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We review measurements of alt;supagt;4alt;/supagt;He near the superfluid transition ina#13; arrangements whereby an array of weak links couple relatively larger, more bulk-a#13; like alt;supagt;4alt;/supagt;He regions. In contrast to experiments which focus on the dependence ofa#13; the superflow on the chemical potential difference across the links, these studiesa#13; focus on the specific heat of both the weak links and that of the larger coupleda#13; regions, as well as the behavior of the superfluid fraction within the weak links.a#13; The data show unexpected results which reflect a very long range coupling asa#13; well as modification of the weak link itself due to the proximity to bulk-likea#13; helium. One finds that while the three-dimensional correlation length ξ(t), wherea#13; t = |1 − T/Talt;subagt;λalt;/subagt;|, is involved in these long-range effects, the distance over whicha#13; these can be seen is of the order of 100 to 1000 times ξ(t). These results call intoa#13; question our understanding of the meaning of the correlation length at a criticala#13; point as the 'range' over which information can propagate. These studies area#13; the first to measure the thermodynamic properties of weak links for a criticala#13; system where fluctuations are important. They differ in essential ways witha#13; expectations from mean-field considerations. We compare results with other alt;supagt;4alt;/supagt;Hea#13; measurements, with superconductors and the theoretical calculations of the Isinga#13; model.

Previous measurements of the superfluid density ${\ensuremath{\rho}}_{s}$ and specific heat for $^{4}\mathrm{He}$ have identified effects that are manifest at distances much larger than the correlation length ${\ensuremath{\xi}}_{3\text{D}}$ [Perron et al., Nat. Phys. 6, 499 (2010); Perron and Gasparini, Phys. Rev. Lett. 109, 035302 (2012); Perron et al., Phys. Rev. B 87, 094507 (2013)]. We report here measurements of the superfluid density which are designed to explore this phenomenon further. We determine the superfluid fraction ${\ensuremath{\rho}}_{s}/\ensuremath{\rho}$ from the resonance of 34-nm films of varying widths $4\ensuremath{\le}W\ensuremath{\le}100\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\text{m}$. The films are formed across a Corbino ring separating two chambers where a thicker 268-nm film is formed. This arrangement is realized using lithography and direct Si-wafer bonding. We identify two effects in the behavior of ${\ensuremath{\rho}}_{s}/\ensuremath{\rho}$: one is hydrodynamic, for which we present an analysis, and the other is a correlation-length effect which manifests as a shift in the transition temperature ${T}_{c}$ relative to that of a uniform 34-nm film uninfluenced by proximity effects. We find that one can collapse both ${\ensuremath{\rho}}_{s}/\ensuremath{\rho}$ and the quality factor of the resonance onto universal curves by shifting ${T}_{c}$ as $\mathrm{\ensuremath{\Delta}}{T}_{c}\ensuremath{\sim}{W}^{\ensuremath{-}\ensuremath{\nu}}$. This scaling is a surprising result on two counts: it involves a very large length scale $W$ relative to the magnitude of ${\ensuremath{\xi}}_{3\text{D}}$ and the dependence on $W$ is not what is expected from correlation-length finite-size scaling which would predict $\mathrm{\ensuremath{\Delta}}{T}_{c}\ensuremath{\sim}{W}^{\ensuremath{-}1/\ensuremath{\nu}}$.

We report measurements of superfluid density and specific heat of a 33.6 nm film near the superfluid transition. The film is formed between two patterned and directly bonded silicon wafers. These measurements were undertaken with the primary purpose of understanding coupling and proximity effects in a situation when the film was in contact with helium in a larger confinement (Perron et al. in Nat. Phys. 6:499, 2010; Perron and Gasparini in Phys. Rev. Lett. 109:035302, 2012). However, these data are also relevant to issues of correlation-length finite-size scaling. This is the thinnest hard-wall confined film for which such scaling has been tested for the specific heat and superfluid density. One expects that at some small thickness such scaling should fail. We compare our results with previous data of helium in a similar confinement but at larger thickness. We find good agreement with scaling in regions where previous data scaled, and confirm the lack of scaling where previously reported. In our analysis we consider a native oxide growth between the etching and bonding steps of cell fabrication and its effect on our scaling analysis.

