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Universal Behaviour of the Superfluid Fraction and Tc of He-3 in 99.5% Open Aerogel

We have investigated the superfluid transition of He-3 in a 99.5% porosity silica aerogel. This very dilute sample shows behaviour intermediary between bulk He-3 and He-3 confined to the denser aerogels previously studied. We present data on both the superfluid transition temperature and the superfluid density and compare our results with previous measurements. Finally, we show that the suppression of the superfluid transition temperature and suppression of the superfluid density of He-3 in aerogel follow a universal relation for a range of aerogel samples.

The properties of bulk 3 He are well understood. The extreme purity of 3 He at low temperatures makes it an ideal system to study the agreement between theoretical and experimental results on non-conventional Cooper pairing in the absence of disorder. Disorder plays a crucial role in suppressing the pairing interaction in high T c superconductors, the other well established non swave paired system. The superfluid transition of 3 He confined to a sample of very porous silica aerogel was first reported four years ago [1,2]. The aerogel provides a structural disorder background [4] to the liquid. 3 He is compressible, and the density can be continuously tuned by ≈30% while maintaining a fixed disorder. The 3 He zero temperature coherence length ξ 0 , defined as ξ 0 =hv f /k B T c , varies from 180Å to over 700Å as a function of density. Because the Cooper pairs in 3 He form in a p-wave state, quasiparticle scattering from the aerogel strands is pairbreaking [3]. Thus the 3 He-in-aerogel system is well suited to the exploration of the effect of impurity scattering and disorder on the superfluid transition and phase diagram.
The superfluidity of 3 He in silica aerogel has been studied using torsional oscillators [1,[4][5][6], NMR [2,[6][7][8][9] and sound propagation [10,11] techniques. These measurements show that both the superfluid transition temperature (T c ) and superfluid density (ρ s ) of the 3 He are suppressed by the disorder, but that the transition remains sharp [1]. This suppression is sensitive to both the density and the microstructure of the aerogel sample.
The simplest model for the effect of impurity scattering on the 3 He superfluid transition is the homogeneous scattering model (HSM) which is based on the Abrikosov-Gorkov model for a superconductor with magnetic impurities that induce pair-breaking via spinflip scattering [12]. This mechanism is similar to that of diffuse scattering of Cooper paired 3 He from a surface [13], and is unable to explain the observed behaviour. Specifically, the observed suppression of the superfluid density is much greater than predicted by this model. More sophisticated models, such as the isotropic inhomogeneous scattering model (IISM) [14][15][16] proposed by Thuneberg and coworkers are able to quantitatively predict the superfluid transition temperature of 3 He in aerogel (for small suppressions) and have had success at qualitatively explain-ing the observed superfluid densities.
In this Letter we present data from several different experiments on 3 He in aerogel, including new results on 3 He confined to a 99.5% porosity sample. This sample is a factor of four more dilute than any previously investigated [17] and is crucial for understanding the evolution from bulk 3 He to a regime where impurity scattering dominates. In comparing these different samples we find evidence that the relation between superfluid density and the superfluid transition temperature of 3 He in aerogel follows a universal behaviour, independent of the aerogel sample. This is significant because both of these quantities are individually sensitive to the microstructure of the aerogel, and vary greatly from sample to sample. We also present evidence that the suppression of T c can be related to the correlation length (ξ a ) of the aerogel sample.
The aerogels used in the experiments discussed in this Letter were grown under basic conditions [18][19][20][21]. Under these conditions gelation is the result of diffusion limited aggregation of small (≈30Å diameter) primary silica particles. The aerogels are characterized by a fractal dimension (D f ) related to the real space correlations, and a long length scale cutoff to these correlations (ξ a ) above which the sample appears homogeneous. The fractal exponent depends only on the gelation process, while the cutoff length is also dependent on the average density. Simulations based on the diffusion-limited clustercluster aggregation (DLCA) algorithm predict that the fractal exponent should lie between 1.7 and 1.9, which is in good agreement with small-angle X-ray scattering (SAXS) measurements (Table I). We note that ξ a in the most dilute sample, D, could not be inferred from the data as the SAXS did not extend to sufficiently small q. Samples A and C have the same density, but were made under different conditions and have slightly different fractal dimension and significantly different ξ a . Both samples A and C have been studied with SAXS. Sample B was made under conditions very similar to sample C. We do not have direct information on its microstructure from SAXS, but assume here that samples B and C are essentially identical. The correlation length for sample D is within the range obtained from simulations based on the DLCA algorithm and is consistent with the SAXS data of Mulders on a 99.6% sample (ξ a ≥2000Å). For a more extensive discussion the reader is referred to reference [4].
