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Anomalous transport due to Weyl fermions in the chiral antiferromagnets Mn3X, X = Sn, Ge

Abstract

The recent discoveries of strikingly large zero-field Hall and Nernst effects in antiferromagnets Mn3X (X = Sn, Ge) have brought the study of magnetic topological states to the forefront of condensed matter research and technological innovation. These effects are considered fingerprints of Weyl nodes residing near the Fermi energy, promoting Mn3X (X = Sn, Ge) as a fascinating platform to explore the elusive magnetic Weyl fermions. In this review, we provide recent updates on the insights drawn from experimental and theoretical studies of Mn3X (X = Sn, Ge) by combining previous reports with our new, comprehensive set of transport measurements of high-quality Mn3Sn and Mn3Ge single crystals. In particular, we report magnetotransport signatures specific to chiral anomalies in Mn3Ge and planar Hall effect in Mn3Sn, which have not yet been found in earlier studies. The results summarized here indicate the essential role of magnetic Weyl fermions in producing the large transverse responses in the absence of magnetization.

Introduction

In the field of quantum materials, topology—a mathematical concept that describes the robustness of form—has inspired a new era by introducing novel topological phases of matter1,2,3. Until recently, most of the associated discoveries take place in weakly interacting materials in which electron correlation effects have only a minor role. Now the hunt for new topological phases is shifting toward strongly correlated electron systems. The notion of strong electronic correlation provides a common thread linking a wide variety of fascinating emergent phenomena in quantum materials, ranging from high-Tc superconductivity to fractional quantum Hall effect. Its convergence with topology affords an exceptional venue to explore topological order in magnetism. A key discovery along this direction is the surprisingly large anomalous Hall effect (AHE) in chiral antiferromagnets4,5,6, which has triggered extensive research efforts in various fields, ranging from topological condensed-matter physics, spintronics, and energy harvesting technology7,8,9,10,11,12,13,14,15,16. A serious challenge facing condensed-matter and material physicists is revolutionizing information technology that breaks through the fundamental limit of Moore’s law, namely, developing strategies that enhance the processing speed and energy efficiency simultaneously. With this regard, spintronics has provided cutting-edge technologies, introducing nonvolatile forms of logic and memory devices. Traditionally focused on ferromagnetic (FM) materials, recent studies have revealed further benefits using antiferromagnets: antiferromagnets possess ultrafast dynamics and are insensitive to stray fields that perturb neighboring cells, allowing them to exceed the capacity of the FM counterparts17,18. Thus, the large AHE in antiferromagnets observed at room temperature may serve as a readout for a spintronic device1,16,17,18. Another remarkable discovery is the substantial anomalous Nernst effect (ANE) accompanying the large AHE without net magnetization9,13,15,19. This feature indicates that the control of Berry curvature is essential for enhancing the transverse thermoelectric effect, which has inspired new development in thermoelectric converters9,20,21,22,23.

Understanding of the mechanism behind the unusually large AHE and ANE in antiferromagnets serve as the basis for their application in modern technologies; this is where the novel topological quasiparticles—Weyl fermions—come into play. In a metallic or semimetallic material, the double electronic band degeneracy enforced by the Kramers theorem is lifted with the breaking of either time-reversal or spatial inversion symmetry (IS). The linearly dispersing touching points of two nondegenerate bands are known as Weyl nodes, around which the Weyl Hamiltonian $${H_{W}} = \pm \hbar v_F{\boldsymbol{k}} \cdot {\upsigma}$$ describes the electronic states24,25. According to the dimensionality of the electronic band touching, the topological states can be classified into topological insulators, nodal points, and nodal lines; the Weyl node is a typical example of the nodal point. In this case, the emergent quasiparticle represents a condensed-matter physics analog of the long-anticipated relativistic Weyl fermion that remains unobserved in high-energy physics experiments. The Weyl nodes carry quantized magnetic charges that act as a source or drain of Berry curvature24. The Berry curvature can be regarded as a fictitious magnetic field in momentum space, as it behaves in the same manner as a real magnetic field under symmetry operations26,27. In contrast with Dirac fermions in graphene, the Weyl node is a three-dimensional object; namely, the corresponding Weyl Hamiltonian involves all three Pauli matrices that form the basis of the Hilbert space. As a result, any perturbation in the form of a linear superposition of the Pauli matrices cannot annihilate the Weyl points but merely shift its position in momentum space. The magnetic charge is protected as in the Gauss’s law for an electronic charge in real space. This topological protection of Weyl nodes against perturbation renders them appealing potential sources of real-life applications.

The Weyl semimetal (WSM) state occurs in two situations, requiring either a broken time-reversal symmetry (TRS) or a broken spatial IS; the latter has been established in weakly interacting electron systems with spectroscopic evidence of Weyl fermions. The experimental quest for a TRS-breaking WSM is far more challenging, yet it features advantageous properties from both fundamental and practical standpoints. The TRS-breaking or magnetic WSM provides the possibility of realizing a model system that exhibits macroscopic spontaneous responses due to Weyl fermions in zero field, which are absent in the IS-breaking WSM. For instance, the substantial net Berry curvature inherent to the Weyl nodes, combined with the magnetism, may yield unusually large anomalous transport and optical properties in TRS-breaking WSMs. Moreover, the magnetic texture provides a handle for manipulating the Weyl fermions4,5,8,28, making the magnetic WSMs particularly attractive in spintronic applications16.

In noncollinear antiferromagnets, the combination of the symmetry of the spin structure and the underlying band topology allows for the realization of a TRS-breaking WSM8,28,29. When the Weyl nodes reside close to the Fermi energy, they may manifest through large intrinsic AHE and ANE due to the significant enhancement in the Berry curvature. In fact, in addition to the chiral anomaly in magnetotransport, the observation of AHE and ANE in antiferromagnets has provided first crucial experimental signatures of magnetic Weyl fermions8. Here, we provide an overview of transport measurements in chiral antiferromagnet Mn3X (X = Sn, Ge) that features unusually large AHE and ANE. Combined with spectroscopic studies and first-principles calculations, the comprehensive set of transport signatures in Mn3X (X = Sn, Ge) serves as an effective probe of magnetic Weyl fermions, paving an important avenue for future identification of the WSM state in strongly correlated materials.

Basic properties of Mn3X (X = Sn, Ge)

Kagome-based metals have recently become a new focus in the field of quantum materials for their topological electronic structures, as well as large electromagnetic and transport responses4,5,6,9,11,12,15,23,30,31. Mn3Sn and Mn3Ge are hexagonal chiral antiferromagnets (space group P63/mmc) that comprise breathing-type kagome layers of Mn atoms in the ab-plane (Fig. 1a); these kagome planes are stacked along the c-axis32. Neutron diffraction studies in the magnetically ordered phase revealed an inverse triangular spin structure, in which Mn moments are aligned within the kagome layers, forming a 120° ordering with a negative vector chirality and a q = 0 ordering wave vector; the ordered moment is about 3 μB/Mn (μB, Bohr magneton) in Mn3Sn and about 2.5 μB/Mn in Mn3Ge33,34,35,36. This chiral spin structure reduces the hexagonal crystal structure to orthorhombic in both compounds. Within each triangular motif, only one of the three Mn moments points along the local easy axis ($$\left[ {2\bar 1\bar 10} \right]$$ for Mn3Sn (Fig. 1a) and $$\left[ {01\bar 10} \right]$$ for Mn3Ge (Fig. 1b)). Small canting of the remaining two spins in the ab-plane induces weak ferromagnetism with a tiny net moment of about 3 mμB/Mn in Mn3Sn and about 7 mμB/Mn in Mn3Ge33,35. The second-order nature of the magnetic transition suggests that the spin structure of Mn3X (X = Sn, Ge) belongs to the D6h point group, which allows four distinct 120° spin structures with q = 0. Recent polarized neutron diffraction studies on single crystals constrain the ground-state spin structure to E1g symmetry type, for which the weak ferromagnetism is inevitable37,38. This information is essential for verifying the theoretical proposals of Weyl nodes based on band structure calculations8,28,39,40. Owing to the competition between the single-ion anisotropy and the Dzyaloshinskii–Moriya interaction, the in-plane anisotropic energy of Mn is absent up to fourth order in Mn3X (X = Sn, Ge), and six-order anisotropy is necessary for selecting a unique ground-state spin structure33,34,41,42,43. With such small anisotropic energy, the coupling between a magnetic field and the weak FM moment takes on a key role that enables magnetic-field control of the antiferromagnetic (AFM) spin structure.

The Mn moments residing on an octahedron made of two triangles of neighboring kagome bilayers constitute a cluster magnetic octupole; the inverse triangular spin structure can be viewed as a ferroic order of cluster magnetic octupoles (Fig. 1b)7,44,45. Mn3Sn serves as a textbook example for the developing cluster multipole (CMP) theory, which identifies the symmetry equivalence of noncolinear AFM spin structures with FM states by considering CMP as the order parameter of noncollinear AFM orders44,45. The large magneto-optical Kerr effect has been discovered in Mn3Sn as the first example in the AFM metal and enables visualization of the magnetic octupole domains and their switching under an applied magnetic field7. The interplay between CMP in real-space magnetic domains and the Weyl points in momentum space may yield intriguing topological electric transport that remains to be explored46.

Below the Néel temperature TN ≈ 430 K, Mn3Sn enters the inverse triangular spin state, which is taken over by helical ordering and a cluster glass phase at low temperatures4,9,34,43,47,48,49, depending on the Mn concentration. In contrast, Mn3Ge retains the inverse triangular spin structure down to at least 0.3 K, as is evident from the magnetization M curves of our newly synthesized Mn3Ge single crystals (Fig. 2e): M undergoes a sharp transition at the Néel temperature TN ≈ 372 K, and no additional transition occurs down to 0.3 K. The in-plane magnetization reaches a saturation value of about 7 mμB/Mn at B = 0.1 T and T < 100 K, which is approximately ten times greater than the magnetization for B || [0001].

The hexagonal structure of bulk Mn3Sn and Mn3Ge is stable only in excess Mn4,5,33,35, leading to Mn-substituted Sn/Ge sites. The off-stoichiometry is reflected in a short electronic mean free path of the order several nm at room temperature, comparable to the lattice parameters13,19. This compositional disorder may alter the magnetic and anomalous transport properties of the material and deserves detailed studies. We have recently investigated the behavior of Si-doped Mn3Ge crystals and find that Si doping into the Ge sites provides a means for systematically examining how the physical properties evolve with excess Mn. By doping Si into Ge sites, the level of extra Mn x is enhanced concomitantly; the lattice parameters and the unit cell volume exhibit a slight linear decrease with increasing amount of extra Mn. The volume reduction rate dV/dx (=−39 ~ −48Å3/Mn) in Mn3+x(GeSi)1-x is nearly identical to the value found in Mn3+xGe1-x, implying that Si acts merely as an impurity ion at the Ge site. The magnetization of Si-doped Mn3Ge samples shows temperature variation similar to that of undoped Mn3Ge, while the Néel temperature decreases monotonically with increasing amount of Si doping (Fig. 2e, inset).

Observation of anomalous transport in Mn3X (X = Sn, Ge)

Anomalous Hall effect (AHE)

The AHE is traditionally considered to be proportional to net magnetization, and therefore is expected to arise only in FM materials50. In the past decade, theoretical advances in understanding the mechanism of AHE has established that the intrinsic contribution to the anomalous Hall conductivity (AHC) is the sum of the Berry curvature of all the occupied bands26,27,51, such that:

$$\sigma _{xy} = \frac{{e^2}}{\hbar }\mathop {\int}\limits_{\rm{BZ}} {\frac{{d{\boldsymbol{k}}}}{{\left( {2\pi } \right)^3}}{\Omega}_{n,z}\left( {\boldsymbol{k}} \right)f_{n,k}} ,$$
(1)

where Ωn,z (k) is the out-of-plane component of the Berry curvature with the band index n and fn, k is the Fermi–Dirac distribution function. Namely, the intrinsic AHE is driven not by the magnetization but by the Berry curvature determined entirely by the band topology of a perfect crystal. That is to say, the sizable AHE can be realized even in the absence of net magnetization, such as in spin liquids and antiferromagnets, as long as the system sustains a substantial net Berry curvature in the momentum space24,26,27,28,39,44,52. In fact, the first experimental detection of this kind was made in 2009 in the spin liquid compound Pr2Ir2O7, where a large zero-field Hall conductivity of about 10 Ω−1 cm−1 is observed in the absence of magnetization in a spin-ice state29,53,54. Systems harboring Weyl points offer an ideal physical setting that promotes strongly enhanced AHE without net magnetization, owing to the divergent Berry curvature in the immediate vicinity of the Weyl points. Following the observation of AHE in Pr2Ir2O7, the seminal paper by Wan et al.24 proposed that TRS-breaking Weyl nodes can be stabilized by the all-in all-out magnetic order, another type of magnetic cluster octupole, in the family of pyrochlore iridates R2Ir2O7 (R = Y and lanthanides), which brought the concept of magnetic WSM to the center stage of condensed-matter physics.

In 2015, the first observation of the large AHE in an antiferromagnet was achieved using the chiral AFMs Mn3X (X = Sn, Ge)4,5. At room temperature, the Hall resistivity ρH of Mn3Sn measured under in-plane magnetic fields $${\it{B}}\parallel [2\bar 1\bar 10]$$ and $${\it{B}}\parallel [01\bar 10]$$ displays a sharp hysteresis loop with a small coercivity of approximately 100 Oe (Fig. 2a)4. The magnetic-field dependence of ρH is anisotropic; the sweep of the in-plane fields generates a substantial zero-field Hall component and a narrow hysteresis in ρyz and ρzx, whereas the out-of-plane field $$B\parallel \left[ {0001} \right]$$ yields a linear-in-B response in ρxy without hysteresis. The in-plane-field responses, ρyz and ρzx, are unusually large for an AFM material and are comparable to the values found in strong ferromagnets26. By contrast, the magnetization isotherms M (B) observed under the in-plane magnetic fields show a vanishingly small spontaneous (zero field) magnetization M of about 3 mμB/Mn originated from spin canting (Fig. 2b)4. The similar hysteretic and anisotropic behavior of ρH (B) and M (B) indicates that the rotation of AFM domains leads to the sign change of the Hall effect. Figure 2c depicts the temperature dependence of the zero-field Hall conductivity, σyz, σzx, and σxy, obtained under $$B\parallel \left[ {2\bar 1\bar 10} \right]$$, $$B\parallel \left[ {01\bar 10} \right]$$, and $$B\parallel \left[ {0001} \right]$$, respectively4. The zero-field values of σyz and σzx gradually increase from the room-temperature value of about 20 Ω−1 cm−1 on cooling, reaching a maximum value of about 130 Ω−1 cm−1 for −σzx and about 70 Ω−1 cm−1 for −σyz before dropping sharply upon entering the low temperature spin-glass phase at T < 50 K. In contrast, σxy becomes finite only in the low temperature spin-glass phase and its magnitude increases strongly on cooling for T < 50 K (Fig. 2c)4.

The negligible σxy at high temperatures is in line with the in-plane polarization of the magnetic octupole due to coplanar 120° spin structure, whereas the drastic increase in σxy observed at T < 50 K may involve a topological Hall effect4. For T < 20 K, the Hall measurements in polycrystalline Mn3Sn samples reveal sharp peaks of ρxy (B) at about ±0.8 T, suggesting a topological Hall contribution $$\rho _{xy}^T$$ that dominates the Hall signal in the low-T regime55. The spontaneous magnetization at T < 20 K increases by over an order of magnitude relative to the high-temperature value; the coercivity also rises dramatically from only a few hundred oersteds at high temperatures to about 1 T at 2 K. Similar low-T behavior in both ρxy and M is observed in polycrystals of a sister material Mn3Ga56. The topological Hall effect arises from scalar chirality subtended by three neighboring spins of a noncoplanar spin texture26,53,54,57,58,59. Recent studies have highlighted the topological Hall effect as a transport signature of a magnetic skyrmion state, as have been seen in various noncentrosymmetric skyrmion-hosting materials MnSi, FeGe, and MnGe60,61,62. Therefore, the above experiments indicate that Mn3Sn may undergo a transition from the high-T coplanar 120° spin structure with zero scalar spin chirality to a low-T noncoplanar spin ordering, in which the sizeable scalar spin charily acts as a fictitious magnetic field in real space, thereby inducing nontrivial transport responses. Given the strongly anisotropic behavior of AHE and M observed in Mn3X systems, detailed single-crystal studies are necessary for an in-depth exploration of the topological Hall effect and the magnetic skyrmion scenario.

In contrast with Mn3Sn, we show that the Hall conductivity σyz of our Mn3Ge single crystals increases on cooling and saturates at ~300 Ω−1 cm−1 below about 100 K (Fig. 2f) owing to the absence of additional magnetic phase transition at low temperatures5,6. Contrary to the FM case, neither of σyz nor σzx follows the behavior of M (T) (Fig. 2e, f), which are consistent with earlier reports5. With a slight increase in the extra Mn concentration, the saturated value of the AHC significantly declines (Fig. 2f), suggesting that the excess Mn controls the number of the conduction electrons and thereby tunes the Fermi level EF, as the electron states at EF are constituted solely by the 3d electrons8,28.

Anomalous Nernst effect (ANE)

The ANE is another key quantity that measures the Berry curvature. Unlike the AHE that involves the Berry curvature over all occupied bands, the ANE is determined by the Berry curvature lying close to the Fermi energy EF9,27,63. The Nernst signal Sji is obtained using the same electrodes as in the Hall effect measurements, by applying temperature gradient instead of charge current; namely, an induced transverse voltage Vj perpendicular to both the magnetic field B and the heat current $$Q_i \propto - \nabla _iT$$ is measured at various temperatures. The Nernst coefficient $$S_{xy} = E_y/( - \nabla _xT)$$ is related to the anomalous transverse thermoelectric conductivity αxy such that $$S_{xy} = \alpha _{xy}\rho _{xx} - \rho _{xx}\sigma _{xy}S_{yy}$$, where ρxx, σxy, and Syy are the longitudinal resistivity, the Hall conductivity, and the Seebeck coefficient, respectively4. The connection of αxy with the Berry curvature near EF is well established, namely:

$$\alpha _{xy} = \frac{e}{{Th}}\mathop {\int}\limits_{\rm{BZ}} {\frac{{d{\boldsymbol{k}}}}{{\left( {2\pi } \right)^3}}{\mathrm{{\Omega}}}_{n,z}\left( {\boldsymbol{k}} \right)\left\{ \left({\it{\epsilon }}_{n,{\boldsymbol{k}}} - \mu \right)f_{n,{\boldsymbol{k}}} + k_BT{\mathrm{log}}\left[1 + e^{ - \beta \left( {{\it{\epsilon }}_{n,{\boldsymbol{k}}} - \mu } \right)}\right]\right\} }$$
(2)

where Ωn,z (k) is the out-of-plane component of the Berry curvature, $${\it{\epsilon }}_{n,{\boldsymbol{k}}}$$ is the band energy, μ is the chemical potential, and fn, k is the Fermi–Dirac distribution function, with n the band index63.

