Introduction

The well-known Wiedemann–Franz (WF) law is one of the fundamental properties of a Fermi liquid. It relates the electronic thermal conductivity κe, the electrical conductivity σ, and the absolute temperature T through a simple formula, κe/σT = L, where L is called the Lorenz ratio and is given by the Sommerfeld value L0 = 2.44 × 10−8 W Ω K−2. This formula relies on the single-particle description of the transport properties and a purely elastic electron scattering. Although the WF law is usually not obeyed at T ≠ 0 because of the importance of inelastic scattering, it must be obeyed at T = 0 in the Fermi liquid where the electrons are elastically scattered by static disorders. Thus, it had been believed that the validity (invalidity) of WF law at zero-temperature limit is the smoking gun of Fermi liquid (non-Fermi liquid). Out of the frame of Fermi liquid, the WF law is commonly not obeyed1, as observed in strongly correlated electron systems2,3. For example, in some heavy-fermion materials, a violation of the WF law even at zero-temperature limit was found near a magnetic quantum critical point4, although some later studies indicated very high requirements for the experimental testing of the WF law in such case5. Furthermore, the violation of WF law has recently been reported in some two-dimensional materials or semimetals, like graphene, WP2, and Sb, etc6,7,8,9,10,11,12,13. Interestingly, these were found to be due to the peculiar hydrodynamic electron transport and do not imply any non-Fermi liquid behavior.

As the basis of Landau’s Fermi-liquid theory14, quasiparticles can transport not only charge but also entropy, and the WF law is held when quasiparticles keep their energy during collisions. Due to the weakness of interactions among low-energy excitations, both currents and momenta relax slowly in the Fermi liquid. Whereas the strongly interacting non-Fermi liquids have fundamentally different kinematics of conductivities with only momenta relaxing slowly15. Thus, there are enough reasons to expect the violation of the WF law. Additional entropy carriers like gapless neutral collective degrees of freedom can only transport heat16, and there may be inelastic scattering between charged and neutral degrees of freedom that affect the transport of heat and charge differently. In the Fermi-liquid framework, the inapplicability of the WF law characterized as the Lorenz ratio L deviating from L0 can also happen, but it does not necessarily attribute to the breakdown of the underlying quasiparticle picture and indicate a failure of the Fermi-liquid description for the relevant physics. In the 1960s, Gurzhi pointed out that electron hydrodynamic transport, as opposed to conventional single-particle diffusive transport, is feasible in solids17,18. In that case, the WF law is expected to deviate with the L number lower (unipolar hydrodynamics) or higher (ambipolar hydrodynamics) than L012,19. However, it is worth noting that all the deviations induced by electron hydrodynamic occur in moderate temperature regime7,8,9,12,13,20,21,22,23. It has been rarely explored, and it is not clear whether the WF law is still obeyed at zero-temperature limit in these compensated semimetals. Some theoretical results for the electron hydrodynamics actually suggested that the L number finally recovers to L0 at the zero-temperature limit12. This indicates that the nature of these materials is essentially Fermi liquid. Strictly speaking, the WF law is valid for Fermi liquid system only at zero-temperature limit, and studying the validity of the WF law at finite temperature is actually not conclusive for the nature of electron systems6,7,8,9,10,12,13,19,20,21,23,24. The experimental study of the validity of the WF law at zero-temperature limit for the compensated semimetals is still called for.

TaAs2, NbAs2, and NdSb have been found to be topological compensated semimetals. Note that NdSb is a magnetic material with an antiferromagnetic ground state below TN = 15 K while the others are nonmagnetic. All these topological semimetals exhibit extremely large magnetoresistance (MR) due to the compensated carriers and high mobility25,26,27,28,29,30,31,32,33,34. While the magnetotransport properties of these materials have been investigated in detail, the heat transport of them remains to be studied, which is very useful for characterizing the nature of the electron system. In a word, it is a fundamental question to test the validity of WF law at the zero-temperature limit for these materials. In this work, we study the charge and heat transport of TaAs2, NbAs2, and NdSb single crystals at very low temperatures. The strong suppression of the thermal conductivity in magnetic fields, as well as the large positive MR, reveals that the heat carriers are mainly electrons and holes at low temperatures and zero fields. One striking finding is that the thermal conductivity exhibits peculiar T4 behavior at ultralow temperatures down to several tens of millikelvin, while the resistivity is T-independent, indicating a strong violation of WF law. In addition, the thermal conductivity displays giant quantum oscillations, which are antiphase with that derived from WF law.

