Anomalous and spin Hall effects in hcp cobalt from GGA+U calculations

We have calculated the intrinsic anomalous and spin Hall conductivities, spin and orbital magnetic moments, and also spin polarization of Hall currents in hexagonal cobalt within the density functional theory with the generalized gradient approximation (GGA) plus onsite Coulomb interaction (GGA + U). The accurate all-electron full-potential linearized augmented plane-wave method is used. We find that the anomalous Hall conductivity (AHC) (sigma(A)(xy)) and orbital magnetic moment (m(o)) of cobalt with the magnetization being along the c axis (m parallel to c) calculated in the GGA + U scheme with U = 1.6 eV and J = 0.9 eV agree rather well with the corresponding experimental values while, in contrast, the sigma(A)(xy) and m(o) from the GGA calculations are significantly smaller than the measured ones. This suggests that moderate U = 1.6 eV and J = 0.9 eV are appropriate for Co metals. The calculated AHC and spin Hall conductivity (SHC) (sigma(S)) are highly anisotropic, and the ratio of the Hall conductivity for m parallel to c to that for the magnetization being in plane (m parallel to ab) from the GGA + U calculations, is similar to 16.0 for the AHC and similar to 6.0 for the SHC. For m parallel to c, the spin-up and spin-down Hall currents are found to flow in the same direction. The magnitude of the calculated Hall current spin polarization (P-H) is large and P-H = -0.61. Remarkably, for m parallel to ab, the spin-up and spin-down Hall currents are predicted to flow in the opposite directions. This indicates that the Hall current contains both spin-polarized charge current and pure spin current, resulting in the magnitude of the Hall current spin polarization (P-H = -1.59) being larger than 1.0.