Randomness-induced redistribution of vibrational frequencies in amorphous solids

Abstract

Much of the discussion in the literature of the low frequency part of the density of states of amorphous solids was dominated for years by comparing measured or simulated density of states to the classical Debye model. Since this model is hardly appropriate for the materials at hand, this created some amount of confusion regarding the existence and universality of the so- called “Boson Peak” which results from such comparisons. We propose that one should pay attention to the different roles played by different aspects of disorder, the first being disorder in the interaction strengths, the second positional disorder, and the third coordination disorder. These have different effects on the low-frequency part of the density of states. We examine the density of states of a number of tractable models in one and two dimensions, and reach a clearer picture of the softening and redistribution of frequencies in such materials. We discuss the effects of disorder on the elastic moduli and the relation of the latter to frequency softening, reaching the final conclusion that the Boson peak is not universal at all. The study of the density of states of solid materials started with attempts to understand the temperature dependence of the specific heat at low temperatures, say CV ≡ (∂U/∂T )V where U is the energy and T the temperature of the system. This called for a microscopic theory for solids, and the first one was developed by Einstein, assuming that in d dimensions each atom is represented as a d-dimensional harmonic oscillator [1] (in the original paper the case d = 3 was considered). In this article Planck’s quantization assumption, which was originally applied to radiation, was extended to solid vibrations [2]. In the case of dN linear oscillators each with its own frequency ω i, Einstein’s result can be expressed as