Robust anomalous metallic states and vestiges of self duality in two-dimensional granular In-InOx composites

With the application of perpendicular magnetic field, the zero-resistance superconducting state in the In/InOx system is rapidly terminated at a magnetic-field-tuned QSMT, and an anomalous metallic phase is identified for $0\lesssim\mu_{0}H<0.04$ T. Fig. 2(a) shows resistivity that saturates as $T\rightarrow 0$ at a level orders of magnitude lower than the Drude value. As magnetic field increases, resistivity initially rises as a power-law that is essentially temperature-independent below 300 mK for both S2 and S2G (see Fig. 2(b)). Then, at $H\approx 0.04$ T, resistivity quickly picks up, leading to a giant magneto-resistance (MR) peak. The transition between saturating and diverging temperature dependence of resistivity is marked in the inset of Fig. 2(b) at 0.052 T and 0.043 T, for S2 and S2G respectively. This is reminiscent of a true H-SIT, with a transition to boson-dominated insulating ground state [27, 28]. At $H\approx 0.08$ T, resistivity peaks at $40\ k\Omega/\square$ at the highest. Beyond the peak, resistivity slowly decreases up to 8 T, but remains orders of magnitude larger than the normal state value, indicating the persistence of superconducting correlation well into the insulating state [28, 31]. Qualitatively, S0 and S1 behave similarly as shown in Supplementary Figure 2, but have much higher resistivity up to $0.5M\Omega/\square$ at the peak.

Starting with the low-field regime, where a failed superconducting phase yields a saturated resistance, the temperature-independent power-law MR before the rapid upturn, $\rho(H)\propto H^{0.66}$, fully overlaps with the anomalous metallic regime. Exponentiating this behavior, this is equivalent to a quantum-tunneling expression with $\rho(H)\propto\exp(-U(H)/k_{B}T_{\text{eff}})$, previously observed in amorphous MoGe films [5], where the logarithmic field dependence of $U(H)$ was attributed to activation of free dislocations in a vortex lattice with short spatial correlations [32]. Since this mechanism should strongly depend on the morphology of the film, it is no surprise that the magneto-resistance curves in the anomalous metallic phase for both S2 and S2G films collapse on the same curve in Fig. 2(b). Slight differences between the two samples are observed as the resistance increases above saturation towards a putative crossing point, which we will dub as “avoided true H-SIT.” We will come back to analyze the form of the resistance increase.

Beyond the resistivity upturn at $H\approx 0.04\ T$, isotherm crossing points that separate positive MR on the low-field regime from positive MR on the high-field regime are observed (marked by full circles) in the inset of Fig. 2(b). This type of behavior would describe a true H-SIT if indeed a true superconducting phase would be attained, accompanied by divergence of the resistance on the insulating side. Since resistance saturation is observed in the low temperature limit of both regimes, we may consider this avoided criticality as a crossover and attempt scaling at higher temperatures (here $T>100$ mK), before the system is dominated by the anomalous metallic phase. Similar approach was demonstrated previously [6, 33] to lead to scaling exponents of order $z\nu\approx 1.5$ when we scale the data according to $\rho(T,H)=\rho_{c}\mathcal{F}(|H-H_{c}|T^{-1/z\nu})$ in Fig. 2(c). It is also interesting to note that such a behavior with a similar critical exponent was previously observed in a 2D system of aluminum islands coupled by 2DEG [11], thus can be added to the list of common features between systems of induced superconducting granularity in otherwise presumed homogeneous materials such as amorphous-MoGe [5, 6] and systems with extreme granularity in ordered [11] and disordered (this manuscript) films.

Examination of the resistance trends on both sides of the crossing point reveals an intriguing behavior which may indicate the existence of vestiges of duality despite it being an “avoided transition.”

Starting with the “insulating regime,” again S2 and S2G show similar behavior, exhibiting anomalous logarithmic divergence of resistivity that is much weaker than the commonly observed activation or variable-range hopping trends. As shown in Figs. 3(a) and 3(b), where resistivity is plotted versus logarithm of temperature, within a broad range of temperature, the data can be fit by a straight line in the form $\Delta R/R(1\text{K})=A\cdot R(1\text{K})\ln(1/T)$, where the slope $A=(8.44\pm 0.12)\times 10^{-5}\ \Omega^{-1}$ for S2 at 4 T. Similar magnitude slopes are found for other magnetic fields, and for the different anneal stages. The data and particularly the slopes of the log-$T$ behavior are neither compatible with Kondo-effect behavior, nor are they compatible with weak localization corrections. Similar logarithmic divergence has been observed in the high-field insulating state of underdoped La_2-xSr_xCuO₄ [34], and in amorphous InOx [24]. Invoking inhomogeneous microstructure for these otherwise considered homogeneous materials, a possible connection between granularity and logarithmic divergence of the temperature-dependent resistivity has been previously discussed (see e.g. Ref. [25]). We further note that for magnetic fields close to the crossing point, the logarithmic divergence tends to saturation, while increasing the magnetic field beyond the magnetoresistance peak seems to recover a “normal” insulating behavior commensurate with the initial 2D resistivity of the film before superconductivity sets in. Since the high field resistivity is much smaller than the peak in magnetoresistance, we interpret the saturation on the insulating side as an increase in local phase coherence due to remnant superconductivity.

