Main

Conventional zero-temperature (T=0) metallic states do not exist in 2D systems possessing any disorder, because of Anderson localization8,9. To reconcile this fact with experimental evidence for T=0 metals in 2D, it has been proposed that the experimental observations do not pertain to conventional metals, but rather to spatially inhomogeneous superconducting (or, more generally, correlated) states3,4,10. Inhomogeneity is thought to arise in some of these systems because of phase separation; however, it can also be tunably engineered, for example, in hybrid superconductor–normal–superconductor (SNS) systems, such as the arrays studied here. In arrays of SNS junctions, the diffusion of electron pairs from the superconductor into the normal metal11,12,13—known as the proximity effect—gives rise to global superconductivity, through a transition typically described using the phenomenological theory of Lobb, Abraham and Tinkham (LAT)14. According to the LAT theory, the T=0 state is always superconducting, and no zero-temperature metallic state should appear.

Most previous studies of SNS arrays used islands much larger than the superconducting coherence length ξSC (that is, having well-defined superconductivity)1; however, there is evidence that arrays of mesoscopic islands (that is, islands of dimensions comparable to ξSC) exhibit behaviour that deviates from the LAT theory5,15, and might therefore possess non-superconducting T=0 states. Furthermore, the dependence of the superconducting transition on key parameters—such as island spacing and size—has not previously been studied systematically. In this Letter, we present transport measurements on arrays of mesoscopic niobium (Nb) islands having systematically varying inter-island spacings, placed on patterned gold (Au) films. We observe that the device resistance drops to zero in two steps as the temperature is lowered. The lower-temperature drop, at temperature T2, is associated with superconducting phase-locking across the array; the data show that the dependence of T2 on island spacing and thickness deviates strongly from LAT theory. Surprisingly, the higher-temperature drop, at T1, traditionally associated with the superconducting transition of each island, also depends strongly on the island spacing, and seems to extrapolate to zero at finite spacings. This observation implies that superconductivity on individual islands is fragile, and that a T=0 metallic state might be realizable for very weakly coupled islands.

Our samples consist of 10 nm-thick Au, patterned for four-point transport measurements, on Si/SiO2 substrates (see Methods for fabrication details). The Au patterns are overlaid with triangular arrays of 260 nm diameter Nb islands, as shown in Fig. 1. Each array contains more than 10,000 Nb islands. The data in this Letter are from two sets of devices: having 87-nm (±2 nm)- and 145-nm (±2 nm)-thick Nb islands respectively. The devices in each set are identical, except for varied island spacing. X-ray diffraction and scanning electron microscopy of the Nb revealed columnar grains 30 nm in diameter, typical of evaporated Nb (ref. 16); thus, each Nb island contains 50–100 grains. The superconducting coherence length of Nb is estimated to be 27 nm (see Methods), comparable to the grain size but smaller than the island size.

Figure 1: AFM topography of arrays of Nb islands on Au and SEM image of device.
figure 1

a,b, AFM images of arrays of 87-nm-thick Nb islands (red) on 10-nm-thick Au underlayer (yellow). Each array has an edge-to-edge spacing of 140 nm (a) and 340 nm (b). The scale bar is 500 nm. c, False colour SEM image of island array (red rectangle) overlapping Au four-probe pattern (yellow), with the measurement schematic indicated.

