Anomalous transport properties of the antiferromagnetic Weyl semimetals Mn3 X (X = Sn, Ge)

Noncollinear antiferromagnets Mn3 X (X = Sn, Ge) are characterized by a large anomalous Hall effect originating from a large Berry curvature despite a vanishingly small magnetization. From recent first-principle theories, the large Berry curvature is predicted to be induced by a existence of Weyl nodes broken time-reversal symmetry. The large anomalous Nernst effect is also contributed by the magnetic Weyl state around the Fermi level EF, and likely shares its origin with the anomalous Hall effect. The thermoelectric transport S(T) and thermomagnetic transport Sji(T) are thus investigated in single crystals of Mn3 X. Here, Mn3 X exhibits a large anomalous Nernst effect; in particular, the signal magnitude of Mn3Ge exceeds 1μV/K, which is 1.5 times that of Mn3Sn. The Weyl properties are discussed by analyzing the thermal conductivity, specific heat, and Seebeck and Nernst effects. We also evaluate the zero-field Nernst-driven thermoelectric figure of merit for device applications in the antiferromagnets Mn3 X.


Introduction
The large anomalous Hall effect (AHE) observed in the noncollinear antiferromagnets (AFMs) Mn 3 X (X = Sn and Ge) is known to be given by the intrinsic contribution due to large Berry curvature generated by time-reversal-breaking Weyl metals [1,2]. The existence of the magnetic Weyl nodes at Fermi level E F are suggested by recent first-principles calculations of the electronic band structure [3,4,5,6]. Additionally, the planar Hall effect accompanied by positive longitudinal magnetoconductance has been observed recently in magnetotransport measurements. Such a phenomenon is considered to be induced by the chiral anomaly corroborated with the magnetic Weyl semimetal owing to the clear evidence of Weyl nodes, which is provided by angle-resolved photoemission spectroscopy and the corresponding theoretical band structure.
The sum of the Berry curvature over the occupied states below the E F theoretically constitutes the intrinsic contribution of AHE. When the E F are tuned just at the Weyl  [7]. The Berry curvature is also detected using the thermoelectric counterpart of the AHE, the so-called anomalous Nernst effect (ANE). In fact, the observed anomalous Nernst effect is stronger than the effect reported for ferromagnets in the antiferromagnets Mn 3 X [7,8]. When the finite Berry curvature at Weyl nodes appears only in the k B T energy region around E F , the ANE is provided by the contribution of the Weyl nodes in the k-space because the ANE intrinsically determines the spectrum of the Berry curvature at E F [7]. The ANE thus provides us important properties on the Berry curvature and its relation to the Weyl nodes near E F [6,7].
Here, a comprehensive study of the large spontaneous ANE are performed in the chiral antiferromagnets Mn 3 X. In antiferromagnet Mn 3 Sn, the inverse-triangular spin structure is only magnetically stable from the spin-glass transition temperature T s ( ∼ = 50 K) to the antiferromagnetic Néel temperature T N ( ∼ = 430 K). Although Mn 3 Ge has the same type of spin structure, it is stable above 300 mK and has no additional transition below ∼ = 380 K. The low-temperature Nernst effect S ji is affected by the spectrum of the Berry curvature at the effective E F and reveals a unique property of Weyl metals. It is thus essential to investigate the low-temperature ANE of Mn 3 Ge in elucidating the magnetic Weyl phenomenon for future application to evaluate the thermodynamic properties of the Nernst-type modules as well as our previous studies of Mn 3 Sn [9, 10].

Experimental and Results
Figures 1a and 1b present the specific heats C ab (T ) and C ab (T )/T of Mn 3 Ge (Mn 3 Sn). The peak is observed at 65 K (70 K) for Mn 3 Ge (Mn 3 Sn). According to the Einstein-Debye equation of C ab /T = γ + βT 2 , where γ and β are respectively parameters of the electronic and lattice contributions to the specific heat, the best parameter fittings to the Mn 3 Ge (Mn 3 Sn) data are obtained in the low-temperature regime of 2-15 K as γ = 22.5 mJ/mol K 2 (γ =32.5 mJ/mol K 2 ) and β = 156 µJ/mol (β = 300 µJ/mol). Here, the Debye temperature is evaluated as T θ = 291 K (T θ = 234 K) using C ab /T = (12π 4 N k )/(5T 3 θ )T 2 and Boltzmann's constant N k [5]. In Figure 1c, the positive Seebeck coefficient S ii is shown at 300 K and indicates that hole carriers are dominated by electron carriers [5,10]. S ii (T ) changes sign at ∼ 100 K and peaks at ∼ 50 K in both the Q||[2110] and Q||[0110] of Mn 3 Ge. Additionally, we obtain from the estimated Debye temperatures a crossover temperature of T θ /5 = 58.6 and 46.8 K for the minimum Seebeck coefficient of Mn 3 Ge and Mn 3 Sn respectively. Figure 1d displays the zero-field thermal conductivity κ(T ) in the warming process after annealing at a temperature above T N , which aligns the domain along B. The value κ(300 K) ∼ 8-12 W/Km of Mn 3 Sn is slightly more conductive than that of Mn 3 Ge. The spontaneous Nernst voltage can be applied to thermoelectric modules that convert only a temperature gradient into electrical energy and ordinary Seebeck-type at room temperature [11]. As shown in Fig. 2a, the temperature dependence of S ji is maximized at 100 K (200 K) for Mn 3 Ge (Mn 3 Sn). We find that the Nernst coefficient is largest in Mn 3 Ge as the antiferromagnet. The Seebeck-driven thermoelectric figure of merit ZT = (S 2 /ρκ)T is a quantity used to characterize the performance of a device and defined for a maximally efficient thermoelectric generator. Similarly, the Nernst-driven Z N T = (S 2 ji /ρκ)T is defined in Mn 3 Ge and is only ∼ 9 × 10 −6 at 300 K, which indicates higher value compared with the value for Mn 3 Sn (∼ 2 × 10 −6 at 300 K) reported in our previous study [10] and is still quite low for fabricating a Nernst-driven thermoelectric device.

Conclusion
A sizable Nernst coefficient S ji is obtained for antiferromagnetic Mn 3 Ge, which exceeds 1 µV/K at around 100 K. This value is comparable to the Nernst coefficients of ordinal ferromagnets and twice that of Mn 3 Sn. The ANE in Mn 3 Ge is thus assumed to be induced by large Berry curvature, indicating the existence of paired Weyl points. In both Mn 3 Sn and Mn 3 Ge, the AHE and ANE are considered to share the same origin. The large anomalous Nernst coefficient S ji for Mn 3 Ge beyond the magnetization scaling relation are shown in Fig. 2b [8]. Rrom a viewpoint of the ANE, however, the zero-field Nernst-driven thermopower of antiferromagnetic Weyl metals still does not have sufficient ZT for practical application, in contrast with Seebeck-driven thermopower in semiconductors. In future practical applications, antiferromagnetic Weyl semimetals are expected to be available for topological memory besides Nernst-driven thermoelectric modules as discussed here. These Weyl semimetals are suitable for microfabricated devices owing to the weak magnetic leakage due to integration. We need to investigate further their possible use in spintronic applications [12,13] and continue searching for the antiferromagnetic type of such novel materials.