3.8.3 Steady-State Kinetic Equations

Under steady-state condition the GREEN's functions depend on time differences. One usually FOURIER transforms the time difference coordinate, $ \tau=t-t'$, to energy

 G({\bf {r}_1},{\bf {r}_2};E) =...
 G({\bf {r}_1},{\bf {r}_2};\tau) \ .
 \end{array}\end{displaymath} (3.70)

Under steady-state condition the quantum kinetic equations, (3.64), (3.65), and (3.69), can be written as [60]:

 \left[E-\hat{H}_0({\bf {r}_1})...
...2};E) \ = \ \delta_{{\bf {r}_1},{\bf {r}_2}} \ ,
 \end{array}\end{displaymath} (3.71)

 G^\mathrm{\lessgtr}({\bf {r}_1...
...;E) \
 G^\mathrm{a}({\bf {r}_4},{\bf {r}_2};E) \ ,
 \end{array}\end{displaymath} (3.72)

where $ \Sigma$ is the total self-energy. A similar transformation can be applied to self-energies. However, to obtain self-energies one has to first apply LANGRETH's rules and then FOURIER transform the time difference coordinate to energy. We consider the self-energies discussed in Section 3.6. The evaluation of the HARTREE self-energy due to electron-electron interaction is straightforward, since it only includes the electron GREEN's function. However, the lowest-order self-energy due to electron-phonon interaction contains the products of the electron and phonon GREEN's functions. Using LANGRETH's rules (Table 3.1) and then FOURIER transforming the self-energies due to electron-phonon interaction, (3.50) takes the form

 ({\bf q},\hbar\omega_{{\bf q},\lambda}) \ ,
 \end{array}\end{displaymath} (3.73)

To calculate the retarded self-energy, however, it is more straightforward to FOURIER transform the relation $ \Sigma^\mathrm{r}(\tau)=\theta(\tau)[\Sigma^\mathrm{>}
(\tau)-\Sigma^\mathrm{<}(\tau)]$, see (3.52). By defining the broadening function $ \Gamma $

 \Gamma({\bf r_1},{\bf r_2};E)=...
...{m} \Sigma^\mathrm{<}({\bf r_1},{\bf r_2};E) \ , \
 \end{array}\end{displaymath} (3.74)

the retarded self-energy is given by the convolution of $ -i\Gamma(E)$ and the FOURIER transform of the step function [33]

 \Sigma^\mathrm{r}(E) \ = \ -i\...
...c{\delta(E)}{2} \
 + \ \frac{i}{2\pi E}\right) \ ,
 \end{array}\end{displaymath} (3.75)

where $ \otimes $ denotes the convolution. Therefore, the retarded self-energy is given by [116]

 \Sigma^\mathrm{r}({\bf r_1},{\...
 r_1},{\bf r_2};E')}{E-E'} \ ,
 \end{array}\end{displaymath} (3.76)

where $ \mathrm{P}$ stands for principal part.

M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors