Orientation Dependence of the Intrinsic Anomalous Hall Effect in hcp Cobalt

We carry out first-principles calculations of the dependence of the intrinsic anomalous Hall conductivity of hcp Co on the spin magnetization direction. The Hall conductivity drops from 481 to $116\mathrm{S}/\mathrm{cm}$ as the magnetization is tilted from the easy axis ($c$ axis) to the $ab$ plane. These values agree reasonably well with measurements on single crystals, while the angular average of $226\mathrm{S}/\mathrm{cm}$ is in excellent agreement with the value of $205\mathrm{S}/\mathrm{cm}$ measured in polycrystalline films. The strong intrinsic anisotropy is shown to arise from quasidegeneracies near the Fermi level.

Figure 1

Evolution of the components of the anomalous Hall conductivity parallel (${\sigma }_m$) and perpendicular (${\sigma }_{\theta }$) to the spin magnetization [Eq. (3)] as ${\mathbf{M}}_s$ is tilted by $\theta$ from the $c$ axis towards the $a$ axis. The solid lines are fits to the first-principles data, as described in the text. The left and right insets show, respectively, ${\sigma }_m(\pi /2,\phi )$ and ${\sigma }_{\phi }(\theta ,0)$.

Figure 2

Upper panel: Interband MCD spectrum for two magnetization directions. Lower panel: Cumulative contribution to the AHC from the spectrum above energy $\hslash \omega$, $A_m^{\mathrm{AH}}(\omega )=\frac{2}{\pi }{\int }_{\omega }^{{\omega }_{\max }}\frac{1}{{\omega }^{\prime }}\mathrm{Im}{\sigma }_m({\omega }^{\prime })d{\omega }^{\prime }$, truncated at ${\omega }_{\max }=7\mathrm{eV}$. Circles denote the AHC calculated directly from Eq. (1). Inset: Cumulative contribution to the gauge-invariant self-rotation per atom from the spectrum below energy $\hslash \omega$, $B_m^{\mathrm{SR}}(\omega )=\frac{V_c\hslash }{2\pi ec}{\int }_0^{\omega }\mathrm{Im}{\sigma }_m^a({\omega }^{\prime })d{\omega }^{\prime }$ ($V_c$ is the cell volume).

Figure 3

Upper panel: Energy bands close to the Fermi level for ${\mathbf{M}}_s\parallel c$ axis ($\theta =0$) and ${\mathbf{M}}_s\parallel a$ axis ($\theta =\pi /2$). Middle and lower panels: Berry curvature ${\mathbf{\Omega }}_{\mathbf{k}}$ for $\theta =0$, $\theta =\pi /2$, and $\theta =\pi /4$. The full height of the resonance peaks near the $L$ point is of the order of ${10}^3\mathrm{a}.\mathrm{u}.$ The high-symmetry points are given in lattice coordinates.

Figure 4

Evolution of ${\mathit{\sigma }}^a$ as the magnetization is rotated in the $ac$ plane, plotted as ${\sigma }_z^a(\cos \theta )$ and ${\sigma }_x^a(\sin \theta )$. Inset: Anisotropy ratio ${\sigma }_z^a(\theta =0)/{\sigma }_x^a(\theta =\pi /2)$ versus the reduced magnetization, according to the spin-fluctuation model [17]. The dotted line denotes the approximate location of the $\mathrm{hcp}\to \mathrm{fcc}$ transition.