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# Anisotropic band flattening in graphene with one-dimensional superlattices

## Abstract

Patterning graphene with a spatially periodic potential provides a powerful means to modify its electronic properties1,2,3. In particular, in twisted bilayers, coupling to the resulting moiré superlattice yields an isolated flat band that hosts correlated many-body phases4,5. However, both the symmetry and strength of the effective moiré potential are constrained by the constituent crystals, limiting its tunability. Here, we have exploited the technique of dielectric patterning6 to subject graphene to a one-dimensional electrostatic superlattice (SL)1. We observed the emergence of multiple Dirac cones and found evidence that with increasing SL potential the main and satellite Dirac cones are sequentially flattened in the direction parallel to the SL basis vector, behaviour resulting from the interaction between the one-dimensional SL electric potential and the massless Dirac fermions hosted by graphene. Our results demonstrate the ability to induce tunable anisotropy in high-mobility two-dimensional materials, a long-desired property for novel electronic and optical applications7,8. Moreover, these findings offer a new approach to engineering flat energy bands where electron interactions can lead to emergent properties9.

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## Data availability

Presented measurement data are available from the corresponding author upon reasonable request.

## Code availability

The computer code for calculating band structures and simulating Rxx and Ryy values is available from the corresponding author upon reasonable request.

## References

1. Park, C. H., Yang, L., Son, Y. W., Cohen, M. L. & Louie, S. G. Anisotropic behaviours of massless Dirac fermions in graphene under periodic potentials. Nat. Phys. 4, 213–217 (2008).

2. Park, C. H., Son, Y. W., Yang, L., Cohen, M. L. & Louie, S. G. Landau levels and quantum Hall effect in graphene superlattices. Phys. Rev. Lett. 103, 046808 (2009).

3. Brey, L. & Fertig, H. A. Emerging zero modes for graphene in a periodic potential. Phys. Rev. Lett. 103, 046809 (2009).

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

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

6. Forsythe, C. et al. Band structure engineering of 2D materials using patterned dielectric superlattices. Nat. Nanotechnol. 13, 566–571 (2018).

7. Xia, F., Wang, H., Hwang, J. C. M., Neto, A. H. C. & Yang, L. Black phosphorus and its isoelectronic materials. Nat. Rev. Phys. 1, 306–317 (2019).

8. Tian, H. et al. Low-symmetry two-dimensional materials for electronic and photonic applications. Nano Today 11, 763–777 (2016).

9. Shi, L. K., Ma, J. & Song, J. C. W. Gate-tunable flat bands in van der Waals patterned dielectric superlattices. 2D Mater. 7, 015028 (2019).

10. Geim, A. K. & Novoselov, K. S. The rise of graphene. Nat. Mater. 6, 183–191 (2007).

11. Novoselov, K. S., Mishchenko, A., Carvalho, A. & Neto, A. H. C. 2D materials and van der Waals heterostructures. Science 353, aac9439 (2016).

12. Yankowitz, M. et al. Emergence of superlattice Dirac points in graphene on hexagonal boron nitride. Nat. Phys. 8, 382–386 (2012).

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

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

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

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

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

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

19. Barbier, M., Vasilopoulos, P. & Peeters, F. M. Extra Dirac points in the energy spectrum for superlattices on single-layer graphene. Phys. Rev. B 81, 075438 (2010).

20. Dubey, S. et al. Tunable superlattice in graphene to control the number of Dirac points. Nano Lett. 13, 3990–3995 (2013).

21. Drienovsky, M. et al. Towards superlattices: lateral bipolar multibarriers in graphene. Phys. Rev. B 89, 115421 (2014).

22. Drienovsky, M. et al. Few-layer graphene patterned bottom gates for van der Waals heterostructures. Preprint at https://arxiv.org/abs/1703.05631 (2017).

23. Drienovsky, M. et al. Commensurability oscillations in one-dimensional graphene superlattices. Phys. Rev. Lett. 121, 026806 (2018).

