Transport in Metals: Effect of the Nonequilibrium Phonons

Abstract

The coupled Boltzmann equations for phonons and electrons in a metal are solved simultaneously using first-order perturbation theory. Use is made of the Kohler variational principle to put the transport coefficients in terms of determinants. The final results are in the form of integrals involving the phonon-phonon relaxation time $\ensuremath{\tau}(\ensuremath{\sigma})$ and the electron-electron relaxation time. The details of how a finite electrical conductivity exists by means of umklapp processes even when $\ensuremath{\tau}(\ensuremath{\sigma})\ensuremath{\rightarrow}\ensuremath{\infty}$ (the Peierls equilibrium problem) is revealed in the electrical conductivity expression. In general, the effects may be described as "phonon relaxation" effects, altering all the transport coefficients somewhat when $\ensuremath{\tau}(\ensuremath{\sigma})$ gets large, and "phonon-drag" or drift effects, altering appreciably only the thermoelectric power.