Limits on transverse zero sound in Fermi liquid ³He

Abstract

Landau predicted that transverse sound propagates in a Fermi liquid with sufficiently strong interactions such as for liquid ³He, unlike a classical fluid that cannot support shear oscillations. Previous attempts to observe this unique collective mode yielded inconclusive results. We microfabricated acoustic cavities with a micrometer-scale path length that is suitable for direct detection of this sound mode and used the interference fringes from our acoustic Fabry-Pérot cavities to determine both real and imaginary parts of the acoustic impedance. No interference was observed. Either transverse sound does not exist or its attenuation must be very high, well above 2000 per centimeter. We discuss a theoretical framework for such high attenuation.

Limits are established on the validity of Lev Landau’s prediction for existence of transverse sound in a Fermi liquid.

INTRODUCTION

The Fermi liquid theory, proposed by Landau (1) and expanded on later by many others, lies at the foundation of theoretical condensed matter physics and our understanding of the Fermi liquid state. Nearly all the central predictions of Landau’s theory have been verified, including the existence of longitudinal zero sound in liquid ³He (2). Landau also predicted the existence of another propagating sound mode, called transverse zero sound (TZS), where the polarization of the wave is perpendicular to the wave vector. Measurements by Roach and Ketterson (3) were initially believed to demonstrate the existence of TZS in ³He but were later shown by Flowers et al. (4) to be the result of an incoherent quasiparticle excitation.

It was originally expected that transverse sound would be more difficult to detect in the superfluid state as the number of unpaired quasiparticles reduces with temperature. However, it was shown by Moores and Sauls (5) that a coupling to the collective modes of the superfluid state contributes to the stress tensor in a manner much stronger than Landau’s predicted mode in the degenerate Fermi liquid. This coupling supports a propagating transverse sound mode that exhibits an acoustic Faraday effect. Both were experimentally demonstrated by Lee et al. (6). That work also led to the discovery of a previously unknown order parameter collective mode (7) and other works by Davis et al. (8) and Collett (9, 10). At this date, superfluid ³He remains the only fluid where transverse sound waves have been observed (Supplementary Materials). In the present work, we revisit the normal Fermi liquid state to search for the last unverified prediction of Landau, using the results in the superfluid as a quantitative reference.

In a Fermi liquid, low-energy quasiparticle excitations near a degenerate Fermi surface respond to external fields, leading to changes in the quasiparticle distribution function, n_k. Landau showed that as a result, there can be distortions of the Fermi surface in momentum space, as shown in Fig. 1A. The Fermi surface can be thought of as a vibrating membrane, and the interactions can be projected onto a basis of different angular momentum channels for the corresponding distortions of the Fermi surface.

Fig. 1. Landau’s prediction and previous experiment.

(A) Oscillations of the Fermi surface for various l-channels and sound modes for a rightward moving plane wave solution to Eq. 1 at T = 0. (B) Flowers et al. (4) showed that the pressure dependence of the attenuation reported by Roach and Ketterson (3) is consistent with single-particle contributions, not transverse sound.

The dynamics of wave propagation in a Fermi liquid are governed by the Landau-Boltzmann kinetic equation (11)

$$i(v_{\mathrm{F}}\hat{\mathbf{k}}\cdot \mathbf{q}-\mathrm{\omega }){\mathrm{\Phi }}_{\hat{\mathbf{k}}}+{\mathit{iv}}_{\mathrm{F}}\hat{\mathbf{k}}\cdot \mathbf{q}\int \frac{d{\mathrm{\Omega }}_{{\mathbf{k}}^{\prime }}}{4\mathrm{\pi }}\mathbb{F}(\hat{\mathbf{k}}\cdot {\hat{\mathbf{k}}}^{\prime }){\mathrm{\Phi }}_{{\hat{\mathbf{k}}}^{\prime }}=\mathrm{\delta }{I}_{\hat{\mathbf{k}}}$$ (1)

where v_F is the Fermi velocity, q and ω are the wave vector and the frequency of sound, ${\mathrm{\Phi }}_{\hat{\mathit{k}}}$ encodes the variation of the Fermi surface, where $\mathbb{F}(\hat{\mathbf{k}}\cdot {\hat{\mathbf{k}}}^{\prime })$ are the Fermi liquid interactions between quasiparticles with momentum k and k′, and δI_k is the collision integral for binary quasiparticle collisions. The Fermi liquid interactions can be projected in different angular momentum channels for spin-symmetric and spin antisymmetric interactions. Only spin-symmetric interactions are relevant for sound propagation. The strength of each l-channel is given by the Fermi liquid parameters $F_{\mathrm{l}}^s$; s denotes spin symmetric.

In the hydrodynamic regime, the quasiparticle collision rate, 1/τ, is much higher than the frequency of sound so ωτ ≪ 1. At low temperatures well below the Fermi temperature, T ≪ T_F, the collision rate is low (τ⁻¹ ∼ T²) and insufficient to restore oscillations to equilibrium. The Fermi liquid interaction dominates, and the sound mode is collisionless zero sound, ωτ ≫ 1. Landau predicted that for sufficiently strong values of $F_{\mathrm{l}}^s$, not only longitudinal zero sound but also TZS can exist.

While any positive, nonzero value of $F_1^s$ leads to transverse current fluctuations, a transverse sound wave will only propagate if the speed of sound relative to the Fermi velocity is c_t/v_F > 1, thereby reducing Landau damping from the particle-hole continuum and imposing a condition on the Fermi liquid parameters (12, 13)

$$\frac{F_1^s}{3}+\frac{F_2^s}{1+F_2^s/5}>2$$ (2)

a condition satisfied by liquid ³He at all pressures. Therefore, TZS can be expected to propagate.

Roach and Ketterson (3) measured the transverse acoustic impedance of ³He as a function of temperature from the hydrodynamic regime (∼100 mK) into the zero sound regime (∼2 mK), with features that were thought to be consistent with zero sound; however, this was shown to be otherwise by Flowers et al. (4), as displayed in Fig. 1B.

To distinguish between TZS and an incoherent quasiparticle excitation, an interference experiment must be performed. In the present work, we have substantially extended the sensitivity range of acoustic Fabry-Pérot interferometers that have been used previously to study transverse sound in the superfluid state (6, 8, 9) (Supplementary Materials). There are also limitations to accuracy in the measurement of very low attenuation that we established at very low temperature and pressure in the superfluid, ∼500 cm⁻¹.

RESULTS

We used an acoustic cavity formed between a piezoelectric transducer and a reflecting plate, shown in Fig. 2A; details are in the Supplementary Materials. Depending on the speed of sound and path length D, the reflected signal will interfere destructively or constructively at the transducer surface, creating an oscillating signal corresponding to interference fringes in the acoustic impedance as the velocity of sound is varied by sweeping pressure or temperature. In this figure for the superfluid, each oscillation corresponds to a change of one wavelength in a distance of 2D. The amplitude of the interference pattern is proportional to the attenuation coefficient, $\mathrm{\delta }\mathcal{A}=e^{-\mathrm{\alpha }2D}$, easily observed in the superfluid ³He B-phase with cavities of ∼30 μm. Using microfabrication techniques, we have reduced D to 5.2 μm to increase signal sensitivity by ∼400 to accommodate expected high attenuation in the normal Fermi liquid, $\mathcal{O} (1000 {\mathrm{cm}}^{-1})$ (14).

Fig. 2. Transverse sound in superfluid ³He.

(A) Interference fringes obtained during a pressure sweep in the superfluid state at 0.95 mK. Transverse sound results from off-resonant coupling to the J = 2 − Higgs mode in the superfluid (17, 18). The inset shows the acoustic cavity formed between a piezoelectric ac-cut quartz transducer (top, gold disk) and a microfabricated silicon reflecting plate (bottom disk). (B) Interference fringes obtained during a temperature sweep at 10.5 bar and 103.9 MHz in the superfluid state are shown used to calculate the speed of sound showing good agreement with the theory of Moores and Sauls (5) for a cavity size of D = 5.2 μm.

We first consider experiments in the superfluid to characterize the quality of the cavities and calibrate sensitivity in measurement of attenuation. The amplitude of the interference fringes is proportional $\mathrm{\delta }\mathcal{A}$. If the amplitude of interference can be determined for a known value of α, then it can be used to estimate the cavity detection sensitivity to highly attenuated signals. From past experiments, the smallest $\mathrm{\delta }\mathcal{A}$ detected was ∼0.03 (15).

However, this estimate does not include surface Andreev bound states that contribute to attenuation (15). Otherwise, α is expected to decrease to zero at T = 0, and there could be attenuation from spontaneous emission due to vacuum fluctuations. We found that α saturates around 400 to 500 cm⁻¹, independent of pressure (15), and there is no observed decrease in α below 0.5 T_c which provides a good background reference for the present work. By cooling to ∼0.5 T_c and performing a pressure sweep, one can obtain the expected amplitude of the interference fringe for transverse sound in the superfluid propagating with 500 cm⁻¹ of attenuation, where T_c is the superfluid transition temperature. This provides a quantitative reference for the background to the attenuation measurements in the Fermi liquid.

