content_image/355290.jpg content_image/355288.jpg content_image/355287.jpg content_image/355289.jpg content_image/355285.jpg content_image/355286.jpg content_image/355284.jpg content_image/355283.jpg
where χ is a correlation function arising from the electron diffusion, given as ( V is the system volume) 

<img src='content_image/355290.jpg'>

Here σ$_{±}$ ≡ eD$_{±}$ν$_{±}$ is the spin-resolved Boltzmann conductivity divided by e , and the corre- lation length is given as 

The lifetime of the spin σ electron reads τ$_{s}$$_{,σ}$ ≡ τ$_{σ}$/ ( κ$_{⊥}$γ$_{σ}$ ). Defining µ$_{s}$ = µ$_{+}$ − µ$_{−}$ , we see that spin-resolved effective potential satisfies 

In three-dimensions, the correlation function reads 

<img src='content_image/355288.jpg'>

<img src='content_image/355287.jpg'>

<img src='content_image/355289.jpg'>

The local part of the spin current arises also from the anomalous spin current due to the spin-orbit interaction, defined in Eq. (16). This contribution is calculated by evaluating the last two diagrams in Fig. 2 as (see Appendix C) 

where 

<img src='content_image/355285.jpg'>

<img src='content_image/355286.jpg'>

This spin-orbit correction to the spin conductivity is anisotropic, resulting in a spin version of the anisotropic magnetoresistance (AMR) effect, namely, spin AMR effect. 

From Eqs. (24)(28)(35), the leading contribution to the spin current for small q and Ω is obtained as the sum of the local part driven by the electric field and the diffusive part as 

<img src='content_image/355284.jpg'>

where σ$_{s}$ ≡ σ 0 s + δσ$_{s}$ and n is the unit vector along the magnetization. In terms of the angle θ defined by cos θ ≡ ( n · E ) /E , the magnitude of the field-driven (local) current reads 

<img src='content_image/355283.jpg'>

