Experimental observation of hydrodynamic-like behavior in 3D topological semimetal ZrTe 5

Hydrodynamic fluidity in condensed matter physics has been experimentally demonstrated only in a limited number of compounds because of the stringent conditions that must be satisfied. Herein, we demonstrate the existence of hydrodynamic-like properties driven by the collective excitation of the Dirac fluid in the three-dimensional topological semimetal ZrTe 5 . By measuring the electrical and thermal properties in a wide temperature range, we find a regime satisfying phononic hydrodynamic-like characteristics with two representative experimental evidences: a faster evolution of the thermal conductivity than in the ballistic regime and the existence of a local maximum of the effective mean free path. In contrast to phononic hydrodynamics, the Wiedemann-Franz law is violated by about a factor of 100. Moreover, phonon-dragged anomalies are observed, which serve as a signature of the Dirac fluidity in this system.


Introduction
Phonon-dominant heat conduction is described by Fourier's law, in which phonons scatter from other phonons, impurities, and interfaces [1][2][3] . This process takes place through the momentum-relaxing process known as Umklapp scattering (hereafter U-scattering). During this process, heat currents are dissipated and the crystal momentum is not conserved [1][2][3] . On the other hand, Fourier's law is no longer valid when the temperature T is sufficiently low in which the crystal momentum is conserved due to dominant Normal scattering (hereafter N-scattering) [4][5][6] . These two types of scattering mechanisms are known for a diffusive and a ballistic regime, respectively, and have been confirmed in many materials for over 50 years [4][5][6] .
Meanwhile, unusual transport phenomena between the ballistic and the diffusive regimes have recently been reported on low-dimensional materials [7][8][9][10][11][12] , in which thermal conductivity evolves more rapidly than in the ballistic regime 8,12 . Such an intermediate region is called hydrodynamic because it is analogous to macroscopic transport phenomena in water fluids 13,14 .
Two different characteristics are known in the phonon hydrodynamic system: the Poiseuille flow and the second-sound [14][15][16] . The former is characterized by a steady-state phonon flow where thermal resistance diffuses due to the boundary scattering combined with N-scattering.
In comparison, the latter is the wave-propagation of a T gradient without significant attenuation.
The phonon Poiseuille flow could be confirmed by the T and sample width dependencies of 8,9,12 .
Despite the fascination of hydrodynamics in solid state systems, it has been observed only in a narrow T range and required remarkably low-T with abundant N-scattering as well as a suitable sample size. For example, the reported T range for phonon Poiseuille flow is 0.5 -1.0 K in suspended graphene 7,16 . This is because U-scattering overwhelms N-scattering in almost every T range except for significantly low temperatures. In this T range, electrons rather than phonons are the primary heat carriers. For these reasons, phonon-hydrodynamic behavior has been experimentally demonstrated in less than a handful of compounds, such as black P 9 , SrTiO3 17 , and thin-graphite 8 . Therefore, the search for new materials in which hydrodynamics contributed by phonons or other collective excitations can be observed is of great interest to the condensed matter community.
In this study, we performed thermal and electrical transport experiments for ZrTe5 single crystals to investigate the hydrodynamic property. In fact, the ZrTe5 study is initiated decades ago due to its considerable thermoelectric performance and its resistivity anomaly. Recently, it has gained renewed attention due to non-trivial topological phenomena such as a 3D quantum Hall effect 18 , a quantum spin Hall effect on a monolayer 19 , and a chiral magnetic effect 20 .
Moreover, bulk ZrTe5 has been reported to sit at the boundary between a weak-and strongtopological insulator so that external perturbation easily influences its topology [21][22][23] . Herein, Dirac-Fluid (DF) hydrodynamics could be added to a new list of exotic physical properties in ZrTe5. We present two experimental signatures for phonon hydrodynamics: a faster evolution than in the ballistic regime and a local maximum of the effective mean-free-path (MFP). In contrast to conventional phonon hydrodynamics, we additionally observe the phonon-dragged anomalies near the onset of a hydrodynamic regime with an exceptionally strongly violated Lorentz ratio. These observations have important implications for ongoing research on the various possible types of quasiparticle hydrodynamics, especially in a threedimensional topological semimetal material.

