For many years scientists have tried different strategies to improve superconductivity. The recent development in the growth of thin films and grains at the nanoscale has allowed to measure in certain materials critical temperatures that differ from the bulk values. At the same time these results have motivated theoretical studies that try to reproduce them using first-principles calculations and different theories about superconductivity.

We aim to contribute in this area with an analytical study of MgB$2$ superconducting thin films. We choose to study this material since it is a conventional superconductor with the highest critical temperature: 39 K among these kind of superconductors.

Another motivation to carry out this research is that in recent experiments they have managed to grow MgB$2$ films with an acceptable quality and a thickness of less than 10 nm. Even though further optimization of the growing techniques needs to be performed, this demonstrates that the effects we have studied theoritically should not be far from being measurable.

MgB$2$ is a two-band superconductor. To our knowledge there are not many studies regarding this kind of superconductors at the nanoscale. We have studied a range of thicknesses going from 2 nm to 20 nm for infinite films (lateral sizes much larger than its thickness) and finite films (lateral size of the order of its thickness). We have used a mean field approach (Bardeen- Cooper-Schrieffer or BCS theory) and tried to extend the model presented by Thompson and Blatt of a one-band superconductor thin film to the case of a two-band superconductor thin film.

The most relevant results of this research are: for an infinite thin film with a thickness of less than 10 nm the critical temperature ”oscillates” between $Tc=62.4$ K and $Tc=31.2$ K, showing that shape resonances induce both enhancement and suppression as the thickness changes. We have also seen that superconductivity can be in average enhanced or suppressed depending on the type of bands and sign of the effective masses. Finally, we have obtained that the equations derived are robust under changes of the parameters used.

It is important to mention that due to the limitations of the theory used and the simplifications made: neglecting quantum fluctuations, defects, the influence of substrate, etc. our results must not be taken fully quantitative, specially for thicknesses smaller 4 nm. Yet, we believe they grasp most of the relevant physical phenomena of the described nano-scale system.