Theory of the thermoelectric power or Seebeck coefficient: The case of phonon scattering for a degenerate free-electron gas

In order to illustrate the flexibility of the variational solution of the Boltzmann equation for transport coefficients, and in particular for the thermoelectric power of solids, we use a simple model of free electrons and consider their scattering from a spherical Fermi surface by acoustic phonons alone, without any mass enhancement effect or umklapp process. We recall the variational method, pointing out that formal constraints exist; we take two trial functions only, and we calculate the Seebeck effect in such a framework. The low-temperature behavior of this effect is known to be very complicated. Sign reversals occur which cannot be explained by a model based upon free or nearly free electrons. As shown hereby, when the inelasticity of the scattering of quasifree electrons by phonons is taken into account, nontrivial behavior in the intermediate (but still low) temperature range can be found. The Appendixes contain analytical expressions.