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Imaging hydrodynamic electrons flowing without Landauer–Sharvin resistance

Abstract

Electrical resistance usually originates from lattice imperfections. However, even a perfect lattice has a fundamental resistance limit, given by the Landauer1 conductance caused by a finite number of propagating electron modes. This resistance, shown by Sharvin2 to appear at the contacts of electronic devices, sets the ultimate conduction limit of non-interacting electrons. Recent years have seen growing evidence of hydrodynamic electronic phenomena3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18, prompting recent theories19,20 to ask whether an electronic fluid can radically break the fundamental Landauer–Sharvin limit. Here, we use single-electron-transistor imaging of electronic flow in high-mobility graphene Corbino disk devices to answer this question. First, by imaging ballistic flows at liquid-helium temperatures, we observe a Landauer–Sharvin resistance that does not appear at the contacts but is instead distributed throughout the bulk. This underpins the phase-space origin of this resistance—as emerging from spatial gradients in the number of conduction modes. At elevated temperatures, by identifying and accounting for electron–phonon scattering, we show the details of the purely hydrodynamic flow. Strikingly, we find that electron hydrodynamics eliminates the bulk Landauer–Sharvin resistance. Finally, by imaging spiralling magneto-hydrodynamic Corbino flows, we show the key emergent length scale predicted by hydrodynamic theories—the Gurzhi length. These observations demonstrate that electronic fluids can dramatically transcend the fundamental limitations of ballistic electrons, with important implications for fundamental science and future technologies.

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Fig. 1: Landauer–Sharvin bulk geometrical resistance and the experimental setup for its measurement.
Fig. 2: Imaging the Landauer–Sharvin bulk resistance in a Corbino disk.
Fig. 3: Observation of the perfect elimination of Landauer–Sharvin bulk resistance by hydrodynamic electron flow.
Fig. 4: Imaging spiralling magneto-hydrodynamic electron flow and its Gurzhi boundary layer.

Data availability

 Source data are provided with this paper. Additional data that support the plots and other analysis in this work are available from the corresponding author upon request.

References

  1. Landauer, R. Spatial variation of currents and fields due to localized scatterers in metallic conduction. IBM J. Res. Dev. 1, 223–231 (1957).

    MathSciNet  Article  Google Scholar 

  2. Sharvin, Y. V. A. Possible method for studying Fermi surfaces. Sov. Phys. JETP 21, 655 (1965).

    ADS  Google Scholar 

  3. de Jong, M. J. M. & Molenkamp, L. W. Hydrodynamic electron flow in high-mobility wires. Phys. Rev. B 51, 389–402 (1995).

    Google Scholar 

  4. Bandurin, D. A. et al. Negative local resistance caused by viscous electron backflow in graphene. Science 351, 1055–1058 (2016).

    ADS  CAS  Article  Google Scholar 

  5. Crossno, J. et al. Observation of the Dirac fluid and the breakdown of the Wiedemann-Franz law in graphene. Science 351, 1058–1061 (2016).

    ADS  CAS  Article  Google Scholar 

  6. Moll, P. J. W., Kushwaha, P., Nandi, N., Schmidt, B. & Mackenzie, A. P. Evidence for hydrodynamic electron flow in PdCoO2. Science 351, 1061–1064 (2016).

    ADS  CAS  Article  Google Scholar 

  7. Krishna Kumar, R. et al. Superballistic flow of viscous electron fluid through graphene constrictions. Nat. Phys. 13, 1182–1185 (2017).

    CAS  Article  Google Scholar 

  8. Gooth, J. et al. Thermal and electrical signatures of a hydrodynamic electron fluid in tungsten diphosphide. Nat. Commun. 9, 4093 (2018).

    ADS  CAS  Article  Google Scholar 

  9. Braem, B. A. et al. Scanning gate microscopy in a viscous electron fluid. Phys. Rev. B 98, 241304 (2018).

    ADS  CAS  Article  Google Scholar 

  10. Berdyugin, A. I. et al. Measuring Hall viscosity of graphene’s electron fluid. Science 364, 162–165 (2019).

    ADS  CAS  Article  Google Scholar 

  11. Tan, C. et al. Realization of a universal hydrodynamic semiconductor in ultra-clean dual-gated bilayer graphene. Sci Adv. 8, eabi8481 (2022).

