content_image/327931.jpg content_image/327930.jpg content_image/327929.jpg content_image/327928.jpg
out to be the first diagram in Fig. 1, which reads (see Appendix B for details) α 2 2 e ˙ A ˙ 

<img src='content_image/327931.jpg'>

where g r k σ and g a k σ are the retarded and advanced electron Green’s functions, respectively, carrying the wave vector k and spin σ with zero frequency. As we see, only z component of the torque is finite. The Green’s functions include the lifetime arising from the self-energy process due to normal impurities and the spin-orbit interaction. The inverse lifetime for the electron with spin σ (= ± ) is given as 

<img src='content_image/327930.jpg'>

where ν$_{σ}$ is the spin-resolved electron density of states, n$_{i}$ and v$_{i}$ are the concentration and the potential strength of the impurities, κ$_{z,σ}$ ≡ 1 3 n$_{so}$λ$_{so}$ 2 n$_{i}$v 2 i k 4 F σ and κ$_{⊥}$ ≡ 2 3 n$_{so}$λ$_{so}$ 2 n$_{i}$v 2 i k 2 F $_{+}$k 2 F − are dimensionless ratios of the spin-orbit interaction to the normal impurity scattering ( k$_{F σ}$ is the spin-dependent Fermi wavelength), and γ$_{σ}$ ≡ ν $^{−}$σ ν$_{σ}$ . The total relaxation torque is therefore given by (Eq. (5)), 

where 

<img src='content_image/327929.jpg'>

<img src='content_image/327928.jpg'>

and δγ ≡ γ/ 3. The parameter γ is proportional to the spin flip rate due to the spin-orbit interaction. Our result indicates that the relaxation torque is zero in uniform ferromagnet when uniform electric field is applied. Thus the spin current does not decay in this case. In fact, the static solution of Eq. (1) with T z = 0 is j z s = constant. 

The degree of the asymmetry, δγ/γ , is not universal but is model dependent. For instance, in the case of junction with weak electron hopping at point-like leads, δγ vanishes [20]. 

## IV. SPIN CURRENT 

We here calculate the spin current within the same formalism. Within the linear response theory, the spin current is calculated by estimating the Feynman diagrams shown in Figs. 

