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

Multiband superconductivity is realized in a plethora of systems, from high-temperature superconductors to very diluted superconductors. While several properties of multiband 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 paper, we investigate how impurity scattering ${\ensuremath{\tau}}^{\ensuremath{-}1}$ affects the superconducting transition temperature ${T}_c$ across a Lifshitz transition, in the regime where intraband pairing is dominant and interband pairing is subleading. This is accomplished by deriving analytic asymptotic expressions for ${T}_c$ and $\ensuremath{\partial}{T}_c/\ensuremath{\partial}{\ensuremath{\tau}}^{\ensuremath{-}1}$ in a two-dimensional two-band system. When the interband pairing interaction is repulsive, we find that, despite the incipient nature of the band crossing the Fermi level, interband impurity scattering is extremely effective in breaking Cooper pairs, making $\ensuremath{\partial}{T}_c/\ensuremath{\partial}{\ensuremath{\tau}}^{\ensuremath{-}1}$ quickly approach the limiting Abrikosov-Gor'kov value of the high-density regime. In contrast, when the interband 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 light on the nontrivial role of impurity scattering in multiband superconductors.