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In the case of superconductors, coupling and proximity effects have been well understood for many years. The large correlation-length amplitude ξ0 makes coupling between two superconductors easy to achieve. Weak links with dimensions of the order of ξ0, which can be a metal, an insulator or another superconductor, enable measurements of the familiar Josephson effects (see for instance ref. 9). With 4He these effects are much more difficult to achieve because ξ0 is of the order of interatomic dimensions. Furthermore, weak links and coupling between two regions of 4He cannot be achieved through any material other than 4He. We can, however, take advantage of the growth of the correlation length with temperature T and critical exponent ν, ξ = ξ0|1 − T/Tλ|−ν ≡ ξ0 t−ν near the transition temperature Tλ and construct apertures whereby analogous Josephson effects can be realized10,11.

We are unaware of any theory that describes the effects on thermodynamic properties of a ‘grain’ or dot of helium due to a coupling to another dot through a weak link, or, for that matter, how this coupling is affected by the geometry or ‘strength’ of this link. This question arose while attempting to verify finite-size scaling for zero-dimensional crossover12. We had measured the heat capacity of helium in arrays of (1 μm)3 and (2 μm)3 boxes linked by channels of 1 μm × 1 μm × 0.019 μm and 2 μm × 2 μm × 0.010 μm respectively. These data did not scale anywhere in the critical region13. The lack of scaling is unexpected, because for both two- and one-dimensional crossover, at least for T>Tλ, other data do scale14. More significantly, an analysis of the zero-dimensional data suggested that the smaller boxes had a higher heat capacity than expected13. This might result from a coupling among these boxes, which, before the present work, had been an open question.

The experimental cell used in the present study is described and shown schematically in Fig. 1. The cell confines 4He to a array of (2 μm)3 boxes, 6 μm centre to centre, with a 31.7 ± 0.1 nm 4He film linking them. The uncertainty in the film thickness is the standard deviation associated with the measurement of the thermal oxide before the patterning of this oxide into the posts and wall shown in Fig. 1. The rationale for the design at the centre of the cell and the eventual staging of the cell to measure heat capacity have been discussed previously16. The superfluid density ρs of the film is obtained by using adiabatic fountain resonance17.

Figure 1: The confinement cell.
figure 1

A cut-away view, not to scale, of a cell of (2 μm)3 boxes connected by a 31.7 nm planar region. This planar region is formed when the top wafer is bonded to the bottom wafer at the separation determined by the support posts and outer wall. This cell consists of two oxidized and patterned wafers of silicon 5 cm in diameter. One wafer has an array of 3.4×107 boxes of (2 μm)3 volume patterned in the SiO2. The top wafer has a series of square support posts formed from a (31.7±0.1) nm oxide growth. The centre of the top wafer has a series of 8-mm-long, 15-μm-wide channels that take the helium from the centre filling hole to the rest of the cell. The cell is formed by direct wafer bonding15. The resulting structure is an array of boxes spaced 6 μm centre to centre and connected through a planar opening 31.7 nm high.

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The heat capacity of 4He confined in this cell is shown as filled circles in Fig. 2. These data are for T<Tλ (upper branch) and T>Tλ (lower branch). The dominant peak near is the maximum for the transition of 4He confined in the boxes. It is shifted below Tλ as expected from finite-size effects18. Because the connecting film is normal at this temperature, the order parameter in each box has an arbitrary phase. The second feature at is associated with the transition in the 31.7 nm film. The superfluid onset for the film (see below) takes place at a yet lower temperature tc. It is at tc that a global phase coherence must be established. However, because of the confinement, the magnitude of ρs will be different in the boxes and the film. This observation of a two-step transition into global coherence for 4He mirrors observations with granular superconductors4,19, and superconducting arrays20. This was also observed earlier in a less-well-controlled geometry, where proximity and coupling effects could not be identified21. We also note that calculations of the specific heat of coupled regions of two-dimensional Ising spins result in a two-peak structure: one associated with local ordering and another with long-range order22.

Figure 2: Heat-capacity signatures of a two-step transition.
figure 2

The total measured heat capacity for the helium in the cell described in Fig. 1 (filled circles), and the calculated heat capacity for a 31.7 nm uniform film (open circles). This calculation is done using the scaling functions for planar confinement, Equations (1) and (2), and using 1.22 μmol for the amount of helium in the film. The inset shows an expanded view of the data in the region where the film orders. Here the vertical line indicates the expected temperature Tm of the specific heat maximum if the film were not influenced by the presence of boxes14. This is obtained by a least-squares fit of the data for planar confinement14 and yields tm (31.7 nm)=(1.90±0.07)×10−3.

