Universal scaling of the critical temperature for thin films near the superconducting-to-insulating transition

Thin superconducting films form a unique platform for geometrically confined, strongly interacting electrons. They allow an inherent competition between disorder and superconductivity, which in turn enables the intriguing superconducting-to-insulating transition and is believed to facilitate the comprehension of high-$T_c$ superconductivity. Furthermore, understanding thin film superconductivity is technologically essential, e.g., for photodetectors and quantum computers. Consequently, the absence of established universal relationships between critical temperature $\left(T_c\right)$, film thickness $\left(d\right)$, and sheet resistance $\left(R_s\right)$ hinders both our understanding of the onset of the superconductivity and the development of miniaturized superconducting devices. We report that in thin films, superconductivity scales as $d{T}_c\left(R_s\right)$. We demonstrated this scaling by analyzing the data published over the past 46 years for different materials (and facilitated this database for further analysis). Moreover, we experimentally confirmed the discovered scaling for NbN films, quantified it with a power law, explored its possible origin, and demonstrated its usefulness for nanometer-length-scale superconducting film-based devices.

Figure 1

Metallic and superconducting behavior of thin NbN films. (a) Critical temperature of NbN films as a function of the film thickness $\left(d\right)$ and (b) sheet resistance $\left(R_s\right)$ indicate no clear correlation with general functions of the form $T_c\left(R_s\right)$ or $T_c\left(d\right)$. (c) Resistivity (ρ ≡ $R_{s}d)$ of the NbN films vs thickness reveals a scattered data set. Chemical treatment of the substrate prior to deposition is suspected of influencing the properties of the red solid points here and in Fig. 3, while their high resistivity is outside the range presented in (c) but is discussed in the Supplemental Material [38].

Figure 2

TEM micrographs of epitaxial NbN film on an MgO substrate. (a) Atomic structure of an NbN film on an MgO substrate, demonstrating epitaxial growth of long-range cubic structure and good lattice matching with the substrate. The lattice constants of both MgO and NbN are 4.35 ± 0.1 Å. (b) Selective area electron diffraction from an area within the MgO substrate only and (c) from an area that spans the MgO substrate, NbN film, and the glue layer (that is used to protect the film from the top) demonstrates high crystallinity of both the MgO and NbN and a good lattice matching between these substances. The bright spot in the center of (c) is due to the amorphous glue.

Figure 3

Fitting our NbN on MgO data to the power law $d{T}_c\left(R_s\right)$. Plotting $d{T}_c$ vs $R_s$ for the NbN films reduces the scattering significantly with respect to the curves in Fig. 1. The blue line is the best fit to Eq. (1) $(A$ = 9448.1 and $B$ = 0.903). Red solid points are discussed in the Supplemental Material [38].

Figure 4

Scaling of superconducting and metallic properties in thin films. (a) and (b) are the dependences of $d{T}_c$ on $R_s$ for various superconductors (data sets are arbitrarily split into two panels to prevent indistinguishability). The linearity of the data on a log-log scale is in good agreement with Eq. (1). (c) Resistivity dependence on thickness for representative materials (log-log scale). (d) $T_c$ as a function of thickness and (e) of sheet resistance of these materials on log-log scales. The following symbols were used in (a–e): -NbN from Wang et al. [15, 16, 17]; -NbN by Semenov et al. [19]; -NbN by Kang et al. [18]; -our NbN films (from Fig. 1); -Mo [50]; -Bi and -Pb by Haviland, Liu, and Goldman [20]; -Al from Cohen and Abeles [11]; -Nb [51]; -disordered TiN by Klapwijk et al. [21, 22]; -disordered TiN by Baturina et al. [52]; -$\alpha {\mathrm{Nb}}_3\mathrm{Ge}$ [23]; -αMoGe from Graybeal and Beasley [53]; -αMoGe from Graybeal et al. [24, 25, 26]; -αMoGe by Yazdani and Kapitulnik [27]; -αReW by Raffy et al. [28]; while is Al, and , , , , , and are Pb films, corresponding to the same symbols used by Strongin et al. [12]. A complete list of the data is given in the Supplemental Material [38].

Figure 5

Universality of the scaling of $d{T}_c\left(R_s\right)$. (a) Plot of intercept vs slope $[A$ vs $B$ with respect to Eq. (1)] of the best linear fits for the graphs in Figs. 4 and 4 suggests that the parameters $A$ and $B$ are correlated (for details, see S7 and S17 in the Supplemental Material [38]). Legend corresponds to Fig. 4, while the only material outside the trend is molybdenum. (b) Histogram of the exponent $B$ of the different materials with a mean value at $B\approx 0.95$.