Anomalous spin-Hall effect in 2D Cr-based materials

We study the THE in various systems on 2D magnetic materials (CrIS₃, CrCl₃, and CrBr₃). From the estimations a skyrmion size is about 30-40 nm. The computations of a spin Hall effect and magnetoresistance require the calculations of the nonequilibrium distribution function, which we will determine from the kinetic Boltzmann equation. The distribution function describes by a 4 × 4 matrix because of different spin projections and the spin-dependent momenta of conducting electrons. The matrix elements contain various parameters such as effective masses, exchange constants, etc., that will be found from the first principle calculations. The reason why such system exhibit topological Hall effect is because the scattering transition probability matrix has an antisymmetric part with respect to the scattering angle. The scattering asymmetry acts as an effective magnetic field, which sign can be either the same for both spin projections of an incident electron, hence leading to a topological charge Hall effect, or opposite for different electron spin projections, leading to the topological spin Hall effect