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 bands^26,27,51, such that:

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 ρ_H of Mn₃Sn measured under in-plane magnetic fields $$ B\parallel \left[2\overline{1}\overline{1}0\right]$$ and $$ B\parallel \left[01\overline{1}0\right]$$ displays a sharp hysteresis loop with a small coercivity of approximately 100 Oe (Fig. 2a)⁴. The magnetic-field dependence of ρ_H is anisotropic; the sweep of the in-plane fields generates a substantial zero-field Hall component and a narrow hysteresis in ρ_yz and ρ_zx, whereas the out-of-plane field $$ B\parallel \left[0001\right]$$ yields a linear-in-B response in ρ_xy without hysteresis. The in-plane-field responses, ρ_yz and ρ_zx, 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 mμ_B/Mn originated from spin canting (Fig. 2b)⁴. The similar hysteretic and anisotropic behavior of ρ_H (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, σ_yz, σ_zx, and σ_xy, obtained under $$ B\parallel \left[2\overline{1}\overline{1}0\right]$$, $$ B\parallel \left[01\overline{1}0\right]$$, and $$ B\parallel \left[0001\right]$$, respectively⁴. The zero-field values of σ_yz and σ_zx 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 −σ_zx and about 70 Ω⁻¹ cm⁻¹ for −σ_yz before dropping sharply upon entering the low temperature spin-glass phase at T < 50 K. In contrast, σ_xy 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 σ_xy 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 σ_xy 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 ρ_xy (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 ρ_xy 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 texture^{26,53,54,57–59}. 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 MnGe^60–62. 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 σ_yz 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 temperatures^5,6. Contrary to the FM case, neither of σ_yz nor σ_zx 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 E_F, as the electron states at E_F are constituted solely by the 3d electrons^8,28.

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 E_F^9,27,63. The Nernst signal S_ji 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 V_j perpendicular to both the magnetic field B and the heat current $$ Q_i\propto -{\nabla }_{i}T$$ is measured at various temperatures. The Nernst coefficient $$ S_{xy}=E_y/\left(-{\nabla }_{x}T\right)$$ is related to the anomalous transverse thermoelectric conductivity α_xy such that $$ S_{xy}={\alpha }_{xy}{\rho }_{xx}-{\rho }_{xx}{\sigma }_{xy}{S}_{yy}$$, where ρ_xx, σ_xy, and S_yy are the longitudinal resistivity, the Hall conductivity, and the Seebeck coefficient, respectively⁴. The connection of α_xy with the Berry curvature near E_F is well established, namely:

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 ρ_H and the magnetization M, the field dependence of −S_ji is strongly anisotropic; the in-plane field components −S_yz and −S_zx 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 −S_xy remains zero within experimental resolution without showing any hysteresis. The comparison between the field dependence of −S_ji and M helps unveil the driving mechanism behind the ANE. Although −S_yz and M measured under $$ B\parallel \left[2\overline{1}\overline{1}0\right]$$ display nearly identical hysteresis loops in the low-field regime, −S_yz 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 octupole^44,45. The temperature dependence of the in-plane zero-field components, −S_yz and −S_zx, peaks at around 200 K with a maximum value of about 0.6 μVK⁻¹ for Mn_3.06Sn_0.94. With a slightly more Mn in Mn_3.09Sn_0.94, the peak in −S_yz and −S_zx 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 −S_yz 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 −S_yz and −S_zx 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 −S_yz and −S_zx are nearly identical, with magnitude consistent with our finding. Under a high magnetic field of 14 T, −S_yz and −S_zx 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 −S_xy is negligibly small compared to −S_yz and −S_zx in the studied temperature range. The behavior of S_ji 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 S_ji provides a rough estimate of the lowest Weyl node energy relative to E_F⁶⁴. 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, S_ii (T), of Mn_3.04Ge_0.96 for $$ Q\parallel \left[0001\right]$$ and $$ Q\parallel \left[01\overline{1}0\right]$$ configurations (Fig. 2h, inset). The behavior of S_ii (T) is again strongly anisotropic; while S_zz remains negative in the entire measured temperature range and is weakly dependent on temperature, S_yy 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 Mn₃Sn^9,19.

