3.9.4 Current Conservation

The current is conserved as long as the right-hand-side of (3.85) approaches zero as $ 2\rightarrow 1$

\begin{displaymath}\begin{array}{l}\displaystyle \lim_{2\rightarrow 1}
 \Sigma^\mathrm{a}(32) \ \right] \ = 0 \ ,
 \end{array}\end{displaymath} (3.90)

The current is obviously conserved if there is no interaction, whereas the situation is different in the interacting case. As described in Section 3.6, the interactions are described in terms of appropriate self-energies. However, self-energies can often be obtained approximately only. Therefore, one could choose an approximation which violates the continuity equation, which, of course, is not physical. It is straightforward to show that the approximated self-energy due to electron-phonon interaction within the self-consistent BORN approximation (3.50) preserves the current continuity. M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors