Phonon residual resistance of pure crystals

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%.