Abstract
Hydrodynamics, which generally describes the flow of a fluid, is expected to hold even for fundamental particles such as electrons when interparticle interactions dominate^{1}. Although various aspects of electron hydrodynamics have been revealed in recent experiments^{2,3,4,5,6,7,8,9,10,11}, 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 singleelectron transistor^{12}, we image the Hall voltage of electronic flow through channels of highmobility 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 electronelectron scattering length is available upon request.
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Acknowledgements
We thank G. Falkovich, A. Shytov, L. Levitov, D. Bandurin and R. KrishnaKumar for discussions. We further acknowledge support from the Helmsley Charitable Trust grant, the ISF (grant number 712539), WISUK collaboration grant, the ERCCog (See1DQmatter number 647413), the Deloro Award, the Minerva Foundation, and the Emergent Phenomena in Quantum Systems initiative of the Gordon and Betty Moore Foundation.
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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.
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Correspondence to Shahal Ilani.
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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 momentumrelaxing mean free path across the phase diagram of flow regimes and the electron mobility. a, Boltzmann calculation of l_{tr} versus l_{MR} and l_{ee}. The twodimensional map shows the ratio of the finitefieldtransport mean free path, l_{tr}(B) = h/[2e^{2} (πn)^{1/2}ρ_{xx}(B)], and the bulk mean free path, l_{tr}/l_{MR}, calculated using Boltzmann theory at W/R_{c} = 3.2 for a channel with diffusive walls, as a function of l_{ee}/W and l_{MR}/W. b, Electron mobility μ measured with scanning SET, equivalent to the l_{MR} 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 straightline pattern, whereas the right half is patterned with a sawtoothed 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 sawtooth etches. c, Zoom of magnetoresistance data from Fig. 1c plotted as l_{tr}(B) for varying density. At the transition where the double peak disappears, l_{tr}(B = 0) ≈ W/(1 – p), allowing estimation of specularity p. d,Theoretical scaling of l_{tr}(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 E_{y} using a fit to the centre of the channel, as a function of magnetic field plotted in units of W/R_{c} ∝ B. The blue trace is measured at T = 4 K and hole density of n = −6.06 × 10^{11} 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 × 10^{12} cm^{−2} on device B, and the yellow trace is measured at T = 150 K and a hole density of n = −3.15 × 10^{11} cm^{−2} on device A, which is the device used throughout the main text. Two distinct regimes are apparent: Below W/R_{c} ≈ 2, the curvature is nearly independent of W/R_{c}, whereas above it varies noticeably, acquiring large values. b, Curvature as a function of W/R_{c} extracted from a Boltzmann simulation of E_{y} as described in the main text. Coloured curves correspond to values of l_{MR} and l_{ee} that best match experiment. This figure verifies that by imaging at W/R_{c} = 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 E_{y}, is plotted as a function of the excitation amplitude V_{ex} applied between the contacts of the channel. Error bars correspond to standard deviation of κ from the leastsquares fit of a parabola to the data. The blue trace shows T = 7.5 K and n = −1.5 × 10^{11} cm^{−2}; the purple trace shows T = 75 K and n = −3.3 × 10^{11} 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 E_{y}, κ, measured as a function of W/R_{c}. 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/R_{c} 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 L_{eff}/W, which is inversely proportional to the resistivity, on l_{ee}/W, for fixed values of l_{b}/W = l_{MR}/W. The purple and redcoloured 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 l_{ee}/W is fairly weak when l_{MR}/W is not much larger than 1. Figure reprinted with permission from de Jong and Molenkamp^{2}; 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 E_{F} = k_{B}T, 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 j_{x} and comparison of theoretical E_{y} and j_{x} across phase diagram.
a, Curvature of ballistic current profile versus l_{MR}/W. The analytic solution (blue curve) is based on de Jong and Molenkamp^{2} and the red curve is a Monte Carlo billiard ball simulation result. The two methods agree perfectly until l_{MR} exceeds the channel length used in the billiard ball simulation, beyond which the solutions begin to deviate. b, Curvature κ of E_{y}, as in Fig. 4b, calculated by Boltzmann simulation (see Methods), as a function of l_{ee}/W and l_{MR}/W for W/R_{c} = 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 j_{x}, extracted from the same simulation as a. For j_{x}, the curvature in the ballistic regime is essentially constant at κ ≈ 0.31 and so the curvature of j_{x} is less discriminating between the hydrodynamic and ballistic regimes than the curvature of E_{y}, which becomes negative. In the other regimes, the curvatures of j_{x} and E_{y} 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 E_{y} 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 E_{y} is normalized as in the main text by the classical Hall field E_{cl} = (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/s4158601917889
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