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# Visualizing Poiseuille flow of hydrodynamic electrons

## Abstract

Hydrodynamics, which generally describes the flow of a fluid, is expected to hold even for fundamental particles such as electrons when inter-particle interactions dominate1. Although various aspects of electron hydrodynamics have been revealed in recent experiments2,3,4,5,6,7,8,9,10,11, the fundamental spatial structure of hydrodynamic electrons—the Poiseuille flow profile—has remained elusive. Here we provide direct imaging of the Poiseuille flow of an electronic fluid, as well as a visualization of its evolution from ballistic flow. Using a scanning carbon nanotube single-electron transistor12, we image the Hall voltage of electronic flow through channels of high-mobility graphene. We find that the profile of the Hall field across the channel is a key physical quantity for distinguishing ballistic from hydrodynamic flow. We image the transition from flat, ballistic field profiles at low temperatures into parabolic field profiles at elevated temperatures, which is the hallmark of Poiseuille flow. The curvature of the imaged profiles is qualitatively reproduced by Boltzmann calculations, which allow us to create a ‘phase diagram’ that characterizes the electron flow regimes. Our results provide direct confirmation of Poiseuille flow in the solid state, and enable exploration of the rich physics of interacting electrons in real space.

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## Data availability

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

## Code availability

Computer code for reproducing the Boltzmann simulations and computing the electron-electron scattering length is available upon request.

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## Acknowledgements

We thank G. Falkovich, A. Shytov, L. Levitov, D. Bandurin and R. Krishna-Kumar for discussions. We further acknowledge support from the Helmsley Charitable Trust grant, the ISF (grant number 712539), WIS-UK collaboration grant, the ERC-Cog (See-1D-Qmatter number 647413), the Deloro Award, the Minerva Foundation, and the Emergent Phenomena in Quantum Systems initiative of the Gordon and Betty Moore Foundation.

## Author information

Authors

### Contributions

J.A.S., L.E. and S.I. conceived the experiments. J.A.S., L.E., A.R., D.D. and S.I. performed the experiments. J.A.S., L.E., A.R. and S.I. analysed the data. J.B., D.J.P. and M.B.-S. fabricated the graphene devices. K.W. and T.T. supplied the hBN crystals. T.S., T.H., R.Q., A.P., A.R. and A.S. performed theoretical calculations. J.A.S., L.E. and S.I. wrote the manuscript, with input from all authors.

### Corresponding author

Correspondence to Shahal Ilani.

## 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.

Peer review information Nature thanks Klaus Ensslin, Boris Narozhny and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

## Extended data figures and tables

### Extended Data Fig. 1

Relation between transport and momentum-relaxing mean free path across the phase diagram of flow regimes and the electron mobility. a, Boltzmann calculation of ltr versus lMR and lee. The two-dimensional map shows the ratio of the finite-field-transport mean free path, ltr(B) = h/[2e2 (π|n|)1/2ρxx(B)], and the bulk mean free path, ltr/lMR, calculated using Boltzmann theory at W/Rc = 3.2 for a channel with diffusive walls, as a function of lee/W and lMR/W. b, Electron mobility μ measured with scanning SET, equivalent to the lMR data presented in Fig. 1d.

### Extended Data Fig. 2 Diffusivity of etched channel walls in the experiment.

a, Illustration of a channel used to assess the diffusivity of etched walls by direct comparison to lithographically roughened walls. The walls of the left half of the channel are patterned with a typical straight-line pattern, whereas the right half is patterned with a saw-toothed pattern to introduce roughness. The region enclosed by the dashed box in the upper right is an AFM image of the etched walls. The red dashed line marks the spatial region spanning both wall patterns along which the potential drop is measured. b, Measured potential drop along the centre of the channel (red dashed line in a). Away from the voltage steps at the contacts, the potential drops linearly across the device, with no observable change in slope when the walls transition from straight line to saw-tooth etches. c, Zoom of magnetoresistance data from Fig. 1c plotted as ltr(B) for varying density. At the transition where the double peak disappears, ltr(B = 0) ≈ W/(1 – p), allowing estimation of specularity p. d,Theoretical scaling of ltr(B = 0) with n for varying p superimposed with the experimental data (bold black line), indicating that p is nearly zero (fully diffusive walls).

### Extended Data Fig. 3 Dependence of Hall field profile curvature on magnetic field.

a, Measured traces of κ, extracted from Ey using a fit to the centre of the channel, as a function of magnetic field plotted in units of W/Rc B. The blue trace is measured at T = 4 K and hole density of n = −6.06 × 1011 cm−2 on device B (see Methods and Extended Data Fig. 5). The orange trace is measured at T = 50 K and n = −1.02 × 1012 cm−2 on device B, and the yellow trace is measured at T = 150 K and a hole density of n = −3.15 × 1011 cm−2 on device A, which is the device used throughout the main text. Two distinct regimes are apparent: Below W/Rc ≈ 2, the curvature is nearly independent of W/Rc, whereas above it varies noticeably, acquiring large values. b, Curvature as a function of W/Rc extracted from a Boltzmann simulation of Ey as described in the main text. Coloured curves correspond to values of lMR and lee that best match experiment. This figure verifies that by imaging at W/Rc = 1.3 as in the main text, the profiles are not influenced by the magnetic field.

