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# Imaging viscous flow of the Dirac fluid in graphene

## Abstract

The electron–hole plasma in charge-neutral graphene is predicted to realize a quantum critical system in which electrical transport features a universal hydrodynamic description, even at room temperature1,2. This quantum critical ‘Dirac fluid’ is expected to have a shear viscosity close to a minimum bound3,4, with an interparticle scattering rate saturating1 at the Planckian time, the shortest possible timescale for particles to relax. Although electrical transport measurements at finite carrier density are consistent with hydrodynamic electron flow in graphene5,6,7,8, a clear demonstration of viscous flow at the charge-neutrality point remains elusive. Here we directly image viscous Dirac fluid flow in graphene at room temperature by measuring the associated stray magnetic field. Nanoscale magnetic imaging is performed using quantum spin magnetometers realized with nitrogen vacancy centres in diamond. Scanning single-spin and wide-field magnetometry reveal a parabolic Poiseuille profile for electron flow in a high-mobility graphene channel near the charge-neutrality point, establishing the viscous transport of the Dirac fluid. This measurement is in contrast to the conventional uniform flow profile imaged in a metallic conductor and also in a low-mobility graphene channel. Via combined imaging and transport measurements, we obtain viscosity and scattering rates, and observe that these quantities are comparable to the universal values expected at quantum criticality. This finding establishes a nearly ideal electron fluid in charge-neutral, high-mobility graphene at room temperature4. Our results will enable the study of hydrodynamic transport in quantum critical fluids relevant to strongly correlated electrons in high-temperature superconductors9. This work also highlights the capability of quantum spin magnetometers to probe correlated electronic phenomena at the nanoscale.

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## Data availability

The data that support the plots and other analysis in this work are available upon request.

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## Acknowledgements

We thank B. Narozhny for helpful discussions, M. J. Turner for annealing diamond samples and S. Y. F. Zhao for assisting with the magneto-transport measurement. This material is based on work supported by, or in part by, the United States Army Research Laboratory and the United States Army Research Office under contract/grant number W911NF1510548 and number W911NF1110400, as well as the Quantum Technology Center (QTC) at the University of Maryland. A.T.P. and Y.X. were primarily supported by the US Department of Energy, Basic Energy Sciences Office, Division of Materials Sciences and Engineering under award DE-SC0001819. T.X.Z., A.H. and U.V. were partly supported by ARO grant number W911NF-17-1-0023 and the Gordon and Betty Moore Foundations EPiQS Initiative through grant number GBMF4531. Fabrication of samples was supported by the US Department of Energy, Basic Energy Sciences Office, Division of Materials Sciences and Engineering under award DE-SC0019300. A.Y. also acknowledges support from ARO grants W911NF-18-1-0316 and W911NF-1-81-0206 and the STC Center for Integrated Quantum Materials, NSF grant number DMR-1231319. Part of this work was supported under the NSF grant number EFMA 1542807; and the Elemental Strategy Initiative conducted by MEXT, Japan, and JSPS KAKENHI grant JP15K21722 (K.W. and T.T.). Finally, this work was partly carried out at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. F.C. acknowledges partial support from the Swiss National Science Foundation grant number P300P2-158417. P.K. acknowledges support from ARO (W911NF-17-1-0574). M.M.F. acknowledges support from the Office of Naval Research grant N00014-18-1-2722. A.T.P. acknowledges support from the Department of Defense through the National Defense Science and Engineering Graduate Fellowship (NDSEG) Program. Y.X. acknowledges partial support from the Harvard Quantum Initiative in Science and Engineering. This research used resources of the Center for Functional Nanomaterials, which is a US DOE Office of Science Facility, at Brookhaven National Laboratory under contract number DE-SC0012704. This work was performed, in part, at the Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Infrastructure Network, which is supported by the NSF under award no. ECS-0335765. CNS is part of Harvard University.

## Author information

Authors

### Contributions

M.J.H.K., A.Y. and R.L.W. conceived the project. M.J.H.K., T.X.Z., A.Y. and R.L.W. designed the experiments. M.J.H.K. and T.X.Z. performed scanning NV measurements and room-temperature electrical transport measurements and analysed the data. T.X.Z. fabricated the scanning probes and built the scanning probe setup. M.J.H.K. and Q.L. built the NV wide-field imaging setup, performed the measurement, and analysed the data. M.J.H.K., T.X.Z., L.E.A., A.H. and U.V. performed the magneto-transport experiment and analysed the data. M.J.H.K., Y.J.S., J.K.S., C.B., A.T.P., Y.X. and H.Z. fabricated the devices. M.M.F. provided the theory on the current profile of non-interacting electrons. K.W. and T.T. grew the hBN single crystals. M.J.H.K., T.X.Z., Q.L., F.C., P.K., A.Y. and R.L.W. contributed to the discussion. M.J.H.K., T.X.Z., A.Y. and R.L.W. wrote the manuscript with input from all authors. P.K., A.Y. and R.L.W. supervised the project.

### Corresponding authors

Correspondence to Amir Yacoby or Ronald L. Walsworth.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Low-mobility graphene devices.

a, b, Optical image of the two bare monolayer graphene devices (a and b), with experimental results reported in panels c and d. The width of each device is W = 1 μm. c, Conductivity σ as a function of the gate voltage Vg of the device shown in a. Data are shown in red. The blue line is a linear fit, which allows for extraction of the mobility. d, Current profiles of two bare monolayer graphene devices. Grey is for the device shown in a. Pink is for the device shown in b. The data are shown in comparison to the parabolic current profile measured for the encapsulated graphene device shown in Fig. 2e. Error bars correspond to the relative deviation of Jy that generates 2χ2, where χ2 is the cost function.

### Extended Data Fig. 2 Scanning NV measurement of current profiles in a W = 1.5 μm channel.

a, Measurement at the CNP. b, Data from a compared with data from Fig. 2e. In this panel, the horizontal axis is x in micrometres, where for all the other panels the horizontal axis is x/W. c, d, Measurement in the estimated carrier density range n = 0.08 × 1012–0.30 × 1012 cm−2 (c) and n = 0.30 × 1012–0.53 × 1012 cm−2 (d). In a, c and d, the black dashed line is the ideal uniform profile, the blue dashed curve is the ideal Poiseuille profile and the green curve is a fit to the data with equation (1).

## Supplementary information

### Supplementary Information

This file contains Supplementary Figures 1–21.

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Ku, M.J.H., Zhou, T.X., Li, Q. et al. Imaging viscous flow of the Dirac fluid in graphene. Nature 583, 537–541 (2020). https://doi.org/10.1038/s41586-020-2507-2

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