Anisotropy of the spin Hall effect in a Dirac ferromagnet

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.

Figure 1

(a) Band dispersions and (b) Fermi contours ($\mu /\mathrm{\Delta }=3$, positive energy branch) of the Dirac ferromagnet. The red colored bands correspond to $\eta =+1$, and blue colored bands correspond to $\eta =-1$. The magnetization is set to $M/\mathrm{\Delta }=0.4$.

Figure 2

Chemical potential dependence of intrinsic SHC contributions: (a) ${\sigma }_{21}^{3,\mathrm{iso}}$ and (b) ${\sigma }_{21}^{3,\widehat{m}}$. The strength of magnetization $\tilde{M}$ is set to $0.1,0.5,0.8$. The energy cutoff is set to ${\mathrm{\Lambda }}_{\varepsilon }=100$. The conductivity is in the unit of ${\sigma }_s^0=e\mathrm{\Delta }/8{\pi }^2$. The chemical potential is normalized by $\mathrm{\Delta }$, i.e., $\tilde{\mu }\equiv \mu /\mathrm{\Delta }$.

Figure 3

The angular dependence $\left(m_1\right)$ of intrinsic SHC ${\sigma }_{21}^3$ for various chemical potentials: (a) $\tilde{\mu }\equiv \mu /\mathrm{\Delta }=0$, (b) $\tilde{\mu }=1$, and (c) $\tilde{\mu }=2$. The black solid line (*) shows the results for the pristine Dirac Hamiltonian [29]. The strength of the magnetization $\tilde{M}\equiv M/\mathrm{\Delta }$ is set to $0.1,0.5,\mathrm{and}0.8$. The energy cutoff is set to ${\mathrm{\Lambda }}_{\varepsilon }=100$. The conductivity is in the unit of ${\sigma }_s^0=e\mathrm{\Delta }/8{\pi }^2$.

Figure 4

The magnetization strength $\tilde{M}$ dependence of intrinsic SHC ${\sigma }_{21}^3$ for various chemical potential $\tilde{\mu }$'s. The circled line groups correspond to the magnetization direction parallel with or perpendicular to $\widehat{x}$. The energy cutoff is set to ${\mathrm{\Lambda }}_{\varepsilon }=100$. The conductivity is in the unit of ${\sigma }_s^0=e\mathrm{\Delta }/8{\pi }^2$.

Figure 5

Comparison between (a) $C^{\mathrm{iso}}$ and $-D_{\mathrm{\Lambda }}^{\mathrm{iso}}$ (dashed); (b) $C^{\mathrm{iso}}$ and $-D_{{\mathrm{\Lambda }}_{\varepsilon }}^{\mathrm{iso}}$(dashed); (c) $C^{\widehat{m}}$ and $-D_{\mathrm{\Lambda }}^{\widehat{m}}$ (dashed); (d) $C^{\widehat{m}}$ and $-D_{{\mathrm{\Lambda }}_{\varepsilon }}^{\widehat{m}}$ (dashed). $M$ is set to be $0.1,0.5,0.8$, corresponding to color sequence from red to purple. The cutoff momentum/energy is set to be $\mathrm{\Lambda }={\mathrm{\Lambda }}_{\varepsilon }=100$, according to the limit of chemical potential ${\tilde{\mu }}_{\min }=-100$. The divergent parts are taken in opposite sign for comparison. The conductivity is in the unit of ${\sigma }_s^0=e\mathrm{\Delta }/8{\pi }^2$.