Superconducting Pair Correlations in an Amorphous Insulating Nanohoneycomb Film

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Science  23 Nov 2007:
Vol. 318, Issue 5854, pp. 1273-1275
DOI: 10.1126/science.1149587


The Cooper pairing mechanism that binds single electrons to form pairs in metals allows electrons to circumvent the exclusion principle and condense into a single superconducting or zero-resistance state. We present results from an amorphous bismuth film system patterned with a nanohoneycomb array of holes, which undergoes a thickness-tuned insulator-superconductor transition. The insulating films exhibit activated resistances and magnetoresistance oscillations dictated by the superconducting flux quantum h/2e. This 2e period is direct evidence indicating that Cooper pairing is also responsible for electrically insulating behavior.

The 50-year-old Bardeen Cooper Schrieffer (BCS) theory (1) provides a microscopic description of superconductivity in metals that also generally applies to superfluidity in other systems of fermions, including liquid 3He (2) and ultracold atomic gases (3). BCS theory introduced a novel phase of matter consisting of weakly bound Cooper pairs of conduction electrons that condense into a single quantum state. This phase exhibits zero dc resistance because correlations among the Cooper pairs inhibit the scattering processes normally responsible for electrical dissipation (1). Counterintuitively, Cooper pair formation has also been invoked to account for recently uncovered high-resistance states of matter (48). Moreover, these states appear in amorphous ultrathin films (5, 79), where the factors favoring insulating behavior can compete with Cooper pair formation (10).

The question of whether Cooper pairing occurs in systems with a resistive ground state is at the heart of discussions of the insulator–to-superconductor quantum phase transition (IST). Numerous systems, including ultrathin amorphous films (11) and wires (12) and high-temperature superconducting oxides (8), can be tuned from an insulating phase, which has a resistance that asymptotically approaches infinity in the zero temperature limit, to a superconducting phase, which has zero resistance. The tuning parameters, which include film thickness (11), magnetic field (6), and magnetic impurities (13), determine the quantum ground state of the system. Consequently, the transformation is deemed a quantum, rather than a thermal, phase transition (4). Models of ISTs broadly separate into those that do or do not presume that Cooper pairs exist in the insulating phase. If Cooper pairs are present, then models include only Bose degrees of freedom and Cooper pairs and vortices, and the IST occurs through Bose condensation (4). If not, then the microscopic interactions among the fermionic degrees of freedom, electrons, must be considered, and the very formation of pairs can dominate the IST (14). Intermediate models of bosons within a dissipative background of fermions have also been developed (15). However, opinions differ widely on which scenario applies for amorphous film systems, because some experiments support a boson-only picture and others do not. Electron tunneling measurements suggest that Cooper pairs first appear at the same thickness as superconductivity in thin elemental films, in support of fermionic models (16). On the other hand, signatures in the magnetoresistance of other amorphous films suggest that vortices (17) and/or Cooper pairs (57) can persist well into an insulating phase.

We have studied a set of films that undergo an IST and can be probed directly for Cooper pairs in their insulating phase. Our films consist of an ultrathin amorphous metal patterned with a nanometer honeycomb (NHC) array of holes, which are tuned to pass from an insulating to a superconducting state by increasing their thickness. The resistive phases of systems that also exhibit a superconducting phase have been scrutinized for Cooper pairs using Nernst-effect measurements of vortex motion (18), electron tunneling measurements of the density of states (16), and ac conductivity measurements (19). These reveal at least the presence of pairing fluctuations. It is desirable, however, to devise a detection scheme that is sensitive to the spatial phase coherence of the pair wavefunction and thus to long-lived Cooper pairs. We constructed a simple but effective platform, which relies on the magnetic field dependence of the pair wavefunction phase (2021). In a magnetic vector potential Embedded Image, the phase varies in space according to Embedded Image, where Embedded Image is the superfluid current and λ is the penetration depth. This relation plus the requirement that the phase be single-valued at any point leads to magnetic flux quantization. For superconductors patterned with an ordered array of holes, this combination causes their free energy and transport properties to oscillate in an applied magnetic field with a period of h/2eS, where S is the area of a unit cell. Nonsuperconducting systems containing Cooper pairs can also exhibit magneto-oscillations, provided that the pair phase-coherence length exceeds the hole spacing. Correspondingly, patterning the holes as closely spaced as possible enhances the sensitivity to Cooper pairs.

To create films with a nanometer-scale hole array, we used a templated growth method analogous to that used to form metal nanowires on carbon nanotubes (12). We quenched deposit from vapor Sb followed by Bi onto a cooled [temperature (T) = 8 K] anodized aluminum oxide substrate previously structured with an NHC array of holes (23). The data presented here primarily come from a series of films with hole radius (rhole) = 27 nm and thicknesses (dBi) 1.09 < dBi < 1.3 nm (23).

With increasing Bi thickness, both the unpatterned and NHC films pass from insulating to superconducting states (Fig. 1, A and B, respectively). In each case, the sheet resistance R(T) transitions from insulating-like (dR/dT <0) to superconducting-like (dR/dT >0) atthe lowest temperatures, but the transitions differ in important details. The IST of the unpatterned film occurs first at dBi = 0.72 nm. Nearest its IST, the conductance of its insulating phase decreases logarithmically with decreasing temperature, in agreement with previous results on these amorphous films (24). In contrast, the IST for NHC films occurs at dBi = 1.27 nm, and its superconducting state emerges from an insulator whose resistance rises exponentially to the lowest measurement temperatures (Fig. 1C). In the thinnest films, this rise is monotonic, whereas closer to the IST, a minimum intercedes. The lowest-temperature data fit well to R = R0exp(T0/T)χ, with χ = 1 and a single R0 ∼6.2 kΩ. Data from the series of NHC films with rhole = 23 nm show that the activated behavior extends to at least 0.1 K. The activation energies, T0, decrease from 4 K in the thinnest film to nearly zero at the IST.

Fig. 1.

Electronic transport. (A) IST of unpatterned a-Bi/Sb films produced simultaneously with NHC films. From top to bottom, dBi = 0.57, 0.59, 0.63, 0.65, 0.67, 0.69, 0.7, 0.74, 0.81, and 0.95 nm. There is no deviation in this data from that in the literature. (B) IST of a-Bi/Sb NHC films with dBi = 1.09, 1.13, 1.16, 1.17, 1.20, 1.24, 1.27, 1.3, and 1.32 nm from top to bottom. Lines are guides to the eye. (C) Arrhenius plot of resistance versus inverse temperature for the insulating films in (B) and fits to the Arrhenius form (gray dashed lines). The hatched area reflects the range of transition temperatures for the unpatterned films for this range of thickness. Films 1 to 7 have activation energies T0 = 4.05, 3.30, 2.30, 1.62, 1.05, 0.30, and 0.08 K, respectively.

The magnetoresistances, R(H), of insulating NHC films near the IST oscillate with a well-defined period, HM = 0.23 T, which we have used to normalize the H axis in Fig. 2A. Each R(H) exhibits six or more discernible oscillations that decay into a rising background. The oscillations first appear at dBi ≈ 1.16 nm, near where the minimum in the R(T) is developed. However, experiments on the smaller hole size indicate that the oscillations need not be associated with the development of the minimum. Figure 2, B and C, show the R(T) at H = 0, HM/2, and HM for a film with and without a minimum, respectively, each showing that the R(T) maintains an activated form H ≠ 0, but T0 varies and is larger at f = 1/2 than at f = 0 or 1, where f = H/HM. By comparison, the prefactor R0 varies only slightly with field. Thus, the R(H) oscillations primarily reflect T0 oscillations and correspondingly, their amplitude grows exponentially as T decreases. In addition, Fig. 2, B and C, show that the amplitude of the magneto-oscillations decreases with increasing normal state resistance. Finally, Fig. 3 exhibits the variation of T0(0), T0(1/2) and the mean field transition temperature of the reference film Tc0 (24) with dBi. It illustrates the range over which the combination of oscillations and exponential localization has been observed and suggests that the IST occurs as T0(0) → 0 with metallicity [R(T, f = 0) = RCR0] at the critical point.

Fig. 2.

Magnetic field effects on transport. (A) Normalized magnetoresistance oscillations for one of the NHC films (no. 6) at three different temperatures. The frustration axis has been normalized by h/2eS, where S is the area of a unit cell (see Fig. 1B). (B) Arrhenius plot of resistance for the same film, showing that the magnetoresistance oscillations in (A) arise as a result of oscillations of the activation energy with magnetic field. Dashed gray lines are fits to the data. f = 0 data are shown as solid circles, f = 1/2 as open circles, and f = 1 as squares. (C) Difference in resistance observed in different fields as a function of temperature in a film with rhole = 23 nm and no minimum in the R(T,H = 0) shown as an inset.

Fig. 3.

Energy scales derived from the data (rhole = 27 nm). Activation energies in f = 0 are shown as solid circles. For those films exhibiting magnetoresistance oscillations, activation energies for f = 1/2 are shown as open circles. The critical temperature for unpatterned films is shown as diamonds.

