Superfluidity in thin 4He films adsorbed on graphite

PHYSICA ELSEVIER

Physica B 197 (1994) 269-277

Superfluidity in thin

4He films adsorbed on graphite

P.A. Crowell, J.D. Reppy* Laboratory of Atomic and Solid State Physics and Materials Science Center, Cornell University, Ithaca, N Y 14853-2501, USA

Abstract

We have used the torsional oscillator technique to study superfluidity in 4He films adsorbed on the basal plane of graphite for coverages between zero and seven atomic layers. The development of superfiuidity in this system is significantly different from that observed on disordered substrates such as Mylar and porous glasses. The most unusual phenomenon is a reentrant superfluid phase in the second layer of adsorbed helium. Superfluidity appears at a coverage just above one and one-half layers and then disappears before the completion of the second layer. We observe superfluidity again just above second layer completion, but its evolution with increasing coverage continues to be unusual. Plateaus in the superfluid transition temperature and superfluid signal occur over parts of the third and fourth layers. A periodic structure appears in the measured superfluid mass through the completion of the seventh layer.

1. Introduction

One of the important questions concerning superfluid 4He films on surfaces is the role of disorder. Given the level of interest in this problem, [1] it is surprising that very little is known about thin-film superfluidity in ordered systems. Almost all experimental studies of superfluidity in 4He films have used strongly disordered substrates such as Mylar or porous glasses [2,3]. There have been several studies of superfluidity in 4He films adsorbed on graphite [4,5,6,7] but these have been confined to films thicker than 3.5 layers and with transition temperatures greater than 700mK. The effects of the ordered graphite adsorption potential are much stronger at lower coverages, where heat capacity measurements have revealed a variety of structural phases not seen in 4He films adsorbed on disordered substrates [8,9,10]. * Corresponding author.

In this paper we report on torsional oscillator studies of superfluidity in 4He films adsorbed on graphite for coverages between zero and seven atomic layers. The development of superfluidity in this system is distinct from that seen for 4He films adsorbed on disordered substrates. Structural phase transitions in the film significantly affect the superfluidity observed in the second, third and fourth layers of adsorbed helium. The most dramatic evidence of this relationship is the suppression of superfluidity in the second layer by the solidification of the film. This review will discuss our results in the context of the large body of work on structural phase transitions in 4He films adsorbed on graphite. A short report of our results was published previously [11]. 2. Background

4He and 3He films adsorbed on graphite have proven to be ideal systems for the study of phase transitions in two dimensions. The phase dia-

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P.A. Crowell, J.D. Reppy / Physica B 197 (1994) 269-277

270

gram of submonolayer films for both isotopes was mapped out in an elegant series of experiments in the early 1970's [8]. In Fig. 1, we show the phase diagram for submonolayer 4He based on the results of these experiments [12,13]. The dominant feature of this phase diagram is the commensurate solid phase, labeled C, in which the 4He atoms form a triangular lattice in registry with the underlying graphite honeycomb lattice. At low densities, there is a two-phase region, labeled C + G, at low temperatures. The conventional view [13] is that the two coexisting phases are the commensurate solid phase and a two-dimensional (2D) surface gas. At high densities, near layer completion,, the system forms an incommensurate solid (IC). This hexagonal close-packed solid forms a crystalline 'pseudosubstrate' for the second layer of adsorbed helium. The other features of this diagram will not be considered here. Given this rich phase diagram, it is not surprising that the 4He-graphite system continues to be an object of experimental and theoretical attention. One point of interest is the phase diagram lO

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of higher layers. This has been addressed in heat capacity studies by Polanco and Bretz [9] and more recently by Greywall and Busch [10]. Both of these studies found evidence for a gas-liquid coexistence region in the second layer at low densities. Greywall and Busch (GB) also explored the second layer at higher coverages. Their data suggest that the second layer forms a commensurate solid at intermediate densities and that an incommensurate solid forms in the vicinity of layer completion. In the third layer, GB find evidence for another gas-liquid coexistence region at low densities. The data are less conclusive at higher third-layer densities. They observe no clear melting peak like that seen for the first- and second-layer solids. We will be considering the second- and third-layer phase diagrams in more detail below. Although there are many open questions concerning the growth of 4He films on graphite, this system clearly affords the opportunity to study superfluidity under conditions that are entirely different from those found on other substrates. The underlying substrate potential is ordered and each of the first two layers is a crystalline solid upon completion. Superfluidity in either the second or third layer would thus occur under ideal circumstances: a single layer of 4He atoms on an ordered 4He substrate. Furthermore, it has been shown that 4He films on graphite continue to grow in a layer-by-layer fashion through at least seven layers [6]. Third sound measurements have established that the layered structure has a significant effect on the superfluid properties of films thicker than 3½ layers [5,6]. The effects of layering should be even more pronounced in thinner films. Perhaps the most interesting possibility is that of superfluidity in the submonolayer regime. The phase diagram shown in Fig. 1 suggests that this is unlikely since a solid forms at the lowest temperatures over the entire first layer. Greywall and Busch, however, have proposed an alternative phase diagram in which the condensed phase at low densities is a liquid instead of a solid [10]. This is essentially the same interpretation as that accepted for the low-density regions of the second and third layers. They go on