Figure 2d shows the key findings of the substantial room-temperature ANE in Mn3Sn reported by Ikhlas et al.9. Similar to the Hall conductivity ρH and the magnetization M, the field dependence of −Sji is strongly anisotropic; the in-plane field components −Syz and −Szx exhibit sharp hysteretic jump with a substantial zero-field component of about 0.3 μVK−1, more than three orders of magnitude higher than the expected value according to the scaling relation for FM materials (see below), while −Sxy remains zero within experimental resolution without showing any hysteresis. The comparison between the field dependence of −Sji and M helps unveil the driving mechanism behind the ANE. Although −Syz and M measured under $$B\parallel [2\bar 1\bar 10]$$ display nearly identical hysteresis loops in the low-field regime, −Syz levels off at higher fields rather than following the linear increase of M. Given that the B-linear component of M arises from the field-induced spin canting, the distinction between field dependence of ANE and M indicates that the ANE is insensitive to the spin canting, but instead is governed by the Berry curvature tied in with the order parameter, namely, the cluster magnetic octupole44,45. The temperature dependence of the in-plane zero-field components, −Syz and −Szx, peaks at around 200 K with a maximum value of about 0.6 μVK−1 for Mn3.06Sn0.94. With a slightly more Mn in Mn3.09Sn0.94, the peak in −Syz and −Szx shifts to a higher temperature of about 250 K with a suppressed maximum value of about 0.3 μVK−1. The dramatic change in ANE with only 1% of variation in Mn concentration indicates the proximity of the Fermi energy to the Weyl points, which is discussed in more detail in the next section.

Similar to the case of Mn3Sn, our newly obtained Nernst coefficient −Syz of Mn3Ge single crystals shows a step-like profile with a sizeable zero-field component at room temperature, as shown in Fig. 2g. On cooling, the zero-field components of −Syz and −Szx behave similarly and reach a maximum of about 1.35 μVK−1 near 100 K (Fig. 2h), which is more than twice larger than the value reported for Mn3Sn9. The recent studies by Wuttke et al.15 also find that the temperature dependence of zero-field −Syz and −Szx are nearly identical, with magnitude consistent with our finding. Under a high magnetic field of 14 T, −Syz and −Szx are further enhanced to about 1.5 μVK−1, while the broad peak remains at around 100 K (see below). The out-of-plane field component −Sxy is negligibly small compared to −Syz and −Szx in the studied temperature range. The behavior of Sji observed in Mn3X (X = Sn and Ge) shares remarkable similarities with the low-temperature field-induced ANE of prototypical nonmagnetic WSMs TaAs and TaP, in which the peak position of Sji provides a rough estimate of the lowest Weyl node energy relative to EF64. For T > 50 K, the ANE observed in TaAs and TaP gives way to the B-linear ordinary contribution64, whereas the ANE in Mn3X (X = Sn and Ge) dominates the Nernst signal over a much wider temperature range up to room temperature.

To comprehensively characterize the thermoelectric properties, we have newly measured the longitudinal Seebeck coefficient, Sii (T), of Mn3.04Ge0.96 for $$Q\parallel \left[ {0001} \right]$$ and $$Q\parallel \left[ {01\bar 10} \right]$$ configurations (Fig. 2h, inset). The behavior of Sii (T) is again strongly anisotropic; while Szz remains negative in the entire measured temperature range and is weakly dependent on temperature, Syy exhibits a sign change from positive to negative below about 200 K and peaks negatively at about 60 K. The sign change is presumably related to the phonon drag effect65, as found in the case of Mn3Sn9,19.

Anomalous thermal Hall effect (ATHE)

As the thermal counterpart of the AHE, the anomalous thermal Hall conductivity, $$\kappa _{ij}^A$$, provides essential clues about the driving mechanism behind the dramatically enhanced room-temperature AHE in Mn3X (X = Sn, Ge). Similar to the behavior of the Hall conductivity and the Nernst signal, the thermal Hall conductivities, κxz (B) and κzy (B), in Mn3Sn show a sharp hysteretic jump corresponding to the anomalous component19,66 (Fig. 3a, b). The anomalous Wiedemann–Franz (WF) law is defined as follows:

$$L_{ij}^A = \frac{{\kappa _{ij}^A}}{{T\sigma _{ij}^A}} = L_0 = \frac{{\pi ^2}}{3}\left( {\frac{{k_B}}{e}} \right)^2,$$
(3)

which relates the ratio between anomalous thermal and electrical Hall conductivities with the Sommerfeld number L0. In conventional ferromagnets such as Fe, Ni, and NiCu alloys, the magnitude of $$L_{ij}^A$$ displays a downward deviation from L0 in the high-temperature regime due to inelastic scattering and approaches L0 only in the zero-temperature limit19,65,66,67,68 (Fig. 3c). In Mn3Sn, however, the $$L_{ij}^A$$ is nearly temperature independent and satisfies the WF law in the entire studied temperature range from room temperature down to about 50 K19,66,69 (Fig. 3c). This behavior suggests a negligible effect of inelastic scattering on the anomalous transverse transport of Mn3Sn, excluding phonon contributions or skew scattering by magnetic excitations as a possible cause of the room-temperature AHE and ATHE68,70. The validity of the WF law over a wide temperature range in Mn3Sn further supports the intrinsic Berry-curvature mechanism for its large anomalous transverse effects4,9,26,56,63,66,71.

In contrast, the $$L_{zx}^A$$ of Mn3Ge follows the anomalous WF law $$L_{zx}^A \approx L_0$$ below 100 K down to 0.3 K but deviates from L0 above 100 K13 (Fig. 3d). The high-temperature violation of the anomalous WF law in Mn3Ge is unlikely to result from inelastic scattering, but instead may come from a strong energy distribution of the Berry curvature near EF that results in a mismatch between its electrical and thermal summations over the Fermi surface. The difference in the valid T range of the anomalous WF law suggests that the Weyl points play a vital role in Mn3Sn in shaping the behavior of the observed temperature dependence of $$L_{zx}^A$$. As for Mn3Ge, aside from the contribution from Weyl points, its Berry spectrum near EF is influenced by an anti-crossing gap induced by spin-orbit coupling (SOC), which serves as the primary source of the downward deviation in $$L_{zx}^A$$ above 100 K13.

Evidence for magnetic Weyl fermions

Following the experimental discoveries of the large AHE and ANE, Mn3X (X = Sn, Ge) is proposed as the magnetic version of the WSM (Weyl magnet) based on first principle calculations8,28. The coplanar 120° spin structure of Mn3Sn and Mn3Ge (Fig. 1a, b) lowers the original hexagonal crystal symmetry to orthorhombic, which possesses a nonsymmorphic symmetry and two mirror reflections, MxT and MyT, with time reversal T added. These symmetry properties set strong constraints on the positions of the Weyl points. That is, the mirror symmetries ensure that the Weyl points appear along a k-direction parallel to the local easy magnetization axis, namely, the x-axis for Mn3Sn and y-axis for Mn3Ge; the corresponding k-space location of Weyl points are shown in Fig. 1d, e, respectively7,8. By applying a rotating magnetic field in the xy-plane, one may shift the locations of the Weyl points governed by the underlying spin texture along a hypothetical nodal ring obtained in the absence of the SOC, as illustrated in Fig. 1d.

