Critical Spin Fluctuation Mechanism for the Spin Hall Effect

We propose mechanisms for the spin Hall effect in metallic systems arising from the coupling between conduction electrons and local magnetic moments that are dynamically fluctuating. Both a side-jump-type mechanism and a skew-scattering-type mechanism are considered. In either case, dynamical spin fluctuation gives rise to a nontrivial temperature dependence in the spin Hall conductivity. This leads to the enhancement in the spin Hall conductivity at nonzero temperatures near the ferromagnetic instability. The proposed mechanisms could be observed in $4d$ or $5d$ metallic compounds.

Figure 1

Scattering processes involving (a) ${\mathcal{F}}_{0,1}$ terms, (b) ${\mathcal{F}}_2$ terms, and (c) ${\mathcal{F}}_3$ terms. Yellow arrows indicate conduction electrons, and green arrows indicate local moments. In the ${\mathcal{F}}_{2(3)}$ scattering processes, the electron deflection depends on the direction of the local moment (the electron spin), leading to the side-jump-type (skew-scattering-type) contribution to ${\sigma }_{SH}$.

Figure 2

Diagrammatic representation for the side-jump contribution. Solid (wavy) lines are the electron Green’s functions (the spin fluctuation propagators). Squares (circles) are the spin (charge) current vertices, with filled symbols representing the velocity correction with ${\mathcal{F}}_2$, i.e., side jump. Filled triangles are the interaction vertices with ${\mathcal{F}}_{0,1}$.

Figure 3

Diagrammatic representation for the skew-scattering contribution. Filled pentagons are the interaction vertices with ${\mathcal{F}}_3$. The definitions of the other symbols or lines are the same as in Fig. 2.

Figure 4

Schematic temperature dependence of (a) ${\tau }_{sf}^{-1}$ and $I(T,\delta )$, (b) ${\tau }_{\mathrm{dis}}^{-1}$ (dashed line), ${\tau }_{ee}^{-1}$ (dotted line), and ${\tau }_{ep}^{-1}$ (dash-dotted line), and (c) ${\sigma }_{SH}^{\text{skew scat}}$. Red lines and blue lines correspond to the clean system and the dirty system, respectively. At a low (intermediate, high) temperature regime, ${\tau }_c^{-1}$ is dominated by ${\tau }_{\mathrm{dis}}^{-1}$ (${\tau }_{sf}^{-1}$, ${\tau }_{ee}^{-1}$ or ${\tau }_{ep}^{-1}$), creating ${\sigma }_{SH,\max }^{\text{skew scat}}$ at $T_{\max }$.