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 magnetoelectric 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 viscosityinduced 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.
Introduction
In an overwhelmingly large group of conducting materials, charge transport can be described by the rather simple model of a freeelectron gas. Its basis is that the carriers move unimpededly until they scatter with phonons or defects. Such collisions usually relax both the momentum and the energy currents, and consequently impose a resistance to charge and heat flow alike. In most conventional conductors, electrical and thermal transport are therefore related via the Wiedemann–Franz law, which states that the product of the electrical resistivity ρ and the thermal conductivity κ, divided by the temperature T is a constant L = ρκ/T, yielding the Sommerfeld value L_{0} = 2.44 × 10^{−8} W Ω K^{−2}. ρ and κ are intrinsic material properties and independent of the size and geometry of the conducting bulk. However, the conventional freeelectron model fails to describe transport in strongly interacting electron systems^{1}. The difficulty is to find a theoretical framework that captures the frequent interparticle collisions that define the interaction within the manybody system. Recently, it has been rediscovered that the theory of hydrodynamics, which is normally applied to explain the behavior of classical liquids like water, could be used to describe the collective motion of electrons in such a system^{2,3,4,5,6,7,8,9}.
In contrast to the freeelectron gas, the energy dissipation in a hydrodynamic electron fluid is dominated by momentumconserving electron–electron scattering or smallangle electron–phonon scattering^{10}. A signature of the hydrodynamic nature of transport emerges when the flow of the electrons is restricted to channels^{11,12}. The electrical resistance of a hydrodynamic electron liquid is then proportional to its shear viscosity, and therefore paradoxically increases with increasing mean free path of the electrons^{2,13}. Viscosityinduced shear forces at the channel walls cause a nonuniform velocity profile, so that the electrical resistivity becomes a function of the channel width. Moreover, the electrical resistivity will become small with increasing width, because momentumrelaxing processes within the bulk are strongly suppressed. The thermal conductivity is instead dominated by faster momentumconserving collisions. Consequently, a strong violation of the Wiedemann–Franz law is predicted^{3,4,5,14}.
Despite the significant difference in the microscopic mechanisms behind momentum and energycurrentrelaxing collisions, both processes should be limited by the quantum indeterminacy in the energy dissipation with a timescale larger than τ_{ℏ} = ℏ/(k_{B}T)^{3,15,16,17}, where τ_{ℏ} is determined only by the Boltzmann constant k_{B}, the reduced Planck constant ℏ and the temperature T. This concept of, sometimes called Planckian dissipation, follows directly from the uncertainty principle when one applies equipartition of energy and any degree of freedom only carries k_{B}T^{3}.
For quantum hydrodynamic fluids, it has been developed in the frameworks of a string theory, known as antideSitter space/conformal field theory correspondence (AdS/CFT)^{1,18}. Holographic models successfully predicted the universal bound on the momentumrelaxation time τ_{mr} in a strongly interacting neutral plasma^{19}. The momentumrelaxation bound τ_{mr} ≥ τ_{ℏ} can also be expressed as the ratio of the shear viscosity to the entropy density, and is not only supported by experiments on the quarkgluon plasma and on the ultracold Fermi gas^{20,21}, but is also wellrespected in classical fluids such as water.
Likewise, maximally dissipative processes matching the timescale τ_{ℏ} have recently been proposed to underpin the Tproportional resistivity of metals^{3,15}, and are believed to be at the root of hightemperature superconductivity^{17,22}. However, whether the AdS/CFT predictions prove to be relevant and provide a true bound for hydrodynamic electron systems is still an open question. In the context of charge transport described by quasiparticles, τ_{mr} represents the characteristic scattering time to randomize the excess forward momentum of a quasiparticle. Momentum relaxation in most conductors is determined by the way in which the electrons couple via Umklapp processes to the lattice or to the disorder of the host solid. Thus, τ_{mr} of the electron system is determined by extrinsic coupling parameters and is not generally universal^{3}. In the hydrodynamic regime, the momentum and energycurrent relaxations are independent processes, which in principle enables the isolation of the momentumconserving scattering time τ_{mc}. The characteristic time τ_{er} needed to dissipate the excess energy of a quasiparticle includes both momentumrelaxing and conserving scattering processes. As such, to a first approximation the momentumconserving scattering time is 1/τ_{er} = 1/τ_{mc} + 1/τ_{mr}^{5}.