  12. Sulpizio, J. A. et al. Visualizing Poiseuille flow of hydrodynamic electrons. Nature 576, 75–79 (2019).

    ADS  CAS  Article  Google Scholar 

  13. Ku, M. J. H. et al. Imaging viscous flow of the Dirac fluid in graphene. Nature 583, 537–541 (2020).

    ADS  CAS  Article  Google Scholar 

  14. Jenkins, A. et al. Imaging the breakdown of ohmic transport in graphene. Preprint at https://arxiv.org/abs/2002.05065 (2020).

  15. Keser, A. C. et al. Geometric control of universal hydrodynamic flow in a two-dimensional electron fluid. Phys. Rev. X 11, 031030 (2021).

    CAS  Google Scholar 

  16. Gupta, A. et al. Hydrodynamic and ballistic transport over large length scales in GaAs/AlGaAs. Phys. Rev. Lett. 126, 076803 (2021).

    ADS  CAS  Article  Google Scholar 

  17. Krebs, Z. J. et al. Imaging the breaking of electrostatic dams in graphene for ballistic and viscous fluids. Preprint at https://arxiv.org/abs/2106.07212 (2021).

  18. Vool, U. et al. Imaging phonon-mediated hydrodynamic flow in WTe2. Nat. Phys. https://doi.org/10.1038/s41567-021-01341-w (2021).

  19. Shavit, M., Shytov, A. & Falkovich, G. Freely flowing currents and electric field expulsion in viscous electronics. Phys. Rev. Lett. 123, 026801 (2019).

    ADS  CAS  Article  Google Scholar 

  20. Stern, A. et al. Spread and erase – how electron hydrodynamics can eliminate the Landauer-Sharvin resistance. Preprint at https://arxiv.org/abs/2110.15369?context=cond-mat.str-el (2021).

  21. Gurzhi, R. N. Minimum of resistance in impurity free conductors. Sov. Phys. JETP 17, 521 (1963).

    Google Scholar 

  22. Nagaev, K. E. & Ayvazyan, O. S. Effects of electron-electron scattering in wide ballistic microcontacts. Phys. Rev. Lett. 101, 1–4 (2008).

    Article  Google Scholar 

  23. Nagaev, K. E. & Kostyuchenko, T. V. Electron-electron scattering and magnetoresistance of ballistic microcontacts. Phys. Rev. B 81, 1–9 (2010).

    Article  Google Scholar 

  24. Andreev, A. V., Kivelson, S. A. & Spivak, B. Hydrodynamic description of transport in strongly correlated electron systems. Phys. Rev. Lett. 106, 256804 (2011).

    ADS  CAS  Article  Google Scholar 

  25. Torre, I., Tomadin, A., Geim, A. K. & Polini, M. Nonlocal transport and the hydrodynamic shear viscosity in graphene. Phys. Rev. B 92, 165433 (2015).

    ADS  Article  Google Scholar 

  26. Levitov, L. & Falkovich, G. Electron viscosity, current vortices and negative nonlocal resistance in graphene. Nat. Phys. 12, 672–676 (2016).

    CAS  Article  Google Scholar 

  27. Scaffidi, T., Nandi, N., Schmidt, B., Mackenzie, A. P. & Moore, J. E. Hydrodynamic electron flow and Hall viscosity. Phys. Rev. Lett. 118, 226601 (2017).

    ADS  Article  Google Scholar 

  28. Guo, H., Ilseven, E., Falkovich, G. & Levitov, L. S. Higher-than-ballistic conduction of viscous electron flows. Proc. Natl Acad. Sci. USA 114, 3068–3073 (2017).

    ADS  CAS  Article  Google Scholar 

  29. Narozhny, B. N., Gornyi, I. V., Mirlin, A. D. & Schmalian, J. Hydrodynamic approach to electronic transport in graphene. Ann. Phys. 529, 1700043 (2017).