Results

Figure 1 a–c presents the temperature-dependent resistivity of TaAs2, NbAs2, and NdSb, respectively. The charge transport of these materials has been previously studied at low temperatures down to 2 K. Here, the resistivity measurements of the single crystals of these materials have been extended to lower temperatures (300 mK). In a zero magnetic field, the resistivity ρ(T) of TaAs2 and NbAs2 decreases monotonically with lowering temperature until it reaches a finite value (residual resistivity) at T < 2 K, which is characterized as a metallic behavior. The ρ(T) curve of NdSb exhibits a sharp peak at 15 K, which is attributed to the antiferromagnetic transition. At low temperatures, the resistivity of the three clearly shows the T-independent behavior. With applying magnetic fields up to 14 T, the resistivity is strongly enhanced, as shown in Supplementary Fig. 1.

Fig. 1: Electrical and thermal transport of TaAs2, NbAs2, and NdSb.
figure 1

a–c Temperature dependence of the electrical resistivity of TaAs2, NbAs2, and NdSb under zero field, respectively. The insets show the electrical resistivity below 10 K plotted with a logarithmic scale. The electric current (J) is parallel to the b-axis (TaAs2 and NbAs2) or a-axis (NdSb). d–f Ultralow-temperature thermal conductivity of TaAs2, NbAs2, and NdSb under different magnetic fields, respectively. The heat current is parallel to the b-axis (TaAs2 and NbAs2) or a-axis (NdSb). The magnetic field is perpendicular to the heat current. The solid red lines indicate that the κ follows a T4 temperature dependence at the low-temperature limit.

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Figure 1 d–f shows the thermal conductivity κ of TaAs2, NbAs2, and NdSb measured at very low temperatures and in magnetic fields up to 14 T. The most remarkable phenomenon is that the κ in zero field follows a T4 behavior at very low temperatures (<200 mK). This is distinctly different from the usual T3 and T dependence of κ for phonons and electrons at ultralow temperatures, respectively. By applying magnetic fields, the magnitude of κ is strongly suppressed, which has good correspondence to the large positive MR effect. This clearly indicates that the main heat carriers are electrons and holes. At the same time, the temperature dependence of κ at the lowest temperature becomes weaker with increasing field. In the highest field of 14 T, the κ follows a roughly T2.5 behavior at very low temperatures (<200 mK), which means that the heat transport of charges is significantly suppressed and phonon heat transport remains. For TaAs2, with the analysis of specific heat, the phononic thermal conductivity at very low temperatures can be estimated to be κph = 0.219 T3 W K−1 m−1, which is indeed very small compared to the total κ (Supplementary Figs. 2 and 3). Therefore, the electronic thermal conductivity dominates at ultralow temperatures, and its T4-dependence and the low-temperature T-independent behavior of resistivity clearly indicate a strong violation of WF law. Considering the high quality of the crystals used here (residual resistivity ratio RRR = 355 for TaAs2), we also measured the κ of two other TaAs2 samples with lower RRR values. As shown in Supplementary Fig. 4, the quality of the crystal has a significant impact on the behavior of κ. The ultralow-temperature κ of samples with lower RRR values has weaker temperature dependence, which is close to T3 at the lowest temperatures. This is also unique because the electronic thermal conductivity also dominates at low temperatures in these samples (Supplementary Fig. 4).

The L/L0 ratio has been calculated to describe the violation of WF law. Here, L is the experimentally measured Lorenz number, and L0 is the Sommerfeld value. As shown in Fig. 2a–c, the Lorenz numbers of TaAs2, NbAs2, and NdSb all exhibit a drastic decrease at ultralow temperatures. The downward deviation from WF law remains at the zero-temperature limit, where the Lorenz number L is much smaller than the Sommerfeld value L0. In fact, we also examined the L/L0 ratio in another material, NbSb2, and observed a similar deviation from the WF law (Supplementary Figs. 5 and 6). Obviously, the current results reveal the non-Fermi liquid ground state of these materials. Recently, the study of nuclear quadrupole resonance on TaSb2 also indicates that the magnetic excitations cannot be captured by Fermi liquid theory35. The non-Fermi liquid ground state seems to be a commonality in these compensated topological semimetals.

