We solve the kinetic equations in mode-space, (4.25) and (4.26), to obtain the GREEN's functions. The required elements for calculating the GREEN's functions are the HAMILTON ian, and the electron-phonon and contact self-energies. As discussed in Section 4.4, diagonal elements of the HAMILTONian are potential energies, which can be obtained from the solution of the POISSON equation, and off-diagonal elements represent the coupling between adjacent rings of carbon atoms in the CNT. Given the contact properties and the contact-device coupling parameters, the contact self-energy can be calculated once at the start of the simulation (see Section 4.5).
The calculation of the electron-phonon self-energy is presented in Section 4.6. Within the self-consistent BORN approximation of the electron-phonon self-energy (Section 3.6), the non-interacting GREEN's function is replaced by the full GREEN's function . However, the full GREEN's is the not known and has to be calculated. As a result, a coupled system of equations is achieved which can be solved by iteration
In semi-classical simulations, the coupled system of the transport and POISSON equations is solved by GUMMEL's or NEWTON's method . Both GUMMEL's method  and a variation of NEWTON's method  can be employed in self-consistent quantum mechanical simulations. While GUMMEL's method has a fast initial error reduction, for NEWTON's method it is very important that the initial guess is close to the solution. The computational cost per iteration of NEWTON's method can be much higher than that for GUMMEL's method.
We employed GUMMEL's method, where after convergence of the electron-phonon self-energy the POISSON equation is solved once. Based on the updated electrostatic potential, the GREEN's functions and the electron-phonon self-energy are iterated again. These two iterations continue until a convergence criterion is satisfied. Finally, the total current through the device is calculated.
M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors