Phase diagram of the strongly disordered $s$-wave superconductor NbN close to the metal-insulator transition

We present a phase diagram as a function of disorder in three-dimensional NbN thin films as the system enters the critical disorder for destruction of the superconducting state. The superconducting state is investigated using a combination of magnetotransport and scanning tunneling spectroscopy measurements. Our studies reveal three disorder regimes. At low disorder ($k_{F}l$ ∼ 10–4), the system follows the mean field Bardeen-Cooper-Schrieffer behavior, where the superconducting energy gap vanishes at the temperature at which electrical resistance appears. For stronger disorder ($1<k_{F}l$ < 4), a “pseudogap” state emerges, where a gap in the electronic spectrum persists up to temperatures much higher than $T_c$, suggesting that Cooper pairs continue to exist in the system even after the zero resistance state is destroyed. Finally, at even stronger disorder ($k_{F}l$ < 1), the global superconducting ground state is destroyed even though superconducting correlations continue to survive, as evidenced from a pronounced magnetoresistance peak at low temperatures.

Figure 1

(a) ρ vs $T$ for NbN films with different $k_F$$l$; the inset shows the expanded view at low temperatures. (b) Variation of $T_c$ with $k_F$$l$. (c) Variation of $n$ (purple stars), ρ(285 K) (blue triangles) and ${\rho }_m$ (black squares) with $k_F$$l$. (d) Conductivity σ vs $T$ at low temperature for the samples with $k_F$$l$ ∼ 0.82, 0.49, and 0.42. The dotted lines are extrapolations to σ($T$ → 0).

Figure 2

(a)–(f) Normalized tunneling spectra at different temperatures for NbN films with different levels of disorder. The spectrum shown in the thick line corresponds to the temperature at which the low-bias feature in the tunneling conductance disappears. Each tunneling spectrum is averaged over 32 equally spaced points along a 150 nm line on the sample surface. (g)–(l) The spectra corresponding to (a)–(f) after subtracting the V-shaped background.

Figure 3

Intensity plots of $G_{\mathrm{sub}}$($V$) as a function of temperature and applied bias for six samples (upper part of each panel), along with resistance vs temperature in the same temperature range (lower part of each panel). The vertical dashed lines in the upper panels correspond to $T_c$.

Figure 4

$G_{\mathrm{sub}}$($V$) vs $V$ spectra along a 150 nm line (measured at 2.6 K) for six NbN thin films with different levels of disorder.

Figure 5

Spatial variation of $G_{\mathrm{sub}}$($V$) vs $V$ spectra recorded along a 190 nm line at different temperatures for an NbN thin film with $T_c$ ∼ 2.7 K. Large inhomogeneity in the tunneling DOS is observed as we enter the pseudogap state.

Figure 6

Resistivity as a function of magnetic field at different temperatures for four strongly disordered NbN thin films with (a) $k_F$$l$ ∼ 0.42, (b) $k_F$$l$ ∼ 0.49, (c) $k_F$$l$ ∼ 0.82, and (d) $k_F$$l$ ∼ 1.23. The samples with $k_F$$l$ < 1 show a pronounced peak in ρ–$H$. (e) and (f) Expanded view of ρ($H$) − ρ($H$ $=$ 0) vs magnetic field close to temperatures at which the peak in the MR vanishes.

Figure 7

Phase diagram of strongly disordered NbN, showing $T_c$ (green squares) and $T^{*}$ (blue circles) as functions of $k_F$$l$. $T_c$ is obtained from transport measurements, while $T^{*}$ is the crossover temperature at which the low-bias feature disappears from the observed tunneling conductance. The samples with $k_F$$l$ < 1 remain nonsuperconducting down to 300 mK. The three regimes with increasing disorder are shown as $\mathbf{I}$, $\mathbf{II}$, and $\mathbf{III}$. A pseudogap (PG) state emerges between $T_c$ and $T^{*}$ for samples with $T_c\lesssim 6$ K (regime $\mathbf{II}$). The temperature at which the peak in the MR vanishes for the strongly disordered samples (regime $\mathbf{III}$) is also shown (purple stars).

Figure 8

Superfluid stiffness ($J$/$k_B$) and penetration depth λ($T$ → 0) for NbN films with different $T_c$. The solid line corresponds to $J$/$k_B$ $=$ $T_c$. Regimes $\mathbf{I}$ and $\mathbf{II}$ corresponding to the phase diagram are delineated by the dashed vertical line.

Figure 9

Variation of $H_c$${}_2$(0) (for $k_F$$l$ > 1) and $H_p$ (for $k_F$$l$ < 1) as a function $k_F$$l$.