Superfluid 4He shares with superconductors a transition into a low temperature state where the order parameter is a wave function. For the low temperature superconductors, which have a large zero temperature correlation length, this results in well known Josephson effects reflecting the overlap of the wave function across barriers and weak links. Similar phenomena are harder to realize for 4He because the zero temperature correlation length is of the order of interatomic dimensions. The fact that for 4He the critical region, where the correlation length diverges, is accessible experimentally leads to a possible new kind of coupling. This differs from that of a superconductor in the sense that critical fluctuations are important. We have seen such coupling whereby two regions of confined 4He interact and influence their respective thermodynamic behavior (Perron et al. in Nat. Phys. 6:499–502, 2010). This interaction extends over length scales which are much larger than the correlation length. We describe measurements of heat capacity and superfluid density which illustrate this behavior.

Some effects that have long been observed in superconductors are difficult to see in superfluid helium-4, despite the similar physics governing these systems, owing to the small correlation length in helium-4. Proximity effects have now been observed in that system, using an array of micrometre-size boxes linked by a thin helium film.

We have studied the superfluid transition of 4He in situations where the growth of the correlation length is limited by a uniform spatial confinement. Under these conditions the critical behavior attained in the thermodynamic limit is greatly modified. One expects that data for similar confinement should scale with the correlation length. This is found to be true for the specific heat in planar confinement, 2D crossover, except in the region of the specific heat maximum and on the superfluid side. The modifications due to confinement depend on the details of the geometry. Studies of the specific heat with 1D and 0D crossover show the important role played by the lower dimension. Far from the transition the details of the confining geometry reveal contributions to the specific heat which can be attributed to surfaces and edges. Comparison of experimental resugts with theoretical calculations show agreement in some areas and disagreement in others.

We report studies of a 4He film confined between two silicon wafers separated by 3189 A. The film is connected to a bulk helium reservoir via small channels 100 A high, 8 µm wide by 2000 µm long. This cell design has allowed us to study the heat capacity in a planar confinement (a film of ∞ lateral size), and the superfluid density in the connecting channels (a film of finite lateral size). This work is relevant to finite-size scaling of the specific heat for 2D confinement and it is compared with earlier data. It is also relevant to finite-size 2D behavior for the superfluid density which is related to the recent theory of Sobnack and Kusmartsev. Analysis of the data is presented as well as a discussion of future cell designs to address, in particular, the behavior of laterally confined films.

We report measurements of the specific heat near the superfluid transition of 4He confined in uniform channels of 1 µm square cross-section. This system undergoes a crossover from three dimensional behavior (3D) to 1D as the transition is approached. This resugts in a substantial rounding of the specific heat maximum as well as a shift to colder temperatures relative to the bulk system. We compare these data to previous measurements where crossovers from 3D to 2D and 0D were studied with the smallest confining dimension being the same (1 µm) in each case. We also compare these resugts in the context of finite-size scaling to previous studies where crossover to 1D was measured in cylindrical geometries. We identify regions where surface and edge effects dominate the specific heat, and compare these amplitudes to theory, where available. The realization of the confining geometry in this work is achieved with a combination of silicon lithography and direct wafer bonding.

When helium is confined to a uniform small dimension L, its behavior is modified due to the limitation of the growth of the correlation length. This has been explored for planar confinement over a wide range of L. Less is known about confined mixtures, especially in the case of the specific heat. Here two principal effects come into play: the magnitude of the correlation length ξ increases, and the measured heat capacity Cpx must be converted to Cpφ, where φ=μ3−μ4, before finite-size scaling predictions can be verified. The increase in ξ makes a given confinement L “look smaller” as the concentration x is increased. This, as well as changes of L itself, can be used to test predictions of correlation-length scaling. We report measurements of the specific heat of confined mixtures and compare with the well established scaling for the pure system.

The second-order phase transition of 4He from a normal fluid to a superfluid is ideally suited for studies of critical behaviour. In particular, effects of confinement have been studied recently to verify theoretical predictions of correlation-length scaling and calculations of specific scaling functions. These predictions are summarized for the specific heat and the superfluid density. The method of achieving confinement is discussed, as well as the measuring technique. The specific heat and the superfluid density in planar confinement are examined. It is found that the specific heat scales well on the normal side, and just as well on the superfluid side until the region of the specific heat maximum is reached. Here deviations from scaling are seen. It is possible that this behaviour is associated with the specific crossover in two dimensions. The superfluid fraction, which has been measured for the same type of confinement in two different ways, does not scale. Results of a calculation for the superfluid density to assess the role of the inhomogeneity induced by the van der Waals attraction at the confining walls are presented.

