Phonon residual resistance of pure crystals

TL;DR

This paper investigates the phonon residual resistance in perfect metallic crystals caused by electron-phonon interactions due to zero-point vibrations, deriving its dependence on crystal thickness and electric field, with specific calculations for copper, silver, and gold.

Contribution

It provides a theoretical analysis of phonon residual resistance in perfect crystals, including conditions where phonon scattering dominates and how it relates to crystal purity and dimensions.

Findings

Phonon residual resistance scales as d^{-5}E^{-3}.

In pure crystals, phonon scattering can dominate over impurity scattering under certain conditions.

Calculated phonon residual resistivity for Cu, Ag, Au at specific purity levels.

Abstract

Using the Boltzmann transport equation, we study phonon residual resistance of perfect metallic crystals of a finite thickness $d$ along which a weak constant electric field $E$ is applied. This resistance which is $\propto d^{-5}E^{-3}$, is due to scattering of electric field-heated electrons with emission of long-wave acoustic phonons. This electron-phonon interaction is caused by zero-point vibrations of the atoms in the perfect crystal lattice sites. Consideration is carried out for Cu, Ag and Au single crystals with the thickness of about 1 cm, in the fields of the order of 1 mV/cm. Following the Matthiessen rule, the resistance of the pure crystals the thicknesses of which are much larger than the electron mean free path, is represented as the sum of the impurity and phonon residual resistances. The condition on the thickness $d$ and the field $E$ is found at which the phonon scattering of the field-heated electrons dominates. Under this condition, the low-temperature resistances of pure crystals do not depend on the their purity and determine the phonon residual resistivity of the ideal crystals. The calculations are performed for Cu with a purity of at least 99.9999%.