Revealing the Geometrical and Vibrational Properties of the Defects Driving the Boson Peak

Abstract

In amorphous solids, the vibrational density of states shows an excess of modes over the Debye model, known as the boson peak, whose origin remains unclear. Studies suggest a link to quasi-localized nonphononic vibrations or 'defects,' but identifying them is challenging due to hybridization with phonons that renders methods based on localization properties, such as the participation ratio, unreliable. We introduce a practical method to separate hybridized phonons from localized vibrations and find that boson peak phonons hybridize with compact, two-dimensional defects exhibiting oscillatory pure shear deformations. These two-dimensional defects are also exposed by the procedure recently employed to identify stringlets (Nature Physics volume 18, pages 669-677 (2022)), suggesting that these may not be one-dimensional objects as speculated. Our work demonstrates the presence of localized defects at the boson peak frequency and provides a comprehensive characterization of their vibrational and geometric properties, resolving the tension between the concepts of quasi-localized quadrupolar defects and stringlets.

Figures (10)

• Figure 1: a Colour map of the magnitude $|{\bm e}_{k,i}|^2$ of the particle displacement associated with an eigenmode with $\omega_k=\omega_{\rm bp}$. Swirls characterizing hybridization with plane waves are observed. b The cage-relative displacement, $|{\bm e}_{k,i}^{\rm cr}|^2$ reveals the presence of many vibrational defects not apparent in the standard measure. In these plots, the magnitudes are scaled so that their average value is $1$.
• Figure 2: a Probability distribution of the number of particles belonging to each defect in 3D and 2D (inset). b Average coordination number of the particles in clusters of size $n$. c We normalise ($\sum \sigma_i = 1$) and order ($\sigma_i > \sigma_{i+1}$) the standard deviations of the particles' positions along the principal axis of each 'defect', considering defects with at least $n=3$ particles. In two dimensions, the probability distribution of $P(\sigma_1)$ peaks at intermediate values and drops at large ones, indicating that the defects are not one-dimensional ($\sigma_1 = 1$), but two-dimensional objects, albeit not precisely circular $(\sigma_1=1/2)$. d In three dimensions, the probability distribution $P(\sigma_1,\sigma_2)$ clarifies that clusters are not one-dimensional ($\sigma_1 = 1$) or two-dimensional $(\sigma_1+\sigma_2 = 1)$. We thus assume they are three-dimensional, albeit non-spherical $(\sigma_1 = \sigma_2 = 1/3)$. Data for $x_c = 1.7$ are shifted vertically for clarity. e Radius of gyration, $R_g$, as a function of cluster size $n$. 3D data are shifted by a factor $3$ for clarity. f The characteristic length scale $\lambda$ of defects governs the boson peak frequency in two and three dimensions. In all panels, symbols identify the values of $x_c$.
• Figure 3: 1 Heat map of the magnitude of the coarse-grained cage-relative displacement, normalised by its average value, with superimposed quadrupolar displacements of the defects. The figure refers to $x_c =1.7$ and depicts 1/12th of the system's area. b Density map of the two largest (in modulus) eigenvalues of the deviatoric strain associated with the defects, normalised to its maximum value. The density is higher close to the $\Upsilon_2=-\Upsilon_1$ line, implying that the quadrupoles correspond to pure shear deformations. c We represent the quadrupolar deformations by attaching to each defect's centre of mass the displacements along the principal axis $\pm \Upsilon_1{\bm u}_1$ (blue), $\pm \Upsilon_2{\bm u}_2$ (green) and $\pm \Upsilon_3{\bm u}_3$ (red). Since quadrupoles mostly correspond to pure-shear deformations, ${\bf u}_3$ is barely visible.
• Figure 4: a The coordination number of vibrability clusters increases with $n$, following a trend similar to that of clusters formed based on cage-relative displacement (see Fig.3b). b Particles exhibiting high cage-relative displacement at the boson peak (red) and high vibrability (blue), both in the top 5%, for $x_c = 1.7$. Strong correlations between the two are observed. c The fraction of particles displaying both large cage-relative displacement and high vibrability significantly exceeds the expectation under random conditions.
• Figure 5: a Eigenvector at the boson peak frequency for a two-dimensional system with $x_c = 1.7$. It is arduous to spot in similar modes isolated quadrupolar-looking defects lerner2023bosonMoriel2024. b Defects identified by our approach. c A zoom on a region corresponding to 1/12th of the system's area.