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arXiv:1205.4737 · Physical Review B

Superfluid stiffness renormalization and critical temperature enhancement in a composite superconductor

Gideon Wachtel, Assaf Bar-Yaacov, Dror Orgad

Published Oct 31, 2012• 14 citations
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Abstract

We study a model of a composite system constructed from a ``pairing layer'' of disconnected attractive-$U$ Hubbard sites that is coupled by single-particle tunneling, ${t}_{\ensuremath{\perp}}$, to a disordered metallic layer. For small interlayer tunneling the system is described by an effective long-range $XY$ phase model whose critical temperature, ${T}_{c}$, is essentially insensitive to the disorder and is exponentially suppressed by quantum fluctuations. ${T}_{c}$ reaches a maximum for intermediate values of ${t}_{\ensuremath{\perp}}$, which we calculate using a combination of mean-field, classical, and quantum Monte Carlo methods. The maximal ${T}_{c}$ scales as a fraction of the zero-temperature gap of the attractive sites when $U$ is smaller than the metallic bandwidth, and is bounded by the maximal ${T}_{c}$ of the two-dimensional attractive Hubbard model for large $U$. Our results indicate that a thin, rather than a thick, metallic coating is better suited for the enhancement of ${T}_{c}$ at the surface of a phase fluctuating superconductor.