We report measurements of the superfluid fraction rho(s)/rho of films of (3)He-(4)He mixtures confined between silicon wafers at 0.0483 microm separation. The data obtained using adiabatic fountain resonance (AFR) can be used to test for the first time expectations of correlation-length scaling in the case of planar mixtures. For the mixtures, the data for rho(s)/rho collapse well on a universal function. The dissipation associated with AFR can also be scaled, and indicates two-dimensional crossover. These results are in contrast to pure (4)He, where over a wider range of confinements, the data for rho(s)/rho are found not to scale.

We report an analysis of a superfluid Helmholtz resonance in the case of helium confined in a superleak. The resonance of the superfluid is achieved under nearly adiabatic conditions. Equations are derived for the resonance frequency, the temperature oscillations of the superleak and the phase relation of this signal relative to an ac heat input. The resonance frequency yields the superfluid fraction of the confined helium. Data are analyzed as function of frequency and temperature and yield parameters such as the dissipation and thermal conductivity which determine the resonance line shape. Estimates are made of the thermodynamic parameters in the resonance equation by using derivatives along the pressure-temperature-concentration lambda surface. These parameters are compared with results from the analysis of the resonance.

We report measurements of the specific heat of 4He near the superfluid transition while confined between silicon wafers at 0.9869 μm separation. These data are analyzed to check on the behavior expected from correlation-length scaling. Comparison is also made with other data for planar confinement, as well as data for cylindrical confinement. These represent different lower-dimensional crossovers. We find that the present data scale very well above the bulk transition temperature, and in the region immediately below it. Near the specific heat maximum however, the data for planar confinement do not collapse on a universal curve. We compare these results with specific theoretical scaling functions. In particular we find that on the normal side, and for large enough values of the scaling variable, one can describe the data well using the concept of the surface specific heat. The locus of the data in this region agrees well with the most recent theoretical calculations.

We report heat capacity measurements of confined films of4He. These studies were undertaken to test predictions of correlation-length scaling. They are the first measurements for completely confined films over a range of confinements, and represent a geometry where criticality changes from 3-dimensions (3D) to 2D. The finite system is realized with a4He film confined between two, 2″ diameter, silicon wafers, which are separated by a small gap. A new technique was developed to bond these wafers at a uniform separation. The gap size, which determines the film thickness, ranges from 0.05 to 0.7 μm in the present work, and has better than 1% uniformity. The bonded cells are used to conduct high precision heat capacity measurements using a modified ac technique. This involves oscillating the sample temperature, as in conventional ac calorimetry, but with simultaneous dc regulation of the average temperature. The data are analyzed using a modified Sullivan–Seidel equation, which takes into account in an empirical way the finite conductivity of the cell. This procedure yields heat capacity data with good absolute accuracy and high resolution. Scaling analysis of the data both above and below the bulk transition temperature shows collapse onto universal curves determined only by the ratio of the correlation length to the confinement size. This is true everywhere except near the heat capacity maximum. Here, and into the superfluid side there is lack of scaling which might be associated with 2D crossover. We compare this result with calculations of scaling functions and find that these tend to underestimate the effect of confinement. Comparison with earlier results for cylindrical confinement shows differences which are most striking in the region of the specific heat maximum. The cylindrical and planar confinement data follow similar trends above the superfluid transition of bulk helium. Below the transition, however, the present data show much more structure. Fits of the scaled planar data above the transition to an empirical scaling function yield a correlation length exponent of νeff=0.674±0.001.

We report new specific heat data for4He confined between two silicon wafers at 0.0483 μm separation. This extends our work by a factor of two in the range of confinements studied, and allows us a better check on scaling predictions. The present data have been obtained with a new design for the SiO2pattern which is used to achieve the wafer's separation. With this, we have obtained reliable data in the region where the confined helium becomes superfluid. We find that for T > Tλthe new data collapse well onto a universal function. Below Tλ, but above the specific heat maximum, the data also collapse well. Near the maximum, however, there is lack of collapse which persists into the superfluid region. It seems likely that this is connected with the two-dimensional crossover for the geometry of these cells

We describe a new resonance of superfluid 4 He when confined in a superleak. In our experiment the superleak is sealed except at a filling hole which communicates with bulk liquid. The resonance is driven and detected thermally. The restoring force is provided by the compressibility of the helium under conditions which are very nearly adiabatic. We derive expressions for the resonance frequency, the lineshape and phaseshift and compare these with measurements on a 0.2113μm superleak.