The calculated band dispersion in both Mn3Sn and Mn3Ge reveals multiple pairs of type-II Weyl nodes at various energies28, which possess finite density of state at the nodal point owing to the touching between electron and hole pockets. Among those, the most relevant to transport properties are the pairs sitting close to the Fermi level EF. In the case of Mn3Sn, the two pairs of Weyl points lying in the vicinity of EF arise from electron-hole band crossings along the K-M-K direction8, as shown in Fig. 1c. Each pair consists of two Weyl points of opposite chirality, with $${\mathrm{W}}_1^ +$$ and $${\mathrm{W}}_2^ -$$ lying at E ~ EF + 60 meV and their partners $${\mathrm{W}}_1^ -$$ and $${\mathrm{W}}_2^ +$$ appearing at a slightly higher energy E ~ EF + 90 meV. The experimental quest for the Weyl fermions involves various experimental probes combined with theoretical analyses, as presented below. In this section, we summarize the evidence of Weyl points obtained for Mn3Sn and Mn3Ge and present our updated analyses of the transverse thermoelectric coefficient, αji, observed in Mn3Ge based on the density functional theory (DFT) calculations.

Analysis of the anomalous transverse coefficients and carrier doping effect

We begin by discussing the temperature variation of the anomalous transverse thermoelectric conductivity72, $$\alpha _{xz} = (\rho _{xx}S_{xz} - \rho _{xz}S_{zz})(\rho _{xx}\rho _{zz} + \rho _{xz}^2)$$. The −αxz observed in Mn3Sn and Mn3Ge behaves similarly, namely, −αxz gradually increases with decreasing temperature, forms a broad maximum, and then declines at lower temperatures (Fig. 4b, h, i)15. In both materials, the anomalous Nernst signal Sxz dominates αxz (Fig. 4c)15. The analysis presented in Wuttke et al.15 is based on a minimal model describing a single pair of Weyl points near EF formed by a crossing of linearized bands. The resulting fitting formula for −αzx involves three key parameters, that is, the Berry curvature strength, $${\tilde{\mathrm{{\Omega}}}}$$, near EF, the k-space nodal separation, Δk, between the two Weyl points, and the Weyl point energy, E, relative to EF. For a system with multiple pairs of Weyl points, the E value provides an estimate of the lowest possible Weyl point energy. For both Mn3Sn and Mn3Ge, such a model well describes the experimental −αzx (solid lines in Fig. 4b, d)15, and the E value of Weyl points yielded by the best fit is in reasonable agreement with the predicted energy given by band structure calculations8,28.

We next present a comparison between the experimentally observed −αji in Mn3Sn and Mn3Ge with our DFT calculations. The Mn3Sn and Mn3Ge samples usually contain extra Mn, as mentioned earlier. Since the states at the EF are occupied solely by 3d electrons, a small amount of Mn doping may generate a sizeable shift of the chemical potential relative to the EF of the stoichiometric case. A 1% Mn-doping in Mn3Sn may shift up the chemical potential by about 6 meV from EF; thus, the two Mn compositions, Mn3.06Sn0.94 and Mn3.09Sn0.91, have the chemical potential of E = EF + 36 meV and E = EF + 54 meV, respectively, as confirmed by angle-resolved photoemission spectroscopy (ARPES) measurements7. Moreover, the ARPES study finds a significant bandwidth renormalization and quasiparticle damping. Thus, in the calculation of the anomalous transverse thermoelectric conductivity, we take both the real and imaginary parts of the self-energy into account, such that:

$${\mathrm{{\Sigma}}} = \left( {1 - \frac{1}{{Z}}} \right){\it{\epsilon }} - {iA}{\it{\epsilon }}^2 - i{\upgamma}$$
(4)

The Z value is estimated to be 0.2 for Mn3Sn according to the renormalization factor estimated by the ARPES and specific heat measurements7,8. Given the larger bandwidth of Mn3Ge in comparison with Mn3Sn44,45, a larger Z value of 0.5 is chosen. The coefficient 1/ZA determines the energy window within which the quasiparticle is well defined. We assume a window size of 40 (100) meV for Mn3Sn (Mn3Ge) based on the ARPES measurement7 of Mn3Sn so that the size covers the energy scale for Weyl fermions relevant for both compounds. Thus, AZ2 = 5 eV−1 for both Mn3Sn and Mn3Ge. The solid curves in Fig. 4h are the calculated −αzx/T for Mn3.06Sn0.94 and Mn3.09Sn0.91 that are dominated by the Berry curvature stemming from the low-energy Weyl points. They reproduce the overall temperature dependence of the experimental −αzx/T yet overestimate the magnitude by nearly a factor of 2. Note that we limit the temperature range for the comparison of theory and experiment up to 200 K; the temperature variation of the magnetization M is negligible in this T regime, and thus the spin fluctuations should have only a minor effect on the transport properties.

In the case of Mn3Ge, our DFT calculations predict the presence of several Weyl points near E – EF = 78 meV (Fig. 4f) and 63 meV (Fig. 4g) under $$B\parallel [01\bar 10]$$ on the kz = 0 and ±0.137 (Å−1) planes, respectively, where the energy gap between the electron and hole bands becomes vanishingly small within the Brillouin zone (BZ). These Weyl points are located at non-high-symmetric positions in the BZ and harbor type-II Weyl fermions; their energies and positions in the k-space are in good agreement with the W1 and W3 pairs reported in Yang et al.28. The theoretical calculation suggests that a 1% Mn-doping may shift the chemical potential up by about 17 meV in Mn3Ge, and, as a result, the experimental Hall conductivity, σji shown in Fig. 2f decreases from 300 Ω−1 cm−1 in Mn3.03Ge0.97 to 170 Ω−1 cm−1 in Mn3.04Ge0.96. The calculated energy spectrum of σji undergoes such a sharp change at E – EF ≈ 60 meV; correspondingly, the calculated −σji and −αji with the chemical potential of E – EF = +50 meV and E – EF = +68 meV roughly reproduces the T dependence and magnitudes observed for Mn3.03Ge0.97 and Mn3.04Ge0.96 (Fig. 4h, i).

