Fig. 1: Prediction of the anomalous Hall in-plane Hall effect in the Weyl ferromagnet Fe₃Sn.

a Left panel: shown is the top view of the lattice structure of Fe₃Sn within the two-dimensional kagome plane (xy-plane). The hexagonal unit cell (black solid line) and definition of Cartesian coordinates with respect to the crystallographic axes are indicated. Right panel: shown is an isometric view of the bilayer stacking of the kagome layer along the z-direction. b, c shown are the schematic distributions of the total Berry curvature within the xz-plane at k_y = 0 for an ideal in-plane ferromagnet with magnetization vector M = (M_x,  0,  0) and an in-plane ferromagnet with out-of-plane canting M = (M_x,  0,  M_z), respectively. The mirror plane \({{{\mathcal{M}}}}_{{{\rm{x}}}}\) is indicated. In case without canting, \({{{\mathcal{M}}}}_{{{\rm{z}}}}t{{\mathcal{T}}}\) is preserved and the total Berry curvature integrates to zero. A finite spin canting M_z breaks \({{{\mathcal{M}}}}_{{{\rm{z}}}}t{{\mathcal{T}}}\) resulting in a finite total Berry curvature. d, e shown are the energy-integrated calculated Berry curvature obtained from a Wannierized ab initio band structure of Fe₃Sn within the first hexagonal Brillouin zone (black solid line) at k_z = 0 without and with out-of-plane spin canting, respectively. f Shown is the calculated anomalous in-plane Hall conductivity σ_IPHE(ϕ_B). The azimuthal angle ϕ_B is defined as the angle between the direction of an in-plane magnetic field B_∥ and the in-plane component of the magnetization M_x, as schematically indicated. Without canting, M = (M_x,  0,  0), the symmetry \({{{\mathcal{M}}}}_{{{\rm{z}}}}t{{\mathcal{T}}}\) is preserved, resulting in σ_IPHE = 0 (blue solid line). Breaking of this symmetry by a finite out-of-plane canting M_z ≠ 0 and M = (M_x,  0,  M_z) permits a finite σ_IPHE(ϕ_B) that is modulated by B_∥ (solid red dots). Shown is the result for M_z/M_x = 0.1. Details of the model calculations are presented in the “Methods” section.