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Nanoscale imaging of equilibrium quantum Hall edge currents and of the magnetic monopole response in graphene

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Although the recently predicted topological magnetoelectric effect1 and the response to an electric charge that mimics an induced mirror magnetic monopole2 are fundamental attributes of topological states of matter with broken time-reversal symmetry, so far they have not been directly observed in experiments. Using a SQUID-on-tip3, acting simultaneously as a tunable scanning electric charge and as an ultrasensitive nanoscale magnetometer, we induce and directly image the microscopic currents generating the magnetic monopole response in a graphene quantum Hall electron system. We find a rich and complex nonlinear behaviour, governed by the coexistence of topological and non-topological equilibrium currents, that is not captured by the monopole models2. Furthermore, by imaging the equilibrium currents of individual quantum Hall edge states, we reveal that the edge states, which are commonly assumed to carry only a chiral downstream current, in fact carry a pair of counterpropagating currents4, in which the topological downstream current in the incompressible region is counterbalanced by a non-topological upstream current flowing in the adjacent compressible region. The intricate patterns of the counterpropagating equilibrium-state orbital currents provide insights into the microscopic origins of the topological and non-topological charge and energy flow in quantum Hall systems.

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Fig. 1: Magnetic monopole response and the topological and non-topological magnetoelectric effects.
Fig. 2: Topological and non-topological equilibrium currents in graphene in the presence of charge disorder.
Fig. 3: Topological and non-topological QH edge state currents in a p–n junction.
Fig. 4: Mixed magnetoelectric effect.

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Data availability

The data represented in Figs. 2b–d,h–j, 3b–j and 4b–i are available with the online version of this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Change history

  • 24 January 2020

    In the version of this Letter originally published online, owing to a technical error, the ORCID number for the author Aviram Uri was missing; it should have been 0000-0002-5172-2535. This has now been corrected in all versions.


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We thank L.S. Levitov and A.V. Shytov for stimulating discussions and M.E. Huber for the SOT readout set-up. This work was supported by the European Research Council (ERC) under the EU Horizon 2020 programme grant no. 785971, by the Israel Science Foundation grant no. 921/18, by NSF/DMR-BSF Binational Science Foundation (BSF) grant no. 2015653, and by the Leona M. and Harry B. Helmsley Charitable Trust grant no. 2018PG-ISL006. J.H.S. is grateful for financial support from the Graphene Flagship. C.K.L. acknowledges support from the STC Center for Integrated Quantum Materials (CIQM) under NSF award 1231319. C.K.L. and E.Z. acknowledge the support of the MISTI (MIT International Science and Technology Initiatives) MIT–Israel Seed Fund. Y.K. thanks the Humboldt Foundation for support. The growth of hBN crystals was sponsored by the Elemental Strategy Initiative conducted by the 497 MEXT, Japan and the CREST (JPMJCR15F3), JST.

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Authors and Affiliations



A.U. and E.Z. designed the experiment. A.U. and S.G. performed the measurements and the data analysis. Y.K. and J.S. designed and fabricated the samples. K.B. fabricated the SOTs. C.K.L. performed the quantum mechanical calculations and contributed to the theoretical analysis. N.A. performed the Comsol simulations and contributed to the theoretical analysis. E.O.L. designed and built the scanning SOT microscope. Y.M. fabricated the tuning forks. T.T. and K.W. grew the hBN crystals. A.U., S.G. and E.Z. wrote the manuscript with input from J.S. and Y.K. All authors participated in discussions and in writing the manuscript.

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Correspondence to Aviram Uri or Eli Zeldov.

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Supplementary information

Supplementary Information

Supplementary Figs. 1–12 and discussions.

Supplementary Video 1

Rendering of the local y-component of the current density, J y, of the quantum Hall edge states along a p–n junction. As the movie evolves, a larger carrier density difference is imposed between the left and right sides of the junction as indicated by the filling factors ν L and ν R. At first, a single strip of topological current, denoted T, appears on each side of the junction, in the incompressible strips with integer filling ν = ±2. As the carrier density difference grows, a compressible strip carrying counterpropagating nontopological current, denoted NT, appears on each side of the junction, followed by additional topological and nontopological strips.

Supplementary Video 2

Rendering of the local y-component of the current density, J y, of the quantum Hall edge states along an artificial gate-induced edge. As the movie evolves, the carrier density on the right half is increased, while the left half is kept depleted. The movie reveals that each edge state is comprised of a countepropagating pair of topological and nontopological currents, denoted T and NT respectively.

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Uri, A., Kim, Y., Bagani, K. et al. Nanoscale imaging of equilibrium quantum Hall edge currents and of the magnetic monopole response in graphene. Nat. Phys. 16, 164–170 (2020).

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