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# Berry curvature dipole senses topological transition in a moiré superlattice

## 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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## 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.

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## 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.

## Author information

Authors

### Contributions

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.

### Corresponding authors

Correspondence to Subhajit Sinha, Amit Agarwal or Mandar M. Deshmukh.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature Physics thanks Eduardo Castro 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 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.

## Supplementary information

### Supplementary Information

Supplementary Figs. 1–25 and Sections I–XIV.

## Source data

### Source Data Fig. 1

Source data for Fig. 1b–e,g,h.

### Source Data Fig. 2

Source data for Fig. 2.

### Source Data Fig. 3

Source data for Fig. 3a–h.

### Source Data Fig. 4

Source data for Fig. 4.

### Source Data Extended Data Fig. 1

Source data for Extended Data Fig. 1.

### Source Data Extended Data Fig. 2

Source data for Extended Data Fig. 2.

### Source Data Extended Data Fig. 3

Source data for Extended Data Fig. 3.

### Source Data Extended Data Fig. 4

Source data for Extended Data Fig. 4b–e.

### Source Data Extended Data Fig. 5

Source data for Extended Data Fig. 5.

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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. (2022). https://doi.org/10.1038/s41567-022-01606-y

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• DOI: https://doi.org/10.1038/s41567-022-01606-y