The 99.5% porosity aerogel used for our experiment was grown inside the (on average) 100 µm large pores of a coarse silver sinter. Previous torsional oscillator experiments have been affected by the presence of spurious resonances resulting from composite modes of 3 He and aerogel whose frequency crosses the resonant frequency of the cell [1,6]. The strength of these resonances grow as the porosity of the aerogel sample increases, affecting the quality of data [1,5,6]. In our cell the aerogel is clamped to the silver sinter, and the effect of spurious resonances is strongly reduced.
We operated our torsional oscillator in self-resonant mode near 483 Hz. The temperature in the cell was measured using a lanthanum diluted cerous magnesium nitrate ac susceptibility thermometer, thermally connected to the sample through a shared reservoir of 3 He. Data was collected while the temperature increased at a rate of 20 µK per hour. The period shift of the oscillator as the superfluid 3 He decoupled from the torsion head provided both the transition temperature of the 3 He in aerogel and the superfluid density. The pressure is plotted versus Tc for different aerogel samples. The legend is as follows: filled squares are cell A (98% aerogel) [1], filled triangles are cell B (98% aerogel) [4], filled circles are for cell C (98% aerogel) [5] and empty circles are cell D (99.5% aerogel). The solid line is the superfluid transition curve for bulk 3 He. Figure 1 shows the superfluid transition temperature for several different aerogel samples. Three of these samples (A, B and C) have a nominal porosity of 98%, the fourth one (D) is our 99.5% aerogel. All of these measurements were done by monitoring the period shift in torsional oscillators filled with the aerogel and 3 He. The difference in transition temperature between samples A, B and C arises from differences in the microstructure of the aerogels (see table I and reference [4]). The relative suppression of the transition temperature (1-T c /T c0 ) (with T c0 the transition temperature in bulk 3 He) is larger at lower pressures (larger ξ 0 ) than at higer pressures (smaller ξ 0 ). In view of the fact that all the aerogels used in these experiments have a similar fractal dimension and primary particle size, one would expect them to be mutually self similar on length scales shorter than the fractal cutoff length. Thus for temperature dependent 3 He coherence lengths ξ(T ) shorter than ξ a , the ratio of density of silica sampled at two different ξ(T ) is independent of aerogel density. Experimentally one observes a strong dependence of the suppression of T c on the aerogel density. In figure 2 we show the dependence of the relative suppression of T c as a function of ξ 0 . At short coherence lengths, the relative suppression is small for all samples. However, when the pressure is reduced, increasing ξ 0 , this suppression shows a marked dependence on the microstructure of the aerogel. The transition temperature shows an evolution from a strong impurity scattering regime when the 3 He is confined to a 98% porosity aerogel sample towards the behaviour of bulk 3 He in the 99.5% sample.
Motivated by figure 2 we plot the relative suppression of T c against ξ 0 scaled by the aerogel correlation length ξ a . The aerogel-limited mean free path (l g ) would be another natural choice to compare against ξ 0 . However, this length scale has not been independently measured. If we use the HSM to determine l g from T c /T c0 , the values of l g show a strong pressure dependence [22]. Figure 3 shows that the scaled transition temperature depends solely on the ratio of ξ 0 to ξ a independent of the aerogel density. The error bars for sample D result from the high and low estimates (2000Å and 3000Å respectively) for the correlation length for a 99.5% porosity sample from DLCA simulations. We determined the temperature-dependent superfluid density of 3 He in 99.5% aerogel using the shift in resonant frequency of our torsional oscillator upon warming. A small, temperature dependent contribution due to bulk 3 He in the cell was subtracted. The remaining shift was scaled by the period change due to filling the cell with 3 He at 50 mK and by the tortuosity (measured with 4 He) to obtain the superfluid density. Figure 4 shows the bare superfluid density (ρ b s /ρ) plotted versus the reduced temperature T/T c for different aerogel samples and pressures. The bare superfluid density is obtained from ρ s by stripping away the Fermi liquid factor [23] according to: and is equivalent to 1-Y(T) where Y(T) is the temperature dependent Yosida function for bulk 3 He. We plot the bare superfluid density versus reduced temperature for cell A (98%), filled symbols, and cell D (99.5%), empty symbols.