Figure 2 shows resistance measurements for the devices, as well as an illustration of the two-step development of superconductivity. The data in Fig. 2a,b show that both T1 and T2 decrease with increasing island spacing. The resistance exhibits an abrupt change in slope at T1, but not the sharp drop seen for larger islands1. Figure 3 shows a plot of T1 versus island spacing. It is evident that T1 decreases more rapidly for the shorter islands, but seems to decrease linearly with spacing for both the shorter and taller islands, extrapolating to zero at 840 nm and 2,600 nm, respectively. The resulting T=0 states would thus be metallic in that they would have finite resistance at finite island spacing. The data in Fig. 2 also show that T2 is more strongly depressed for shorter islands than for taller islands. As schematized in Fig. 2c, these trends can be understood using a model of coupled islands, each composed of grains, having two characteristic energy scales: (1) J, the coupling between grains on an individual island, and (2) J′ (<J), the coupling between grains on neighbouring islands. According to this scheme, for T>T1, the separate grains on each island have incoherent superconducting phases; at T1, intra-island phase coherence develops, and the system’s resistance decreases. For very large islands, T1 would depend only on J, which grows with island height but is spacing-independent. For mesoscopic islands, however, the T1 of an isolated island is depressed (possibly to T=0) by phase fluctuations among the grains; the inter-island coupling J′ serves to reduce these fluctuations by increasing the effective ‘dimensionality’ of the island system, thereby stabilizing superconductivity. Thus, T1 decreases for larger spacings (that is, as J′ decreases). Below T1, the intra-island phase coherence strengthens continuously (Fig. 2c, region II); thus, the system resistance continuously decreases rather than steeply dropping at T1. Region III of Fig. 2c shows the familiar proximity behaviour; here, the normal-metal coherence length17ξN increases until it becomes comparable to the island spacing. Then, inter-island phase coherence begins to emerge (Fig. 2c, region IV), and at T2 the system undergoes a Berezinskii–Kosterlitz–Thouless transition to a fully superconducting state1,2.

Figure 2: Superconductivity in Nb island arrays.
figure 2

a,b, Temperature dependent resistive transitions in arrays having different edge-to-edge island spacings. The island diameter is 260 nm for all arrays. The islands are 87 nm thick (a) and 145 nm thick (b). Black arrows in a mark T1 and T2 for the islands spaced 140 nm apart. The data are normalized to the resistance at 10 K. Note that T1 and T2 occur at higher temperatures for thicker islands. In a, the lowest temperature curves are cut off by the minimum attainable temperature of our apparatus. c, The curve illustrates two-step resistance versus temperature behaviour, with the island transition marked at T1, and film transition marked at T2. Pictures show three islands, each limited to four grains for simplicity. In region I, the Nb islands are normal metals. In region II, the phase of the grains (represented by arrows) starts to become coherent throughout each island (although there is not yet inter-island phase coherence). At T1, Cooper pairs diffuse from the Nb into the Au, and the resistance drops. The grains have intra-island Josephson coupling J and nearest-neighbour inter-island coupling J’(represented by red squiggly lines). In region III, J has saturated, but J’ continues to increase as the normal metal coherence length ξN increases. In region IV, ξN becomes comparable to the island spacing, and the entire system of film and islands progresses towards having global phase coherence. As the temperature is further decreased, the film undergoes a transition to a superconducting state at T2.

Figure 3: Dependence of T1 on array geometry.
figure 3

The first transition temperature T1 is plotted versus island spacing for 87-nm-thick islands (black squares) and 145 nm thick islands (red circles). Solid lines are linear fits and dashed curves are fits to equation (3), that is, the coupled-XY-chain model, in which the coupling J0 and the length scale α are treated as fit parameters (see text and Methods). J0 depends strongly on island height, but α is approximately constant. The points for the smallest island spacings are not shown, as the transitions for this spacing do not show two steps; for a similar reason, the 140 nm spacing for the thicker islands is shown, but not included in the fit.

The inset to Fig. 4 shows how T2 decreases with increasing island spacing. For each device, T2 was extracted by measuring the temperature at which current–voltage (IV) curves became nonlinear (see Supplementary Information). The dependence of T2 on array parameters deviates from LAT theory both quantitatively (that is, T2 decreases more rapidly with island spacing than predicted) and qualitatively (that is, T2 depends strongly on island height). Figure 4 also shows the systematic dependence of ξN(T2) on island spacing, where and the normal-metal diffusion constant D≈94 cm2 s−1(see Methods). We observe ξN(T2) to vary approximately linearly with island spacing.

Figure 4: Dependence of T2 on array geometry.
figure 4

The normal-metal coherence length at T2, ξN(T2), is shown as a function of island spacing. The temperature T2 for each device was extracted from the temperature-dependence of IV curves (see Supplementary Information). The error bars primarily result from uncertainty in the diffusion constant (due to the standard deviation in the Au resistance at 10 K). Solid lines are linear fits and dashed lines are fits to LAT theory (ref. 14). The point for the closest spaced islands is excluded from the plot, as for them the transition shows only one step. The inset shows T2 for each device versus edge-to-edge spacing, for 87-nm-thick islands (black squares) and 145-nm-thick islands (red circle). The open circles mark the Tc of the unpatterned bilayers (8.75 K and 9.1 K, respectively).

