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Sci Adv. 2017 September; 3(9): e1700612.

Published online 2017 September 15. doi: 10.1126/sciadv.1700612

PMCID: PMC5600526

Received 2017 February 28; Accepted 2017 August 16.

Copyright © 2017 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is **not** for commercial advantage and provided the original work is properly cited.

Electrons confined to two dimensions display an unexpected diversity of behaviors as they are cooled to absolute zero. Noninteracting electrons are predicted to eventually “localize” into an insulating ground state, and it has long been supposed that electron correlations stabilize only one other phase: superconductivity. However, many two-dimensional (2D) superconducting materials have shown surprising evidence for metallic behavior, where the electrical resistivity saturates in the zero-temperature limit; the nature of this unexpected metallic state remains under intense scrutiny. We report electrical transport properties for two disordered 2D superconductors, indium oxide and tantalum nitride, and observe a magnetic field–tuned transition from a true superconductor to a metallic phase with saturated resistivity. This metallic phase is characterized by a vanishing Hall resistivity, suggesting that it retains particle-hole symmetry from the disrupted superconducting state.

Conventionally, possible ground states of a disordered two-dimensional (2D) electron system at zero temperature include superconducting, quantum Hall liquid, or insulating phases. However, transport studies near the magnetic field–tuned superconductor-insulator transition in strongly disordered films suggested the emergence of anomalous metallic phases that persist in the zero-temperature limit, with resistances (ρ_{xx}) much lower than their respective nonsuperconducting state values (ρ_{N}) (extrapolated from above the superconducting transition temperature *T*_{c}). Initial studies of these phases on amorphous MoGe films (*1*–*3*) were followed by similar observations in amorphous indium oxide (*4*, *5*), tantalum (*6*), and indium-gold alloy (*7*) films, as well as in crystalline materials (*8*, *9*) and hybrid systems consisting of a superconducting metal in contact with a 2D electron gas, such as tin-graphene (*10*). Metallic phases were also observed in weakly disordered 2D superconductors at zero field when either disorder or carrier density is tuned (*10*–*13*).

Despite the ubiquitous appearance of this metallic phase, progress in understanding its origin has been slow. Early theoretical treatments explored quantum fluctuations in the presence of a dissipative bath, presumably due to residual fermionic excitations (*14*–*18*), a Bose metal phase (*19*, *20*), and an exotic non–Fermi liquid vortex metal phase (*21*, *22*). Finally, noting that the distribution of the superconducting order parameter is highly inhomogeneous in the presence of disorder, Spivak *et al*. (*23*) examined a metallic phase that is stabilized by quantum fluctuations while showing significant superconducting correlations. To date, there has been no conclusive evidence that distinguishes any one of these scenarios.

Here, we present evidence that the anomalous metallic phase can be described as a “failed superconductor,” where particle-hole symmetry, reminiscent of the superconducting state, plays a major role in determining its properties. This conclusion is a result of extensive Hall effect measurements on amorphous tantalum nitride (TaN_{x}) and indium oxide (InO_{x}) films that are weakly disordered (*24*). Specifically, we find that ρ_{xx} in both systems becomes finite at the transition from a “true superconductor” to an anomalous metal at a magnetic field *H*_{M1}. The Hall resistance ρ_{xy}, zero in the superconductor because of electron-hole symmetry, remains zero for a wide range of magnetic fields, before becoming finite at a field *H*_{M2} well below *H*_{c2}, the superconducting critical field. This apparent electron-hole symmetric behavior may herald the appearance of what has been termed the “elusive” Bose metal (*19*, *20*).

Figure 1 depicts a set of resistive transitions at increasing magnetic fields measured on amorphous TaN_{x} and InO_{x} films (pictured in Fig. 1A). Sample growth and characterization details are described in the Supplementary Materials and in previous studies (*25*, *26*). For TaN_{x}, the transition to saturated resistance evolves smoothly, such that for magnetic fields above ~1 T, saturation of the resistance is apparent; the lower field transitions seem to continue at lower temperatures in an activated fashion as observed in MoGe (*2*). However, for InO_{x}, the transition from an activated behavior with a true superconducting state to a state with saturation of the resistance is more dramatic. Here, the resistance of the sample becomes immeasurably small below ~1 K for magnetic fields below 1.2 T (*T*_{c} ≈ 2.6 K for this sample). In both materials, the saturation persists to high fields and resistances comparable to the normal-state resistance. However, Hall effect measurements indicate a sharp boundary at *H*_{M2} between the anomalous metallic phases with ρ_{xx} << ρ_{N} and the metallic behavior that persists at higher fields.