We used both temperature and pressure sweeps to calibrate our cavities. A demagnetization cooling temperature sweep at 103.9 MHz and 10.5 bar is shown in Fig. 2B. Changing temperature affects the superfluid gap Δ (16) and the J = 2 − Higgs order parameter collective mode frequency, Nguyen et al. (17, 18). As the resonance with the mode is approached, the speed of sound increases substantially. The interference fringes were used to calculate the speed of sound, which compares well with the theory of Moores and Sauls (5), Fig. 2B, and our previous work.

After verifying the performance of the cavities in the superfluid state and establishing a reference for α, we performed a series of pressure sweeps in the Fermi liquid at temperatures hovering above T_c, as noted quantitatively in the next section. The attenuation of TZS in the Fermi liquid state is expected to decrease as α ∼ T², calculated in that section and displayed in Fig. 3 for illustration. Clearly, it is best to search for interference at the lowest temperature by varying the pressure.

Fig. 3. Simulation.

(A) Calculated attenuation coefficient, α, for TZS as a function of temperature, for different relaxation times, α ∼ T². Measurements were at ∼2 mK where temperatures are in the zero sound regime in the Fermi liquid below 5 mK (blue shaded region). Solid and dashed lines are for $F_2^s=0$ and 1, respectively. (B) Imaginary part of the transverse impedance Z″ (Eq. 3) under a pressure sweep at 2.3 mK (red line) and temperature sweep at various fixed pressures (black lines). In the simulation, diffusive boundary conditions were used [s₁ = s₂ = 0; (20)]. Parameters in the simulation were tabulated in (22), except the viscous relaxation time τ_v taken from Wheatley (28). The attenuation at 2.3 mK was chosen to be 1000 cm⁻¹.

DISCUSSION

Flowers and Richardson (4, 19) considered an oscillating semi-infinite plate with two relaxation times to account for high attenuation of TZS. Using this approach, Kuorelahti et al. (20) solved the Landau-Boltzmann equation for a cavity. Following their procedure, we calculated the attenuation for the transverse sound in Fig. 3A and the imaginary part of the acoustic impedance in Fig. 3B for a 5.2-μm cavity with a frequency of 103.9 MHz.

In addition, it is known that without plating ⁴He on the surface, ³He atoms are adsorbed on the surface, forming a thin ³He solid layer. In this instance, the scattering of ³He quasiparticles on the solid layer can be regarded as being diffusive (21) for all interfaces in the simulation (20).

The temperature enters the zero sound regime in the Fermi liquid below 5 mK. With reduced attenuation, as shown in Fig. 3A, transverse sound is expected to propagate. Its return to the transducer affects the acoustic impedance (22) of the transducer, Z (Eq. 3), producing interference fringes in the imaginary part, Z″, during a pressure sweep and various temperature sweeps. To compare the numerical result with experimental data, Z is converted into a frequency shift from the relations

$$Z=Z^{\prime }+i{Z}^{″}=\frac{1}{4}n\mathrm{\pi }{Z}_{\mathrm{q}}\mathrm{\Delta }{Q}^{-1}+i\left(\frac{1}{2}n\mathrm{\pi }{Z}_{\mathrm{q}}\frac{\mathrm{\Delta }{f}_0}{f_0}\right)$$ (3)

where n is the nth-order harmonic, Z_q is the impedance of quartz, Q is the Q-factor, and f₀ is the nth harmonic frequency. The changes in Q⁻¹ and f₀ are defined as the differences between loaded and unload values (3, 20).

Pressure sweeps at 21st (103.9 MHz at T = 1.998 ± 0.007 mK) and 29th (143.7 MHz at T = 2.39 ± 0.06 mK) harmonics of the transducer are shown in Fig. 4. For Fig. 4A, the radio frequency spectrometer was tuned to a single frequency (103.9 MHz) while tracking the reflected signal from the transducer. The red and green curves show the expected amplitude and periodicity of the interference pattern for TZS with an attenuation coefficient of α = 500 and 2000 cm⁻¹, respectively. The amplitudes of these curves were scaled from the result in the superfluid state for the background signal in this cavity with α ∼ 500 cm⁻¹. It is evident that there are no interference fringes consistent with TZS.

Fig. 4. Pressure sweep.

Data in the Fermi liquid state ∼2 mK compared with the expected interference pattern obtained from numerically solving the Landau-Boltzmann equation in a 5.2-μm cavity. (A) Pressure sweep at a single fixed frequency using our continuous wave spectrometer at fixed frequency, 103.9 MHz. The red curve shows the expected amplitude with an attenuation of α = 500 cm⁻¹. The blue curve is for α = 2000 cm⁻¹. (B) Pressure sweep of the full spectrum of the 29th at 143.7 MHz from which the central frequency was determined. Neither sweep indicates evidence of an interference pattern consistent with a propagating transverse mode.

