47 pages, 13 figures. Contributed chapter to "Low-Temperature Thermal and Vibrational Properties of Disordered Solids" (A Half-Century of universal "anomalies" of glasses), Ed. M. A. Ramos, to appear
The boson peak (BP) is an excess of vibrational states over the Debye law appearing at terahertz frequencies. It is found in all glasses and marks the crossover between the long-wavelength behavior, where the solid can be considered as an isotropic continuum, and the region where the wavelength of the sound wave starts to experience the microscopic details of the structure. This chapter is devoted to a review of the main experimental observations regarding the vibrational dynamics of amorphous solids, as detected by neutron and X-ray scattering techniques. A first part of the chapter is devoted to the measurements of the BP and its evolution as a function of external parameters, such as temperature, pressure, or density. The second part of the chapter reviews the wavevector evolution of the dynamic structure factor, which provides evidence of a pseudo-acoustic propagating mode up to frequencies comparable to those of the BP. This longitudinal mode has a sound attenuation which follows the Rayleigh scattering law and a negative dispersion that can partially explain the deviation from the Debye law. At higher frequencies, the inelastic spectrum presents a complex pattern of vibrations, with evidences of two peaks in various systems. To conclude, we will highlight the information that can be gained on the nature of the glass vibrational modes from a comparison with the dynamics of the corresponding polycrystal.
We use a local projectional analysis method to investigate the effect of topological disorder on the vibrational dynamics in a model glass simulated by molecular dynamics. Evidence is presented that the vibrational eigenmodes in the glass are generically related to the corresponding eigenmodes of its crystalline counterpart via disorder-induced level-repelling and hybridization effects. It is argued that the effect of topological disorder in the glass on the dynamical matrix can be simulated by introducing positional disorder in a crystalline counterpart.
Low-frequency vibrations of glassy and crystalline orthoterphenyl are studied by means of neutron scattering. Phonon dispersions are measured along the main axes of a single crystal, and the corresponding longitudinal and transversal sound velocities are obtained. For glassy and polycrystalline samples, a density of vibrational states is determined and cross-checked against other dynamic observables. In the crystal, low-lying zone-boundary modes lead to an excess over the Debye density of states. In the glass, the boson peak is located at even lower frequencies. With increasing temperature, bo…
We present a novel approach for parameter-free modeling of the structural, dynamical and electronic properties of non-crystalline materials based on ab-initio Molecular Dynamics, improved signal processing technique and computer visualization. The method have been extensively tested by investigating hydrogen and silicon dynamics in hydrogenated amorphous silicon (a-Si:H). By comparing the theoretical and experimental vibrational spectra we demonstrate how to relate vibrational properties to the structural stability, bonding and hydrogen diffusion. We extracted microscopic characteristics that …
We investigate the high-frequency behavior of the density of vibrational states in three-dimensional elasticity theory with spatially fluctuating elastic moduli. At frequencies well above the mobility edge, instanton solutions yield an exponentially decaying density of states. The instanton solutions describe excitations, which become localized due to the disorder-induced fluctuations, which lower the sound velocity in a finite region compared to its average value. The exponentially decaying density of states (known in electronic systems as the Lifshitz tail) is governed by the statistics of a…
The anharmonic decay rates of atomic vibrations in amorphous silicon (a-Si) and paracrystalline silicon (p-Si), containing small crystalline grains embedded in a disordered matrix, are calculated using realistic structural models. The models are 1000-atom four-coordinated networks relaxed to a local minimum of the Stillinger-Weber interatomic potential. The vibrational decay rates are calculated numerically by perturbation theory, taking into account cubic anharmonicity as the perturbation. The vibrational lifetimes for a-Si are found to be on picosecond time scales, in agreement with the prev…
We show that viscoelastic effects play a crucial role in the damping of vibrational modes in harmonic amorphous solids. The relaxation of a given plane wave is described by a memory function of a semi-infinite one-dimensions mass-spring chain. The initial vibrational energy spreads from the first site of the chain to infinity. In the beginning of the chain, there is a barrier, which significantly reduces the decay of vibrational energy below the Ioffe-Regel frequency. To obtain the parameters of the chain, we present a numerically stable method, based on the Chebyshev expansion of the local vi…
We propose theoretical approach based on combination of graph theory and generalized Ising model (GIM), which enables systematic determination of extremal structures for crystalline solids without any information about interactions or constituent elements. The conventional approach to find such set of structure typically employs configurational polyhedra (CP) on configuration space based on GIM description, whose vertices can always be candidates to exhibit maximum or minimum physical quantities. We demonstrate that the present approach can construct extended CP whose vertices not only include…
Disordered solids exhibit unusual properties of their vibrational states and thermal conductivities. Recent progresses have well established the concept of "elastic heterogeneity", i.e., disordered materials show spatially inhomogeneous elastic moduli. In this study, by using molecular-dynamics simulations, we gradually introduce "disorder" into a numerical system to control its modulus heterogeneity. The system starts from a perfect crystalline state, progressively transforms into an increasingly disordered crystalline state, and finally undergoes structural amorphisation. We monitor independ…
Our recent study elucidate that information of density of states in configuration space (CDOS) for non-interacting system, characterized by spatial constraint on the system, plays essential role to determine thermodynamically equilibrium properties for interacting systems. Particularly for disordered states, variance of CDOS along all possible coordinations plays significant role. Despite this fact, even for binary system of crystalline solids, analytic expression for variance of CDOS as a function of composition has not been clarified so far. Here we successfully derive variance of CDOS as a …
