Skip to main content

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.

Imaging tunable quantum Hall broken-symmetry orders in graphene

Abstarct

When electrons populate a flat band their kinetic energy becomes negligible, forcing them to organize in exotic many-body states to minimize their Coulomb energy1,2,3,4,5. The zeroth Landau level of graphene under a magnetic field is a particularly interesting strongly interacting flat band because interelectron interactions are predicted to induce a rich variety of broken-symmetry states with distinct topological and lattice-scale orders6,7,8,9,10,11. Evidence for these states stems mostly from indirect transport experiments that suggest that broken-symmetry states are tunable by boosting the Zeeman energy12 or by dielectric screening of the Coulomb interaction13. However, confirming the existence of these ground states requires a direct visualization of their lattice-scale orders14. Here we image three distinct broken-symmetry phases in graphene using scanning tunnelling spectroscopy. We explore the phase diagram by tuning the screening of the Coulomb interaction by a low- or high-dielectric-constant environment, and with a magnetic field. In the unscreened case, we find a Kekulé bond order, consistent with observations of an insulating state undergoing a magnetic-field driven Kosterlitz–Thouless transition15,16. Under dielectric screening, a sublattice-unpolarized ground state13 emerges at low magnetic fields, and transits to a charge-density-wave order with partial sublattice polarization at higher magnetic fields. The Kekulé and charge-density-wave orders furthermore coexist with additional, secondary lattice-scale orders that enrich the phase diagram beyond current theory predictions6,7,8,9,10. This screening-induced tunability of broken-symmetry orders may prove valuable to uncover correlated phases of matter in other quantum materials.

Your institute does not have access to this article

Access options

Buy article

Get time limited or full article access on ReadCube.

$32.00

All prices are NET prices.

Fig. 1: Landau level tunnelling spectroscopy in graphene.
Fig. 2: Quantum Hall ferromagnetic gap at charge neutrality.
Fig. 3: Tunable broken-symmetry states of charge-neutral graphene.
Fig. 4: KB order in unscreened charge-neutral graphene.
Fig. 5: CDW order in moderately screened charge-neutral graphene.

Data availability

All data described here are available at Zenodo51.

References

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

    ADS  CAS  PubMed  Google Scholar 

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

    ADS  CAS  PubMed  Google Scholar 

  3. Wong, D. et al. Cascade of electronic transitions in magic-angle twisted bilayer graphene. Nature 582, 198–202 (2020).

    ADS  CAS  PubMed  Google Scholar 

  4. Zondiner, U. et al. Cascade of phase transitions and Dirac revivals in magic-angle graphene. Nature 582, 203–208 (2020).

    ADS  CAS  PubMed  Google Scholar 

  5. Saito, Y. et al. Hofstadter subband ferromagnetism and symmetry-broken Chern insulators in twisted bilayer graphene. Nat. Phys. 17, 478–781 (2021).

    CAS  Google Scholar 

  6. Nomura, K. & MacDonald, A. H. Quantum Hall ferromagnetism in graphene. Phys. Rev. Lett. 96, 256602 (2006).

    ADS  PubMed  Google Scholar 

  7. Alicea, J. & Fisher, M. P. A. Graphene integer quantum Hall effect in the ferromagnetic and paramagnetic regimes. Phys. Rev. B 74, 075422 (2006).

    ADS  Google Scholar 

  8. Herbut, I. F. Theory of integer quantum Hall effect in graphene. Phys. Rev. B 75, 165411 (2007).

    ADS  Google Scholar 

  9. Jung, J. & MacDonald, A. H. Theory of the magnetic-field-induced insulator in neutral graphene sheets. Phys. Rev. B 80, 235417 (2009).

