Multi-temperature mixture of phonons and electrons and nonlocal thermoelectric transport in thin layers

Nonlocal transport equations for the phonon and the electron contributions to the heat flux, and for the electron contribution to the electric current, including thermoelectric effects, are used to discuss heat and electric transport in two-dimensional systems, for distances smaller than the meanfree path of the heat carriers. The flux of phonons and electrons through the crystal lattice is regarded as a mixture of gases, each of which with its own temperature. The compatibility of the transport equations with second law of thermodynamics is investigated in the framework Extended Irreversible Thermodynamics. The constitutive equations for the specific Helmholtz free-energy and for the overall entropy flux are obtained as well.