On the correct formula for the lifetime broadened superconducting density of states

TL;DR

This paper critiques the widely used Dynes formula for superconducting density of states, proposing a new microscopically justified formula with a finite imaginary energy gap component, and validating it with experimental data.

Contribution

The paper introduces a new formula for the superconducting density of states that is microscopically justified, replacing the phenomenological Dynes formula.

Findings

The new formula accurately fits experimental data for Pb$_{0.9}$Bi$_{0.1}$.

The decay rate at the gap edge is given by twice the imaginary part of the energy gap.

Both formulas fit data well, but only the new one has a solid microscopic foundation.

Abstract

We argue that the well known Dynes formula [Dynes R C {\it et al.} 1978 {\it Phys. Rev. Lett.} {\bf 41} 1509] for the superconducting quasiparticle density of states, which tries to incorporate the lifetime broadening in an approximate way, cannot be justified microscopically for conventional superconductors. Instead, we propose a new simple formula in which the energy gap has a finite imaginary part $-\Delta_2$ and the quasiparticle energy is real. We prove that in the quasiparticle approximation 2$\Delta_2$ gives the quasiparticle decay rate at the gap edge for conventional superconductors. This conclusion does not depend on the nature of interactions that cause the quasiparticle decay. The new formula is tested on the case of a strong coupling superconductor Pb$_{0.9}$Bi$_{0.1}$ and an excellent agreement with theoretical predictions is obtained. While both the Dynes formula and the one proposed in this work give good fits and fit parameters for Pb$_{0.9}$Bi$_{0.1}$, only the latter formula can be justified microscopically.