Giant Magnetic Quantum Oscillations and Chiral Anomaly in the Thermal Conductivity of a Weyl Semimetal

Giant quantum oscillations of magneto-thermal conductivity amounting to two orders of magnitude of the estimation based on the Wiedemann-Franz law have been observed in the prototypical Weyl semimetal TaAs. The characteristic oscillation frequency ($F$ $\approx$ 7 T) agrees well with that confirmed for a small hole-type Fermi pocket enclosing a Weyl node. A comparative analysis of various potential scenarios suggests a significant electron-phonon coupling that strongly modulates the phonon mean free path through Landau quantization of the electronic density of states to be at the heart. Resembling the chiral-anomaly induced positive magneto-electrical conductivity, an increase of the thermal conductivity in parallel magnetic field has also been observed. Our findings pose the question whether these are characteristic also for other recently discovered topological electronic materials, calling for more intensive investigations along this line.

In comparison to the electrical transport that reveals an abundance of novel features like chiral anomaly [1][2][3][4], thermal transport in particular thermal conductivity of Weyl semimetals, remains largely unexplored. This is partially ascribable to the technical difficulties in precise thermal management at low temperatures. Also, such work is seemingly insignificant because, for most electrical conductors, the Wiedemann-Franz (WF) law has already provided an easy approach to the electronic thermal conductivity from the electrical conductivity σ in the elastic scattering limit of Fermi liquid picture. At least in the zero-temperature limit, the WF law is theoretically expected to be fulfilled for various topological materials [5][6][7]. Recent experimental results, however, suggest its violation by Dirac/Weyl fermions in particular circumstances, e.g., on the surface of a topological insulator [8], in the hydrodynamic transport regime [9], and in the presence of magnetic field [10], attracting renewed attention.
On the other hand, magnetic quantum oscillations (MQOs) of thermal transport like thermopower and Nernst effect have turned out to be useful complements to the conventional techniques like the Shubnikov-de Haas (SdH) effect of σ(B) [11][12][13][14]. However, MQOs in the isothermal magneto-thermal conductivity κ(B) have been rarely reported thus far. For, an estimate of the oscillation amplitude based on both the SdH effect and the WF law (denoted as κ e,WF (B) in Fig. 1b) is generally too small to be experimentally detected, as is the case of TaAs (10 −3 −10 −1 W/Km), cf. Fig. 1 and Supplementary Information (SI). The few reported examples of MQOs in κ(B) include GaAs/AlGaAs heterostructures containing twodimensional electron system (2DES) [15] and the Weyl semimetal NbP [16]. An enhanced electron-phonon (e-p) interaction that modulates the lattice thermal conductance, or a large electronic bipolar-diffusion effect, were discussed as possible origin, respectively.
In this Letter, we report on an experimental observation of surprisingly large MQOs in magneto-thermal conductivity of the prototypical Weyl semimetal TaAs. The oscillation amplitude is the largest so far found for this quantity, exceeding the expectation of the WF law [κ e,WF (B)] by two orders of magnitude. A comparative analysis involving different possible scenarios indicates that the lattice thermal conductivity, through significant e-p coupling, plays a dominant role in the observed MQOs. In addition, an unprecedented enhancement of κ(B) was observed only when the field was aligned parallel to heat current (B dT ), reminiscent of the chiral-anomaly induced, positive magneto-electrical conductivity.
Our results provide a new avenue for studying quantum transport of chiral fermions. We employed two TaAs samples cut from one single crystal grown by chemical vapor transport [17]. They were rectangular shaped, with the long axis either along the c axis chromel-AuFe 0.07% thermocouple for detecting the temperature difference dT , see SI. Prior to our measurements, the thermocouple had been carefully calibrated in magnetic fields by using two Cernox CX1050 thermometers; a similar field dependence as previously reported [18] has been observed, cf. Fig. S2 in SI. The employment of a thin thermocouple ensures a rapid thermal relaxation when scanning the field, which is of uttermost importance to detect the MQOs in κ(B). This can be confirmed from the recorded dT (B) curve at T = 2 K, cf. Fig. S3, which is almost field-symmetric with reproducible MQOs. comparable oscillation magnitude (dκ/κ 0T ≈ 30%), with however a more complicated FFT spectrum compared to that of TaAs [16].
The common description of κ for nonmagnetic conductors considers an electronic (κ e ) and a lattice contribution (κ l ). For semimetals and semiconductors a bipolar contribution κ bi has to be taken into account, too. This is due to thermally excited electron-hole pairs which diffuse to the cold end where they recombine, releasing their excitation energies [22].
