Topological Hall effect in ferromagnetic Weyl semimetal Mn$_5$Ge$_3$ originating in competing dipolar interaction and magnetocrystalline anisotropy

Abstract: We report the anomalous and topological Hall effect of the ferromagnetic Weyl semimetal Mn$_5$Ge$_3$. We observe a significant anisotropic anomalous Hall effect (AHE) due to nonzero Berry curvature in the momentum space, such that the anomalous Hall conductivity (AHC) is 965 S/cm for the $xy$-plane and 233 S/cm for the $zx$-plane of the single crystal. The band structure calculations predict several Weyl and nodal points span across the momentum space, gapped out under the spin-orbit coupling effect, leading to significant $k$-space Berry curvature and large AHC. Experimentally, we also demonstrate a sizeable topological Hall effect that is originated by the non-coplanar chiral spin structure due to the competition between the out-of-plane uniaxial magnetocrystalline anisotropy and the dipole-dipole interaction between two Mn sublattices. This study hints at the importance of dipole-dipole interactions in producing the skyrmion lattice in Mn$_5$Ge$_3$.