First-principles calculations on the spin anomalous Hall effect of ferromagnetic alloys

The spin anomalous Hall effect (SAHE) in ferromagnetic metals, which can generate spin-orbit torque to rotate the magnetization of another ferromagnetic layer through a non-magnetic spacer in magnetic junctions, has attracted much attention. We theoretically investigated the spin anomalous Hall conductivity (SAHC) of the L1$_0$-type alloys $X$Pt($X$=Fe,Co,Ni) on the basis of first-principles density functional theory and linear response theory. We found that the SAHC of FePt is much smaller than the anomalous Hall conductivity (AHC), leading to very small polarization for the anomalous Hall effect $\zeta$=SAHC/AHC of around 0.1. On the other hand, the SAHC increases with increasing number of valence electrons($N_{\rm v}$), and CoPt and NiPt show relatively large values of $|\zeta|$, greater than 1. The negative contribution of the spin-down-down component of AHC is the origin of the large SAHC and $\zeta$ in CoPt and NiPt, which is due to the anti-bonding states of Pt around the Fermi level in the minority-spin states.