We describe a technique for performing high resolution calorimetry on samples with very small heat capacity (∼10μJ/K). This involves using ac calorimetry, in which the sample is subjected to temperature oscillations, with simultaneous dc regulation of the average temperature. This method works well for small samples of4He, where adiabatic calorimetry would be impossible. We study the frequency response of our system using a modified Sullivan-Seidel equation, which allows for small temperature inhomogeneities at high frequencies. These procedures enable one to obtain heat capacity data possessing a high degree of absolute accuracy.

Measurements of the heat capacity and superfluid fraction of confined (4)He have been performed near the lambda transition using lithographically patterned and bonded silicon wafers. Unlike confinements in porous materials often used for these types of experiments(3), bonded wafers provide predesigned uniform spaces for confinement. The geometry of each cell is well known, which removes a large source of ambiguity in the interpretation of data. Exceptionally flat, 5 cm diameter, 375 µm thick Si wafers with about 1 µm variation over the entire wafer can be obtained commercially (from Semiconductor Processing Company, for example). Thermal oxide is grown on the wafers to define the confinement dimension in the z-direction. A pattern is then etched in the oxide using lithographic techniques so as to create a desired enclosure upon bonding. A hole is drilled in one of the wafers (the top) to allow for the introduction of the liquid to be measured. The wafers are cleaned(2) in RCA solutions and then put in a microclean chamber where they are rinsed with deionized water(4). The wafers are bonded at RT and then annealed at ~1,100 °C. This forms a strong and permanent bond. This process can be used to make uniform enclosures for measuring thermal and hydrodynamic properties of confined liquids from the nanometer to the micrometer scale.

We report measurements of the specific heat of thick helium films to test scaling predictions near the superfluid transition, Tl. These films, bounded by two silicon wafers, range in thickness from 0.107 to 0.692 mm. The specific heat of the films is suppressed relative to that of bulk helium, and the difference of the data from bulk helium scales well with the exponent of the correlation length. We propose an empirical scaling function which fits the data well for T . Tl, and yields the surface specific heat. We also compare the data with various theoretical calculations and experiments. We find that the calculations underestimate the effect of confinement. [S0031-9007(97)02843-3] The motivation for this work was to study effects of confinement on the heat capacity of liquid 4 He near the superfluid transition temperature, Tl. Compared to bulk helium, the heat capacity of a finite sample is suppressed, and the maximum is rounded and shifted below Tl. The magnitude of these effects depends on the confinement size L, boundary conditions, and, if appropriate, the lower crossover dimension of the confined system.

Proximity and coupling effects between contiguous regions of 4He are expected to occur over a distance on the order of the temperature-dependent correlation length ξ (t). We present recent measurements of 4He confined in arrays of (2 μm)3 boxes linked through a 4He film. The specific heat and superfluid fraction of the 4He in these geometries reveal effects which occur at distances much larger than expected. These effects include enhancements in specific heat and superfluid fraction as well as shifts in the temperatures of the superfluid onset and specific-heat maximum. Our analysis gives evidence for the relevance of the finite-size correlation length ξ (t,L) in explaining these effects even though the effects occur at distances over an order of magnitude larger than ξ (t,L). These results can be used to deduce the finite-size scaling locus for ξ (t,L). The spatial distance over which coupling and proximity effects are observed raises questions regarding the physical mechanism for these effects and the interpretation of ξ (t). Lastly, these results are discussed in connection to other measurements of 4He in confined geometries, and the relevance to other critical systems.

We present sorption measurements for3He and4He in the temperature range of 1.5 K to 4.1 K, and for20Ne in the temperature range of 22 K to 27 K by crystals of C60, C70 and crystals of the mixture of these two molecules, 80% C60, 20% C70 We analyze these data by taking into account the non-ideality of the gas in equilibrium with the adsorbate. We calculate chemical potentials and isosteric heats. We find that there is no obvious evidence of intercalation of helium in these crystals at low temperatures. At higher temperatures there are some anomalies in the helium isotherms, and indication of excess sorption. The isosteric heat shows a minimum in this region which can be interpreted as penetration of the helium into a region of repulsive potential. We also find that levels of sorption, at the same chemical potential difference from saturation, are higher for4He than for3He. They are also higher for4He on C70 than for the other crystals. For neon our work is concentrated around the triple point. We find that the isotherms indicate the formation of liquid or solid films. Below the triple point, and above a few atomic layers, the neon film does not grow uniformly.

