Unifying Description of the Vibrational Anomalies of Amorphous Materials

The vibrational density of states $D(\omega )$ of solids controls their thermal and transport properties. In crystals, the low-frequency modes are extended phonons distributed in frequency according to Debye’s law, $D(\omega )\propto {\omega }^2$. In amorphous solids, phonons are damped, and at low frequency $D(\omega )$ comprises extended modes in excess over Debye’s prediction, leading to the so-called boson peak in $D(\omega )/{\omega }^2$ at ${\omega }_{\mathrm{bp}}$, and quasilocalized ones. Here we show that boson peak and phonon attenuation in the Rayleigh scattering regime are related, as suggested by correlated fluctuating elasticity theory, and that amorphous materials can be described as homogeneous isotropic elastic media punctuated by quasilocalized modes acting as elastic heterogeneities. Our numerical results resolve the conflict between theoretical approaches attributing amorphous solids’ vibrational anomalies to elastic disorder and localized defects.

Figure 1

(a) Dependence of elastic disorder correlation length ${\xi }_e\propto {\gamma }^{1/3}$ and disorder parameter $\gamma$ on $x_c$, a parameter controlling the extension of the attractive well. Error bars are smaller than the symbol size. (b) Reduced $D(\omega )$, normalized by its maximum value, as a function of $\omega /{\omega }_{\mathrm{bp}}$. We found ${\omega }_{\mathrm{bp}}\simeq 4.5c_s/{\xi }_e$ (inset).

Figure 2

(a) The frequency dependence of the scaled attenuation rate is consistent with corr-FET as concern the limiting low-frequency value, $\propto {\gamma }^2$. We combine data for $N=32\mathrm{k}$, 64 k, 256 k, 512 k, 2048 k, and 8192 k. Symbols are as in Fig. 1. (b) Analogous results are obtained investigating the scaled sound attenuation rate of amorphous solid configurations prepared minimizing the energy of ultrastable liquids in equilibrium at temperature $T_p$, below the mode coupling one. Data are from Refs. [29, 55], to which we refer for further details. Symbols identify the system size: 192 k (squares), 96 k (circles), 48 k (triangles).

Figure 3

(a) Scaling of the low-frequency density of states. (b) QLMs volume estimated by the low-frequency limit of $Ne$ with $e$ the participation ratio and $N$ the system size. Representative error bars are shown. The full lines correspond to ${\xi }_e^3\propto \gamma$. For both panels $N=4000$ and data are averaged over at least ${10}^4$ realizations. Symbols indicate different $x_c$ values as in Fig. 1.

Figure 4

Probability distribution of the local shear modulus ${\mu }_w/{\mu }_0-1$ coarse grained over a length scale $w$, ${\mu }_0$ being the average modulus. In (a) the coarse graining length scale equals the defect size, $w={\xi }_d={\xi }_e$, whose $x_c$ dependence is in Fig. 1. In panel (b), $w=4a_0$. For each $x_c$, results are obtaining averaging over 50 independent $N=256000$ particles configurations. Symbols are as in Fig. 1.