Local inversion-symmetry breaking controls the boson peak in glasses and crystals

It is well known that amorphous solids display a phonon spectrum where the Debye $\sim {\omega }^2$ law at low frequency melds into an anomalous excess-mode peak (the boson peak) before entering a quasilocalized regime at higher frequencies dominated by scattering. The microscopic origin of the boson peak has remained elusive despite various attempts to put it in a clear connection with structural disorder at the atomic/molecular level. Using numerical calculations on model systems, we show that the microscopic origin of the boson peak is directly controlled by the local breaking of center-inversion symmetry. In particular, we find that both the boson peak and the nonaffine softening of the material display a strong correlation with a new order parameter describing the local inversion symmetry of the lattice. The standard bond-orientational order parameter, instead, is shown to be inadequate and cannot explain the boson peak in randomly-cut crystals with perfect bond-orientational order. Our results bring a unifying understanding of the boson peak anomaly for model glasses and defective crystals in terms of a universal local symmetry-breaking principle of the lattice.

Figure 1

(a) One realization of our random network for $Z=6$. The dots represent atoms which are connected by harmonic springs. Local inversion-symmetry is evidently broken by the randomness and lack of correlation in bond orientations. (b) Schematic 2D picture of a regular lattice where locally the removal of a bond breaks the inversion-symmetry on atom $i$. The main consequence of inversion-symmetry breaking induced by the cutting of the bond is the imbalance of NN forces (arrows) acting on atom $i$ when it reaches its affine position under strain. The net force acting on atom $i$ in its affine position has to be released through an additional nonaffine displacement. The atom $i$ also acts as a scattering and quasilocalization center [8] for incoming vibrational excitations.

Figure 2

Shear modulus as a function of connectivity. (a) Shear modulus for the RN model glass. (b) Shear modulus of the FCC crystal with randomly cut bonds.

Figure 3

Vibrational density of states $D\left(\omega \right)$ calculated for the RN glass (solid line) and for the randomly-cut FCC crystal (dotted line), for four different values of atomic connectivity $Z=6,7,8,9$. The solid arrow indicates the approximate position of the boson peak frequency, ${\omega }_{\mathrm{BP}}$, while the dashed arrow indicates the position of the lowest van Hove peak, ${\omega }_{\mathrm{VH}}$. For $Z=6,{\omega }_{\mathrm{BP}}\approx 0$. The low-energy part of the spectrum, including the boson peak, appears practically identical for the RN glass and for the randomly cut FCC crystal.

Figure 4

Vibrational density of states normalized by the Debye behavior, $D\left(\omega \right)/{\omega }^2$, for the same data of Fig. 3.

Figure 5

Crossover frequency ${\omega }^{*}$ (which marks the start of the linear regime in $\omega$), and the boson peak frequency ${\omega }_{\mathrm{BP}}$, both plotted as a function of the average connectivity $Z$. The data points refer to both random network and defective FCC systems, since both systems have exactly the same values of ${\omega }^{*}$ and ${\omega }_{\mathrm{BP}}$. The dashed lines are guides to the eye.

Figure 6

Comparison between the inversion-symmetry order parameter $\left(F_{\mathrm{IS}}\right)$ and the standard bond-orientational order parameter $\left(F_6\right)$ for the RN glasses and for the randomly-cut FCC crystals as a function of connectivity $Z$. There is a perfect collapse of $F_{\mathrm{IS}}$ for the two systems onto a master curve as a function of $Z$.

Figure 7

Participation ratios calculated for the RN glass and for the randomly-cut FCC crystal, for different values of average connectivity $Z$.