Turning to the transition to the “anomalous metal regime” below the putative crossing point, we find an anomalous logarithmic divergence of the conductivity (calculated as the inverse resistivity since the Hall contribution is negligible), that is much weaker than the commonly observed activation or variable-range hopping commonly found where a true H-SIT is observed [2, 26]. As shown in Figs. 3(c) and 3(d), where conductivity is plotted versus logarithm of temperature, within a broad range of temperature, the data can be fit by a straight line. Similar to the insulating regime, this logarithmic divergence exhibits large prefactor, which again deems it as a crossover to the exponential divergence rather than a weak correction to a normal state conductivity.

Fig. 3 is a striking demonstration of vestiges of the H-SIT quantum critical point in this “avoided transition.” It further makes the connection between the resistance saturation on the superconducting side and that on the insulating side, where in our highly granular films it can be attributed to vestiges of local phase coherence in strongly fluctuating superconducting grains.

Finally, we turn to making a more microscopic connection between the logarithmic divergence and the morphology of our films. We will then argue that this connection can be generalized to samples where strong inhomogeneity emerges in the superconducting state following amplification of local disorder [15, 16, 17]. Focusing on the insulating side, here, with a carefully analyzed microstructure, we are able to show that the observed logarithmic divergence is indeed consistent with originating from the extreme granular inhomogeneity, leading to weak resistivity divergence over an intermediate range of temperature. As shown in Fig. 1(b), the granular structure consists of a hierarchy of grain sizes, spanning over two orders of magnitude, from microscopic interstitial ones to “large grains” — a fraction of a $\mu$m. Hence, in the insulating state, the system consists of a wide range of tunneling barriers between the metallic (or superconducting) islands. These barriers are determined by a competition between normal or Josephson tunneling and charging energy mediated by the dielectric background environment. Assuming a log-uniform distribution of barrier’s energy $\Delta$: $p(\Delta)=1/\Delta$ and summing parallel conductances, we show that the resultant resistance can exhibit weak power law or logarithmic divergence. At much lower temperature, however, the resistivity will diverge much faster in an activated fashion. A more detailed account of our analysis is given in the supplementary. This emerging behavior further supports our intuition that transport is determined by a collective effect of the In grains coupled within the InOx matrix.

It was recently demonstrated that where true H-SIT is observed, the quantum phase transition exhibits the emergence of self duality with an exponential diverging variable range hoping behavior of the resistivity above, and the conductivity below, the transition [29, 26]. The insulating phase was then identified as a Hall insulator, while the superconducting phase would be an equivalent phase for vortices. Moreover, the assembly of the critical behavior effects were found to be similar to that of quantum Hall to insulator transition [35], particularly the critical exponents and the emergence of self duality [29, 26]. In fact, in attempting to summarize the above results into a phase diagram we observe further similarity between the two phenomena.

Fig. 4 is a phase diagram for the In/InOx system based on the experiments of Hen et al. [26] observing a true H-SIT and the present data showing an intervening metallic phase. The vertical axis represents an external parameter that controls the nature of the critical behavior. While in uniform films it could be the disorder, in the presence of strong granularity it may represent the distribution of the intergrain couplings. Where material parameters are tuned to observe an anomalous metal (e.g. weak intergrain couplings), a putative crossing point is observed as well with a non-universal critical resistance and different critical exponents, typically $z\nu\approx 1.5$. The fact that the resistance above the crossing resembles the conductance below the crossing, where both lead to a regime of saturated resistance suggests that vestiges of the self-duality line that emerges at the true H-SIT remain also in the regime where a metallic phase emerges, hence marks two dual regimes of anomalous metallicity. This observation is marked with the dash-dotted line in the phase diagram. Thus, while for true H-SIT a $\sigma_{xx}\sim e^{(\Delta_{s}/T)^{\delta}}$ on the superconducting side transforms to $\rho_{xx}\sim e^{(\Delta_{I}/T)^{\delta}}$ in the insulating side [29] (here $\Delta_{s}$ and $\Delta_{I}$ are activation gap scales for the superconducting and insulating regimes respectively). At weaker control parameter a $\sigma_{xx}\sim{\rm ln}(\Delta_{s}/T)$ that appears following the resistance saturation crosses over to $\rho_{xx}\sim{\rm ln}(\Delta_{I}/T)$ with increasing magnetic field. The insulating side then exhibits further increase of the resistance before pairing is quenched and Fermi-dominated insulating behavior is recovered.