Measurements of the heat capacity in two-dimensional helium-3 adsorbed on graphite provide further evidence of an unexpected liquid state at temperatures near absolute zero.

We report measurements of the superfluid fraction ${\ensuremath{\rho}}_{s}/\ensuremath{\rho}$ and specific heat ${c}_{p}$ near the superfluid transition of $^{4}\mathrm{He}$ when confined in an array of $(2\text{ }\text{ }\ensuremath{\mu}\mathrm{m}{)}^{3}$ boxes at a separation of $S=2\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$ and coupled through a 32.5 nm film. We find that ${c}_{p}$ is strongly enhanced when compared with data where coupling is not present. An analysis of this excess signal shows that it is proportional to the finite-size correlation length in the boxes $\ensuremath{\xi}(t,L)$, and it is measurable as far as $S/\ensuremath{\xi}\ensuremath{\sim}30\ensuremath{-}50$. We obtain $\ensuremath{\xi}(0,L)$ and the scaling function (within a constant) for $\ensuremath{\xi}(t,L)$ in an ${L}^{3}$ box geometry. Furthermore, we find that ${\ensuremath{\rho}}_{s}/\ensuremath{\rho}$ of the film persists a full decade closer to the bulk transition temperature ${T}_{\ensuremath{\lambda}}$ than a film uninfluenced by proximity effects. This excess in ${\ensuremath{\rho}}_{s}/\ensuremath{\rho}$ is measurable even when $S/\ensuremath{\xi}g100$, which cannot be understood on the basis of mean field theory.

We report measurements of the superfluid fraction ρ(s)/ρ and specific heat c(p) near the superfluid transition of 4He when confined in an array of (2 μm)3 boxes at a separation of S=2 μm and coupled through a 32.5 nm film. We find that c(p) is strongly enhanced when compared with data where coupling is not present. An analysis of this excess signal shows that it is proportional to the finite-size correlation length in the boxes ξ(t,L), and it is measurable as far as S/ξ∼30-50. We obtain ξ(0,L) and the scaling function (within a constant) for ξ(t,L) in an L3 box geometry. Furthermore, we find that ρ(s)/ρ of the film persists a full decade closer to the bulk transition temperature T(λ) than a film uninfluenced by proximity effects. This excess in ρ(s)/ρ is measurable even when S/ξ100, which cannot be understood on the basis of mean field theory.

We report sorption isotherms and optical measurements for fullerite C 60 /C 70 crystals and C 60 films exposed to helium. The low temperature isotherms show excess sorption above that expected for surface films. We interpret this as intercalation of helium in the fullerite lattice. Our observations suggest that helium is mobile within the crystal, and opens up the possibility of realizing a new type of three-dimensional quantum fluid. Raman measurements are inconclusive on the issue of intercalation

We report measurements of the convective thermal conductance of3He-4He mixture films near the Kosterlitz-Thouless transition. The thickness of our4He films is 14.7 and 19.1 A above the inert layer and the3He concentration ranged from 0.033 to almost 2%. The thermal response is tested for the critical behavior as observed in pure films, and it is found to be preserved in the mixture films case. However, the parametersb, D/a2 andTc exhibit a strong dependence on the concentration. The mixture film conductance at fixedT-Tc is found to decrease upon addition of3He implying a decrease in the 2D correlation length. Mixture films thus exhibit 2D behavior over a narrower temperature range than pure films. Further, for temperaturesT

We report measurements of the convective conductance of {sup 4}He films near the Kosterlitz-Thouless transition. Data for 14 films are analyzed over a range of critical temperatures from 1.28 to 2.16 K, and thicknesses 11.7 to 156 A. We obtain good agreement using two different methods of analysis with the predicted behavior of the dynamic theory. The exponential divergence is observed up to six decades in conductance and one decade in reduced temperature. We find that the parameter {ital b}, which determines the sharpness of the divergence, and also the cusp in the superfluid density increases with film thickness. This is consistent with the growth of the three-dimensional correlation length. We find also that the ratio of diffusion constant to the square of the vortex core radius, {ital D}/{ital a}{sup 2}, is a decreasing function of film thickness. We compare this with measurements in which {ital D} is obtained directly. We also report measurements of the nonlinear dependence of the conductance below the transition. These are shown to be consistent with measurements above the transition.

