= \int I_{ch}(E) dE$. Note that the current is constant along the channel at each cross section, however, its energy is not constant, i.e. the charge carriers can gain or lose energy as they propagate. This happens in the presence of inelastic scattering (optical phonons). Thus, the energy of the current is position dependent and it is its scaled integral in the channel which provides the overall value of the Seebeck coefficient. In experimental settings, one extracts the Seebeck coefficient from the open circuit voltage upon the application of a thermal gradient along the channel, as $S = \Delta V / \Delta T$, which equivalently can also be computed by $S = I_{ch (\Delta V = 0)} / G \Delta T$. In [@Kim11], it was validated that the two methods of extracting the Seebeck coefficient are equivalent, which makes it easier in time consuming simulations (as the ones we undertake) to only run the $\Delta V \neq 0$ case and still be able to extract the Seebeck coefficient by integrating the energy of the current flow over the length of the channel, and that is how we extract the Seebeck coefficient in this work.

Note that from NEGF we obtain 