Non-equilibrium phonon distribution caused by an electrical current

Abstract

In an attempt to explain flash sintering experiments, it had been proposed that the electron–phonon coupling leads to a proliferation of short wave-length lattice vibrations in an electric field. In this paper we investigate this by solving two coupled Boltzmann equations, describing a free electron gas in an electric field scattering from a crystal lattice coupled to a heat bath. The electric field imposes cylindrical symmetry and drives the electrons and the phonons into a non-equilibrium steady state. We find that the phonon distribution shows a strong excess population at the Brillouin zone edge in the direction of the electric field. We argue analytically, that this can be traced back to the shifted Fermi sphere for the electrons. Furthermore, not only energy but also momentum is exchanged in the electron–phonon system, which defies any attempt at describing the system by a two-temperature model.