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In the inset of Fig. 2 we indicate with a vertical line the expected position of the specific-heat maximum for a 31.7 nm planar film without the influence of the boxes. This is well known from measurements over a wide range of film thickness14. The signature of the film’s maximum in the present data is shifted, warmer than expected, an indication of the influence of the already superfluid boxes on the heat-capacity maximum of the normal film. From this observation we conclude that the boxes and film, at least near tm,f, act as a collective, rather than separate, thermodynamic system. To see the full extent of this behaviour, we can subtract from the total heat capacity that of a 31.7 nm uniform film, which can be calculated using empirical scaling functions established for films in other studies14. Finite-size correlation-length scaling18 predicts that the thermodynamic response of a finite system of small dimension L should be related to its behaviour in the thermodynamic limit by functions that depend on the variable L/ξ, or, equivalently, after introducing ξ=ξ0t−ν, the variable t l1/ν with l≡L/ξ0, ξ0=0.143 nm for 4He and T>Tλ. In the case of the specific heat, a convenient function to use is the difference at the same temperature between the specific heat in the thermodynamic limit and that of the finite system C(t,L). This difference should be related to a scaling function g1(t l1/ν) as14

where the exponents ν=0.6705 (ref. 23) and α=−0.0115 are consistent with the hyperscaling relation α=2−3ν. Measurements for films over a wide range of L support this expected scaling behaviour up to the neighbourhood of the specific-heat maximum. Thus the specific heat of any film can be calculated from the scaling function g1(t l1/ν). For temperatures above Tλ we can represent g1 with the function24

where the fitting parameters are given in refs 14 and 24. Similarly, below Tλ, and up to , we can use13

This function, with parameters in J mol−1 K−1 A−=−45.72±0.18, b−=−0.3191±0.053, c−=−1.244±0.028, d−=31.21±0.25, fits the scaled planar data with random deviations in the region t≲0.8tm,f. For larger t, scaling fails14. The value for the maximum can be deduced independently of equation (3) from existing data14. The calculated heat capacity of a 1.22 μmol uniform film 31.7 nm thick is shown at the bottom of Fig. 2 as open circles. This can be subtracted from the total heat capacity to obtain the specific heat of the helium in the boxes. These data are shown as the filled circles in Fig. 3. The solid lines are (ref. 16), and our data have been normalized to this in the region t ≥ 10−3 (T>Tλ). Also on this plot, we show the specific heat from an earlier set of measurements where the (2 μm)3 boxes were spaced 4 μm centre to centre and were connected with channels of 2 μm×2 μm×0.010 μm (ref. 12). In contrast to the present cell, the helium in these 10-nm-high connecting channels represents only 0.5% of the helium in the cell and contributes negligibly to the heat capacity. The good agreement between the two sets of data in Fig. 3 (apart from the region where the film orders) leads to the conclusion that, in these two rather different arrangements of the boxes, the data represent very nearly the specific heat of isolated (2 μm)3 dots of helium.

Figure 3: Isolated boxes and enhanced film.
figure 3

The specific heat for helium confined in (2 μm)3 boxes with very different connecting regions, and therefore different couplings, is plotted. For the filled circles the data represent that of Fig. 1 with the calculated contribution of a uniform 31.7 nm film subtracted. For the data shown as the open boxes such correction is not needed because the film contributes a negligible amount. The relatively good agreement between these two sets of data in most of the temperature region indicates that they represent the specific heat of isolated (2 μm)3 boxes of helium. The inset is an expanded view of the region where the 31.7 nm film becomes superfluid. The calculated contribution of a uniform film in this region still leaves an excess specific heat, which is an enhancement owing to the proximity of the film to the helium in the boxes. An analogous effect of proximity will be seen in the superfluid density in Fig. 4. The solid lines represent the behaviour of bulk helium.

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Shown in the inset of Fig. 3 is the region where the film orders. The calculated subtraction for the contribution of a uniform film leaves the data above those for the (2 μm)3 boxes unaffected by a film. This indicates that the film correction, using the scaling function, as well as the individual point for the film’s maximum, underestimates the contribution of the 31.7 nm film, or, equivalently, that the specific heat of the film in the presence of the superfluid boxes has been enhanced.

