For these reasons we believe that a workshop that brings together experts working in this field is timely. Moreover the resulting  collaborations could lead to high impact in both communities.

A novel aspect of the proposal is the idea of inviting experts working in the areas of

high Tc superconductivity and nanoscience in order to stimulate research on the physics of these materials at these length scales. 

Despite its interest, from a theoretical and practical point of view, we are not aware of any previous meeting that targets this audience. Another innovative feature of the proposal is the aim to combine the expertise of material scientists, condensed matter theorists and experts in computational techniques with the purpose to explore the frontiers in experimental control and theoretical modelling. This is key to make substantial progress in the field and to understand the conditions and limits of enhancement of superconductivity in the nanoscale.

Finally we would like to stress that this workshop directly addresses two of the recom-

mended topics, ”Computational Materials Science and Nanoscience” and ”Simulations of

Phase Transitions and Critical Phenomena”, mentioned in the CECAM 2012 call.

This proposal is motivated by recent technological developments in the synthesis and characterization of high quality nano-structures and in the theoretical techniques needed to model these materials. Another stimulus  is the recent discovery of intrinsic high Tc superconductivity in materials such as iron-pnictides, MgB2 , fullerides and carbon nanotubes together with the first experimental studies of finite size effects in cuprates high Tc superconductors. A good experimental control at the nanoscale makes it feasible to quantitatively compare experimental findings with theoretical calculations, a key ingredient for a deeper understanding of these systems. Likewise the interplay between the physics of high Tc superconductors and nanoscale effects promises to open new avenues of theoretical and experimental research. These advances have the potential to boost dramatically the interest in nanoscale superconductivity in both the physics and material science communities. 

For these reasons we believe that a workshop that brings together experts working in this field is timely. Moreover the resulting  collaborations could lead to high impact in both communities.

A novel aspect of the proposal is the idea of inviting experts working in the areas ofhigh Tc superconductivity and nanoscience in order to stimulate research on the physics of these materials at these length scales. 
Despite its interest, from a theoretical and practical point of view, we are not aware of any previous meeting that targets this audience. Another innovative feature of the proposal is the aim to combine the expertise of material scientists, condensed matter theorists and experts in computational techniques with the purpose to explore the frontiers in experimental control and theoretical modelling. This is key to make substantial progress in the field and to understand the conditions and limits of enhancement of superconductivity in the nanoscale.

The experimental and theoretical study of superconductivity in the nanoscale started inthe early sixties, shortly after the discovery of the BCS theory. The earliest experimentsperformed on granular thin films of elemental superconductors [1] showed that the superconducting properties were quite different from their bulk counterparts. In some materials Tc was greatly enhanced (Al, Sn, In) but in others (Pb, Hg) the differences were rather small.[1]

Theoretical papers [2] confirmed that, even within a BCS approach, finite size effect had a profound impact on the superconducting state. No direct comparison between theory and experiment was possible as it was difficult to have good control on the size and shape of the nanoparticles in granular films. Besides, the particle size distribution and the inter-grain coupling which could not be quantified precisely played an important role in determining the superconducting properties.

After an initial burst of activity, the field entered a phase of steady progress. With advancements in experimental techniques, new ways to grow low dimensional superconductors emerged. Thin films and nanoparticles were followed by the synthesis of one dimensional nanowires. It was found [3] that the superconducting transition in nanowires became broader due to thermal and quantum fluctuations of the superconducting order parameter.  This broadening increases as the wire diameter decreases, a result which was previously observed in the case of small particles [4]. 

Since the thermodynamic properties of nanoparticles [5] were well understood theoretically, models were also proposed to explain the results obtained in nanowires quantitatively [6]. The field received an impetus in the mid nineties after the study of Ralph et al on single Al nanoparticles [7] where it was shown for the first time that superconductivity survived in single particles down to a few nanometers. Besides, their experiments of single electron tunneling into the nanoparticles also showed that the critical particle diameter for the destruction of superconductivity depends on the even or odd number of electrons present in the nanoparticle. Although this important ”parity effect” was postulated earlier [8], a large number of theoretical work in ultrasmall particles followed which tried to understand these results quantitatively [9]. 

