Acoustic properties of amorphous metals at very low temperatures: Applicability of the tunneling model

Studies of the relative change of sound velocity $\Delta v/v$ and internal friction $Q^{-1}$ of a metallic glass (PdSiCu) in the temperature range $0.2\mathrm{mK}<~T<\mathrm{}0.5\mathrm{}\mathrm{K}$ and at kHz frequencies reveal significant deviations from the predictions of the tunneling model for both acoustic properties. At the lowest temperatures, the logarithmic temperature dependence of $\Delta v/v$ is no longer observed, and in addition, the sound-velocity variation strongly depends on the applied acoustic intensity. In clear contrast to the predicted “plateau” in the internal friction, we observe $Q^{-1}\propto \ln T$ throughout the whole temperature range investigated, the magnitude of $Q^{-1}$ being dependent on the applied strain, too. The apparent discrepancy between experimental data and tunneling model is traced back to different considerations of strain-dependent effects: in contrast to the experiment, the tunneling model is linear in strain and thus does not account for any strain dependence. As a consequence, comparison between experimental results and tunneling model should be possible for zero-strain values of $\Delta v/v$ and $Q^{-1}$ which have been determined by detailed investigations of the acoustic intensity dependence of both acoustic properties at different constant temperatures. The main result of this work is that the tunneling-model behavior is unveiled only in the limit of zero strain. Thus, for the first time the behavior expected within the tunneling model for an amorphous metal has been scrutinized: we report a logarithmic temperature dependence for $\Delta v/v$ at $T>\mathrm{}0.2\mathrm{}\mathrm{mK}$ and a temperature-independent plateau of $Q^{-1}$ at $T>\mathrm{}0.5\mathrm{}\mathrm{mK}.$