Spectroscopic investigations of the electronic band structure

Identification of the band structure and the locations of Weyl points in the momentum space provides direct evidence for the Weyl fermions. In weakly interacting electron systems with broken IS, clear spectroscopic evidence of Weyl points and surface Fermi arcs has been established1,73. The search of Weyl fermions in magnetic materials that breaks TRS is far more challenging because strong correlation effects due to magnetism inevitably lead to diffusive character of the Weyl excitations and prevent their detection by spectroscopic probes. While recent ARPES experiments find some evidence for magnetic Weyl fermions in FM Co-based materials74,75, the present Mn3X (X = Sn, Ge) compounds feature much stronger electron correlations than the Co-based systems, as reflected by the large renormalization factor of five. Such dramatic bandwidth renormalization is comparable to that of high-temperature superconductors and heavy-fermion compounds76, therefore posing an intrinsic challenge for direct observation of Weyl fermions. Figure 5a–c summarizes the ARPES spectra of Mn3Sn reported by Kuroda et al.8. The kz-dispersion of ARPES intensity along the H-K-H high-symmetry direction reveals a quasiparticle peak near EF at a photon energy of $$h\upsilon = 103\,{\mathrm{eV}}$$ (the blue arrow in Fig. 5a). This feature indicates that $$h\upsilon = 103\,{\mathrm{eV}}$$ selectively measures the kz = 0 band dispersion involving the low-energy Weyl points. The ARPES constant-energy surface at EF obtained using $$h\upsilon = 103\,{\mathrm{eV}}$$ captures six elliptical contours on the kxky-plane, in qualitative agreement with the topological features of the electron-type Fermi pocket predicted by the DFT calculation (Fig. 5b). More importantly, this Fermi pocket is derived from the electron band that generates the Weyl points at its crossing with a hole band at $$E \sim E_F + 60\,{\mathrm{meV}}$$. The evolution of this electron band is traced by the Ekx cuts measured at various ky along the K-M-K line (Fig. 5c), consistent with the theoretical band dispersion after renormalizing the energy scale by a factor of five. That is, the electron band (the red line) approaches the hole band (the blue line) with increasing ky, form the Weyl nodes, and again moves apart from the hole band. The momentum distribution curve measured at 60 K displays two additional anomalies only for the kx cut made exactly along K-M-K, corresponding to the linear band crossing points between the electron and hole bands (Fig. 5c). Although the strong correlation effects hinder the observation of direct signatures of Weyl points and surface Fermi arcs, these findings in APRES measurements are essential, as they identify the two bands that form the low-energy pair of Weyl nodes (Fig. 1c, d) and verify the calculated band structure.

Chiral anomaly in Mn3X (X = Sn, Ge)

The gapless nature of the Weyl fermions renders their acute responses in low-energy probes, with the hallmark feature in magnetotransport known as the chiral anomaly. In the presence of a high magnetic field, the Weyl crossings are quantized into Landau levels (LLs), with a chiral lowest LL for each Weyl node. Namely, electrons occupying the lowest LL are divided into left- and right-moving groups of opposite chirality. Hypothetically, these two groups do not mix, leading to separate number conservation of the three-dimensional left- and right-handed Weyl fermions. Once an electric field $$E\parallel B$$ is applied, the conservation of chirality is no longer valid, as a result of the mixing between the left-and-right movers77,78. This phenomenon is known as the chiral anomaly. In such a case, charge pumping between the Weyl nodes of opposite chirality generates the positive longitudinal magnetoconductivity (LMC) or the negative magnetoresistivity (LMR) only for the $$E\parallel B$$ configuration. Meanwhile, the absence of the chiral anomaly for the perpendicular configuration EB leads to a negative LMC or a positive LMR due to the ordinary scattering mechanism. Thus, the combination of the positive LMC for $$E\parallel B$$ and the negative LMC for EB provides the transport evidence for Weyl fermions and has been widely studied for both nonmagnetic and magnetic WSM candidates8,21,29,79,80,81,82.

In this section, we discuss the Weyl-fermion-induced magnetotransport anomalies in Mn3Sn and Mn3Ge. The magnetoconductance obtained for Mn3Sn at 60 K is strongly anisotropic (Fig. 5d)8. When the magnetic field B is applied parallel to the electric field E, a positive magnetoconductance is observed, while a negative magnetoconductance appears in the EB configuration. In the low-field regime, the positive LMC for $$E\parallel B$$ shows a linear increase with B, consistent with the chiral anomaly of a type-II WSM83,84. In magnetic conductors, the external magnetic field generally suppresses the charge carrier scattering due to thermal spin fluctuations, leading to a positive magnetoconductance irrespective of the angle between E and B. While this effect may complicate the identification of the chiral anomaly, especially at high temperatures, such positive isotropic LMC should cease on cooling. In contrast, the positive LMC observed in Mn3Sn for $$E\parallel B$$ rises dramatically with decreasing temperature from 300 K down to 60 K before entering into a cluster glass phase at T < 50 K (Fig. 5e)8. This behavior excludes the suppression of spin fluctuations or weak-localization as possible causes for the observed positive LMC, verifying that the chiral anomaly of the Weyl fermions is the primary driver of the significantly anisotropic magnetotransport in Mn3Sn.

Unlike the case of Mn3Sn, the ferro-octupole phase in Mn3Ge remains intact further down to 0.3 K, which provides an advantage for discerning the intrinsic effects of Weyl fermions from that of the magnetic spin fluctuations. Figure 5f shows the field dependence of the magnetoconductivity $${\it{{\Delta}}}\sigma \left( B \right) = \left( {\sigma \left( B \right) - \sigma \left( 0 \right)} \right)$$ observed in Mn3Ge at 0.3 K. The sharp dip in the magnetoconductivity for $$B \le 2\,{\mathrm{T}}$$ is most likely caused by magnetic domain effects, as the magnetization curve shows hysteretic behavior in the same field region at 2 K. The domain walls are most likely chiral and their effect on transport is an intriguing subject for future studies11. For B > 2T, the magnetoconductivity measured in $$B\parallel \left[ {01\bar 10} \right]$$ becomes negative in the EB configuration, indicating that the effect due to spin fluctuations is fully suppressed by lowering the temperature to approximately one-thousandth of the Néel temperature. In contrast, the magnetoconductivity in the $$E\parallel B$$ configuration is positive and is nearly quadratic in B (Fig. 5f), which is distinct from the B-linear behavior observed in Mn3Sn. The B2 dependence of the positive LMC is likely a result of strongly titled type-II Weyl cones with the applied magnetic field perpendicular to the tilt axis84.

Apart from the chiral anomaly, other conventional mechanisms may yield positive LMC; one such well-known effect is the current jetting85. High-mobility semimetals may exhibit a very large transverse magnetoresistance relative to the LMR owing to the orbital effect. In such cases, the strongly anisotropic MR may restrict the current to the direction parallel to the field, while the current flow transverse to the field is largely suppressed. This field-induced steering of the current may lead to positive LMC that depends strongly on the sample dimension and geometry. Recent studies have shown that the current-jetting effect serves as the primary mechanism behind the positive LMC observed in several Dirac and WSM candidates85. The carrier mobility of Mn3X (X = Sn, Ge) is considerably lower than that of weakly correlated semimetals and zero-gap semiconductors, with a nearly isotropic magnetoresistance. Therefore, any current-jetting effect is unlikely to occur. Nevertheless, experimental tests were performed to address this concern. The magnetoconductivity measured at different voltage contacts on the sample are nearly identical, and all measured samples display qualitatively the same behavior, indicating that the observed positive LMC of Mn3X (X = Sn, Ge) is intrinsic rather than arising from current inhomogeneities.

