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,
, to energy
![\begin{displaymath}\begin{array}{l}\displaystyle
G({\bf {r}_1},{\bf {r}_2};E) =...
...iE\tau/\hbar}
G({\bf {r}_1},{\bf {r}_2};\tau) \ .
\end{array}\end{displaymath}](img662.png) |
(3.70) |
Under steady-state condition the
quantum kinetic equations, (3.64), (3.65), and
(3.69), can be
written as [60]:
![\begin{displaymath}\begin{array}{l}\displaystyle
\left[E-\hat{H}_0({\bf {r}_1})...
...2};E) \ = \ \delta_{{\bf {r}_1},{\bf {r}_2}} \ ,
\end{array}\end{displaymath}](img663.png) |
(3.71) |
![\begin{displaymath}\begin{array}{l}\displaystyle
G^\mathrm{\lessgtr}({\bf {r}_1...
...;E) \
G^\mathrm{a}({\bf {r}_4},{\bf {r}_2};E) \ ,
\end{array}\end{displaymath}](img664.png) |
(3.72) |
where
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
![\begin{displaymath}\begin{array}{ll}\displaystyle
\Sigma_\mathrm{el-ph}^{\gtrle...
...mbda}
({\bf q},\hbar\omega_{{\bf q},\lambda}) \ ,
\end{array}\end{displaymath}](img665.png) |
(3.73) |
To calculate the retarded self-energy, however,
it is more straightforward to FOURIER transform the relation
, see (3.52). By defining the broadening
function
![\begin{displaymath}\begin{array}{l}\displaystyle
\Gamma({\bf r_1},{\bf r_2};E)=...
...{m} \Sigma^\mathrm{<}({\bf r_1},{\bf r_2};E) \ , \
\end{array}\end{displaymath}](img667.png) |
(3.74) |
the retarded self-energy is given by the convolution of
and
the FOURIER transform of the step function [33]
![\begin{displaymath}\begin{array}{l}\displaystyle
\Sigma^\mathrm{r}(E) \ = \ -i\...
...c{\delta(E)}{2} \
+ \ \frac{i}{2\pi E}\right) \ ,
\end{array}\end{displaymath}](img669.png) |
(3.75) |
where
denotes the convolution. Therefore, the retarded self-energy is
given by [116]
![\begin{displaymath}\begin{array}{l}\displaystyle
\Sigma^\mathrm{r}({\bf r_1},{\...
...}\frac{\Gamma({\bf
r_1},{\bf r_2};E')}{E-E'} \ ,
\end{array}\end{displaymath}](img670.png) |
(3.76) |
where
stands for principal part.
M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors