Twofold and Fourfold Symmetric Anisotropic Magnetoresistance Effect in A Model with Crystal Field

We theoretically study the twofold and fourfold symmetric anisotropic magnetoresistance (AMR) effects of ferromagnets. We here use the two-current model for a system consisting of a conduction state and localized d states. The localized d states are obtained from a Hamiltonian with a spin--orbit interaction, an exchange field, and a crystal field. From the model, we first derive general expressions for the coefficient of the twofold symmetric term ($C_2$) and that of the fourfold symmetric term ($C_4$) in the AMR ratio. In the case of a strong ferromagnet, the dominant term in $C_2$ is proportional to the difference in the partial densities of states (PDOSs) at the Fermi energy ($E_{\rm F}$) between the $d\varepsilon$ and $d\gamma$ states, and that in $C_4$ is proportional to the difference in the PDOSs at $E_{\rm F}$ among the $d\varepsilon$ states. Using the dominant terms, we next analyze the experimental results for Fe$_4$N, in which $|C_2|$ and $|C_4|$ increase with decreasing temperature. The experimental results can be reproduced by assuming that the tetragonal distortion increases with decreasing temperature.