Intrinsic anomalous Hall effect in nickel: A GGA + $U$ study

The electronic structure and intrinsic anomalous Hall conductivity of nickel have been calculated based on the generalized gradient approximation (GGA) plus on-site Coulomb interaction (GGA + $U$) scheme. The highly accurate all-electron full-potential linearized augmented plane-wave method is used. It is found that the intrinsic anomalous Hall conductivity (${\sigma }_{xy}^H$) obtained from the GGA + $U$ calculations with $U=1.9$ eV and $J=1.2$ eV is in nearly perfect agreement with that measured recently at low temperatures while, in contrast, the ${\sigma }_{xy}^H$ from the GGA calculations is about 100$\%$ larger than the measured one. This indicates that, as for the other spin-orbit interaction (SOI)-induced phenomena in 3$d$ itinerant magnets, such as the orbital magnetic magnetization and magnetocrystalline anisotropy, the on-site electron-electron correlation, though moderate only, should be taken into account properly in order to get the correct anomalous Hall conductivity. The intrinsic ${\sigma }_{xy}^H$ and the number of valence electrons ($N_e$) have also been calculated as a function of the Fermi energy ($E_F$). A sign change is predicted at $E_F=-0.38$ eV ($N_e=9.57$), and this explains qualitatively why the theoretical and experimental ${\sigma }_{xy}^H$ values for Fe and Co are positive. It is also predicted that fcc Ni${}_{(1-x)}$Co(Fe,Cu)${}_x$ alloys with $x$ being small would also have the negative ${\sigma }_{xy}^H$ with the magnitude being in the range of 500–1400 ${\Omega }^{-1}$cm${}^{-1}$. The most pronounced effect of including the on-site Coulomb interaction is that all the $d$-dominant bands are lowered in energy relative to the $E_F$ by about 0.3 eV, and consequently, the small minority spin $X_2$ hole pocket disappears. The presence of the small $X_2$ hole pocket in the GGA calculations is attributed to be responsible for the large discrepancy in the ${\sigma }_{xy}^H$ between theory and experiment.

Figure 1

Calculated anomalous Hall conductivity ${\sigma }_{xy}^H$ as a function of the inverse of the number of $k$ points in the Brillouin zone ($N_k$). The dashed lines are a polynomial fit to the calculated values to get the extrapolated value of ${\sigma }_{xy}^H$ at $N_k=\infty$. For comparison, the theoretical value by Wang et al. (Ref.10) and the experimental value by Ye et al. (Ref.15) are also shown as the horizontal dotted and dot-dashed lines, respectively.

Figure 2

Relativistic GGA and GGA + $U$ band structures. The GGA + $U$ band structure was obtained using $U=1.9$ eV and $J=1.2$ eV. The Fermi level (dotted horizontal line) is at 0 eV. The thick curves denote band 8’s, which are also indicated by the arrowed lines.

Figure 3

Relativistic band structure (a), anomalous Hall conductivity (${\sigma }_{xy}^H$) (b), and number of valence electrons ($N_e$) (c), calculated with (solid blue lines) and without (dashed red lines) on-site Coulomb interaction $U$. Both ${\sigma }_{xy}^H$ and $N_e$ were calculated using the fine $k$-point mesh of $106\times 106\times 106$. The GGA + $U$ results were obtained using $U=1.9$ eV and $J=1.2$ eV. The Fermi level (dotted horizontal line) is at 0 eV. In (a), the thick curves denote band 8’s, which are also indicated by the arrowed lines.