Umklapp scattering and electron pairing cutoff in BCS superconductors

TL;DR

This paper proposes that umklapp scattering limits electron pairing in BCS superconductors, leading to more accurate predictions of superconducting properties by considering a reduced phonon cutoff frequency.

Contribution

It introduces a novel insight that umklapp scattering overlaps with normal scattering, effectively lowering the phonon cutoff frequency and improving BCS theory predictions for various superconductors.

Findings

Accurate $2\\Delta/k_BT_c$ ratios for 12 metals using reduced cutoff frequency.

The cutoff frequency is about 14.8% of the Debye frequency.

Cancellation effects between scattering types may influence $T_c$ more than electron-phonon strength.

Abstract

In a superconductor electrons form pairs for which the end states of normal and umklapp scattering may overlap. This cuts electron pairing off at a phonon frequency, $\omega_c$, low compared with the Debye frequency, $\omega_D$. Using this insight, correct values of $2\Delta/k_BT_c$ (average error 8.9%) for 12 superconductive metals, including Hg and Pb, are achieved from simple BCS formalism with an average $\omega_c/\omega_D$ of 0.148: Landau's idea of a Fermi liquid may cover strong-coupling superconductors. The cancellation between normal and umklapp scattering may be more important than a strong electron-phonon interaction in reaching a high critical temperature $T_c$.