Theoretical Study of Anisotropic Magnetoresistance Effect: Magnetization Direction Dependence in Ferromagnet with Crystal Field of Cubic Symmetry

We theoretically study the magnetization direction dependence of anisotropic magnetoresistance (AMR) effect of a ferromagnet with a crystal field of cubic symmetry. We first extend our theoretical model [1] to a model including the crystal field effect. Using the model, we next obtain an analytical expression of the AMR ratio; that is, ${\rm AMR}(\theta)$=$C_0 + C_2 \cos (2 \theta) + C_4 \cos (4 \theta)$, where $\theta$ is the relative angle between the magnetization direction and the electric current direction. The coefficients $C_0$, $C_2$, and $C_4$ are expressed by a spin-orbit coupling constant, an exchange field, a crystal field, and s-s and s-d scattering resistivities. Using this expression, we analyze the following experimental results for Fe$_4$N [2] and Co$_2$MnSi Heusler alloy [3]: As for Fe$_4$N [2], $|C_2|$ and $|C_4|$ increase with decreasing temperature. In contrast, Co$_2$MnSi Heusler alloy [3] has a small temperature dependence of $C_2$ and $C_4 \sim 0$. \\ $[1]$ S. Kokado {\it et al}., J. Phys. Soc. Jpn. {\bf 81} (2012) 024705. \\ $[2]$ M. Tsunoda {\it et al}., Appl. Phys. Express {\bf 3} (2010) 113003. \\ $[3]$ F. J. Yang {\it et al}., Phys. Rev. B {\bf 86} (2012) 020409.