Abstract
Topological aspects of the electron wave function—including the Berry curvature and Chern number—play a crucial role in determining the physical properties of materials. Although the Berry curvature and its effects in materials have been studied1,2, detecting changes in the Chern number can be challenging, particularly changes in the valley Chern type. In this regard, twisted double bilayer graphene3,4,5,6,7 has emerged as a promising platform to gain electrical control over the Berry curvature hotspots8 and the valley Chern numbers of topological flat bands9,10. In addition, strain-induced breaking of the threefold rotation symmetry leads to a non-zero first moment of Berry curvature (called the Berry curvature dipole)11. Here we show that a sign change in the Berry curvature dipole detects topological transitions in the bands. In twisted double bilayer graphene, the perpendicular electric field simultaneously tunes the valley Chern number and Berry curvature dipole, providing a tunable system to probe the topological transitions. Furthermore, we find hysteresis in the transport response that is caused by switching of the electric polarization. This holds promise for next-generation Berry-curvature-based memory devices. Our technique can be emulated in three-dimensional topological systems to probe topological transitions governed by parameters such as pressure or anisotropic strain.
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Spontaneous time-reversal symmetry breaking in twisted double bilayer graphene
Nature Communications Open Access 29 October 2022
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Data availability
Source data are provided with this paper. The experimental data used in the figures of the main text are available at Zenodo47. Additional data related to this study are available from the corresponding authors upon reasonable request.
Code availability
The code that supports the findings of this study is available from the corresponding authors upon reasonable request.
References
Xiao, D., Chang, M.-C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010).
Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P. Anomalous Hall effect. Rev. Mod. Phys. 82, 1539–1592 (2010).
Burg, G. W. et al. Correlated insulating states in twisted double bilayer graphene. Phys. Rev. Lett. 123, 197702 (2019).
Shen, C. et al. Correlated states in twisted double bilayer graphene. Nat. Phys. 16, 520–525 (2020).
Adak, P. C. et al. Tunable bandwidths and gaps in twisted double bilayer graphene on the verge of correlations. Phys. Rev. B 101, 125428 (2020).
Cao, Y. et al. Tunable correlated states and spin-polarized phases in twisted bilayer–bilayer graphene. Nature 583, 215–220 (2020).
Liu, X. et al. Tunable spin-polarized correlated states in twisted double bilayer graphene. Nature 583, 221–225 (2020).
Sinha, S. et al. Bulk valley transport and Berry curvature spreading at the edge of flat bands. Nat. Commun. 11, 5548 (2020).
Koshino, M. Band structure and topological properties of twisted double bilayer graphene. Phys. Rev. B 99, 235406 (2019).
Zhang, Y.-H., Mao, D., Cao, Y., Jarillo-Herrero, P. & Senthil, T. Nearly flat Chern bands in moiré superlattices. Phys. Rev. B 99, 075127 (2019).
Fu, L. & Sodemann, I. Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials. Phys. Rev. Lett. 115, 216806 (2015).
Ma, Q., Grushin, A. G. & Burch, K. S. Topology and geometry under the nonlinear electromagnetic spotlight. Nat. Mater. 20, 1601–1614 (2021).
Du, Z. Z., Lu, H.-Z. & Xie, X. C. Nonlinear Hall effects. Nat. Rev. Phys. 3, 744–752 (2021).
Facio, J. I. et al. Strongly enhanced Berry dipole at topological phase transitions in BiTeI. Phys. Rev. Lett. 121, 246403 (2018).
Hu, J.-X., Zhang, C.-P., Xie, Y.-M. & Law, K. T. Nonlinear Hall effects in strained twisted bilayer WSe2. Preprint at https://arxiv.org/abs/2004.14140 (2020).
Song, J. C. W., Samutpraphoot, P. & Levitov, L. S. Topological Bloch bands in graphene superlattices. Proc. Natl Acad. Sci. USA 112, 10879–10883 (2015).
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).
Ma, Q. et al. Observation of the nonlinear Hall effect under time-reversal-symmetric conditions. Nature 565, 337–342 (2019).
Shvetsov, O. O., Esin, V. D., Timonina, A. V., Kolesnikov, N. N. & Deviatov, E. V. Nonlinear Hall effect in three-dimensional Weyl and Dirac semimetals. JETP Lett. 109, 715–721 (2019).
Xiao, J. et al. Berry curvature memory through electrically driven stacking transitions. Nat. Phys. 16, 1028–1034 (2020).
Kumar, D. et al. Room-temperature nonlinear Hall effect and wireless radiofrequency rectification in Weyl semimetal TaIrTe4. Nat. Nanotechnol. 16, 421–425 (2021).
Son, J., Kim, K.-H., Ahn, Y., Lee, H.-W. & Lee, J. Strain engineering of the Berry curvature dipole and valley magnetization in monolayer MoS2. Phys. Rev. Lett. 123, 036806 (2019).
Ho, S.-C. et al. Hall effects in artificially corrugated bilayer graphene without breaking time-reversal symmetry. Nat. Electron. 4, 116–125 (2021).
