block matrix

$$(\omega -H) G = I$$

consider L,C and R region and rewrite it

$$(w-H) = \left[ \begin{array}{ccc} \omega -H_{L} & -H_{LC} &... ... -H_{CR} \\ 0 & -H_{RC} & \omega -H_{R} \end{array} \right]$$ (54)

$G_{C}$, the Green function at the C region, can be calculates as¹

$$G_{C} = \{ \omega -H_{C} - H_{CR} (\omega -H_{R})^{-1} H_{RC} - H_{CL} (\omega -H_{L})^{-1} H_{LC} \}^{-1}$$ (55)

$$= \{ \omega -H_{C} -\Sigma_R -\Sigma_L \}^{-1}$$ (56)

$$\Sigma_R = H_{CR} (\omega -H_{R})^{-1} H_{RC} = H_{CR} G_R H_{RC}$$ (57)

$$\Sigma_L = H_{CL} (\omega -H_{L})^{-1} H_{LC} = H_{CL} G_L H_{LC}$$ (58)

$G_R$ and $G_L$ corresponds to the surface Green function, $G_{00}$, in the previous section.

Note that $G_{C}(\omega )$ is calculated for $G_{C}^R(\omega )=G_{C}(\omega +i\delta )$ and $G_{C}^A(\omega )=G_{C}(\omega -i\delta )$.

define for the later convenience

$$\Gamma _R = i ( \Sigma_R^R - \Sigma_R^A )$$ (59)

$$\Gamma _L = i ( \Sigma_L^R - \Sigma_L^A )$$ (60)

In the equilibrium condition, ²

$$G^< = i A f(E-\mu)$$ (61)

$$A = i (G^R - G^A )$$ (62)

$$G^{<} = -(G^R-G^A) f(E-\mu)$$ (63)

while

$$G^< = G^R \Sigma^< G^A$$ (64)

therefore

$$\Sigma^< = -{G^R}^{-1} (G^R -G^A) f(E-\mu) {G^A}^{-1}$$ (65)

$$= -( {G^A}^{-1} - {G^R}^{-1} ) f(E-\mu)$$ (66)

$$= -( \Sigma^R -\Sigma^A ) f(E-\mu)$$ (67)

How are $\Sigma_R$ and $\Sigma_L$? They are defined in non-equilibrium region, C. but near the region L or R, they approximately satisfy equilibrium condition.

$$\Sigma_R^{<} = - ( \Sigma_R^{R} - \Sigma_R^{A} ) f(E-\mu_R)$$ (68)

$$\Sigma_L^{<} = - ( \Sigma_L^{R} - \Sigma_L^{A} ) f(E-\mu_L)$$ (69)