These observations imply that the insulating phase near the IST consists of localized Cooper pairs that are phase-coherent over distances exceeding the interhole spacing. The period HMh/2eS corresponds to one superconducting flux quantum, ϕ0 = h/2e, per unit cell area, S, of the array (fig. S1B), consistent with Cooper pairing (20, 21). In the range of dBi where oscillations are observed, Tc0 is comparable to the temperature at which R(T) begins to decrease, strongly suggesting that pairing fluctuations begin at a relatively high temperature (9) to enhance the conductance. Subsequently, pairs form, becoming locally phase-coherent and spatially localized at low temperatures. We should add that localized Cooper pairs might also be present but not detectable in thinner films if their localization length is less than the interhole spacing.

The data indicate that Cooper pairs not only influence the transport process in the insulating films but, in fact, are the primary charge carriers. First, the continuous variation and size of T0 near the IST tends to rule out a quasi-particle transport mechanism of the type exhibited in some granular films (25). The superconducting energy gap Δ0 ≅ 1.7 kBTc0 (1) is the expected activation energy for quasi-particle tunneling. Here Δ0 is too large (1.7 kBTc0 >> T0) and varies too little over the dBi range of the oscillations to be associated with T0. Second, the amplitude and phase of the magnetoresistance oscillations support a Cooper pair transport process. Oscillations with an h/2eS period can also occur in both weakly and strongly localized normal metal systems. In both cases, their amplitude is very small (<1%) compared with those observed here (26). This difference reflects the much shorter phase-coherence length for single electrons as compared to that for Cooper pairs. Moreover, the phase of the oscillations in the strongly localized case is such that R(H) first decreases (26), opposite to that in Fig. 2A. Instead, the phase here is consistent with field-induced screening currents reducing the effective Josephson coupling in the films (27) and thus decreasing the coherence of the Cooper pairs.

The activated resistances and T0(H) oscillations are reminiscent of those observed in lithographically defined Josephson junction arrays (JJAs) (28), where Cooper pair localization (CPL) naturally occurs if Coulomb interactions dominate Josephson coupling processes (4, 28). This qualitative agreement suggests that a similar competition drives CPL in NHC films. In JJAs, well-defined inter-island capacitances control the repulsive charging energies, EC, and inter-island tunneling rates control the Josephson coupling energy, EJ. Tuning EJ < EC creates a localized Cooper pair phase characterized by an activation energy that depends on EC and oscillates with magnetic field. The field dependence suggests that an additional “phase” or EJ term also contributes to the activation energy (27). In NHC films, the links between nodes could dictate EJ. An estimate of EC, however, indicates the necessity to consider more than a “lumped element” model to describe the NHC film CPL. Specifically, taking the nodal regions in the honeycomb array to be islands of diameter L ≈ 50 nm, the estimated charging energy is EC = 4e20ϵL ≈ 104 K/ϵ, where ϵ is the dielectric constant and ϵ0 is the vacuum permittivity. Even for ϵ as large as 100, EC far exceeds all pairing-energy scales (for example, Tc0 ≈ 2K) and thus cannot be directly relevant to this competition. Alternatively, this calculation implies that the pairs localize to islands that encompass a large number (thousands) of holes and thus are not associated with any obvious structure.

The data suggest that superconducting islands, without relation to any structure, may spontaneously form near the IST as experiments (24, 29) and theory (14) suggest, through disorder-induced fluctuations in the amplitude of the order parameter (14). This tendency to form islands and their influence on the IST could be accentuated by the NHC substrate. The reduction in film area caused by the holes can inhibit “percolation,” and the small thickness variations, δdBi/dBi = 20% due to the undulations in the substrate (23), may enhance order-parameter variations. Using Tc0(dBi) at 2 K (Fig. 3), we estimate this contribution to order-parameter variations, δTc0/Tc0, to be as large as 40%.

The coincidence of the 2e-based oscillations with activated resistances and “quasi-reentrance” suggests the latter two observations as possible signatures of boson localization in unpatterned films. Interestingly, activated resistances (7) and quasi-reentrance (6) appear in the magnetic field–tuned IST of indium oxide, which those authors suggest is driven by CPL. These similarities across materials and ISTs hint that the NHC patterning and the application of magnetic fields induce similar fluctuations of the order-parameter amplitude. Moreover the present results both provide the clearest example of Cooper pair localization in an amorphous system to date and suggest a general approach to directly detecting their presence in other nonsuperconducting systems such as underdoped high-Tc superconductors above Tc (18).

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