Huang, M. et al. Giant nonlinear Hall effect in twisted WSe2. Preprint at https://arxiv.org/abs/2006.05615 (2021).
Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).
Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).
Sharpe, A. L. et al. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene. Science 365, 605–608 (2019).
Serlin, M. et al. Intrinsic quantized anomalous Hall effect in a moiré heterostructure. Science 367, 900–903 (2020).
Das, I. et al. Symmetry-broken Chern insulators and Rashba-like Landau-level crossings in magic-angle bilayer graphene. Nat. Phys. 17, 710–714 (2021).
Zhang, C.-P. et al. Giant nonlinear Hall effect in strained twisted bilayer graphene. Preprint at https://arxiv.org/abs/2010.08333 (2020).
Pantaleón, P. A., Low, T. & Guinea, F. Tunable large Berry dipole in strained twisted bilayer graphene. Phys. Rev. B 103, 205403 (2021).
He, Z. & Weng, H. Giant nonlinear Hall effect in twisted bilayer WTe2. npj Quantum Mater. 6, 101 (2021).
He, M. et al. Symmetry breaking in twisted double bilayer graphene. Nat. Phys. 17, 26–30 (2021).
Chebrolu, N. R., Chittari, B. L. & Jung, J. Flat bands in twisted double bilayer graphene. Phys. Rev. B 99, 235417 (2019).
McGilly, L. J. et al. Visualization of moiré superlattices. Nat. Nanotechnol. 15, 580–584 (2020).
Kazmierczak, N. P. et al. Strain fields in twisted bilayer graphene. Nat. Mater. 20, 956–963 (2021).
He, P. et al. Quantum frequency doubling in the topological insulator Bi2Se3. Nat. Commun. 12, 698 (2021).
Xiao, C., Zhou, H. & Niu, Q. Scaling parameters in anomalous and nonlinear Hall effects depend on temperature. Phys. Rev. B 100, 161403 (2019).
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).
Mannaï, M. & Haddad, S. Twistronics versus straintronics in twisted bilayers of graphene and transition metal dichalcogenides. Phys. Rev. B 103, L201112 (2021).
Li, Y. et al. Unraveling strain gradient induced electromechanical coupling in twisted double bilayer graphene moiré superlattices. Adv. Mater. 33, 2105879 (2021).
Varma Sangani, L. D. et al. Facile deterministic cutting of 2D materials for twistronics using a tapered fibre scalpel. Nanotechnology 31, 32LT02 (2020).
Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proc. Natl Acad. Sci. USA 108, 12233–12237 (2011).
He, W.-Y., Goldhaber-Gordon, D. & Law, K. T. Giant orbital magnetoelectric effect and current-induced magnetization switching in twisted bilayer graphene. Nat. Commun. 11, 1650 (2020).
Yasuda, K., Wang, X., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Stacking-engineered ferroelectricity in bilayer boron nitride. Science 372, 1458–1462 (2021).
Zheng, Z. et al. Unconventional ferroelectricity in moiré heterostructures. Nature 588, 71–76 (2020).
Sinha, S. et al. Experimental data for Berry curvature dipole senses topological transition in a moiré superlattice. Zenodo https://doi.org/10.5281/zenodo.5211285 (2021).
Acknowledgements
We thank J. Song, A. Pasupathy, S. Todadri, B. Datta, S. Ghosh and S. Mandal for helpful discussions and comments. We thank S. Kanthi R. S., J. Sarkar, K. Maji and R. Dhingra for experimental assistance. M.M.D. acknowledges Nanomission grant SR/NM/NS-45/2016 and DST SUPRA SPR/2019/001247 grant along with the Department of Atomic Energy of Government of India 12-R&D-TFR-5.10-0100 for support. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan (grant no. JPMXP0112101001), and JSPS KAKENHI (grant nos. 19H05790 and JP20H00354). A.A. acknowledges IIT Kanpur (India), Science Engineering and Research Board (SERB) (India), and the Department of Science and Technology (DST) (India) for financial support. A.C. acknowledges the Institute Post-Doctoral fellowship of IIT Kanpur. K. Das acknowledges IIT Kanpur for the Senior Research Fellowship. A.A., A.C. and K. Das also thank CC-IIT Kanpur, for use of the high-performance computing facility. K. Debnath is grateful to the Jawaharlal Nehru Centre for Advanced Scientific Research, India, for a research fellowship. U.V.W. acknowledges support from a JC Bose National Fellowship of SERB-DST.
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S.S., P.C.A. and L.D.V.S. fabricated the devices. S.S. and P.C.A. performed the measurements and analysed the data. A.C., K. Das and A.A. calculated the band structure and performed the BCD and Chern number calculations. K. Debnath and U.V.W. performed the polarization calculations. K.W. and T.T. grew the hBN crystals. S.S., P.C.A., A.A. and M.M.D. wrote the manuscript with inputs from all the authors. M.M.D. supervised the project.
