We solve the kinetic equations in modespace, (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 electronphonon and contact selfenergies. 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 offdiagonal elements represent the coupling between adjacent rings of carbon atoms in the CNT. Given the contact properties and the contactdevice coupling parameters, the contact selfenergy can be calculated once at the start of the simulation (see Section 4.5).
The calculation of the electronphonon selfenergy is presented in Section 4.6. Within the selfconsistent BORN approximation of the electronphonon selfenergy (Section 3.6), the noninteracting 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 semiclassical simulations, the coupled system of the transport and POISSON equations is solved by GUMMEL's or NEWTON's method [226]. Both GUMMEL's method [253] and a variation of NEWTON's method [254] can be employed in selfconsistent 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 electronphonon selfenergy the POISSON equation is solved once. Based on the updated electrostatic potential, the GREEN's functions and the electronphonon selfenergy 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 NanotubeBased Transistors