Insight into the exotic nature of the anomalous metallic phase is obtained when we examine the behavior of the Hall effect at low temperatures, depicted for both TaN_{x} and InO_{x} films in Fig. 2. Whereas in strongly disordered materials, the Hall resistance was found to be zero below the superconductor-insulator transition crossing point at *H*_{c} [realized by InO_{x} films (*27*, *28*)], the weakly disordered films here show ρ_{xy} = 0 up to a field *H*_{M2} < *H*_{c2} (circled in Fig. 2). Furthermore, the Hall resistance is found to be zero (to our noise limit δρ_{xy}, below which we cannot rule out a finite but very small ρ_{xy}) in a wide range of magnetic fields, *H*_{M1} < *H* < *H*_{M2}, where saturation of the longitudinal resistance is also observed. For TaN_{x}, the upper limit is δρ_{xy} ~ 3 × 10^{−4} ohms, whereas for InO_{x}, the upper limit is δρ_{xy} ~ 5 × 10^{−4} ohms.

To further elucidate the fact that there is a phase transition (or a sharp crossover at zero temperature) in the vortex state that appears at *H*_{M1}, we examine the nature of the vortex resistivity tensor in the entire field range below *H*_{c2} in Fig. 3. Vortex motion should obey the scaling relation ρ_{xy} (ρ_{xx}^{2}/*H*) tan θ_{H} (*29*), whether exhibiting flux flow, thermally assisted flux flow, or vortex glass (creep) behaviors.

Figure 3 shows the longitudinal resistance ρ_{xx}(*H*), the Hall resistance ρ_{xy}(*H*), ρ_{xx} and ρ_{xy} with a log scale, the ratio ρ_{xx}^{2}/ρ_{xy}, and the Hall conductivity σ_{xy} as a function of the magnetic field for InO_{x} at temperatures below 1 K. (Data for TaN_{x} are presented in the Supplementary Materials.) These curves capture all three field-tuned transitions in the films. First, the longitudinal resistance ρ_{xx} shows the transition to the metallic state at *H*_{M1} (Fig. 3A), above which the Hall resistance ρ_{xy} is still zero (Fig. 3B). Above *H*_{M2}, the Hall resistance is finite, and both ρ_{xx} and ρ_{xy} show ρ ~ exp(*H*/*H*_{0}) scaling (Fig. 3C), previously associated with the metallic phase in MoGe. In addition, above *H*_{M2}, ρ_{xx} and ρ_{xy} obey scaling of ρ_{xx}^{2}/ρ_{xy} *H* (Fig. 3D), indicating a state of dissipating vortex motion (*29*). This scaling fails just above *H*_{M2} at the lowest temperature, where ρ_{xx}^{2}/ρ_{xy} decreases below the expected field-linear behavior. As a result, the Hall conductivity σ_{xy} (Fig. 3E) starts to decrease with decreasing field and extrapolates to zero at *H*_{M2}, further supporting the picture of a particle-hole symmetric state. Because ρ_{xx} extrapolates to a finite value at zero temperature in fields above H_{M2}, we identify this regime with a pure flux flow resistance. Finally, as we increase the field beyond *H*_{c2} both ρ_{xx} and ρ_{xy} recover their normal-state values.

The identification of zero ρ_{xy} in the anomalous metallic regime needs to be tested against the possibility that it is too low to measure because of the appearance of local superconducting “puddles.” In particular, because the anomalous metal–superconductor system is expected to be inhomogeneous (*14*, *23*), we may have a system of superconducting islands (for which σ_{xx}^{S} → ∞ and σ_{xy}^{S} = 0) embedded in a metal (characterized by σ_{xx}^{M} and σ_{xy}^{M}). If the metal percolates, then for any dilution of the system by superconducting “islands” the measured Hall conductivity satisfies σ_{xy} = σ_{xy}^{M} (*30*); this behavior would persist until the superconductivity is quenched. In Fig. 3E, we plot σ_{xy}^{M} (thick lines) along with σ_{xy} calculated by inverting the resistivity tensor: σ_{xy} = −ρ_{xy}/[ρ_{xx}^{2} + ρ_{xy}^{2}]. Above *H*_{c2}, σ_{xy} shows normal-state behavior. However, just below *H*_{c2} and well above *H*_{M2}, σ_{xy} has departed from σ_{xy}
^{M}, indicating that the anomalous metallic state (as well as the vortex liquid phase above it) is not a matrix of superconducting “puddles” embedded in a metal matrix.

Before we discuss the resulting phase diagram for these 2D disordered films, several points need to be emphasized. First, in the absence of superconducting attractive interactions, these films are expected to be weakly localized and insulating in the limit of zero temperature, although this limit is impossible to observe in finite-sized films with good metallic conduction. Second, the phases that we probe are all identified at finite magnetic fields and finite temperatures. In principle, in the presence of a finite magnetic field, there are no true finite-temperature superconducting phases in two dimensions in the presence of disorder (*31*), whereas in practice, the superconducting phase that we identify exhibits zero resistance. The transition to this phase, either as a function of temperature or magnetic field through *H*_{M1}, is therefore understood as a sharp crossover to a state with immeasurably low resistance. In a similar way, we understand the anomalous metallic phase that exhibits a zero Hall resistance. At low temperatures, this Hall resistance persists to be zero through *H*_{M1} (presumably continuing to manifest electron-hole symmetry) but abruptly becomes finite above *H*_{M2}.

