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Physical Review B

Superfluid density of a pseudogapped superconductor near the superconductor-insulator transition

M. V. Feigel’man, L. B. Ioffe

Published Sep 29, 2015• 15 citations
Publisher

Abstract

We analyze the critical behavior of the superfluid density ${\ensuremath{\rho}}_{s}$ in strongly disordered superconductors near the superconductor-insulator transition and compare it with the behavior of the spectral gap $\mathrm{\ensuremath{\Delta}}$ for collective excitations. We show that, in contrast to conventional superconductors, the superconductors with preformed pairs display an unusual scaling relation ${\ensuremath{\rho}}_{s}\ensuremath{\propto}{\mathrm{\ensuremath{\Delta}}}^{2}$ close to the superconductor-insulator transition. This relation has been reported in very recent experiments.

DOI
10.1103/physrevb.92.100509

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We analyze the critical behavior of the superfluid density \ensuremathρs{\ensuremath{\rho}}_{s}\ensuremathρs​ in strongly disordered superconductors near the superconductor-insulator transition and compare it with the behavior of the spectral gap \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ for collective excitations. We show that, in contrast to conventional superconductors, the superconductors with preformed pairs display an unusual scaling relation \ensuremathρs\ensuremath∝\ensuremathΔ2{\ensuremath{\rho}}_{s}\ensuremath{\propto}{\mathrm{\ensuremath{\Delta}}}^{2}\ensuremathρs​\ensuremath∝\ensuremathΔ2 close to the superconductor-insulator transition. This relation has been reported in very recent experiments.
We analyze critical behavior of superfluid density ρsρ_sρs​ in strongly disordered superconductors near superconductor-insulator transition and compare it with the behavior of the spectral gap ΔΔΔ for collective excitations. We show that in contrast to conventional superconductors, the superconductors with preformed pairs display unusual scaling relation ρs∝Δ2ρ_s \propto Δ^2ρs​∝Δ2 close to superconductor-insulator transition. This relation have been reported in very recent experiments.
In two dimensions there is a direct superconductor-to-insulator quantum phase transition driven by increasing disorder. We elucidate, using a combination of inhomogeneous mean field theory and quantum Monte Carlo techniques, the nature of the phases and the mechanism of the transition. We make several testable predictions specifically for local spectroscopic probes. With increasing disorder, the system forms superconducting blobs on the scale of the coherence length embedded in an insulating matrix. In the superconducting state, the phases on the different blobs are coherent across the system whereas in the insulator long range phase coherence is disrupted by quantum fluctuations. As a consequence of this emergent granularity, we show that the single-particle energy gap in the density of states survives across the transition, but coherence peaks exist only in the superconductor. A characteristic pseudogap persists above the critical disorder and critical temperature, in contrast to conventional theories. Surprisingly, the insulator has a two-particle gap scale that vanishes at the SIT, despite a robust single-particle gap.
It is well known that the metal-insulator transition in two dimensions for non-interacting fermions takes place at infinitesimal disorder. In contrast, the superconductor-to-insulator transition takes place at a finite critical disorder (on the order of Vc ~ 2t), where V is the typical width of the distribution of random site energies and t is the hopping scale. In this article we compare the localization/delocalization properties of one and two particles. Whereas the metal-insulator transition is a consequence of single-particle Anderson localization, the superconductor-insulator transition (SIT) is due to pair localization – or, alternatively, fluctuations of the phase conjugate to pair density. The central question we address is how superconductivity emerges from localized single-particle states. We address this question using inhomogeneous mean field theory and quantum Monte Carlo techniques and make several testable predictions for local spectroscopic probes across the SIT. We show that with increasing disorder, the system forms superconducting blobs on the scale of the coherence length embedded in an insulating matrix. In the superconducting state, the phases on the different blobs are coherent across the system whereas in the insulator long-range phase coherence is disrupted by quantum fluctuations. As a consequence of this emergent granularity, we show that the single-particle energy gap in the density of states survives across the transition, but coherence peaks exist only in the superconductor. A characteristic pseudogap persists above the critical disorder and critical temperature, in contrast to conventional theories. Surprisingly, the insulator has a two-particle gap scale that vanishes at the SIT despite a robust single-particle gap.
Strongly disordered superconductors undergo a transition to insulation upon increasingdisorder, driven by the localization of preformed Cooper pairs. A key and implicit question in this transition is the fate of the superfluid density as the critical disorder approaches.By measuring the kinetic inductance of superconducting resonators ofamorphous indium oxide, we show that the transition to superconductivity is bosonicwith a critical temperature ruled by superconducting phase fluctuations. Upon reachingthe critical disorder terminating superconductivity, the superfluid stiffness remainssurprisingly finite at the transition to insulation. This indicates an unexpected first-ordernature of the disorder-driven quantum phase transition to insulator, opening anew paradigm for superfluid disordered systems of quantum particles.An important by-product of the interplay between localization and superconductivity is the obtention of large kinetic inductance materials, allowing the design of the sought-after superinductances which play a major role in recent quantum circuit developments.
We show, using a Bogoliubov-de Gennes (BdG) mean field theory, that the local pairing amplitude Δ(r) becomes highly inhomogeneous with increasing disorder in an s-wave superconductor. The probability distribution P(Δ) is peaked about the BCS value at low disorder, but with increasing disorder, progressively develops into a broad distribution with significant build up of weight near Δ≈0. At high disorder, the system is found to form superconducting "islands" separated by a non-superconducting sea. Surprisingly, a finite energy gap persists into the highly disordered state in spite of many sites having negligible pairing amplitude and is understood in detail within the BdG framework. Once the pairing amplitude becomes inhomogeneous, the role of quantum phase fluctuations becomes crucial in driving a superconductor-insulator transition at a critical disorder. The insulator is unusual as it has a finite gap for all disorder strengths in marked contrast to the Anderson insulator in non-interacting systems. We treat the phase fluctuations within a self consistent harmonic approximation and obtain the superfluid stiffness as a function of disorder, which agrees well with our earlier quantum Monte Carlo studies.
We study a superfluid on a lattice close to a transition into a supersolid phase and show that a uniform superflow in the homogeneous superfluid can drive the roton gap to zero. This leads to supersolid order around the vortex core in the superfluid, with the size of the modulated pattern around the core being related to the bulk superfluid density and roton gap. We also study the electronic tunneling density of states for a uniform superconductor near a phase transition into a supersolid phase. Implications are considered for strongly correlated superconductors.