Vibrational density of states of amorphous solids with long-ranged power-law-correlated disorder in elasticity

Bingyu Cui,Alessio Zaccone

doi:10.1140/epje/i2020-11995-2

Abstract

.A theory of vibrational excitations based on power-law spatial correlations in the elastic constants (or equivalently in the internal stress) is derived, in order to determine the vibrational density of states D(omega) of disordered solids. The results provide the first prediction of a boson peak in amorphous materials where spatial correlations in the internal stresses (or elastic constants) are of power-law form, as is often the case in experimental systems, leading to a logarithmic enhancement of (Rayleigh) phonon attenuation. A logarithmic correction of the form sim -omega^{2}lnomega is predicted to occur in the plot of the reduced excess DOS for frequencies around the boson peak in 3D. Moreover, the theory provides scaling laws of the density of states in the low-frequency region, including a simomega^{4} regime in 3D, and provides information about how the boson peak intensity depends on the strength of power-law decay of fluctuations in elastic constants or internal stress. Analytical expressions are also derived for the dynamic structure factor for longitudinal excitations, which include a logarithmic correction factor, and numerical calculations are presented supporting the assumptions used in the theory.Graphical abstract

Highlights

Understanding the physics of vibrational spectra of disordered systems is a classical topic in condensed-matter physics [1,2,3,4]
We reveal that the boson peak picks up a logarithmic correction which is most evident in the excess density of states (DOS)
We developed a theory of vibrational excitations in disordered media with long-ranged power-law– correlated disorder to extract the density of states (DOS)

Summary

Introduction

Understanding the physics of vibrational spectra of disordered systems is a classical topic in condensed-matter physics [1,2,3,4]. Following numerical evidence of a logarithmic enhancement correction of the form Γ (k) ∼ −kd+1 ln(k) to the Rayleigh scattering law (with wavenumber k, in d-dimension) [59], it has been shown analytically that long-ranged power-law spatial correlations in elasticity, or equivalently in the internal stresses, are the cause of such enhancement [57]. [57], by finding the rigorous solution to the self-consistent anistropic wave propagation problem, it was possible to derive the logarithmic Rayleigh scattering law, which is ubiquitously observed in experiments and simulations [61,62,63,64,65,66,67,68], and to show that it is the direct result of the power-law correlation in internal stresses or elastic constants. We examine the asymptotic scaling behavior of the DOS in the frequency regimes where modes are quasi-localized due to the disorder ( undergoing diffusive-like propagation instead of ballistic propagation typical of standard phonons, as demonstrated for glasses in earlier works [52])

Theory for longitudinal excitations in 2D

General theory for amorphous solids in 3D

Conclusions