Absence of the anomalous Hall effect in planar Hall experiments

Recently, the planar Hall effect has attracted tremendous interest. In particular, an in-plane magnetization can induce an anomalous planar Hall effect with a $2\ensuremath{\pi}/3$ period for hexagon-warped energy bands. This effect is similar to the anomalous Hall effect resulting from an out-of-plane magnetization. However, this anomalous planar Hall effect is absent in the planar Hall experiments. Here, we explain its absence, by performing a calculation that includes not only the Berry curvature mechanism, as those in the previous theories, but also the disorder contributions. The conventional $\ensuremath{\pi}$-period planar Hall effect will occur if the mirror-reflection symmetry is broken, which buries the anomalous one. This is because the anomalous planar Hall effect is of the higher order with respect to the small $h/({E}_{F}\ensuremath{\tau})$, when compared to the conventional planar Hall effect, with ${E}_{F}$ being the Fermi energy and $\ensuremath{\tau}$ the relaxation time. We show that an in-plane strain can enhance the anomalous Hall conductivity and changes the period from $2\ensuremath{\pi}/3$ to $2\ensuremath{\pi}$. We propose a scheme to extract the hidden anomalous planar Hall conductivity from the experimental data. Our work will be helpful in detecting the anomalous planar Hall effect and could be generalized to understand mechanisms of the planar Hall effects in a wide range of materials.