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Third-order nonlinear Hall effect in a quantum Hall system

Nature Nanotechnology (2024)Cite this article

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Abstract

In two-dimensional systems, perpendicular magnetic fields can induce a bulk band gap and chiral edge states, which gives rise to the quantum Hall effect. The quantum Hall effect is characterized by zero longitudinal resistance (Rxx) and Hall resistance (Rxy) plateaus quantized to h/(υe2) in the linear response regime, where υ is the Landau level filling factor, e is the elementary charge and h is Planck’s constant. Here we explore the nonlinear response of monolayer graphene when tuned to a quantum Hall state. We observe a third-order Hall effect that exhibits a nonzero voltage plateau scaling cubically with the probe current. By contrast, the third-order longitudinal voltage remains zero. The magnitude of the third-order response is insensitive to variations in magnetic field (down to ~5 T) and in temperature (up to ~60 K). Moreover, the third-order response emerges in graphene devices with a variety of geometries, different substrates and stacking configurations. We term the effect third-order nonlinear response of the quantum Hall state and propose that electron–electron interaction between the quantum Hall edge states is the origin of the nonlinear response of the quantum Hall state.

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Fig. 1: Schematics of the linear Hall effect and nonlinear Hall effect in the classic and quantum regimes.
Fig. 2: Observation of the third-order nonlinear Hall plateaus within the QHSs υ = ±2.
Fig. 3: Cubic current dependence of the third-order Hall effect within QHSs.
Fig. 4: The effects of magnetic field and temperature on the third-order nonlinear response of QHS.

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

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

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Acknowledgements

We acknowledge Y. Zhang, Q. Niu and A. H. MacDonald for helpful discussions. P.H. was sponsored by the National Key Research and Development Program of China (grant numbers 2020YFA0308800 and 2022YFA1403300), National Natural Science Foundation of China (grant numbers 12174063 and U23A2071), Natural Science Foundation of Shanghai (grant numbers 21ZR1404300 and 23ZR1403600) and the start-up funding from Fudan University. J.H. is supported by the NRF Singapore under its NRF-ISF joint programme (grant number NRF2020-NRF-ISF004-3518) and the A*STAR under its MTC Young Investigator Research Grant (YIRG) (grant number M23M7c0124). J.S. is sponsored by the National Natural Science Foundation of China (grant number 11427902). N.N. is supported by JSPS KAKENHI grant numbers JP24H00197 and JP24H02231. N.N. was supported by the RIKEN TRIP initiative.

Author information

Author notes

  1. Hiroki Isobe

    Present address: Department of Physics, Kyushu University, Fukuoka, Japan

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

Authors and Affiliations

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

    Pan He, Jingru Li & Jian Shen

  2. Shanghai Branch, Hefei National Laboratory, Shanghai, China

    Pan He & Jian Shen

  3. RIKEN Center for Emergent Matter Science (CEMS), Wako, Japan

    Hiroki Isobe & Naoto Nagaosa

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

    Gavin Kok Wai Koon, Jun You Tan & Junxiong Hu

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

    Junxiong Hu

  6. Fundamental Quantum Science Program, TRIP Headquarters, RIKEN, Wako, Japan

    Naoto Nagaosa

  7. Department of Physics, Fudan University, Shanghai, China

    Jian Shen

  8. Shanghai Research Center for Quantum Sciences, Shanghai, China

    Jian Shen

  9. Zhangjiang Fudan International Innovation Center, Fudan University, Shanghai, China

    Jian Shen

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Contributions

P.H. and J.S. planned the study. P.H. performed the transport measurements and the data analysis. G.K.W.K., J.Y.T., J.H. and J.L. fabricated devices. H.I. and N.N. performed theoretical studies. P.H., H.I., N.N. and J.S. wrote the paper. All authors commented on the paper.

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Correspondence to Pan He, Hiroki Isobe, Junxiong Hu, Naoto Nagaosa or Jian Shen.

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Extended data

Extended Data Fig. 1 Observation of the third-order nonlinear response of QHS in graphene on hBN (Device 2).

a, Schematic structure of the graphene on hBN. b, Optical image of a Hall bar device of graphene on hBN (Device 2), with a channel width W = 1 µm and a source to drain length L = 13 µm (distance between closest arms is 2 µm), respectively. The bar indicates 10 μm. c, e, The first harmonic voltage along the longitudinal \({V}_{x}^{1\omega }\) (c) and transverse directions \({V}_{y}^{1\omega }\) (e) at around the QHSs with υ = ±2. d, f, The third harmonic voltage along the longitudinal \({V}_{x}^{3\omega }\) (d) and transverse directions \({V}_{y}^{3\omega }\) (f) at around the QHSs with υ = ±2. The results in (c-f) were measured under H = 9 T, T = 2 K and I = 1.5 μA. The plateau is highlighted by the orange box.

Source data

Extended Data Fig. 2 Cubic current dependce of the third-order nonlinear response of QHS in Device 2.

a, b, The \({V}_{y}^{3\omega }\) (a) and \({V}_{x}^{3\omega }\) (b) as a function of Vg at around the QHSs (υ = ±2) under different a.c. currents. c, The \({V}_{y}^{3\omega }\) plateau magnitude extracted from (a) as a function of I. The solid curve shows a cubic fitting to the \({V}_{y}^{3\omega }\) (I) curve. d, The plateau of \({V}_{x}^{3\omega }\) extracted from (b) as a function of I. The dashed line indicates the vanishing \({V}_{x}^{3\omega }\). The error bars represent the s.d. Data in (c, d) are presented as mean values ± s.d. The results in (a-d) were measured under H = 9 T and T = 1.7 K in Device 2.