Another magnetotransport signature of the chiral anomaly is the planar Hall effect (PHE), which is formulated as follows:

$${\Delta}\sigma = {\Delta}\sigma _{{\mathrm{chiral}}}{\mathrm{cos}}^2\theta$$
(5)
$${\Delta}\sigma _{H}^{{\mathrm{PHE}}} = {\Delta}\sigma _{{\mathrm{chiral}}}{\mathrm{sin}}\,\theta {\mathrm{cos}}\theta$$
(6)

where Δσ and $${\Delta}\sigma _{H}^{{\mathrm{PHE}}}$$ are the LMC and the planar Hall conductivity, respectively86. Unlike the conventional Hall effect measurements with applied E- and B-fields being mutually perpendicular, the E- and B-fields in the PHE measurement are coplanar during the field angle (θ) rotation, with the B-field lying within the kagome plane (see schematic in Fig. 5g). According to Eq. (6), the PHE reaches extrema at 45° and 135°, and its amplitude is given by the chiral anomaly induced positive magnetoconductivity. The angular dependence of Δσ and $${\Delta}\sigma _{H}^{{\mathrm{PHE}}}$$ observed in Mn3Sn are well described by the theoretical forms for the chiral anomaly (Fig. 5g), and further verifies the presence of magnetic Weyl fermions.

Scaling behavior in Mn3X (X = Sn, Ge)

The results summarized above provide firm evidence for the existence of Weyl nodes near the Fermi energy, which play a central role in generating the AHE and ANE. When the AHE comes from the intrinsic mechanism, the Hall conductivity is independent of the longitudinal conductivity, following the universal scaling law26,87,88,89. Figure 6a presents the transverse Hall conductivity, σH, as a function of the longitudinal conductivity, σ, for various single crystals of Mn3+xX1-x (X = Sn, Ge) with different composition in comparison with those of FM materials. Here, σ and σH refer to the values obtained at the low temperature limit of the phase where the inelastic scattering is minimized. Notably, for all studied samples, the longitudinal σ is in the range of 5 × 103 Ω−1 cm−1, and the σH value is nearly independent of σ (Fig. 6a, inset). Both features indicate that Mn3X (X = Sn, Ge) sits in the “good metal” regime, where the AHE is governed by the intrinsic mechanism.

Summary and perspectives

The combination of the experimental and theoretical studies on the chiral antiferromagnets Mn3X (X = Sn, Ge) has revealed a comprehensive set of evidence that confirms the presence of Weyl fermions, in particular, the large Berry-curvature-driven AHE and ANE, positive LMC and PHE originated from the chiral anomaly. The large Berry curvature stemming from the Weyl nodes leads to significantly enhanced AHE and ANE in the absence of net magnetization. To further demonstrate this feature, we show a summary comparison of the magnetization dependence of the spontaneous Hall conductivity σH (M) and the anomalous Nernst coefficient Syx (M) obtained in Mn3X (X = Sn and Ge) with those observed for various conventional FM materials (Fig. 6b, c)4,5,6,9,15,21,23,89. The large σH (M) and Syx (M) of Mn3X manifest themselves beyond the linear relation for the ordinary ferromagnets (red shaded region). Other known members of this regime are the FM cobalt-based WSMs Co2MnGa21 and Co3Sn2S223; in both compounds, recent ARPES studies reveal evidence of Weyl fermions74,75, which are identified as the dominant cause of the unusually large AHE and ANE.

The research on Mn3X (X = Sn, Ge) therefore paves the path for designing and investigating new Weyl magnets with strong electron correlation. The realization of Weyl magnets triggers further exploration of their intriguing properties emerging from the interplay between magnetism and topology, which may cast new light on the fundamental science behind topological quantum states of matter. The great tunability and robustness of Weyl fermions, combined with the unique advantages of the AFM spin structure over FM counterparts, render Mn3X (X = Sn, Ge) remarkable potential to spark advances in spintronics and energy harvesting technologies. Intensive efforts are underway to make the novel properties of Weyl magnets useful for future applications16,17,18,20,21,22.

Methods

Single-crystal growth

Single crystals of Mn3Ge were synthesized by the Bridgman and Bi self-flux methods, using high-purity starting materials (Bismuth—99.999%, Manganese and Germanium—99.999%) with an optimized ratio. The Bridgman growth follows a procedure similar to that described in previous studies of Mn3Sn9. The initial step was to obtain polycrystalline samples by melting mixtures of manganese and germanium in an arc furnace under argon atmosphere. Starting from these polycrystalline materials, the single crystals of Mn3Ge were grown using a single-zone Bridgman furnace, by applying a maximum temperature of 1080 °C and a growth speed of 1.5 mm/h. In the case of Si-doped samples, the mixed germanium and silicon are melted with pure manganese in an arc furnace with the stoichiometric ratio. The subsequent Bridgman growth of single crystals was achieved with a lower maximum temperature of 1050 °C and a slower growth speed of 0.5 mm/h. The crystals were then annealed at 740 °C for 3 days.

Crystal characterization

The crystals were characterized by single-crystal X-ray diffraction (RAPID, Rigaku) at room temperature. The lattice parameters are obtained by Rietveld refinement. All the samples were shown to be single phase, with lattice parameters consistent with previous work5. According to the energy dispersive X-ray analysis with a scanning electron microscope, the compositions of crystals obtained by Bi-flux and Bridgman methods are Mn3.03Ge0.97 and Mn3.04Ge0.96, respectively. The oriented single crystals were cut into a bar shape by spark erosion for transport and magnetization measurements.

Magnetization measurements

The magnetization measurements on orientated samples were conducted using a commercial SQUID magnetometer (MPMS, Quantum Design) in the temperature range of 2–400 K.

Electrical resistivity and Hall effect studies

The longitudinal and Hall voltage signals were measured simultaneously in six-probe geometry; contacts to the crystals were made by gluing 20 μm gold wires with silver epoxy or attaching them by spot welding. Measurements at temperatures of 2–400 K were performed in the PPMS system (Quantum Design), and a helium-3 sample-in-vacuum insert system (HelioxVT, Oxford Instruments) was employed for low-temperature measurements at 0.3 K. The Hall contributions to the longitudinal resistivity and vice versa were eliminated by adding and subtracting the resistivity data taken at positive and negative magnetic fields. The uncertainties of the longitudinal resistivity and Hall resistivity are about 1-2%.

