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arXiv:2501.15428 · Physical review. B./Physical review. B

Optimizing the critical temperature and superfluid density of a metal-superconductor bilayer

Yutan Zhang, Philip M. Dee, Benjamin Cohen-Stead, Thomas A. Maier, Steven Johnston, Richard Scalettar

Published Jul 2, 2025• 4 citations
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Abstract

A promising path to realizing higher superconducting transition temperatures $T_c$ 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 $T_c$ 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 $-|U|$. In the strongly interacting regime, we find that increasing the interlayer hybridization $t_\perp$ results in a nonmonotonic enhancement of $T_c$, with an optimal value comparable to the maximum $T_c$ observed in the single-layer attractive Hubbard model, confirming trends inferred from other approaches. In the intermediate coupling regime, when $-|U|$ is close to the value associated with the maximum $T_c$ of the single-layer model, increasing $t_\perp$ tends to decrease $T_c$, 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)].

DOI
10.1103/lcgr-bqcv

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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.
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The relationship between crystal structures and superconducting critical temperatures has attracted considerable attention as a clue to designing higher-TcT_{\rm c}Tc​ superconductors. In particular, the relationship between the number nnn of CuO2_22​ layers in a unit cell of cuprate superconductors and the optimum superconducting transition temperature TcoptT_{\rm c}^{\rm opt}Tcopt​ is intriguing. As experimentally observed in layered cuprates, TcoptT_{\rm c}^{\rm opt}Tcopt​ increases when nnn is increased, up to n=3n=3n=3, and, then, decreases for larger nnn. However, the mechanism behind the nnn dependence of TcoptT_{\rm c}^{\rm opt}Tcopt​ remains elusive although there have been many studies on the nnn dependence. In this paper, we studied a bilayer ttt-t′t't′ Hubbard model to clarify the effects of the adjacent CuO2_22​ layers on the stability of the superconductivity by using a many-variable variational Monte Carlo method. We calculate the superconducting correlation at long distance and zero temperature, and the amplitude of the superconducting gap functions estimated from the momentum distribution as the observables correlated with TcoptT_{\rm c}^{\rm opt}Tcopt​. It is found that the in-plane superconducting correlation is not enhanced in comparison with that in the single-layer ttt-t′t't′ Hubbard model. The superconducting correlations of the bilayer Hamiltonian are significantly small in the overdoped region in comparison with those of single-layer Hamiltonian, which is attributed to the van Hove singularity. In addition, we found that the amplitude of the superconducting gap functions is also similar in both the single-layer and bilayer ttt-t′t't′ Hubbard model at the optimal doping. Therefore, we conclude that the adjacent Hubbard layers are not relevant to the enhancement of TcoptT_{\rm c}^{\rm opt}Tcopt​ in the bilayer cuprates. Possible origins of the enhanced TcoptT_{\rm c}^{\rm opt}Tcopt​ other than the adjacent layers are also 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.
Inspired by the recent experimental report on the pressure-induced superconductor-insulator transition in cuprate superconductors as well as the superconductivity of the Ruddlesden-Popper-phase La3Ni2O7{\mathrm{La}}_{3}{\mathrm{Ni}}_{2}{\mathrm{O}}_{7}La3​Ni2​O7​ under high pressure, we systematically investigated the single-orbital bilayer Hubbard model in the regime of large interlayer hybridization to mimic the pressure effects. We map out the phase diagram of interlayer hybridization versus density in the regime of intermediate to strong hybridization. In particular, we found that sufficiently strong hybridization can destroy the ${s}^{\ifmmode\pm\else\textpm\fi{}}$-wave pairing and induces its transition to correlated metallic, pseudogap, and Fermi liquid phases depending on the doping regime. The phase diagram hosted by the bilayer model implies its role as a versatile platform to explore the pressure effects on the two-dimensional to three-dimensional crossover physics of Hubbard-type models.
The interaction-driven insulator-to-metal transition has been reported in the ionic Hubbard model (IHM) for moderate interaction UUU, while its metallic phase only occupies a narrow region in the phase diagram. To explore the enlargement of the metallic regime, we extend the ionic Hubbard model to two coupled layers and study the interplay of interlayer hybridization VVV and two types of intralayer staggered potentials $\mathrm{\ensuremath{\Delta}}$: one with the same (in-phase) and the other with a $\ensuremath{\pi}$-phase shift (antiphase) potential between layers. Our determinant quantum Monte Carlo (DQMC) simulations at lowest accessible temperatures demonstrate that the interaction-driven metallic phase between Mott and band insulators expands in the $\mathrm{\ensuremath{\Delta}}\text{\ensuremath{-}}V$ phase diagram of bilayer IHM only for in-phase ionic potentials; while antiphase potential always induces an insulator with charge density order. This implies possible further extension of the ionic Hubbard model from the bilayer case here to a realistic three-dimensional model.