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.
In an overwhelmingly large group of conducting materials, charge transport can be described by the rather simple model of a free-electron 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 L0 = 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 free-electron model fails to describe transport in strongly interacting electron systems1. The difficulty is to find a theoretical framework that captures the frequent inter-particle collisions that define the interaction within the many-body 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 system2,3,4,5,6,7,8,9.
In contrast to the free-electron gas, the energy dissipation in a hydrodynamic electron fluid is dominated by momentum-conserving electron–electron scattering or small-angle electron–phonon scattering10. A signature of the hydrodynamic nature of transport emerges when the flow of the electrons is restricted to channels11,12. 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 electrons2,13. Viscosity-induced 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 momentum-relaxing processes within the bulk are strongly suppressed. The thermal conductivity is instead dominated by faster momentum-conserving collisions. Consequently, a strong violation of the Wiedemann–Franz law is predicted3,4,5,14.
Despite the significant difference in the microscopic mechanisms behind momentum- and energy-current-relaxing collisions, both processes should be limited by the quantum indeterminacy in the energy dissipation with a timescale larger than τℏ = ℏ/(kBT)3,15,16,17, where τℏ is determined only by the Boltzmann constant kB, 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 kBT3.
For quantum hydrodynamic fluids, it has been developed in the frameworks of a string theory, known as anti-de-Sitter space/conformal field theory correspondence (AdS/CFT)1,18. Holographic models successfully predicted the universal bound on the momentum-relaxation time τmr in a strongly interacting neutral plasma19. The momentum-relaxation bound τmr ≥ τℏ can also be expressed as the ratio of the shear viscosity to the entropy density, and is not only supported by experiments on the quark-gluon plasma and on the ultracold Fermi gas20,21, but is also well-respected in classical fluids such as water.
Likewise, maximally dissipative processes matching the timescale τℏ have recently been proposed to underpin the T-proportional resistivity of metals3,15, and are believed to be at the root of high-temperature superconductivity17,22. 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 quasi-particles, τmr represents the characteristic scattering time to randomize the excess forward momentum of a quasi-particle. 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, τmr of the electron system is determined by extrinsic coupling parameters and is not generally universal3. In the hydrodynamic regime, the momentum and energy-current relaxations are independent processes, which in principle enables the isolation of the momentum-conserving scattering time τmc. The characteristic time τer needed to dissipate the excess energy of a quasi-particle includes both momentum-relaxing and conserving scattering processes. As such, to a first approximation the momentum-conserving scattering time is 1/τer = 1/τmc + 1/τmr5.
Hydrodynamic effects have been postulated to have a role in the T-linear resistivity of high-temperature superconductors above their critical temperature, but purely hydrodynamic transport is not directly applicable to most of those systems. Extracting τmr and τer separately from experimental data has remained challenging because of the strong momentum-relaxation contribution. In fact, as momentum-relaxation processes are always present in a real material system, momentum can only be quasi-conserved. This, however, does not mean that hydrodynamic signatures are not observable in transport experiments. Hydrodynamic effects become dominant, when the momentum-conserving scattering length lmc = vFτmc provides the smallest spatial scale in the system, lmc ≪ w ≪ lmr, where w is the sample width, lmr = vFτmr the momentum-relaxing scattering length and vF the Fermi velocity11,23.
Inspired by pioneering experiments on semiconductor wires24, signatures for hydrodynamic electron flow were recently reported in ultraclean PdCoO223 and graphene25,26,27. 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 (WP2). 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.
Tungsten diphosphide bulk characterization
For our study, we have chosen the semimetal WP228 because momentum-relaxing Umklapp scattering is anomalously strongly suppressed in this material. WP2 exhibits a space-group symmetry with two pairs of twofold degenerated Weyl points close to the intrinsic Fermi level. WP2 contains a mirror plane perpendicular to the a-axis, a c-glide perpendicular to the b-axis and a twofold screw axis along the c-axis. Owing to the high crystalline anisotropy, typical WP2 crystals are needle-shaped with an orientation along the a-axis. Moreover, the magneto-transport 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 plane28, similarly to bulk PdCoO229. We note that although WP2 is referred to as a semimetal, it exhibits a finite density of free charge carriers at the Fermi energy (Supplementary Fig. 1). The best single-crystalline bulk samples of WP2 exhibit a T-linear electrical resistivity above 150 K that is dominated by electron–phonon Umklapp scattering and a temperature-independent resistivity below 20 K as previously observed in PdCoO223. Its residual resistivity is only 3 nΩ cm, i.e., a remarkable three times lower value than that of PdCoO2 in the hydrodynamic regime. At temperatures between 20 and 150 K, ρ increases exponentially with increasing T, which has been attributed to the phonon-drag effect. Phonon drag is considered beneficial for reaching the hydrodynamic regime, because it provides another source of momentum-conserving scattering23. At 4 K, the bulk samples exhibit a mean free path of lmr ≈ 100 μm in the a–c direction (see Supplementary Note 1 and Supplementary Figure 2 for details). The lmr of WP2 exceeds the momentum-relaxing scattering length of hydrodynamic PdCoO2 and that of graphene25,26 by one and two orders of magnitude, respectively. These properties make WP2 an ideal material for investigating hydrodynamic effects and the associated dissipative bounds in its strongly correlated electron system.