P.A. Crowell, J.D. Reppy / Physica B 197 (1994) 269-277

arguing that the condensed first-layer liquid is superfluid at low temperatures and that the heat capacity peak traditionally associated with condensation is in fact a signature of a KosterlitzThouless transition in the liquid [14]. They propose a similar scenario for the second and third layers. The proposed superfluid would coexist with a 2D surface gas in all three cases.

3. Experimental details A strong motivation exists for undertaking a study of superfluidity in 4He films adsorbed on graphite for coverages below three atomic layers. In pursuing this goal, we have had to address some of the limitations encountered in previous studies of superfluidity in this system. Thermal transport, which has been used to study thicker films [4], is ineffective at low coverages and temperatures because of the depletion of the vapor phase. The attenuation of third sound has prevented its use for studying films thinner than 31 layers. [6] Even the torsional oscillator technique [15], which has proven to be the most effective tool for studying extremely thin films, encounters some obstacles in the case of exfoliated graphite substrates such as Grafoil [16]. Takano and Reppy [17] found that the superfluid signal in a torsional oscillator study of 4He on Grafoil was about 1% of that expected for an ideal substrate. This drop is the cost of exfoliation, which leaves a large number of crystallites separated by steps and voids. The poor connectivity of the substrate reduces the superfluid signal. In spite of these difficulties, Takano and Reppy were able to resolve a superfluid transition at 0.7 K in a film with a total thickness of 31 layers. This suggested that with some improvements, torsional oscillator studies could be extended to lower coverages. We have fabricated two torsional oscillators for our experiments. The first contains 0.11 grams of UCAR graphite foam [16] and the second 16 disks of Grafoil, 12.2mm in diameter and 0.25mm thick. Graphite foam and Grafoil are both exfoliated basal-plane graphite substrates. The surface crys-

271

tallites have a characteristic size of order 900/~ and 150 ~ for foam and Grafoil respectively [18]. The substrate surface areas of the two cells are 2.33 ___0.7 m 2 (foam) and 12.8 -+ 0.3 m 2 (Grafoil). The Grafoil cell has been particularly satisfactory and was used for most of the measurements discussed below. A Kapton diaphragm strain gauge [19] was connected to each cell, permitting vapor pressure measurements to be made in situ. In each case, a 4He vapor pressure isotherm at 0.9 K showed layer-by-layer growth through six layers, indicating a high-quality surface. We have conducted several types of measurements. The bulk of the data discussed below were obtained with the Grafoil cell by incrementing the temperature in steps while recording the resonant period and amplitude of the oscillator. The period stability of the foam cell was poor, so we used it only fi3r dissipation measurements. These were taken while the temperature drifted slowly through the superfluid transition. For both types of measurements, 4He samples were added to the cell from a calibrated dosing volume and annealed at temperatures varying from 10 K for the first layer down to 0.9 K for coverages above 3 layers. Each coverage was then cooled slowly to low temperature. The first step in analyzing the resonant period data was to subtract the background period from the data. A baseline was then established by the period measured above the superftuid transition. The data were subtracted from this baseline, giving the superftuid period shift AP(T), which is proportional to the superfluid mass. The low temperature period shift AP(0) was determined by evaluating AP(T) at the lowest temperature point, typically 20 mK. We chose this definition of AP(0) because it was not possible to extrapolate the period to T = 0 for some coverages. At constant drive, the dissipation Q-~ is inversely proportional to the amplitude of the oscillator. Several irregular features in the background dissipation made a subtraction impractical. The only quantitative information derived from the dissipation data was the temperature Tpeak of the dissipation peak at the superfluid transition. We were able to resolve a dissipation

P.A. Crowell, J.D. Reppy / Physica B 197 (1994) 269-277

272

peak in the Grafoil data for coverages above 22 a t o m s / n m 2 and in the foam data for coverages above 2 6 a t o m s / n m z. The peaks were located using spline fits of the dissipation data.

