Homogeneously disordered superconducting TiN thin films close to the superconductor-insulator transition (SIT) have been investigated by scanning tunneling spectroscopy [1]. At low temperature, we observed spatial fluctuations of the superconducting gap $Delta$ in agreement with theoretical predictions [2,3]. When the sample gets closer to the critical disorder, its superconducting critical temperature $T_c$ is significantly depressed towards zero while these inhomogeneities are reinforced with a local strong $Delta/T_c$ ratio. This non vanishing spectral gap demonstrates the persistence of local superconducting pairing across the disorder-tuned SIT. At higher temperature, a pseudogap state has been revealed above $T_c$. The thermal evolution of the density of states at the Fermi level in this pseudogap regime, analysed in the framework of disorder-enhanced two-dimensional superconducting fluctuations, gives a clear signature of short-lived Cooper pairs well above Tc. The interplay of these superconducting fluctuations with the Coulomb interaction will be discussed. [1] B. Sac’ep’e, textit{et al.} Physical Review Letters textbf{101}, 157006 (2008). [2] A. Ghosal, M. Randeria, and N. Trivedi, Phys. Rev. Lett. textbf{81}, 3940 (1998); Phys. Rev. B textbf{65}, 014501 (2001). [3] M. A. Skvortsov, and M. V. Feigel’man, Phys. Rev. Lett. textbf{95}, 057002 (2005).
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