## Abstract

Vortices are the hallmarks of hydrodynamic flow. Strongly interacting electrons in ultrapure conductors can display signatures of hydrodynamic behaviour, including negative non-local resistance^{1,2,3,4}, higher-than-ballistic conduction^{5,6,7}, Poiseuille flow in narrow channels^{8,9,10} and violation of the Wiedemann–Franz law^{11}. Here we provide a visualization of whirlpools in an electron fluid. By using a nanoscale scanning superconducting quantum interference device on a tip^{12}, we image the current distribution in a circular chamber connected through a small aperture to a current-carrying strip in the high-purity type II Weyl semimetal WTe_{2}. In this geometry, the Gurzhi momentum diffusion length and the size of the aperture determine the vortex stability phase diagram. We find that vortices are present for only small apertures, whereas the flow is laminar (non-vortical) for larger apertures. Near the vortical-to-laminar transition, we observe the single vortex in the chamber splitting into two vortices; this behaviour is expected only in the hydrodynamic regime and is not anticipated for ballistic transport. These findings suggest a new mechanism of hydrodynamic flow in thin pure crystals such that the spatial diffusion of electron momenta is enabled by small-angle scattering at the surfaces instead of the routinely invoked electron–electron scattering, which becomes extremely weak at low temperatures. This surface-induced para-hydrodynamics, which mimics many aspects of conventional hydrodynamics including vortices, opens new possibilities for exploring and using electron fluidics in high-mobility electron systems.

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## Data availability

The data that support the findings of this study are available from the corresponding authors on request.

## Code availability

The current reconstruction codes used in this study are available from the corresponding authors on request.

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## Acknowledgements

We thank M. Shavit and V. Steinberg for discussions. This work was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (grant no. 785971), the German–Israeli Foundation for Scientific Research and Development (GIF; grant no. I-1505-303.10/2019) and the Israel Science Foundation (ISF; grant no. 994/19). G.F. was supported by the Scientific Excellence Center at WIS, the Simons Foundation (grant no. 662962), the EU Horizon 2020 programme (grant no. 873028), the US–Israel Binational Science Foundation (BSF; grant no. 2018033) and NSF–BSF (grant no. 2020765). B.Y. acknowledges financial support by the ERC (Consolidator NonlinearTopo Project, grant no. 815869) and the ISF (grant no. 2932/21). E.Z. acknowledges the support of the Andre Deloro Prize for Scientific Research. L.S.L. and E.Z. acknowledge the support of the Sagol Weizmann–MIT Bridge Program. M.H. and E.Z. acknowledge the support of the Leona M. and Harry B. Helmsley Charitable Trust grant no. 2018PG-ISL006 and 2112-04911. A.K.P. acknowledges postdoctoral fellowship support from the Council for Higher Education, Israel, through the Study in Israel program.

## Author information

### Authors and Affiliations

### Contributions

A.A.-S., T.V., A.K. and E.Z. conceived the experiments. A.K.P. and M.H. grew and characterized the bulk WTe_{2} crystals. A.K. fabricated and characterized the devices. T.V. and A.A.-S. conducted the SOT magnetic imaging measurements and data analysis. I.R. and Y.M. fabricated the SOTs and the tuning fork feedback. A.Y.M. developed the current density reconstruction method. M.E.H. designed and built the SOT readout system. Y.W., T.H. and B.Y. performed the band structure and electron–electron scattering calculations. T.H. developed the vortex stability model. A.A.-S. performed the finite-element numerical simulations. G.F., L.S.L. and E.Z. developed the para-hydrodynamic model. A.A.-S., T.V., T.H., A.K.P., E.Z., G.F. and L.S.L. wrote the manuscript with contributions from the rest of the authors.

### Corresponding author

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## Extended data figures and tables

### Extended Data Fig. 1 Transport characterization of bulk WTe_{2} single crystals.

**a**, Resistivity, \(\rho \), as a function of temperature of our highest purity crystal. At \(T=\) 2 K, the resistivity is \(\rho \,=\) 0.23 µΩ·cm corresponding to \(RRR\,\cong \) 3,250. Inset: optical image of crystals from the optimized quality growth. **b**, Magnetoresistance, \(MR=\frac{\rho (B)-\rho (0)}{\rho (0)}\), as a function of magnetic field at 2 K showing \(MR\,\cong \) 62,000 at 9 T. **c**, \(MR\) vs. \(RRR\) at \(T=\) 2 K and \(B\,=\) 9 T of our different crystals synthesized by flux growth (black dots) in comparison to reported values (open circles) in the literature^{10,62,63,64}. The black line is a guide to the eye. **d**, Longitudinal and transverse conductivities \({\sigma }_{xx}\) and \({\sigma }_{xy}\) vs. magnetic field at 4.2 K and their fit to the two band model with resulting parameters \({n}_{e}\,=\) 2.4 × 10^{19} cm^{−3}, \({n}_{h}=\) 2.3 × 10^{19} cm^{−3}, \({\mu }_{e}=\) 5.1 × 10^{5} cm^{2}/Vs, and \({\mu }_{h}=\) 2.7 × 10^{5} cm^{2}/Vs.