For a Fermi liquid, κ e can be obtained from σ via the WF law, with the Sommerfeld value of Lorenz number L 0 ≡ π 2 3 ( k B 2 ) 2 = 2.44 × 10 −8 W·Ω·K −2 . The WF law is valid for a wide variety of materials, as long as the dominating charge scattering process is elastic. This is often realized at low enough temperatures under dominating impurity scattering. On the other hand, a potential bipolar term κ bi will increase with T due to the enhanced thermal excitation of electron-hole pairs, see ref. [23]. to be violated upon applying field due to the linear band dispersions [25]. However, the resulting oscillation amplitude of κ e (B) is very small, i.e., less than 1% of the zero-field value up to B = 6 T, in striking contrast to that of TaAs, which amounts to more than 300%, cf. Figs. 1-2.
An alternative explanation for the giant MQOs in κ(B) relies on the bipolar-diffusion term κ bi (B). This effect has been intensively investigated for graphene which fulfills all the requirements for a significant κ bi (B), e.g., a zero band gap and charge neutrality [23]. There, bipolar-diffusion enhances L by a factor 2−4 at room temperature; this factor diminishes with lowering T due to decreasing thermal excitations. The enhancement was experimentally found to be only 35% at subkelvin region [26]. For TaAs, the estimated κ e,WF (T ) amounts to 1/4 (TAc) and 70% (TAa) of the measured κ(T ) at room temperature [27]. There, the approximate T −1 profile of κ(T ) indicates a sizable contribution due to the lattice contribution κ l . An enhancement of L due to the bipolar diffusion, if any, should be no larger than 4 fold and decreases upon cooling. Therefore, this effect appears to be negligible in current discussion on TaAs. In addition, the dominant one-frequency (F = 7 T) oscillation also argues against the bipolar scenario for the oscillating κ(B) in this material.
Given that neither κ e (B) nor κ bi (B) can provide a satisfactory explanation for the observed MQOs in κ(B) of TaAs, whether the Landau quantized electronic DOSs can trigger a quantum oscillating phonon contribution κ l (B) appears to be interesting. Indeed, the oscillating κ(B) observed in GaAs/AlGaAs heterostructures was explained by considering the e-p coupling [15]. There, the heat-carrying phonons couple to the Landau-quantized 2D-DOSs confined to the interface and the phonon mean free path is consequently modulated 8 by quantizing field. The non-monotonic T -dependence of the oscillation amplitude shown in Fig. 3 strongly supports such a scenario for TaAs: The MQOs tend to disappear at absolute zero because there κ l vanishes. We note the great difference of the oscillations between the two cases. In the 2D heterostructures, dT oscillates within only 2% of its zero-field value, which is two orders of magnitude smaller than that of TaAs.
As acoustic phonons dominate κ(T ) of TaAs [28], our findings suggest a significant interaction between Weyl fermions and acoustic phonons. This is likely for TaAs with a Weyl node of extremely low energy (∼ 2 meV, cf. ref. [20]), allowing for acoustic phonons to be scattered by low-energy Weyl fermions of similar wavevectors. A different but intimately related consequence of e-p interaction in a quantizing field is known as 'magnetophonon oscillation' [29], i.e., resonance scattering of quantized electrons by phonon of a distinct frequency. First-principles calculations of phonon dispersion for TaAs reveal a negligible field dependence of the phonon DOSs (see SI). This suggests that the modulated phonon mean free path, rather than the phonon frequency, is at the heart of the giant MQOs in κ(B) which, therefore, occur in the lattice heat conductivity κ l (B) rather than its electronic counterpart κ e (B).
Apart from the giant MQOs, yet another unprecedented feature stands out, namely, the apparently different trends of κ(B) measured in different field configurations. Figure  When T is sufficiently increased, the difference shrinks and finally the two curves fall almost on top of each other (Fig. 4 inset).
Generally, the isothermal κ(B) of a nonmagnetic conductor decreases with B because of the classical, positive orbital magnetoresistance and the consequently diminishing κ e (B), cf.
Eq. 1 and ref. [30]. This term, however, is far too insufficient to explain the decreasing κ ⊥ (B), cf. Fig. 4 [31]. Instead, a field-induced, local diamagnetism that yields an enhanced anharmonic magnetic force on the lattice vibrations is probably involved in the decrease of κ ⊥ (B), as has been reported for InSb [32].
A much more intriguing observation, as shown in Fig. 4, is an increasing κ (B) in a paral- This provides a qualitative explanation to the different trends between κ ⊥ (B) and κ (B). A quantitative comparison requires the values of the prefactors α and β of the chiral anomaly terms, which depend on the band structure and can be obtained by k-space integrals of Boltzmann transport coefficients with Berry curvature and chiral anomaly effects taken into account [6].
In conclusion, we have observed giant MQOs in the thermal conductivity κ(B) of the prototypical Weyl semimetal TaAs. Electronically triggered magnetic quantum oscillations of the lattice thermal conductivity κ l (B) through corresponding oscillations of the electronphonon coupling can explain our observations, at least qualitatively. In addition, the clear difference between κ (B) and κ ⊥ (B) very likely manifests the, as yet never found, evidence of