Break-up of long-range coherence due to phase fluctuations in ultrathin   superconducting NbN films
  Using scanning tunneling spectroscopy (STS), we address the problem of the superconductor-insulator phase transition (SIT) in homogeneously disordered ultrathin (2-15 nm) films of NbN. Samples thicker than 8 nm, for which the Ioffe-Regel parameter $k_F l \geq 5.6$, manifest a conventional superconductivity : A spatially homogeneous BCS-like gap, vanishing at the critical temperature, and a vortex lattice in magnetic field. Upon thickness reduction, however, while $k_F l$ lowers, the STS revealed striking deviations from the BCS scenario, among which a progressive decrease of the coherence peak height and spatial inhomogeneities. The thinnest film (2.16 nm), while not being exactly at the SIT ($T_C \approx 0.4 T_{C-bulk}$), showed astonishingly vanishing coherence peaks and the absence of vortices. In the quasi-2D limit, such clear signatures of the loss of long-range phase coherence strongly suggest that, at the SIT the superconductivity is destroyed by phase fluctuations. 
Y. Noat, T. Cren, C. Brun, F. Debontridder, V. Cherkez, K. Ilin, M.   Siegel, A. Semenov, H.-W. H\"ubers, and D. Roditchev
cond-mat.supr-con cond-mat.mes-hall