The magnetic Weyl semimetallic state can lead to intriguing magnetotransport, such as chiral anomaly and the layered quantum Hall effect. Mn3X (X = Sn, Ge) is a noncollinear antiferromagnetic semimetal where a Weyl semimetallic state is stabilized by time-reversal symmetry breaking. Compared to the well-studied Mn3Sn, the Weyl fermion-induced magnetotransport in Mn3Ge has been merely studied. Here, we report an in-depth study on the magnetotransport in a microfabricated Mn3Ge single crystal from room temperature to 10 K. We reveal an anomalous anisotropic magnetoresistance with fourfold symmetry and a positive high-field longitudinal magnetoresistance below the critical temperature (160–170 K). The possible origin is the temperature-induced tilting of the Weyl nodes. Our study helps to understand the magnetotransport properties in the Weyl fermion system.

Three-dimensional gapless topological excitation in Weyl and Luttinger semimetals can manifest novel properties in condensed-matter systems.^{1–5} In the Weyl semimetallic state, the nondegenerate energy bands linearly touch near the Fermi level and form the Weyl nodes due to the underlying symmetry breaking.^{6} A pair of Weyl nodes with opposite chirality act as the source and sink of the large Berry curvature, leading to nontrivial magnetotransport.^{7,8} The fingerprint of Weyl fermions is the chiral anomaly.^{9} Under the electric field applied parallel to the magnetic field, the charge pumping between two Weyl nodes with opposite chirality results in a negative longitudinal magnetoresistance (LMR). Generally, symmetry constraints involving either the inversion or time-reversal symmetry (TRS) topologically protect the Weyl fermion state.^{10} Therefore, the Weyl nodes cannot annihilate but tilt in the momentum space via any perturbation, respecting the translational symmetry.^{2} Moreover, the tilting of Weyl nodes may result in additional magnetotransport responses where the conduction electrons acquire the shifted distribution and anomalous velocity.^{11–14}

Among the Weyl fermion systems, the noncollinear Weyl semimetals Mn_{3}X (X = Sn, Ge) have been intensively studied because of their giant spontaneous anomalous Hall effect (AHE) even with vanishingly small net magnetization.^{15–18} These compounds host a hexagonal crystal structure with an antichiral spin texture.^{19} The time-reversal symmetry breaking stabilizes the magnetic Weyl fermion state. The correlated electronic structure has been theoretically calculated^{20} and experimentally verified by angle-resolved photoemission spectroscopy.^{3} Despite the similar crystal structure, Mn_{3}Ge exhibits many more pairs of Weyl nodes than Mn_{3}Sn, connected by the Fermi arcs on the Fermi surface.^{20} It may result in a more complicated magnetotransport in Mn_{3}Ge. Pioneering work on the chiral anomaly and anomalous Nernst effect has been reported for Mn_{3}Ge.^{21,22} Still, the study of magnetotransport driven by Weyl fermions in Mn_{3}Ge is far less than the well-studied Mn_{3}Sn.^{3,21,23} Here, we report the magnetotransport in a Mn_{3}Ge single crystal device microfabricated by focus ion beam (FIB) from room temperature down to 10 K. In addition to the twofold magnetotransport consistent with the chiral anomaly, it is discovered that a fourfold symmetric AMR behavior occurs below 160 K. It is in good agreement with the theoretical results, considering the tilting Weyl node.^{13} Meanwhile, the LMR under a high magnetic field regime (>2 T) changes the sign from negative to positive below 170 K.

The bulk Mn_{3}Ge single crystal is grown by the bismuth flux method shown in our previous report.^{24} The chemical composition is determined to be 3.03:0.97 by energy dispersive x-ray (EDX) measurement. We use the FIB to microfabricate the device, with the configuration shown in Fig. 1(a) for AMR measurement. The device is about 50 *µ*m in length, 8 *µ*m in width, and 2 *µ*m in thickness. The applied magnetic field is rotated in the Kagome plane with an angle *φ*. Before the AMR measurement, we confirmed the FIB fabricated sample quality by the temperature dependence of the longitudinal resistivity, as shown in Fig. 1(b). It shows the metallic–semimetallic transition behavior at ∼230 K, the same as the bulk. Figure 1(c) shows the AMR signals as a function of *φ* measured from 340 K down to 10 K under a magnetic field of 1.5 T. We further plot the AMR ratio = (*ρ*_{‖} − *ρ*_{⊥})/*ρ*_{⊥} × 100% and Δ*ρ*_{xx} = *ρ*_{‖} − *ρ*_{⊥} as a function of the temperature in Fig. 1(d), where ‖ and ⊥ refer to the parallel and perpendicular directions of the magnetic field relative to the applied current, respectively. Mn_{3}Ge shows a negative AMR, varying from −0.1% to −0.5%. The amplitude of the AMR ratio increases and then decreases with decreasing temperature, reaching the maximum at 100 K.

In general, the AMR signal follows $\rho xx\phi =\rho 0+\Delta \rho xx\u2061cos2\u2061\phi $, which exhibits a twofold symmetry. However, $\rho xx\phi $ deviates from the twofold function at low temperatures, as shown in Fig. 1(c). To understand the origin, we performed AMR measurements from 340 K down to 10 K under magnetic fields of 1.5, 4, 7, and 9 T. Figure 2(a) shows the angular dependence of −Δ*ρ*_{xx} under various magnetic fields at 280 K. The AMR signals have similar twofold symmetric angular dependence, but the amplitude increases when increasing the magnetic field from 1.5 to 9 T. It can be well-fitted with the cos^{2} *φ* function, as shown in Fig. 2(b). We also performed the angular dependence measurement at 10 K, as shown in Fig. 2(c). Surprisingly, none of them follow the cos^{2} *φ* function from 1.5 to 9 T. The AMR begins to have an additional fourfold symmetric contribution (cos^{4} *φ*), besides the cos^{2} *φ* component at 10 K. Therefore, we decompose the angular dependence curve under 9 T into cos^{2} *φ* and cos^{4} *φ* contributions, as shown in Fig. 2(d). Noticeably, the magnitude of the cos^{4} *φ* component becomes as large as the cos^{2} *φ* component at 10 K, a clear contrast to the 280 K case. We summarize both cos^{2} *φ* and cos^{4} *φ* contributions extracted from the angular variation in −Δ*ρ*_{xx} as a function of the magnetic field at 280 and 10 K cases in Fig. 2(e). It is clear that the magnitude of the cos^{2} *φ* contribution is proportional to the magnetic field at 280 K but is not at 10 K. The magnitude of the cos^{4} *φ* contribution is nearly zero at 280 K while it is finite and independent of the magnetic field at 10 K. It indicates that the fourfold symmetric AMR signal is caused by the temperature rather than the magnetic field. To verify the temperature-caused fourfold symmetric AMR effect, we plot the cos^{2} *φ* and cos^{4} *φ* contributions of −Δ*ρ*_{xx} as a function of temperature under magnetic fields of 1.5 and 9 T, as shown in Fig. 2(f). The cos^{4} *φ* contribution becomes observable as the temperature decreases below 160 K under magnetic fields of 1.5 and 9 T. It is clear that the temperature induces the additional fourfold symmetric AMR. In addition, the cos^{2} *φ* component changes the sign when the temperature is lower than 130 K.

Here, we try to analyze the possible origin of the temperature-induced anomalous magnetotransport. As is shown in the supplementary material, there is no distinct difference in the magnetization curves with temperature. We thus exclude the effect of magnetism on the magnetotransport. Generally, a slight change in the magnetic structure can affect magnetotransport. Therefore, we mainly focus on the temperature-driven magnetic structure variation. While both E_{1g} and B_{2g} symmetries with opposite chirality are energetically equivalent in Mn_{3}Ge,^{25} the neutron diffraction has demonstrated that the ground-state spin structure with E_{1g} symmetry in Mn_{3}Ge is stable due to the Dzyaloshinskii–Moriya interaction over the whole temperature range.^{19,26,27} Thus, the temperature should not change the chirality of the magnetic structure. The previous theoretical report shows that the tilting Weyl nodes under E_{1g} symmetry lead to a higher fold symmetric AMR and planar Hall contribution, consistent with our experimental observation.^{13} As shown in Figs. 3(b) and 3(a), the tilting vector ** t** = (cos

*θ*

*,*sin

*θ*

*,*0) is assumed relative to the reference position under E

_{1g}symmetry. Meanwhile, the spins are rotated by −

*φ*when the magnetic field

**=**

*B**B*(cos

*φ*

*,*sin

*φ*

*,*0) is applied as shown in Fig. 3(a). As a result, the Weyl nodes acquire a new tilting angle

*θ*–

*φ*, as shown in Fig. 3(b). The ultimate longitudinal conductivity is expressed as $\sigma xx=\u2211n=06a\u0304n\u2061cos(n\phi )$, where the sign and magnitude of coefficient $a\u0304n$ are determined by chirality and

*θ*. For the chirality of E

_{1g}symmetry, only even

*n*components are nonzero. Thus, the fourfold symmetric component can be expected. As for the role of temperature, it may slightly change the magnetic structure, for example, by modifying the in-plane lattice constant parameter

^{27}and may affect the tilting angle

*θ*. Thus, the coefficient $a\u0304n$ for the fourfold component may vanish at a high-temperature regime.

