Evidence for a Finite Temperature Insulator

In superconductors the zero-resistance current-flow is protected from dissipation at finite temperatures (T) by virtue of the short-circuit condition maintained by the electrons that remain in the condensed state. The recently suggested finite-T insulator and the"superinsulating"phase are different because any residual mechanism of conduction will eventually become dominant as the finite-T insulator sets-in. If the residual conduction is small it may be possible to observe the transition to these intriguing states. We show that the conductivity of the high magnetic-field insulator terminating superconductivity in amorphous indium-oxide exhibits an abrupt drop, and seem to approach a zero conductance at T<0.04 K. We discuss our results in the light of theories that lead to a finite-T insulator.

were also able to estimate the T dependence of the e-ph scattering rate, τ e−ph , on the high B side of the insulating peak and found a rather strong dependence of τ e−ph ∼ T −4 , which is in agreement with the modified dirty metal model 29,31 . The success of this theoretical description provides an essential indication that, in our regime of measurements, the electrons are decoupled from the phonons.
The realization that our samples exhibit a strongly T-dependent insulating behavior with diminishing e-ph coupling motivated us to conduct a systematic study of their Ohmic transport at very low T (T < 0.3 K). In order to achieve that, we had to greatly improve our ability to measure very high sheet resistance (R). While our earlier studies 21 were limited to R up to 10 9 Ω , several improvements (described in the supplementary materials) extended the range of our measurements to 10 12 Ω . These improvements enabled the results that follow.
The data presented here are obtained from the sample S1aHiR, a thin film of a:InO, patterned in Hall bar geometry, 0.5 × 0.25 mm 2 in size. The sample is superconducting at B = 0 with a T c ≈ 1.1 K (see left inset of Fig. 1) and undergoes a B-driven SIT. In Fig. 1 we show two isotherms of R in the insulating region, as a function of B from 0.5 to 12 T, at T = 0.08 and 0.1 K. Both show the insulating peak at 5 T. Due to technical reasons we were unable to pinpoint the B c of our sample but located it to be between 0.16 and 0.4 T. The sample exhibited the thermal bi-stability in the insulating phase as evident by a typical I-V characteristic 30 , at B = 0.55 T and T = 13 mK, shown in top right inset of Fig. 1.
Our main results are presented in Fig. 2 where we plot the T-dependence of R at various B's, from 0.5-12 T, spanning the insulating peak. Depending on the R-range, measurements were done using two different techniques. For the moderate-R range (R < 10 8 Ω ) data were obtained by continuous two-terminal measurements (solid lines), whereas for R > 10 8 Ω each datum (marker) was obtained from a full I-V scan (see methods). The dashed lines joining the markers are guides to the eye.
Based on earlier studies which were limited to a much lower R-range, we were anticipating activated behavior 21,26 and adopted an Arrhenius form to present our data. However, the broad range of R in this study brings about the observation of clear deviations from activated transport. While the low R (R < 10 6 Ω ) data are still consistent with activated behavior (for reference we added a dashed black straight line, indicating activated behavior in Fig. 2) the high R data, offering several orders of magnitude broader range, clearly are not.
The deviations, seen in all B values of Fig. 2, crucially differ depending on the value of B. At the high B's, the convex shape of the curve indicates sub-activation behavior. This behavior is illustrated in Fig. 3(a) where R(B = 12 T) is plotted (in red), using a logarithmic scale, vs. T −1/2 . The data convincingly follow a straight line over our full T-range indicating, This is consistent with the Efros-Shklovskii (ES) variable range hopping (VRH) mechanism of transport 32 . T ES and R ES are the ES temperature (T ES = 14.8 K) and pre-factor respectively. This dependence holds, with increasing T ES , for B's down to the peak position (at B = 5 T, T ES = 23.6 K).   The picture changes dramatically at lower B's, approaching the SIT (0.5 < B < 2 T). An attempt, shown in blue in Fig. 3(a), to plot data taken in this B range using the ES form clearly fails. A simple activated form is also inadequate as the data clearly appear concave (see Fig. 2).
The concave curvature evident in the B < 2 T data of Fig. 2 signals an unusual, faster than exponential 33 , R(T) dependence. The anomaly is clearly seen when we plot, in Fig. 3 as a function of T at B = 0.75 T. Focusing on the T < 0.3 K range we see that σ decreases moderately upon cooling until T = 0.1 K and then undergoes a precipitous drop of 6 orders of magnitude to the noise level in our measurement (σ = 10 −12 Ω −1 ). As we stated earlier, our attempts, indicated by the black curve in Fig. 3(b), to fit these data with an Arrhenius form, failed. For reference we add σ(T) taken at B = 12 T where ES dependence holds (shown in red in that figure).