Hydrodynamic effects have been postulated to have a role in the Tlinear resistivity of hightemperature superconductors above their critical temperature, but purely hydrodynamic transport is not directly applicable to most of those systems. Extracting τ_{mr} and τ_{er} separately from experimental data has remained challenging because of the strong momentumrelaxation contribution. In fact, as momentumrelaxation processes are always present in a real material system, momentum can only be quasiconserved. This, however, does not mean that hydrodynamic signatures are not observable in transport experiments. Hydrodynamic effects become dominant, when the momentumconserving scattering length l_{mc} = v_{F}τ_{mc} provides the smallest spatial scale in the system, l_{mc} ≪ w ≪ l_{mr}, where w is the sample width, l_{mr} = v_{F}τ_{mr} the momentumrelaxing scattering length and v_{F} the Fermi velocity^{11,23}.
Inspired by pioneering experiments on semiconductor wires^{24}, signatures for hydrodynamic electron flow were recently reported in ultraclean PdCoO_{2}^{23} and graphene^{25,26,27}. However, as these materials exhibit only relatively weak scattering, experimental evidence of a universal thermal dissipation bound has been elusive. It is therefore desirable to go beyond previous experiments and investigate the dissipative timescales of a material in the limit of strong hydrodynamics.
Here we report the observation of experimental signatures of hydrodynamic electron flow in the Weyl semimetal tungsten diphosphide (WP_{2}). Using thermal and magnetoelectric 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 viscosityinduced 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.
Results
Tungsten diphosphide bulk characterization
For our study, we have chosen the semimetal WP_{2}^{28} because momentumrelaxing Umklapp scattering is anomalously strongly suppressed in this material. WP_{2} exhibits a spacegroup symmetry with two pairs of twofold degenerated Weyl points close to the intrinsic Fermi level. WP_{2} contains a mirror plane perpendicular to the aaxis, a cglide perpendicular to the baxis and a twofold screw axis along the caxis. Owing to the high crystalline anisotropy, typical WP_{2} crystals are needleshaped with an orientation along the aaxis. Moreover, the magnetotransport in bulk single crystals has been shown to be highly anisotropic by 2.5 orders of magnitudes between the a−c and the a−b plane^{28}, similarly to bulk PdCoO_{2}^{29}. We note that although WP_{2} is referred to as a semimetal, it exhibits a finite density of free charge carriers at the Fermi energy (Supplementary Fig. 1). The best singlecrystalline bulk samples of WP_{2} exhibit a Tlinear electrical resistivity above 150 K that is dominated by electron–phonon Umklapp scattering and a temperatureindependent resistivity below 20 K as previously observed in PdCoO_{2}^{23}. Its residual resistivity is only 3 nΩ cm, i.e., a remarkable three times lower value than that of PdCoO_{2} in the hydrodynamic regime. At temperatures between 20 and 150 K, ρ increases exponentially with increasing T, which has been attributed to the phonondrag effect. Phonon drag is considered beneficial for reaching the hydrodynamic regime, because it provides another source of momentumconserving scattering^{23}. At 4 K, the bulk samples exhibit a mean free path of l_{mr} ≈ 100 μm in the a–c direction (see Supplementary Note 1 and Supplementary Figure 2 for details). The l_{mr} of WP_{2} exceeds the momentumrelaxing scattering length of hydrodynamic PdCoO_{2} and that of graphene^{25,26} by one and two orders of magnitude, respectively. These properties make WP_{2} an ideal material for investigating hydrodynamic effects and the associated dissipative bounds in its strongly correlated electron system.