Source data

Extended Data Fig. 3 Observation of the third-order nonlinear response of QHS in a hBN-encapsulated graphene device (Device 3).

a, The longitudinal resistance Rxx (left axis) and the Hall resistance Rxy (right axis) as a function of Vg at T = 1.7 K and H = 9 T. The characteristic plateaus of QHE at filling factor υ = ±2 are well defined with Rxy = ±h/2e2 and Rxx = 0. The device image is shown in the inset. The current channel width W = 1 μm and the source to drain length L = 6 μm. The bar indicates 5 μm. b, c, The \({V}_{x}^{3\omega }\) (b) and \({V}_{y}^{3\omega }\) (c) as a function of Vg at around the QHSs (υ = +2) under different a.c. currents. d, The \({V}_{y}^{3\omega }\) (\({V}_{x}^{3\omega }\)) plateau magnitude extracted from (b, c) as a function of I. The error bars represent the s.d. Data are presented as mean values ± s.d. The solid curve shows a cubic fitting to the \({V}_{y}^{3\omega }\) (I). The dashed line indicates the vanishing \({V}_{x}^{3\omega }\). e, f, The \({V}_{y}^{3\omega }\) as a function of Vg at around the QHS (υ = +2) under different magnetic fields (e) and temperatures (f). The plateaus are marked by a dashed line in (e, f), indicating the same height under different magnetic fields and temperatures.

Extended Data Fig. 4 Observation of the third-order nonlinear response of QHS in a circular graphene device (Device 4).

a, Optical image of the circular graphene device encapsulated by hBN with a diameter D = 6 μm. b, The longitudinal resistance Rxx (left axis) and the Hall resistance Rxy (right axis) as a function of Vg. c, d, The \({V}_{x}^{3\omega }\) (c) and \({V}_{y}^{3\omega }\) (d) as a function of Vg under different a.c. currents at the QHS υ = +2. A finite plateau of \({V}_{y}^{3\omega }\) was found in the region 2 V < Vg < 6 V, where the corresponding \({V}_{x}^{3\omega }\) is zero. e, The magnitude of third-order response plateau as a function of I. The solid curve shows a cubic fitting to \({V}_{y}^{3\omega }(I)\). The dashed line indicates the vanishing \({V}_{x}^{3\omega }\) at different currents. The results in (b–d) were measured at T = 1.7 K and H = 12 T.

Source data

Extended Data Fig. 5 Observation of the third-order nonlinear response of QHS with υ = ±2 and ± 6 in a graphene device SiO2/Si (Device 6).

a, Optical image of the graphene device supported by SiO2/Si substrate with a width of 4 μm. b, The longitudinal resistance Rxx (left axis) and the Hall resistance Rxy (right axis) as a function of Vg. c, d, The \({V}_{y}^{3\omega }\) (c) and \({V}_{x}^{3\omega }\) (d) as a function of Vg under different a.c. currents at the QHS υ = ±2. A finite plateau of \({V}_{y}^{3\omega }\) was found, where the corresponding \({V}_{x}^{3\omega }\) is zero. e, The magnitudes of \({V}_{y}^{3\omega }\) plateau as a function of I. The solid curve shows a cubic fitting to \({V}_{y}^{3\omega }(I)\). f, The vanishing \({V}_{x}^{3\omega }\) at different currents. The error bars represent the s.d. Data in (e, f) are presented as mean values ± s.d. g, h, The Hall resistance Rxy (left axis) and the \({V}_{y}^{3\omega }\) (right axis) as a function of Vg covering υ = ±2 and ±6 at H = 9 T (g) and H = -9T (h).

Source data

Extended Data Fig. 6 Observation of the third-order nonlinear response of QHS with υ = ±2 and ± 6 in hBN-encapsulated graphene devices (Devices 7 and 9).

a, b, The \({V}_{y}^{1\omega }\) (a) and \({V}_{y}^{3\omega }\) (b) as a function of Vg covering the QHSs with υ = ±2 and ±6 at H = ± 9 T, T = 1.7 K and I = 5 μA in Device 9. The ratio \({V}_{y}^{3\omega }\)(υ = +2)/\({V}_{y}^{3\omega }\)(υ = +6) = -25.5 μV/-2.6 μV = 9.8 at H = +9 T and \({V}_{y}^{3\omega }\)(υ = −2)/\({V}_{y}^{3\omega }\)(υ = -6) = 25.3 μV/2.6 μV = 9.7 at H = -9 T, showing a good agreement with the theory of 1/υ2 dependence. c, d, The \({V}_{y}^{1\omega }\) (c) and \({V}_{y}^{3\omega }\) (d) as a function of Vg covering the QHSs with υ = ±2 and ±6 at H = ± 9 T, T = 1.7 K and I = 5 μA in Device 7. The ratio \({V}_{y}^{3\omega }\)(υ = ±2)/\({V}_{y}^{3\omega }\)(υ = ±6) close to 9 is also observed in this device.

Source data

Extended Data Table 1 Summary of the results in different devices
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Supplementary information

Supplementary Information

Supplementary Figs. 1–18, Notes 1–3 and references.

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He, P., Isobe, H., Koon, G.K.W. et al. Third-order nonlinear Hall effect in a quantum Hall system. Nat. Nanotechnol. (2024). https://doi.org/10.1038/s41565-024-01730-1

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