Fig. 2: The departure from the WF law at ultralow-temperature.
figure 2

a–c Temperature dependent L/L0 ratio of TaAs2, NbAs2 and NdSb under zero field, respectively. L is the calculated Lorenz number based on experimental measurements, and L0 is the Sommerfeld value.

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We further explored the magnetic field dependence of the resistivity and thermal conductivity of TaAs2, NbAs2, and NdSb at ultralow temperatures. Figure 3a–c shows the ρ(B) curves of the three materials at two selected temperatures below 1 K. The ρ(B) curves follow quadratic behavior and remain unsaturated at high fields, indicating the compensation of electrons and holes. Clear SdH oscillation can be observed at high fields, similar to that observed at temperatures above 2 K. Interestingly, the thermal conductivity also exhibits significant quantum oscillations. Figure 3d–f presents the κ(B) curves of the three materials at several selected temperatures below 1 K. With the increase of magnetic field, the κ is strongly suppressed by nearly two orders of magnitude. Apparently, the large negative magneto-thermal conductivity has a direct correlation to the extremely large positive MR, and the phonon thermal conductivity should be independent of the field. From this result, it is further confirmed that the phonon contribution to the total thermal conductivity in zero fields is likely less than several percent. Therefore, the T4 temperature dependence of κ in zero field is a pure electron transport behavior. At higher magnetic fields, the quantum oscillations of thermal conductivity appear in a nearly constant background. The oscillation frequencies of thermal conductivity correspond well to that of electrical resistivity (Supplementary Figs. 7–10).

Fig. 3: Quantum oscillations of electrical resistivity and thermal conductivity in TaAs2, NbAs2, and NdSb at ultralow temperature.
figure 3

a–c Magnetic field-dependent resistivity of TaAs2, NbAs2 and NdSb, respectively. The temperature is fixed at 380 mK and 850 mK. The 850 mK data are shifted upward to distinguish. d–f Magnetic field-dependent normalized thermal conductivity of TaAs2, NbAs2, and NdSb at several selected temperatures, respectively.

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To explore the potential origin of the giant quantum oscillations of κ(B), we compared the experimental κ(B) with the κWF(B) calculated using WF law in Fig. 4a–c. It can be seen that there are significant differences between them. The quantum oscillations observed in the experiment κ(B) are much larger than those derived from the WF law κWF(B). After removing the smooth background, the oscillatory components can be extracted. As shown in Fig. 4d–f, the oscillation amplitude of κWF(B) can only be compared to that of κ(B) after multiplying by 10 or 100. In addition, it should be noted that the oscillation phase of κ(B) is opposite to that of κWF(B). That is, when σ(B) shows a maximum value due to the intersection of Fermi level and Landau level (causing a maximum in the density of states), κ(B) shows a minimum value. A similar antiphase feature has also been observed in several other materials, such as TaAs, NbP, and Sb36,37,38. These observations are clearly contrary to what is expected in the case of electron diffusion transport. A suitable explanation is that the giant thermal quantum oscillations are the result of electron-phonon coupling, through which the phononic thermal conductivity can be influenced by the Landau-quantized electronic density of states (DOS). Since the strength of electron-phonon interaction is proportional to the quantized DOS, the enhancement of electronic DOS will lower the phononic thermal conductivity. Recently, chiral zero sound has been proposed to explain the giant thermal quantum oscillations in Weyl semimetals36,39. Chiral zero sound is an exotic collective mode of Weyl fermions, which propagates along the magnetic field and induces strong thermal quantum oscillations. Although the chiral zero sound is more pronounced in the parallel field configuration, it can also affect thermal conductivity to some extent in the perpendicular field configuration36,39. Considering the existence of Weyl fermions induced by magnetic field in TaAs2, NbAs2, and NdSb31,40, the chiral zero sound may also contribute to the observed giant thermal quantum oscillations.

Fig. 4: Comparison between experimental thermal conductivity κ and calculated thermal conductivity κWF.
figure 4

a–c The normalized κ and κWF of TaAs2, NbAs2, and NdSb at 380 mK. κWF is the calculated thermal conductivity using the WF law. d–f The extracted oscillation amplitude of κ and κWF plotted as a function of 1/B. The oscillation amplitude of κWF has been multiplied by a factor (10 or 100) to compare with that of κ.