We now turn to a more quantitative description of these transitions. From the Ginzburg–Landau perspective, T1 for an isolated island of lateral dimensions comparable to ξSC should equal the transition temperature for a continuous film of the same height, because the suppression of superconductivity due to superconductor–vacuum interfaces at the sides is negligible13,18. This expectation is inconsistent with the pronounced spacing-dependence seen in the data. Because the islands are themselves granular, we attribute the unexpected depression of T1 to spatial fluctuations of the superconducting phase within each island, that is, among the constituent grains, which have lateral dimensions comparable to ξSC. The basic physics of the trend of T1 can be captured by a simple XY model19, governed by the Hamiltonian:

where θiis the superconducting phase of grain i, p indexes islands, 〈i jp denotes nearest-neighbour grains on island p, and 〈p p′〉 denotes nearest-neighbour islands. Each grain on an island is assumed to couple with equal strength to every grain on neighbouring islands; hence the inter-island interaction J′ can be regarded as ‘mean-field’. The temperature dependence of J′ is taken to have the standard proximity form2,14: J′(T)≈J0exp(−d/ξN(T)), where d is the edge-to-edge spacing of the islands and J0 is the coupling amplitude. To reproduce the strong depression of T1 for widely spaced islands, we approximate each island as a one-dimensional chain of XY spins.

Although one-dimensional coupled spins are by no means a fully realistic description of the experimental system, this model captures the following key features of the data: (1) the strong, non-saturating depression of T1, and (2) the two-step character of the transition, that is, the fact that T1>T2 so that intra-island ordering occurs before inter-island ordering (see Methods for details). Fits to the XY-chain model are shown in Fig. 3, and agree reasonably well with the experimental data. The most notable discrepancy involves the largest (340 nm) spacing for the thinner islands, for which T1 is lower than the model predicts; this might suggest that other—possibly quantum—fluctuations are significant in this regime.

We now turn to the spacing- and height-dependence of T2. As seen in Fig. 4, ξN(T2) depends approximately linearly on d; such a relationship implies, in particular, that for large d the transition occurs when d/ξN(T2) is a constant. This observation conflicts with the LAT theory2,14 (presumed to be valid for our measurement regime of d>ξN), which predicts that kBT2J0exp[−d/ξN(T2)] (see fits in Fig. 4). The asymptotic constancy of d/ξN(T2) can be accounted for in one of two ways. The first is to modify the LAT theory by replacing the proximity expression for J′ with the quasiclassical T=0 expression12, J1/d2. This replacement, strictly valid only for dξN(T2), would yield a modified LAT threshold of the form T21/d2, and consequently a linear relationship between d and ξN(T2). Although this modification of LAT theory reproduces the observed linear relationship, it cannot explain the height-dependence of T2. An alternative explanation, which captures both the linear relationship and the height-dependence, is to retain the proximity-effect form of J′ but assume the existence of a mesoscopic energy scale on each island (an effective ‘charging energy’ or, alternatively, the electronic level spacing on a grain (M.V. Feigel’man, personal communication)) that competes with superconductivity. In this scenario T2 would occur when J0exp[−d/ξN(T2)]=κ for some constant κ; this implies the existence of a minimum inter-island coupling κ that must be overcome for superconductivity to be attained, even at T=0, and consequently the possibility of a T=0 metallic state.

Both phase transitions studied in this work occur at temperatures that seem to extrapolate to zero, suggesting the existence of two quantum phase transitions; in particular, our devices may approach a quantum superconductor–metal transition, which has been predicted but not observed3,4,5,10. The tunability of our devices—including the ability to vary island geometry, spacing, material properties and disorder—thus makes them excellent test-beds for exploring such transitions. The unconventional metallic state we observe at temperatures between T1 and T2 is similar to that predicted for phase-separated quantum metals4,5 in that it possesses ‘regional’ phase correlations, that is, correlations at length-scales larger than that of single islands but not global in extent. The ability to stabilize regional correlations, in the absence of long-range ordering, is characteristic of a variety of inhomogeneous correlated systems, including high-temperature superconductors20, coupled magnetic chains21, and strained superconducting films22. The tunability of our system could thus help elucidate open questions in these materials.

