et al. Tunable correlated states and spin-polarized phases in twisted bilayer–bilayer graphene. Nature https://doi.org/10.1038/s41586-020-2260-6 (2020).

34. 34.

Burg, G. W. et al. Correlated insulating states in twisted double bilayer graphene. Phys. Rev. Lett. 123, 197702 (2019).

35. 35.

Jung, J. & MacDonald, A. H. Accurate tight-binding models for the π bands of bilayer graphene. Phys. Rev. B 89, 035405 (2014).

36. 36.

Moon, P. & Koshino, M. Optical absorption in twisted bilayer graphene. Phys. Rev. B 87, 205404 (2013).

37. 37.

Saxena, S. S. et al. Superconductivity on the border of itinerant-electron ferromagnetism in UGe2. Nature 406, 587–592 (2000).

38. 38.

Aoki, D. et al. Coexistence of superconductivity and ferromagnetism in URhGe. Nature 413, 613–616 (2001).

39. 39.

Huy, N. T. et al. Superconductivity on the border of weak itinerant ferromagnetism in UCoGe. Phys. Rev. Lett. 99, 067006 (2007).

40. 40.

Julian, S. Viewpoint: pairing with spin fluctuations. Physics 5, 17 (2012).

41. 41.

Liu, X. et al. Frictional magneto-Coulomb drag in graphene double-layer heterostructures. Phys. Rev. Lett. 119, 056802 (2017).

42. 42.

Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science 342, 614–617 (2013).

## Acknowledgements

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.

## Author information

Authors

### Contributions

X.L. and P.K. conceived the experiment. X.L., Z.H., Y.R., H.Y. and D.H.N. fabricated the samples. X.L. and Z.H. performed the measurements and analysed the data. E.K., J.Y.L. and A.V. conducted the theoretical analysis. X.L., Z.H. and P.K. wrote the paper with input from Y.R., H.Y., E.K., J.Y.L. and A.V. K.W. and T.T. supplied hBN crystals.

### Corresponding authors

Correspondence to Xiaomeng Liu or Philip Kim.

## Ethics declarations

### Competing interests

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.

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

## Extended data figures and tables

### Extended Data Fig. 1 Theoretical band structure of TDBG.

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.

### Extended Data Fig. 2 Hall effect in a device with robust half-filled insulators.

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.

### Extended Data Fig. 3 Field-induced Chern insulator in the θ = 1.26° device.

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.

### Extended Data Fig. 4 Critical behaviours in the θ = 1.26° device.

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, IV curves outside the halo.

### Extended Data Fig. 5 Enhancement of the critical temperature under in-plane magnetic field.

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.

### Extended Data Fig. 6 Origin of a small residual resistivity at T ≪ Tc.

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.

### Extended Data Fig. 8 Effective mass calculation for the 1.33° and 1.26° devices.

ac, 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 ac 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.

### Extended Data Fig. 9 Device characterization.

a, cg, 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.

## Rights and permissions

Reprints and Permissions

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

• Accepted:

• Published:

• Issue Date:

• ### Imaging moiré flat bands in three-dimensional reconstructed WSe2/WS2 superlattices

• Hongyuan Li
• , Shaowei Li
• , Mit H. Naik
• , Jingxu Xie
• , Xinyu Li
• , Jiayin Wang
• , Emma Regan
• , Danqing Wang
• , Wenyu Zhao
• , Sihan Zhao
• , Salman Kahn
• , Kentaro Yumigeta
• , Mark Blei
• , Takashi Taniguchi
• , Kenji Watanabe
• , Sefaattin Tongay
• , Alex Zettl
• , Steven G. Louie
• , Feng Wang
•  & Michael F. Crommie

Nature Materials (2021)

• ### Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene

• Jeong Min Park
• , Yuan Cao
• , Kenji Watanabe
• , Takashi Taniguchi
•  & Pablo Jarillo-Herrero

Nature (2021)

• ### Moiré heterostructures as a condensed-matter quantum simulator

• Dante M. Kennes
• , Martin Claassen
• , Lede Xian
• , Antoine Georges
• , Andrew J. Millis
• , James Hone
• , Cory R. Dean
• , D. N. Basov
• , Abhay N. Pasupathy
•  & Angel Rubio

Nature Physics (2021)

• ### Quantum criticality in twisted transition metal dichalcogenides

• Augusto Ghiotto
• , En-Min Shih
• , Giancarlo S. S. G. Pereira
• , Daniel A. Rhodes
• , Bumho Kim
• , Jiawei Zang
• , Andrew J. Millis
• , Kenji Watanabe
• , Takashi Taniguchi
• , James C. Hone
• , Lei Wang
• , Cory R. Dean
•  & Abhay N. Pasupathy

Nature (2021)

• ### Carrier transport theory for twisted bilayer graphene in the metallic regime

• Girish Sharma
• , Indra Yudhistira
• , Nilotpal Chakraborty
• , Derek Y. H. Ho
• , M. M. Al Ezzi
• , Michael S. Fuhrer
• , Giovanni Vignale

Nature Communications (2021)