Abstract: It is well known that the formation of discrete electron levels strongly influences the pairing in metallic nanograins. Here, we focus on another effect of quantum confinement in superconducting grains that was not studied previously, i.e., spatially nonuniform pairing. This effect is very significant when single-electron levels form bunches and/or a kind of shell structure. We find that, in highly symmetric grains, the order parameter can exhibit variations with position by an order of magnitude. Nonuniform pairing is closely related to a quantum-confinement-induced modification of the pairing-interaction matrix elements and size-dependent pinning of the chemical potential to groups of degenerate or nearly degenerate levels. For illustrative purposes, we consider spherical metallic nanograins and also rectangular shapes. We show that the relevant matrix elements are, as a rule, enhanced in the presence of quantum confinement, which favors spatial variations of the order parameter, compensating the corresponding energy cost. The size-dependent pinning of the chemical potential further increases the spatial variation of the pair condensate. The role of nonuniform pairing is smaller in less symmetric confining geometries and/or in the presence of disorder. However, it always remains of importance when the energy spacing between discrete electron levels δ is approaching the scale of the bulk gap ΔB, i.e., δ>0.10.2 ΔB.

Abstract: We investigate how the interplay of quantum confinement and particle number-parity fluctuations affects superconducting correlations in ultra-small metallic grains. Using the number-parity projected BCS formalism we calculate the critical temperature and the excitation gap as a function of the grain size for grains with even and odd number of confined carriers. We show that the experimentally observed anomalous increase of the coupling ratio ΔE/kBTc with decreasing superconducting grain size can be attributed to an enhancement of the number-parity fluctuations in ultra-small grains.

Abstract: To date, several experimental groups reported measurements of the thickness dependence of T-c of atomically uniform single-crystalline Pb nanofilms. The reported amplitude of the T-c oscillations varies significantly from one experiment to another. Here we propose that the reason for this unresolved issue is an interplay of the quantum-size variations in the single-electron density of states with thickness-dependent oscillations in the phonon-mediated electron-electron coupling. Such oscillations in the coupling depend on the substrate material, the quality of the interface, the protection cover, and other details of the fabrication process, changing from one experiment to another. This explains why the available data do not exhibit one-voice consistency about the amplitude of the T-c oscillations. Our analyses are based on a numerical solution of the Bogoliubov-de Gennes equations for a superconducting slab.

Abstract: Chiral fermions are responsible for low-temperature properties of vortices in fermionic condensates, both superconducting (charged) and superfluid (neutral). One of the most striking consequences of this fact is that the core of a single-quantum vortex collapses at low temperatures, T -> 0 (i.e., the Kramer-Pesch effect for superconductors), due to the presence of chiral quasiparticles in the vortex-core region. We show that the situation changes drastically for fermionic condensates confined in quasi-one-dimensional and quasi-two-dimensional geometries. Here quantum confinement breaks the chirality of in-core fermions. As a result, instead of the ultimate shrinking, the core of a single-quantum vortex extends at low temperatures, and the condensate profile surprisingly mimics the multiquantum vortex behavior. Our findings are relevant for nanoscale superconductors, such as recent metallic nanoislands on silicon, and also for ultracold superfluid Fermi gases in cigar-shaped and pancake-shaped atomic traps.

Abstract: From a numerical solution of the Bogoliubov-de Gennes equations, we investigate an interplay of the transverse discrete modes with a longitudinal supercurrent in a metallic cylindrical superconducting nanowire. The superconductor-to-normal transition induced by a longitudinal superflow of electrons is found to occur as a cascade of jumps in the order parameter (supercurrent and superfluid density) as a function of the superfluid velocity for diameters d<1015 nm (for Al parameters) and sufficiently low temperatures T<0.30.4Tc, with Tc the critical temperature. When approaching Tc, the jumps are smoothed into steplike but continuous drops. A similar picture occurs for d>1520 nm. Only when the diameter exceeds 5070 nm the quantum-size cascades are fully washed out, and we arrive at the mesoscopic regime. Below this regime the critical current density jc exhibits the quantum-size oscillations with pronounced resonant enhancements: the smaller the diameter, the more significant is the enhancement. Thickness fluctuations of real samples will smooth out such oscillations into an overall growth of jc with decreasing nanowire diameter.

Abstract: We analyze the paradigmatic competition between intraband and crossband Cooper-pair formation in twoband superconductors, neglected in most works to date. We derive the phase-sensitive gap equations and describe the crossover between the intraband-dominated and the crossband-dominated regimes, delimited by a “gapless” state. Experimental signatures of crosspairing comprise notable gap splitting in the excitation spectrum, non-BCS behavior of gaps versus temperature, as well as changes in the pairing symmetry as a function of temperature. The consequences of these findings are illustrated on the examples of MgB2 and Ba0.6K0.4Fe2As2.

Abstract: This Letter presents a generalized Ginzburg-Landau equation for the superconducting order parameter which includes the terms resulting from the confining interaction associated with the specimen boundary. While the original Ginzburg-Landau theory had been developed for a bulk superconductor, this generalization is meant for study of a meso-superconductor. (C) 2004 Elsevier B.V. All rights reserved.

Abstract: Conventional superconductivity is inevitably suppressed in ultra-small metallic grains for characteristic sizes smaller than the Anderson limit. Experiments have shown that above the Anderson limit the critical temperature may be either enhanced or reduced when decreasing the particle size, depending on the superconducting material. In addition, there is experimental evidence that whether an enhancement or a reduction is found depends on the strength of the electronphonon interaction in the bulk. We reveal how the strength of the e-ph interaction interplays with the quantum-size effect and theoretically obtain the critical temperature of the superconducting nanograins in excellent agreement with experimental data. We demonstrate that strong e-ph scattering smears the peak structure in the electronic density-of-states of a metallic grain and enhances the electron mass, and thereby limits the highest T-c achievable by quantum confinement.

Abstract: Recent progress in materials synthesis enabled fabrication of superconducting atomically flat single-crystalline metallic nanofilms with thicknesses down to a few monolayers. Interest in such nano-thin systems is attracted by the dimensional 3D-2D crossover in their coherent properties which occurs with decreasing the film thickness. The first fundamental aspect of this crossover is dictated by the Mermin-Wagner-Hohenberg theorem and concerns frustration of the long-range order due to superconductive fluctuations and the possibility to track its impact with an unprecedented level of control. The second important aspect is related to the Fabri-Perot modes of the electronic motion strongly bound in the direction perpendicular to the nanofilm. The formation of such modes results in a pronounced multiband structure that changes with the nanofilm thickness and affects both the mean-field behavior and superconductive fluctuations. Though the subject is very rich in physics, it is scarcely investigated to date. The main obstacle is that there are no manageable models to study a complex magnetic response in this case. Full microscopic consideration is rather time consuming, if practicable at all, while the standard Ginzburg-Landau theory is not applicable. In the present work we review the main achievements in the subject to date, and construct and justify an efficient multiband mean-field formalism which allows for numerical and even analytical treatment of nano-thin superconductors in applied magnetic fields.