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Extended data
Extended Data Fig. 1 Variation of longitudinal resistance with temperature.
a-c, Variation of longitudinal resistance Rxx with temperature T for filling ν = 0.125 (a), ν = 0 (b) and ν = − 4 (c). The color of the line-plots indicates the corresponding displacement field.
Extended Data Fig. 2 Scaling of normalized nonlinear Hall voltage \({V}_{xy}^{2\omega }/{({V}_{xx}^{\omega })}^{2}\) with the square of longitudinal conductivity (\({\sigma }_{xx}^{2}\)) for different fillings ν with displacement field as parameter.
a, d, g, j, The variation of nonlinear Hall voltage \({V}_{xy}^{2\omega }\) (blue-colored data points corresponding to the left axis) and longitudinal voltage \({V}_{xx}^{\omega }\) (orange-colored data points corresponding to the right axis) as a function of the displacement field D/ϵ0 for four different fillings. b, e, h, k, The corresponding variation of normalized nonlinear Hall voltage \({V}_{xy}^{2\omega }/{({V}_{xx}^{\omega })}^{2}\) (black-colored data points corresponding to the left axis) and square of longitudinal conductivity \({\sigma }_{xx}^{2}\) (red-colored data points corresponding to the right axis) as a function of the displacement field D/ϵ0, extracted for the same fillings used in a, d, g, and j, respectively. c, f, i, l, The variation of normalized nonlinear Hall voltage \({V}_{xy}^{2\omega }/{({V}_{xx}^{\omega })}^{2}\) with square of longitudinal conductivity \({\sigma }_{xx}^{2}\) plotted parametrically as a function of the displacement field D/ϵ0, using b, e, h, and k, respectively. The displacement field value of data points in V nm−1 is indicated by the color (color bar is shown in top right). The dashed green line and dashed blue line indicate fits to linear scaling in regime-I and regime-II respectively, used to extract BCD. The fillings shown here are ν = 0.112 (a-c), 0.138 (d-f), 0.150 (g-i), and 0.175 (j-l). The light green background and light blue background correspond to regime-I and regime-II, respectively, as discussed in Fig. 3a of the main manuscript. The measurements were performed using a current of 100 nA with a frequency of 177 Hz at a temperature of 1.5 K.
Extended Data Fig. 3 Scaling of normalized nonlinear Hall voltage \({V}_{xy}^{2\omega }/{({V}_{xx}^{\omega })}^{2}\) with square of longitudinal conductivity (\({\sigma }_{xx}^{2}\)) with temperature as parameter.
a, The variation of nonlinear Hall voltage \({V}_{xy}^{2\omega }\) (blue-colored data points corresponding to the left axis) and longitudinal voltage \({V}_{xx}^{\omega }\) (orange-colored data points corresponding to the right axis) as a function of temperature T for ν = 0.125. b, The corresponding variation of normalized nonlinear Hall voltage \({V}_{xy}^{2\omega }/{({V}_{xx}^{\omega })}^{2}\) (black-colored data points corresponding to the left axis) and square of longitudinal conductivity \({\sigma }_{xx}^{2}\) (red-colored data points corresponding to the right axis) as a function of T, extracted for the same filling used in a. c, The variation of \({V}_{xy}^{2\omega }/{({V}_{xx}^{\omega })}^{2}\) with \({\sigma }_{xx}^{2}\) plotted parametrically as a function of T, using the results in b. The temperature value of data points in Kelvin is indicated by the color. The dashed line indicates a linear fit till 7 K.
Extended Data Fig. 4 Evolution of Berry curvature dipole (BCD).
a, Dependence of the y-component of BCD on the Energy (E) and inter-layer potential (Δ) at the conduction band side. b, c, Energy dispersion along high symmetry k-paths for Δ = 25 meV (b) and Δ = 30 meV (c) before transition. d, e, Similar energy dispersion for Δ = 38 meV (d) and Δ = 43 meV (e) after transition. The color map shows the Berry curvature value for the flat bands.
Extended Data Fig. 5 Metastable states and polarization in TDBG.
a, Electronic structure of Graphene-Graphene-hBN shows a band gap of 26 meV at K point. The inset shows the spatial distribution of the wave functions of bands labelled 2 and 3. b, Electric field (E) calculated from the slope of the average macroscopic potential of Graphene-Graphene-hBN in vacuum. c, Evolution and crossing of band 2 and 3 at E = − 0.0039 V/Å and E = 0.0039 V/Å of upper and lower trilayer of hBN-TDBG-hBN as a function of electric field using our rigid band model. d, metastable states for (i) E < − 0.0039 V/Åand (ii) E > 0.0039 V/Å, with nonzero polarization in hBN-TDBG-hBN that are accessible with the electric field.
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Supplementary Figs. 1–25 and Sections I–XIV.
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Sinha, S., Adak, P.C., Chakraborty, A. et al. Berry curvature dipole senses topological transition in a moiré superlattice. Nat. Phys. 18, 765–770 (2022). https://doi.org/10.1038/s41567-022-01606-y
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DOI: https://doi.org/10.1038/s41567-022-01606-y
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Spontaneous time-reversal symmetry breaking in twisted double bilayer graphene
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