Phase Fluctuations in a Strongly Disordered $s$-Wave NbN Superconductor Close to the Metal-Insulator Transition

We explore the role of phase fluctuations in a three-dimensional $s$-wave superconductor, NbN, as we approach the critical disorder for destruction of the superconducting state. Close to critical disorder, we observe a finite gap in the electronic spectrum which persists at temperatures well above $T_c$. The superfluid density is strongly suppressed at low temperatures and evolves towards a linear-$T$ variation at higher temperatures. These observations provide strong evidence that phase fluctuations play a central role in the formation of a pseudogap state in a disordered $s$-wave superconductor.

Figure 1

(a) $\sigma$ vs $T$ for films with $k_{F}l\sim 10.12$, 8.82, 8.13, 8.01, 5.5, 4.98, 3.65, 3.27, 2.21, 1.68, 1.58, and 1.24. The inset shows an expanded view of $\sigma (T)$ for the film with $k_{F}l\sim 1.24$ in zero field and in magnetic field of $5T$. (b) ${\sigma }_0$ and $T_c$ as a function of $k_{F}l$.

Figure 2

(a)–(c) Intensity plots of $G(V)/G_N$ as a function of temperature and applied bias for three different films (upper panels), and the resistance vs temperature ($R\mathrm{\text{−}}T$) in the same temperature range (lower panels); vertical dotted lines in the upper panels correspond to $T_c$. Insets of panels (a),(b) and left inset of panel (c) show representative tunneling spectra in the superconducting state (blue), in the pseudogap state (green) and above the pseudogap temperature (red). The right inset of panel (c) shows the AA background subtracted spectra at 2.65 K.

Figure 3

(a),(b) ${\lambda }^{-2}(T)$ for NbN films with different disorder; the solid lines are the expected temperature variation from dirty-limit BCS theory. (c) ${\lambda }^{-2}(0)$ and ${\lambda }^{-2}(0{)}_{\mathrm{BCS}}$ as function of $T_c$; the inset shows the variation $\Delta (0)$ as function of $T_c$. (d) Temperature variation of $[\lambda (T)/\lambda (0){]}^{-2}$ for the film with $T_c=2.27\mathrm{K}$; the solid lines are fits to the $T^2$ dependence of ${\lambda }^{-2}(T)/{\lambda }^{-2}(0)$ at low temperature ($T\le 0.65\mathrm{K}$) and the $T$ dependence at higher temperature; the inset shows an expanded view of the $T^2$ dependence at low temperatures.