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

Abstract

The doping dependent isotope effect on the critical temperature (Tc) is\ncalculated for multi-band multi-condensate superconductivity near a 2.5\nLifshitz transition. We focus on multi-band effects that arises in\nnano-structures and in density wave metals (like spin density wave or charge\ndensity wave) as a result of the band folding. We consider a superlattice of\nquantum stripes with finite hopping between stripes near a 2.5 Lifshitz\ntransition for appearing of a new sub-band making a circular electron-like\nFermi surface pocket. We describe a particular type of BEC (Bose-Einstein\nCondensate) to BCS (Bardeen-Cooper-Schrieffer condensate) crossover in\nmulti-band / multi-condensate superconductivity at a metal-to-metal transition\nthat is quite different from the standard BEC-BCS crossover at an\ninsulator-to-metal transition. The electron wave-functions are obtained by\nsolving the Schr\\"odinger equation for a one-dimensional modulated potential\nbarrier. The k-dependent and energy dependent superconducting gaps are\ncalculated using the k-dependent anisotropic Bardeen-Cooper-Schrieffer (BCS)\nmulti-gap equations solved joint with the density equation, according with the\nLeggett approach currently used now in ultracold fermionic gases. The results\nshow that the isotope coefficient strongly deviates from the standard BCS value\n0.5, when the chemical potential is tuned at the 2.5 Lifshitz transition for\nthe metal-to-metal transition. The critical temperature Tc shows a minimum due\nto the Fano antiresonance in the superconducting gaps and the isotope\ncoefficient diverges at the point where a BEC coexists with a BCS condensate.\nOn the contrary Tc reaches its maximum and the isotope coefficient vanishes at\nthe crossover from a polaronic condensate to a BCS condensate in the new\nappearing sub-band.\n