Thermal and electrical signatures of a hydrodynamic electron fluid in tungsten diphosphide

In stark contrast to ordinary metals, in materials in which electrons strongly interact with each other or with phonons, electron transport is thought to resemble the flow of viscous fluids. Despite their differences, it is predicted that transport in both conventional and correlated materials is fundamentally limited by the uncertainty principle applied to energy dissipation. Here we report the observation of experimental signatures of hydrodynamic electron flow in the Weyl semimetal tungsten diphosphide. Using thermal and magneto-electric transport experiments, we find indications of the transition from a conventional metallic state at higher temperatures to a hydrodynamic electron fluid below 20 K. The hydrodynamic regime is characterized by a viscosity-induced dependence of the electrical resistivity on the sample width and by a strong violation of the Wiedemann–Franz law. Following the uncertainty principle, both electrical and thermal transport are bound by the quantum indeterminacy, independent of the underlying transport regime.

To account for all band contributions in the electrical transport, we calculated the harmonic mean giving the average effective mass m * = 4(1/m + 1/m + 1/m + 1/m) -1 = 1.22 m0 and the mean Fermi velocity vF = (∑vFi)/4 = 2.67×10 5 ms -1 . The lmr in the a-c plane is then determined as a function of temperature from the T-dependent mobility  = evFlmr/m * .

Supplementary Note 2: Interface resistance at the metal/semimetal junction
The interface resistance is an important issue, dealing with low sample electrical resistances in the range of m. We therefore have carried out three independent cross-checks to evaluate interface resistance in our devices. All three methods show independently that the interface resistance at the metal/semimetal junction is negligible small in our experiment, despite the low sample resistance: A common method to evaluate the contact resistance is to compare four 4-and 2-point (or quasi-4-point) measurements. Quasi-4-terminal (red) and actual 4-terminal (blue) resistivity of the 9 m-wide WP2 ribbon as a function of temperature are shown in Supplementary Fig. 3. The deviation between the two curves is below 1 % at 300 K and rises to 10 % at 4 K. We note, however, that this enhancement at low temperature is within the measurement error, due to the low resistivity in this temperature range. The 9 m-wide has the lowest resistance and should therefore be most sensitive to the interface resistance.
Further, we have used a Focused Ion Beam to cut out four-terminal devices. Two devices of 0.8 m and 3.5 m width have been fabricated ( Supplementary Fig. 4). The resistivity data points obtained from these devices fit in perfectly to the width-dependent series.
Next, we have checked the residual resistivity 0 of the width-dependent resistivity, fitting the data by  = 0 + 1w  . As shown exemplarily in Supplementary Fig. 4 for the 4 K data, we find An important cross-check that the evaluated 0 itself does not introduce the width dependence is to determine 0 from the magnetic field-dependent resistivity data of each width independently. We therefore fit each magnetoresistance curve individually ( Supplementary Fig.   5) by the  = 0 + 1,aw  /(1+(1,bB) 2 ), where B is the magnetic field and  is extracted from the width-dependent analysis above. As shown in Supplementary Fig. 6, all individual curves result in a 0(4 K) of around 4, in full agreement with the previous analysis.
Therefore, we conclude that within our measurement precision, the metal/semimetal interface perfectly transmits charge carriers and does not represent a noticeable resistance. This could indicate a different mechanism for resistances at hydrodynamic/normal metal electron interfaces. resistivity of the 9 m-wide WP2 ribbon as a function of temperature. The deviation between the two curves is below 1 % at 300 K and rises to 10 % at 4 K. We note, however, that this enhancement at low temperature is within the measurement error, due to the low resistivity in this temperature range.  Fig. 1 (d). shown. Resistance of heater (blue) (a) and sensor (red) (b) for the given heating and sensing currents. Temperature rise above 100 K for heater (c) and sensor (d). The symbols denote measurement data, the lines linear fits.

Supplementary Note 4: Considerations of Thermal Contact Resistance
To minimize the influence of thermal contact resistance, we fabricated electrical contacts to the sample with a contact area to the metal of 16 m 2 . For the phonon contribution of the thermal boundary conductance we expect values of the order of 10 -8 to 10 -7 Km 2 W -1 . Therefore, the thermal resistance of the contacts will be on the order of 10 4 -10 5 KW -1 , which is small compared to the measured overall resistance of 1 to 4×10 6 KW -1 . Moreover, if we include the electron contribution to thermal conductance into this consideration, we expect the difference will be an order of magnitude further apart. Note, that although WP2 is classified as a semimetal, we have an additional band at the Fermi energy contributing to the sample carrier density. The large contact size, however, increases the systematic error in the sample dimension length used in the analysis.

Supplementary Note 5: Additional magneto-hydrodynamic analysis
Employing the hydrodynamic model for  = 2, 1 can be expresses as 1(B) = m * /(e 2 n)·12(B)w -2 , with (B) = 0/(1+(2erc) 2 ). From this expression, we can extract er as a function of temperature from the data in Supplementary Fig. 11 and calculate the viscosity  = vF 2 er/4 as a function of temperature. As shown in Supplementary Fig. 12, the viscosities extracted from the width-dependent zero field data ( Supplementary Fig. 7) and extracted from the fielddependent data in of Supplementary Fig. 11 are in excellent agreement. This agreement between the viscosities extracted from independent dependencies is an important cross-check of our interpretation and shows the consistency of our results.
With vF =ler/er from the bulk analysis above, we can extract the momentum conserving length le for each width independently and compare it to the momentum relaxing mean free oath from the Hall analysis. As exemplarily shown in Supplementary Fig. 13 at 4 K, we find an excellent agreement between the mean free paths of the different widths. The obtained size-regime validates the application of the hydrodynamic model for the data where  = 2.

Supplementary Note 6: Additional key quantities
To guide future interpretations of the data, we have calculated additional key quantities that are connected to the strength of the electron-electron interaction. In Supplementary Fig. 15 and Supplementary Fig. 16, we plot the ratio of dynamic viscosity and number density (D/n) in units of ℏ as a function of temperature and magnetic field, respectively. D/n is directly related to the momentum diffusivity.
Also, lmrkF is a useful dimensionless quantity that characterizes the strength of interactions. lmrkF