A new Si‐SiO2 bonding process has been developed to achieve a uniform spacing between two silicon wafers of 2 and 3 in. diam. Spacings between 0.1 and 3.9 μm have been obtained so far. Hydrostatic pressure is used to force the two wafers into intimate contact at points where bonding is desired. The bonding is performed at a temperature of ∼1150 °C. The uniformity of bonding and spacing between the wafers is checked by a Fabry–Perot interferometer technique at room temperature and by measurements of superfluid density of He II at low temperature. These results are compared with ellipsometer and stylus measurements of the oxide thickness which is designed to govern the wafers’ spacing. We find that these different techniques yield consistent results.

The specific heat of helium homogeneously confined in one or more dimensions is expected to collapse onto a scaling function which depends only on the ratio of the smallest dimension of confinement to the correlation length, written as L/ξ. This may be rewritten to explicitly show the temperature dependence of the correlation length as L/ξ0t−ν, where the constant ξ0 is the prefactor of the correlation length, t is a dimensionless temperature difference from the superfluid transition, and ν is the critical exponent associated with the correlation length. Thus, in principle, one should be able to obtain the exponent ν from the scaling of thermodynamic measurements of confined helium for various L’s. This would represent an independent determination of ν distinct from what is obtained using the behavior of the bulk superfluid density, or via the bulk specific heat and the hyperscaling relation. In practice, this analysis is hampered by the lack of a theoretical expression for the scaling function. We present...

The universality of phase transitions is an important prediction of theories of critical behavior. Simply stated, microscopically different systems near a critical point may be described by universal quantities if their dimensionality is the same and the order parameter has the same degrees of freedom. One way to test this idea is to measure the thermodynamic response of a set of systems to an input where the response changes with a variation in some quantity like spatial confinement, impurity concentration, or even pressure. While the response of each system is different, the behavior may still be described by a common critical exponent if the idea of universality is correct. Confined mixtures of 3He‐4He and pure 4He are believed to satisfy these requirements. Here, amplitudes such as the magnitude of the correlation length and the temperature of the transition both depend upon the concentration of the mixture and the extent of confinement. However, universality predicts the value of the critical exponen...

We report on recent measurements of the specific heat of helium confined in pill‐boxes 2 μm across and 2 μm deep made lithographically on a silicon wafer. The experimental cells distribute liquid from a bulk reservoir to ∼ 108 boxes by an array of very shallow fill‐channels (0.019 μm and 0.010 μm) which represent a negligible volume compared to that of the boxes. Since the channels are so shallow, the helium in them becomes superfluid at a much lower temperature than the liquid in the boxes. Therefore, during the course of the heat capacity measurements, the liquid in the channels in always normal, and the cell would be expected to behave as a system of uncoupled boxes. We compare these measurements with one previously made of a cell where the confinement was to 1 μm boxes with an equivalent fill arrangement. While the shift in the position of the specific heat maximum relative to the 1 μm cell is what one would expect on the basis of finite‐size scaling, there are discrepancies in the specific heat amplitude between the 2 μm cell utilizing different depth fill‐channels, and with the 1 μm cell. It is possible that the channels, even though normal and of negligible volume, provide a weak coupling between the boxes leading to a collective rather than single‐box behavior.

We report measurements of the superfluid fraction of 4He confined in small channels 9.4 nm high, 19 μm wide by 2000 μm long. This confinement corresponds to a film of finite lateral extent. The data show a shift in the transition to a lower temperature which is larger than the logarithmic dependence expected from finite‐size scaling and Berezinskǐ ‐Kosterlitz‐Thouless theory. This shift however is smaller than the one proposed by Sobnack and Kusmartsev for this kind of geometry. When examining the behavior of the shift for confinement at several widths, we found that the shift favors a power law with a larger exponent than predicted by Sobnack and Kusmartsev.