DOI
10.1103/physrevb.86.134531

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We study a model of a composite system constructed from a ``pairing layer'' of disconnected attractive-UUU Hubbard sites that is coupled by single-particle tunneling, t\ensuremath⊥{t}_{\ensuremath{\perp}}t\ensuremath⊥​, to a disordered metallic layer. For small interlayer tunneling the system is described by an effective long-range XYXYXY phase model whose critical temperature, Tc{T}_{c}Tc​, is essentially insensitive to the disorder and is exponentially suppressed by quantum fluctuations. Tc{T}_{c}Tc​ reaches a maximum for intermediate values of t\ensuremath⊥{t}_{\ensuremath{\perp}}t\ensuremath⊥​, which we calculate using a combination of mean-field, classical, and quantum Monte Carlo methods. The maximal Tc{T}_{c}Tc​ scales as a fraction of the zero-temperature gap of the attractive sites when UUU is smaller than the metallic bandwidth, and is bounded by the maximal Tc{T}_{c}Tc​ of the two-dimensional attractive Hubbard model for large UUU. Our results indicate that a thin, rather than a thick, metallic coating is better suited for the enhancement of Tc{T}_{c}Tc​ at the surface of a phase fluctuating superconductor.
A promising path to realizing higher superconducting transition temperatures TcT_cTc​ is the strategic engineering of artificial heterostructures. For example, quantum materials could, in principle, be coupled with other materials to produce a more robust superconducting state. In this work, we add numerical support to the hypothesis that a strongly interacting superconductor weakened by phase fluctuations can boost its TcT_cTc​ by hybridizing the system with a metal. Using determinant quantum Monte Carlo (DQMC), we simulate a two-dimensional bilayer composed of an attractive Hubbard model and a metallic layer in two regimes of the interaction strength −∣U∣-|U|−∣U∣. In the strongly interacting regime, we find that increasing the interlayer hybridization t⊥t_\perpt⊥​ results in a nonmonotonic enhancement of TcT_cTc​, with an optimal value comparable to the maximum TcT_cTc​ observed in the single-layer attractive Hubbard model, confirming trends inferred from other approaches. In the intermediate coupling regime, when −∣U∣-|U|−∣U∣ is close to the value associated with the maximum TcT_cTc​ of the single-layer model, increasing t⊥t_\perpt⊥​ tends to decrease TcT_cTc​, implying that the correlated layer was already optimally tuned. Importantly, we demonstrate that the mechanism behind these trends is related to enhancement in the superfluid stiffness, as was initially proposed by Kivelson [Physica B: Condensed Matter 318, 61 (2002)].
It has been proposed that the superconducting transition temperature Tc{T}_{\text{c}}Tc​ of an unconventional superconductor with a large pairing scale but strong phase fluctuations can be enhanced by coupling it to a metal. However, the general efficacy of this approach across different parameter regimes remains an open question. Using the dynamical cluster approximation, we study this question in a system composed of an attractive Hubbard layer in the intermediate coupling regime, where the magnitude of the attractive Coulomb interaction ∣U∣|U|∣U∣ is slightly larger than the bandwidth WWW, hybridized with a noninteracting metallic layer. We find that while the superconducting transition becomes more mean-field-like with increasing interlayer hopping, the superconducting transition temperature Tc{T}_{\text{c}}Tc​ exhibits a nonmonotonic dependence on the strength of the hybridization ${t}_{\ensuremath{\perp}}$. This behavior arises from a reduction of the effective pairing interaction in the correlated layer that outcompetes the growth in the intrinsic pair-field susceptibility induced by the coupling to the metallic layer. We find that the largest Tc{T}_{\text{c}}Tc​ inferred here for the composite system is comparable to the maximum value currently estimated for the isolated negative-UUU Hubbard model.
We have considered the half-filled disordered attractive Hubbard model on a square lattice, in which the on-site attraction is switched off on a fraction fff of sites, while keeping a finite UUU on the remaining ones. Through quantum Monte Carlo simulations for several values of fff and UUU and for system sizes ranging from $8\ifmmode\times\else\texttimes\fi{}8$ to $16\ifmmode\times\else\texttimes\fi{}16$, we have calculated the configurational averages of the equal-time pair structure factor Ps{P}_{s}Ps​ and, for a more restricted set of variables, the helicity modulus ${\ensuremath{\rho}}_{s}$, as functions of temperature. Two finite-size scaling Ans\"atze for Ps{P}_{s}Ps​ have been used: one for zero temperature and the other for finite temperatures. We have found that the system sustains superconductivity in the ground state up to a critical impurity concentration fc{f}_{c}fc​, which increases with UUU, at least up to U=4U=4U=4 (in units of the hopping energy). Also, the normalized zero-temperature gap as a function of fff shows a maximum near $f\ensuremath{\sim}0.07$ for $2\ensuremath{\lesssim}U\ensuremath{\lesssim}6$. Analyses of the helicity modulus and of the pair structure factor led to the determination of the critical temperature as a function of fff for U=3U=3U=3, 4, and 6: they also show maxima near $f\ensuremath{\sim}0.07$, with the highest Tc{T}_{c}Tc​ increasing with UUU in this range. We argue that, overall, the observed behavior results from both the breakdown of charge-density-wave-superconductivity degeneracy and the fact that free sites tend to ``push'' electrons toward attractive sites; the latter effect being more drastic at weak couplings.
We investigate the disorder-driven superconductor to insulator quantum phase transition (SIT) in an interacting fermion model using determinantal quantum Monte Carlo (QMC) methods. The disordered superconductor is modeled by an attractive Hubbard model with site disorder chosen randomly from a uniform distribution. The superconducting state which exists for small disorder is shown to evolve into an insulating phase beyond a critical disorder. The transition is tracked by the vanishing of (a) the superfluid stiffness, and (b) the charge stiffness or the delta function peak in the optical conductivity at zero frequency. We also show the behavior of the charge, spin, pair, and current correlations in the presence of disorder. Results for the temperature dependence of the dc conductivity, obtained by an approximate analytic continuation technique, are also presented both in the metallic phase above Tc{T}_{c}Tc​ and the insulating phase. We discuss some of the complications in extracting the resistance at the transition point.
We consider the attractive Hubbard model with on-site disorder as a prototype of a disordered superconductor. We solve the Bogoliubov-de Gennes equations on two-dimensional finite clusters at zero temperature and evaluate the electromagnetic response to a vector potential. We find that the standard decoupling between transverse and longitudinal response does not apply in the presence of disorder. Moreover, the superfluid density is strongly reduced by the relaxation of the phase of the order parameter already at mean-field level when disorder is large. We also find that the anharmonicity of the phase fluctuations is strongly enhanced by disorder. Beyond mean field, this provides an enhancement of quantum fluctuations inducing a zero-temperature transition to a nonsuperconducting phase of disordered preformed pairs. Finally, the connection of our findings with the glassy physics for extreme dirty superconductors is discussed.
The attractive Hubbard model has become a model readily realizable with ultracold atoms on optical lattices. However, the superconducting (superfluid) critical temperatures Tc{T}_{c}Tc​'s are still somewhat smaller than the lowest temperatures achieved in experiments. Here, we consider two possible routes, generically called layering, to increase Tc{T}_{c}Tc​: a bilayer and a simple cubic lattice, both with tunable hopping tz{t}_{z}tz​ between attractive Hubbard planes. We have performed minus-sign--free determinant quantum Monte Carlo simulations to calculate response functions such as pairing correlation functions, uniform spin susceptibility, and double occupancy, through which we map out some physical properties. We have found that by a judicious choice of fillings and intensity of on-site attraction, a bilayer can exhibit Tc{T}_{c}Tc​'s between 1.5 and 1.7 times those of the single layer; for the simple-cubic lattice the enhancement can be 30% larger than the maximum for the single layer. We also check the accuracy of both a BCS-like estimate for Tc{T}_{c}Tc​ in the attractive Hubbard model as well as of an upper bound for Tc{T}_{c}Tc​ based on the superfluid density.