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

## TL;DR

This paper presents a relativistic and non-relativistic analysis of spin-polarized currents, spin-transfer torque, and spin Hall effect considering electromagnetic non-minimal coupling, revealing novel spin component fractionalization in 2D systems.

## Contribution

It introduces a relativistic framework for non-minimally coupled electrons and derives non-relativistic spin interactions, highlighting new effects on spin currents and Landau levels in 2D systems.

## Key findings

- Relativistic total angular momentum includes external electromagnetic torque effects.
- Spin-orbit interaction naturally emerges in the non-relativistic limit.
- Spin components become spatially separated due to non-minimal coupling in 2D.

## Abstract

In this contribution, we start off from a fully relativistic description of a single electron non-minimally coupled to an external electromagnetic field. Making direct use of the field equation, instead of canonically deriving from the Lagrangian density, the relativistic total angular momentum is attained, where the effect of the relativistic torque due to the external sources is also taken into account. 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 as well. In addition, to inspect the non-relativistic regime, a perturbative expansion of the field equation up to the leading order in $\left(v/c\right)$ is carried out, where spin-orbit interaction terms naturally 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 of the planar regime and the non-minimal coupling, a peculiar sort of fractionalization of the spin-up and -down components emerges with the different spin components localized in distinct positions.