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

The spin anomalous Hall effect (SAHE) in ferromagnetic metals, which can generate a spin-orbit torque to rotate the magnetization of another ferromagnetic layer through a nonmagnetic spacer in magnetic junctions, has attracted much attention. We theoretically investigated the spin anomalous Hall conductivity (SAHC) of the $L{1}_0$-type alloys $X\mathrm{Pt}$ ($X=\mathrm{Fe},\mathrm{Co},\mathrm{Ni}$) 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 $\zeta =\mathrm{SAHC}/\mathrm{AHC}$ of around 0.1. On the other hand, the SAHC increases with an increasing number of valence electrons ($N_{\mathrm{v}}$), and CoPt and NiPt show relatively large values of $\left|\zeta \right|$, 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 antibonding states of Pt around the Fermi level in the minority-spin states.

Figure 1

Schematic viewgraph of (a) spin anomalous Hall effect (SAHE: magnetization $\widehat{\mathit{m}}$ is perpendicular to electric field $\widehat{E}$) and (b) magnetization-independent spin Hall effect (SHE: $\widehat{\mathit{m}}$ is parallel to $\widehat{E}$) in ferromagnets. (c) Crystal structure and coordinate system of $L{1}_0\text{−}X\mathrm{Pt}$ ($X=\mathrm{Fe},\mathrm{Co},\mathrm{Ni}$).

Figure 2

(a) Anomalous Hall conductivity (AHC) ${\left(\mathrm{\Omega }\mathrm{cm}\right)}^{-1}$ and (b) spin anomalous Hall conductivity (SAHC) $\left(\frac{\hslash }{e}\right){\left(\mathrm{\Omega }\mathrm{cm}\right)}^{-1}$ of $L{1}_0$-FePt, CoPt, and NiPt as a function of the number of valence electrons $N_{\mathrm{v}}$. The AHC and SAHC listed in Table 1 correspond to the values of $N_{\mathrm{v}}=18$ for FePt, $N_{\mathrm{v}}=19$ for CoPt, and $N_{\mathrm{v}}=20$ for NiPt, respectively.

Figure 3

Spin-decomposed AHC (a) ${\sigma }_{xy}^{\uparrow \uparrow }$, (b) ${\sigma }_{xy}^{\uparrow \downarrow }$, (c) ${\sigma }_{xy}^{\downarrow \uparrow }$, and (d) ${\sigma }_{xy}^{\downarrow \downarrow }$ ${\left(\mathrm{\Omega }\mathrm{cm}\right)}^{-1}$ for FePt, CoPt, and NiPt as a function of the number of valence electrons $N_{\mathrm{v}}$.

Figure 4

Color map of Berry curvatures ${\mathrm{\Omega }}_{xy}^{\mathrm{charge}}\left(\mathit{k}\right)$ in Eq. (2) for (a) FePt, (b) CoPt, and (c) NiPt and the nodal line $N\left(\mathit{k}\right)$ in Eq. (6) for (d) FePt, (e) CoPt, and (f) NiPt plotted on the Fermi surface in the three-dimensional Brillouin zone of the tetragonal unit cell, which are visualized with fermisurfer [31].

Figure 5

Projected density of states (DOS) onto each atomic orbital (a) $d\left(yz,zx\right)$, (b) $d(3z^2-r^2)$, (c) $d\left(xy\right)$, and (d) $d(x^2-y^2)$ for FePt, CoPt, and NiPt as a function of energy relative to the Fermi energy.