Microbeam characterization
For our experiments, we produced a series of WP_{2} microribbons by milling chemical vapor transport grown single crystals. The high anisotropy of the crystals results in rectangular microsamples that retain the needleshaped orientation of the grown crystal. Using a micromanipulator, the milled microribbons were transferred to a predefined metallic line structure (Fig. 1a). The solely mechanical fabrication method prevented chemical contamination and damage of the ultrapure source material. Electronbeam deposition of platinum was used to provide electrical contacts by connecting the ribbon ends to the underlying metal lines. Electron microscopy was used to determine the distance between the contact lines, i.e., the length l of the microribbon along the aaxis of the crystal, its thickness t along the baxis, and its width w along the caxis. The transport direction in our samples matches the crystal’s aaxis. The thickness is approximately t = w/2. However, the high anisotropy of the magnetotransport yields a mean free path in the a–b direction that is about 250 times lower than that in the a–c direction^{29}. Therefore, w is the characteristic length scale of the samples^{28}, justifying the use of twodimensional models for the inplane transport properties in a–c. We investigated four microribbons with widths of 0.4, 2.5, 5.6, and 9 μm, all satisfying w ≪ l_{mr} at low temperatures. Note that to observe the surface effects of all walls in equal strength in the transport experiments, one would have to design the sample such that its width is 250 times larger than its height.
Widthdependence of the electrical resistivity
In a first set of transport experiments, we studied the T and wdependence of the electrical resistivity ρ = V/I·wt/l of the microribbons (Fig. 1b). For this purpose, we measured the voltage response V to an ACcurrent bias I, using the standard lowfrequency lockin technique (see Methods for details). The elongated geometry of the microribbons with contact lines wrapping around the whole cross section of the samples was chosen to ensure homogenous current distributions. Because of the low resistivity of the bulk sample, special care must be taken in extracting the intrinsic ρ of the WP_{2} ribbons. We therefore compare fourterminal with quasifourterminal resistivity measurements. The quasifourterminal measurements exclude the resistance of the contact lines, but in principle include the interface resistances at the metal/semimetal junction. The electrical contact resistances are found, however, to be negligibly small, and not measurable in our experiment (see Fig. 1d and the Supplementary Note 2 and Supplementary Figure 3–6 for a detailed analysis). Conventionally, ρ = ρ_{0} = m*/(e^{2}nτ_{mr}) is an intrinsic bulk property and does not depend on w. According to the Drude model, ρ only depends on the effective mass m* of the charge carriers, the elementary charge e and the carrier concentration n. However, when boundary scattering becomes significant, ρ can turn into a powerlaw function of the sample size ρ ~ w^{β}^{27}. The exponent β characterizes different transport regimes. In the wellestablished ballistic regime (w ≪ l_{er}, l_{mr}), for example, a power of 0 > β ≥ −1 occurs and the electrical resistivity in the limit of fully diffusive scattering is given by is given by ρ ~ w^{−1}. Further, in a hydrodynamic fluid (l_{er} ≪ w ≪ l_{mr}), the flow resistance is determined solely by the interaction with the sample boundaries, reducing the average flow velocity of the electron fluid (Fig. 1c). As a consequence, a power of −1 > β ≥ −2 is indicative of hydrodynamics. Recent theory predicts that the electrical resistivity in the Navier–Stokes flow limit is modified as ρ = m^{*}/(e^{2}n)·12ηw^{−2}, where electron–electron and smallangle electron–phonon scattering are parameterized in the shear viscosity η (6−8).
As shown in Fig. 1b, all ribbons investigated consistently exhibit a constant ρ = ρ_{0} above 150 K. In accordance with the bulk measurements^{28}, ρ increases linearly with increasing T, as expected for dominant electron–phonon scattering. At lower temperatures, however, ρ becomes a nonmonotonic function of T and increases with decreasing w. The change of slope in ρ(T) is more pronounced in narrower ribbons, corroborating the importance of the sample’s spatial boundaries in this temperature regime. As in real materials, the momentum is only quasiconserved, ρ always contains a widthindependent Drude offset ρ_{0} in addition to the widthdependent powerlaw component ρ_{1}w^{β}. To extract the power β from the experimental data (Fig. 1d and Supplementary Fig. 4), we have subtracted ρ_{0} from ρ at all temperatures, fitting the experimental data with ρ = ρ_{0} + ρ_{1}w^{β}. The exponents obtained were then crosschecked by a logarithmic analysis of ρ − ρ_{0} = ρ_{1}w^{β} (Supplementary Fig. 7 and Supplementary Note 3). As shown in Fig. 1d, we found that the lowtemperature regime (T < 20 K) is well described by an inverse quadratic relation ρ − ρ_{0} ∼ w^{−2}, in agreement with the Navier–Stokes description of hydrodynamic flow^{11,12,13}. The residual resistivity ρ_{0} of about 4 nΩ cm obtained at 4 K matches the bulk resistivity excellently.
