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The factor Γ$_{σσ}$ ′ ( q , Ω) contains all the vertex corrections due to the normal impurities and the spin-orbit interaction shown in Fig. 3. The equation of motion for Γ$_{σσ}$ ′ is derived in the same manner as in Ref. [21] carried out in the context of quantum correction (the diffusion without the spin-orbit interaction was considered in Ref. [22]). The equation is obtained as 

where 

<img src='content_image/374697.jpg'>

<img src='content_image/374696.jpg'>

where D$_{σ}$ ≡ ( k$_{F σ}$ ) 2 3 m 2 τ$_{σ}$ is the diffusion constant. Neglecting quantities of order of ( κ$_{z}$, κ$_{⊥}$ ) $^{2}$, Eq. (25) is solved as 

<img src='content_image/374695.jpg'>

Using Eq. (26), we obtain Γ$_{σσ}$ ′ as (assuming the rotational symmetry for the wave vectors when averaging over the spin-orbit potential) 

<img src='content_image/374694.jpg'>

By use of Eq. (28) and summing over the wave vectors, the normal spin current, Eq. (24), reads 

<img src='content_image/374693.jpg'>

where σ 0 s ≡ e ∑ $_{±}$( ± ) D$_{±}$ν$_{±}$ is the bare spin conductivity divided by e . The first term of Eq. (29) is the field-driven contribution. The second gradient term is a diffusive contribution (vertex corrections), arising from the spin accumulation. The effective potential describing the spin accumulation, µ$_{s}$ , reads 

<img src='content_image/374692.jpg'>

