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

TL;DR

This paper introduces an exact solution method for the linearized Boltzmann equation in metals at low temperatures, accurately deriving transport coefficients for electron scattering by phonons and magnons without uncontrolled approximations.

Contribution

It develops a novel technique for exact solutions of the Boltzmann equation, eliminating previous approximations and providing precise prefactors for temperature-dependent transport laws.

Findings

Confirmed the T^5 law for electrical resistivity due to phonons and antiferromagnons.

Derived the T^2 law for thermal resistivity and T law for thermopower with exact prefactors.

Established the T^{5/2}, T^{1/2}, and T laws for helimagnons' contributions.

Abstract

We present a technique for an exact solution of the linearized Boltzmann equation for the electrical and thermal transport coefficients in metals in the low-temperature limit. This renders unnecessary an uncontrolled approximation that has been used in all previous solutions of the integral equations for the transport coefficients. Applications include electron-phonon scattering in nonmagnetic metals, as well as the magnon contribution to the electrical and thermal conductivities, and to the thermopower, in metallic ferromagnets, antiferromagnets, and helimagnets. In this paper, the first of a pair, we set up the technique and apply it to the scattering of electrons by phonons, antiferromagnons, and helimagnons. We show that the Bloch $T^5$ law for the electrical resistivity, the $T^2$ law for the thermal resistivity, and the $T$ law for the thermopower due to phonon and antiferromagnon scattering are exact, and determine the prefactors exactly. The corresponding exact results for helimagnons are $T^{5/2}$, $T^{1/2}$, and $T$, respectively. In a second paper we will consider the scattering by ferromagnons.