### Extended Data Fig. 2 Dependence of the reconstructed current densities on the assumed SOT scanning height.

**a**, Numerical simulation of \({J}_{x}(x,y)\) normalized by the average current density \({I}_{0}/W\) in the strip in \(\theta ={35}^{\circ }\) sample for \(D/W=\) 0.28 and \(\xi \,=\) 200 nm. The span of the color scale is \(\pm \)0.05. **b–e**, Current densities \({J}_{x}(x,y)\) reconstructed from the inversion of the measured \({B}_{z}(x,y)\) in WTe_{2} sample A with \(\theta ={35}^{\circ }\) assuming effective SOT scanning heights of \(h\,=\) 20 nm (**b**), 50 nm (**c**), 100 nm (**d**) and 150 nm (**e**). The nominal scanning height was 50 nm. The span of the color scale is \(\pm \)0.05. **f–j**, Same as **a–e**, but for \({J}_{y}(x,y)\) on color scale of \(\pm \)1. **k–o**, Same as **f–j**, but on expanded color scale of \(\pm \)0.05. The \({J}_{y}\) vortex counterflow current (light blue) is resolved in the chambers on a large artificial ringing background outside the strip edges.

### Extended Data Fig. 3 Current profiles in narrow Au and WTe_{2} strips.

**a**, A uniform current density \({J}_{y}(x)\) in \(W=\) 550 nm strip (light green line) from which \({B}_{z}(x)\) is computed at a height \(h\,=\) 150 nm. The \({J}_{y}(x)\) (green dotted symbols) is then reconstructed by inversion of the calculated \({B}_{z}(x)\), showing the unavoidable distortions and ringing. The \({J}_{y}(x)\) reconstructed from the experimental \({B}_{z}(x)\) in the Au strip (black line) shows consistency with a uniform current distribution in the ohmic regime. **b**, Same as (**a**) for a Poiseuille current profile (light blue) with \(D/W=\) 0.28 and no-slip boundary conditions. The reconstructed \({J}_{y}(x)\) from the experimentally measured \({B}_{z}(x)\) in WTe_{2} strip (black) is inconsistent with the theoretically reconstructed \({J}_{y}(x)\) (dotted blue) from \({B}_{z}(x)\) corresponding to the Poiseuille profile. **c**, Same as (**b**) for hydrodynamic flow with \(D/W=\) 0.28 and slip length \(\xi \,=\) 200 nm (light blue) showing good correspondence between the theoretically reconstructed \({J}_{y}(x)\) (dotted blue) and the experimentally derived \({J}_{y}(x)\) (black) in accord with the conclusions in the main text. **d**, Same as (**b**) for hydrodynamic flow with \(D/W=\) 0.28 and no-stress boundary conditions (light blue). The reconstructed theoretical \({J}_{y}(x)\) (dotted blue) underestimates the experimentally derived \({J}_{y}(x)\) (black) supporting the conclusion of a finite slip length. **e**–**g**, Comparison between theoretically calculated current profiles in the hydrodynamic regime with \(\xi =\) 200 nm (light blue line) and in the ballistic flow (light red line) with boundary reflectivity coefficients of \(r\,=\) 0 (fully diffuse) (**e**), \(r\,=\) 0.5 (**f**), and \(r\,=\) 1 (specular) (**g**). The solid lines show \({J}_{y}(x)\) calculated from Eqs. 2 and 3 while the dotted lines are the current profiles reconstructed from the calculated corresponding \({B}_{z}(x)\). These results demonstrate the difficulty in using reconstructed current profiles in strip geometry for distinguishing between the hydrodynamic flow with finite slip length and the ballistic transport, in contrast to vastly different vortex stability phase diagrams in these two regimes.

