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Graphene moiré superlattices with giant quantum nonlinearity of chiral Bloch electrons

Nature Nanotechnology volume 17, pages 378–383 (2022)Cite this article

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Abstract

Graphene-based samples have shown a plethora of exotic characteristics and these properties may help the realization of a new generation of fast electronic devices. However, graphene’s centrosymmetry prohibits second-order electronic transport. Here, we show giant second-order nonlinear transports in graphene moiré superlattices at zero magnetic field, both longitudinal and transverse to the applied current direction. High carrier mobility and inversion symmetry breaking by hexagonal boron nitride lead to nonlinear conductivities five orders of magnitude larger than those in WTe2. The nonlinear conductivity strongly depends on the gate voltage as well as on the stacking configuration, with a giant enhancement originating from the moiré bands. Longitudinal nonlinear conductivity cannot originate from Berry curvature dipoles. Our theoretical modelling highlights skew scattering of chiral Bloch electrons as the physical origin. With these results, we demonstrate nonlinear charge transport due to valley-contrasting chirality, which constitutes an alternative means to induce second-order transports in van der Waals heterostructures. Our approach is promising for applications in frequency-doubling and energy harvesting via rectification.

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Fig. 1: hBN-encapsulated graphene with broken inversion symmetry and nonlinear transport property.
Fig. 2: Nonlinear transports in hBN-encapsulated graphene (Device 1).
Fig. 3: Tunability of nonlinear transport with gating (Device 1).
Fig. 4: Temperature dependence of the nonlinear transport (Device 1).

Data availability

The data that support the findings of this study are available within the paper and the Supplementary Information. Source data are provided with this paper. Other relevant data are available from the corresponding authors on reasonable request.

Code availability

The codes that support this study are available from the corresponding authors on reasonable request.

References

  1. Novoselov, K. S. et al. Electric field effect in atomically thin carbon films. Science 306, 666–669 (2004).

    Article  CAS  Google Scholar 

  2. Novoselov, K. S. et al. Two-dimensional gas of massless Dirac fermions in graphene. Nature 438, 197–200 (2005).

    Article  CAS  Google Scholar 

  3. Zhang, Y., Tan, Y.-W., Stormer, H. L. & Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438, 201–204 (2005).

    Article  CAS  Google Scholar 

  4. Shen, Y.-R. The Principles of Nonlinear Optics (Wiley-Interscience, 1984).

  5. Yankowitz, M., Ma, Q., Jarillo-Herrero, P. & LeRoy, B. J. van der Waals heterostructures combining graphene and hexagonal boron nitride. Nat. Rev. Phys. 1, 112–125 (2019).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  8. Ma, Q. et al. Observation of the nonlinear Hall effect under time-reversal-symmetric conditions. Nature 565, 337–342 (2019).

    Article  CAS  Google Scholar 

  9. Kang, K., Li, T., Sohn, E., Shan, J. & Mak, K. F. Nonlinear anomalous Hall effect in few-layer WTe2. Nat. Mater. 18, 324–328 (2019).

    Article  CAS  Google Scholar 

  10. Sodemann, I. & Fu, L. Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials. Phys. Rev. Lett. 115, 216806 (2015).

    Article  CAS  Google Scholar 

  11. Du, Z. Z., Wang, C. M., Li, S., Lu, H.-Z. & Xie, X. C. Disorder-induced nonlinear Hall effect with time-reversal symmetry. Nat. Commun. 10, 3047 (2019).

    Article  CAS  Google Scholar 

  12. Isobe, H., Xu, S.-Y. & Fu, L. High-frequency rectification via chiral Bloch electrons. Sci. Adv. 6, eaay2497 (2020).

    Article  CAS  Google Scholar 

  13. He, P. et al. Quantum frequency doubling in the topological insulator Bi2Se3. Nat. Commun. 12, 698 (2021).

    Article  CAS  Google Scholar 

  14. Tokura, Y. & Nagaosa, N. Nonreciprocal responses from non-centrosymmetric quantum materials. Nat. Commun. 9, 3740 (2018).

