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Hall effects in artificially corrugated bilayer graphene without breaking time-reversal symmetry


Strain can be used to modify the band structure—and thus the electronic properties—of two-dimensional materials. However, research has focused on the use of monolayer graphene with a limited lowering of spatial symmetry and considered only the real-space pseudo-magnetic field. Here we show that lithographically patterned strain can be used to create a non-trivial band structure and exotic phase of matter in bilayer graphene. The approach creates artificially corrugated bilayer graphene in which real-space and momentum-space pseudo-magnetic fields (Berry curvatures) coexist and have non-trivial properties, such as Berry curvature dipoles. This leads to the appearance of two Hall effects without breaking time-reversal symmetry: a nonlinear anomalous Hall effect that originates from the Berry curvature dipole, previously only observed in the Weyl semimetal WTe2, and a linear Hall effect that originates from a warped band dispersion on top of Rashba-like valley–orbit coupling and is similar to the recently proposed Magnus Hall effect.

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Fig. 1: Artificially corrugated BLG and its electronic properties.
Fig. 2: Nonlinear AHE and Berry curvature dipole.
Fig. 3: Pseudo-PHE and pseudo-magnetoresistance anisotropy.
Fig. 4: Electronic band structure of corrugated BLG.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author on reasonable request.

Code availability

The code that supports the theoretical plots within this paper is available from the corresponding author upon reasonable request.


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We thank T.-R. Chang, Y.-C. Chen, C.-J. Chung, S.-Z. Ho, Y.-D. Liou, L. W. Smith and J. I.-J. Wang for helpful discussions and/or technical assistance. This work is supported by the Ministry of Science and Technology in Taiwan (grant numbers MOST-108-2638-M-006-002-MY2, MOST-105-2628-M-006-003-MY3 and MOST-107-2112-M-006-025-MY3), and the Higher Education Sprout Project, Ministry of Education to the Headquarters of University Advancement at the National Cheng Kung University (NCKU). C.-H.C. acknowledges support from a Yushan Young Scholar Program, under the Ministry of Education (MOE), Taiwan. C.O. acknowledges support from a VIDI grant (project 680-47-543) financed by the Netherlands Organization for Scientific Research (NWO).

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Authors and Affiliations



S.-C.H. performed the measurements and analysed the data with help from S.-T.L. and Y.-C.H. C.-H.C. developed the theory and performed the calculations with assistance from B.H., C.O. and T.-M.C. S.-C.H., Y.-C.H. and T.-H.-Y.V. developed and performed the sample fabrication. S.-C.H. and T.-M.C. designed the experiment. S.-C.H., C.-H.C. and T.-M.C. wrote the manuscript with input from all authors. T.-M.C. supervised the project.

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Correspondence to Tse-Ming Chen.

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Extended data

Extended Data Fig. 1 Berry curvature dipole of corrugated BLG.

Calculated Berry curvature of the tilted mini-Dirac cone shown in Fig. 1c in the main article. Large scale inhomogeneity (for more than three orders of magnitude) of the Berry curvature distribution around the warped Fermi circle is clearly observed in such a BLG corrugation. The Fermi energy is set at 13.25 meV.

Extended Data Fig. 2 Fabrication processes of corrugated BLG.

a-e, Schematic illustrations showing a series of our fabrication process steps. The colour representation: the SiO2 substrate is highlighted in grey, hBN in purple, BLG in green, and the Cr/Au contacts in yellow. Fabrication details for each step are described in Materials and Methods.

Extended Data Fig. 3 90-degree Hall angle and reproducibility of nonlinear AHE.

The second-harmonic longitudinal (\({E}_{xx}^{2\omega }\) and \({E}_{yy}^{2\omega }\), both in grey) and transverse (\({E}_{yx}^{2\omega }\) and \({E}_{xy}^{2\omega }\), both in red) responses measured as a function of gate voltage Vg when the current is applied (a) perpendicular to (that is along the x direction) and (b) parallel to (that is along the y direction) the corrugation direction at T = 5 K. The absence of longitudinal nonlinear response regardless of the driving current direction (that is, a 90-degree Hall angle) is the hallmark of the Berry curvature dipole induced nonlinear Hall effect. (c) The current-voltage characteristic of \({E}_{yx}^{2\omega }\) and \({E}_{xx}^{2\omega }\) for Vg − VNP = 20.5 V (black arrow in (a)). The second-harmonic Hall voltage are well fitted by a quadratic function shown in orange curve. The quadratic I-V characteristics for the nonlinear AHE are consistently observed at various Vg.

