Superconducting properties of metallic nanowires can be entirely different from those of bulk superconductors because of the dominating role played by thermal and quantum fluctuations of the order parameter. For superconducting wires with diameters below $ \sim 50$ nm quantum phase slippage is an important process which can yield a non-vanishing wire resistance down to very low temperatures. Further decrease of the wire diameter, for typical material parameters down to $\sim 10$ nm, results in proliferation of quantum phase slips causing a sharp crossover from superconducting to normal behavior even at T=0. A number of interesting phenomena associated both with quantum phase slips and with the parity effect occur in superconducting nanorings. We review recent theoretical and experimental activities in the field and demonstrate dramatic progress in understanding of the phenomenon of superconductivity in quasi-one-dimensional nanostructures.Comment: 62 pages, 47 figures Misprints corrected, several equations are adapted to the experimentally relevant diffusive limi

A hundred years after discovery of superconductivity, one fundamental prediction of the theory, the coherent quantum phase slip (CQPS), has not been observed. CQPS is a phenomenon exactly dual1 to the Josephson effect: whilst the latter is a coherent transfer of charges between superconducting contacts 2,3 , the former is a

Below a certain temperature T(c) (typically cryogenic), some materials lose their electric resistance R entering a superconducting state. Following the general trend toward a large scale integration of a greater number of electronic components, it is desirable to use superconducting elements in order to minimize heat dissipation. It is expected that the basic property of a superconductor, i.e., dissipationless electric current, will be preserved at reduced scales required by modern nanoelectronics. Unfortunately, there are indications that for a certain critical size limit of the order of approximately 10 nm, below which a "superconducting" nanowire is no longer a superconductor in a sense that it acquires a finite resistance even at temperatures close to absolute zero. In the present paper we report experimental evidence for a superconductivity breakdown in ultranarrow quasi-1D aluminum nanowires.

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...