Theoretical Study on Anisotropic Magnetoresistance Effects of I//[100], I//[110], and I//[001] for Ferromagnets

We theoretically study anisotropic magnetoresistance (AMR) effects for ferromagnets with a tetragonal crystal field. Here, the tetragonal distortion exists in the [001] direction, the magnetization lies in the (001) plane, and the current I direction, i, is i=I//[100], I//[110], or I//[001]. We extend our previous model [1] to a model including the crystal field effect [2]. Using the model, we obtain analytical expressions of the resistivity and the AMR ratio for each i. In particular, the AMR ratio is expressed as AMRⁱ(φ)=C₀ⁱ+C₂ⁱcos(2φ)+C₄ⁱcos(4φ), where φ is the relative angle between the magnetization direction and the current direction and C₀ⁱ, C₂ⁱ, and C₄ⁱ are coefficients of the constant term, twofold symmetric term, and fourfold symmetric term for i. The coefficients are composed of a spin-orbit coupling constant, an exchange field, a crystal field, and s-s and s-d scattering resistivities. As a result, we confirm the relation of C₄^I//[100]=-C₄^I//[110], which was experimentally observed for Fe₄N [3] and Ni [4].
[1] S. Kokado et al., J. Phys. Soc. Jpn. 81, 024705 (2012).
[2] S. Kokado et al., J. Phys. Soc. Jpn. 84, 094710 (2015).
[3] M. Tsunoda et al., Appl. Phys. Express 3, 113003 (2010).
[4] G. Dedié, J. Phys. F 5, 706 (1975).