This result enables a quantitative extraction of the kinematic shear viscosity of the electron liquid as η = 3.8 × 10^{−2} m^{2} s^{−1} at 4 K (see Supplementary Note 3). Multiplication by the mass density M = nm* yields a dynamic viscosity of about η_{D} = 1 × 10^{−4} kg m^{−1} s^{−1} at 4 K, which is on the order of that of liquid nitrogen at 75 K. The observed w^{−2} dependence provides strong evidence of hydrodynamic electron flow in WP_{2}, whereas the regime between 20 and 150 K can be explained by a smooth transition to a hybrid state in which viscositystimulated boundary scattering mixes with momentumrelaxing electron–phonon collisions^{23}.
Estimation of the Lorenz number
Next, we investigated the Lorenz number L = κ ρ/T in WP_{2}, which is widely considered to be an important observable for characterizing thermal and charge transport characteristics^{5,14,26,30}. Therefore, we determined the thermal conductivity κ of the 2.5μmwide WP_{2} sample. The measurements were performed with open boundary conditions, prohibiting electric current flow (see Methods for details). Zero electric current forces zero momentum flux, which decouples the heat flow from the momentum drag in the hydrodynamic regime^{14}. The ribbon was mounted on a microsystem platform^{30,31} with two integrated heater/sensors as shown in Fig. 2a. The sensor device is thermally insulated through 1.2mmlong silicon nitride bars operated in vacuum to 1.6 × 10^{−5} K W^{−1} at room temperature. The fabrication and characterization of the platform are described in detail in the Methods and in the Supplementary Figures 8–10. Two gold resistors, serving as both thermometers and heaters, were calibrated at each temperature and used to measure both the temperature bias along and the heat flux through the WP_{2} sample. Although used routinely for the thermal characterization of microscale and nanoscale samples, this method often suffers from thermal contact resistance effects, in particular when applied to the characterization of nanoscale structures. To minimize such effects, we chose large dimensions for both the sample as well as for the electrical contacts. Nevertheless, we calculate relatively large expected systematic errors as shown as error bars in the plot (see Supplementary Note 4). The experimentally extracted κ as a function of T is given in Fig. 2(b).
As shown in Fig. 2c, the Wiedemann–Franz relationship holds above 150 K, as the Lorenz number assumes the Sommerfeld value L ≈ L_{0}. With decreasing temperature, however, L is strongly reduced by more than one order of magnitude to a value below 0.05 L_{0} at 20 K. We note, that our thermal conductance measurements are in agreement with independent measurements on macroscopic WP_{2} crystals leading to L ≈ 0.25 L_{0} at T = 11 K, despite the full phonon contribution being included^{32}. The interaction between the channel boundaries and the carriers should also influence the heat transport, because heat currents are not only relaxed by momentum conserving, but also by momentumrelaxing scattering events. This result implies, that either the phonon contribution is relatively small due to the metallic character of the WP_{2}, or that the electroncontribution is below L_{0} even above 150 K. In any case the phonon thermal conductivity appears to be relatively small, which may be due to the heavy Watoms that cause slow phonon velocities.
