Orientation and temperature dependence of the anomalous Hall effect in hcp cobalt

We calculate from first-principles the evolution of the intrinsic anomalous Hall conductivity vector $\vec\sigma^a$ of hcp Co as the spin magnetization direction $\hat{\bf M}$ is tilted away from the $c$-axis. We find that $\vec\sigma^a$ varies smoothly with the tilt angle $\theta$, and that its magnitude is strongly reduced, by a factor of about four, between $\theta=0$ and $\theta=\pi/2$, in good agreement with the measured anisotropy ratio of about three.\footnote{N. V. Volkenshtein {\it et al.}, Fiz. Metal. Metalloved. {\bf 11}, 152 (1961).} In addition to the anisotropic linear magnetization dependence ($\sigma^a_z/M_z\not=\sigma^a_x/M_x$) expected for any uniaxial crystal, there is a considerable nonlinearity in the dependence of $\sigma_x^a$ on $M_x=M\sin\theta$, while the relation between $\sigma_z^a$ and $M_z=M\cos\theta$ is essentially linear, as in ${\rm Mn}_5{\rm Ge}_3$.\footnote{C. Zeng {\it et al.}, Phys. Rev. Lett. {\bf 96}, 037204 (2006).} The overall angular dependence of $\vec\sigma^a$ is well-described by an expansion in terms of $l=1$ and $l=3$ spherical harmonics. From Zener's model for the influence of thermal fluctuations of $\hat{\bf M}({\bf r})$ on the temperature dependence of magnetic anisotropies,\footnote{C. Zener, Phys. Rev. {\bf 96}, 1335 (1954).} we predict that the $l=3$ terms give rise to an appreciable increase with temperature of the anisotropy ratio.