For our experiments, we produced a series of WP2 micro-ribbons by milling chemical vapor transport grown single crystals. The high anisotropy of the crystals results in rectangular micro-samples that retain the needle-shaped orientation of the grown crystal. Using a micro-manipulator, the milled micro-ribbons were transferred to a pre-defined metallic line structure (Fig. 1a). The solely mechanical fabrication method prevented chemical contamination and damage of the ultra-pure source material. Electron-beam 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 micro-ribbon along the a-axis of the crystal, its thickness t along the b-axis, and its width w along the c-axis. The transport direction in our samples matches the crystal’s a-axis. The thickness is approximately t = w/2. However, the high anisotropy of the magneto-transport yields a mean free path in the a–b direction that is about 250 times lower than that in the a–c direction29. Therefore, w is the characteristic length scale of the samples28, justifying the use of two-dimensional models for the in-plane transport properties in a–c. We investigated four micro-ribbons with widths of 0.4, 2.5, 5.6, and 9 μm, all satisfying w ≪ lmr 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.
Width-dependence of the electrical resistivity
In a first set of transport experiments, we studied the T- and w-dependence of the electrical resistivity ρ = V/I·wt/l of the micro-ribbons (Fig. 1b). For this purpose, we measured the voltage response V to an AC-current bias I, using the standard low-frequency lock-in technique (see Methods for details). The elongated geometry of the micro-ribbons 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 WP2 ribbons. We therefore compare four-terminal with quasi-four-terminal resistivity measurements. The quasi-four-terminal 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 = ρ0 = m*/(e2nτmr) 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 power-law function of the sample size ρ ~ wβ27. The exponent β characterizes different transport regimes. In the well-established ballistic regime (w ≪ ler, lmr), 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 (ler ≪ w ≪ lmr), 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*/(e2n)·12ηw−2, where electron–electron and small-angle 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 = ρ0 above 150 K. In accordance with the bulk measurements28, ρ increases linearly with increasing T, as expected for dominant electron–phonon scattering. At lower temperatures, however, ρ becomes a non-monotonic 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 quasi-conserved, ρ always contains a width-independent Drude offset ρ0 in addition to the width-dependent power-law component ρ1wβ. To extract the power β from the experimental data (Fig. 1d and Supplementary Fig. 4), we have subtracted ρ0 from ρ at all temperatures, fitting the experimental data with ρ = ρ0 + ρ1wβ. The exponents obtained were then cross-checked by a logarithmic analysis of ρ − ρ0 = ρ1wβ (Supplementary Fig. 7 and Supplementary Note 3). As shown in Fig. 1d, we found that the low-temperature regime (T < 20 K) is well described by an inverse quadratic relation ρ − ρ0 ∼ w−2, in agreement with the Navier–Stokes description of hydrodynamic flow11,12,13. The residual resistivity ρ0 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 m2 s−1 at 4 K (see Supplementary Note 3). Multiplication by the mass density M = nm* yields a dynamic viscosity of about ηD = 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 WP2, whereas the regime between 20 and 150 K can be explained by a smooth transition to a hybrid state in which viscosity-stimulated boundary scattering mixes with momentum-relaxing electron–phonon collisions23.
