Quantum Metallicity on the High-Field Side of the Superconductor-Insulator Transition

We investigate ultrathin superconducting TiN films, which are very close to the localization threshold. Perpendicular magnetic field drives the films from the superconducting to an insulating state, with very high resistance. Further increase of the magnetic field leads to an exponential decay of the resistance towards a finite value. In the limit of low temperatures, the saturation value can be very accurately extrapolated to the universal quantum resistance $h/e^2$. Our analysis suggests that at high magnetic fields a new ground state, distinct from the normal metallic state occurring above the superconducting transition temperature, is formed. A comparison with other studies on different materials indicates that the quantum metallic phase following the magnetic-field-induced insulating phase is a generic property of systems close to the disorder-driven superconductor-insulator transition.

Figure 1

Temperature dependence of $R_{\square }$. The two TiN films differ slightly in their normal state resistance $R_{\square }(10\mathrm{K})=8.24\mathrm{k}\Omega$ (sample A) and $8.74\mathrm{k}\Omega$ (sample B), respectively. The resistance reaches a maximum at $R_{\square }(T_{\mathrm{m}}=0.48\mathrm{K})=29.4\mathrm{k}\Omega$ (sample A) and $R_{\square }(T_{\mathrm{m}}=0.72\mathrm{K})=18.6\mathrm{k}\Omega$ (sample B). All data in this work were taken at a measurement frequency of $0.4–2\mathrm{Hz}$ with an ac current $0.01–1\mathrm{nA}$.

Figure 2

Sheet resistance $R_{\square }$ in perpendicular magnetic field. Top: (sample A) $T=60$, 80, 95, 120, 140, 180, 300, 450, 650, 850, 990 mK. The inset shows a close-up view of the $R(B)$ curves measured at temperatures corresponding $dR/dT<0$ at zero magnetic field. Bottom: (sample B) $T=60$, 75, 90, 100, 130, 180, 220, 260, 300, 360, 480, 625 mK. $R_{\square }(T=60\mathrm{mK})$ reaches a maximum at $B_m\sim 1.2\mathrm{T}$ (sample A) and at $B_m\sim 1.6\mathrm{T}$ (sample B).

Figure 3

Scaling plot of the data in Fig. 2. For certain values of $R_{\mathrm{sat}}$, $\ln [1/R_{\mathrm{sat}}-G_{\square }(B)]$ varies linearly versus $B$, with a $T$-independent slope. The linear slope corresponds to a characteristic field $B^{*}\simeq 10.7\mathrm{T}$ (sample A) and $\simeq 6.8\mathrm{T}$ (sample B), respectively.

Figure 4

Temperature dependence of $R_{\mathrm{sat}}$ and $\beta$. (a) $R_{\mathrm{sat}}(T)$, (b) $R_{\mathrm{sat}}^{-1}(T^{1/3})$, and (c) $\beta (T)$ for samples A and B and the data from Fig. 1(a) of Ref. 12. For the latter, we obtained $B^{*}\simeq 4.5\mathrm{T}$ and $\beta \simeq 2e^2/h$, nearly independent of $T$.