Inhomogeneous
s-wave superconductivity is studied in the two-dimensional, square lattice attractive Hubbard Hamiltonian using the Bogoliubov-de Gennes (BdG) mean field approximation. We find that at weak coupling, and for densities mainly below half-filling, an inhomogeneous interaction in which the on-site interaction
Ui takes on two values,
Ui=0,2U results in a larger zero temperature pairing amplitude, and that the superconducting
Tc can also be significantly increased, relative to a uniform system with
Ui=U on all sites. These effects are observed for stripe, checkerboard, and even random patterns of the attractive centers, suggesting that the pattern of inhomogeneity is unimportant. Monte Carlo calculations which reintroduce some of the fluctuations neglected within the BdG approach see the same effect, both for the attractive Hubbard model and a Hamiltonian with
d-wave pairing symmetry.
The effect of random on-site disorder on s-wave (BCS) superconductors described by a two dimensional negative-U Hubbard model is studied using Bogoliubov-de Gennes (BdG) method. The mean field pairing amplitudes become spatially inhomogeneous at large values of disorder where the system breaks up into superconducting islands with large pairing amplitudes, separated by insulating strips. The amplitude fluctuations are correctly accounted for via BdG calculations, however it misses phase fluctuations which are inherent to low dimensions. The phase fluctuations affect superconducting properties strongly, and the effect is more pronounced in the limit of large disorder. We provide a close estimate of the actual transition temperature, Tcby incorporating phase fluctuations about the inhomogeneous BdG state. This is obtained by relating the jump in renormalized Ds(obtained from a self consistent Harmonic approximation on a phase-only Hamiltonian) at the Kosterlitz-Thouless (KT) transition and the KT transition temperature, TKTobtained from Ds(T) = (2/π)TKTas temperature tends to TKTfrom below. This yields opening of a large region sandwiched between [Formula: see text] (obtained by the vanishing of [Formula: see text]) and TKTwhere there is no phase coherence between the pairs, however amplitude correlations continue to exist, reminiscent of a pseudogap phase in high-Tcsuperconductors which is marked by short ranged preformed pairs without any definite phase relation between them. Further, the appearance of the superconducting islands from a homogeneous phase indicates the evolution of a system consisting of large and overlapping pairs to one that contains short and tightly bound pairs - a scenario termed as BCS-BEC crossover. We have investigated in details the crossover phenomenon as a function of disorder and confirmed it's existence at small values of electron concentration, however, absent at larger densities.
We study
s-wave superconductivity in the two-dimensional square-lattice attractive Hubbard Hamiltonian for various inhomogeneous patterns of interacting sites. Using the Bogoliubov--de Gennes mean-field approximation, we obtain the phase diagram for inhomogeneous patterns in which the on-site attractive interaction
Ui between the electrons takes on two values
Ui=0 and $\ensuremath{-}U∕(1\ensuremath{-}f)$ (with
f the concentration of noninteracting sites) as a function of average electron occupation per site,
n, and study the evolution of the phase diagram as
f varies. In certain regions of the phase diagram, inhomogeneity results in a larger zero-temperature average pairing amplitude (order parameter) and also a higher superconducting critical temperature
Tc, relative to a uniform system with the same mean interaction strength (${U}_{i}=\ensuremath{-}U$ on all sites). These effects are observed for stripes, checkerboard, and even random patterns of the attractive centers, suggesting that the pattern of inhomogeneity is unimportant. The phase diagrams also include regions where superconductivity is obliterated due to the formation of various charge-ordered phases. The enhancement of
Tc due to inhomogeneity is robust as long as the electron doping per site,
n, is less than twice the fraction of interacting sites $[2(1\ensuremath{-}f)]$ regardless of the pattern. We also show that for certain inhomogeneous patterns, when $n=2(1\ensuremath{-}f)$, increasing temperature can work against the stability of existing charge-ordered phases for large
f and, as a result, enhance
Tc.
The temperature dependence of an isolated quantum vortex, embedded in an otherwise homogeneous fermionic superfluid of infinite extent, is determined via the Bogoliubov--de Gennes (BdG) equations across the BCS-BEC crossover. Emphasis is given to the BCS side of this crossover, where it is physically relevant to extend this study up to the critical temperature for the loss of the superfluid phase, such that the size of the vortex increases without bound. To this end, two techniques are introduced. The first one solves the BdG equations with ``free boundary conditions,'' which allows one to determine with high accuracy how the vortex profile matches its asymptotic value at a large distance from the center, thus avoiding a common practice of constraining the vortex in a cylinder with infinite walls. The second one improves on the regularization procedure of the self-consistent gap equation when the interparticle interaction is of the contact type, and permits us to considerably reduce the time needed for its numerical integration by drawing elements from the derivation of the Gross-Pitaevskii equation for composite bosons starting from the BdG equations.
Elemental good metals, including noble metals (Cu, Ag, Au) and several
s-block elements, do not exhibit superconductivity in bulk at ambient pressure, mainly due to weak electron-phonon coupling that cannot overcome Coulomb repulsion. Quantum confinement in ultra-thin films reshapes the electronic spectrum and the density of states near the Fermi level, producing strong, often non-monotonic, thickness dependencies of the critical temperature in established superconductors. Here, we examine whether confinement alone, or combined with proximity effects, can induce superconductivity in metals that are non-superconducting in bulk form. We review recent theoretical progress and introduce a unified framework based on a confinement-generalized, isotropic one-band Eliashberg theory, where the normal density of states becomes energy dependent and key parameters (
EF,
λ, $μ^$) acquire explicit thickness dependence. By numerically solving the Eliashberg equations using ab initio or experimentally determined electron-phonon spectral functions
α2F(Ω) and Coulomb pseudopotentials $μ^$, and without adjustable parameters, we compute the critical temperature
Tc as a function of film thickness for representative noble, alkali, and alkaline-earth metals. The results predict that superconductivity emerges only in selected cases and within extremely narrow thickness windows, typically at sub-nanometer scales (
L∼0.4−0.6 nm), indicating strong fine-tuning requirements for confinement-induced superconductivity in good metals. We also consider layered superconductor/normal-metal systems where confinement and proximity effects coexist. In these heterostructures, a substantial enhancement of the critical temperature is predicted, even when the constituent materials are non-superconducting or weak superconductors in bulk form.
The properties of a two-dimensional low density (n<<1) electron system with strong onsite Hubbard attraction U>W (W is the bandwidth) in the presence of a strong random potential V uniformly distributed in the range from -V to +V are considered. Electronic hoppings only at neighboring sites on the square lattice are taken into account, thus W=8t. The calculations were carried out for a lattice of 24x24 sites with periodic boundary conditions. In the framework of the Bogoliubov - de Gennes approach we observed an appearance of inhomogeneous state of spatially separated Fermi-Bose mixture of Cooper pairs and unpaired electrons with the formation of bosonic droplets of different size in the unpaired fermionic matrix. We observed an increase in the droplet size (from individual bielectronic pairs to larger droplets and finally to the percolation cluster) when we increase the electron density at fixed values of the Hubbard attraction and random potential.