Correlation between AMR and PHE in LSMO thin films

In magnetic conductors, the dependence of resistivity on the angle $\theta$ between the current and magnetization, a phenomenon known as anisotropic magnetoresistance (AMR), is given by $E_{x} = \Delta \rho_{\perp} j_{x} + (\Delta \rho_ {\parallel} - \Delta \rho_{\perp}) j_{x} \cos^{2} \theta$, where $\Delta \rho_{\parallel}$ and $\Delta \rho_{\perp}$ are the resistivities parallel and perpendicular to the magnetization, respectively. In addition, a transverse voltage, known as planar Hall effect (PHE), is generated with the angular dependence given by $E_{y} = (\Delta \rho_{\parallel} - \Delta \rho_{\perp}) j_{x} \sin \theta \cos \theta$. Doped manganite La$_{1-x}$Sr$_{x}$MnO$_{3}$ (LSMO) thin films have been demonstrated to exhibit both AMR and PHE with the expected angular dependencies\footnote{Y. Bason, L. Klein, J.-B. Yau, X. Hong, and C. H. Ahn, Appl. Phys. Lett. \textbf{84}, 2593 (2004)}. However, while $\Delta \rho_{AMR}$ and $\Delta \rho_ {PHE}$, the resistivities extracted from AMR and PHE with $\Delta \rho = (\rho_{\parallel} - \rho_{\perp})$, are expected to be identical, we find that $\Delta \rho_{AMR} \geq \Delta \rho_{PHE}$ throughout our measurements. Further investigation of this apparent discrepancy reveals that for a fixed magnetic field, $\Delta \rho_{AMR}$ and $\Delta \rho_{PHE}$ become nearly equal at temperatures around (below) $T_{c}$. Moreover, the discrepancy between $\Delta \rho_{AMR}$ and $\Delta \rho_ {PHE}$ becomes more substantial with increasing doping concentrations. We will discuss possible mechanisms responsible for this behavior.