Thermoelectric studies

The thermoelectric properties were measured by the one-heater and two-thermometer configuration using the PPMS system (Quantum Design). Bar-shaped samples with a typical dimension of 10 × 2 × 2 mm3 were used for the measurements. By applying a temperature gradient $$- \nabla T$$ parallel to the long direction of the sample, the thermoelectric longitudinal and transverse emf voltages Vi and Vj were obtained in an open circuit condition. The thermal gradient $$- \nabla T$$ was monitored by two thermometers, which were 5 mm apart and were linked to the sample via strips of 0.5-mm-wide copper-gold plates. The magnitude of the transverse voltage $$\nabla V$$ was linearly related to the applied temperature difference $$\nabla T$$, which was typically set to be 1.5% and 2.0% of the sample temperature for Seebeck and Nernst measurements, respectively. The Seebeck coefficient Sii and Nernst signal Sji were derived as $$S_{ij} = E_i/\nabla T$$ and $$S_{ji} = E_j/\nabla T$$ where Ei and Ej are the longitudinal and transverse electric fields. The magnetic field dependence of the Nernst signal was obtained after removing the longitudinal component, which is roughly field independent. The uncertainties of the Seebeck and Nernst signals are dominated by the uncertainties of the corresponding geometrical factors and are estimated to be 10–20%.

Density functional theory (DFT) calculations of the band structure

The electronic structure calculations were performed using the Quantum ESPRESSO package90. The exchange–correlation energy functionals were considered within the generalized gradient approximation, following the Perdew–Burke–Ernzerhof scheme91. A 7 × 7 × 7 k-point grid and projector augmented wave pseudopotentials92 were applied for the calculations. The cut-off energies of 80 and 320 Ry were chosen for the wave functions and the charge density, respectively. By using the Wannier90 code93,94,95, the Wannier basis set was constructed from the Bloch states obtained in the DFT calculation, which includes 292 bands. The Wannier basis mentioned above consists of localized (s,p,d)-character orbitals at each Mn site, and (s,p)-character orbitals at Ge site, which gives 124 orbitals/u.c. in total. The Berry curvature, along with the transverse thermoelectric conductivity63, was calculated using a Wannier-interpolated band structure94 with 70 × 70 × 70 k-point grid and additional adaptive k-point grid of 3 × 3 × 3 in regions where the Berry curvature is large.

Thermoelectric conductivity

To calculate the thermoelectric conductivity, we begin with the calculation of the AHC, σAH, at finite temperature:

$$\sigma _{{\mathrm{AH}}} = \sigma _{{\mathrm{AH}}}^I + \sigma _{{\mathrm{AH}}}^{\rm{II}}$$
(7)
$$\sigma _{{\mathrm{AH}}}^I = \frac{{ - \hbar }}{{2\pi {\mathrm{V}}}}{\Sigma}_k\mathop {\smallint }\limits_{ - \infty }^\infty d{\it{\epsilon }}\frac{{\partial f({\it{\epsilon }})}}{{\partial {\it{\epsilon }}}}{\mathrm{Tr}}\left[ {\hat J_XG^R\hat J_YG^A} \right]$$
(8)
$$\sigma _{{\mathrm{AH}}}^{\rm{II}} = \frac{{ - \hbar }}{{4\pi {\mathrm{V}}}}\mathop {\sum }\limits_k \mathop {\smallint }\limits_{ - \infty }^\infty d{\it{\epsilon }}f({\it{\epsilon }}){\mathrm{Tr}}\left[ {\hat J_X\frac{{\partial \hat G^R}}{{\partial {\it{\epsilon }}}}\hat J_Y\hat G^R - \hat J_XG^R\hat J_Y\frac{{\partial \hat G^R}}{{\partial {\it{\epsilon }}}} - < R \to A > } \right]$$
(9)

here $$\hat J_X$$ and $$\hat J_Y$$ denote the current operator in the x and y direction, and $$\hat G^R$$and $$\hat G^A$$ represent the retarded and advanced Green’s function, respectively. V is the volume of the system. In the regarded Green’s function, we consider the self-energy:

$${\Sigma} = \left( {1 - \frac{1}{{\mathrm{Z}}}} \right){\it{\epsilon }} - {\mathrm{iA}}{\it{\epsilon }}^2 - i\gamma$$
(10)

We obtained the thermoelectric conductivity, αji, following the Mott relation, that is:

$$\alpha _{ji} = - \frac{{\pi ^2}}{{3|e|T}}(k_BT)^2\frac{{\partial \sigma _{ji}({\it{\epsilon }})}}{{\partial {\it{\epsilon }}}}$$
(11)

Data availability

The data that support the findings of this study are available on reasonable request from the corresponding author (S.N.).

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Acknowledgements

This work is partially supported by CREST (JPMJCR18T3), by JST-Mirai Program Grant (JPMJMI20A1), Japan Science and Technology Agency, by Grants-in-Aids for Scientific Research on Innovative Areas (15H05882, 15H05883, and 15K21732) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan, by New Energy and Industrial Technology Development Organization, by Grants-in-Aid for Scientific Research (19H00650) from the Japanese Society for the Promotion of Science (JSPS). The work at the Institute for Quantum Matter, an Energy Frontier Research Center was funded by DOE, Office of Science, Basic Energy Sciences under Award # DE-SC0019331. The work for first-principles calculation was supported in part by JSPS Grant-in-Aid for Scientific Research on Innovative Areas (18H04481 and 19H05825) and by MEXT as a social and scientific priority issue (Creation of new functional devices and high-performance materials to support next-generation industries) to be tackled by using post-K computer (hp180206 and hp190169). M.F. acknowledges support from the Japan Society for the Promotion of Science (Postdoctoral Fellowship for Research in Japan (Standard)). The use of the facilities of the Materials Design and Characterization Laboratory at the Institute for Solid State Physics, The University of Tokyo, is gratefully acknowledged.

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S.N. conceived the project and planned the experiments. T.C., T.T., M.F., I.M., and S.N. performed the experiments and analyzed the data. S.M., T.K., M.K., F.I., and R.A. performed the first-principles calculations. D.N.H. and R.I. performed the TEM and ICP measurements and analyses. S.N., M.F., T.T., S.M., and T.C. wrote the paper. All authors discussed the results and commented on the paper.

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Correspondence to Satoru Nakatsuji.

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Chen, T., Tomita, T., Minami, S. et al. Anomalous transport due to Weyl fermions in the chiral antiferromagnets Mn3X, X = Sn, Ge. Nat Commun 12, 572 (2021). https://doi.org/10.1038/s41467-020-20838-1

• Accepted:

• Published:

• Anisotropic magnetic interactions in hexagonal AB -stacked kagome lattice structures: Application to Mn3X ( X=Ge,Sn,Ga ) compounds

• A. Zelenskiy
• , T. L. Monchesky
• , M. L. Plumer
•  & B. W. Southern

Physical Review B (2021)