Confinement of superconducting fluctuations due to emergent electronic inhomogeneities

The microscopic nature of an insulating state in the vicinity of a superconducting state in the presence of disorder is a hotly debated question. While the simplest scenario proposes that Coulomb interactions destroy the Cooper pairs at the transition, leading to localization of single electrons, an alternate possibility supported by experimental observations suggests that Cooper pairs instead directly localize. The question of the homogeneity, granularity, or possibly glassiness of the material on the verge of this transition is intimately related to this fundamental issue. Here, by combining macroscopic and nanoscale studies of superconducting ultrathin NbN films, we reveal nanoscopic inhomogeneities that emerge when the film thickness is reduced. For the thinnest films, scanning tunneling spectroscopy at low temperature unveils inhomogeneities in the superconducting properties, of typical size L-i, that are not correlated to any structural inhomogeneity and that are found to persist above the critical temperature in the form of a pseudogap. Remarkably enough, while the thickest films display a purely two-dimensional behavior in the superconducting fluctuations above the critical temperature, paraconductivity in the pseudogap regime of the thinnest samples demonstrates fluctuations of the amplitude of the order parameter, corresponding to zero-dimensional fluctuating regions of size precisely Li. We propose that an anomalous diffusion slowing-down process is at play at long wave vectors, leading to some confinement of the superconducting fluctuations, which allows us to explain the simultaneous paradoxical presence of a pseudogap and zero-dimensional amplitude fluctuations of the order parameter. These findings call for further theoretical investigation to understand this intermediate state where Cooper pairs continuously evolve from a bound state of fermionic objects into localized bosonic entities.