Extended Data Fig. 4 Comparison of nonlinear AHE responses in corrugated BLG, simple BLG, and corrugated MLG.

The second harmonic Hall voltage response \({V}_{y}^{2\omega }\) as a function of the back-gate voltage is measured using standard lock-in technique in the corrugated BLG, simple BLG and corrugated MLG for comparison. The signal in the corrugated BLG is well reproducible with the phases of the second-harmonic responses being locked at ± 90 with respect to the driving reference, which is in agreement with the expectations for the nonlinear measurements. In contrast, the small, fluctuant \({V}_{y}^{2\omega }\) in simple BLG and corrugated MLG are noise signal because they are irreproducible and the phase cannot be locked, showing the absence of nonlinear AHE in these two systems.

Extended Data Fig. 5 Temperature dependence of the linear longitudinal conductivity and the nonlinear Hall effect.

a, The linear longitudinal resistivity as a function of gate voltage. b, Temperature dependence of the linear longitudinal conductivity versus the nonlinear Hall voltage for comparison. The nonlinear Hall effect decreases substantially with increasing temperature and disappear at only 10K, whereas the linear longitudinal conductivity σxx only changes a little bit (for less than 2%) in this temperature interval. The fact that they are distinctly different suggests that the nonlinear Hall effect is irrelevant to scattering.

Extended Data Fig. 6 Comparison of linear transverse responses in corrugated BLG, simple BLG, and corrugated MLG.

a, The linear transverse resistivity ρyx as a function of the back-gate voltage is measured in the corrugated BLG, simple BLG and corrugated MLG. In comparison with the ρyx in corrugated BLG, the signal is negligibly small for simple BLG and corrugated MLG except near the charge neutrality point. The transverse response near the NP may be due to the charge inhomogeneity because of the electron-hole puddles or the imperfection of the Hall bar that were commonly observed in graphene near the charge neutrality point55,56, and is not the major focus of this work. In corrugated BLG, uniquely, can be seen over a very wide range of Fermi energy and well explained by the model of pseudo-PHE. b-d, Colourmaps of the transverse resistivity ρyx as a function of Vg and B in the corrugated BLG (b), simple BLG (c) and corrugated MLG (d). In corrugated BLG, interplay between the zero-magnetic-field pseudo-PHE and the classical Hall effect can be clearly seen, exhibiting an anti-crossing structure in the colourmap. In contrast, for the simple BLG and corrugated MLG, a standard colour plot of the Hall effect in graphene with four quadrants crossing at zero magnetic field and the neutrality point is shown.

Extended Data Fig. 7 Reproducibility of pseudo-PHE.

a, The linear Hall resistivity ρyx measured as a function of gate voltage Vg in another corrugated BLG device. b, The linear Hall resistivity that is theoretically calculated as a function of Fermi level position ϵF in the corrugated BLG with ϕ = 10, where ϕ is the angle between the graphene zigzag axis and the corrugation direction.

Extended Data Fig. 8 Time reversal invariant pseudo-PHE.

a, The planar Hall resistivities ρyx (see top insets in b for measurement setup) and ρxy (see c, top insets) measured as a function of Vg at zero external magnetic field B = 0. Although the current directions and measurement setups for ρyx and ρxy are very different with respect to the corrugation and device geometry, the fact the ρxy = ρyx indicates that the resistivity is a symmetric tensor and the observed pseudo-PHE is indeed time reversal symmetric. b, c, Colour rendition of ρyx(B, Vg) (b) and ρxy(B, Vg) (c). The top insets show representative current trajectories for each region labelled in the colour plot. Interplay between the pseudo-PHE and the HE results in an asymmetric overall Hall resistivity, for example, the Hall resistivity ρyx(B = + 0.9T) (region labelled ii in b) is completely different to ρyx(B = − 0.9T) (region labelled iii in b). However, ρyx(B = + 0.9T) (region labelled ii in b) is nearly the same as ρxy(B = − 0.9T) (region labelled iii in c). d, ρyx(B = + 0.9T) and ρxy(B = − 0.9T) are explicitly plotted as a function of Vg to demonstrate the equivalence.

Extended Data Fig. 9 Electronic band structure of corrugated MLG.

a, Calculated band dispersion of the corrugated monolayer graphene for K (orange) and K\(^{\prime}\) (green) valley along ky direction at kx = 0. b, The band dispersion along kx direction at ky = 5 (corresponding to dashed line in a) shows a conventional symmetric Brillouin zone.

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Ho, SC., Chang, CH., Hsieh, YC. et al. Hall effects in artificially corrugated bilayer graphene without breaking time-reversal symmetry. Nat Electron 4, 116–125 (2021).

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