    Article  Google Scholar 

  30. Holder, T. et al. Ballistic and hydrodynamic magnetotransport in narrow channels. Phys. Rev. B 10, 245305 (2019).

    ADS  Article  Google Scholar 

  31. Levchenko, A. & Schmalian, J. Transport properties of strongly coupled electron–phonon liquids. Ann. Phys. 419, 168218 (2020).

    MathSciNet  CAS  Article  Google Scholar 

  32. Hong, Q., Davydova, M., Ledwith, P. J. & Levitov, L. Superscreening by a retroreflected hole backflow in tomographic electron fluids. Preprint at https://arxiv.org/abs/2012.03840 (2020).

  33. Honig, M. et al. Local electrostatic imaging of striped domain order in LaAlO3/SrTiO3. Nat. Mater. 12, 1112–1118 (2013).

    ADS  CAS  Article  Google Scholar 

  34. Ella, L. et al. Simultaneous voltage and current density imaging of flowing electrons in two dimensions. Nat. Nanotechnol. 14, 480–487 (2019).

    ADS  CAS  Article  Google Scholar 

  35. Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science 342, 614–617 (2013).

    ADS  CAS  Article  Google Scholar 

  36. Ben Shalom, M. et al. Quantum oscillations of the critical current and high-field superconducting proximity in ballistic graphene. Nat. Phys. 12, 318–322 (2016).

    Article  Google Scholar 

  37. Efetov, D. K. & Kim, P. Controlling electron-phonon interactions in graphene at ultrahigh carrier densities. Phys. Rev. Lett. 105, 256805 (2010).

    ADS  Article  Google Scholar 

  38. Waissman, J. et al. Realization of pristine and locally tunable one-dimensional electron systems in carbon nanotubes. Nat. Nanotechnol. 8, 569–574 (2013).

    ADS  CAS  Article  Google Scholar 

Download references

Acknowledgements

We thank L. Ella, G. Falkovich, L. Levitov, M. Polini, M. Shavit, A. Rozen, A. V. Shytov and U. Zondiner for useful discussions. Work was supported by the Leona M. and Harry B. Helmsley Charitable Trust grant, ISF grant (no. 1182/21), Minerva grant (no. 713237), Hydrotronics (no. 873028) and the ERC-Cog (See-1D-Qmatter, no. 647413). T.S. acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), in particular the Discovery Grant (no. RGPIN-2020-05842), the Accelerator Supplement (no. RGPAS-2020-00060) and the Discovery Launch Supplement (no. DGECR-2020-00222). During the preparation of this manuscript, we became aware of a partially related STM work17, which images voltage drops in flows across a constriction.

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Contributions

C.K., J.B., J.A.S, A.K.G. and S.I. designed the experiment. C.K., J.B. and J.A.S. performed the experiments. J.B. and D.P. fabricated the devices. C.K., J.B., J.A.S. and S.I. analysed the data. T.S., A.S. and S.I. wrote the theoretical model. K.W. and T.T. supplied the hBN crystals. C.K., J.B., A.S. and S.I. wrote the manuscript with input from other authors.

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Correspondence to S. Ilani.

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Nature thanks Yonglong Xie and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary Information

Supplementary Sections 1–18: Device fabrication; transport measurements; angular symmetry of the measured flow; measurement of the point-spread function (PSF) of the imaging experiments; determining the contact transparency from the measured resistance profile; determining the momentum-relaxing mean-free path over the full temperature range; additional data at a different carrier density; imaging measurements on a second Corbino device; comparing resistance profiles at different temperatures but with similar lMR; the Irrelevance of bulk magneto-resistance contributions; measurement of the Hall angle profile at T = 180 K; temperature dependence of lee; derivation of equation (1) in the main text; Boltzmann simulations of interacting flow in a Corbino geometry; temperature dependence of the outer contact resistance; the dependence of the number of conduction modes on radius in a Corbino device; the physical significance of the resistance function R(r) and accuracy in R(r) measurements from Nanotube SET sensitivity. Supplementary Figures 1–12 and additional references.

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Kumar, C., Birkbeck, J., Sulpizio, J.A. et al. Imaging hydrodynamic electrons flowing without Landauer–Sharvin resistance. Nature 609, 276–281 (2022). https://doi.org/10.1038/s41586-022-05002-7

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