To interpret, we recall that in conventional conductors, the Wiedemann–Franz law holds, i.e., L equals the Sommerfeld value L_{0} = π^{2} k_{B}^{2}/(3e^{2}). Violations of the Wiedemann–Franz law typically are an indication of invalidity of the quasiparticle picture, strong difference between τ_{er} and τ_{mr} in quasiparticle scattering, or ambipolar physics. Phonon contributions to κ enhance L. Conversely, small (O(1)) deviations below L_{0} are usually observed in metals near 0.1 of the Debye temperature. Many ultrapure metals at low temperatures, for example, can exhibit reduced values of L/L_{0} < 0.5. Rhenium^{33} and Silver^{34} are among the most extreme cases with L/L_{0} at and below 0.1. This effect is due to electron–phonon scattering and a transition from inelastic large angle to smallangle scattering processes^{13,35}. In hydrodynamic systems, L can become arbitrarily small because of the difference between the two relaxation timesτ_{mr} and τ_{er} governing electrical and thermal transport, respectively^{3,5,14,36}. Hence, we expect the ratio between τ_{mr} and τ_{er} in WP_{2} to be at least one order of magnitude^{5}. However, we note that this can only be a lower bound of the difference between the scattering times in the electron system because the residual phonon contributions are not subtracted from κ. The observed maximum in thermal conductance at around 10 K is consistent with this notion. Nevertheless, the consideration of a separate heatconduction channel via the crystal lattice through phonons must be made with care in correlated materials. A recent claim argues that the hydrodynamic fluid may comprise both phonons and electrons and it is sometimes referred to as an electron–phonon soup^{37}. In any case, the measured thermal conductance shown here includes both phonon and electron contributions.
The Lorenz value we obtained belongs to the lowest such values ever reported^{30}. This is an indication for strong inelastic scattering, and, in combination with the charge transport data shown above, an independent evidence of a hydrodynamic electron fluid below 20 K.
Magnetohydrodynamic transport
Exploiting the welljustified conjecture of the hydrodynamic nature of transport in WP_{2} at low temperatures, we can now manipulate and tune the viscosity of the electron fluid to obtain further information on the magnitude of τ_{mc}. For this, the resistivity of the microribbons is measured as a function of the magnetic field B at fixed T. B = B is set along the baxis of the crystal and thus perpendicular to the direction of current flow. In an electron liquid, the viscosity is defined by the internal friction between layers of different velocities^{11}, mediated by the exchange of electrons (Fig. 3a). The strength of the friction is given by the penetration depth λ_{p} of the electrons, which is on the order of the mean free path between collisions, λ_{p} ≈ l_{mc}, at zero magnetic field. When B is turned on, however, this penetration depth is limited by the cyclotron radius r_{c} = mv_{F}/eB. Thus, in a strong magnetic field, the viscosity should tend to zero, providing a mechanism for a large negative magnetoresistivity. Solving the corresponding magnetohydrodynamic equations results in the magnetic fielddependent viscosity along the aaxis of the crystal η(B) = η_{0}/(1 + (2τ_{mc}ω_{c})^{2}), where ω_{c} = v_{F}/r_{c} is the cyclotron frequency (see Supplementary Note 5 and Supplementary Figures 11–13 for details)^{11,12,35}.
As shown in Fig. 3b, we observe a large negative magnetoresistivity at low temperatures. When ρ(B) − ρ_{0} is normalized by w^{−2}, all experimental data below 20 K collapse onto a single curve, matching the magnetic fielddependent viscosity model excellently. The average viscosity of the single traces at zero field η_{0} equals the viscosity η obtained from the w^{−2} fits above. This agreement is an important crosscheck, confirming our results and the hydrodynamic interpretation. Furthermore, in accordance with the theory, η(B) tends to zero as the magnetic field is enhanced.
Extraction of the relaxation times
The τ_{mc} extracted at individual T values are plotted in Fig. 3c. Throughout the hydrodynamic regime, we find that the momentumconserving scattering time in the electron liquid is tied to the Planckian limit^{3,16,17} as τ_{mc} ~ τ_{ℏ} = ℏ/(k_{B}T). As an important crosscheck, we calculated the kinematic shear viscosity^{11,12} from τ_{mc} as η = v_{F}^{2}τ_{mc}/4, and obtained consistent values with those extracted above (Supplementary Fig. 12), confirming the underlying dissipation bound.
