Anisotropy of the spin Hall effect in a Dirac ferromagnet

TL;DR

This paper investigates how ferromagnetic ordering affects the anisotropy of the intrinsic spin Hall effect in a Dirac Hamiltonian system, revealing persistent anisotropy even at zero magnetization and discontinuities at phase transitions.

Contribution

It introduces a minimal Dirac model with ferromagnetism to analyze the anisotropic spin Hall effect and uncovers how ferromagnetic ordering modifies interband transition rules and SHC behavior.

Findings

Spin Hall conductivity becomes axially anisotropic due to ferromagnetism.

Anisotropy persists even when magnetization approaches zero.

Discontinuity in SHC occurs at the transition between ferromagnetic and pristine Dirac phases.

Abstract

We study the intrinsic spin Hall effect of a Dirac Hamiltonian system with ferromagnetic exchange coupling, a minimal model combining relativistic spin-orbit interaction and ferromagnetism. The energy bands of the Dirac Hamiltonian are split after introducing a Stoner-type ferromagnetic ordering which breaks the spherical symmetry of pristine Dirac model. The totally antisymmetric spin Hall conductivity (SHC) tensor becomes axially anisotropic along the direction of external electric field. Interestingly, the anisotropy does not vanish in the asymptotic limit of zero magnetization. We show that the ferromagnetic ordering breaks the spin degeneracy of the eigenfunctions and modifies the selection rules of the interband transitions for the intrinsic spin Hall effect. The difference in the selection rule between the pristine and the ferromagnetic Dirac phases causes the anisotropy of the SHC, resulting in a discontinuity of the SHC as the magnetization, directed orthogonal to the electric field, is reduced to zero in the ferromagnetic Dirac phase and enters the pristine Dirac phase.