Spectroscopic evidence for strong correlations between local superconducting gap and local Altshuler-Aronov density of states suppression in ultrathin NbN films

Disorder has different profound effects on superconducting thin films. For a large variety of materials, increasing disorder reduces electronic screening which enhances electron-electron repulsion. These fermionic effects lead to a mechanism described by Finkelstein: when disorder combined to electron-electron interactions increases, there is a global decrease of the superconducting energy gap $\mathrm{\Delta }$ and of the critical temperature $T_c$, the ratio $\mathrm{\Delta }$/$k_B{T}_c$ remaining roughly constant. In addition, in most films, an emergent granularity develops with increasing disorder and results in the formation of inhomogeneous superconducting puddles. These gap inhomogeneities are usually accompanied by the development of bosonic features: a pseudogap develops above the critical temperature $T_c$ and the energy gap $\mathrm{\Delta }$ starts decoupling from $T_c$. Thus the mechanism(s) driving the appearance of these gap inhomogeneities could result from a complicated interplay between fermionic and bosonic effects. By studying the local electronic properties of an NbN film with scanning tunneling spectroscopy (STS), we show that the inhomogeneous spatial distribution of $\mathrm{\Delta }$ is locally strongly correlated to a large depletion in the local density of states (LDOS) around the Fermi level, associated to the Altshuler-Aronov effect induced by strong electronic interactions. By modeling quantitatively the measured LDOS suppression, we show that the latter can be interpreted as local variations of the film resistivity. This local change in resistivity leads to a local variation of $\mathrm{\Delta }$ through a local Finkelstein mechanism. Our analysis furnishes a purely fermionic scenario explaining quantitatively the emergent superconducting inhomogeneities, while the precise origin of the latter remained unclear up to now.

Figure 1

(a) Square resistance of a 2.1 nm thick NbN film, measured in the same stage where STM is performed. $T_c$ is defined when $R_{\square }$ reaches zero, leading to $T_c=3.8$ K. (b) Typical local $dI/dV\left(V\right)$ spectrum measured at $T=300$ mK. Set point for spectroscopy $V=-10$ mV, $I=150$ pA. (c) Local $dI/dV\left(V\right)$ spectrum measured at the same location and temperature as in (b) but on a larger voltage scale. The large depletion of the LDOS seen around $E_F$ has a characteristic V-shape due to electron-electron interactions enhanced by disorder. The red (lighter) curve shows a power-law fit of this V-shape dependence, according to the relation $dI/dV\left(V\right)=b\times V^{\alpha }$, where $b$ is a constant.

Figure 2

Spatial cross-correlations between variations of the local superconducting gap $\mathrm{\Delta }\left(\mathbf{r}\right)$ and the Altshuler-Aronov exponent $\alpha \left(\mathbf{r}\right)$. Measurements done at 300 mK in zero magnetic field. (a) STM topography of a $300\times 250{\mathrm{nm}}^2$ area of the NbN sample (height variation $Z$ in nm). Scanning parameters: $V=-50$ mV and $I=50$ pA. (b) Color-coded map showing the local energy gap $\mathrm{\Delta }\left(\mathbf{r}\right)$ (in meV) measured in area (a) by STS. (c) Color-coded map showing the local exponent $\alpha \left(\mathbf{r}\right)$ in area (a). (d) Color-coded map showing the cross-correlation between the topography $Z(x,y)$ shown in (a) and the gap map $\mathrm{\Delta }(x,y)$ shown in (b). No cross-correlations are found. (e) Histogram of $\mathrm{\Delta }\left(\mathbf{r}\right)$ values occurring in (b). (f) Histogram of $\alpha \left(\mathbf{r}\right)$ values occurring in (c). (g) Color-coded map showing the cross-correlation between the gap map $\mathrm{\Delta }(x,y)$ shown in (b) and the exponent map $\alpha (x,y)$ shown in (c). A strong spatial anticorrelation is found: where local electron-electron interactions are stronger [larger $\alpha (x,y)]\mathrm{\Delta }(x,y)$ is smaller and vice versa. (h) Representative $dI/dV$ spectra of the regions seen in (b). Brighter (respectively darker) spectra are measured in brighter (darker) regions of (b). (i) Same spectra as in (h) but over an energy scale 20 to 30 times larger than $\mathrm{\Delta }$. Brighter (respectively darker) spectra are measured in brighter (darker) regions of (c).

Figure 3

Spatial decay of the correlation functions calculated from the radial averaging of the 2D autocorrelation or cross-correlation functions from the data shown in Fig. 2. Each profile starts at the center of each map (corresponding to the origin 0) and extends up to 100 nm. (a) Spatial dependence of ${\rho }_{\alpha }\left(r\right)$ defined as the correlation function of the Altshuler-Aronov exponent $\alpha$(r). (b) Spatial dependence of ${\rho }_{\mathrm{\Delta }}\left(r\right)$ defined as the correlation function of the superconducting energy gap $\mathrm{\Delta }$(r). (c) Spatial dependence of ${\rho }_{\text{cross}}\left(r\right)$ defined as the correlation function of the 2D cross-correlation between the local Altshuler-Aronov exponent $\alpha \left(\mathbf{r}\right)$ and superconducting energy gap $\mathrm{\Delta }\left(\mathbf{r}\right)$.

Figure 4

Linecut of twelve $dI/dV\left(\mathbf{r}\right)$ spectra (shown by black curves) measured along the line indicated by a white rectangle in Fig. 2. This linecut starts at a bright puddle with larger local gap and smaller exponent. 33 nm away it ends at a dark puddle with smaller gap and larger exponent. Each spectrum is spaced by 3 nm from the previous one. Vertically each spectrum is offset by one for clarity including the first spectrum which is the bottom one. Each $dI/dV\left(\mathbf{r}\right)$ curve is superposed with its fitted curve (shown by red lines) using a power law $b\times V^{\alpha \left(\mathbf{r}\right)}$. Set point for spectroscopy $V=-10$ mV and $I=150$ pA.