Boson peak as a consequence of rigidity: A perturbation theory approach

Some evidence is provided that the boson peak and floppy modes share a common origin. In the particular case of periodic systems, we show how a boson peak occurs as a consequence of a reduction of constraints in an overconstrained lattice, in contrast to floppy modes, which occur for a reduction of constraints in a flexible or isostatic lattice. In fact, the present approach allows us to follow the transformation of the boson peak into a floppy mode when a system goes from rigid to flexible. We use perturbation theory and Green’s functions to see how resonances appear in the low-frequency region of the local vibrational density of states. For overconstrained lattices, we found that the boson peak frequency depends on the square root of the coordination of the lattice, and is at most 0.3 of the Debye frequency, a value close to the observed experimental ratio of 0.1. We also obtain the expected Rayleigh scattering for overconstrained networks, while we predict a different scattering for isostatic networks due to their critical nature. As an example, the effects of removing constraints are analyzed in a face-center-cubic lattice, and the same consequences are observed in a square network with and without diagonal links.

Figure 1

Examples of two kinds of rigid lattices; (a) isostatic (using cyclic boundary conditions), and (b) overconstrained. The diluted bond constraint $k\equiv k_0-\alpha k_0$ is also shown. When $\alpha =1$, the bond is removed. $M$ is the particles’ mass.

Figure 2

Evolution of the reduced vibrational density of sates $[{\rho }_d(\omega ;\alpha )/{\omega }^2]$ with respect to the perfect FCC ($\alpha =0$). When a bond is cut, $\alpha =1$(circles, green on line), a peak arises in ${\rho }_d(\omega ;\alpha =1)/{\omega }^2$ on a minimal frequency.

Figure 3

Real and Imaginary parts of $G_0(1)$ (blue and green lines, respectively) for the FCC lattice, and the function $f(\omega ;\alpha )=[1-2/\alpha ]/{\omega }^2$. The resonant frequencies (${\omega }_R$) are found at the intersection of the curves $\mathrm{Re}{G}_0(1)$ and $f(\omega ;\alpha )$, over different alpha values. Notice how $\mathrm{Im}{G}_0$ goes to zero as the frequency is reduced.

Figure 4

RVDOS $[{\rho }_d(\omega /{\omega }_m;\alpha )/(\omega /{\omega }_m)]$ for a square lattice with crossed diagonals. The case $\alpha =0$ corresponds to the pure network, and then gradually a bond is removed up to $\alpha =1$.

Figure 5

RVDOS $[{\rho }_d(\omega /{\omega }_m;\alpha )]$ of a large chain versus the normalized frequency for different $\alpha$ values of a diluted bond. The points and lines represent the numerical and theoretical results from perturbation theory, denoted by NR and PT, respectively. We can see an increase of low, frequency modes and a smoothing of the spectrum. The differences between PT and NR at very low frequencies are due to finite size effects.