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Discussion

The most important and interesting discovery is the T4 behavior of electronic thermal conductivity at ultra-low temperatures and the resulting unusual violation of the WF law. Qualitatively, the strong downward deviation of WF law in these topological semimetals at zero-temperature limit points to a breakdown of the Fermi liquid. Though the violation of the WF law in some particular cases for Fermi-liquid systems was attributed to the electron hydrodynamic effect, the present work displays two exotic results that cannot be explained by this theory. First, some theoretical results for the electron hydrodynamics indicated that while the strong violation of WF law occurs at finite temperature, the L number finally recovers to L0 at the zero-temperature limit12,19. This is in contrast to the experimental result of the present work. Second, till now there are no theoretical results on the electron hydrodynamics predicting the unusual T4 dependence of electronic thermal conductivity at very low temperatures. Therefore, it is likely that the present experimental results point to some other unknown transport mechanism within the non-Fermi liquid framework.

In fact, the non-Fermi liquid behaviors have been proposed for several topological semimetals such as the Luttinger semimetal, nodal line semimetal, and double Weyl semimetals41,42,43,44. For the conventional Fermi liquid metals, the long-range Coulomb interaction is screened down to a short-range interaction such that the Landau quasiparticle description preserves at low temperatures. In contrast, in topological semimetals, the suppressed density of states near the Fermi level can only partially screen the long-range Coulomb interaction. The residual long-range Coulomb interaction is still singular and hence converts the system into non-Fermi liquids. Along this line of reasoning, the controlled analysis of the interaction effects on the topological semimetals was developed, where the physical properties of the relevant non-Fermi liquids were obtained45,46. For the current non-Fermi liquid behaviors in our compensated topological semimetal systems, we expect that similar physics of the partially screened Coulomb interaction should also occur. The presence of the small electron and hole pockets, however, may complicate the theoretical understanding. More theoretical exploration for the violation of the WF law and the nature of these topological semimetals with partially screened Coulomb interaction is called for.

In summary, the thermal conductivity and resistivity of topological compensated semimetals TaAs2, NbAs2, and NdSb have been studied at ultralow temperatures. The thermal conductivity of these materials presents an unusual T4 dependence, revealing the strong violation of WF law and the non-Fermi liquid ground state. The Lorenz ratio is much smaller than the Sommerfeld value even when approaching the zero-temperature limit. At ultralow temperatures and high magnetic fields, giant quantum oscillations of thermal conductivity have been observed. The thermal quantum oscillations are antiphase with the oscillating electronic DOS, and the oscillation amplitudes are much larger than that derived from WF law. The electron-phonon coupling and chiral zero sound may contribute to the observed giant thermal quantum oscillations.

Methods

Sample preparation and characterizations

The single crystals of TaAs2, NbAs2 and NbSb2 were grown by chemical vapor transport method25, and the single crystals of NdSb were grown from Sn flux32. By using the X-ray Laue photograph, all these crystals were cut precisely along the crystallographic axes with typical dimensions of 1.7 × 0.5  × 0.1 mm3, where the longest dimension is along the b-axis (TaAs2, NbAs2, and NbSb2) or a-axis (NdSb). The electric and heat currents were applied along the longest dimension while the magnetic fields were applied perpendicular to the longest dimension.

Thermal conductivity measurements

Thermal conductivity was measured by using a conventional “one heater, two thermometers” technique47,48,49. Data at 50 mK - 1 K regime were measured in a 3He/4He dilution refrigerator. Data at 300 mK - 30 K regime were measured similarly in a 3He refrigerator. The RuO2 chip resistors were used as heaters and thermometers and were connected to the gold contacts by using gold wires and silver epoxy. The RuO2 thermometers were pre-calibrated for use in magnetic fields. Data at 8–100 K regime were measured in a 4He pulse-tube refrigerator and two Cernox sensors were used for thermometers. The leads for heater and resistor thermometers on samples are long and thin NbTi superconducting wires (0.01 mm diameter). The typical error of thermal conductivity data is smaller than 5%, while the relative error in field dependence of data at a certain temperature is much smaller.

Electrical resistivity measurements

Resistivity was measured by using the standard four-probe method. Note that these two transport measurements for each material were done on the same piece of a single crystal with the same contacts for direct checking the validity of WF law.

Specific heat measurements

The specific heat was measured with the relaxation method using a commercial physical property measurement system (PPMS, Quantum Design) equipped with a 3He insert.