Phonon temperature and electron temperature in thermoelectric coupling

We develop a mesoscopic model of thermoelectric coupling in nanosystems, allowing for different phonon and electron temperatures, and mutual energy exchange. Its compatibility with the second law of thermodynamics is proved. By comparisons with other theoretical proposals, the different coefficients involved in the model are identified. We consider two illustrations: (a) for systems where the electron mean-free path is considerably shorter than the phonon mean-free path, the non-equilibrium phonon temperature may be different with respect to the local-equilibrium temperature of electrons; (b) for systems with large electron mean-free path, one may have the so-called “hot electrons,” namely, electrons having a higher temperature than that of the phonons.