the spin, and [ ] < represents the lesser components, and 〈 〉 i represents the averaging over the random spin-orbit impurities. Taking the lesser component, we obtain The retarded and advanced electron Green’s functions at zero frequency are represented by g r and g $^{a}$, respectively. The leading contribution for small 1 / ( ϵ$_{F}$ τ ) is given by Expanding q in the Green’s function, we obtain the leading contribution as Using the rotational symmetry, i.e., 〈 k$_{i}$k$_{j}$ 〉 = k 2 3 δ$_{ij}$ and 〈 k$_{i}$k$_{j}$k$_{l}$k$_{k}$ 〉 = k 4 $_{15}$( δ$_{ij}$δ$_{kl}$ + δ$_{ik}$δ$_{jl}$ + δ$_{il}$δ$_{jk}$ ) (the average here denotes the angular average), we obtain We carry out the average over the impurity spin-orbit interaction as Eq. (10) to obtain The asymmetric part of the trace is calculated by use of ( A ( ≡ 1 $_{2}$( A$_{+}$ + A$_{−}$ + σ$_{z}$ ( A$_{+}$ − A$_{−}$ ))) and B ( ≡ 1 $_{2}$( B$_{+}$ + B$_{−}$ + σ$_{z}$ ( B$_{+}$ − B$_{−}$ ))) are any diagonal 2 × 2 matrices) tr[( σ$_{β}$Aσ$_{γ}$ − σ$_{γ}$Aσ$_{β}$ ) B ] = − 2 iϵ$_{βγz}$ ∑ σ σA$_{σ}$B$_{−}$$_{σ}$. (B7) T z so = 4 i 45 π n$_{so}$λ$_{so}$ 2 e m 2 ∫ d Ω 2 π ∑ kk ′ ∑ q ∑ βγ ϵ$_{zβγ}$e − i q · $^{x}$e i Ω $^{t}$Ω( q · A − q$_{γ}$A$_{γ}$/ 2) k $^{2}$( k $^{′}$) $^{4}$tr[ σ$_{β}$g r $_{k}$σ$_{γ}$ g r k ′ ( g a k ′ ) $^{2}$] + c . c . (B6) T z so = − i π e m 2 ∫ d Ω 2 π ∑ kk ′ ∑ q ∑ klβγ ϵ$_{zβk}$e − i q · $^{x}$e i Ω t 〈 v ( k ) so ( k − k $^{′}$) v ( γ ) so ( k ′ − k ) 〉 i × 1 45 A$_{l}$ Ω k $^{2}$( k $^{′}$) $^{4}$( − 4 δ$_{kγ}$q$_{l}$ + δ$_{lγ}$q$_{k}$ + δ$_{kl}$q$_{γ}$ )tr[ σ$_{β}$g r $_{k}$σ$_{γ}$ g r k ′ ( g a k ′ ) $^{2}$] + c . c . (B5) T z so = − i π e m 2 ∫ d Ω 2 π ∑ kk ′ ∑ q ∑ ijklo ∑ mnβγ ϵ$_{ijk}$ϵ$_{zβk}$ϵ$_{mnγ}$e − i q · $^{x}$e i Ω t 〈 v ( k ) so ( k − k $^{′}$) v ( γ ) so ( k ′ − k ) 〉 i × A$_{l}$ Ω k$_{i}$k ′ $_{j}$k ′ $_{m}$k$_{n}$ k ′ $_{l}$k ′ $_{o}$q$_{o}$ tr[ σ$_{β}$g r $_{k}$σ$_{γ}$ g r k ′ ( g a k ′ ) $^{2}$] + c . c . (B4) T z so = − i π e m ∫ d Ω 2 π ∑ kk ′ ∑ q ∑ ijkl ∑ mnβγ ϵ$_{ijk}$ϵ$_{zβk}$ϵ$_{mnγ}$e − i q · $^{x}$e i Ω t 〈 v ( k ) so ( k − k $^{′}$) v ( γ ) so ( k ′ − k ) 〉 i × A$_{l}$ Ω k$_{i}$k ′ $_{j}$k ′ $_{m}$k$_{n}$ k ′ $_{l}$tr[ σ$_{β}$g r $_{k}$σ$_{γ}$ g r k ′ g a k $^{′}$+ $_{q}$] + c . c . (B3) T z so = − i 2 π e m ∫ d Ω 2 π ∑ kk ′ ∑ q ∑ ijkl ∑ mnβγ ϵ$_{ijk}$ϵ$_{zβk}$ϵ$_{mnγ}$e − i q · $^{x}$e i Ω t 〈 v ( k ) so ( k − k $^{′}$) v ( γ ) so ( k ′ − k ) 〉 i × A$_{l}$ Ω( k − k $^{′}$)$_{i}$ ( k + k ′ + q )$_{j}$ k ′ $_{m}$k$_{n}$ ( k ′ + q 2 ) $_{l}$tr[ σ$_{β}$g r $_{k}$σ$_{γ}$ g r k ′ g a k $^{′}$+ $_{q}$] + c . c . (B2) 17