Electrical conductivity and nuclear magnetic resonance relaxation rate of Eliashberg superconductors in the weak-coupling limit

TL;DR

This paper compares electrical conductivity and NMR relaxation in weak-coupling Eliashberg superconductors to BCS theory, highlighting fundamental differences and deriving analytical expressions in the dirty limit.

Contribution

It provides a detailed analysis of conductivity differences between Eliashberg and BCS theories, including the impact of the imaginary gap component in the weak-coupling limit.

Findings

Eliashberg theory conductivity differs from BCS due to an imaginary gap component.

Analytical expressions for low-frequency conductivity in the dirty limit are derived.

NMR relaxation rate is analytically obtained at zero frequency.

Abstract

Electrical conductivity is an important transport response in superconductors, enabling clear signatures of dynamical interactions to be observed. Of primary interest in this paper is to study characteristics of the electron-phonon interaction in weak-coupling Eliashberg theory (Eth), and to note the distinctions with Bardeen-Cooper-Schrieffer (BCS) theory. Recent analysis of weak-coupling Eth has shown that while there are modifications from the BCS results, certain dimensionless ratios are in agreement. Here we show that the conductivities in BCS theory and Eth fundamentally differ, with the latter having an imaginary gap component that damps a divergence. We focus on the dirty limit, and for both BCS theory and Eth we derive expressions for the low-frequency limit of the real conductivity. For Eth specifically, there are two limits to consider, depending on the relative size of the frequency and the imaginary part of the gap. In the case of identically zero frequency, we derive an analytical expression for the nuclear magnetic resonance relaxation rate. Our analysis of the conductivity complements the previous study of the Meissner response and provides a thorough understanding of weak-coupling Eth.