Acoustic properties of metallic glasses at low temperatures: Tunneling systems and their dephasing

The low temperature acoustic properties of bulk metallic glasses measured over a broad range of frequencies rigorously test the predictions of the standard tunneling model. The strength of these experiments and their analyses is mainly based on the interaction of the tunneling states with conduction electrons or quasiparticles in the superconducting state. A series of experiments at kHz and GHz frequencies on the same sample material essentially confirms previous measurements and their discrepancies with theoretical predictions. These discrepancies can be lifted by considering more correctly the linewidths of the dominating two-level atomic-tunneling systems. In fact, dephasing caused or mediated by interaction with conduction electrons may lead to particularly large linewidths and destroy the tunneling sytems' two-level character in the normal conducting state.

Figure 1

Temperature dependence of the transverse sound velocity of the metallic glass ${\text{Zr}}_{59}{\text{Ti}}_3{\text{Cu}}_{20}{\text{Ni}}_8{\text{Al}}_{10}$ measured at 969 MHz. The superconducting transition is at $T_{\text{c}}=1.39\mathrm{K}$. Data marked normal conducting is obtained in a magnetic field of $4\mathrm{T}$. Lines show numerical calculations as explained in the text.

Figure 2

Temperature dependence of the sound velocity of a metallic glass as calculated with the standard tunneling model for a measuring frequency of $200\mathrm{MHz}$. Bottom curves comprise energy relaxation only by phonons (red) and additionally by quasiparticles (blue) and electrons (green). Constant linewidth $h{\tau }_2^{-1}\ll E$ is assumed. Sound velocities of normal- and superconducting states merge at low temperatures. Including the lifetime limited linewidth ${\tau }_2^{-1}={\tau }_1^{-1}/2$ with plausible and temperature dependent ${\tau }_1^{-1}$ rates already produces a crossing of $\delta v/v$ of the normal state and the superconducting state (curves next to bottom). Addition of a dephasing process ${\tau }_{\varphi }^{-1}\propto T$ shifts this crossing to higher temperatures. Finally, a modified dephasing model ${\tau }_{\varphi }^{-1}\propto f\left(T\right)$ (see text) reproduces the experimental data (see Figs. 1 and 3) very well (topmost curves).

Figure 3

Relative temperature variation of the transverse sound velocity of glassy ${\text{Zr}}_{59}{\text{Ti}}_3{\text{Cu}}_{20}{\text{Ni}}_8{\text{Al}}_{10}$ measured at frequencies between $200\mathrm{MHz}$ and $1.63\mathrm{GHz}$. Blue crosses are data without magnetic field where the sample is superconducting below $T_{\mathrm{c}}=1.39\mathrm{K}$. A magnetic field of $4\mathrm{T}$ keeps the sample normal conducting at all temperatures (green $\times$). Dashed lines result from numerical calculations using always the same parameters as detailed in the text. Sets of data and curves for different frequencies are shifted for clarity.

Figure 4

Energy relaxation rate of TS in the metallic glass as extracted from the onset of the relaxation process related to respective maxima of $\delta v/v$. The energy relaxation rates ${\tau }_1^{-1}$ of the fastest, symmetric thermal TS due to interaction with phonons (red), thermally activated quasiparticles (blue), and electrons in the normal conducting state (green) are adjusted to the data. The dashed (orange) line indicates the energy splitting of thermal TS with $E\approx k_{\text{B}}T$. Low frequency data are determined by vibrating reed experiments, high frequency by pulsed ultrasound. Full symbols denote data of samples machined from a bulk metallic glass rod of ${\text{Zr}}_{59}{\text{Ti}}_3{\text{Cu}}_{20}{\text{Ni}}_8{\text{Al}}_{10}$; open symbols result from of a splat quenched sample of ${\mathrm{Zr}}_{46.8}{\mathrm{Ti}}_{8.2}{\mathrm{Cu}}_{7.5}{\mathrm{Ni}}_{10}{\mathrm{Be}}_{27.5}$. For crosses see text.

Figure 5

Temperature dependence of the longitudinal sound velocity at $1.44\mathrm{GHz}$ (upper panel) and of the Young's modulus velocity measured by a vibrating reed [30] at $1.1\mathrm{kHz}$ (lower panel). Arrow indicates the transition from phonon to quasiparticle dominated relaxation. Dashed lines show numerical calculations.

Figure 6

Temperature dependence of the attenuation $tan\delta$ of transverse ultrasound in the GHz frequency range. Values are determined by the squared ratio of the amplitudes of the first transmitted ultrasound pulse and a fixed reference signal. Solid lines are calculated for weak excitation using values for $W_{\text{el}}$ and ${\gamma }_{\text{t}}$ derived from the temperature dependence of the sound velocity. For $P_0$ values see text. The acoustic intensity for the experiment was not at the weak excitation level. Dash-dotted lines show calculations of only the relaxation contribution of the superconduction sample. Experimental data of both frequencies is vertically shifted to adjust for the zero line of the calculated pure relaxation contribution. $969\mathrm{MHz}$ data are shifted by another $15\times {10}^{-5}$.