Rigorous results for the electrical conductivity due to electron-phonon scattering

Abstract

We present a rigorous solution of the Boltzmann equation for the electron-phonon scattering problem in three spatial dimensions in the limit of low temperatures. The different temperature scaling of the various scattering rates turns the temperature into a control parameter that is not available in classical kinetic theory and allows for a rigorous proof of Bloch's T 5 law. The relation between the Boltzmann equation and the Kubo formula is also discussed, as well as implications for the scattering of electrons by excitations other than phonons. Contents

Exact solution of the Boltzmann equation for low-temperature transport coefficients in metals II: Scattering by ferromagnons

“…Our technique for doing so is modeled after the analysis of the electron-phonon scattering problem in Paper I, which in turn is based on a mathematically rigorous treatment that was given in Ref. 9. What makes the exact solution possible is the zero eigenvalue of the collision operator Λ 0 , see Eq.…”

“…Only very recently has it been shown that a mathematically rigorous solution of the integral equation does indeed yield the Bloch T 5 law for the case of electron-phonon scattering. 9 In a previous paper 10 (to be referred to as Paper I) we have simplified and extended the method of Ref. 9.…”

Exact solution of the Boltzmann equation for low-temperature transport coefficients in metals I: Scattering by phonons, antiferromagnons, and helimagnons

“…The assumptions implicit in this exact solution can be eliminated, and the procedure made rigorous, if at the expense of substantially increased mathematical complexity, see Ref. 8. In Sec.…”

“…Only very recently has it been shown that a mathematically rigorous solution of the integral equation does indeed yield the Bloch T 5 law for the phonon contribution to the electrical resistivity ρ ∝ T 5 . 8 The purpose of the present paper is four-fold: First, we show that the method of Ref. 8 can be simplified substantially by making some assumptions about the collision operator that are common in theoretical physics, chiefly, the existence of a spectral representation.…”