The AMR effect is attributable to the chiral anomaly in the Weyl fermion system. Therefore, we expect the corresponding temperature-driven anomalous chiral anomaly behavior in Mn_{3}Ge. A negative LMR usually evidences the chiral anomaly. The Landau levels shown in Fig. 4(a) characterize the Weyl semimetallic states. The zeroth Landau levels form the linear dispersions with opposite chirality (*χ* = ±1) in the Weyl fermion system. Once the electric field *E* is applied parallel to the magnetic field *B*, the electrons will be pumped from one Landau level with *χ* = +1 to the other with *χ* = −1.^{28} Consequently, the population difference of electrons will cause chemical potential imbalance and facilitate conduction. Meanwhile, the imbalance relaxes via the scattering between two nodes and finally reaches an equilibrium state.^{29}^{,} Figure 4(b) shows the LMR for the magnetic field under *B* // *E* // *x* and *B* ⊥ *E* configurations at 280 K. Clearly, the negative LMR occurs under the *B* // *E* configuration but not under the *B* ⊥ *E* configuration. This negative LMR was also reported in Mn_{3}Sn in the whole temperature.^{3,23} However, when the temperature decreases down to 10 K, Mn_{3}Ge exhibits a negative LMR under the low magnetic field regime (<2 T) while it exhibits a positive LMR at the high magnetic field regime (>2 T), as shown in Fig. 4(c). Interestingly, Chen *et al.*^{21} measured the chiral anomaly of Mn_{3}Ge bulk single crystals under the *B* // *E* // *z* configuration at 0.3 K, but there is no sign change. It indicates that the temperature may only affect the in-plane magnetic structure. To study the temperature-induced anomalous LMR, we plot *dρ*/*dB* as a function of temperature, as shown in Fig. 4(d). We can clearly see that the LMR starts to change the sign from negative to positive when the temperature is lowered below 170 K. This fact corresponds to the appearance of the fourfold symmetric AMR effect due to the tilting Weyl nodes. Noticeably, the positive LMR under a high magnetic field may also originate from the Coulomb interaction among the electrons.^{30} As temperature decreases, the relaxation rate between two nodes slows down. The increasing population of conduction electrons within one node enhances the Coulomb interaction under the high magnetic field regime, which may contribute to the positive LMR. In addition, a recent theory^{31} suggested that the fourfold symmetric AMR in ferromagnetic metals may be caused by the relaxation time anisotropy brought on by variations in the Fermi-level density of states. It may also play a similar role in antiferromagnetic Weyl semimetals. However, such a contribution should be small in a hexagonal system, such as Mn_{3}Ge.

In conclusion, we have reported temperature-induced anomalous magnetotransport in the Weyl semimetal Mn_{3}Ge. An additional fourfold symmetric AMR and a positive LMR occur below ∼160 K. The possible origin is the temperature-induced tilting of Weyl nodes. The fourfold symmetric AMR has been experimentally observed in various material systems, but the origin is still poorly understood.^{31} Our study may promote exploration of the microscopic mechanism underlying fourfold symmetric AMR.

*Note added in proof*. While preparing the article, we became aware of a similar work, Ref. 27. They proposed another possible origin: the high-field positive LMR is driven by the metallic–semimetallic transition near 200 K. Their neutron diffraction demonstrates that there is no observable magnetic structure change with temperature but there is a small vibration of the in-plane lattice constant. We think that this lattice constant vibration may lead to Weyl node tilting with temperature. This can be the reason that the sign change in the chiral anomaly occurs under the *B* // *E* // *x* configuration but does not occur under the *B* // *E* // *z* configuration with decreasing temperature.

## SUPPLEMENTARY MATERIAL

See the supplementary material for the magnetization characterization of Mn3Ge at room and low temperatures.

We thank S. Kurosawa for the helpful comments. This work was financially supported by JST-CREST (Grant No. JPMJCR18T3), the JST-Mirai Program (Grant No. JPMJMI20A1), the Japan Science and Technology Agency. M.W. would like to acknowledge the support from the JSPS Research Program for Young Scientists (Grant No. 21J21461). The work at the Institute for Quantum Matter, an energy frontier research center, was funded by the DOE, Office of Science, Basic Energy Sciences, under Award No. DE-SC0019331.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Mingxing Wu**: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal). **Kouta Kondou**: Methodology (supporting). **Taishi Chen**: Investigation (supporting); Methodology (supporting). **Satoru Nakatsuji**: Funding acquisition (equal); Writing – review & editing (supporting). **Yoshichika Otani**: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.