We have measured the heat capacity of (3)He-(4)He mixtures confined in films of thickness 48.3 and 986.9 nm. The confinement is defined by direct bonding of two silicon wafers. The heat capacity is measured using an ac technique and then transformed to correct for exponent renormalization effects. The data address the expected universal critical behavior along the lambda line as function of (3)He concentration. We discuss the results of several analyses of the data, and we show that a universal collapse can be achieved for all the mixtures. However, this is on a locus which differs from that of the pure system. An alternative analysis is also presented which yields collapse of all the data under certain assumptions. We believe these data are the first to test universality of finite-size scaling for the specific heat along a locus of transitions.

We have measured the heat capacity of (3)He-(4)He mixtures confined in films of thickness 48.3 and 986.9 nm. The confinement is defined by direct bonding of two silicon wafers. The heat capacity is measured using an ac technique and then transformed to correct for exponent renormalization effects. The data address the expected universal critical behavior along the lambda line as function of (3)He concentration. We discuss the results of several analyses of the data, and we show that a universal collapse can be achieved for all the mixtures. However, this is on a locus which differs from that of the pure system. An alternative analysis is also presented which yields collapse of all the data under certain assumptions. We believe these data are the first to test universality of finite-size scaling for the specific heat along a locus of transitions.

The critical behavior at a second order phase transition is characterized by the divergence of the correlation length xi. We have studied the superfluid transition of 4He in a series of experimental cells in which this divergence of xi is modified due to finite-size confinement. In particular, the design of these cells is such that the smallest dimension is kept the same, 1 microm, but the geometry is such that one obtains crossover to dimensionality of 2, 1, and 0. This corresponds to films, channels, and boxes filled with helium. We measure the specific heat and compare these results with theoretical expectations. We identify surface and line specific heat contributions by analyzing the deviation of the specific heat from its behavior in the thermodynamic limit. The design of these cells is made possible by a combination of silicon lithography and direct wafer bonding.

We have obtained the superfluid fraction of 4 He and mixtures confined in channels 0.0185 μm ×1.08 μm ×∞ . We compare this with film data L×∞×∞, and data for channels 0.0483 μm ×3.0 μm ×∞ . The behavior of the data in the smallest channels is quite different from what one might expect for a planar film of a given thickness. The transition is shifted to a lower temperature than expected; and, the overall behavior of ρs does not follow the trend observed with other confined films. This suggests that the lateral dimension plays a significant role in the behavior. We believe our observations are consistent with the recent proposal of Sobnack and Kusmartsev about a new mechanism for vortex unbinding in 2D films.

We have measured the heat capacity of helium mixtures confined to lithographically created cylindrical boxes whose height, 1.08 μm , equals their diameter. The specific heat at constant concentration x, C px , is renormalized due to the 3 He impurity. Thus, in order to observe critical behavior, a conversion must be made to a specific heat at constant φ=μ 3 −μ 4 , C pφ . The confined system's specific heat, near it's maximum, has values which rise above the bulk system's specific heat at the same temperature. This is unexpected and is true for both Cpx and Cpφ. This enhancement, not observed with pure 4 He , becomes more dramatic as x and the correlation length increase. The shift of the maximum with x is much larger for the boxes than for 2D confined mixtures. These observations might be related to a “band structure” effect associated with the 18.5 nm channels which connect the cylinders.

We report measurements of the specific heat near the superfluid transition of 4 He confined to 1 μm 3 cylindrical boxes patterned in SiO2. This system crosses from a 3D behavior to a 0D behavior near the transition. This has a marked effect on the specific heat as seen by a pronounced rounding of the maximum and a shift to a temperature much lower than the transition of the bulk system (and systems with 2D or 1D crossover). We plot the data according to correlation-length scaling theory and compare this to a planar system with the same smallest confinement. Compared to our previous studies of planar systems, the 0D cell has 3× the surface to volume ratio as well as ∼750× as much edge length. We examine the regions where surface and edge effect contributions can be separated. We find that the data do not reach the expected value for the surface region. There is also evidence for a region where the term associated with edge contributions dominates.

Recently, it was shown that two confined regions of liquid 4He exhibit a proximity effect over distances much larger than the correlation length ξ[1; 2]. Here we report measurements of the superfluid fraction ρs/ρ and specific heat cp of a 33.6 nm film. Comparison with previous data from a 31.7 nm film in contact with an array of 34 × 106 (2 μm)3 boxes of 4He allows us to show quantitatively the enhancement in ρs/ρ and cp due to the presence of the boxes in the temperature region where the film orders. The enhancement in ρs/ρ is observed up to distances 650 times the bulk correlation length. This anomalously large length scale is analogous to a giant proximity effect observed in High-Tc superconductors (HTSC)[3].