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, κ_xz (B) and κ_zy (B), in Mn₃Sn show a sharp hysteretic jump corresponding to the anomalous component^19,66 (Fig. 3a, b). The anomalous Wiedemann–Franz (WF) law is defined as follows:

Fig. 3

Anomalous transverse Wiedemann–Franz (WF) law found in Mn₃X (X = Sn, Ge).

a, b Field dependence of the thermal Hall conductivity κ_xz (a) and κ_zy (b). c Temperature dependence of the anomalous Lorentz number $$ L_{zx}^{\mathrm{AH}}={\left(e/k_B\right)}^{2}T{\kappa }_{zx}^{\mathrm{AH}}/{\sigma }_{zx}^{\mathrm{AH}}$$ for Mn_3.06Sn_0.94 (filled blue squares) and Mn_3.09Sn_0.91 (filled red squared). The horizontal axis indicates the temperature normalized by the Debye temperature, θ_D. The θ_D of the two samples is about 300 K. The dashed line marks $$ L_0=\frac{{\pi }^2}{3}$$. The $$ L_{xy}^{\mathrm{AH}}$$ of conventional ferromagnetic metals Ni with θ_D ≈ 450 K (open purple circles) and Fe with θ_D ≈ 470 K (open green triangles) are also included for comparison^67,68. Inset: temperature dependence of $$ L_{xy}^{\mathrm{AH}}$$ observed in Ni and NiCu alloys⁶⁸. The definition of $$ L_{zx}^{\mathrm{AH}}$$ and L₀ here is different from that in Eq. (3). d Temperature dependence of the anomalous Hall conductivity, $$ -{\sigma }_{zx}^{\mathrm{AH}}$$ (top), the anomalous thermal Hall conductivity divided by temperature, $$ -{\kappa }_{zx}^{\mathrm{AH}}/T$$ (middle), and the anomalous Lorenz number, $$ L_{zx}^{\mathrm{AH}}=T{\kappa }_{zx}^{\mathrm{AH}}/{\sigma }_{zx}^{\mathrm{AH}}$$ (bottom) for Mn₃Ge. This panel is replotted with data extracted from ref. ¹³. Adapted from ref. ¹⁹, APS (a, b), ref. ⁶⁶ (c), and ref. ¹³, AAAS (d).

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 limit^19,65–68 (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 K^19,66,69 (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 ATHE^68,70. 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 effects^{4,9,26,56,63,66,71}.

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 E_F 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 E_F 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 calculations^8,28. 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, M_xT and M_yT, 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, respectively^7,8. 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 E_F. In the case of Mn₃Sn, the two pairs of Weyl points lying in the vicinity of E_F 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 ~ E_F + 60 meV and their partners $$ {\mathrm{W}}_1^{-}$$ and $$ {\mathrm{W}}_2^{+}$$ appearing at a slightly higher energy E ~ E_F + 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, α_ji, 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, $$ {\alpha }_{xz}=\left({\rho }_{xx}{S}_{xz}-{\rho }_{xz}{S}_{zz}\right)\left({\rho }_{xx}{\rho }_{zz}+{\rho }_{xz}^2\right)$$. The −α_xz observed in Mn₃Sn and Mn₃Ge behaves similarly, namely, −α_xz 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 S_xz dominates α_xz (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 E_F formed by a crossing of linearized bands. The resulting fitting formula for −α_zx involves three key parameters, that is, the Berry curvature strength, $$ \tilde{\mathrm{\Omega }}$$, near E_F, the k-space nodal separation, Δk, between the two Weyl points, and the Weyl point energy, E, relative to E_F. 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 −α_zx (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 calculations^8,28.

Fig. 4

Transverse thermoelectric effects in Mn₃X (X = Sn, Ge) and their theoretical analyses.

a–c Temperature dependence of the Nernst coefficient, $$ -S_{ij}$$, measured under B = 14 T. $$ S_{zx}$$ (red circle), $$ S_{yz}$$ (blue diamond), and $$ S_{xy}$$ (black triangle). The label definitions are modified from the original ones in ref. ¹⁵ so that they become consistent with the results obtained by other groups. a Zero-field anomalous transverse thermoelectric conductivity, $$ -{\alpha }_{xz}$$, with the best fit to the minimal model of Weyl fermions (solid line) (b), and the two contributions to $$ -{\alpha }_{xz}$$ (c) in Mn₃Ge. d Temperature dependence of $$ -{\alpha }_{\mathit{xz}}$$ in Mn_3.06Sn_0.94 and Mn_3.09Sn_0.91 from ref. ⁹. e k-space distribution of the Weyl points (W₁–W₉) under $$ B\parallel \left[0\overline{1}10\right]$$ predicted by the DFT calculation in ref. ³⁰. The red and blue frames mark the W₁ and W₃ pairs that are consistent with the WP1 (f) and WP3 (WP′3) (g) pairs revealed by our calculation. Contour plots of the theoretical Berry-curvature spectrum |Ω_z| in the k_x–k_y-plane under $$ B\parallel \left[01\overline{1}0\right]$$ at k_z = 0 (Å⁻¹) (f) and at k_z = 0.137 (Å⁻¹) (g). The arrows indicate the Berry curvature arising from the WP1 (f) and WP3 (WP′3) (g) pairs. Each pair consists of two Weyl points with opposite chirality. Experimental $$ -{\alpha }_{zx}/T$$ (solid symbols) vs. theoretical curves obtained from the DFT calculations for Mn₃Sn (h) and Mn₃Ge (i). Calculations are made with the energy shifts in the chemical potential corresponding to the extra Mn in the off-stoichiometric samples, relative to the Fermi level E_F of the stoichiometric case (i.e., E = E_F). A convergence factor of γ = 0.02 and a coefficient AZ² = 5.0 eV⁻¹ are used in the calculation. The renormalized temperature T* = T/5 for Mn₃Sn is set accordingly to the ARPES results. A smaller renormalization factor T* = T/2 is chosen for Mn₃Ge given its larger bandwidth relative to that of Mn₃Sn. The inset of (i) shows the calculated Hall conductivity $$ -{\sigma }_{zx}$$ for Mn_3.06Ge_0.94 and Mn_3.03Ge_0.97 using the same parameters. Adapted from ref. ¹⁵, APS (a–d).