Our inability to fit the data using an exponential or stretched exponential dependence along with the e-ph decoupling we observe in our samples point in the direction of a finite-T insulator 5 . To test this possibility we fit our data with the following phenomenological form: which describes the vanishing of the conductivity at finite T = T * . The result of our fit is plotted using the black dashed line in Fig. 3(b), from which we obtain T 0 = 0.138 K and T * = 0.031 K. The data follow this functional form down to T = 0.042 K and σ = 1.3 × 10 −10 Ω −1 , where deviation larger than our measurement accuracy develop. In any real system σ = 0 is not a realistic expectation. This is because when σ becomes very small other, parallel, channels will carry the electronic current and contribute to σ. Each such channel will lead to the measured σ being higher, and can account for the deviations we observe at σ < 1.3 × 10 −10 Ω −1 . These can be due to physical processes within the sample or, possibly, due to leakage currents elsewhere in the measurement circuit. More recently, a theoretical paper utilizing a mean field description to a system near the MBL transition 34 suggested such deviations should be expected. By using Eq. (2) we do not intend to adhere to a specific theoretical model 2 . It is merely a phenomenological description intended to highlight the unusual aspect of our data: σ(T) exhibits a dramatic drop at T < 0.1 K and appear to approach σ = 0 at a finite T = T * . The B-dependence of T * and T 0 obtained by fitting our data using Eq. (2) are plotted as the inset in Fig. 3(b). The shaded region indicates the approximate location of the SIT in this sample. It is worth noting that both T * and T 0 seem to approach zero in this region.
Another way to illustrate the abrupt nature of the conductivity drop near T * is to compare it to the superconductivity transition in one of our disordered a:InO films. In Fig. 4 we plot σ vs. T at B = 0.75 T for this sample, whereas in the inset we plot R vs. T for sample MInOLa4 at B = 0 T. Despite the different T-range their appearance is remarkably similar: both quantities exhibit a sharp drop over a rather narrow T-range.
It is important to discuss one alternative to Eq. (2) that, on first sight, appears to agree with our results. At least some of the lower B data of Fig. 2 can be described, at T < 0.05 K, by an Arrhenius form indicating activated transport, which results from a mobility gap in the spectrum. A quantitative analysis clearly renders this view inadequate for the following reason. Fitting the B = 0.75 T data using an Arrhenius form leads to an activation T of 0.91 K. If a mobility gap of such magnitude existed in our system we would expect a much sharper increase in R at 0.91 > T > 0.05 K, as seen in the fit presented in the supplementary material. This drop is clearly missing in our data rendering an activated interpretation highly unlikely unless the 0.91 K gap only opens at T < 0.1 K. We are not aware of a theoretical work predicting such a possibility.
While the new results presented here appear to be in contradiction with earlier findings 21,26 of activated transport in the peak region, this is not the case: the activation behavior is seen at T's higher than 0.2 K, below which deviations from activation are seen (see Fig. 2). For these higher T's, where activation is seen, the maximum value of the activation energy is close to T C (B = 0), confirming earlier observations.
The data we are showing here is consistent with transition into a finite-T insulating state. It is tempting to associate this state with the MBL state suggested theoretically [1][2][3][4] . Some of the ingredients are certainly present: our system is highly disordered, strongly interacting and, at the relevant T, the electrons decouple from the phonons.
There are other tests that are needed to fully establish the link between our observations and the MBL state chief among which is showing that our electrons are ineffective in reaching equilibrium 1,2 . This is usually indicated by the presence of long relaxation times in transport. So far, in our experiments, we have not seen such effects but Ovadyahu's group, who study similar materials in a different regime, reported such slow relaxation phenomena 35,36 .
On the other hand, we recall that the systems in which we observe the transition to the finite-T insulating state are superconductors at low B and only becomes insulating as B is increased beyond the SIT. Furthermore Cooper-pairing is still dominant in transport even within the insulating regime. While the possible role of Cooper-pairs in forming the finite-T insulator was not considered within the framework of the MBL theories, it was explicitly considered by Vinokur 28 et al., in accordance with the suggested duality 37 nature of the 'superinsulating' state and, more recently, by Feigel'man et al. 38 who considered the fractal nature of the electronic wave function near a mobility edge and suggested that, if an attractive interaction near the SIT is considered, a finite-T insulator become feasible. More detailed experiments are needed to test the relevance of these theories.
In summary, we have been able to observe an abrupt drop in σ by several orders of magnitude occurring at T < 0.1 K in a:InO thin film near B induced SIT. This has been found to occur at T and B where the electrons decouple from the host lattice phonons. The measured data cannot be explained using ES model but fit well with the finite-T electron localization down to a certain conductivity.