Phonons, Diffusons, and the Boson Peak in Two-Dimensional Lattices with Random Bonds

Abstract

Within the model of stable random matrices possessing translational invariance, a two-dimensional (on a square lattice) disordered oscillatory system with random strongly fluctuating bonds is considered. By a numerical analysis of the dynamic structure factor S(q, ω), it is shown that vibrations with frequencies below the Ioffe-Regel frequency ω_IR are ordinary phonons with a linear dispersion law ω(q) ∝ q and a reciprocal lifetime б ~ q³. Vibrations with frequencies above ω_IR, although being delocalized, cannot be described by plane waves with a definite dispersion law ω(q). They are characterized by a diffusion structure factor with a reciprocal lifetime б ~ q², which is typical of a diffusion process. In the literature, they are often referred to as diffusons. It is shown that, as in the three-dimensional model, the boson peak at the frequency ωb in the reduced density of vibrational states g(ω)/ω is on the order of the frequency ω_IR. It is located in the transition region between phonons and diffusons and is proportional to the Young’s modulus of the lattice, ω _b ≃E.