    ADS  Google Scholar 

  10. Kharitonov, M. Phase diagram for the ν = 0 quantum Hall state in monolayer graphene. Phys. Rev. B 85, 155439 (2012).

    ADS  Google Scholar 

  11. Young, A. F. et al. Spin and valley quantum Hall ferromagnetism in graphene. Nat. Phys. 8, 550–556 (2012).

    CAS  Google Scholar 

  12. Young, A. F. et al. Tunable symmetry breaking and helical edge transport in a graphene quantum spin Hall state. Nature 505, 528–532 (2014).

    ADS  CAS  PubMed  Google Scholar 

  13. Veyrat, L. et al. Helical quantum Hall phase in graphene on SrTiO3. Science 367, 781–786 (2020).

    ADS  CAS  PubMed  Google Scholar 

  14. Li, S.-Y., Zhang, Y., Yin, L.-J. & He, L. Scanning tunneling microscope study of quantum Hall isospin ferromagnetic states in the zero Landau level in a graphene monolayer. Phys. Rev. B 100, 085437 (2019).

    ADS  CAS  Google Scholar 

  15. Checkelsky, J. G., Li, L. & Ong, N. P. Zero-energy state in graphene in a high magnetic field. Phys. Rev. Lett. 100, 206801 (2008).

    ADS  PubMed  Google Scholar 

  16. Checkelsky, J. G., Li, L. & Ong, N. P. Divergent resistance at the Dirac point in graphene: evidence for a transition in a high magnetic field. Phys. Rev. B 79, 115434 (2009).

    ADS  Google Scholar 

  17. Ezawa, Z. F. Quantum Hall Effects (World Scientific, 2013).

  18. Herbut, I. F. SO(3) symmetry between Néel and ferromagnetic order parameters for graphene in a magnetic field. Phys. Rev. B 76, 085432 (2007).

    ADS  Google Scholar 

  19. Kharitonov, M., Juergens, S. & Trauzettel, B. Interplay of topology and interactions in quantum Hall topological insulators: U(1) symmetry, tunable Luttinger liquid, and interaction-induced phase transitions. Phys. Rev. B 94, 035146 (2016).

    ADS  Google Scholar 

  20. Zhang, Y. et al. Landau-level splitting in graphene in high magnetic fields. Phys. Rev. Lett. 96, 136806 (2006).

    ADS  CAS  PubMed  Google Scholar 

  21. Abanin, D. A. et al. Dissipative quantum Hall effect in graphene near the Dirac point. Phys. Rev. Lett. 98, 196806 (2007).

    ADS  PubMed  Google Scholar 

  22. Andrei, E. Y., Li, G. & Du, X. Electronic properties of graphene: a perspective from scanning tunneling microscopy and magnetotransport. Rep. Prog. Phys. 75, 056501 (2012).

    ADS  PubMed  Google Scholar 

  23. Dial, O. E., Ashoori, R. C., Pfeiffer, L. N. & West, K. W. High-resolution spectroscopy of two-dimensional electron systems. Nature 448, 176–179 (2007).

    ADS  CAS  PubMed  Google Scholar 

  24. Luican, A., Li, G. & Andrei, E. Y. Quantized Landau level spectrum and its density dependence in graphene. Phys. Rev. B 83, 041405(R) (2011).

    ADS  Google Scholar 

  25. Chae, J. et al. Renormalization of the graphene dispersion velocity determined from scanning tunneling sprectroscopy. Phys. Rev. Lett. 109, 116802 (2012).

    ADS  PubMed  Google Scholar 

  26. Jung, S. et al. Evolution of microscopic localization in graphene in a magnetic field from scattering resonances to quantum dots. Nat. Phys. 7, 245–251 (2011).

    CAS  Google Scholar 

  27. Liu, X. et al. Visualizing broken symmetry and topological defects in a quantum Hall ferromagnet. Science 375, 321–326 (2021).

    ADS  PubMed  Google Scholar 

  28. Motruk, J., Grushin, A. G., de Juan, F. & Pollmann, F. Interaction-driven phases in the half-filled honeycomb lattice: an infinite density matrix renormalization group study. Phys. Rev. B 92, 085147 (2015).

