Anomalous isotope effect near a 2.5 Lifshitz transition in a multi-band multi-condensate superconductor made of a superlattice of stripes

TL;DR

This paper investigates how the isotope effect on superconductivity varies near a 2.5 Lifshitz transition in a multi-band superconductor, revealing anomalous behavior linked to a BEC-BCS crossover in nano-structured materials.

Contribution

It introduces a detailed theoretical analysis of isotope effects near a Lifshitz transition in multi-band superconductors, highlighting novel BEC-BCS crossover phenomena in superlattice structures.

Findings

Isotope coefficient deviates from 0.5 near the transition

Critical temperature shows a minimum due to Fano antiresonance

Isotope coefficient diverges where BEC coexists with BCS

Abstract

The doping dependent isotope effect on the critical temperature (Tc) is calculated for multi-band multi-condensate superconductivity near a 2.5 Lifshitz transition. We focus on multi-band effects that arises in nano-structures and in density wave metals (like spin density wave or charge density wave) as a result of the band folding. We consider a superlattice of quantum stripes with finite hopping between stripes near a 2.5 Lifshitz transition for appearing of a new sub-band making a circular electron-like Fermi surface pocket. We describe a particular type of BEC (Bose-Einstein Condensate) to BCS (Bardeen-Cooper-Schrieffer condensate) crossover in multi-band / multi-condensate superconductivity at a metal-to-metal transition that is quite different from the standard BEC-BCS crossover at an insulator-to-metal transition. The electron wave-functions are obtained by solving the Schr\"odinger equation for a one-dimensional modulated potential barrier. The k-dependent and energy dependent superconducting gaps are calculated using the k-dependent anisotropic Bardeen-Cooper-Schrieffer (BCS) multi-gap equations solved joint with the density equation, according with the Leggett approach currently used now in ultracold fermionic gases. The results show that the isotope coefficient strongly deviates from the standard BCS value 0.5, when the chemical potential is tuned at the 2.5 Lifshitz transition for the metal-to-metal transition. The critical temperature Tc shows a minimum due to the Fano antiresonance in the superconducting gaps and the isotope coefficient diverges at the point where a BEC coexists with a BCS condensate. On the contrary Tc reaches its maximum and the isotope coefficient vanishes at the crossover from a polaronic condensate to a BCS condensate in the new appearing sub-band.