The Heat Capacity of ³He-B in Silica Aerogel

Heat capacity measurements provide key signatures of the transition from the normal to the superfluid phase of liquid ³He. In pure ³He the magnitude of the jump in specific heat provides a direct measure of strong-coupling corrections to weak-coupling BCS theory for the superfluid phases (1), as well as direct evidence of spin-triplet pairing (2). For ³He confined in sub-micron cavities, or embedded in low-density random solids, scattering of quasiparticles inhibits pair-formation, suppresses both $T_{c}$ and the discontinuity in the heat capacity, $\Delta C/\gamma_{\text{S}}T_{c}$, both of which determine the magnitude of the superfluid order paramter in the region near the transition (3; 4). The low temperature heat capacity also reflects the spectrum of quasiparticle states. For pure bulk ³He-B the quasiparticle spectrum is gapped everywhere on the Fermi surface leading to exponential suppression of the quasiparticle contribution to the specific heat, $C_{v}\propto e^{-\Delta/k_{\text{B}}T}$ (5). But for pure bulk ³He-A, e.g. stabilized by a magnetic field, the excitation gap has nodes at two symmetry protected points on the Fermi surface, i.e. $\hat{\bf p}=\pm\hat{\bf z}$, $|\Delta({\bf p})|=\Delta\,\left(1-(\hat{\bf z}\cdot\hat{{\bf p}})^{2}\right)^{1/2}$. Thus, the spectrum is gapless at these two points on the Fermi surface, and the corresponding quasiparticle spectrum at excitation energies well below the maximum gap in the A-phase, $\epsilon\ll\Delta_{\text{0}}$, scales as $N(\epsilon)\propto N_{f}\,(\epsilon/\Delta_{\text{0}})^{2}$. This spectrum implies a low-temperature specific heat that is a power law with $C_{v}\propto(k_{\text{B}}T/\Delta_{0})^{3}$ (6).

For ³He confined in cavities, or for ³He films, boundary scattering of quasiparticles is pair-breaking and leads to the formation of sub-gap quasiparticle states that modify the the thermodynamic properties of bulk superfluid ³He (4; 7). Evidence from specific heat measurements of subgap quasiparticles (surface Andreev bound states) from surface pair breaking was reported for superfluid ³He confined in silver powder (8), and for ³He confined in a cooper cavity coated with ⁴He to provide specular scattering (9).

Sub-gap quasiparticles are also generated by scattering from the random potential when ³He is infused into low-density random solids (10). Indeed specific heat measurements for the B phase of superfluid ³He confined in high porosity silica aerogel were interpreted to imply the existence of a finite density of quasiparticle states at the Fermi level (11). In this report I present analysis of the specific heat data from Choi et al. (11) based on a free energy functional of the superfluid order parameter that incorporates pair-breaking from quasiparticle scattering off silica aerogel strands and clusters, and is valid at over the full temperature range below $T_{c}$.

The order parameter of superfluid ³He in globally isotropic 98.2% porous silica aerogel is identified as the Balian-Werthamer (BW) state, but modified by the pair-breaking effect of quasiparticle scattering off the random potential over the entire pressure range of the phase diagram (12; 13; 14). Here I report results and analysis of the specific heat of superfluid ³He-B based on a free energy functional which extends the GL theory to all temperatures for superfluid ³He embedded in an homogeneous isotropic random medium as a model for quasiparticle scattering off the random potential of silica aerogel. This functional is based on a reduction of the Luttinger-Ward functional to leading order in the expansion parameters of Fermi-liquid theory (15).

In the weak-coupling limit for spatially uniform equilibrium states the functional reduces to

and $\epsilon_{n}=(2n+1)\pi k_{\text{B}}T$ are the Fermion Matsubara energies. The random potential is parametrized by the quasiparticle-impurity scattering rate $1/\tau$, or mean-free path, $\ell=v_{f}\tau$. The sum in Eq. (1) is cutoff for $|\epsilon_{n}|\geq\varepsilon_{c}$, where $k_{\text{B}}T_{c}\ll\varepsilon_{c}\ll E_{f}$. The cutoff defining the bandwidth of attraction, $\varepsilon_{c}$, as well as the strength of the pairing interaction, $v_{1}$, in the p-wave channel can be eliminated in favor of the measured transition temperature of pure, bulk superfluid ³He via the eigenvalue of the linearized gap equation, $k_{\text{B}}T_{c_{0}}=1.13\,\varepsilon_{c}\,e^{-1/v_{1}}$. The functional in Eq. (1) reduces to the weak-coupling BCS free energy functional for pure ³He (Eq. 5.16 of Ref. (16)) for $\tau\rightarrow\infty$.

