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# Graphene moiré superlattices with giant quantum nonlinearity of chiral Bloch electrons

## 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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• ### Strong room-temperature bulk nonlinear Hall effect in a spin-valley locked Dirac material

Nature Communications Open Access 23 January 2023

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## 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.

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## 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

Authors

### 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.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

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 J (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.

### 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.

### 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.

### 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.

## Supplementary information

### Supplementary Information

Supplementary Figs. 1–11 and Table 1.

## Source data

### Source Data Fig. 2

Statistical Source Data

### Source Data Fig. 3

Statistical Source Data

### Source Data Fig. 4

Statistical Source Data

### Source Data Table 1

Statistical Source Data

### Source Data Extended Data Fig. 2

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### Source Data Extended Data Fig. 3

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### Source Data Extended Data Fig. 4

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### Source Data Extended Data Fig. 5

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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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• DOI: https://doi.org/10.1038/s41565-021-01060-6

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