Estimation of the Lorenz number
Next, we investigated the Lorenz number L = κ ρ/T in WP2, which is widely considered to be an important observable for characterizing thermal and charge transport characteristics5,14,26,30. Therefore, we determined the thermal conductivity κ of the 2.5-μm-wide WP2 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 regime14. The ribbon was mounted on a microsystem platform30,31 with two integrated heater/sensors as shown in Fig. 2a. The sensor device is thermally insulated through 1.2-mm-long 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 WP2 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 ≈ L0. With decreasing temperature, however, L is strongly reduced by more than one order of magnitude to a value below 0.05 L0 at 20 K. We note, that our thermal conductance measurements are in agreement with independent measurements on macroscopic WP2 crystals leading to L ≈ 0.25 L0 at T = 11 K, despite the full phonon contribution being included32. 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 momentum-relaxing scattering events. This result implies, that either the phonon contribution is relatively small due to the metallic character of the WP2, or that the electron-contribution is below L0 even above 150 K. In any case the phonon thermal conductivity appears to be relatively small, which may be due to the heavy W-atoms 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 L0 = π2 kB2/(3e2). Violations of the Wiedemann–Franz law typically are an indication of invalidity of the quasi-particle picture, strong difference between τer and τmr in quasi-particle scattering, or ambipolar physics. Phonon contributions to κ enhance L. Conversely, small (O(1)) deviations below L0 are usually observed in metals near 0.1 of the Debye temperature. Many ultra-pure metals at low temperatures, for example, can exhibit reduced values of L/L0 < 0.5. Rhenium33 and Silver34 are among the most extreme cases with L/L0 at and below 0.1. This effect is due to electron–phonon scattering and a transition from inelastic large angle to small-angle scattering processes13,35. In hydrodynamic systems, L can become arbitrarily small because of the difference between the two relaxation timesτmr and τer governing electrical and thermal transport, respectively3,5,14,36. Hence, we expect the ratio between τmr and τer in WP2 to be at least one order of magnitude5. 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 heat-conduction 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 soup37. 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 reported30. 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.
Exploiting the well-justified conjecture of the hydrodynamic nature of transport in WP2 at low temperatures, we can now manipulate and tune the viscosity of the electron fluid to obtain further information on the magnitude of τmc. For this, the resistivity of the micro-ribbons is measured as a function of the magnetic field B at fixed T. |B| = B is set along the b-axis 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 velocities11, mediated by the exchange of electrons (Fig. 3a). The strength of the friction is given by the penetration depth λp of the electrons, which is on the order of the mean free path between collisions, λp ≈ lmc, at zero magnetic field. When B is turned on, however, this penetration depth is limited by the cyclotron radius rc = mvF/eB. Thus, in a strong magnetic field, the viscosity should tend to zero, providing a mechanism for a large negative magneto-resistivity. Solving the corresponding magnetohydrodynamic equations results in the magnetic field-dependent viscosity along the a-axis of the crystal η(B) = η0/(1 + (2τmcωc)2), where ωc = vF/rc 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 magneto-resistivity at low temperatures. When ρ(B) − ρ0 is normalized by w−2, all experimental data below 20 K collapse onto a single curve, matching the magnetic field-dependent viscosity model excellently. The average viscosity of the single traces at zero field η0 equals the viscosity η obtained from the w−2 fits above. This agreement is an important cross-check, 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 τmc extracted at individual T values are plotted in Fig. 3c. Throughout the hydrodynamic regime, we find that the momentum-conserving scattering time in the electron liquid is tied to the Planckian limit3,16,17 as τmc ~ τℏ = ℏ/(kBT). As an important cross-check, we calculated the kinematic shear viscosity11,12 from τmc as η = vF2τmc/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. τmr is extracted from combined resistivity and Hall measurements on the bulk sample28 (Supplementary Fig. 1a). With the carrier concentration n (Supplementary Fig. 1b) obtained from Hall measurements and Shubnikov-de Haas oscillations28, we calculated the mobility μ = (ρen)−1 (Supplementary Fig. 1c). Using the average effective mass of the charge carriers m* = 1.21m0 (m0 is the free-electron mass) and μ = eτmr/m*, we eventually obtained τmr as a function of the temperature (Fig. 3c). At low temperatures, τmr is three orders of magnitudes larger than τmc and thus also than the Planckian bound. Consequently, lmc < w ≪ lmr (Supplementary Fig. 13), validating the hydrodynamic treatment of transport in WP2 in this temperature regime.
While our experiments allow for the extraction of the relaxation times τmr and τmc, the microscopic origin of the hydrodynamic transport regime remains elusive. However, recent ab initio calculations of the scattering time-resolved Fermi surfaces in WP2 suggest that phonon related processes, rather than purely electron–electron processes, have a critical role in the emergence of the hydrodynamic behavior38.
