Planar Aerogel and Superfluid ³He, Structure and Transitions

As a p-wave superfluid, ³He is subject to pairbreaking via scattering from surfaces and impurity sites[1, 2]. As a consequence, the orbital component of the ³He superfluid order parameter couples strongly to anisotropic scattering[3]. Dilute impurities in the form of aerogel, grown with a structural anisotropy, orient and alter the order parameter. Anisotropy has been previously shown to strongly influence the relative stability of the superfluid phases, with compressed aerogel favoring the B phase[4] with a nematic-like structure, and stretched aerogel favoring the A phase[5] with a planar-like structure[6]. The more extreme case of nematic Nafen aerogel has likewise been shown to stabilize the polar phase[7]. When imbibed in anisotropic silica aerogel an orbital flop transition has been observed[8], where the axis of the orbital angular momentum $\bm{\ell}$ spontaneously reorients.

The stretched aerogel is grown intrinsically strained, with the strain characterized by the ratio of the diameter of the sample after supercritical methanol extraction, compared to the inner diameter of the tube in which it was grown[9]. The strain axis of the aerogel sample, $\bm{\epsilon}$ was measured via optical birefringence, and found to coincide with the cylindrical axis. The $16\%$ stretched aerogel sample is situated in a Stycast epoxy fixture attached to the silver heat exchanger by a fill line, and is surrounded by a quantity of pure, bulk ³He.

The identity and orientation of the superfluid phase of ³He can be established from the NMR resonance frequency and its dependence on tipping-angle caused by the nuclear dipole-dipole interaction[10], providing information on the structure and orientation of the superfluid order parameter[11]. Additionally, the total spectral weight of the NMR spectrum is proportional to the susceptibility, measured as a function of temperature. In these experiments, temperature sweeps were performed by delivering intermittent NMR pulses as the sample warmed. The temperature was monitored with a ³He melting curve thermometer and ${}^{195}\mathrm{Pt}$ NMR thermometry.

The measurements of spectra and susceptibility for $16\%$ stretched aerogel in Fig. 1 a, c were performed for a range of pressures and fields from $74.5\,\mathrm{mT}$ to $195\,\mathrm{mT}$ and from $19.0$ to $26.55\,\mathrm{bar}$[12]. The sample gas had a small amount of ⁴He added such that the surface of the aerogel was coated with $\approx 3.5$ atomic layers (commonly referred to as pre-plating), suppressing the formation of paramagnetic solid ³He[13, 14]. The sample was oriented such that $\bm{\epsilon}$ was aligned with the static magnetic field $\bm{H_{0}}$. We compare with measurements from a $19\%$ compressed aerogel Fig. 1 b, d, performed and analyzed in the same way[8]. It is significant that this aerogel has the opposite direction of strain and consequently a nematic structure as compared with the planar structure of the $16\%$ stretched sample.

In Fig. 1, NMR spectra for small tipping angles from both the (a, c) $16\%$ stretched and (b, d) $19\%$ compressed aerogels are compared. The upper panels Fig. 1 a, b display the NMR spectra, while the lower panels c, d display the integrated total spectral weight of these spectra, normalized to just above the pure superfluid transition temperature $T_{c0}$. Each series of NMR spectra display five transition temperatures - two correspond to the transitions in the pure superfluid surrounding the aerogel sample. At $T_{c0}$, the pure liquid becomes the superfluid A phase and is seen to have a positive frequency shift separating it from the spectral component of the liquid ³He in the aerogel, which is still in the normal state. At $T_{ab0}$ the pure liquid enters into the B phase and the satellites disappear and contribute minimally to the overall spectral weight. The commensurate decrease in susceptibility between both samples at $T_{ab0}$ reflects the similar geometry of both sample arrangements.

The fraction of the superfluid imbibed in the aerogel constitutes the majority of the spectral weight of the NMR response. Below the aerogel superfluid transition temperature $T_{c}$, the superfluid imbibed in either compressed or stretched aerogels is in the A phase. In the stretched aerogel (a), the orbital angular momentum vector $\bm{\ell}\parallel\bm{\epsilon}\parallel\bm{H_{0}}$, producing a negative frequency shift comparable to what is expected in a slab confined system[15, 16, 17]. In contrast, in the compressed aerogel (b) the orbital angular momentum is 2D-disordered with $\bm{\ell}\bot\bm{\epsilon}\parallel\bm{H_{0}}$, producing a small positive frequency shift[18].

Below the aerogel A to B transition temperature, denoted $T_{ab}$, both samples transition to the B phase. This transition preserves the orientation of the orbital angular momentum relative to the axis of anisotropy. As a consequence in the stretched aerogel, with $\bm{\ell}\parallel\bm{\epsilon}$, the spectrum is effectively unshifted while in the compressed aerogel with $\bm{\ell}\bot\bm{\epsilon}$ there is a large positive frequency shift.

At the orbital-flop temperature $T_{x}$ the orbital quantization axis $\bm{\ell}$ reorients by $90^{\circ}$ relative to the strain axis $\bm{\epsilon}$. As a consequence the different response of the NMR frequency shift to the structure of the aerogel is swapped between the compressed and stretched aerogel, with the compressed aerogel now effectively unshifted while the stretched aerogel experiences a strong positive frequency shift. Although the orbital flop transition has been previously identified in stretched aerogel samples in the A phase without preplating[5, 19, 6], it is noteworthy that in $16\%$ stretched aerogel it also occurs in the B phase in an orientation-temperature order exactly reversed from the similar transition occurring in the B phase in compressed aerogel.

