A. SCIENTIFIC BACKGROUND

A.1. Superconducting nanofilms and nanowires.

Recent advances in nanofabrication resulted in high-quality metallic nano-structures like single-crystalline films with thickness down to a few monolayers and wires (both single-crystalline and made of strongly coupled grains) with diameters down to 5 - 10 nm. In particular, ultrathin flat islands of Pb can now be routinely fabricated with atomically uniform thickness down to few monolayers [1-11]. Such single-crystalline nanofilms are formed on a disordered wetting layer (about a few monolayers of Pb) on a silicon substrate, and this disordered interface controls the electron mean free path. Effects of such a disorder on the electron motion perpendicular to the nanofilm are minor, and clear signatures of the formation of the quantum-well states and, so, of splitting of the conduction band into a series of single-electron subbands can be observed in tunneling spectra (see, e.g., Refs. 1, 6, 8, 11). In this case the superconducting condensate is facilitated by multiple quantum channels that appear due to transverse quantum confinement. The high-quality single-crystalline nanofilms are low resistive and show no significant suppression of superconductivity driven by fluctuations or disorder (see, e.g., discussion in Ref. 6). In particular, it was recently found [11] that superconductivity survives in Pb films with thickness down to 2 monolayers, i.e., in the presence of only one quantum channel for the longitudinal electron motion. A similar stability of superconductivity against dimensional reduction was observed in high-quality nanowires, both single-crystalline (Sn, see Refs. 12-14) and made of strongly coupled grains (Al, see Refs. 15-18). Here signatures of the superconducting state were found for thicknesses down to 5 - 10nm [16, 17].

Transverse quantization of electron motion in high-quality superconducting nanofilms and nanowires opens the unique possibility to investigate the multiband superconductivity governed by quantum confinement. The multiple-channel structure can in general suggest a wealth of interesting superconducting phenomena, such as, e.g., Leggett's collective mode [20], Tanaka's solution [21-23] and fractional flux [22, 24]. The multichannel structure induced by quantum-size effects can result in additional nontrivial issues because the lower edges (bottoms) of single-electron subbands formed due to quantum confinement move in energy with changing sample thickness. In particular, superconducting properties exhibit quantum-size oscillations accompanied by significant enhancements each time when the bottom of a new single-electron subband passes through the Fermi surface, i.e., quantum-size superconducting resonance [25]. Recently, quantum-size oscillations of the critical superconducting temperature and critical magnetic field have been observed in single-crystalline Pb nanofilms [1, 4, 9]. Furthermore, the results of a numerical solution of the Bogoliubov-de Gennes (BdG) equations for metallic nanowires in Ref. 26 showed that the quantum-size effects are responsible for a systematic shift of Tc to higher values as recently observed in aluminum and tin nanorods with decreasing thickness [12, 13, 15-17]. Novel effects due to quantum-size induced multiband superconductivity have been predicted: the formation of new Andreev-type states induced by quantum confinement [26] and a cascade structure of the superconductor-to-normal transition driven by a magnetic field or supercurrent [27].

Another important aspect is that the superconducting properties of ultra-narrow nanowires can be significantly influenced by quantum fluctuations of the order parameter. For superconducting wires with diameters below 30 - 50 nm quantum phase slippage can yield a non-vanishing wire resistance down to very low temperatures. This resistance is almost negligible for diameters larger than 15 - 20 nm. However, it is expected [19, 28, 29] that a further decrease of the wire diameter, for typical material parameters down to 5 - 10 nm, results in a proliferation of quantum phase slips causing a sharp crossover from superconducting to normal behavior even at T = 0. Such a crossover was recently observed in aluminum nanowires with diameter of about 8 nm [16]. However, no signature of such a crossover was found in similar aluminum nanowires with cross-section 5.2 nm x 6.1 nm and length of about 100 micrometer [17]. Despite recent progress in the understanding of quantum fluctuations, there are still open questions, e.g., influence of granularity on the interpretation of the experiments, the impact of the transverse quantization on quantum tunneling of the order parameter and, so, quantum-slippage scenario, etc.