24. Kuiri, M., Gupta, G. K., Ronen, Y., Das, T. & Das, A. Large Landau-level splitting in a tunable one-dimensional graphene superlattice probed by magnetocapacitance measurements. Phys. Rev. B 98, 035418 (2018).

25. Dean, C. R. et al. Boron nitride substrates for high-quality graphene electronics. Nat. Nanotechnol. 5, 722–726 (2010).

26. Allen, P. B.in Quantum Theory of Real Materials (eds Chelikowsky, J. R. & Louie, S. G.) viii, 549 (Kluwer Academic Publishers, 1996).

27. Madsen, G. K. H. & Singh, D. J. BoltzTraP. A code for calculating band-structure dependent quantities. Comput. Phys. Commun. 175, 67–71 (2006).

28. Shore, K. A. Introduction to graphene-based nanomaterials: from electronic structure to quantum transport. Contemp. Phys. 55, 344–345 (2014).

29. Weiss, D., Vonklitzing, K., Ploog, K. & Weimann, G. Magnetoresistance oscillations in a two-dimensional electron gas induced by a submicrometer periodic potential. Europhys. Lett. 8, 179–184 (1989).

30. Gerhardts, R. R., Weiss, D. & Vonklitzing, K. Novel magnetoresistance oscillations in a periodically modulated two-dimensional electron gas. Phys. Rev. Lett. 62, 1173–1176 (1989).

31. Beenakker, C. W. J. Guiding-center-drift resonance in a periodically modulated two-dimensional electron gas. Phys. Rev. Lett. 62, 2020–2023 (1989).

32. Endo, A. & Iye, Y. Measurement of anisotropic transport using unidirectional lateral superlattice with square geometry. J. Phys. Soc. Jpn. 71, 2067–2068 (2002).

33. Qiao, J. S., Kong, X. H., Hu, Z. X., Yang, F. & Ji, W. High-mobility transport anisotropy and linear dichroism in few-layer black phosphorus. Nat. Commun. 5, 4475 (2014).

34. Wu, S., Killi, M. & Paramekanti, A. Graphene under spatially varying external potentials: Landau levels, magnetotransport, and topological modes. Phys. Rev. B 85, 195404 (2012).

35. Xu, H. et al. Oscillating edge states in one-dimensional MoS2 nanowires. Nat. Commun. 7, 12904 (2016).

## Acknowledgements

This research was supported primarily by the Office of Naval Research (ONR) Young Investors Program (no. N00014-17-1-2832). P.M. was supported by the Science and Technology Commission of Shanghai Municipality (grant no. 19ZR1436400) and NYU-ECNU Institute of Physics at NYU Shanghai. This research was carried out using the High Performance Computing resources at NYU Shanghai.

## Author information

Authors

### Contributions

Y.L., P.M. and C.R.D. conceived the experiment. Y.L. fabricated the samples, performed transport measurements and analysed transport data. P.M. provided theoretical modelling of the system and helped interpret transport data. S.D. and C.F. performed preliminary studies on a van der Pauw sample. Y.L., P.M. and C.R.D. co-wrote the paper. K.W. and T.T. provided hBN material for device fabrication.

### Corresponding author

Correspondence to Cory R. Dean.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature Nanotechnology thanks the anonymous reviewers for their contribution to the peer review of this work.

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## Supplementary information

### Supplementary Information

Supplementary Figs. S1–S7 and Discussions S1–S8.

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Li, Y., Dietrich, S., Forsythe, C. et al. Anisotropic band flattening in graphene with one-dimensional superlattices. Nat. Nanotechnol. 16, 525–530 (2021). https://doi.org/10.1038/s41565-021-00849-9

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• DOI: https://doi.org/10.1038/s41565-021-00849-9

• ### Band conductivity oscillations in a gate-tunable graphene superlattice

• Robin Huber
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• ### An image interaction approach to quantum-phase engineering of two-dimensional materials

• Valerio Di Giulio
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• F. Javier García de Abajo

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