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# Imaging phonon-mediated hydrodynamic flow in WTe2

## Abstract

In the presence of interactions, electrons in condensed-matter systems can behave hydrodynamically, exhibiting phenomena associated with classical fluids, such as vortices and Poiseuille flow1,2,3. In most conductors, electron–electron interactions are minimized by screening effects, hindering the search for hydrodynamic materials; however, recently, a class of semimetals has been reported to exhibit prominent interactions4,5. Here we study the current flow in the layered semimetal tungsten ditelluride by imaging the local magnetic field using a nitrogen-vacancy defect in a diamond. We image the spatial current profile within three-dimensional tungsten ditelluride and find that it exhibits non-uniform current density, indicating hydrodynamic flow. Our temperature-resolved current profile measurements reveal a non-monotonic temperature dependence, with the strongest hydrodynamic effects at approximately 20 K. We also report ab initio calculations showing that electron–electron interactions are not explained by the Coulomb interaction alone, but are predominantly mediated by phonons. This provides a promising avenue in the search for hydrodynamic flow and prominent electron interactions in high-carrier-density materials.

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## Data availability

Data for figures that support the manuscript are available at https://doi.org/10.7910/DVN/ULRIZG. All other data that support the findings of this paper are available from the corresponding authors upon request.

## Code availability

Source codes for the ab initio model and Boltzmann transport equation calculation are available from the corresponding authors upon request.

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## Acknowledgements

We thank M. Ku and A. S. Jermyn for fruitful discussions. We also thank P. Gumann and R. Stöhr for the initial design and construction of the microscope. T.X.Z. thanks R. Walsworth and M. Turner for diamond-processing assistance. Funding: this work was primarily supported by ARO grant no. W911NF-17-1-0023 and the Gordon and Betty Moore Foundations EPiQS Initiative through grant no. GBMF4531. Fabrication of samples was supported by the US Department of Energy (DOE), Basic Energy Sciences Office, Division of Materials Sciences and Engineering, under award DE-SC0019300. A.Y. acknowledges support from ARO grants W911NF-18-1-0316 and W911NF-1-81-0206, NSF grant DMR-1708688, and the STC Center for Integrated Quantum Materials, NSF grant no. DMR-1231319, and the Aspen Center of Physics supported by NSF grant PHY-1607611. G.V., Y.W. and P.N. acknowledge support from the Army Research Office MURI (Ab-Initio Solid-State Quantum Materials) grant no. W911NF-18-1-0431, which supported the development of computational methods to describe the microscopic, temperature-dependent dynamics in low-dimensional materials. G.V. acknowledges support from the Office of Naval Research grant on High-Tc Superconductivity at Oxide–Chalcogenide Interfaces (grant no. N00014-18-1-2691), which supported the theoretical and computational methods for phonon-mediated interactions. Y.W. is partially supported by the STC Center for Integrated Quantum Materials, NSF grant no. DMR-1231319, for the development of computational methods for topological materials. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the US Department of Energy under contract no. DE-AC02-05CH11231 as well as resources at the Research Computing Group at Harvard University. 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. P.N. is a Moore Inventor Fellow and gratefully acknowledges support through grant no. GBMF8048 from the Gordon and Betty Moore Foundation. C.A.C.G. acknowledges support from the NSF Graduate Research Fellowship Program under grant no. DGE-1745303. A.T.P. acknowledges support from the Department of Defense (DOD) through the National Defense Science and Engineering Graduate Fellowship (NDSEG) Program.

## Author information

Authors

### Contributions

J.G., P.N. and A.Y. conceived the project. U.V., A.H., T.X.Z. and Y.D. designed and developed the experimental setup under the guidance of A.Y. N.K. and J.G. grew the bulk WTe2 crystals under the guidance of C.F. U.V., A.H., Z.Q. and A.T.P fabricated and characterized the WTe2 flakes. G.V. implemented the numerical BTE methods under the guidance of P.A. and P.N. G.V., Y.W. and C.A.C.G. developed and implemented the ab initio theoretical methods under the guidance of P.N. U.V., A.H., G.V., Y.W., P.N. and A.Y. analysed the data and discussed the results. All the authors contributed to the writing of the manuscript.