Furthermore, the stationarity condition for the impurity-averaged functional, $\delta\Delta\bar{\Omega}/\delta\vec{\Delta}(\hat{\bf p})^{*}=0$, generates the impurity-renormalized gap equation,

The renormalized Matsubara energy encodes the effect of de-pairing by the scattering of pair-correlated quasiparticles off the distribution of impurities. This is evident from the suppression of the transition temperature (3),

The dimensionless pair-breaking parameter $\alpha$ is the product of the mean scattering rate, $1/\tau$, and the Cooper pair formation time, $\hbar/2\pi k_{\text{B}}T_{c_{0}}$, i.e. $\alpha=\hbar/2\pi\tau\,k_{\text{B}}T_{c_{0}}$, The pair-breaking parameter can also be expressed as the ratio of the ballistic coherence length, $\xi_{0}=\hbar v_{f}/2\pi k_{B}T_{c_{0}}$, of pure ³He and the transport mean free path, $\ell=v_{f}\tau$, i.e. $\alpha=\xi_{0}(p)/\ell$. The range of disorder from weak to strong pair breaking can be explored by varying the pressure and/or aerogel density. The solution for the universal curve for $T_{c}/T_{c_{0}}$ versus $\xi_{0}/\ell$ is shown in Fig. 6 of Ref. (17). The disorder critical point where $T_{c}\rightarrow 0$ is $\alpha_{c}\simeq 0.281$. Since $\xi=\hbar v_{f}/2\pi k_{\text{B}}T_{c}\rightarrow\infty$ as $\alpha\rightarrow\alpha_{c}$ the homogeneous scattering model (HSM) prediction for the critical point is robust. Indeed the disorder driven quantum critical point has been observed for ³He in 98.2% porous aerogels, (18) and provides an accurate result for the quasiparticle-impurity mean free path. However, away from the quantum critical point the HSM does not accurately account for $T_{c}$ vs. pressure for ³He-aerogel due to the correlated nature of the random potential for silica aerogels (3). The effects of correlations on $T_{c}$ and other observable properties has been studied in Refs. (19; 14). In order to incorporate aerogel correlations and compare the weak-coupling predictions for the heat capacity, $C/T$, with experimental data reported in Ref. (11), I fit the pairbreaking parameter, $\alpha$, at each pressure to the measured transition temperature for ³He-aerogel. Fermi liquid data, c.f. $N_{f}=3n/4E_{f}$, $E_{f}=\frac{\mbox{\small 1}}{\mbox{\small 2}}v_{f}p_{f}$, $v_{f}$, $p_{f}=\hbar k_{f}$, $n=k_{f}^{3}/3\pi^{2}$, are obtained from the tabulated data of Ref. (20). In addition, I used the published value for the volume of ³He in the aerogel sample of Ref. (11), $V_{a}=1.028\,\mbox{cm}^{3}$. The resulting normal-state value of $C/T$ agrees nearly perfectly with the reported values in Ref. (11) except for a slight discrepancy at $p=20.06\,\mbox{bar}$ as shown in Fig. 3.

One might expect reasonable agreement between theory and experiment for the heat capacity of the superfluid phase at low pressures since both strong-coupling effects and deviations from the HSM are relatively small. Indeed the theoretical calculations of $C/T$ at $p=11.31$, $14.11$ and $20.06$ bar (Figs. 1, 2 and 3) are all in good agreement with the experimenal data, although there is broadening of the transition visible at all pressures which I expect is associated with large-scale inhomogeneity of the aerogel sample. At higher pressures, $p=24.90$ bar and $p=29.02$ bar (Figs. 4 and 5), the slope of $C/T$ is still reasonably well reproduced, but there is an increase in the theoretical value of $C/T$ compared to the experimental data. This shift almost certainly reflects limitations of the HSM based on point impurities. The correlated nature of the aerogel structure, as well as the finite size of the silica strands and clusters, become more important at higher pressures where the Cooper pair correlation length is smallest and approaches the aerogel correlation length. Note that strong-coupling enchancement of the experimental heat capacity jump is expected to be largest at high pressures, but is suppressed significantly relative to that of pure bulk ³He at the same pressure.

Finally, I note that the density of states at the Fermi level for each pressure, extracted from $\lim_{T\rightarrow 0}C/T\sim N(0)$, is predicted to be independent of pressure (see Fig. 6). This is in contrast to the density of states for normal ³He which varies substantially with pressure as shown in Fig. 6. For the superfluid phase $N(0)$ reflects the density of sub-gap states confined on the aerogel strands and clusters. Thus, it seems reasonable to assume that the values of $N(0)$ are determined by the surface area of aerogel strands that host surface Andreev bound states, and thus is insensitive to pressure and ³He density. I also note that the non-monotonic behavior of $C/T$ that is predicted at higher pressures is due to the finite bandwidth of the impurity-induced DOS centered at the Fermi energy (15).

Superfluid ³He-B embedded in high-porosity aerogel is predicted to exhibit gapless behavior at low temperatures over the entire pressure range. Analysis of the specific heat data for pressures $11\,\mbox{bar}\leq p\leq 29\,\mbox{bar}$ points directly to a band of sub-gap quasiparticle bound states with a finite density of states at the Fermi level. Measurements of $C/T$ on these samples down to $300\,\mu\mbox{K}$ could provide definitive and direct measurements of a finite density of surface Andreev bound states confined on the aerogel structure.