    ADS  Google Scholar 

  29. Capponi, S. & Läuchli, A. M. Phase diagram of interacting spinless fermions on the honeycomb lattice: a comprehensive exact diagonalization study. Phys. Rev. B 92, 085146 (2015).

    ADS  Google Scholar 

  30. Alba, E., Fernandez-Gonzalvo, X., Mur-Petit, J., Pachos, J. K. & Garcia-Ripoll, J. J. Seeing topological order in time-of-flight measurements. Phys. Rev. Lett. 107, 235301 (2011).

    ADS  CAS  PubMed  Google Scholar 

  31. Peterson, M. R. & Nayak, C. Effects of Landau level mixing on the fractional quantum Hall effect in monolayer graphene. Phys. Rev. Lett. 113, 086401 (2014).

    ADS  CAS  PubMed  Google Scholar 

  32. Feshami, B. & Fertig, H. A. Hartree–Fock study of the ν = 0 quantum Hall state of monolayer graphene with short-range interactions. Phys. Rev. B 94, 245435 (2016).

    ADS  Google Scholar 

  33. Das, A., Kaul, R. K. & Murthy, G. Coexistence of canted antiferromagnetism and bond-order in ν = 0 graphene. Phys. Rev. Lett. 128, 106803 (2021).

  34. Takei, S., Yacobi, A., Halperin, B. I. & Tserkovnyak, Y. Spin superfluidity in the ν = 0 quantum Hall state of graphene. Phys. Rev. Lett. 116, 216801 (2016).

    ADS  PubMed  Google Scholar 

  35. Wei, D. S. et al. Electrical generation and detection of spin waves in a quantum Hall ferromagnet. Science 362, 229–233 (2018).

    ADS  CAS  PubMed  Google Scholar 

  36. Stepanov, P. et al. Long-distance spin transport through a graphene quantum Hall antiferromagnet. Nat. Phys. 14, 907–911 (2018).

    CAS  Google Scholar 

  37. Assouline, A. et al. Unveiling excitonic properties of magnons in a quantum Hall ferromagnet. Nat. Phys. 17, 1369–1374 (2021).

    CAS  Google Scholar 

  38. Knothe, A. & Jolicoeur, T. Edge structure of graphene monolayers in the ν = 0 quantum Hall state. Phys. Rev. B 92, 165110 (2015).

    ADS  Google Scholar 

  39. Atteia, J., Lian, Y. & Goerbig, M. O. Skyrmion zoo in graphene at charge neutrality in a strong magnetic field. Phys. Rev. B 103, 035403 (2021).

    ADS  CAS  Google Scholar 

  40. Hou, C.-Y., Chamon, C. & Mudry, C. Electron fractionalization in two-dimensional graphene-like structures. Phys. Rev. Lett. 98, 186809 (2007).

    ADS  PubMed  Google Scholar 

  41. Nomura, K., Ryu, S. & Lee, D.-H. Field-induced Kosterlitz–Thouless transition in the N = 0 Landau level of graphene. Phys. Rev. Lett. 103, 216801 (2009).

    ADS  PubMed  Google Scholar 

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

    ADS  CAS  PubMed  Google Scholar 

  43. Li, X.-X. et al. Gate-controlled reversible rectifying behaviour in tunnel contacted atomically-thin MoS2 transistor. Nat. Commun. 8, 970 (2017).

    ADS  PubMed  PubMed Central  Google Scholar 

  44. Choi, Y. et al. Electronic correlations in twisted bilayer graphene near the magic angle. Nat. Phys. 15, 1174–1180 (2019).

    CAS  Google Scholar 

  45. Sakudo, T. & Unoki, H. Dielectric properties of SrTiO3 at low temperatures. Phys. Rev. Lett. 26, 851–853 (1971).

    ADS  CAS  Google Scholar 

  46. Hemberger, J., Lunkenheimer, P., Viana, R., Böhmer, R. & Loidl, A. Electric-field-dependent dielectric constant and nonlinear susceptibility in SrTiO3. Phys. Rev. B 52, 13159 (1995).

