Acoustic Properties of Amorphous Silica between 1 and 500 mK

We have made reliable measurements of the sound velocity $\delta v/v_0$ and internal friction $Q^{-1}$ in vitreous silica at 1.03, 3.74, and 14.0 kHz between 1 mK and 0.5 K. In contrast with earlier studies that did not span as wide a temperature and frequency range, our measurements of $Q^{-1}$ reveal a crossover (as $T$ decreases) only near 10 mK from the $T^3$ dependence predicted by the standard tunneling model to a $T$ dependence predicted if interactions are accounted for. We find good fits at all frequencies using a single interaction parameter, the prefactor of the interaction-driven relaxation rate, in contrast to earlier claims of a frequency dependent power law. We also show that the discrepancy in the slopes $d(\delta v/v_0)/d({log}_{10}T)$ below and above the sound velocity maximum ($1:-1$ observed, $1:-2$ predicted) can be resolved by assuming a modified distribution of tunneling states.

Figure 1

Response of the 14.0 kHz mode at three different temperatures. For each temperature two drive levels were chosen to demonstrate the linear response. Inset: Finite element visualization of the oscillator displacement and strain energy density (linear scale) at 14.0 kHz for 0.4 rad peak displacement. The white broken line indicates the upper edge of the dry clamp.

Figure 2

Measurements of the relative change in sound velocity $\delta v/v_0$ (data points) and predictions of the STM [$b=0$] (curves) for each of the experimental frequencies (offset vertically). The curves correspond to a tunneling strength $C=2.4\times {10}^{-4}$ and the indicated values of $a$, the prefactor of the single-phonon relaxation rate. The best fit at each frequency is represented by a solid curve. These curves are indistinguishable from those corresponding to the same $C$ and $a$, but with the prefactor $b=1.0\times {10}^5{\mathrm{K}}^{-1}{\mathrm{s}}^{-1}$ (see text).

Figure 3

Measured internal friction $Q^{-1}$ (data points) and predictions (curves) for each of the experimental frequencies. The curves correspond to a tunneling strength $C=2.4\times {10}^{-4}$, a single-phonon relaxation prefactor $a=9.0\times {10}^7{\mathrm{K}}^{-3}{\mathrm{s}}^{-1}$, and various values of $b$. The solid black curves correspond to the best fit value of $b$ and agree well with the measured $Q^{-1}$. For comparison, the prediction of the noninteracting model ($b=0$) is shown for $\omega /2\pi =1.03$ and 14.0 kHz.

Figure 4

Measured $\delta v/v_0$ (offset vertically) and $Q^{-1}$ at 1.03 kHz along with tunneling model predictions assuming an interaction-driven relaxation rate $b=1.0\times {10}^5{\mathrm{K}}^{-1}{\mathrm{s}}^{-1}$ and TLS distribution parameter $\mu =0.09$.