A shape-dependent superconducting resonance can be expected when an energy level associated with the transverse motion in a wire passes through the Fermi surface. We show that the recently observed widthdependent increase of T c in Al and Sn nanowires is a consequence of this shape-resonance effect.Increasing the critical temperature ͑T c ͒ of a superconductor ͑SC͒ has been a major challenge. On the one hand one can look for different materials that exhibit a higher T c . Such a search has been very successful over the last 20 years. On the other hand microstructuring of a superconductor is a different and new road that is able to modify T c ͑i.e., increase and/or decrease it͒ and may also give us further insight in the basic mechanism of superconductivity.In earlier works on microstructuring of SCs in the mesoscopic regime, enhancement of the critical current ͑j c ͒ was demonstrated to occur due to trapping of vortices. Also a large increase of the critical magnetic field ͑H c ͒ was realized through such mesoscopic structuring, which is mostly a consequence of surface superconductivity. But in both cases the zero-magnetic-field critical temperature was unaltered. The enhancement of j c and H c could be accurately described by phenomenological theories such as the London approach and the ͑time-dependent͒ Ginzburg-Landau theory.In the present Brief Report we are interested in modifying a SC on the nanoscale. We deal with systems where the electron motion is limited to quasi-one-dimension ͑1D͒. During the last decade nanowires have attracted much attention in the context of phase fluctuations of the order parameter ͑i.e., quantum phase slips͒. 1 But quantization of the electron motion in the transverse direction was not investigated in much detail. However, very recently numerical investigation of the Bogoliubov-de Gennes equations has shown 2 that this quantization results in significant shape-dependent superconducting resonances with a profound effect on the nanowire T c . Such systems have been the subject of recent experimental studies, 3-5 and we demonstrate here that the widthdependent increase of T c found in these experiments is a manifestation of these shape resonances.More than 40 years ago, Blatt and Thompson 6 calculated a remarkable sequence of peaks in the thickness dependence of the energy-gap parameter of single-crystalline superconducting nanofilms in the clean limit. They called these spikes shape resonances. At that time it was not possible to produce high-quality SCs with nanoscale dimensions ͑only very recently were the thickness-dependent oscillations of T c observed experimentally in ultrathin Pb films 7 ͒. For decades atomic nuclei were the only systems where the interplay between quantum confinement and pairing of fermions could be studied experimentally and where the expectations of Blatt and Thompson were confirmed as a series of size resonances in the pairing energy gap of nuclei. 8 Very recently high-quality nanowires have become available, where this resonance effect is expected 2 to b...

In the nearest vicinity of the critical temperature, types I and II of conventional single-band superconductors interchange at the Ginzburg-Landau parameter κ = 1/ √ 2. At lower temperatures this point unfolds into a narrow but finite interval of κ's, shaping an inter-type (transitional) domain in the (κ, T )-plane. In the present work, based on the extended Ginzburg-Landau formalism, we show that the same picture of the two standard types with the transitional domain in between applies also to multi-band superconductors. However, the inter-type domain notably widens in the presence of multiple bands and can become extremely large when the system has a significant disparity between the band parameters. It is concluded that many multi-band superconductors, such as recently discovered borides and iron-based materials, can belong to the inter-type regime.

Recent observation of unusual vortex patterns in MgB(2) single crystals raised speculations about possible "type-1.5" superconductivity in two-band materials, mixing the properties of both type-I and type-II superconductors. However, the strict application of the standard two-band Ginzburg-Landau (GL) theory results in simply proportional order parameters of the two bands-and does not support the "type-1.5" behavior. Here we derive the extended GL formalism (accounting all terms of the next order over the small τ=1-T/T(c) parameter) for a two-band clean s-wave superconductor and show that the two condensates generally have different spatial scales, with the difference disappearing only in the limit T→T(c). The extended version of the two-band GL formalism improves the validity of GL theory below T(c) and suggests revisiting the earlier calculations based on the standard model.