A natural question that arises from this observation is whether Planckian dissipation is exclusive to the hydrodynamic regime in WP2. In the T-linear regime above 150 K, where electron–phonon Umklapp scattering yields L = L0 (because τmc = τmr), we found that, despite their fundamental difference, electron–phonon Umklapp and electron-defect processes are also tied to τℏ. This observation indicates that the dissipation in WP2 is generally tied to a characteristic timescale τℏ, regardless of the transport regime and the details of the underlying scattering mechanisms.
Our analysis suggests that WP2 behaves like a typical Fermi Liquid in some respects and different in others. For example, the existence of quasi-particles is suggested by the Shubnikov-de 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 EF = 5.6 eV (estimated from DFT calculations) and kBT at 4 K of about 1.7 × 104 (Supplementary Note 6 and Supplementary Figures 14–16 for details). However, a Fermi liquid is expected to have a T2 scattering rate when clearly below the Debye temperature (for WP2 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 liquids39.
The T-linearity 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 WP2 earned its right to participate in this universality?
In conclusion, our experiments strongly support the existence of a hydrodynamic electron fluid in WP2. The accompanying independence of τmc and τmr 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 WP2 generates entropy in a very simple and universal way in which the only relevant scale is the temperature.
WP2 single-crystal growth of the bulk sample
Crystals of WP2 were prepared by chemical vapor transport method. Starting materials were red phosphorous (Alfa-Aesar, 99.999%) and tungsten trioxide (Alfa-Aesar, 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 two-zone 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 metallic-needle crystals were characterized by X-ray diffraction.
Electrical transport measurements on the bulk sample
The electrical resistance of the bulk sample is determined within a four-probe configuration under isothermal conditions with an AC bias current of r.m.s. 3 mA at 93.0 Hz, using standard lock-in technique. As for the microbeams, the current is applied along the a-axis 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 b-axis. More details on the bulk characterization can be found in Kumar et al.28 All measurements are carried out in a temperature-variable cryostat, equipped with a superconducting ±9 T magnet (PPMS Dynacool).
Electrical transport measurements on the microbeams
All transport measurements are performed in a temperature-variable 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 quasi-four-probe configuration under isothermal conditions with AC bias currents of r.m.s. 100 μA max at 6.1 Hz, using standard lock-in technique. The quasi-four-terminal 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 nm-thick layer of low-stress silicon nitride (SiMat, Germany). Gold lines with Cr adhesion layer were patterned using optical lithography, metal evaporation and a standard lift-off process. The under-etched regions were defined using optical lithography and etching of the silicon nitride. Finally, the devices were released using wet etching and critical-point drying.
The devices consist of two MEMS platforms each carrying a four-probe resistor of 430 and 700 Ω, respectively, and two electrical leads to contact the sample, such that the WP2 sample can be measured electrically using a four-probe geometry. Each platform is connected to the wafer via four 1.2-mm-long 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 temperature-dependent 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 WP2 sample of 3 × 10−7 W K−1 at room temperature.
Sample mounting on MEMS platform
The sample was mounted using a micro-manipulator under an optical microscope. After positioning the sample, electrical contacts were deposited using electron-beam induced metal (Pt) deposition. Care was taken not to expose the WP2 sample to the electron beam at any point in time except in the contact region during metal deposition. After successful mounting, the WP2 sample bridges the gap between the two platforms, each equipped with a heater/sensor and two electrical leads to the WP2 sample. We note that in favor of large electrical contacts and thereby good thermal coupling, the pairs of electrodes at either end of the WP2 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 micro-heaters and thermometers. The resistors are read out using the four-probe technique and are fabricated on structured silicon nitride membranes to avoid thermal cross-talk via heat conduction through the substrate. The measurements are performed in vacuum to avoid thermal cross-talk 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 phase-sensitive 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 2nd-order polynomial fit, for the lower temperatures we use a 5th-order 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 temperatures41. 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 WP2 sample was zero and the leads floating. Both sides were probed using phase-sensitive 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 Peff is calculated. From the measured resistance values RHeater1 and RHeater2, the temperature rise, ΔT1 and ΔT2 of both heater/sensors is calculated using the temperature calibration (Supplementary Fig. 9). The thermal conductance of the MEMS device and the WP2 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).
The data that support the findings of this study are available from the corresponding author upon request.
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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 copy-editing. Fabian Menges gratefully acknowledges the support from the Society in Science—The Branco Weiss Fellowship and a Swiss National Science Foundation postdoctoral fellowship.