# Impact of disorder on the superconducting transition temperature near a Lifshitz transition

## TL;DR

This paper analyzes how impurity scattering influences the superconducting transition temperature near a Lifshitz transition in multi-band systems, revealing that disorder can significantly alter $T_c$ behavior depending on inter-band pairing interactions.

## Contribution

It provides analytic expressions for $T_c$ and its derivative with respect to impurity scattering in a two-band model near a Lifshitz transition, highlighting the impact of inter-band interactions.

## Key findings

- Impurity scattering can suppress $T_c$ near a Lifshitz transition.
- Repulsive inter-band interactions lead to strong pair-breaking effects.
- Attractive inter-band interactions result in mild $T_c$ suppression.

## Abstract

Multi-band superconductivity is realized in a plethora of systems, from high-temperature superconductors to very diluted superconductors. While several properties of multi-band superconductors can be understood as straightforward generalizations of their single-band counterparts, recent works have unveiled rather unusual behaviors unique to the former case. In this regard, a regime that has received significant attention is that near a Lifshitz transition, in which one of the bands crosses the Fermi level. In this work, we investigate how impurity scattering $\tau^{-1}$ affects the superconducting transition temperature $T_{c}$ across a Lifshitz transition, in the regime where intra-band pairing is dominant and inter-band pairing is subleading. This is accomplished by deriving analytic asymptotic expressions for $T_{c}$ and $\partial T_{c}/\partial\tau^{-1}$ in a two-dimensional two-band system. When the inter-band pairing interaction is repulsive, we find that, despite the incipient nature of the band crossing the Fermi level, inter-band impurity scattering is extremely effective in breaking Cooper pairs, making $\partial T_{c}/\partial\tau^{-1}$ quickly approach the limiting Abrikosov-Gor'kov value of the high-density regime. In contrast, when the inter-band pairing interaction is attractive, pair-breaking is much less efficient, affecting $T_{c}$ only mildly at the vicinity of the Lifshitz transition. The consequence of this general result is that the behavior of $T_{c}$ across a Lifshitz transition can be qualitatively changed in the presence of strong enough disorder: instead of displaying a sharp increase across the Lifshitz transition, as in the clean case, $T_{c}$ can actually display a maximum and be suppressed at the Lifshitz transition. These results shed new light on the non-trivial role of impurity scattering in multi-band superconductors.