### Extended Data Fig. 4 Dependence of Hall field profile curvature on voltage excitation.

κ, the normalized curvature of Ey, is plotted as a function of the excitation amplitude Vex applied between the contacts of the channel. Error bars correspond to standard deviation of κ from the least-squares fit of a parabola to the data. The blue trace shows T = 7.5 K and n = −1.5 × 1011 cm−2; the purple trace shows T = 75 K and n = −3.3 × 1011 cm−2. This plot verifies that by choosing appropriate values for the excitation, as was done for the experiments in the main text, electron heating effects are negligible.

### Extended Data Fig. 5 Comparison of Hall field profile curvature for different devices.

a, Top, optical image of graphene device (device A) patterned into the geometry of a channel, with W = 4.7 μm and L = 15 μm, studied in the main text. Bottom, normalized curvature of Ey, κ, measured as a function of W/Rc. b, Top, optical image of an additional graphene device (device B) used for similar measurements, with W = 5 μm and L = 42 μm. This device was measured in a separate cryostat with a different scanning microscope and different SET. Colour differences between optical images are due to lighting conditions. Bottom, κ versus W/Rc measured for device B, showing a result highly consistent with that in a.

### Extended Data Fig. 6 Distinguishing electron flow regime from transport.

The graph shows the dependence of Leff/W, which is inversely proportional to the resistivity, on lee/W, for fixed values of lb/W = lMR/W. The purple- and red-coloured regions correspond to the parameter ranges of our experiment for T = 75 K and T = 150 K, respectively. It is evident from these curves that the dependence of the resistivity on lee/W is fairly weak when lMR/W is not much larger than 1. Figure reprinted with permission from de Jong and Molenkamp2; copyright 2019 by the American Physical Society.

### Extended Data Fig. 7 Phase diagram demonstrating that the experiment falls within the Fermi liquid regime.

The black line shows the equality EF = kBT, which separates the temperature–density plane into two distinct regimes. Above this line is the Dirac fluid regime in which electrons and holes are both present and thus electron–hole scattering must be considered. Below this line is the degenerate Fermi liquid regime in which only one charge carrier is present. The blue, purple and red lines correspond to the experiments presented in Fig. 4 at T = 7.5 K, T = 75 K and T = 150 K, respectively, and show that our experiments are categorically within the Fermi liquid regime.

### Extended Data Fig. 8 Curvature of ballistic jx and comparison of theoretical Ey and jx across phase diagram.

a, Curvature of ballistic current profile versus lMR/W. The analytic solution (blue curve) is based on de Jong and Molenkamp2 and the red curve is a Monte Carlo billiard ball simulation result. The two methods agree perfectly until lMR exceeds the channel length used in the billiard ball simulation, beyond which the solutions begin to deviate. b, Curvature κ of Ey, as in Fig. 4b, calculated by Boltzmann simulation (see Methods), as a function of lee/W and lMR/W for W/Rc = 1.3. Curvature is calculated over the centre of the channel. Green lines divide the panel into flow regimes as in Fig. 4b. c, Curvature κ of jx, extracted from the same simulation as a. For jx, the curvature in the ballistic regime is essentially constant at κ ≈ 0.31 and so the curvature of jx is less discriminating between the hydrodynamic and ballistic regimes than the curvature of Ey, which becomes negative. In the other regimes, the curvatures of jx and Ey are very similar, and the differences between them diminish as each of the length scales becomes much smaller than W. In the hydrodynamic regime the curvature saturates on the maximal possible value for a strictly parabolic profile, and in the porous regime it follows the length scale $${D}_{\nu }=\frac{1}{2}\sqrt{{l}_{{\rm{MR}}}{l}_{{\rm{ee}}}}$$ as expected.

### Extended Data Fig. 9 Comparisons of representative imaged Ey profiles to the Boltzmann simulated profiles.

Boltzmann simulation profiles are plotted in green, whereas the experimental data is plotted with the same colour scheme as in main text based on temperature (blue for T = 7.5 K, purple for T = 75 K and red for T = 150 K). The field Ey is normalized as in the main text by the classical Hall field Ecl = (B/ne)(I/W).

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Sulpizio, J.A., Ella, L., Rozen, A. et al. Visualizing Poiseuille flow of hydrodynamic electrons. Nature 576, 75–79 (2019). https://doi.org/10.1038/s41586-019-1788-9

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