Spin-polarized current, spin-transfer torque and spin Hall effect in presence of an electromagnetic non-minimal coupling

To start, a fully relativistic description of a single electron non-minimally coupled to an external electromagnetic field is provided. Making use of the field equation, the relativistic total angular momentum is attained, where the effect of the relativistic torque due to the external sources is also considered. Both the spin density and the orbital angular momentum tensors are identified in the relativistic-covariant approach. Furthermore, the symmetric and gauge invariant energy-momentum tensor is derived. To inspect the non-relativistic regime, a perturbative expansion of the field equation is carried out, where spin-orbit interaction terms emerge in the non-relativistic Hamiltonian as a consequence of the non-minimal coupling. Features related to the spin currents, the spin-transfer torque and their dependence on both magnetic and electric external fields are then calculated and discussed. Considering that spin-orbit coupled systems are of particular interest in the study of spin Hall effect, a two-dimensional (planar) scenario of the system is contemplated, where the contributions to the Landau levels are re-assessed. As a consequence, a peculiar sort of fractionalization of the spin-up and -down components emerges with the different spin components localized in distinct positions.