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is anisotropic like the AMR effect for the charge current (we call the effect as spin AMR effect). The diffusive contribution is given as a gradient of a spin chemical potential, µ$_{s}$ . We will derive the linear-response expression for the spin chemical potential. Our microscopic study on the spin current demonstrates the validity of the half-phenomenological treatments [2]. 

## II. MODEL 

We consider the conduction electron system taking account of the spin-orbit interaction, the impurity scattering without spin flip, and the applied electric field. The Hamiltonian of the system is given as H = H$_{0}$ + H$_{so}$ + H$_{em}$ + H$_{imp}$ , where H$_{0}$ is the free electron Hamiltonian including the uniform magnetization, H$_{so}$ is the spin-orbit interaction, H$_{em}$ is the interaction with the gauge field representing the applied electric field, and H$_{imp}$ is the spin-independent impurity scattering. The free part reads 

<img src='content_image/364116.jpg'>

where the electron creation and annihilation operators are denoted by c † and c , respectively, ϵ$_{k}$$_{σ}$ ≡ k 2 2 m − ϵ$_{F}$ − σM , ϵ$_{F}$ is the Fermi energy, M is the spin splitting due to the magnetization and σ ≡ ± represents the spin. The spin-orbit interaction is represented by H$_{so}$ = H$_{so}$ 0 + H$_{so}$ $^{A}$, where ( σ$_{k}$ ( k = x, y, z ) is the Pauli matrix) 

<img src='content_image/364114.jpg'>

<img src='content_image/364115.jpg'>

(We suppress the spin index when obvious, namely, c = ( c$_{+}$, c$_{−}$ ). ) The spin-orbit potential v ( k ) so is assumed to arise from random impurities and to depend on the spin direction ( k ). The averaging over the spin-orbit potential is carried out as 

where n$_{so}$ and λ$_{so}$ are the concentration of the spin-orbit impurities and the strength of the interaction, respectively. The average of the spin-orbit potential at the linear order is zero in our model, and thus we do not take account of the the anomalous Hall and spin Hall effects. 

<img src='content_image/364113.jpg'>