Despite extensive filtering at different stages of our dilution refrigerator, the anomalous metallic phase appears robust. Nonetheless, fragility of the underlying superconducting state can cause this inhomogeneous system to be extremely sensitive to environmental perturbation. In a given magnetic field, barriers associated with collective vortex effects are established throughout the sample, leading to a saturated resistance commensurate with that field. However, it is important to explore whether the observed resistance is due to non-equilibrium effects where, for example, electrons fail to thermalize with the lattice due to poor electron-phonon coupling, or external noise heats the electrons, thus obstructing coherence in weaker Josephson couplings and disrupting global superconductivity. We already discussed the unique microstructure of our samples, where the large In grains ensure good thermalization of the local pair amplitudes. Indeed, Fig. 5(a) shows linear response of the resistance measurements in the anomalous metallic regime over a wide range of current bias. Non-linearities are only observed at higher currents, and are presumably associated with junctions locally exceeding their critical current.

Having established linear response, we further need to explore the possible influence of external radiation on the occurrence of anomalous metallicity. To test for this effect, we may introduce external noise by removing successive layers of filtering in a controlled fashion, or by injecting radio-frequency (RF) signal into the sample and measure its response in resistivity.

We emphasize that our objective is not to study straight forward effects of well coupled radiation to the sample (see discussion below), but rather to inject radiation into various ports that lead to the sample or sample space in order to mimic poor decoupling of the low temperature measurement from the external world. This is important in view of recent studies that suggest that some observations of anomalous metallic state are a consequence of poor filtering and enhanced coupling to the external world.

In Fig. 6, we show Arrhenius plot of resistivity under different experimental conditions. By either introducing 2 GHz signal or bypassing the room temperature low-pass filter (see Fig. 1(d)), resistivity is lowered by as much as $\sim$ 25% at 0.03 T. This is contrary to the electron heating scenario, where an elevated electron temperature would raise saturation resistance. In an extreme case, at a field strength of 0.0002 T or 2 Gauss, the anomalous metallic behavior is suppressed completely and true superconductivity is recovered, where resistivity drops below our measurement sensitivity (See the lower panel in Fig. 6). It is interesting to note that in a completely different system of nanopore-modulated YBCO thin films, where anomalous metallic state has been observed, the removal of filtering resulted in a similar reduction of the saturated resistance at low temperatures [36]. This result was taken as a proof of true cooling of electrons in that highly inhomogeneous system. However, the introduction of deliberate microwave radiation in addition to removal of parts of the filtering system can further illuminate the nature of the robustness of the metallic phase and in particular the effect of re-entrant superconductivity.

Discovered over fifty years ago, microwave-enhanced superconductivity has been studied experimentally in constraint-type [37, 38] and superconductor-normal-superconductor (SNS) Josephson junctions [39, 40]. Dubbed as “microwave-stimulated superconductivity,” [41] this phenomenon was found to be ubiquitous in studies of SNS weak links. Initial theoretical understanding of this phenomenon invoked a non-equilibrium gap-enhancement proposed by Eliashberg [42]. However, more recent analyses, especially taking into account proximity-effect under external radiation, concluded that much of the enhancement of critical current in such SNS junctions arises from enhanced phase coherence (for a recent review see Klapwijk and de Visser [41].)

As demonstrated above, global superconductivity in the the In/InOx system is established via phase coherence among the In grains, coupled through the underlying InOx layer. In the presence of magnetic field, pinned vs. mobile collective vortex effects reflect the restoration or loss of local phase coherence respectively. Thus, a metallic state appears over a zero-resistance superconducting state when a vortex-induced global phase slip appears — marking the loss of global phase coherence. However, the emerging metallic phase retains much of the character of the superconducting phase, where significant superconducting correlations are present, thus establishing a resistance much lower than the respective “Drude resistance.” The microwave enhanced conductivity of this metallic state, and the restoration of the superconducting state at very low fields are a manifestation of the robustness of this failed superconducting state.

We return to Fig. 5, which shows non-linear differential resistance $dV/dI$ in presence of a direct current bias. In zero magnetic field, superconductivity is quenched by a critical current of 1 $\mu$A, which is slightly enhanced by application of RF signal. In an external magnetic field, $dV/dI$ yields a zero-bias minimum value in low field, and a zero-bias peak in relatively large field, more clearly shown in full linear scale in Fig. 5(b).

