Abstract
In twodimensional 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 (R_{xx}) and Hall resistance (R_{xy}) plateaus quantized to h/(υe^{2}) 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 thirdorder Hall effect that exhibits a nonzero voltage plateau scaling cubically with the probe current. By contrast, the thirdorder longitudinal voltage remains zero. The magnitude of the thirdorder response is insensitive to variations in magnetic field (down to ~5 T) and in temperature (up to ~60 K). Moreover, the thirdorder response emerges in graphene devices with a variety of geometries, different substrates and stacking configurations. We term the effect thirdorder 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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Data availability
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.
Code availability
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 startup funding from Fudan University. J.H. is supported by the NRF Singapore under its NRFISF joint programme (grant number NRF2020NRFISF0043518) 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.
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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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Nature Nanotechnology thanks SuYang Xu 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 Observation of the thirdorder 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 (cf) were measured under H = 9 T, T = 2 K and I = 1.5 μA. The plateau is highlighted by the orange box.
Extended Data Fig. 2 Cubic current dependce of the thirdorder 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 V_{g} 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 (ad) were measured under H = 9 T and T = 1.7 K in Device 2.
Extended Data Fig. 3 Observation of the thirdorder nonlinear response of QHS in a hBNencapsulated graphene device (Device 3).
a, The longitudinal resistance R_{xx} (left axis) and the Hall resistance R_{xy} (right axis) as a function of V_{g} at T = 1.7 K and H = 9 T. The characteristic plateaus of QHE at filling factor υ = ±2 are well defined with R_{xy} = ±h/2e^{2} and R_{xx} = 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 V_{g} 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 V_{g} 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 thirdorder 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 R_{xx} (left axis) and the Hall resistance R_{xy} (right axis) as a function of V_{g}. c, d, The \({V}_{x}^{3\omega }\) (c) and \({V}_{y}^{3\omega }\) (d) as a function of V_{g} 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 < V_{g} < 6 V, where the corresponding \({V}_{x}^{3\omega }\) is zero. e, The magnitude of thirdorder 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.
Extended Data Fig. 5 Observation of the thirdorder nonlinear response of QHS with υ = ±2 and ± 6 in a graphene device SiO_{2}/Si (Device 6).
a, Optical image of the graphene device supported by SiO_{2}/Si substrate with a width of 4 μm. b, The longitudinal resistance R_{xx} (left axis) and the Hall resistance R_{xy} (right axis) as a function of V_{g}. c, d, The \({V}_{y}^{3\omega }\) (c) and \({V}_{x}^{3\omega }\) (d) as a function of V_{g} 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 R_{xy} (left axis) and the \({V}_{y}^{3\omega }\) (right axis) as a function of V_{g} covering υ = ±2 and ±6 at H = 9 T (g) and H = 9T (h).
Extended Data Fig. 6 Observation of the thirdorder nonlinear response of QHS with υ = ±2 and ± 6 in hBNencapsulated graphene devices (Devices 7 and 9).
a, b, The \({V}_{y}^{1\omega }\) (a) and \({V}_{y}^{3\omega }\) (b) as a function of V_{g} 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/υ^{2} dependence. c, d, The \({V}_{y}^{1\omega }\) (c) and \({V}_{y}^{3\omega }\) (d) as a function of V_{g} 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.
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He, P., Isobe, H., Koon, G.K.W. et al. Thirdorder nonlinear Hall effect in a quantum Hall system. Nat. Nanotechnol. (2024). https://doi.org/10.1038/s41565024017301
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DOI: https://doi.org/10.1038/s41565024017301
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