Results
Since our primary interest is the hydrodynamic-like system, it is essential to verify that the samples used in this study are clean enough. Otherwise, it is hard to observe even at low T due to the lack of N-scattering. Figure 1a presents the electrical resistivity in a temperature range from 0.3 to 300 K at zero magnetic field. A characteristic peak around ~90 K (Tp) is seen in both samples (Sample #1: 84 K and Sample #2: 89 K), indicating that the Fermi level moves from valence band to conduction band as T decreases 24 . In Fig. 1b and c, we depict the Tdependent dominant charge carrier density n and mobility , respectively. Both quantities are extracted from a two-band model fit (T > 40 K) and the linear fit near the zero-field data of the Hall measurement (T < 40 K). There is an obvious sign change of n with the exceptionally low density (~10 16 to ~10 17 cm -3 ) and the gradual increase of with ultrahigh values (~10 5 to ~10 6 cm 2 V -1 s -1 ) at low T. All values are comparable to our previous observation 18 , which guarantees the excellent crystalline quality of the samples and allows us to examine a hydrodynamic-like regime.
In the following, let us examine the transport evidence for the hydrodynamics in our ZrTe5.
The first clue is to find a that evolves faster than T 3 . To test this, we plot the T dependence In the high T regime (between about 30 K to 300 K), it is governed by the perfect 1/T dependence in all the samples (see supplementary Fig. S2), implying that the U-scattering is the most prominent process in this range. After passing through the maximum, it starts to decrease, indicating the N-scattering process begins to dominate. For Sample #2, the slope of gradually increases towards low T and exceeds a T 5 dependence below ~2 K (dashed line in Fig. 2). Sample #1 behaves similarly with a slightly slower increase in slope. It should be mentioned that shows an irregular behavior at sufficiently low T, where it must converge to , since the thermal energy at low T is mainly transferred from the charge carriers.
However, we see no convergence up to the experimental low T-limit. Below 1 K, is still smaller than by a factor of a few 100. We will discuss more details later in the context of the large-violation of the WF-law.
Near the hydrodynamic-like regime, we find additional anomalies. In Fig. 3a and b, we present the enlarged curves of the electrical resistivity and versus T for Sample #2. At first glance, exhibits a typical T 2 Fermi-liquid (FL) behavior in the 6 -30 K range. Upon further cooling, begins to deviate from the FL behavior below ~6 K (Ta), with a pronounced downward curvature, but remaining at a finite value. A power-law scaling (i.e., = 0 + 2 ) between 6 and 30 K yields a residual resistivity 0 ≈ 0.154 Ω and the pre-factor ≈ 0.098 Ω −2 . In the versus T data (Fig. 3b), a step-like anomaly occurs at a temperature that is in perfect agreement with Ta In fact, such non-FL behavior is only seen in Sample #2 as shown in Fig. 3a and is likely triggered by slight differences in initial crystal growth conditions.
ZrTe5 is a highly sensitive material to growth conditions and slight differences may somewhat alter its electronic and thermal properties 18,22,26 .
On the other hand, a common anomaly for our samples is found in the T-dependent / 3 and Lorenz ratio (L/L0), as shown in Fig. 3c and d. These anomalies occur near the onset of a hydrodynamic-like regime. It is reasonable to conclude that such a peaked anomaly could originate from the phonon-drag effect since no sign is seen in the vs T data (solid lines in Fig. 2), representing the purely electronic contribution. In Ref 27 , a pronounced phonon-drag peak in the low T thermopower was also reported in the Dirac semimetal PtSn4, where the peak was observed exclusively in the thermal transport data, similar to the present study.
Next, we turn to the effective quasiparticle MFP, another critical signature of phonon hydrodynamics. In most previous studies, the magnitude of the MFP has been estimated by the simple relation ℎ = 1 3 ℎ ℎ , where ℎ , , and ℎ denote the phonon specific heat, sound velocity, and phonon MFP, respectively 17 . Instead of taking a conventional route, we attempt to measure the thermal Hall effect, as this could be a direct probe to study quasiparticle dynamics, but it has rarely been performed in topological materials due to the difficulty of obtaining high-quality data. where Δ and P denote the T gradient between two points along the transverse direction and the thermopower, respectively) in a narrow B-field range from -1.0 to 1.0 T. In the main text, only the case of Sample #1 is presented. The value of is close to zero regardless of the measured temperatures. However, in a weak field region (|B| < 0.1 T), an asymmetric thermal Hall feature is found, which becomes stronger as T decreases.
To evaluate the degree of heat deviation, the thermal Hall angle tan (= ) is plotted as a function of B-fields at various temperatures (Fig. 4b).
The trend is not different from vs B. It shows a significant deviation when B-fields are applied and is abruptly suppressed and eventually disappears in the region of higher B-fields. In Fig. 4c, we reveal the zero-field-limit Whereas L/L 0 in our ZrTe 5 deviates from 1 over an entire temperature range and has a large value over a few 100 (see Fig. 3d), the data presented in PtSn4 showed a small value of L/L0 (L/L0 < 1) in a hydrodynamic regime. It has been argued that this is indicative of significant inelastic electron-phonon scattering 11 . In this context, we also rule out the scenario that the electron-phonon interaction induces hydrodynamics in ZrTe5.