We study the shape resonance effect associated with the confined transverse superconducting modes of a cylindrical nanowire in the clean limit. Results of numerical investigations of the Bogoliubov-de Gennes equations show significant deviations of the energy gap parameter from its bulk value with a profound effect on the transition temperature. The most striking is that the size of the resonances is found to be by about order of magnitude larger than in ultrathin metallic films with the same width.PACS numbers: PACS number(s): 74.78.Na Modern rapid miniaturization of electronic circuits requires good understanding of basic mechanisms responsible for the electronic properties of nanoscale structures. The most important point about these structures is that the quantum-confinement effects play the corner-stone role in this case. One can even say in general that recent success in nanofabrication technique has resulted in great interest in various artificial physical systems with unusual phenomena driven by the quantum confinement (quantum dots, nanoscale semiconductors, nanosuperconductors, etc.). The quantum-confined superconductivity is here of special interest due to the macroscopic quantum character: any effect on electron wave functions manifests itself directly in the superconducting order parameter.An obvious consequence of the confinement in a nanoscale superconducting structure is nonuniform spatial distribution of the superconducting condensate because, as it is known since the classical papers by Gor'kov [1] and Bogoliubov [2], the superconducting order parameter can be interpreted as the wave function of the center-of-mass motion of a Cooper pair. It is also known that the Cooper-pair wave function involves important in-medium terms [3]. In the presence of the electron confinement these terms can result in shape resonances in the energy gap parameter, another confinement effect first found and investigated in the paper by Blatt and Thompson [4] for ultrathin metallic films. A shape resonance in the dependence of the energy-gap parameter on the specimen dimensions can occur any time when an electron subband appearing due to the size quantization passes through the Fermi surface [4]. Strong indications for such behaviour are found not only in ultrathin films but also in nanoparticles (see, for example, Refs.[5] and [6]) and superfluid nuclei [7,8] (as it had been predicted by Blatt and Thompson). Recently a new technique of electrodeposition into extended nanopores has been developed [9], which makes it possible to produce single-crystal nanowires of high quality. Thus, the shape resonances in the nanowire superconducting order parameter can be investigated in the clean limit, with a direct link to the microscopic (BSC) theory. In particular, it is of importance to explore the situation where the QPS (quantum phase slips) regime [9,10,11,12]) is expected to generate a new low-temperature metallic phase with proliferating quantum phase slips of the superconducting order parameter (for radii less than 5...

We show that two-band superconductors harbor hidden criticality deep in the superconducting state, stemming from the critical temperature of the weaker band taken as an independent system. For sufficiently small interband coupling γ the coherence length of the weaker band exhibits a remarkable deviation from the conventional monotonic increase with temperature, namely, a pronounced peak close to the hidden critical point. The magnitude of the peak scales as ∝γ-μ, with the Landau critical exponent μ=1/3, the same as found for the mean-field critical behavior with respect to the source field in ferromagnets and ferroelectrics. Here reported hidden criticality of multiband superconductors can be experimentally observed by, e.g., imaging of the variations of the vortex core in a broader temperature range. Similar effects are expected for the superconducting multilayers.

In high-quality nanowires, quantum confinement of the transverse electron motion splits the band of single-electron states in a series of subbands. This changes in a qualitative way the scenario of the magnetic-field induced superconductor-to-normal transition. We numerically solve the Bogoliubovde Gennes equations for a clean metallic cylindrical nanowire at zero temperature in a parallel magnetic field and find that for diameters D 10 ÷ 15 nm, this transition occurs as a cascade of subsequent jumps in the order parameter (this is opposed to the smooth second-order phase transition in the mesoscopic regime). Each jump is associated with the depairing of electrons in one of the single-electron subbands. As a set of subbands contribute to the order parameter, the depairing process occurs as a cascade of jumps. We find pronounced quantum-size oscillations of the critical magnetic field with giant resonant enhancements. In addition to these orbital effects, the paramagnetic breakdown of Cooper pairing also contributes but only for smaller diameters, i. e., D 5 nm.

A procedure to derive the Ginzburg-Landau (GL) theory from the multiband BCS Hamiltonian is developed in a general case with an arbitrary number of bands and arbitrary interaction matrix. It combines the standard Gor'kov truncation and a subsequent reconstruction in order to match accuracies of the obtained terms. This reconstruction recovers the phenomenological GL theory as obtained from the Landau model of phase transitions but offers explicit microscopic expressions for the relevant parameters. Detailed calculations are presented for a three-band system treated as a prototype multiband superconductor. It is demonstrated that the symmetry in the coupling matrix may lead to the chiral ground state with the phase frustration, typical for systems with broken time-reversal symmetry.

We derive the extended Ginzburg-Landau (GL) formalism for a clean s-wave two-band superconductor by employing a systematic expansion of the free-energy functional and the corresponding matrix gap equation in powers of the small deviation from the critical temperature τ = 1 − T /Tc. The two lowest orders of this expansion produce the equation for Tc and the GL theory. It is shown that in agreement with previous studies, the two-band GL theory maps onto the single-band GL model and thus fails to describe the difference in the spatial profiles of the two band condensates. We prove that except for some very special cases, this difference appears already in the leading correction to the GL theory, which constitutes the extended GL formalism. We derive linear differential equations that determine the leading corrections to the band order parameters and magnetic field, discuss the validity of these equations, and consider examples of an important interplay between the band condensates. Finally, we present numerical results for the thermodynamic critical magnetic field and temperature-dependent band gaps (at zero field), which are in a very good agreement with those obtained from the full BCS approach in a wide temperature range. To this end, we emphasize the advantages of our extended GL theory in comparison with the often used two-component GL-like model based on an unreconstructed two-band generalization of the Gor'kov derivation.