    ADS  CAS  Google Scholar 

  47. Sachs, R., Lin, Z. & Shi, J. Ferroelectric-like SrTiO3 surface dipoles probed by graphene. Sci. Rep. 4, 3657 (2014).

    ADS  PubMed  PubMed Central  Google Scholar 

  48. Chen, S., Chen, X., Duijnstee, E. A., Sanyal, B. & Banerjee, T. Unveiling temperature-induced structural domains and movement of oxygen vacancies in SrTiO3 with graphene. ACS Appl. Mater. Interfaces 12, 52915–52921 (2020).

    CAS  PubMed  PubMed Central  Google Scholar 

  49. Groth, C. W., Wimmer, M., Akhmerov, A. R. & Waintal, X. Kwant: a software package for quantum transport. New J. Phys. 16, 063065 (2014).

    ADS  Google Scholar 

  50. Hauschild, J. & Pollmann, F. Efficient numerical simulations with tensor networks: Tensor Network Python (TeNPy). SciPost Phys. Lect. Notes 5, https://doi.org/10.21468/SciPostPhysLectNotes.5 (2018).

  51. Coissard, A. et al. Data for Imaging tunable quantum Hall broken-symmetry orders in graphene. Zenodo https://doi.org/10.5281/zenodo.5838139 (2022).

Download references

Acknowledgements

We thank F. de Juan, H. Fertig, M. Goerbig, G. Murthy and E. Shimshoni for valuable discussions; B. Kousar for careful reading of the manuscript; D. Dufeu, Ph. Gandit, D. Grand, D. Lepoittevin, J.-F. Motte, P. Plaindoux and L. Veyrat for technical assistance in setting up the experimental system. Samples were prepared at the Nanofab facility of the Néel Institute. This work has received funding from the European Union’s Horizon 2020 research and innovation programme ERC grants QUEST No. 637815 and SUPERGRAPH No. 866365, and the Marie Sklodowska-Curie grant QUESTech No. 766025. A.G.G. acknowledges financial support by the ANR under the grant ANR-18-CE30-0001-01 (TOPODRIVE).

Author information

Authors and Affiliations

Authors

Contributions

A.C. and H.V. fabricated the samples. A.C. and D.W. performed the measurements. A.C., D.W., H.S. and B.S. analysed the data. A.G.G. and C.R. conducted the theoretical analysis. K.W. and T.T. supplied the hBN crystals. F.G. provided technical support on the experiment. C.W. and H.C. contributed to the discussion. B.S. conceived the project and wrote the paper with inputs from all co-authors.

Corresponding author

Correspondence to Benjamin Sacépé.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature thanks Christopher Gutiérrez and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Additional information

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 Estimation of the dielectric constant of SrTiO3 and rescaling of the gate map.

a, Line cut at Vb = 0 V, averaged on a range of ±20 mV, of the dIt/dVb gate map in Fig. 2a. b, Estimation from the filling factors obtained in a of the charge carrier density n (blue dots), its polynomial fit (blue curve), and computed values of ϵr ϵSTO (red curve), as a function of gate voltage. The fit yields VCNP 13.5 V. c, Rescaling of the gate map of Fig. 2a as a function of ν.

Extended Data Fig. 2 2D-FT decomposition of the asymmetric Kekulé distortion.

a, 3 × 3 nm2 image showing an asymmetric KB pattern, measured at B = 14 T and Vb = 2 mV. b, 2D Fourier transform (2D-FT) of the STM image in a. Yellow circles indicate peaks of the honeycomb lattice, red and blue circles indicate peaks of the bond-density wave. ch, 1.53 × 3 nm2 Filtered images obtained by considering certain peaks of the FFT as indicated in the top right corner of each panel. The Kekulé lattice is drawn in white for reference. The KB order is mostly retrieved by considering only the yellow and red peaks. The asymmetry of the KB pattern is encoded in the blue peaks whose two of them are twice as high as the others due to the K-CDW order. Scale bar is 500 pm.