We next present a comparison between the experimentally observed −α_ji 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 E_F are occupied solely by 3d electrons, a small amount of Mn doping may generate a sizeable shift of the chemical potential relative to the E_F of the stoichiometric case. A 1% Mn-doping in Mn₃Sn may shift up the chemical potential by about 6 meV from E_F; thus, the two Mn compositions, Mn_3.06Sn_0.94 and Mn_3.09Sn_0.91, have the chemical potential of E = E_F + 36 meV and E = E_F + 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:

The Z value is estimated to be 0.2 for Mn₃Sn according to the renormalization factor estimated by the ARPES and specific heat measurements^7,8. Given the larger bandwidth of Mn₃Ge in comparison with Mn₃Sn^44,45, 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 −α_zx/T for Mn_3.06Sn_0.94 and Mn_3.09Sn_0.91 that are dominated by the Berry curvature stemming from the low-energy Weyl points. They reproduce the overall temperature dependence of the experimental −α_zx/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 – E_F = 78 meV (Fig. 4f) and 63 meV (Fig. 4g) under $$ B\parallel \left[01\overline{1}0\right]$$ on the k_z = 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, σ_ji shown in Fig. 2f decreases from 300 Ω⁻¹ cm⁻¹ in Mn_3.03Ge_0.97 to 170 Ω⁻¹ cm⁻¹ in Mn_3.04Ge_0.96. The calculated energy spectrum of σ_ji undergoes such a sharp change at E – E_F ≈ 60 meV; correspondingly, the calculated −σ_ji and −α_ji with the chemical potential of E – E_F = +50 meV and E – E_F = +68 meV roughly reproduces the T dependence and magnitudes observed for Mn_3.03Ge_0.97 and Mn_3.04Ge_0.96 (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 established^1,73. 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 materials^74,75, 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 k_z-dispersion of ARPES intensity along the H-K-H high-symmetry direction reveals a quasiparticle peak near E_F 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 k_z = 0 band dispersion involving the low-energy Weyl points. The ARPES constant-energy surface at E_F obtained using $$ h\upsilon =103\mathrm{eV}$$ captures six elliptical contours on the k_x–k_y-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 $$ E~E_F+60\mathrm{meV}$$. The evolution of this electron band is traced by the E–k_x cuts measured at various k_y 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 k_y, 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 k_x 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.

Fig. 5

ARPES study and the chiral anomaly in Mn₃X (X = Sn, Ge).

a k_z-dispersion along the H-K-H high-symmetry line (black arrows) obtained at 60 K by varying the incident photon energy hν from 50 to 170 eV. The blue arrow marks the quasiparticle peak developed around K point. b ARPES intensity at E_F in the k_x–k_y-plane obtained with hν = 103 eV. The solid curves represent the Fermi surface predicted by the DFT calculation. c ARPES intensity maps in the E–k_x-plane along the high-symmetry K-M-K direction at various k_y obtained at 60 K. The original spectrum intensity (left) is compared to the intensity divided by the energy-resolution convoluted Fermi–Dirac (FD) function. The solid lines represent the calculated band dispersion, with the red and blue curves marking the electron and hole bands that form the Weyl points. Right: the anomalies in the corresponding momentum distribution curves (MDCs) are marked by the vertical color bars. d Field dependence of the magnetoconductivity in Mn₃Sn for B || I (red) and B ⊥ I (blue) under a magnetic field $$ B\parallel \left[01\overline{1}0\right]$$ at 60 K. e Field dependence of the magnetoconductivity in Mn₃Sn for B || I measured at various temperatures. f Field dependence of the magnetoconductivity for B || I and B ⊥ I and with I || [0001] measured at 0.3 K in Mn₃Ge. The dashed line is a fit to the B² dependence. g Angle dependence of the longitudinal magnetoconductivity (top) and the planar Hall effect (PHE) (bottom) in Mn_3.06Sn_0.94 measured at 300 and 100 K under B = 3 T. The solid lines are the fits to the theoretical forms (see the main text for details). Adapted from ref. ⁸, Springer Nature (a–e).