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One exceptional feature of the data in Fig. 2 is the peak in AP(0) between 17 and 19atoms/ 4.0

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nm z. The temperature dependence of the superfluid signal in this coverage regime is also very unusual as can be seen in the solid circles of Fig. 3. The period shift AP(T) does not approach a constant value as T - * 0 as would be expected for a conventional superfluid. Furthermore, these data do not suggest a well-defined transition temperature. A somewhat more meaningful picture emerges if the period shift is plotted on a logarithmic temperature scale as in Fig. 4. This figure shows a linear relationship between Ap and T over about one order of magnitude in temperature. Although it is still not clear how to assign a superfluid transition temperature, a fairly sharp corner near 400 mK appears in Fig. 4 for one coverage (18.1 atoms/nm2). Unfortunately, most of the data are contaminated by the anomaly at 300 mK. It is clear, however, that the

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temperature at which the signal vanishes is not simply proportional to AP(0). What is the origin of this reentrant superfluid behavior? One indication comes from a comparison with the heat capacity measurements of Greywall [10]. In Fig. 5 we have superimposed our AP(0) data on a map of the heat capacity peaks found by Greywall in his study of the second layer. The maximum in AP(0) occurs at 18.4atoms/nm 2 and superfluidity disappears at 19.1 atoms/nm 2. This coincides roughly with the ir

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273

coverage at which two of the lines of heat capacity peaks end in Fig. 5. The lower branch of peaks marks the boundary between a gasliquid coexistence region at low temperature and a high-temperature fluid phase. The line of peaks near 1.5 K is taken by Greywall to be the melting line separating a commensurate solid phase from the fluid phase. The disappearance of superfluidity thus appears to be associated with the onset of solidification. The interpretation would be simpler if the maximum in AP(0) coincided exactly with the onset of solidification. The discrepancy in Fig. 5 is about 5% of the total coverage. This is consistent with the uncertainties in the surface areas of the two samples. The unusual temperature dependence shown in Fig. 4 is a particular challenge to interpret. Weak links in the superfluid are one possible source of the observed behavior. Given the existing heat capacity data, it appears likely that the superfluid coexists with a 2D surface gas below 18.4atoms/nm 2 and with a 2D solid at higher coverages. The superfluid might be condensed in patches in a background of gas or solid with only a few superfluid bridges connecting the patches. It is also possible that the weak links are caused by defects in the substrate. The superfluid may condense preferentially at corners or step edges, forming a network of superfluid strands. Whatever their origin, we hypothesize that the effective conductivity of the weak links increases with decreasing temperature, leading to the temperature dependence seen in Fig. 4. The conductivity should saturate at lower temperatures. Measurements at lower temperatures will be required to test this model.

6. Third layer Superfluidity reappears just above secondlayer completion, which occurs at 20.4atoms/ nm 2. A dissipation peak at the superfluid transition can be resolved for coverages above 22 atoms/nm 2. As shown in Fig. 2, the size of the superfluid signal AP(0) continues to increase between 22 and 26 atoms/nm 2 but Tp~ak remains fixed at 150 ___30 mK. The superfluid transitions

274

P.A. Crowell, J.D. Reppy / Physica B 197 (1994) 269-277

are very broad in this coverage regime as can be seen in Fig. 3. Both AP(0) and Tpeak increase rapidly above 26 a t o m s / n m 2 while the transition sharpens dramatically. At third layer completion, 28.0 a t o m s / n m 2, the period shift and dissipation have assumed the Kosterlitz-Thouless [14] form. The unusual development of superfluidity in the third layer is likely to be associated with the phase-separation of the film at low coverages. Greywall's heat capacity data [10] indicate that the second layer is solid at completion, so that the third layer starts to fill over a well-ordered solid 'pseudo-substrate.' The third layer is thus a particularly ideal realization of a 2D film since the substrate corrugation is reduced by the underlying solid layers. Calculations by Whitlock et al. [20] and more recently by Clements et al. [21] indicate that a 2D 4He film will be condensed into liquid clusters at T = 0 for densities n below 4 a t o m s / n m 2. As T increases, the clusters coexist with a 2D surface gas until they evaporate at some temperature Tc(n) and the system becomes a uniform fluid. At densities above 4 a t o m s / n m 2, these calculations find that the uniform fluid phase is stable at T = 0. A heat capacity peak that might be associated with the boundary of the liquid-gas coexistence region appears in the data of Greywall at about 0.8 K for coverages between 22 and 2 7 a t o m s / n m e. [101 Dash [22] has discussed the onset of superfluidity inside a gas-liquid coexistence region. A case representative of the third layer of 4He on graphite is shown in Fig. 6. A line of superfluid transitions, which we take to be the KosterlitzT h o u l e s s - N e l s o n ( K T N ) line [23], is shown crossing the coexistence region. Outside the coexistence region, the observed superfluid transition occurs along the KTN line. Inside the coexistence region, the transition occurs at the t e m p e r a t u r e T*, where the coexistence curve crosses the K T N line. The density at the intersection point depends on the shape of the coexistence region, but it should be close to the density at T = 0 , which is expected to be 4 a t o m s / n m 2 as discussed in the previous paragraph. As noted by Dash, the liquid is con-

