Spectroscopic evidence for strong correlations between local resistance and superconducting gap in ultrathin NbN films

Many homogeneously disordered superconducting films follow a Finkelstein mechanism : when disorder combined to electron-electron interactions increase, there is a global decrease of the superconducting energy gap $\Delta$ concomittant with the one of the critical temperature. Simultaneously, in most films an emergent granularity develops with increasing disorder and results in the formation of inhomogeneous superconducting puddles. By studying the local electronic properties of a NbN film with scanning tunneling spectroscopy (STS) we show that the inhomogeneous spatial distribution of $\Delta$ is locally strongly correlated to a large depletion in the local density of states, associated to a zero-bias anomaly (ZBA). By modelling quantitatively the measured ZBA variations, we show that they correspond to local variations of the film resistivity. This local change in resistivity leads to a local variation of $\Delta$ through a local Finkelstein mechanism, furnishing a scenario explaning quantitatively the emerged superconducting inhomogeneities.