Extended Data Fig. 3 Contrast inversion and emergence of the Kekulé bond order.

3 × 3 nm2 STM images during which we changed the bias voltage as shown in the bottom insets (the current color bars are tuned separately for each half of the images). a, We start (bottom) at Vb = 32 mV (LL0+) and switch (top) to Vb = −12 mV (LL0−) to observe a contrast inversion of the KB lattice. b, We start (top) at Vb = 200 mV (LL1) and switch (bottom) to Vb = 20 mV (LL0+) and observe the emergence of the KB order from the honeycomb lattice. Scale bars for both images are 500 pm.

Extended Data Fig. 4 Asymmetry reversal of the Kekulé pattern.

3 × 3 nm2 STM images measured at B = 14 T, Vb = 2 mV and at the same position. The three images were measured successively (scanning time : 1 min). A jump occurs in b at the scan line indicated by the red arrows, leading to an inversion of the asymmetry of the Kekulé pattern. The slow scan axis direction is indicated by the blue arrows on the left of each image. Scale bars for the three images are 500 pm.

Extended Data Fig. 5 Change of the Kekulé asymmetry.

10 × 10 nm2 STM images measured at B = 14 T and Vb = 25 mV. In b, the asymmetry pattern changes at the scan line indicated by the red arrows. Scale bars for both images are 1 nm.

Extended Data Fig. 6 Effect of sublattice charge imbalance and a t2 asymmetry on the zeroth Landau level.

a shows that the effect of a finite charge imbalance Δn = nAnB is to gap the zeroth Landau level of graphene. b shows that a hoping asymmetry Δt2 = t2,At2,B also opens up a gap, that depends on momentum k as we move away from the K and K′ points. The parameters are chosen so that Eg is the same on both plots at the K and K′ points, according to Equation (1). Simulations were performed using the kwant software49 for a 41 × 41 hexagonal lattice with ϕ = 0.003 flux per plaquette, in units of the flux quantum. Energies are measured in units of the nearest-neighbor hopping t. For a, Δt2 = 0 and Δn = 0.045, while for b, Δt2 = 0.015 and Δn = 0.

Extended Data Fig. 7 Induced Δt2 asymmetry by interactions.

a shows that a sublattice charge imbalance Δn = nAnB ≠ 0 develops as V1 increases. b shows the concomitant emergence of a second nearest-neighbor bond asymmetry Δt2 = t2At2B ≠ 0, peaking at intermediate values of V1. The simulations are carried out for cylinder circumferences of Ly = 6, 8, 10 sites, all with bond-dimension χ = 1000, using the tenpy package50.

Extended Data Fig. 8 Disappearance of the charge-density wave at low magnetic field in sample AC23.

a, CDW at B = 9 T. b, CDW at B = 7 T. The Moiré superlattice is visible but does not perturb the CDW pattern. c, d, Honeycomb lattice with no CDW at B = 4 T. e, Honeycomb lattice at B = 4 T with residual traces of CDW, see the zoom in f of the white rectangle. Scale bars for all figures are 500 pm.

Extended Data Fig. 9 Charge-density-wave order in sample AC24.

a, Honeycomb lattice at B = 14 T and Vb = −350 mV observed at ν = 0. b, CDW under the same conditions but at Vb = −18 mV. Scale bars for both figures are 500 pm.

Extended Data Table 1 Geometrical parameters of the four measured samples

Supplementary information

Supplementary Information

This file contains Supplementary Sections 1–6, including Supplementary Figs. 1–6 and references.

Peer Review File

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Coissard, A., Wander, D., Vignaud, H. et al. Imaging tunable quantum Hall broken-symmetry orders in graphene. Nature 605, 51–56 (2022). https://doi.org/10.1038/s41586-022-04513-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41586-022-04513-7

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing