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

Abstract

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 (T c ), film thickness (d), and sheet resistance (R s ) 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 dT c (R s ). 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. Relationships between low-temperature and normal-state properties are crucial for understanding superconductivity. For instance, the Bardeen-Cooper-Schrieffer theory (BCS) successfully associates the normal-to-superconducting transition temperature, T c , with material parameters, such as the Debye temperature ( D ) and the density of states at the Fermi level [N (0)]. Hence, the BCS model allows us to infer superconducting characteristics (i.e., T c ) from properties measured at higher temperatures [1]. In the BCS framework, superconductivity occurs when attractive phonon-mediated electron-electron interactions overcome the Coulomb repulsion, giving rise to paired electrons (Cooper pairs) with a binding energy gap:. Moreover, within a superconductor, all Cooper pairs are coupled, giving rise to a collective electron interaction. Such a collective state is described by a complex global order parameter with real amplitude ( ) and phase (ϕ): = e iϕ .Because superconductivity relies on a collective electron behavior, the onset of superconductivity occurs when the number of participating electrons is just enough to be considered collective, i.e., at the nanoscale [2-5]. Thus, it is known that the superconductivity-disorder interplay varies in thin films and is effectively tuned with the film thickness (d) or with the disorder in the system, which is represented by sheet resistance of the film at the normal state (R s ) [6][7][8][9][10]. The mechanism of superconductivity in thin films has been investigated since the 1930s [6] increase in T c with decreasing thickness in aluminum films in a study that pioneered the currently ongoing research of thin superconducting films. This enhancement of T c , which is still not completely understood, was later confirmed by Strongin et al. [12], who also reported the more common be...

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“…derived the dependence of kinetic inductance on material conductivity and energy gap, or equivalently, on the normal resistance and critical temperature: where d is the wire thickness. Moreover, the relationship between the critical temperature, and the film thickness and sheet resistance has also been formulated empirically for a broad range of superconductors near the SIT limit: , where A and B are material‐dependent constants that help classify the homogeneity of the superconductor (typically, is the limit above which the materials are not crystalline, but amorphous) . This division is in agreement with accumulated data that suggest that the properties of amorphous and crystalline SNSPDs are different (with an emphasize on crystalline NbN and amorphous tungsten silicide, which are shown in Figure ).…”

“…The above distinction between amorphous and crystalline SNSPDs raises a motivation to optimize the amorphousness of a superconductor. Following a recent analysis of the effects of amorphousness on superconducting thin films, such optimization may open a new path for SNSPDs that benefit from the competitive detection properties of amorphous materials as well as from the superior timing performance of crystalline superconductors. It has been suggested recently that using bilayers of amorphous and crystalline materials gives rise to hybridization of the superconducting properties via the proximity effect ( Figure ).…”

Atomic layer deposition of titanium nitride for quantum circuits

“…We then record the critical temperature (T c ) and sheet inductance (L ) for a range of different film thicknesses ( Fig. 1a and Table I that links thickness, T c , and sheet resistance, which has been observed for other thin superconducting films [23]. Moreover, the exponent B = 0.67 ± 0.02 we measure is similar to TiN from other work [24].…”

Amorphous molybdenum silicon superconducting thin films

“…The behaviour can be partially modelled using proximity effect and quantum size effect theories 34,35 which argue that the transition temperature varies with thickness directly, however an alternative analysis shows that there is a correlation between transition temperature and resistivity 36 which has some theoretical explanation. 37 Resistivity is shown as a function of film thickness in figure 4.…”