Fig. 1: Insulating NbTiN film’s sheet resistance evolution with bridge length.

a The Arrhenius plot of sheet resistance R_□ vs. reversed temperature 1/T for fridges of various length L. Dashed line shows \(R\propto \exp (1/T)\). Inset: Experimental setup. The Si substrate with AlN buffer layer is shown with light gray and Hall bridge of NbTiN is dark gray. The square gold contacts are given in yellow. All lateral sizes are given in millimeters. The bridges lengths (i.e., distances between measuring electrodes) shown in the legend in panel b. b Same data as in a but replotted as conductance G = 1/R_□ vs. T in log-line scale. A few curves are omitted to avoid overcrowding. Dotted lines are fits using a two dimensional Coulomb gas model²⁰ that generalizes the Berezinskii–Kosterlitz–Thouless (BKT) formula for conductance \(G\propto \exp [-{(T/{T}_{{\rm{dec}}}-1)}^{1/2}]\) by incorporating a self-consistent solution to the effects of electrostatic screening¹³, where the screening length λ_c and T_dec enter as fitting parameters. For all bridges we find T_dec ≈ 400 mK. c Same data as in b but for temperature renormalized as \({(T/{T}_{{\rm{dec}}}-1)}^{1/2}\). Solid line corresponds to the case of an infinite electrostatic screening length λ_c → ∞. Inset: Screening length as function of the bridge length L. Symbols correspond to BKT-fits of G(T) (dotted lines in b and c) and dashed line is the eye guide. The error bars are obtained from variations of parameters leaving self-consistent solutions of electrostatic equations of ref.  ¹³ unchanged and are much less than the size of symbols.