For comparison, we also determined the timescale of the charge current relaxation related to the electrons’ momentum. τ_{mr} is extracted from combined resistivity and Hall measurements on the bulk sample^{28} (Supplementary Fig. 1a). With the carrier concentration n (Supplementary Fig. 1b) obtained from Hall measurements and Shubnikovde Haas oscillations^{28}, we calculated the mobility μ = (ρen)^{−1} (Supplementary Fig. 1c). Using the average effective mass of the charge carriers m* = 1.21m_{0} (m_{0} is the freeelectron mass) and μ = eτ_{mr}/m*, we eventually obtained τ_{mr} as a function of the temperature (Fig. 3c). At low temperatures, τ_{mr} is three orders of magnitudes larger than τ_{mc} and thus also than the Planckian bound. Consequently, l_{mc} < w ≪ l_{mr} (Supplementary Fig. 13), validating the hydrodynamic treatment of transport in WP_{2} in this temperature regime.
While our experiments allow for the extraction of the relaxation times τ_{mr} and τ_{mc}, the microscopic origin of the hydrodynamic transport regime remains elusive. However, recent ab initio calculations of the scattering timeresolved Fermi surfaces in WP_{2} suggest that phonon related processes, rather than purely electron–electron processes, have a critical role in the emergence of the hydrodynamic behavior^{38}.
A natural question that arises from this observation is whether Planckian dissipation is exclusive to the hydrodynamic regime in WP_{2}. In the Tlinear regime above 150 K, where electron–phonon Umklapp scattering yields L = L_{0} (because τ_{mc} = τ_{mr}), we found that, despite their fundamental difference, electron–phonon Umklapp and electrondefect processes are also tied to τ_{ℏ}. This observation indicates that the dissipation in WP_{2} is generally tied to a characteristic timescale τ_{ℏ}, regardless of the transport regime and the details of the underlying scattering mechanisms.
Discussion
Our analysis suggests that WP_{2} behaves like a typical Fermi Liquid in some respects and different in others. For example, the existence of quasiparticles is suggested by the Shubnikovde Haas oscillations, weakly interacting particles are suggested by the large ratio between the dynamic viscosity and the number density (430 ℏ), and the degeneracy is implied from the ratio of the Fermi energy E_{F} = 5.6 eV (estimated from DFT calculations) and k_{B}T at 4 K of about 1.7 × 10^{4} (Supplementary Note 6 and Supplementary Figures 14–16 for details). However, a Fermi liquid is expected to have a T^{2} scattering rate when clearly below the Debye temperature (for WP_{2} estimated to be above 300 K from heat capacity measurements), which contradicts our results obtained at low temperatures. To find the relaxation time at the Planckian bound and linear in T is unusual, but not in contradiction to the fundamental concepts of Fermi liquids^{39}.
The Tlinearity observed in here could be entered onto the universal plot in of Bruin et al.^{15}, but all the other entries in that plot are either strongly interacting, close to quantum critical points or above the Debye temperature. How has WP_{2} earned its right to participate in this universality?
In conclusion, our experiments strongly support the existence of a hydrodynamic electron fluid in WP_{2}. The accompanying independence of τ_{mc} and τ_{mr} allows the intrinsic thermal current relaxation process to be isolated, which is particularly elusive in other contexts. Remarkably, it turns out that the electron system in WP_{2} generates entropy in a very simple and universal way in which the only relevant scale is the temperature.
Methods
WP_{2} singlecrystal growth of the bulk sample
Crystals of WP_{2} were prepared by chemical vapor transport method. Starting materials were red phosphorous (AlfaAesar, 99.999%) and tungsten trioxide (AlfaAesar, 99.998%) with iodine as a transport agent. The materials were taken in an evacuated fused silica ampoule. The transport reaction was carried out in a twozone furnace with a temperature gradient of 1000 to 900 °C for serval weeks. After reaction, the ampoule was removed from the furnace and quenched in water. The metallicneedle crystals were characterized by Xray diffraction.