As with the transition temperature T c , the ρ b s for these two different samples are similar at high pressures and are both close to the bulk value. At the lowest pressure, there is a factor of five difference in ρ b s /ρ at the same reduced temperature between the 99.5% sample and the 98% sample; ρ b s /ρ is more strongly suppressed by the aerogel than T c . There is also a large difference between the suppression factors of T c and ρ b s in the same aerogel sample. This large suppression in ρ s with respect to T c is consistent with measurements made on other samples [1,4], and cannot be explained with homogenous scattering models for 3 He in aerogel [14]. In order to better understand the superfluid transition of 3 He in aerogel, in figure 5 we plot ρ b s /ρ s at 0.8T c against (T c /T c0 ) 2 for several aerogel samples. The error bars shown for cell B arise from a spurious resonance that introduces uncertainty in the determination of ρ s . As in figure 3, the data collapses on a universal curve. The dashed line is the prediction for a homogeneously scattering model based on the Abrikosov-Gorkov equation [14]. This plot compares aerogels with different densities-there is a factor of four difference in the average impurity density between the 98% samples and the 99.5% aerogel sample. Furthermore, the coherence length of the Cooper pairs varies from 180Å to 600Å over this data, yet the very strong pressure dependence shown in figures 2 and 4 has been factored out in this plot.
The correlations in the aerogel will affect the suppression of T c and the evolution of ρ b s relative to a homogeous disorder, but this suppression apparently depends only on the correlation length of the sample and possibly the fractal exponent. The steep slope of the data in figure 5 is evidence that the fractal nature of the aerogel plays an important role in the development of ρ s [25]. Since all of these samples were base-catalyzed, the behaviour of 3 He in each aerogel is determined mainly by ξ a . As long as ξ 0 is much less than ξ a , the disorder sampled by the ensemble of Cooper pairs should be insensitive to changes in the temperature-dependent coherence length ξ(T), until ξ(T)≈ ξ a . That is, the system has a conformal symmetry normally absent in disordered systems. The evidence of this one-parameter scaling is displayed in figure 5; ρ b s is a function of T c /T c0 (or equivalently ξ 0 /ξ a ) only. This behaviour is reminiscent of the compilation of data from disordered high T c materials by Franz et al [26] with the exception that the relative suppression of T c /T c0 and ρ s /ρ s0 in the high T c materials do not follow a universal behaviour, presumably because the impurities are not fractally correlated.
In order to compare the behaviour of T c and ρ b s of 3 He in aerogel with 3 He in bulk it will be necessary to understand precisely how the fractal disorder affects the superfluid pairing mechanism. One test for models of nonconventional Cooper pairing in the presence of disorder would be to predict the functional form of the universal curve for ρ b s versus (T c ) 2 for 3 He in (base-catalyzed) aerogel. The IISM of Thuneberg et al. predicts a relation between ρ b s /ρ b s0 and T c /T c0 similar to the trend illustrated in figure 5, showing behaviour very different than Abrikosov-Gorkov model [14,15]. This model does not explicitly consider the fractal nature of the aerogel, but shows how inhomogeneities in the disorder can lead to a large suppression of ρ b s relative to T c . In this letter we have presented data from our measurements on 3 He in a very dilute 99.5% porous silica aerogel. The values of T c and ρ b s fall between those of 3 He in bulk and 3 He in denser aerogel samples. We also present T c and ρ b s data for 3 He in aerogel experiments performed at Cornell that show universal behaviour that can be traced to the fractal structure of the aerogel. In order to more fully understand this exciting physical system, more attention must be devoted to understanding the microstructure of the aerogel. Specifically, the universal scaling discussed above depends strongly on the fact that the fractal exponent real-space correlations is similar for all the aerogel samples. It would be interesting to study neutrally catalyzed silica aerogels, which have a different fractal exponent than the base-catalyzed samples [19]. 3 He would be expected to follow different universal behaviour when confined to base-catalyzed and neutrally catalyzed aerogels. As yet, there is no clear theoretical picture for why the transition temperature T c should depend on the ratio ξ 0 /ξ a . The explanation for this behaviour will provide insight into how correlated disorder affects non-conventional Cooper pairing.
We would like to acknowledge helpful conversations with T.L. Ho, S. Yip, E. Thuneberg, J. Beamish, A. Golov and M.H.W. Chan. This research was supported by the NSF under DMR-9705295. *