Though the disorder in high-quality superconducting nanofilms and nanowires is minor, comprehensive study of its effect on the superconducting properties is of great mportance. Since the very early stages of the theory of superconductivity, it has been known [30-32] that the superconducting transition temperature Tc is insensitive to the rate of elastic impurity scattering, i.e., it does not change in the presence of nonmagnetic impurities. This statement is known as the Anderson theorem and is valid provided that (i) the enhanced Coulomb interaction effects [33] and (ii) mesoscopic fluctuations [34] are negligible. These effects become important close to the Anderson localization transition which is sensitive to the geometrical parameters of the sample. In particular, when a nanowire is long enough and its length is larger than the localization length, the Anderson transition occurs [35]. Thus, for the same density of impurities and for the same surface roughness, the role of disorder can significantly vary with the geometrical parameters. In spite of the fact that the impact of disorder on the superconducting state has been investigated during many decades, there are still open and intriguing issues, e.g., a recently reported superinsulator state accompanied by quantum synchronization [36], effects of mesoscopic fluctuations on the magnitude [37] and spatial distribution [38] of the superconducting order parameter, etc. In addition, the interplay of quantum-size induced multiband superconductivity and disorder poses new questions.

A.2. Superconducting metallic nanoparticles.

In superconducting metallic nanoparticles the electron energy is fully quantized and in addition new effects appear due to the finite number of electrons in the system. Superconducting characteristics of such nanosystems can be investigated with powders of metallic nanograins [39-41]. Another possibility is to fabricate films made of metallic nanoparticles embedded in a poorly conducting disordered matrix [42, 43]. Recently, single isolated Pb nanoparticles were fabricated on a silicon substrate [44], which complements the line of research of the classical paper of Black, Ralph, and Tinkham about individual nanometer-scale aluminum nanoparticles [45]. In Ref. 44 significant deviations from the Bardeen-Cooper-Schrieffer scenario have been observed in the temperature dependence of the superconducting energy gap, i.e., thermal fluctuations resulting in nonzero gap above Tc. Several new observations are waiting for an explanation, e.g., a non-monotonic dependence of Tc on the nanoparticle size [41]; coexistence of ferromagnetism and superconductivity in powders of Sn nanoparticles [41]; nontrivial size-dependence of the ratio of the superconducting energy gap to the critical temperature [44]; extremely small coherence length (an order of magnitude smaller than the particle size) [43], etc. Most of the theoretical papers of the last decade were mainly concerned with the problem of the so-called breaking size, i.e., the nanoparticle diameter where superconductivity ceases [45-47].

A.3. Tubular superconductivity.

Another interesting and promising example of nanosystems exhibiting superconducting properties are carbon nanotubes. Their electronic properties can significantly vary, e.g., from metallic to insulating behavior depending on the particular wrapping of the carbon nanotube (the helicity vector) [49]. Under certain circumstances superconductivity appears in nanotubes, both proximity induced and intrinsic (in ropes) [50-55]. Large supercurrents and high suppressing magnetic fields have been measured in single-wall nanotubes and their ropes. By tuning the electronic properties one can change the superconducting behavior substantially, which is of importance for tailoring the superconducting properties for practical applications.

B. COMPUTATIONAL ASPECTS

Superconducting properties of metallic nanofilms and nanowires are strongly dependent on quantum confinement, i.e., on the particular sample geometry. This means that solid conclusions about the equilibrium superconducting quantities can be only found from a numerical solution of the Bogoliubov-de Gennes equations or from a numerical analysis of Gor'kov Green's function formalism [57]. In turn, an interplay of quantum confinement with the main crystal directions results in additional complications, which calls for an accurate treatment of the electron band structure, i.e., via the Bogoliubov-de Gennes equations written in the tight-binding approximation (see, e.g., Refs. 58, 59) or by means of the Kohn-Sham-Bogoliubov-de Gennes equations [60, 61].

The electronic properties of carbon nanotubes depend strongly on the helicity vector [56]. In this case the Bogoliubov-de Gennes approach written in the tight-binding approximation is a relevant tool for investigating superconductivity and, under certain conditions (for single-wall nanotubes), they reduce to the Bogoliubov-de Gennes-Dirac equations [62]. In turn, the Keldysh Green's function formalism [63] is a solid way to investigate transport properties of carbon nanotubes, like supercurrent through a superconductor-carbon-nanotube device. This also implies comprehensive studies based on a complex numerical analysis.

Different computational methods for calculating superconducting properties of nanoscale systems are planned to be demonstrated and discussed at the Workshop. Different representations of the Bogoliubov-de Gennes equations and models based on Green's functions (Gor'kov and Keldysh formalism) will be discussed.

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## Superconductivity in nanosized systems

#### CECAM-HQ-EPFL, Lausanne, Switzerland

## References

**Belgium**

François Peeters ( University of Antwerp ) - Organiser & speaker

**Israel**

Hardy Gross ( The Hebrew University of Jerusalem ) - Organiser & speaker