Electrical transport measurements on the bulk sample
The electrical resistance of the bulk sample is determined within a fourprobe configuration under isothermal conditions with an AC bias current of r.m.s. 3 mA at 93.0 Hz, using standard lockin technique. As for the microbeams, the current is applied along the aaxis of the crystal. Magnetoresistance measurements are performed with a magnetic field applied perpendicular to the direction of current flow, within the b–c plane. The Hall resistance is measured under the same conditions with a magnetic field applied along the baxis. More details on the bulk characterization can be found in Kumar et al.^{28} All measurements are carried out in a temperaturevariable cryostat, equipped with a superconducting ±9 T magnet (PPMS Dynacool).
Electrical transport measurements on the microbeams
All transport measurements are performed in a temperaturevariable cryostat (Dynacool, Quantum Design) in vacuum. The cryostat is equipped with a ±9 T superconducting magnet, swept with a rate of 5 mT s^{−1}. After fabrication, the microbeam devices are mounted on a sample holder and wire bonded. Electrical resistance measurements on the microbeams are carried out in a quasifourprobe configuration under isothermal conditions with AC bias currents of r.m.s. 100 μA max at 6.1 Hz, using standard lockin technique. The quasifourterminal resistivity measurements exclude the resistance of the contact lines, but can in principle include interface resistances at the metal/semimetal junctions. However, we find no evidence for any resistance contribution from this interface (see Supplementary Note 2 for details).
Fabrication of the MEMS platforms for heat transport measurements
The fabrication process of the MEMS devices (Supplementary Fig. 8) is similar to the process of Karg et al.^{31} A silicon wafer was coated with a 150 nmthick layer of lowstress silicon nitride (SiMat, Germany). Gold lines with Cr adhesion layer were patterned using optical lithography, metal evaporation and a standard liftoff process. The underetched regions were defined using optical lithography and etching of the silicon nitride. Finally, the devices were released using wet etching and criticalpoint drying.
The devices consist of two MEMS platforms each carrying a fourprobe resistor of 430 and 700 Ω, respectively, and two electrical leads to contact the sample, such that the WP_{2} sample can be measured electrically using a fourprobe geometry. Each platform is connected to the wafer via four 1.2mmlong silicon nitride suspension legs, two of which carry three gold leads each. Two nitride bridges connecting the two platforms served to stabilize the device while placing the sample and were cut using a focussed ion beam before the measurement. The thermal conductance of each platform was 1.6 × 10^{−5} W K^{−1} at room temperature and showed the expected slight increase in conductance at lower temperatures caused by the temperaturedependent thermal conductivity of silicon nitride and gold. The dimensions of the platform lead to a thermal conductance of the coupling between each heater/sensor platform to the chip carrier of 1.6 × 10^{−5} W K^{−1} at room temperature. This value is 50 times larger than the sample thermal conductance of the WP_{2} sample of 3 × 10^{−7} W K^{−1} at room temperature.
Sample mounting on MEMS platform
The sample was mounted using a micromanipulator under an optical microscope. After positioning the sample, electrical contacts were deposited using electronbeam induced metal (Pt) deposition. Care was taken not to expose the WP_{2} sample to the electron beam at any point in time except in the contact region during metal deposition. After successful mounting, the WP_{2} sample bridges the gap between the two platforms, each equipped with a heater/sensor and two electrical leads to the WP_{2} sample. We note that in favor of large electrical contacts and thereby good thermal coupling, the pairs of electrodes at either end of the WP_{2} sample are not separated to allow subtracting the electrical contact resistance. However, the significant electrical resistance of the long microfabricated leads can be determined and subtracted from the measurement. Given the electrical conductance observed, we estimate a potential systematic error due to electrical contact resistance to be within the scatter of the experiments.
Thermal transport measurements
The method employed in this study is based on the method developed by Li Shi et al.^{40}, and refined since two metal resistors are fabricated using lithography. They serve as both microheaters and thermometers. The resistors are read out using the fourprobe technique and are fabricated on structured silicon nitride membranes to avoid thermal crosstalk via heat conduction through the substrate. The measurements are performed in vacuum to avoid thermal crosstalk through air conduction.
