Universal density of low-frequency states in silica glass at finite temperatures

The theoretical understanding of the low-frequency modes in amorphous solids at finite temperature is still incomplete. The study of the relevant modes is obscured by the dressing of interparticle forces by collision-induced momentum transfer that is unavoidable at finite temperatures. Recently, it was proposed that low-frequency modes of vibrations around the thermally averaged configurations deserve special attention. In simple model glasses with bare binary interactions, these included quasilocalized modes whose density of states appears to be universal, depending on the frequencies as $D\left(\omega \right)\sim {\omega }^4$, in agreement with the similar law that is obtained with bare forces at zero temperature. In this paper, we report investigations of a model of silica glass at finite temperature; here the bare forces include binary and ternary interactions. Nevertheless, we can establish the validity of the universal law of the density of quasilocalized modes also in this richer and more realistic model glass.

Figure 1

Starting from the inherent structure configuration, we perform molecular dynamics monitoring the mean squared displacement. Here we plot mean square displacements (MSDs) vs time. In the considered time window at a large-enough $T$ the cages break and the MSD deviates from the plateau. The final horizontal lines were added to report the final average value of each curve.

Figure 2

Density of states as computed from the effective Hessian at different temperatures, $T=1,4,16,64\mathrm{K}$ for ${\mathrm{SiO}}_2$ glass samples of 1032 [panels (a) and (c)] and 4008 [panels (b) and (d)] atoms. In panels (a) and (b) all the eigenfrequencies are included whereas in panels (c) and (d) only the modes with participation ratios smaller than 0.1 are considered. For all $T$ values, data from over 1000 samples were included. The gray dotted lines report the ${\omega }^3$ and ${\omega }^5$ trends.

Figure 3

Participation ratio [Eq. (3)] calculated for the (left panels) 1032-atoms and (right panels) 4008-atoms system at four different temperatures $T$.

Figure 4

Left panels: Distribution of the minimal vibrational frequency $P\left({\omega }_{min}\right)$ for the largest investigated system size $N=4008$ and four temperatures $T$. The dotted lines are the corresponding Weibull distributions, Eq. (12).

Figure 5

Density of states as computed from the standard bare Hessian computed at the mean coordinates ${\mathit{R}}_i$ at different temperatures $T=1,2,4,\mathrm{and}8\mathrm{K}$ for ${\mathrm{SiO}}_2$ glass samples of $N=1032$ atoms. Only the modes with participation ratio smaller than 0.1 are considered. For all $T$ values, statistics over 1000 samples was accounted.