where χ is a correlation function arising from the electron diffusion, given as ( V is the system volume) 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}$$_{,σ}$ ≡ τ$_{σ}$/ ( κ$_{⊥}$γ$_{σ}$ ). Defining µ$_{s}$ = µ$_{+}$ − µ$_{−}$ , we see that spin-resolved effective potential satisfies In three-dimensions, the correlation function reads The local part of the spin current arises also from the anomalous spin current due to the spin-orbit interaction, defined in Eq. (16). This contribution is calculated by evaluating the last two diagrams in Fig. 2 as (see Appendix C) where This spin-orbit correction to the spin conductivity is anisotropic, resulting in a spin version of the anisotropic magnetoresistance (AMR) effect, namely, spin AMR effect. From Eqs. (24)(28)(35), the leading contribution to the spin current for small q and Ω is obtained as the sum of the local part driven by the electric field and the diffusive part as where σ$_{s}$ ≡ σ 0 s + δσ$_{s}$ and n is the unit vector along the magnetization. In terms of the angle θ defined by cos θ ≡ ( n · E ) /E , the magnitude of the field-driven (local) current reads j loc ,z s = √ ( σ$_{s}$$_{‖}$ ) 2 + (( σ$_{s}$$_{⊥}$ ) 2 − ( σ$_{s}$$_{‖}$ ) $^{2}$) sin 2 θ, (38) j z s = σ$_{s}$ E − δσ$_{s}$ ( n · E ) n − ∇ µ$_{s}$, (37) δσ$_{s}$ ≡ π 9 n$_{so}$λ$_{so}$ 2 e m ∑ ± ( ± )( k$_{F}$$_{±}$ ) $^{4}$( ν$_{±}$ ) $^{2}$τ$_{±}$ . (36) j so ,z s ,i = δσ$_{s}$ (1 − δ$_{i,z}$ ) E$_{i}$, (35) χ ( x ) = 1 4 π | x | ∑ ± ( ± ) σ$_{±}$e −| x | /ℓ$_{s}$$_{,}$ ± . (34) ( −∇ 2 + ( ℓ$_{s}$$_{,σ}$ ) − $^{2}$) µ$_{σ}$ = − σ$_{σ}$ ( ∇ · E ) . (33) ℓ$_{s}$$_{,σ}$ = √ D$_{σ}$τ$_{s}$$_{,σ}$. (32) χ ( x ) ≡ − ∑ ± ( ± ) σ$_{±}$ 1 V ∑ q e − i q · x q 2 + ( ℓ$_{s}$$_{,}$$_{±}$ ) − $_{2}$. (31) 11