Publisher Summary This chapter discusses the expected scaling behavior of helium confined in a uniform, well-defined geometry. By way of example two models are discussed—the two-dimensional (2D) Ising model and the ideal Bose gas model—to see how scaling is actually realized. A description of various experiments designed to test scaling theory, both in the case of films with a free surface and in the case of complete confinement is given. These experiments to other studies of confined helium in powders and porous glasses are compared. With liquid helium, it is easy to realize a situation of a homogeneous film consisting of a fraction of an atomic layer to a few atomic layers. Such a film becomes superfluid at a temperature well below T λ and displays critical behavior quite different from the bulk system. The heat capacity has a power-law behavior in three dimensions with an exponent close to zero. In two dimensions, the heat capacity is expected to have a broad maximum at a temperature above the transition into the superfluid state.

We report new data for 4 He confined between two silicon wafers spaced 9869 A apart. This spacing complements a series of previous measurements which now span a factor of 20 between the smallest and largest confinements. These new data allow us to further check scaling predictions. We find, as reported with previous data, that the present data scale well except near the heat capacity maximum, and below into the superfluid region.

We have previously described a resonance involving the adiabatic motion of 4He from a superleak into a reservoir. We consider here the case of 3He–4He mixtures and show some preliminary results for ρs for confinement to a film with thickness L=0.0483 μm . We relate this to the behavior expected from correlation-length scaling.

Finite-size scaling theory predicts that uniformly small critical systems that have the same dimensionality and belong to the same universality class will scale as a function of the ratio of the spatial length L to the correlation length ζ. This should occur for all temperatures within the critical region. Measurements of the heat capacity of liquid 4He confined to a two-dimensional (2D) planar geometry agree well with this prediction when the 4He is normal but disagree near the specific heat maximum where the confined 4He becomes superfluid. Data for 4He confined to 1D structures show a similar behavior (however the lack of data collapse is not as dramatic). Recent measurements of the heat capacity from two 0D confinements, which differ by a factor of two in size, fail to scale at any temperature within the critical region. This lack of scaling may be due to the interaction of neighboring boxes through the shallow channels used to fill them. This is quite surprising since the liquid in the channels is not superfluid at the temperatures of interest for the helium in the boxes. Furthermore, measurements of the superfluid density of the helium within the channels reveal a critical temperature that is higher than expected suggesting that the normal fluid is affected by the already superfluid regions at each end of these channels. Both of these anomalies might be explained by a proximity effect analogous to what is seen when normal metals are sandwiched between two superconductors.

We report measurements of the specific heat of 3He-4He mixtures near the superfluid transition when confined to channels of 1 /tm square cross section. These data test the universality of finite-size scaling as function of 3He concentration for 1D crossover. The analysis of these data requires that data measured at fixed concentration be converted to a specific heat at constant chemical potential difference = μ3 - μ4. This is carried out according to a procedure performed for planar mixtures by Kimball and Gasparini. We find that, in the most self-consistent analysis of the data, the mixtures define a separate scaling locus from that of pure 4He, both above and below Tλ. An analysis whereby the exponent a is forced to have the same universal value—as opposed to the best-fit value—yields a good collapse of all the data. This is achieved, however, at a cost of self-consistency. These results mirror very closely those obtained for finite-size scaling of confined planar mixtures, i.e. for 2D crossover.

4He at the superfluid transition has been well studied and has yielded valuable checks on aspects of critical behavior ranging from scaling, universality and explicit predictions of critical exponents and amplitude ratios. The most detailed results have been obtained in the thermodynamic limit, i.e., in situations where the smallest confining dimension for the helium sample is much larger than the magnitude of the critical correlation length, ξ. Since this length diverges at the transition, there are many experimental realizations, such as helium films or helium confined in pores, where the effect of confinement to dimensions comparable to ξ manifests itself as a modification of the properties realized in the thermodynamic limit, or as completely new behavior associated with a lower spacial dimension.

We have successfully achieved a uniform sub-μm gap between two flat silicon surfaces. This is done using two 2″ diameter, 0.010″ thick silicon wafers, one of which has a small central hole used later for experiments with liquid4He. The other wafer has a SiO2 pattern made lithographically. The two wafers are bonded together using direct wafer bonding and the uniformity of spacing is found to be better than 1% from IR interference experiments.