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P.A. Crowell, J.D. Reppy / Physica B 197 (1994) 269-277

liquid phase is condensed in narrow channels as opposed to round patches. A second point is that the transitions above 26 atoms/nm 2 are still very broad while according to Fig. 6 they should be of the KT form. A well-defined KT step in the period-shift data only appears above third layer completion. Finally, the value of T* inferred from our data is about 150mK while Fig. 6 implies that it should be approximately 700 mK. In summary, phase-separation apparently plays some role in determining the superfluid properties of the third layer, but we do not find detailed agreement with the percolation model.

275

third, undergoes phase separation at low coverages. One possible origin of the plateau is a structural change in one of the underlying layers. Lauter et al. [24] observe a reconstruction of the second layer in the vicinity of third-layer completion. The lattice constant of the second layer solid increases by about 2% implying that the density increases by at least 4%. The actual density change could be larger if the unreconstructed layer has a large population of vacancies. The increase in the density of the second layer comes at the expense of the fourth, which does not start to fill until the reconstruction is complete.

7. Fourth layer 8. Thicker films

Plateaus in both Tpeak and AP(0) appear in Fig. 2 just above third layer completion. As shown in Fig. 7, the feature in Tpeak Occurs on both Grafoil and graphite foam. The plateau is different from that observed for the third layer in that both AP(0) and Leak are nearly independent of coverage between 28 and 30 atoms/nm 2. In this respect, the data do not agree with the predictions of the percolation model discussed above, although the calculation of Clements et al. [21] predicts that the fourth layer, like the

The low temperature period shift AP(0) enters another plateau region in the vicinity of fourth layer completion as shown in Fig. 2. At this coverage, corrections for desorption are quite significant, introducing a possible systematic error into the determination of AP(0). This shortcoming is avoided by making an isothermal measurement of the period shift. In Fig. 8 we show the period shift AP measured at 20 mK and 500mK as 4He was added to the cell. The

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276

P.A. CroweR, J.D. Reppy / Physica B 197 (1994)269-277

plateau just above third layer completion appears in the 20 mK curve, while a decrease in AP occurs over the same range in the 500 mK curve. A plateau near fourth layer completion occurs in both data sets. Unlike the plateaus in AP(0) and T c at lower coverage, the onset of this feature precedes layer completion. Similar plateaus can be seen in Fig. 8 near fifth and sixth layer completion. The 20 mK measurement had to be abandoned at higher coverages due to a large heat leak. Periodic structure as a function of coverage is also seen in the third sound measurements of Zimmerli et al. [6] It is not readily apparent how to relate the torsional oscillator and third sound measurements since the latter couple to the Van der Waals constant, which is actually a function of the coverage. Both types of measurements demonstrate that the unusual superfiuid properties of the 4He-graphite system extend out to very thick films.

9. Conclusions The development of superfluidity in 4He films adsorbed on graphite is clearly different from that found in the case of disordered substrates. Many open questions, however, will have to be addressed in future experiments. One priority is making a more direct comparison between torsional oscillator and heat capacity experiments in the second and third layers. There is no great impediment to performing these measurements in the same cell, thus eliminating the need to measure the surface area to extremely high accuracy. It should also be possible to study nonexfoliated graphite substrates. Much of the loss in surface area will be compensated for by a decrease in the tortuosity factor. Finally, consideration should be given to studying superfluidity on other crystalline substrates. Two recent studies of 4He adsorbed on H i have found unusual superfiuid behavior in that system [25]. With additional work, we can look forward to a greater understanding of superfluidity in ordered systems.

Acknowledgements We have benefitted from discussions with N.W. Aschcroft, M.H.W. Chan, V. Elser, H. Godfrin, D.S. Greywall and E. Siggia. P.A.C. acknowledges the support of an AT&T Bell Laboratories PhD Scholarship. This work was supported by the National Science Foundation through Grant No. DMR-8921733 and by the MRL Program of the National Science Foundation under Award No. DMR-91-21654; MSC Report No. 7603.

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