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 movers^77,78. 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 E ⊥ B 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 E ⊥ B provides the transport evidence for Weyl fermions and has been widely studied for both nonmagnetic and magnetic WSM candidates^{8,21,29,79–82}.

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 E ⊥ B 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 WSM^83,84. 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 $$ \mathrm{\Delta }\sigma \left(B\right)=\left(\sigma \left(B\right)-\sigma \left(0\right)\right)$$ 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 $$ B\parallel \left[01\overline{1}0\right]$$ becomes negative in the E ⊥ B 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:

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 law^26,87–89. Figure 6a presents the transverse Hall conductivity, σ_H, as a function of the longitudinal conductivity, σ, for various single crystals of Mn_3+xX_1-x (X = Sn, Ge) with different composition in comparison with those of FM materials. Here, σ and σ_H 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 σ_H 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.

Fig. 6

Universal scaling relations for the magnetic Weyl semimetals.

a Universal scaling relation between the Hall conductivity, σ_H, and the longitudinal conductivity, σ, for Mn_3.04Ge_0.96 (2 and 10 K), Mn_3.06Sn_0.94 (60 and 100 K), and Mn_3.09Sn_0.91 (100 K)^4,9, plotted together with the data for various ferromagnets including transition metals (Ni, Gd, Fe, and Co thin films)⁸⁷, Fe single crystals⁸⁷, Co₃Sn₂S₂³¹, MnSi⁸⁹, Fe_1-xCo_xSi⁸⁹, perovskite oxides (SrRuO₃, La_1-xSr_xCoO₃, La_1-x(SrCa)_xMnO₃)⁷¹, spinels (Cu_1-xZn_xCr₂Se₄)⁷¹, pyrochlore (Nd₂(MoNb)₂O₇)⁷¹, and magnetic semiconductor (Ga_1-xMn_xAs, In_1-xMn_xAs, anatase–Co–TiO₂, rutile–Co–TiO₂)⁷¹. Within the good metal regime of $$ 3\times 10^3\le \sigma \le 5\times 10^5{\mathrm{\Omega }}^{-1}{\mathrm{cm}}^{-1}$$ (yellow shaded area), σ_H is predominated by the intrinsic Berry phase contribution. The value of σ_H lies below about 10³ Ω⁻¹ cm⁻¹ (horizontal dotted line) and is nearly constant with σ. In the regime of $$ \sigma \ge 5\times 10^5{\mathrm{\Omega }}^{-1}{\mathrm{cm}}^{-1}$$, the extrinsic skew scattering contribution dominates σ_H. The solid line presents the theoretical prediction by ref. ⁸⁸. Inset: zoomed plot showing the data for the Mn₃Sn and Mn₃Ge single crystals with different compositions, which corresponds to the region framed by the red rectangle in the main panel. b, c Full logarithmic plot of the magnetization (M) dependence of the Nernst coefficient $$ \mid S_{ji}\mid $$ (b) and the Hall conductivity σ_H (c) for Mn₃Sn (blue), Mn₃Ge (red), and other Weyl magnets^4,9,21,31, in comparison with those for ordinary ferromagnets^{9,26,71,87,96}. The red shades mark the regions in which $$ \mid S_{ji}\mid $$ or σ_H is linearly related to M for conventional ferromagnets. The Weyl magnets that violate this linear relation are located in the blue shaded region.

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 Mn_3.03Ge_0.97 and Mn_3.04Ge_0.96, 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 V_i and V_j 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 S_ii and Nernst signal S_ji were derived as $$ S_{ij}=E_i/\nabla T$$ and $$ S_{ji}=E_j/\nabla T$$ where E_i and E_j 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 code^93–95, 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, σ_AH, at finite temperature:

here $$ {\widehat{J}}_X$$ and $$ {\widehat{J}}_Y$$ denote the current operator in the x and y direction, and $$ {\widehat{G}}^R$$and $$ {\widehat{G}}^A$$ 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:

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