Anderson Mobility Gap Probed by Dynamic Coherent Backscattering

We use dynamic coherent backscattering to study one of the Anderson mobility gaps in the vibrational spectrum of strongly disordered three-dimensional mesoglasses. Comparison of experimental results with the self-consistent theory of localization allows us to estimate the localization (correlation) length as a function of frequency in a wide spectral range covering bands of diffuse transport and a mobility gap delimited by two mobility edges. The results are corroborated by transmission measurements on one of our samples.

Figure 1

(a) Sample $\text{L}1$. (b) Bead structure of sample $\text{L}1$. (c) Amplitude transmission coefficient of ultrasonic waves through samples $\text{L}1$ and $\text{L}2$ as a function of frequency.

Figure 2

Dynamic CBS profiles in the diffuse regime (1.65 MHz) (a,b) and in the localized regime (1.22 MHz) (c,d). The results in (a,b) are for sample $\text{L}1$, and in (c,d) for sample $\text{L}2$ (note the different angular scales). In (a,c) theoretical fits (lines) and experimental data (symbols) are shown for three representative times. In (a), the data are fitted using diffusion theory, giving diffusion coefficient $D=D_B=0.7{\mathrm{mm}}^2/\mu \mathrm{s}$ [37], whereas in (c) SC theory is used, giving $\xi =16.5\mathrm{mm}$. Additional examples are shown in Ref. [35]. In (b,d) experimental CBS profiles are shown as a function of both time and angle. The profile narrows quite rapidly in the diffuse regime (b) but is almost constant over the accessible range of times in the localized regime (d).

Figure 3

Experimental results (symbols) and theoretical predictions (lines) for sample $\text{L}1$. Plotted is the reciprocal of the square of the half width at half maximum of the CBS peaks, $\mathrm{\Delta }{\theta }^{-2}(t)$ (error bars are smaller than symbol sizes). Three representative frequencies are shown: $f=1.65\mathrm{MHz}$ (diffuse regime, diffusion coefficient $D_B=0.7{\mathrm{mm}}^2/\mu \mathrm{s}$ extracted from the fit), $f=1.18\mathrm{MHz}$ (subdiffusion as a ME is approached; correlation length $\xi =2.1\mathrm{mm}$), and $f=1.22\mathrm{MHz}$ (Anderson localization; localization length $\xi =12.5\mathrm{mm}$). The inset shows theoretical predictions for longer times.

Figure 4

The ratio of sample thickness $L$ to the localization (correlation) length $\xi$ obtained from fits to experimental CBS profiles (sample $\text{L}1$—red stars, sample $\text{L}2$—green squares) and transverse confinement (TC) data (sample $\text{L}1$—open circles). Error bars represent variations of $L/\xi$ that increase the reduced ${\chi }^2$ by unity; error bars are smaller than symbol size for transmission results. Fuzzy vertical gray lines show our estimates of mobility edges.