    Article  CAS  Google Scholar 

  15. Belinicher, V. I. & Sturman, B. I. The photogalvanic effect in media lacking a center of symmetry. Sov. Phys. Uspekhi 23, 199–223 (1980).

    Article  Google Scholar 

  16. Gorbachev, R. V. et al. Detecting topological currents in graphene superlattices. Science 346, 448–451 (2014).

    Article  CAS  Google Scholar 

  17. Xiao, D., Yao, W. & Niu, Q. Valley-contrasting physics in graphene: magnetic moment and topological transport. Phys. Rev. Lett. 99, 236809 (2007).

    Article  CAS  Google Scholar 

  18. Song, J. C. W., Samutpraphoot, P. & Levitov, L. S. Topological Bloch bands in graphene superlattices. Proc. Natl Acad. Sci. USA 112, 10879–10883 (2015).

    Article  CAS  Google Scholar 

  19. Stepanov, E. A. et al. Direct observation of incommensurate–commensurate transition in graphene-hBN heterostructures via optical second harmonic generation. ACS Appl. Mater. Interfaces 12, 27758–27764 (2020).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  21. He, P. et al. Nonlinear planar Hall effect. Phys. Rev. Lett. 123, 016801 (2019).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  24. DaSilva, A. M., Jung, J., Adam, S. & MacDonald, A. H. Transport and particle-hole asymmetry in graphene on boron nitride. Phys. Rev. B 91, 245422 (2015).

    Article  CAS  Google Scholar 

  25. Wallbank, J. R. et al. Excess resistivity in graphene superlattices caused by umklapp electron–electron scattering. Nat. Phys. 15, 32–36 (2019).

    Article  CAS  Google Scholar 

  26. Wallbank, J. R., Patel, A. A., Mucha-Kruczyński, M., Geim, A. K. & Fal’ko, V. I. Generic miniband structure of graphene on a hexagonal substrate. Phys. Rev. B 87, 245408 (2013).

    Article  CAS  Google Scholar 

  27. Wu, S. et al. Multiple hot-carrier collection in photo-excited graphene Moiré superlattices. Sci. Adv. 2, e1600002 (2016).

    Article  CAS  Google Scholar 

  28. Moriya, R. et al. Emergence of orbital angular moment at van Hove singularity in graphene/h-BN moiré superlattice. Nat. Commun. 11, 5380 (2020).

    Article  CAS  Google Scholar 

  29. Woods, C. R. et al. Commensurate–incommensurate transition in graphene on hexagonal boron nitride. Nat. Phys. 10, 451–456 (2014).

    Article  CAS  Google Scholar 

  30. Yang, W. et al. Epitaxial growth of single-domain graphene on hexagonal boron nitride. Nat. Mater. 12, 792–797 (2013).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  32. Das Sarma, S., Adam, S., Hwang, E. H. & Rossi, E. Electronic transport in two-dimensional graphene. Rev. Mod. Phys. 83, 407–470 (2011).

    Article  CAS  Google Scholar 

  33. Duan, J. et al. High thermoelectricpower factor in graphene/hBN devices. Proc. Natl Acad. Sci. USA 113, 14272–14276 (2016).

    Article  CAS  Google Scholar 

  34. Eckmann, A. et al. Raman fingerprint of aligned graphene/h-BN superlattices. Nano Lett. 13, 5242–5246 (2013).

    Article  CAS  Google Scholar 

  35. Huang, M. et al. Giant nonlinear Hall effect in twisted WSe2. Preprint at arXiv:2006.05615 (2020).

  36. Ortix, C., Yang, L. & van den Brink, J. Graphene on incommensurate substrates: trigonal warping and emerging Dirac cone replicas with halved group velocity. Phys. Rev. B 86, 081405 (2012).

    Article  CAS  Google Scholar 

  37. Yankowitz, M. et al. Dynamic band-structure tuning of graphene moiré superlattices with pressure. Nature 557, 404–408 (2018).