In Fig. 4, we show the resulting phase diagram for the two systems studied here. At a low magnetic field, we observe a narrow superconducting phase, characterized by ρ_{xx} that decreases exponentially with decreasing temperature and zero Hall effect. Upon increasing the magnetic field, the resistance of the sample starts to show saturation in the limit of *T* → 0, whereas the Hall effect does not seem to change from zero. We identify this anomalous metallic phase (“a-Metal” in the figure) already at a finite temperature, where saturation starts to be pronounced, but the important feature here is the extrapolation to zero temperature where true phase transitions manifest at a finite magnetic field. By increasing the magnetic field, we observe a vortex liquid phase, which is commonly observed in measurements on superconducting films at a finite temperature. Although these results suggest the low-temperature metallic phase proposed by Spivak *et al*. (*23*), this proposed phase is highly inhomogeneous and deserves further exploration. Assuming that a metallic phase needs a connection to a dissipative bath, an inhomogeneous state is a likely scenario (*14*, *15*).

The anomalous metallic phase identified above seems to exhibit strong superconducting pairing character but no finite superfluid density on the macroscopic scale. This is seen in the experiments of Liu *et al*. (*5*) where the ac response of 2D low-disorder amorphous InO_{x} films, comparable to those discussed here, exhibited a superconducting response on short length and time scales in the absence of global superconductivity. Similar reasoning leads to the conclusion that vortices can be defined on short length and time scales, similar to the considerations that led to the Kosterlitz-Thouless transition (*32*), where the superfluid density vanishes above the transition at *T*_{KTB}, but vortex-antivortex pairs are observed to proliferate through the system.

The activated part of the resistive transition, just above saturation, fits a 2D collective vortex creep behavior. Hence, it is expected that by lowering the temperature toward *T* = 0, saturation is a consequence of a change in vortex transport, such as a transition to a dissipation-dominated quantum tunneling (*3*, *15*). Finally, by building on recent connections between more strongly disordered films and the quantum Hall liquid-to-insulator transition (*28*), we here observe that the metallic region can be described as an analog to the composite Fermi liquid observed in the vicinity of half-filled Landau levels of the 2D electron gas (*22*).

Disordered InO_{x} films were grown using electron-beam deposition onto cleaned silicon substrates with silicon oxide; careful control of the sample growth resulted in amorphous, nongranular films (*33*). Films of TaN_{x} were deposited using a commercial reactive sputtering tool (AJA International) onto plasma-etched silicon substrates. Film thicknesses (5 to 10 nm) were confirmed by x-ray reflectivity and transmission electron microscopy. In both materials, the films can be considered 2D with respect to superconductivity and localization effects. Film compositions (*x* ≈ 1.5 for InO_{x} and *x* ≈ 1 for TaN_{x}) were checked via x-ray reflectivity, diffraction, and photoemission spectroscopy; we adopted the notation of “InO_{x}” and “TaN_{x}” throughout the text as a reminder that the films are amorphous and nonstoichiometric. Film homogeneity was characterized using transmission electron microscopy and scanning electron microscopy, as well as optically; we found no evidence of inhomogeneity or granularity on any length scale to below the film thickness. Hall bar devices with a width of 100 μm and aspect ratios of either 2 or 4 were fabricated using conventional photolithography techniques, with argon ion milling to define the bar structure and electron beam–evaporated Ti/Au contacts with thicknesses of 10/100 nm.

We measured the longitudinal resistance ρ_{xx} and the Hall resistance ρ_{xy} using conventional four-point low-frequency (≈10 Hz) lock-in techniques; reported values are in the linear response regime. Magnetoresistance and Hall measurements were performed at both positive and negative fields; the Hall resistance was extracted from the transverse voltage by extracting the component antisymmetric in the magnetic field. Measurements on >10 samples for both materials were checked in multiple cryostats; data at temperatures below 2 K used a commercial top-loading dilution refrigerator with a 14 T superconducting magnet. Error bars on the raw data represent ±1σ.

We acknowledge illuminating discussions with B. Spivak and S. Kivelson. **Funding:** Initial work was supported by the NSF (grant NSF-DMR-9508419). This work was supported by the Department of Energy (grant DE-AC02-76SF00515). **Author contributions:** N.P.B. and A.K. conceived and performed the experiments and wrote the manuscript. **Competing interests:** The authors declare that they have no competing interests. **Data and materials availability:** All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/9/e1700612/DC1

Additional data—TaN_{x}

Hall conductivity in the presence of inhomogeneity

fig. S1. Transport regimes in TaN_{x}, comparable to Fig. 3.

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