### Extended Data Fig. 4 Comparison of field and current profiles in Au and WTe_{2} device C.

**a**, \({B}_{z}(x,y)\) in Au sample with \(\theta \,=\) 180° (same as Fig. 1f). **b**, Reconstructed current density \({J}_{y}(x,y)\) normalized by \({I}_{0}/W\) (same as Fig. 1d). **c**, Reconstructed current density \({J}_{x}(x,y)\) (same as Fig. 1g). **d**, \({B}_{z}(x,y)\) in WTe_{2} sample with \(\theta \,=\) 180°. **e**, Reconstructed current density \({J}_{y}(x,y)\). **f**, Reconstructed current density \({J}_{x}(x,y)\). **g**, \({B}_{z}(x,y)\) in Au sample with \(\theta \,=\) 45° (same as Fig. 1l). **h**, Reconstructed current density \({J}_{y}(x,y)\) (same as Fig. 1j). **i**, Reconstructed current density \({J}_{x}(x,y)\) (same as Fig. 1m). **j**, \({B}_{z}(x,y)\) in WTe_{2} sample with \(\theta \,=\) 45°. **k**, Reconstructed current density \({J}_{y}(x,y)\). **l**, Reconstructed current density \({J}_{x}(x,y)\) (same as Extended Data Fig. 9n).

### Extended Data Fig. 5 Visualizing vortices without current inversion in WTe_{2} device A.

**a**, \({B}_{z}(x,y)\) in WTe_{2} sample with \(\theta \,=\) 20°. **b**, Same data after subtraction of the average field along the dashed lines, \({B}_{z}(x,y)-[{B}_{z}(x,y=1.22\,{\rm{\mu }}{\rm{m}})+{B}_{z}(x,y=-1.22\,{\rm{\mu }}{\rm{m}})]/2\), revealing a counterclockwise vortex in the left chamber generating a positive \({B}_{z}\) (red) and a clockwise vortex with negative \({B}_{z}\) (blue) in the right chamber. **c–d**, Corresponding numerical simulations in the hydrodynamic regime showing \({B}_{z}(x,y)\) (**d**) and \({B}_{z}(x,y)\) after the background subtraction (**c**). **e–t**, Same as (**a–d**) for \(\theta \,=\) 35°, 54°, 72°, and 90° samples.

### Extended Data Fig. 6 Comparison between vorticity and vortical stream lines.

**a**, Numerical calculation of vorticity, \({\boldsymbol{\omega }}=\nabla \times {\boldsymbol{J}}\), normalized by \({I}_{0}/{W}^{2}\), in the hydrodynamic regime with \(D\,=\) 155 nm and \(\xi \,=\) 200 nm in \(\theta \,=\) 35° sample. The colors in the strip region, where the normalized vorticity is of the order of 1, are greatly saturated in order to show the vorticity in the chambers. **b**, Calculated laminar (red) and vortical (blue) streamlines in the same geometry. **c–d**, Same as (**a–b**) for \(\theta \,=\) 72° sample.

### Extended Data Fig. 7 Schematic streamlines for purely ohmic, hydrodynamic and ballistic flow.

**a**, If the sample is purely ohmic, the current leaks into the chamber, forming a current dipole decaying as inverse distance squared. **b**, For a purely hydrodynamic flow, no laminar current (red) leaks from the strip into the chamber. Instead, a vortex forms in the vicinity of the aperture in the chamber (blue) in order to decrease the shear due to the gradient in the velocity profile. **c**, In a purely ballistic flow, only the geometry dictates the streamlines, producing a vortex (blue) whose center is positioned near the chamber center.

### Extended Data Fig. 8 Fermi surface and electron-electron mean free path.

**a**, Fermi surface cut for \({k}_{z}=0\). Typical for a compensated semimetal, small electron and hole pockets appear close to the compensation point. If the hole density is slightly larger than the electron density, the Fermi surface features hole pockets near the Gamma point (red) and electron pockets (blue). **b**, \({l}_{ee}\) as calculated from Eq. 14 for 20 bands as a function of temperature (red points). For \(T=\) 145 K, we also show the values for a smaller number of bands. The blue lines denote upper and lower estimates for the \({T}^{-2}\) dependence of \({l}_{ee}\), where the lower one corresponds to the low-temperature asymptotics.