### Corresponding authors

Correspondence to Georgios Varnavides, Prineha Narang or Amir Yacoby.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review informationNature Physics thanks Sergey Dushenko and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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

## Extended data

### Extended Data Fig. 1 Residual resistivity ratio and quantum oscillations.

a, Temperature dependent resistivity at zero magnetic field. The residual resistivity ratio (RRR) between 300 Kand 2 K is approximately 2000, signifying the high quality of the single crystal. b, Magnetic field (applied along the ĉ-axis) dependent resistivity showing Shubnikov-de Haas oscillations at 2 K. c, The fast Fourier transform (FFT) of the background-subtracted data of (b) when plotted against the inverse of the magnetic field. We notice four fundamental peaks corresponding to the extremal orbits of the electron and hole pockets in the Brillouin zone24.

### Extended Data Fig. 2 Magnetic field dependent resistivity.

Fitting of the magnetic field dependent resistivity at 2 K according to the relation $$\frac{\rho (B)-\rho (0)}{\rho (0)}={({\mu }_{av}B)}^{2}$$ where μav is the average mobility24,42. This allows us to extract μav = 2.5 × 105cm2/Vs. Using the relation σ = μne, the average carrier density n = 1.47 × 1020 cm−3 can be extracted.

### Extended Data Fig. 3 Hall resistivity fitted to the two-band model.

a, b, Hall resistivity vs. magnetic field measured at different temperatures. The data is fitted according to the two band model $${\rho }_{xy}=\frac{B}{e}\frac{({n}_{h}{\mu }_{h}^{2}-{n}_{e}{\mu }_{e}^{2})+({n}_{h}-{n}_{e}){\mu }_{h}^{2}{\mu }_{e}^{2}{B}^{2}}{{({n}_{h}{\mu }_{h}+{n}_{e}{\mu }_{e})}^{2}+{({n}_{h}-{n}_{e})}^{2}{\mu }_{h}^{2}{\mu }_{e}^{2}{B}^{2}}$$43,44. c, Carrier density of electrons and holes vs. temperature, extracted from the fits in (a) and (b). d, Mobility of electrons and holes vs. temperature, extracted from the fits in (a) and (b).

### Extended Data Fig. 4 WTe2 device.

Optical image of the WTe2 sample used for the majority of the data in the manuscript, including Main text Fig. 2, 3, and 4. The 100 μm long flake is contacted at its ends with gold contacts. The path along which our magnetic field profiles were taken is shown in white. An additional gold line nearby is used for delivery of RF power to manipulate the NV.

### Extended Data Fig. 5 Diamond scanning probe.

a, An optical image of our scanning tip. A quartz rod is glued to the edge of one of the prongs of a quartz tuning fork. A diamond cube of 50 × 50 × 125 μm is glued to the other side of the rod. b, A closeup image of the diamond cube, with a 3 μm etched pillar visible. This pillar is used as our scanning probe, and an NV defect is located approximately 20 nm away from the pillar edge.

### Extended Data Fig. 6 Multiple pillar sensing.

a, A schematic for multiple pillar sensing. The diamond is tilted at an angle θ (using goniometric motors) so that one pillar is in contact with the substrate, while the scanning pillar maintains a constant height above the sample without touching it. The angle θ controls the height of the NV probe above the sample. b, An optical image of a multiple pillar scanning tip with 7 pillars. The extreme pillars are made to be thicker so they are optimized for durability in contact, while the central pillars are optimized for NV sensing.

## Supplementary information

### Supplementary Information

Supplementary Sections 1–15 and Figs. 1–14.

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Vool, U., Hamo, A., Varnavides, G. et al. Imaging phonon-mediated hydrodynamic flow in WTe2. Nat. Phys. 17, 1216–1220 (2021). https://doi.org/10.1038/s41567-021-01341-w

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