In detail, we proceeded as follows. First, the temperature calibration step was performed. For this, the sample was settled to set temperatures between 4 and 300 K. The resistor on each platform was characterized measuring the voltage drop for a given current for currents between −5 and 5 × 10^{−4} A in increments of 2 μA, in DC, from 5 to 40 mA, in AC using phasesensitive detection modulated at 6 Hz. The electrical resistance extrapolated to the limit of zero current was used for calibration (Supplementary Fig. 10). The values measured using DC and AC operation match within the measurement error, confirming the correct choice of modulation frequency. The resulting resistance versus temperature plot shown in Supplementary Fig. 9 is used for temperature calibration of the two sensors operated as resistive thermometers.
For the subsequent transport measurements, polynomial fits for the resistance versus temperature for the thermometers were used; for temperatures above 50 K, we use a 2ndorder polynomial fit, for the lower temperatures we use a 5thorder polynomial fit. At lower temperatures, the resistance versus temperature plot goes through a minimum at around 10.4 K. This is the expected behavior of gold at these temperatures^{41}. As a consequence, the sensitivity is small at around this temperature and, consequently the systematic error is larger compared to high temperatures. The error bars plotted in Fig. 2b of the main manuscript comprise this variation in sensitivity, as well as known systematic errors and measurement uncertainties.
After the temperature calibration, heat transport measurements were performed. At each cryostat temperature, a small sensing current of 10 nA was applied to heater/sensor 2, while the current was ramped as described above in heater/sensor 1. At all times, the current through the WP_{2} sample was zero and the leads floating. Both sides were probed using phasesensitive detection. For the analysis, we follow the method described by Shi et al.^{40} and refined by Karg et al.^{31} An example is given in Supplementary Fig. 10 for the cryostat temperature of 100 K. In summary, for each current the electrical resistance in the heater and the long leads on the suspension legs is determined. Then, using the current and the resistances, the effective Joule heating power P_{eff} is calculated. From the measured resistance values R_{Heater1} and R_{Heater2}, the temperature rise, ΔT_{1} and ΔT_{2} of both heater/sensors is calculated using the temperature calibration (Supplementary Fig. 9). The thermal conductance of the MEMS device and the WP_{2} sample is then extracted from the slope of plots like Supplementary Fig. 10c and d. The same results were obtained using the symmetry check at one temperature where the roles of heater and sensor side was swapped, which reconfirms the treatment of the two sensors with different resistances.
The uncertainty of the thermal conductance measurements is calculated considering known systematic errors and measurements uncertainties, the largest contributions coming from the effects of device and sample geometry, electrical resistance measurement of the sensors, and the fitting procedure. The sensitivity of electrical resistance of the metal thermometers is strongly reduced at low temperatures. At low temperatures, this is the main contribution to the reported error (see the Supplementary Note 4 for a detailed discussion).
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
Additional information
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Acknowledgements
We thank Steffen Reith and Valentina Troncale for technical support, as well as Stuart Parkin, Philip Moll, Subir Sachdev, Sean Hartnoll, Jan Zaanen, and Andrew Mackenzie for fruitful discussions. We also acknowledge support by Walter Riess and Kirsten Moselund, and thank Charlotte Bolliger for copyediting. Fabian Menges gratefully acknowledges the support from the Society in Science—The Branco Weiss Fellowship and a Swiss National Science Foundation postdoctoral fellowship.
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Contributions
J.G., F.M., C.F., and B.G. conceived the experiment. C.S., N.K., and V.S. synthesized the singlecrystal bulk samples. U.D. fabricated the suspended platforms. J.G., F.M., and B.G. produced the microribbons and processed the devices. J.G. carried out the transport measurements on the microribbons with the help of R.Z. J.G. and B.G. analyzed the data. C.S. and N.K. performed the Hall and resistivity measurements on the bulk samples. Y.S. calculated the band structure. B.G. supervised the project. All authors contributed to the interpretation of the data and to the writing of the manuscript.
Competing interests
The authors declare no competing interests.
Corresponding authors
Correspondence to J. Gooth or B. Gotsmann.
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Further reading

Departure from the Wiedemann–Franz law in WP2 driven by mismatch in Tsquare resistivity prefactors
npj Quantum Materials (2018)
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