    Article  CAS  Google Scholar 

  38. Ribeiro-Palau, R. et al. Twistable electronics with dynamically rotatable heterostructures. Science 361, 690–693 (2018).

    Article  CAS  Google Scholar 

  39. Battilomo, R., Scopigno, N. & Ortix, C. Berry curvature dipole in strained graphene: a Fermi surface warping effect. Phys. Rev. Lett. 123, 196403 (2019).

    Article  CAS  Google Scholar 

  40. & Kumar, D. et al. Room-temperature nonlinear Hall effect and wireless radiofrequency rectification in Weyl semimetal TaIrTe4. Nat. Nanotechnol. 16, 421–425 (2021).

    Article  CAS  Google Scholar 

  41. Finney, N. R. et al. Tunable crystal symmetry in graphene–boron nitride heterostructures with coexisting moiré superlattices. Nat. Nanotechnol. 14, 1029–1034 (2019).

    Article  CAS  Google Scholar 

  42. Zhang, C.-P. et al. Giant nonlinear Hall effect in strained twisted bilayer graphene. Preprint at arXiv:2010.08333 (2020).

  43. Stauber, T., Peres, N. M. R. & Guinea, F. Electronic transport in graphene: a semiclassical approach including midgap states. Phys. Rev. B 76, 205423 (2007).

    Article  CAS  Google Scholar 

  44. Vasko, F. T. & Ryzhii, V. Voltage and temperature dependencies of conductivity in gated graphene. Phys. Rev. B 76, 233404 (2007).

    Article  CAS  Google Scholar 

  45. Hwang, E. H. & Das Sarma, S. Acoustic phonon scattering limited carrier mobility in two-dimensional extrinsic graphene. Phys. Rev. B 77, 115449 (2008).

    Article  CAS  Google Scholar 

  46. Ando, T. Valley Hall conductivity in graphene: effects of higher-order scattering. J. Phys. Soc. Jpn 87, 044702 (2018).

    Article  Google Scholar 

  47. Novikov, D. S. Elastic scattering theory and transport in graphene. Phys. Rev. B 76, 245435 (2007).

    Article  CAS  Google Scholar 

Download references

Acknowledgements

We acknowledge K. Novoselov, A. Rodin and A. Carvalho for discussions. This work was partially supported by the National Research Foundation (NRF) Singapore Investigatorship (grant no. NRFI06-2020-0015). P.H. was sponsored by the National Key Research and Development Program of China (grant no. 2020YFA0308800), the Natural Science Foundation of Shanghai (grant no. 21ZR1404300) and start-up funding from Fudan University. The work at Massachusetts Institute of Technology was supported by the U.S. Army Research Laboratory and the U.S. Army Research Office through the Institute for Soldier Nanotechnologies, under Collaborative Agreement Number W911NF-18-2-0048.

Author information

Author notes

  1. These authors contributed equally: Pan He, Gavin Kok Wai Koon, Hiroki Isobe.

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, National University of Singapore, Singapore, Singapore

    Pan He & Hyunsoo Yang

  2. State Key Laboratory of Surface Physics and Institute for Nanoelectronic Devices and Quantum Computing, Fudan University, Shanghai, China

    Pan He

  3. Shanghai Qi Zhi Institute, Shanghai, China

    Pan He

  4. Centre for Advanced 2D Materials, National University of Singapore, Singapore, Singapore

    Gavin Kok Wai Koon, Jun You Tan, Junxiong Hu & Antonio H. Castro Neto

  5. Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Hiroki Isobe & Liang Fu

  6. Department of Applied Physics, The University of Tokyo, Tokyo, Japan

    Hiroki Isobe

  7. Department of Physics, National University of Singapore, Singapore, Singapore

    Junxiong Hu

  8. Department of Materials Science and Engineering, National University of Singapore, Singapore, Singapore

    Antonio H. Castro Neto

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  1. Pan He
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Contributions

P.H., G.K.W.K. and H.Y. designed the experimental study. P.H. performed the measurements and analysed the data. G.K.W.K., J.Y.T., A.H.C.N. and J.H. fabricated devices. H.I. and L.F. performed theoretical studies. P.H., G.K.W.K., H.I. and H.Y. wrote the manuscript. All authors commented on the manuscript. H.Y. led the project.