The susceptibilities of the superfluid B phase imbibed in the two aerogel samples are substantially different, as seen in the lower panels of Fig. 1 c and d. While the susceptibility of the compressed aerogel B phase displays a gradual decrease with temperature commensurate with the growth of the suppressed order parameter[20, 21], the susceptibility of the ³He in the stretched aerogel remains effectively temperature independent to the lowest temperature, despite the growth of the of the zero-spin projection order parameter amplitude. As a consequence the enhanced susceptibility of planar aerogel is incompatible with both gap-distortion effects and isotropic impurity models[12]. This is also one of the key results of the present work.

Simulations of anisotropic silica aerogels performed with an anisotropically biased diffusion-limited cluster aggregation algorithm show that plane-like structures develop with unidirectionally biased diffusion[6]. The structure factor of such aerogels $S(\bm{q})$, for wavevectors $\bm{q}$, produces an image with similar features as the experimental x-ray scattering cross-section for a stretched aerogel. In these images, more intense scattering at small $\bm{q_{\parallel}}$ in the direction of the axis of anisotropy $\bm{\epsilon}\parallel\bm{q_{\parallel}}$, oriented towards the top of the page, appears as a double-lobed dipole pattern near the center of the image. The structure factor also shows an ellipsoidal pattern, rotated $90\deg$ from the dipole, extending further out in $\bm{q_{\bot}}$. There is a crossover between the dipolar and ellipsoidal regime at an intermediate length of roughly $50\,\mathrm{nm}$. The comparison between the simulated structure factor and the small-angle x-ray scattering validates the simulation from which the structure factor was calculated. Real-space images of the simulated stretched aerogels show the planar-like structures, while compressed aerogels have a nematic-like structure, for details see Ref. [6]. The latter have structures that orient the orbital component of the superfluid order parameter exactly opposite to that of stretched case. This opposite orientation is displayed in the data in Fig. 1, which compares the frequency shift and susceptibility of $16\%$ stretched aerogel[12] to $19\%$ compressed aerogel [8] at comparable pressures and fields.

Calculations of the properties of ³He confined to a planar slab show a strong enhancement of the local magnetic susceptibility due to surface Andreev bound states[23, 24, 25]. In the highest degrees of confinement, it is calculated that the susceptibility averaged over the slab can recover the magnetic susceptibility of the normal liquid[24], although at such confinements the B phase is no longer stable[26]. In the simulated aerogels, the planar structures are similarly closely spaced with a peak in the planar density modulations on the order of $50-100\,\mathrm{nm}$. This length is comparable to the size of the slab necessary to fully recover the normal state susceptibility. As a consequence, the likely cause of the near-constant susceptibility in stretched aerogel is similar bound states formed by planar structures. In contrast to both slab confinement and prior experiments performed in comparably stretched aerogel samples[5, 27], the superfluid displays a large region of stable B phase, Fig. 3. The porosity of the aerogel structure versus the non-porous slabs likewise may play a role in the stability of the B phase at such confinement.

The crossover in the structure seen in the simulations has been identified as being directly responsible for the orbital-flop, a reorientation of the angular momentum axis $\bm{\ell}$, of the superfluid order parameter[6], as studied in compressed aerogel[8]. This transition is driven purely by the structure and, as expected for a transition in the orbital degrees of freedom, does not couple to the magnetic field. This is seen in Fig. 4 and has also been reported in compressed aerogel. Consistently, no change in the magnetic susceptibility would be expected across this magnetic field-independent orbital-flop transition.

In contrast, varying the magnetic field applied to the sample shows suppression of the $B$ to $A$ transition temperature, $T_{ab}$, with increasing field as shown in Fig. 4. For the typically first-order $B$ to $A$ transition, the Clausius-Clapeyron relation requires

For approximately constant susceptibility in a $16\%$ stretched aerogel this transition must be close to isentropic, approximating a second order transition. Similar to the enhanced susceptibility discussed above, this is analogous to the behavior expected in strongly planar-confined ³He. Under this confinement the transition is predicted to be either second-order or weakly first-order, dependent on confinement, magnetic field, and material parameters[30, 31, 25]. Imbibing ³He in aerogel alters the effective material parameters of the superfluid[2, 32, 22, 33], potentially further approaching a second-order $AB$ transition. Additionally, it is noteworthy that the slope of this transition temperature dependence on the square of the magnetic field is very similar to pure ³He, shown in Fig. 4.

Computational studies in conjunction with x-ray scattering experiments have revealed information about the microstructure of anisotropic aerogel. In particular, aerogel with stretched anisotropy has been shown to contain large plane-like structures. The thermodynamics of the A to B transition of superfluid ³He imbibed in such a planar aerogel are qualitatively similar to theoretical expectations for the superfluid transition when imbibed in a thin planar slab. This is manifest both as an enhancement to the magnetic susceptibility and a near-isentropic A to B transition. The structure of the aerogel exhibits a crossover in its preferred direction at intermediate lengths, driving a reorientation transition in the orbital angular momentum of the superfluid when this crossover length is commensurate with the superfluid coherence length. The character of these transitions is a consequence of the anisotropic structure of the aerogel, underscoring the novel physics present in impure unconventional superfluids. We would like to thank T. Mizushima, J.A. Sauls, and A.B. Vorontsov for helpful discussions. This work was supported by NSF DMR-2210112.