For the data in Fig. 4B, a network analyzer was used to acquire the full spectrum of the 29th harmonic around 143.7 MHz while performing a pressure sweep. From the central frequency, both the periodicity and amplitude are directly compared with the numerical simulation. Note that the frequency shifts were derived from Eq. 3. Again, it is evident that there are no interference fringes.

From theory, anomalously higher attenuation α can be understood if there are multiple relaxation times. The attenuation α of the TZS depends on how the collision integral δI is modeled. Lea et al. obtained 500 cm⁻¹ at 2 mK, using a single relaxation time τ (δI ∝ ${\mathrm{\Phi }}_{\hat{\mathbf{k}}}$/τ) (14). However, if we allow each l-channel fluctuation on the Fermi surface to relax at a separate rate τl, the attenuation α can be higher than for a single relaxation time (13).

If the quadrupolar channel (l = 2) has relaxation time τ₂ = τ_ν, with τ_ν being from the viscosous relaxation time of the Fermi liquid, this may differ from τ describing all higher-order Fermi surface distortions (l ≥ 2). Since τ₂ is parametrized by ξ₂, τ₂ = τ_ν = τ/ξ₂ (two relaxation time approximation). The smaller ξ₂ gives rise to a greater attenuation α as shown in Fig. 3A. Estimates from various experiments with ξ₂ = 0.35 (23) and ξ₂ = 0.28 (24) yield an attenuation α of 1300 cm⁻¹. Our experiment indicates that the attenuation α is much higher than these estimations.

In the simple relaxation time approximation, the attenuation of TZS has no frequency dependence. As the sound energy ℏω approaches k_BT, it excites nonthermal quasiparticles that further contribute to the attenuation (11). Landau calls this the quantum limit of TZS corresponding to ℏω > 2πk_BT. Far into the quantum regime where ℏω ≫ 2πk_BT, Landau predicts a simple quadratic frequency dependence of α. For 100-MHz sound at 2 mK, ℏω/2πk_BT ∼ 0.3 is in a regime where some nontrivial frequency dependence might be expected (25). Last, if the full relaxation dynamics of the transport equation is considered, we might expect additional frequency dependence (26).

While it cannot be concluded that transverse sound does not propagate in the Fermi liquid state, if it does exist, the actual attenuation must be substantially higher than theoretically predicted within the relaxation time approximation. At 2 mK, we conclude that the attenuation of sound is substantially greater than 2000 cm⁻¹ and likely to be the case at other pressures and frequencies. This is a marked extension of what has been estimated previously (3, 4, 27).

MATERIALS AND METHODS

To ensure high parallelism between the two surfaces for a well-defined path length, we fabricated a 5.2-μm cavity out of a silicon-on-insulator (SOI) substrate (Fig. 2A). SOI wafers have three layers—a device layer of silicon (Si) (typically micrometer scale), a buried oxide (BOX) layer of SiO₂ (typically ≤1 μm), and lastly a handle layer that is several hundred micrometers thick. The reflecting plate for our cavity is the Si handle layer, while the spacer features are made of the BOX layer and the device layer. The SOI wafer was chosen such that the device layer forms the bulk of the cavity spacing (5 μm) while the BOX layer adds 200 nm, yielding a path length of 5.2 μm. The fabrication from a SOI wafer instead of simply silicon is to ensure a smooth reflecting surface, similar to the surface of the transducer. The dry-etching (Bosch) process of Si would lead to a very rough surface, not suitable for acoustic cavities. The BOX layer of the SOI wafer protects the handle layer’s surface during the dry etching of the device layer. The BOX can be wet etched with hydrofluoric acid after the dry etch process, revealing the pristine handle layer. The transducer is an ac-cut quartz transducer that generates the transverse, thickness shear mode, with gold electrodes on the backside (noncavity side). The fundamental frequency is 4.94 MHz, and the transducer was mainly operated at the 21st (103.9 MHz) and 29th harmonics (143.7 MHz).

However, cavities cannot be made too small since a shorter path length D produces fewer interference fringes. The drive frequency can be increased to compensate for the reduced fringes of a small cavity, but the attenuation also has frequency dependence from two sources. In the relaxation time approximation, TZS has no frequency dependence; however, α ∼ T² [unlike in the hydrodynamic limit where α ∼ ω²/T²; (11)].

Supplementary Materials

This PDF file includes:

Supplementary Text

Fig. S1