Theoretical Study on Four-fold Symmetric Anisotropic Magnetoresistance Effect in Cubic Single-crystal Ferromagnetic Model

Abstract

We theoretically investigated the four-fold symmetric term of the anisotropic magnetoresistance (AMR) effect in cubic ferromagnetic metals (FMs), which has recently been observed experimentally for certain metals, by using fourth-order perturbation theory with respect to the spin-orbit interaction (SOI). The AMR effect is a magnetoresistance effect specific to magnetic metals in which the charge conductivity changes depending on the direction of magnetization in FMs [1], and attracted much attention in the field of spintronics to apply for evaluation of half-metallicity [2] or spin-orbit torque generation [3].Usually, in experiments on bulk polycrystalline FMs, the longitudinal conductivity is proportional to the cos(2θ) where θ is the relative angle between the current direction and the magnetization direction, that is, the AMR effect has two-fold symmetry under the magnetization rotation. Theoretically, this two-fold AMR effect is caused by the SOIs and has been successfully explained by Campbell’s model, an s-d impurity scattering model with considering the second-order perturbation in terms of the SOIs [4].In single-crystal FMs such as Fe4N, however, the AMR measurements exhibited not only the conventional two-fold component but also the unconventional four-fold component proportional to cos(4θ) [5]. While this four-fold AMR effect is expected to stem from the symmetry of the cubic crystal, the microscopic origin is not so straightforward because the magnetization direction dependence of the conduction band is too small to explain this effect.Recently, a possible mechanism of the four-fold AMR effect is theoretically proposed by Kokado [6], wherein the s-d scattering coupled with the SOIs and “tetragonal” symmetric crystal fields is responsible for this effect. This theory is an extension of Campbell’s model in polycrystalline FMs to the single crystal FM model by taking into account the energy level splitting due to the tetragonal crystal fields on the d-states (Fig. 1). Since it is argued that the splitting among d-states is significant for the presence of a four-fold component, this theory requires the extra assumption of existing the tetragonal (uniaxial or planar) distortion to apply to the cubic crystals including the Fe4N. However, the experiment is confronted with the fact that the four-fold component appears even though the sample is almost cubic crystal and has little distortion. The microscopic origin is still controversial; therefore, it is worthwhile to explore a possible mechanism of the four-fold AMR effect in the cubic symmetric single-crystal FMs without tetragonal distortion.In this work, we show the presence of the four-fold AMR effect in cubic single crystal FMs from a microscopic viewpoint [7]. Our main finding is that the higher-order perturbation terms of SOI can also be responsible for the four-fold AMR effect in cubic FMs, instead of the second-order SOI with tetragonal distortion as mentioned in Kokado’s theory. Inspired by the Kokado model, we use the s-d impurity scattering model expressed by an impurity Anderson Hamiltonian with cubic crystal fields and SOI. The conductivity is represented on the basis of the Kubo–Greenwood formula. This approach enables us to consider the non-perturbative role of SOIs on the AMR effects.Firstly, to overlook a physical aspect of this effect, we carried out perturbative analysis with respect to the SOIs. The results indicate that the fourth-order terms of SOI can give rise to the four-fold AMR effect. Furthermore, we extract the two important roles of the SOI: first, to couple the electron’s momentum and magnetization, and second, to induce the orbital splitting among d-states instead of the tetragonal crystal field.Secondly, we performed numerical calculations to evaluate the non-perturbative behavior of the AMR effects. As the results, we successfully obtained both the two-fold and four-fold components of the AMR effect without assuming the tetragonal distortion, which is consistent with the perturbative analysis (Fig. 2). Other parameter dependencies will be presented at the conference. **