Just as the specific heat of the film is affected by the boxes, so is the superfluid fraction ρs/ρ, which is related to the order parameter. To measure this we drive a resonant superflow in the film, which takes place over a ‘bed’ of already superfluid boxes. This flow is driven thermally, thus causing superfluid to move in and out of the cell. The restoring force comes from the compressibility of the helium. The peak mass flow is typically ∼10−12 g m with temperature excursions of less than 10−6 K. Details of the equations of resonance and the hydrodynamic analysis can be found in ref. 17. Data for ρs/ρ are shown in Fig. 4. Two sets of data are plotted here. The open circles are for a uniform film 48.3 nm thick25, and the filled circles are for the 31.7 nm film in the presence of the (2 μm)3 boxes. The behaviour of the bulk ρs/ρ is indicated as a solid line26. The superfluid onset for the uniform 48.3 nm film is consistent with the tc in other uniform films14. The smallest value in ρs/ρ at the transition is also consistent with, but somewhat larger (∼20%) than, the expected Kosterlitz–Thouless (KT) jump 27,28 (indicated by the dashed–dotted horizontal line). For a uniform 31.7 nm film the expected position of tc and expected KT jump are indicated with the crossed solid vertical and dashed horizontal lines respectively. It is at this crossing point that ρs/ρ is expected to vanish for this film. We can see that the presence of the boxes has caused ρs/ρ to vanish at a higher temperature than expected, and with a lower magnitude than the expected KT jump. In none of our previous measurements using adiabatic fountain resonance have we measured a smaller than expected jump in ρs/ρ. Rather, as seen in the 48.3 nm data, measurements will yield a somewhat larger jump simply because, owing to dissipation, the resonance cannot be followed too close to tc. The enhancement of ρs/ρ for confined helium in the presence of bulk liquid has been derived using a mean-field approach29. As the helium in the boxes can be viewed as nearly bulk at the temperature where the film orders, we view this as qualitative support for our observations of the behaviour of the film in the presence of the boxes.

Figure 4: Enhanced superfluid onset and superfluid fraction.
figure 4

Plot of the superfluid fraction for a uniform 48.3 nm film (open circles) and a 31.7 nm film in the presence of helium in (2 μm)3 boxes at 6 μm separation centre to centre (filled circles). The solid line is the bulk behaviour. The horizontal lines represent the lowest values of ρs/ρ expected for uniform films of the respective sizes28. The solid vertical line is at the expected tc for a uniform 31.7 nm film. This is obtained by a least-squares fit of the data for planar confinement and yields tc (31.7 nm) =(2.75±0.04)×10−3. We can see that the film in the presence of the boxes has ρs/ρ that persists to a higher temperature than expected and to a lower value than expected. If these enhancement effects were not present, the filled circles should end at the point where the vertical line and horizontal dashed line cross. The vertical error bars are standard deviations associated with the determination of the resonance frequency from which ρs/ρ is obtained.

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We identify collectively the shift to higher temperature of tm,f and tc and the enhancement of both the specific heat (Fig. 3) and ρs/ρ (Fig. 4) as proximity effects on the 31.7 nm film in the presence of the helium in the (2 μm)3 boxes.

Returning now to the discussion of the zero-dimensional scaling, we can conclude, from the general agreement between the two sets of (2 μm)3 data in Fig. 3, obtained under different conditions of spacing and connectivity, that the previously observed lack of scaling between the (2 μm)3 and (1 μm)3 boxes data is due to the behaviour of the latter. This lack of scaling is manifest in the fact that at the same value of the scaling variable x=t l1/ν the measured quantities on the left-hand side of equation (1) produce two different values for g1(x): g1(x)|2 μm and g1(x)|1 μm. This can be seen in Fig. 47 of ref. 14, and is provided as Supplementary Information. For the (1 μm)3 data we can separate in equation (1) the coupling contribution from the scaling part, or

This yields from equation (1)

This result is plotted in Fig. 5 for the region near Tλ. We can see that the coupling of the (1 μm)3 boxes is significant: it amounts to a 10% increase over the total specific heat at the maximum. The position of the maximum in Ccoupling is at tm,Co=(2.0±0.5)×10−5. This is somewhat closer to Tλ than the maximum of the total specific heat, which is at tm,b=(3.6±0.6)×10−5 (ref. 30); see also Supplementary Information. The coupling of these boxes is surprising if we consider the fact that the correlation length is smaller than the separation of 1 μm (at t=10−4 the bulk correlation length is 0.07 μm and 0.17 μm above and below Tλ respectively), and, as well, that the helium in the connecting channels does not become superfluid until , about three decades colder31. Figure 5 represents the first identification of collective effects on the specific heat of weakly coupled regions of 4He near the superfluid transition. The enhancement of the specific heat may be viewed as a partial crossover to two-dimensional behaviour as opposed to zero dimensional for fully isolated boxes. There is no equivalent observation to this with superconductors because in superconductors the specific heat is not detectably enhanced by critical fluctuations.

Figure 5: Specific heat associated with coupling of helium confined to (1 μm)3 boxes.
figure 5

The excess specific heat Ccoupling is obtained using equation (4). The (1 μm)3 boxes are separated at 2 μm centre to centre and are linked with a channel 1 μm wide and 19 nm high. As noted in the text, the coupling between these boxes extends well into regions where ξ≪1 μm, which is the edge-to-edge separation of the boxes. The error bars for the data represent the combined uncertainty in the value of the experimentally determined functions g1(x)|2 μm and g1(x)|1 μm; see equation (4) and Supplementary Information.

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