DF is our last choice. This type of strange metal was introduced by Crossno et al to describe
the hydrodynamic behavior at the charge neutrality point 11 . As mentioned earlier, the ZrTe5 single crystals used exhibit ultra-high mobility due to their exceptionally high-purity, and their bipolar charge carrier types are more or less compensated at low T. These prerequisites for the realization of the DF scenario are perfectly met 11 . Moreover, the DF is expected to show an enhancement of and the largely violated WF law on the order of hundreds of L/L0, due to the depairing of charge and heat currents in the hydrodynamic regimes, which is in good agreement with our observations. Finally, the T-dependent Dirac-particle thermal conductivity also supports this argument. Since the purely phononic thermal conductivity ℎ is lesssensitive to the magnetic field and the electronic contribution is extremely small in our case, as shown in Fig. 5(a), we can extract the approximation of from (0 T) by subtracting the magneto-thermal conductivity at a sufficiently high B-field. If we then define Δ = (0T) − ( ≠ 0T), it is proportional to , which is presented in Fig. 5(b). When compared with , the Δ is enormously different in the hydrodynamic regime. This is in perfect agreement with the experimental observation of Crossno et al 11 .

Summary
In summary, the main effort of hydrodynamic studies to date has been to find the significant features where either electrons or phonons provide the primary scattering. However, all transport regimesballistic, hydrodynamic and diffusivecan coexist and be coupled, making it more complicated and difficult to differentiate purely quasiparticle hydrodynamic phenomena.
Using ultrahigh-purity single crystals of ZrTe5, we find the transport signature of the hydrodynamic-like behavior, which we attribute to Dirac-quasiparticle excitations. With a significantly violated Lorenz ratio by more than a factor of 100 and a non-trival transport

Author contribution
This work was initiated by L.Z.; C.w.C. carried out the electrical and thermal transport measurements with help of P.W., F.T., M.H. and R.L.; the single crystal samples were provided by G.G.; C.w.C. and L.Z. analyzed the data with the help of S.P., M.H. and R.L.; the manuscript was prepared by C.w.C. and L.Z. with the help of S.P. and R.L., and all authors were involved in discussions and contributed to the manuscript.

Competing financial interests
The authors declare no competing financial interests.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Sample Preparation
Ultra-high quality single crystals of ZrTe5 were grown by the tellurium flux method. Thanks to the relatively large size of the single acicular crystals (l x w x t, Sample #1: 3.20 x 0.30 x 0.08 mm 3 , Sample #2: 2.90 x 0.30 x 0.21 mm 3 ), we were able to perform the electrical and thermal transport experiments on the same bulk samples. Details of the sample growth and structural properties can be found elsewhere 18,20,33 . In the main text, we defined the longest (shortest) dimension is along the a-axis (b-axis), corresponding to the stacking layer direction.

Experimental techniques
For the electrical transport measurement, we performed it by the standard Hall bar method, using an alternating current with an amplitude of 0.01-0.1 mA amplitude and a frequency of 10-20 Hz. The magnetic field was applied in the perpendicular direction to the ac-plane. To measure the thermal transport of such an acicular ZrTe5 crystal, we used a well-known steadystate method with one-heater and three-thermometers. One end of a ~4.0 mm long sample was attached to a copper heat sink, while a small ~100 Ohm resistor and three well-calibrated Cernox thermometers were connected with Ag (100 um) and Pt/W wires (25 um), respectively (see supplementary Fig. S1). Using three lock-in amplifiers and three thermometer chips, we were able to obtain the longitudinal and transverse thermal gradients simultaneously. To eliminate spurious longitudinal (or transverse) components, we measured the magneto-thermal conductivity with opposite field directions and averaged them. Since the sensitivity of the thermometers used in this experiment becomes insensitive towards higher temperatures, we switched to a thermocouple method to record the thermal gradient in the high T regime (T > ~20 K). In the overlapping range (about 10-20 K), we confirmed the consistent results within the error bar; an example for Sample #2 is presented in Supplementary Fig. S2.  T-dependent slope of tan of B in the zero magnetic field limit for two samples. Only the x-axis is presented with a logarithmic scale. In general, this value is proportional to the mean-free-path of the quasiparticles. The inset of (c) shows an initial slope of the electronic Hall angle identical to the thermal Hall angle. The vertical arrows denote the local maxima. is calculated by the WF-law based on electrical resistivity data. ∆ is strongly deviated in a hydrodynamic regime due to the Dirac-Fluid.

Two different method for the thermal conductivity measurement:
thermometers vs thermocouples Figure S2. Thermal conductivity as a function of T for Sample #2. Thermometers were used because of their higher sensitivity at low temperatures due to their semiconducting nature with a large negative slope of their resistance (magenta). On the other hand, thermocouples are more sensitive at higher temperatures (black). In our experiments, we observe the overlapping T range from ~10 to ~20 K. The dashed curve (red) denotes the 1/T dependence curve. Figure S3. T dependence of electrical resistivity at various applied magnetic fields. Data at 0 T are measured during a T sweep, while field data are extracted from magnetoresistance (MR) data. An anomaly at Ta is seen at least up to 8 T, which greatly exceeds the reported critical field of pressure-induced superconductivity in ZrTe5.