Methods

Samples.

Standard photolithographic techniques and electron beam evaporation were used to create the 10-nm-thick four-point pattern of Au with a 4 Å Ti adhesion layer. The Nb islands were then patterned using electron-beam lithography. Before electron beam evaporation of the Nb islands (in an ultrahigh vacuum system at 10−9 torr), the Au surface was Ar+ ion milled to establish a clean interface. For each sample, six arrays were patterned onto a single Si/SiO2 substrate, each with different edge-to-edge spacings: d=90 nm, 140 nm, 190 nm, 240 nm, 290 nm and 340 nm. Four other samples were fabricated; all showed similar data trends, but had limited data ranges (for example, fewer working devices).

The measurement area of every device is 120 μm×30 μm, so the number of islands in each array ranges from 11,400 to 33,516, depending on the island spacing. The large number of islands ensures that discrete percolation paths or individual junction properties do not dominate the conductance. All Nb islands are 260 nm in diameter, which is about 10 times the Nb Ginzburg–Landau dirty-limit coherence length ξSCNb(Tc≈9.1 K)≈27 nm, for an approximate mean free path l≈8 nm. We estimate l from the Einstein relation, using the normal state resistivity ρ(10 K)≈1.12×10−5Ω cm near the transition of an 87-nm-thick unpatterned Nb film. X-ray diffraction spectra of Nb films and scanning electron microscopy of Nb islands showed that the Nb is polycrystalline with growth along the (110) direction, and grain height equivalent to the film thickness. Scanning electron microscope (SEM) images revealed an elongated, columnar grain structure. The Au film resistivity in all devices is ρ(10 K)≈(6.25±0.75)×10−6Ω cm, which is extracted from unpatterned 10-nm-thick films. Using the Einstein relation, we estimate a diffusion constant D≈94.2 cm2 s−1, which yields a mean free path of l≈13 nm and a temperature-dependent coherence length

Measurement.

All measurements above 1.5 K were carried out in a pumped He-4 cryostat, whereas lower temperature measurements were performed in a He-3 refrigerator. Resistance was measured by standard, low-frequency a.c. lock-in techniques using an excitation current of 500 nA. To minimize Joule heating, IV characteristics were measured using rectangular current pulses, with a current-on time of 3.5 ms and current-off time of 3 ms.

Coupled-chain model.

The model of coupled XY spin chains introduced in the main text can be shown to have the following properties: (a) strong, non-saturating depression of T1 for widely spaced islands, and (b) a value of T1 that is nevertheless greater than the inter-island coherence temperature T2. By exploiting a mapping between the XY chain and a quantum mechanical rotor23, one can show that the threshold for an island to acquire a well-defined superconducting phase in the mean field of neighbouring islands is

where z is the coordination number of each island (six for a triangular array) and m is the number of grains on each island. This model satisfies property (a) because equation (2) allows for T1=0 at J′=0 (that is, isolated islands are not superconducting). The model also satisfies property (b) as follows. For temperatures well below T1, the phases of the grains on an individual island are mutually locked; hence, one can neglect the first term of equation (1). Then, T2 is given by z m J′(T2)≈kBT2. Comparing expressions for T1 and T2, we find two separate transitions (that is, T2<T1) provided that ; this condition always holds for large J. If we take Jto have the proximity-effect form JJ0exp(−α/ξN), with α being a constant that is weakly dependent on individual-island parameters, then equation (2) can be rearranged to yield the following dependence of ξN on d:

Figure 3 shows that fits of the data to equation (3) are reasonable. Note that the lack of saturation of T1 seen in the data and reproduced by the model is consistent with a value of T1=0; this admits the possibility of a T=0 metallic state, similar to that described in ref. 4.