FIG. 2: Feynman diagrams representing the local contribution to the spin current (the first two terms in Eq. (37)). The first three diagrams correspond to j (n) s , and the last two diagrams represent the contribution from the anomalous velocity, j so s . Dotted and double dashed lines denote the interaction with the applied electric field and the spin-orbit interaction, respectively. The vertex marked by cross represents the spin current. FIG. 3: Left: The vertex correction contribution to j (n) ,z s , resulting in the diffusive spin current (the last term of Eq. (37)). Right: Γ$_{σσ}$ ′ , which is a ladder process of the successive electron scattering by the normal impurity and the spin-orbit interaction (represented by thick dotted lines) connecting the spin indices σ and σ $^{′}$. 2 and 3, which correspond to the field-induced contribution and the effect of the diffusive electron motion, respectively. We first estimate the normal part of the spin current, j (n) ,z s ,i (Eq. (15)), shown in the first two diagrams in Fig. 2. The contribution j (n) ,z s ,i is defined including the anomalous contribution from the electromagnetic gauge field, j A,z s ,i . Its dominant contribution in the limit of small Ω and small q is calculated as (see Appendix C for detail) In Eq. (24), the first term is the contribution shown in the left of Fig. 2, and the second term is the contribution from the vertex correction (Fig. 3). 9 j (n) ,z s ,i = − i 1 2 π e 2 m 2 ∑ kq e − i q · x ∫ d Ω 2 π e i Ω t ∑ j Ω A$_{j}$ ( q , Ω) ∑ σ σ [ k$_{i}$k$_{j}$g r k − q $_{$_{2}$,σ}$g a k + q $_{2}$,σ + ∑ k $^{′}$σ′ k$_{i}$k ′ $_{j}$g r k − q $_{$_{2}$,σ}$g a k + q $_{$_{2}$,σ}$g r k $^{′}$− q $_{2}$,σ ′ g a k $^{′}$+ q $_{2}$,σ ′ n$_{i}$v 2 i Γ$_{σσ}$ ′ ( q , Ω) ] (24) A $^{v}$so Γ$_{σσ}$ ′ σ σ σ ′ σ ′ = + spin-orbit impurity σ ′ σ ′ σ ′ σ ′ σ ′ σ ′ σ ′′ σ ′′ σ σ σ σ σ σ A