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Superconducting films are usually regarded as type II superconductors even when they are made of a type I material. The reason is the presence of stray magnetic fields that stabilize the vortex matter by inducing long-range repulsive interactions between vortices. While very thin films indeed reach this limit, there is a large interval of thicknesses where magnetic properties of superconducting films cannot be classified as either of the two conventional superconductivity types. Recent calculations revealed that in this interval the system exhibits spontaneous formation of magnetic flux-condensate patterns and superstructures appearing due to the interplay between the long-range stray field effects and proximity to the Bogomolnyi self-duality point. These calculations were based on the periodic in-plane boundary conditions which, as is well known from classical electrodynamics, for systems with long-range interactions can lead to field distortions and considerable discrepancies between results of different calculation methods. Here we demonstrate that similar spontaneous patterns are obtained for superconducting films with open in-plane boundary conditions (vanishing in-plane currents perpendicular to the edges of the finite film) and thus the phenomenon is not an artefact of chosen boundary conditions.

Quantum confinement of the perpendicular motion of electrons in single-crystalline metallic superconducting nanofilms splits the conduction band into a series of single-electron subbands. A distinctive feature of such a nanoscale multi-band superconductor is that the energetic position of each subband can vary significantly with changing nanofilm thickness, substrate material, protection cover and other details of the fabrication process. It can occur that the bottom of one of the available subbands is situated in the vicinity of the Fermi level. We demonstrate that the character of the superconducting pairing in such a subband changes dramatically and exhibits a clear molecule-like trend, which is very similar to the well-known crossover from the Bardeen-Cooper-Schrieffer regime to Bose-Einstein condensation (BCS-BEC) observed in trapped ultracold fermions. For Pb nanofilms with thickness of 4 and 5 monolayers (ML) this will lead to a spectacular scenario: up to half of all the Cooper pairs nearly collapse, shrinking in the lateral size (parallel to the nanofilm) down to a few nanometers. As a result, the superconducting condensate will be a coherent mixture of almost molecule-like fermionic pairs with ordinary, extended Cooper pairs., 7 pages, 3 figures

It is well known that, in bulk, the solution of the Bogoliubovde Gennes equations is the same whether or not the HartreeFock term is included. Here the HartreeFock potential is position independent and so gives the same contribution to both the single-electron energies and the Fermi level (the chemical potential). Thus, the single-electron energies measured from the Fermi level (they control the solution) stay the same. This is not the case for nanostructured superconductors, where quantum confinement breaks the translational symmetry and results in a position-dependent HartreeFock potential. In this case its contribution to the single-electron energies depends on the relevant quantum numbers. We numerically solved the Bogoliubovde Gennes equations with the HartreeFock term for a clean superconducting nanocylinder and found a shift of the curve representing the thickness-dependent oscillations of the critical superconducting temperature to larger diameters.