Subsequently, more efforts were made to get a rather detailed description of the evolution of superconductivity with system size by combining the Richardson formalism (an exact theory incorporating finite size effects to the BCS theory) with numerical and path integral techniques [10]. However direct comparison of these theoretical results with experiments was still not possible since the experiments of Ralph et al did not have a control on the shape of the nanoparticles.

Around the beginning of this decade, technological advancements on both growth and measurement of single nanostructures led to many experiments on low dimensional systems which tested the limits to superconductivity. Berzyadin et al showed that super-conductivity was destroyed in single MoGe nanowires when its resistance approached the quantum resistance [11]. Experiments on Al nanowires showed the breakdown of superconductivity for wire diameters below 10 nm [12]. In two dimensional films of Pb on Si, initial experiments revealed oscillation in the value of Tc with each atomic layer of the films [13].

However, recent experiments reported superconductivity to persit down to a single monolayer with the transition temperature decreasing with the number of atomic layers. This was also explained by theoretical calculations which showed a decrease in the density of electronic states with film thickness [14].

At the same time numerous groups started to investigate the influence of quantum size effects on superconductivity in low dimensional superconductors. Shanenko et al [15] solved numerically the Bogoliubov-de Gennes equations in nanowires and predicted the interesting shape resonance effects which postulates huge oscillations in the transition temperature of the nanowire with  small changes in the wire diameter. Shell effects, which is the analog of shape resonance effects in superconducting nanoparticles with radial o cubic symmetry, were predicted through numerical calculations of the superconducting energy gap and the Richardson formalism [16][17]. An analytical description of superconductivity in superconducting nanoparticles was put forward in [18] by using semiclassical techniques which also predicted the presence of shell effects in nanoparticles with well defined axis of symmetry. Similar techniques were later employed in [19] to estimate the effect of finite size effects in the energy gap as a function of the particle geometry.

These interesting phenomena could be experimentally observed only through measurements on single nanostructures. Recently, this was possible through scanning tunneling spectroscopic measurements on in situ grown single, isolated hemispherical Sn nanoparticles. The observed oscillations in the superconducting energy gap with particle size confirmed the presence of shell effects in small particles[20]. Moreover, in these experiments an accurate determination of the size and shape of the nanoparticle is possible which makes it a model system for quantitative comparison of the theoretical results of the superconducting properties with particle size.

These experimental advancements paves the way for extending the studies of finite size effects on unconventional superconductors such as the high Tc cuprates, newly discovered iron pnictides [21], hexaborides [22], fullerides [23] etc. Work has started in this regard with the observation of intrinsic superconductivity in carbon nanotubes [24] and fullerides [23]. Recently the proliferation of phase slips in cuprate nanowires of diameter 10nm was investigated in [25]. 

On the theoretical side not much work has been done on finite size effects in the cuprates. Only recently Bianconi et al have shown that fractal distribution of dopants can enhance superconductivity in the cuprates [26]. With the discovery of the pnictides, people are also trying to investigate size effects in these systems [27] but the precise dependence of observables on the system size is not known. 

From the interesting effects observed in conventional low dimensional superconductors, we can readily anticipate that the study of unconventional superconductors at the nanoscale is quite promising and will surely be one of the major themes of research in the near future.

Computational superconductivity also has gone through a profound transformation in recent times. The combination of classical techniques such as Density Functional Theory, first principle calculations and the increasing computer performance has led for the first time to the explanation of superconductivity in multiband and strongly coupled superconductors such as: MgB2 [30], iron pnictides [28] and carbon compounds [29]. The adaptation of these ideas and techniques to the nanoscale is key for a truly quantitative description of experimental results which could shed light about the conditions for the enhancement of superconductivity in these materials by finite size effects.