Experimental Evidence for a Collective Insulating State in Two-Dimensional Superconductors

We present the results of an experimental study of the current-voltage characteristics in a strong magnetic field ($B$) of disordered, superconducting, thin films of amorphous indium oxide. As the $B$ strength is increased superconductivity degrades, until a critical field ($B_c$) where the system is forced into an insulating state. We show that the differential conductance measured in the insulating phase vanishes abruptly below a well-defined temperature, resulting in a clear threshold for conduction. Our results indicate that a new collective state emerges in two-dimensional superconductors at high $B$.

Figure 1

(a) Sheet resistance ($\rho$) as a function of $B$ field, of the sample ABa1, measured at four different $T$’s$=0.01$, 0.1, 0.2, and 0.4 K. In the low $B$ range we have plotted four-terminal $\rho$ data measured with ac excitaion currents of 1 nA. In the higher $B$ range (above 3 T) we have plotted two-terminal $\rho$ data measured with ac excitation voltages of $10\mu \mathrm{V}$. The vertical arrow marks $B_c$ ($=1.72\mathrm{T}$), the critical field of the $B$-driven SIT. (b) Evolution of the current-voltage characteristics across the superconductor-insulator transition. Four-terminal differential resistance, of the same film in (a), is plotted against the dc current for $B$ values of 0.5, 1.5, and 2.0 T. All traces are measured at $T=0.01\mathrm{K}$ with ac excitation currents of 10 nA.

Figure 2

Comparison of the current-voltage characteristics of the $B$-driven insulating phase at two $T$’s (0.15 and 0.01 K). The traces show the two-terminal differential conductance measured at $B=2\mathrm{T}$ as a function of dc voltage. The ac excitation voltage applied is $10\mu \mathrm{V}$. The sample used is Ja5 with $B_c=0.4\mathrm{T}$. $V_T$ marks the threshold voltage for conduction at $T=0.01\mathrm{K}$.

Figure 3

Temperature dependence of $dI/dV$ in the $B$-induced insulating phase. $dI/dV$ traces, measured at $B=2\mathrm{T}$, of the sample Ja7 with $B_c=0.3\mathrm{T}$. The ac excitation voltage used is $10\mu \mathrm{V}$. The threshold voltage ($V_T$) for this sample $\sim 1.02\mathrm{V}$. The $V_{\mathrm{d}\mathrm{c}}$ values used are (from bottom to top): 0, 1.02, 1.07, and 1.5 mV. $T^{*}$ is the temperature at which $dI/dV$ vanishes abruptly.

Figure 4

Two-dimensional map of the $dI/dV$ values in the $B-V_{\mathrm{d}\mathrm{c}}$ plane. For the sample of Fig. 2 (Ja5), we have measured $dI/dV$ traces as a function of $V_{\mathrm{d}\mathrm{c}}$ at $B$ intervals of 0.2 T and at $T=0.01\mathrm{K}$. The color scale legend on the right-hand side shows the various colors used to represent the values of $dI/dV$. The horizontal dashed line denotes $B_c$ ($=0.4\mathrm{T}$) of this sample.