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

Abstract

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.

Fig. 1

Effect of the channel width on the electrical resistivity. a False-colored scanning electron-beam microscopy (SEM) image of the device to measure the electrical resistance R = V/I (upper panel) of the WP₂ micro-ribbons, where I is the applied current and V the measured voltage. x, y, and z are the spatial dimensions. The scale bar denotes 10 μm. Ribbons of four different widths w were investigated (lower panels). The error of the measured width is below 5%, including the uncertainty of the SEM and sample roughness. b Temperature (T)-dependent electrical resistivity ρ of the four ribbons. c Sketch of the velocity v_x flow profile for a Stokes flow (left panel, blue line and narrows) and conventional charge flow (right panel, red line and arrows). v_av is the average velocity of the charge-carrier system. β is the exponent of the functional dependence ρ(w) = ρ₀ + ρ₁w^β. d β as a function of T, extracted from power-law fits of the data plotted in b. The error bars denote the errors of the fits. The inset shows a power-law fit at 4 K, where the open and filled symbols represent quasi-four and four-terminal measurements, respectively. The line is a power-law fit, leading to β = −2. The colored background marks the hydrodynamic (light blue) and the normal metallic (Ohmic) temperature regime (light red)

Fig. 2

Violation of the Wiedemann–Franz law. a False-colored SEM image of a microdevice for measuring thermal transport that consist of two suspended platforms bridged by a 2.5-μm-wide WP₂ ribbon. The scale bar denotes 100 μm. b Thermal conductivity κ (left axis) and electrical resistivity ρ (right axis) of the micro-ribbon as a function of temperature. The error bars denote the error of the measurement as described in the Supporting Information. c Lorenz number L = κTρ, calculated from the data in b in units of the Sommerfeld value L₀. The error bars denote the error, coming from the thermal conductivity measurements. The inset shows a zoom of the low-temperature region

Fig. 3

Magnetohydrodynamics and the Planckian bound of dissipation. a The origin of the decrease of the electron viscosity η in a magnetic field B (schematic) perpendicular to the current flow (B = B_z) and the sample width. The viscous friction between two adjacent layers of the electron fluid moving (the arrows point along the flow direction) with different velocities (length of the arrows) is determined by the depth of the interlayer penetration of the charge carriers e⁻ (black dots). In a magnetic field, this depth is limited by the cyclotron radius. b (ρ − ρ₀)/w² as a function of B for all four WP₂ ribbons investigated at 4 K (lines), where ρ₀ is the electrical resistance in zero field. The experimental data have been fitted by the magnetohydrodynamic model in the Navier–Stokes flow limit (gray dots). c Experimentally extracted momentum-relaxing and momentum-conserving relaxation times τ_mr and τ_mc, respectively (symbols, with guide to the eye). The error bars denote the errors of the fits exemplarily displayed for 4 K in a (see Supplementary Information for details). The dashed line marks the Planckian bound on the dissipation time τ_ℏ = ℏ/(k_BT)