The evolution of the differential resistance behavior displayed in Fig. 5 has been observed previously in a 2D system of aluminum islands, coupled with a gated 2D electron gas [11]. Such a system may be considered an ordered array of islands, as shown by the oscillations in the array’s resistance at integer and certain fraction of flux quanta entering their periodic sample. While the overall behavior of the two systems is the same, the random distribution of the In islands in our system blurs any possible ordered oscillation in the magneto-resistance, although it may explain the non-smooth MR data to be discussed in the next section.

Fig. 7 shows the effect of RF radiation on the magneto-resistance. When no RF signal is present, as a result of the sample’s extreme sensitivity to external magnetic field, we cannot clearly identify a lower critical field of a QSMT. Beyond the limitations in measurement sensitivity and magnet field resolution, the main observation is the large fluctuations in MR, up to 30% of the averaged behavior. Such repeatable fluctuations are likely a consequence of flux commensuration effect allowing flux quanta into the granular structure. While such an effect shows precise periodicity in ordered array of coupled grains, such as in the coupled array of Al islands [11], or in studies of square aluminum wire networks in a magnetic field [43], here the area enclosed by loops connecting indium islands through intergrain coupling is random, and the smeared oscillations we observe are a consequence of the mesoscopic nature of the effect, which in turn is also responsible for the observed low critical field. For example, the deep minimum at $\sim$15 G corresponds to a flux quantum fitting in a loop of average diameter $\sim 1.3\mu$m, which in inspecting Figs. 1(a) and 1(b), would correspond to a fundamental loop that includes the larger grains. Indeed, similar effects have been observed in granular cuprates [44], inhomogeneous nanowires [45] and granular lead-film bridges [46].

The fact that the MR fluctuations are related to flux penetration through loops of grains connected by junctions is further evidenced from the effect of radiation. As we demonstrated above, the intergrain coupling in the In/InOx/In junctions in our films is enhanced in the presence of microwave radiation, which in turn should make the films effectively more homogeneous and thus reduce the MR fluctuations. Indeed, Fig. 7 shows the suppression of the MR fluctuations, accompanied by enhancement in critical field in the presence of 0.2 GHz or 2 GHz RF radiation.

Extreme granular inhomogeneity that features a broad distribution of grain size is achieved by depositing poorly wetting In metal onto uniform amorphous InOx thin film. The underlying layer of 300-Å InOx was electron-beam evaporated onto a commercial lithium-ion conductive glass ceramics (Li⁺-ICGC) substrate (MTI corporation) that enables depletion of electrons up to a 2D carrier density of $\sim 10^{14}$ cm^-2 in back-gating configuration, as was also found in a recent study [47]. A deposition rate of 0.32 Å/s in an oxygen partial pressure of $9.5\times 10^{-6}$ Torr yielded a weakly insulating film. Without interrupting the vacuum, at a base pressure of $1\times 10^{-7}$ Torr, indium was evaporated in situ at a rate of 5 Å/s for 100 seconds, yielding a nominal 500-Å layer of granular indium. The granular structure was confirmed by scanning electron microscopy as shown in Fig. 1(a) along with a histogram showing grain size distribution in Fig. 1(b). The composite film was then patterned in Hall bar geometry (200 $\mu$m$\times$100 $\mu$m) using photolithography, before titanium/gold contacts were subsequently patterned onto the sample. Great care was taken to keep the sample below 50 ^∘C at all times to preserve the amorphous nature of the underlying InOx. Effective tuning of disorder, manifested by slow decrease in resistivity, is achieved by annealing the sample in vacuum at room temperature.

Resistance was measured using standard four-point lock-in technique at 3–13 Hz using 0.1–1 nA excitation current. Linear response was verified at various temperatures and magnetic fields. Measurement and filtering schematics can be found in Fig. 1(d).

Extensive filtering minimizes electron heating caused by external radiation and improves electron thermalization. Twisted-pair signal lines are filtered at mixing chamber plate using a commercial QFilter RC/RF filter (QDevil ApS), offering over -8 dB attenuation above 100 kHz and -50 dB above 300 MHz. Additionally, all signal lines are filtered at room temperature by a commercial in-line $\pi$-filter (API technologies corporation), providing an extra -50 dB attenuation above 200 MHz. Gate voltage was applied through a room-temperature-filtered twisted-pair well thermally-anchored at mixing chamber plate. Sample phonon temperature is measured by a calibrated on-chip Ruthenium-Oxide (RuO₂) thermometer positioned close to the sample.

[To be updated by publisher]

The authors declare no competing financial or non-financial interests.