### Extended Data Fig. 9 AFM images of WTe_{2} samples and additional vortical-to-laminar transitions.

**a**, AFM image of sample A analyzed in the main text with \(W=\) 550 nm, \(R\,=\) 900 nm, \(d\,=\) 48 nm, and aperture angles \(\theta \,=\) 20°, 35°, 54°, 72°, 90°, and 120°. **b**, Sample B used for Extended Data Fig. 9d–l with \(W=\) 350 nm, \(R=\) 450 nm, and\(\,d=\) 23 nm. **c**, Sample C with \(W=\) 770 nm and \(d\,=\) 30 nm, and \(R\,=\) 950, 725, and 500 nm (Extended Data Fig. 9m–r) and dual-drive geometry at the bottom part (Extended Data Fig. 10). **d–l**, Transition from single-vortex to two-vortex to laminar flow in sample B. **d–f**, Measurement of single vortex state in device B with \(\theta \,=\) 40° and corresponding simulations in the hydrodynamic regime with \(D\,=\) 123 nm and \(\xi \,=\) 200 nm. **d**, Measured current density \({J}_{y}(x,y)\) normalized by \({I}_{0}/W\) at \({I}_{0}\,=\) 25 µA. **e**, Simulated \({J}_{y}(x,y)\). **f**, Simulated current streamlines showing laminar (red) flow in the central strip and vortical flow (blue) in the chambers. **g–i**, Same as (d–f) for \(\theta \,=\) 60° showing banana-shaped vortex at the transition from a single to double-vortex state. **j–l**, Same as (d–f) for \(\theta \,=\) 100°, showing laminar flow. **m–r**, Vortical flow in sample C with different geometrical parameters. Current density \({J}_{x}(x,y)\) in sample C with \(W\,=\) 770 nm and \(d\,=\) 30 nm and various chamber parameters: \(\theta \,=\) 24° and \(R\,=\) 950 nm (**m**), \(\theta \,=\) 45° and \(R\,=\) 950 nm (**n**), \(\theta \,=\) 60° and \(R\,=\) 950 nm (**o**), \(\theta \,=\) 180° and \(R\,=\) 950 nm (**p**), \(\theta \,=\) 45° and \(R\,=\) 725 nm (**q**) and \(\theta \,=\) 60° and \(R\,=\) 500 nm (**r**). Laminar flow is observed in (p), while vortical flow is present in all the rest of the geometries.

### Extended Data Fig. 10 Vortex–antivortex formation in dual-drive geometry.

**a–f**, Experimentally derived current densities \({J}_{y}(x,y)\) (top row) and \({J}_{x}(x,y)\) (bottom row) in Au and WTe_{2} samples. **a, b**, Current \({I}_{L}\,=\) 50 µA is driven in the up direction in the left strip with no current applied to the right strip resulting in a single vortex in the WTe_{2} chamber in **b**_{2}. **c, d**, Counterpropagating currents \({I}_{L}=\) 50 µA and \({I}_{R}=-\)50 µA applied to the right and left strips, giving rise to a single massive vortex in **d**_{2}. **e, f**, Copropagating currents \({I}_{L}=\) 50 µA and \({I}_{R}\,=\) 50 µA applied to both strips which generates a vortex–antivortex pair in **f**_{2}. **g–l**, Numerical simulations of current densities *J*_{y} (*x*, *y*) (top row), *J*_{x} (*x*, *y*) (middle row) and the corresponding streamlines (bottom row) in the ohmic and hydrodynamic regimes for the three current configurations. The laminar streamlines are marked in red and the vortex streamlines in blue. The experimental data were acquired with pixel size of 10 nm, acquisition time of 40 ms/pixel, and image size of 600×350 pixels/image.

## Supplementary information

### Supplementary Video 1

Simulations of vortical-to-laminar flow transition in the para-hydrodynamic regime versus *θ*. Numerical simulation of the current density *J*_{x}(*x*,*y*) (top right) and the corresponding streamlines (bottom right) in the double-chamber geometry when increasing the aperture angle *θ* for *D*/*W* = 0.28. The left panel shows the vortex stability phase diagram with no-stress boundary conditions as presented in Fig. 3a. The purple dot marks the value of the varying *θ* along the *D*/*W* = 0.28 line. For *θ* ≤ 54°, there is a single vortex in each chamber (blue streamlines). When increasing *θ* further, the laminar flow (red streamlines) splits the single vortex in each chamber into two vortices that are stable up to *θ* ≤ 60°. For *θ* > 60°, the laminar streamlines fill the entire area of the chambers.

### Supplementary Video 2

Simulations of vortical-to-laminar flow transition in the quasi-ballistic regime versus *θ*. Numerical simulation is conducted for the current density *J*_{x}(*x*,*y*) (top right) and the corresponding streamlines (bottom right) in the double-chamber geometry when increasing *θ* for *D*/*W* = 1.5. The left panel shows the vortex stability phase diagram with no-stress boundary conditions as presented in Fig. 3a. The purple dot marks the value of the varying *θ* along the *D*/*W* = 1.5 line. With increasing *θ*, the laminar streamlines (red) gradually penetrate deeper into the chambers, distorting the vortices (blue streamlines) and pushing them towards the outer boundaries. The vortices become extinct at *θ* ≌ 150° without splitting into double vortices, as is the case in the hydrodynamic regime in Supplementary Video 1. For *θ* > 150°, the laminar streamlines fill the entire area of the chambers.

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### Cite this article

Aharon-Steinberg, A., Völkl, T., Kaplan, A. *et al.* Direct observation of vortices in an electron fluid.
*Nature* **607, **74–80 (2022). https://doi.org/10.1038/s41586-022-04794-y

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DOI: https://doi.org/10.1038/s41586-022-04794-y

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