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

Yoshio Miura,Keisuke Masuda

doi:10.1103/physrevmaterials.5.l101402

Abstract

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 $\ensuremath{\zeta}=\mathrm{SAHC}/\mathrm{AHC}$ 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 $|\ensuremath{\zeta}|$, greater than 1. The negative contribution of the spin-down-down component of AHC is the origin of the large SAHC and $\ensuremath{\zeta}$ in CoPt and NiPt, which is due to the antibonding states of Pt around the Fermi level in the minority-spin states.

Highlights

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 found that the spin anomalous Hall conductivity (SAHC) of FePt is much smaller than the anomalous Hall conductivity (AHC), leading to very small polarization for the anomalous Hall effect ζ = SAHC/AHC of around 0.1
The spin current Js generated by the spin Hall effect (SHE) can be given by Js ∝ αSH[s × Jc], where αSH is the spin Hall angle given by the ratio of the spin Hall conductivity σxsypin to the conductivity of the charge current σxx, i.e., σxsypin/σxx, sis the direction of the quantization axis of an electron spin, and Jc is the charge current

Summary

Pt y aX x

Of first-principles calculations and linear response theory in order to clarify the possible origin of the SAHE. NiPt. In Table I, we show the calculation results for AHC σxAyHC, SHC σxSyHC, and SAHC σxSyAHC for L10-type FePt, CoPt, and NiPt alloys. AHC roughly decreases with increasing Nv, and the sign of AHC changes from positive to negative around Nv = 19–20, while the SAHC increases with increasing Nv and reaches a maximum around Nv = 20–21 These results indicate that the differences in AHC and SAHC for FePt, CoPt, and NiPt can be explained by the difference in the number of valence electrons within the rigid-band model. Each spin component σx↑y↑, σx↓y↓, σx↑y↓, and σx↓y↑ indicates the spin component of occupied and unoccupied states in Eq (2), which can be obtained by calculating the eigenvalues of the sz operator for each eigenstate |kn along the spin quantum axis. This means that the spin current of FePt is dominated only by σx↑y↑ according to Eq (5), which is much smaller than σxAyHC

In the case of

FePt is around