Nature Physics  Letter
Coupling and proximity effects in the superfluid transition in ^{4}He dots
 Justin K. Perron^{1}^{, }
 Mark O. Kimball^{1}^{, }
 Kevin P. Mooney^{1}^{, }
 Francis M. Gasparini^{1}^{, }
 Journal name:
 Nature Physics
 Volume:
 6,
 Pages:
 499–502
 Year published:
 DOI:
 doi:10.1038/nphys1671
 Received
 Accepted
 Published online
The superfluid transition in ^{4}He is associated with the onset of an order parameter that is a macroscopic wavefunction. This is similar to the onset of superconductivity in metals and alloys. Yet, some of the hallmark phenomena that are associated with wavefunctions, and are evident in superconductors, have not been identified in ^{4}He. Here we report proximity effects on the specific heat and superfluid density of a thin ^{4}He film in equilibrium with an array of larger boxes of ^{4}He. Comparison with previous data also enables us, by scaling, to deduce an excess specific heat associated with a collective behaviour of weakly coupled boxes. These effects should be relevant to the understanding of experiments where helium has been studied when confined in packed powders, porous materials or other inhomogeneous confinements, where proximity and collective effects must be present but not readily quantifiable^{1, 2}. Our work also forms a point of comparison and contrast with the case of superconductors^{3, 4}, and is relevant to studies of Josephson effects in superfluids, coupling effects in arrays of Bosecondensed atoms^{5, 6}, and magnetic systems where, because of chemical composition, two separate but interacting ordering transitions take place^{7, 8}.
At a glance
Figures
Main
In the case of superconductors, coupling and proximity effects have been well understood for many years. The large correlationlength amplitude ξ_{0} makes coupling between two superconductors easy to achieve. Weak links with dimensions of the order of ξ_{0}, which can be a metal, an insulator or another superconductor, enable measurements of the familiar Josephson effects (see for instance ref. 9). With ^{4}He these effects are much more difficult to achieve because ξ_{0} is of the order of interatomic dimensions. Furthermore, weak links and coupling between two regions of ^{4}He cannot be achieved through any material other than ^{4}He. We can, however, take advantage of the growth of the correlation length with temperature T and critical exponent ν, ξ = ξ_{0}1 − T/T_{λ}^{−ν} ≡ ξ_{0}t^{−ν} near the transition temperature T_{λ} and construct apertures whereby analogous Josephson effects can be realized^{10, 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 finitesize scaling for zerodimensional crossover^{12}. 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 region^{13}. The lack of scaling is unexpected, because for both two and onedimensional crossover, at least for T>T_{λ}, other data do scale^{14}. More significantly, an analysis of the zerodimensional data suggested that the smaller boxes had a higher heat capacity than expected^{13}. 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 ^{4}He to a array of (2 μm)^{3} boxes, 6 μm centre to centre, with a 31.7 ± 0.1 nm ^{4}He 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 previously^{16}. The superfluid density ρ_{s} of the film is obtained by using adiabatic fountain resonance^{17}.
The heat capacity of ^{4}He confined in this cell is shown as filled circles in Fig. 2. These data are for T<T_{λ} (upper branch) and T>T_{λ} (lower branch). The dominant peak near is the maximum for the transition of ^{4}He confined in the boxes. It is shifted below T_{λ} as expected from finitesize effects^{18}. 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 t_{c}. It is at t_{c} that a global phase coherence must be established. However, because of the confinement, the magnitude of ρ_{s} will be different in the boxes and the film. This observation of a twostep transition into global coherence for ^{4}He mirrors observations with granular superconductors^{4, 19}, and superconducting arrays^{20}. This was also observed earlier in a lesswellcontrolled geometry, where proximity and coupling effects could not be identified^{21}. We also note that calculations of the specific heat of coupled regions of twodimensional Ising spins result in a twopeak structure: one associated with local ordering and another with longrange order^{22}.
In the inset of Fig. 2 we indicate with a vertical line the expected position of the specificheat 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 thickness^{14}. 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 heatcapacity maximum of the normal film. From this observation we conclude that the boxes and film, at least near t_{m,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 studies^{14}. Finitesize correlationlength scaling^{18} 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=ξ_{0}t^{−ν}, the variable tl^{1/ν} with l≡L/ξ_{0}, ξ_{0}=0.143 nm for ^{4}He and T>T_{λ}. 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 g_{1}(tl^{1/ν}) as^{14}
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 specificheat maximum. Thus the specific heat of any film can be calculated from the scaling function g_{1}(tl^{1/ν}). For temperatures above T_{λ} we can represent g_{1} with the function^{24}
where the fitting parameters are given in refs 14 and 24. Similarly, below T_{λ}, and up to , we can use^{13}
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≲0.8t_{m,f}. For larger t, scaling fails^{14}. The value for the maximum can be deduced independently of equation (3) from existing data^{14}. 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>T_{λ}). 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 10nmhigh 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.
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 ρ_{s}/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 ρ_{s}/ρ are shown in Fig. 4. Two sets of data are plotted here. The open circles are for a uniform film 48.3 nm thick^{25}, 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 ρ_{s}/ρ is indicated as a solid line^{26}. The superfluid onset for the uniform 48.3 nm film is consistent with the t_{c} in other uniform films^{14}. The smallest value in ρ_{s}/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 t_{c} and expected KT jump are indicated with the crossed solid vertical and dashed horizontal lines respectively. It is at this crossing point that ρ_{s}/ρ is expected to vanish for this film. We can see that the presence of the boxes has caused ρ_{s}/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 ρ_{s}/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 t_{c}. The enhancement of ρ_{s}/ρ for confined helium in the presence of bulk liquid has been derived using a meanfield approach^{29}. 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.
We identify collectively the shift to higher temperature of t_{m,f} and t_{c} and the enhancement of both the specific heat (Fig. 3) and ρ_{s}/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 zerodimensional 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=tl^{1/ν} the measured quantities on the lefthand side of equation (1) produce two different values for g_{1}(x): g_{1}(x)_{2 μm} and g_{1}(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 T_{λ}. 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 C_{coupling} is at t_{m,Co}=(2.0±0.5)×10^{−5}. This is somewhat closer to T_{λ} than the maximum of the total specific heat, which is at t_{m,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 T_{λ} respectively), and, as well, that the helium in the connecting channels does not become superfluid until , about three decades colder^{31}. Figure 5 represents the first identification of collective effects on the specific heat of weakly coupled regions of ^{4}He near the superfluid transition. The enhancement of the specific heat may be viewed as a partial crossover to twodimensional 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.
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Acknowledgements
This work is supported by the NSF, grant no. DMR0605716; the Cornell Nanoscale Science and Technology Facility, project no. 52694, and internal funding from the University at Buffalo.
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Department of Physics, University at Buffalo, The State University of New York, Buffalo, New York 14260, USA
 Justin K. Perron,
 Mark O. Kimball,
 Kevin P. Mooney &
 Francis M. Gasparini
Contributions
F.M.G. proposed the experiment. M.O.K. and J.K.P. built the (2 μm)^{3} confinement cell with the 31.7 nm planar connecting region and collected the data for that cell. K.P.M. and M.O.K. built and measured the previous cells and analysed their data. The analysis of C_{coupling}, the KT jump and shifts in t_{c} and T_{m,f} were carried out by J.K.P. and F.M.G.
Competing financial interests
The authors declare no competing financial interests.
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Justin K. Perron
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