A three-fluid approach in bipolar semiconductors with generation-recombination: constitutive laws and Onsager symmetry

If electrons (e) and holes (h) in metals or semiconductors are heated to the temperatures and greater than the lattice temperature , the electron-phonon interaction causes energy relaxation. In the nonuniform case, a momentum relaxation occurs as well. In view of such an application, a new model based on an asymptotic procedure for solving the kinetic equations of carriers and phonons is proposed, with generation-recombination of electrons and holes, which gives naturally the displaced Maxwellian at the leading order. After that, balance equations for the electron number, hole number, energy densities, and momentum densities are constructed, which constitute now a system of eight equations for the chemical potentials (carriers), the temperatures (carriers and phonons), and the drift velocities (carriers and phonons). In the drift-diffusion approximation the constitutive laws are derived and the Onsager relations recovered.