Corresponding authors

Correspondence to Pan He, Liang Fu or Hyunsoo Yang.

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Nature Nanotechnology thanks Yuval Ronen and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 The second-order nonlinear transports in graphene honeycomb lattice with inversion symmetry breaking.

The second-order electric responses are schematically depicted for an ac electric field Eω (black) applied along a zigzag (a) and an armchair direction (b). The second-order current with double frequency J2ω (red) exists along the armchair direction for both cases. The three-fold rotational symmetry is schematically shown here with a rotation axis perpendicular to the plane and centered at a carbon atom, where the three dashed lines (equivalent crystalline axes) cross.

Extended Data Fig. 2 Tunability of nonlinear transport with gating in another superlattice device (Device 2).

a, b, The longitudinal resistivity ρxx (a) and Hall resistivity ρxy (b) as a function of back gate voltage Vg. c, d, The second-harmonic longitudinal \(V_x^{2\omega }\) (c) and transverse \(V_y^{2\omega }\) (d) voltage as a function of Vg at I = 3 µA. The data in (a), (c) and (d) are measured under zero magnetic field. The data in (b) are obtained at H = 0.5 T. e, f, The second-order nonlinear conductivities along the longitudinal \(\sigma _{{{{xxx}}}}^{\left( 2 \right)}\)(e) and transverse \(\sigma _{{{{yxx}}}}^{\left( 2 \right)}\)(f) directions as a function of Vg. All data are measured at T = 1.7 K. The misalignment angle between graphene and hBN in this device is similar to the one in the main text, as estimated from the Vg where SDP occurs.

Source data

Extended Data Fig. 3 Temperature dependence of the nonlinear transport in superlattice Device 2.

a, ρxx versus Vg measured at different temperatures under zero magnetic field. b, ρxx as a function of temperature for three gate voltages Vg, as marked by arrows in (a). c, The mobility as a function of temperature for three gate voltages. The data are obtained by measuring ρxx (Vg) and ρxy (Vg) simultaneously at H = 0.5 T. d, e, The extracted \(\sigma _{{{{xxx}}}}^{\left( 2 \right)}\)(Vg) (d) and \(\sigma _{{{{yxx}}}}^{\left( 2 \right)}\)(Vg) (e) at different temperatures. At Vg = −62.5 V, \(\sigma _{{{{yxx}}}}^{\left( 2 \right)}\)(Vg) shows a peak in (e). The data in (a), (b), (d) and (e) are obtained under zero magnetic field and I = 3 µA. (f) \(\sigma _{{{{yxx}}}}^{\left( 2 \right)}\) and μ3 as a function of temperature at Vg = −62.5 V.

Source data

Extended Data Fig. 4 The Fermi level dependence of linear conductivity.

The linear conductivity as a function of gate voltage Vg for Device 1 (a) and Device 2 (b). The data were measured at T = 1.7 K and H = 0 T. The arrows indicate the position of primary and secondary Dirac points.

Source data

Extended Data Fig. 5 The cubic scaling of nonlinear conductivity with mobility.

The nonlinear conductivity σyxx versus the cube of mobility for Device 1 (a) and Device 2 (b). The red lines are linear fittings to the experimental data. The error bars represent the standard deviation of the nonlinear conductivity.

Source data

Supplementary information

Supplementary Information

Supplementary Figs. 1–11 and Table 1.

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Source Data Table 1

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He, P., Koon, G.K.W., Isobe, H. et al. Graphene moiré superlattices with giant quantum nonlinearity of chiral Bloch electrons. Nat. Nanotechnol. 17, 378–383 (2022). https://doi.org/10.1038/s41565-021-01060-6

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