Isotropic and anisotropic spin-dependent transport in epitaxial Fe$_3$Si

We investigate spin-dependent transport phenomena in epitaxially grown Fe$_3$Si films, focusing on the anisotropic magnetoresistance (AMR), planar Hall effect (PHE), anomalous Hall effect (AHE), and spin Hall effect (SHE). While the sign and magnitude of the AMR and PHE depend on the current orientation relative to the crystallographic axes, the AHE and SHE remain nearly independent of the current orientation. The anisotropic AMR and PHE are attributed to current and magnetization dependent local band properties, including band crossing/anticrossing at specific $k$ points. In contrast, the isotropic AHE and SHE arise from the Berry curvature integrated over the entire Brillouin zone, which cancels local variations. These findings highlight the interplay between symmetry, band structure, and magnetization in the spin-dependent transport phenomena.