Glasses have a large excess of low-frequency vibrational modes in comparison with crystalline solids. We show that such a feature is a necessary consequence of the geometry generic to weakly connected solids. In particular, we analyze the density of states of a recently simulated system, comprised of weakly compressed spheres at zero temperature. We account for the observed a) constancy of the density of modes with frequency, b) appearance of a low-frequency cutoff, and c) power-law increase of this cutoff with compression. We predict a length scale below which vibrations are very different fr…
Understanding the vibrational and thermal properties of amorphous solids is one of the most discussed and long-standing issues in condensed matter physics. Recent works have made significant steps towards understanding harmonic vibrational states. In particular, it has been established that quasi-localized vibrational modes emerge in addition to phonon-like vibrational modes. In this work, we study the anharmonic properties of these vibrational modes. We find that vibrational modes exhibit anharmonicities that induce particle rearrangements and cause transitions to different states. These anha…
The low-frequency vibrational and low-temperature thermal properties of amorphous solids are markedly different from those of crystalline solids. This situation is counter-intuitive because any solid material is expected to behave as a homogeneous elastic body in the continuum limit, in which vibrational modes are phonons following the Debye law. A number of phenomenological explanations have been proposed, which assume elastic heterogeneities, soft localized vibrations, and so on. Recently, the microscopic mean-field theories have been developed to predict the universal non-Debye scaling law.…
While soft quasilocalized vibrational modes are known to populate the low-frequency spectrum of glassy solids, their contribution to thermal properties is still not fully elucidated. We numerically show that, despite their spatially localized nature, these modes are as effective heat carriers as the delocalized ones and can contribute non-negligible to the total thermal conductivity in the low-temperature regime, especially for T < 0.05 Tg where Tg is the glass transition temperature. We further prove that the mutual coherence between the low-frequency quasilocalized modes and other modes expl…
We present a random matrix approach to study general vibrational properties of stable amorphous solids with translational invariance using the correlated Wishart ensemble. Within this approach, both analytical and numerical methods can be applied. Using the random matrix theory, we found the analytical form of the vibrational density of states and the dynamical structure factor. We demonstrate the presence of the Ioffe-Regel crossover between low-frequency propagating phonons and diffusons at higher frequencies. The reduced vibrational density of states shows the boson peak, which frequency is…
High Entropy Alloys (HEAs) are designed by mixing multiple metallic species in nearly the same amount to obtain crystalline or amorphous materials with exceptional mechanical properties. Here we use molecular dynamics simulations to investigate the role of positional and compositional disorder in determining the low-frequency vibrational properties of CrMnFeCoNi HEAs. Our results show that the expected dependence of the density of states on the frequency as $D(ω)\simω^4$ is recovered for amorphous HEAs and is also observed for partially crystallized alloys with deviations that depend on the de…
In crystals, molecules thermally vibrate around the periodic lattice sites. Vibrational motions are well understood in terms of phonons, which carry heat and control heat transport. The situation is notably different in disordered solids, where vibrational excitations are not phonons and can be even localized. Recent numerical work has established the concept of elastic heterogeneity: disordered solids show inhomogeneous local mechanical response. Clearly, the heterogeneous nature of elastic properties strongly influences vibrational and thermal properties, and it is expected to be the origin …
A random matrix approach is used to analyze the vibrational properties of amorphous solids. We investigated a dynamical matrix M=AA^T with non-negative eigenvalues. The matrix A is an arbitrary real NxN sparse random matrix with n independent non-zero elements in each row. The average values <A_{ij}>=0 and dispersion <A_{ij}^2>=V^2 for all non-zero elements. The density of vibrational states g(w) of the matrix M for N,n >> 1 is given by the Wigner quarter circle law with radius independent of N. We argue that for n^2 << N this model can be used to describe the interaction of atoms in amorphous…
Many plastic crystals, molecular solids with long-range, center-of-mass crystalline order but dynamic disorder of the molecular orientations, are known to exhibit exceptionally high ionic conductivity. This makes them promising candidates for applications as solid-state electrolytes, e.g., in batteries. Interestingly, it was found that the mixing of two different plastic-crystalline materials can considerably enhance the ionic dc conductivity, an important benchmark quantity for electrochemical applications. An example is the admixture of different nitriles to succinonitrile, the latter being …
By the adoption of a quaternion orientational order parameter to describe solidification, the topological origins of the thermal transport properties of crystalline and non-crystalline solid states are considered herein. Global orientational order, achieved by spontaneous symmetry breaking, is prevented at finite temperatures for systems that exist in restricted dimensions (Mermin-Wagner theorem). Just as complex ordered systems exist in restricted dimensions in 2D and 1D, owing to the dimensionality of the order parameter, quaternion ordered systems in 4D and 3D exist in restricted dimensions…
A solid conducts heat through both transverse and longitudinal acoustic phonons, but a liquid employs only longitudinal vibrations. Here, we report that the crystalline solid AgCrSe2 has liquid-like thermal conduction. In this compound, Ag atoms exhibit a dynamic duality that they are exclusively involved in intense low-lying transverse acoustic phonons while they also undergo local fluctuations inherent in an order-to-disorder transition occurring at 450 K. As a consequence of this extreme disorder-phonon coupling, transverse acoustic phonons become damped as approaching the transition temper…
The low-frequency vibrations of glasses are markedly different from those of crystals. These vibrations have recently been categorized into two types: spatially extended vibrations, whose vibrational density of states (vDOS) follows a non-Debye quadratic law, and quasilocalized vibrations (QLVs), whose vDOS follows a quartic law. The former are explained by elasticity theory with quenched disorder and microscopic replica theory as being a consequence of elastic instability, but the origin of the latter is still debated. Here, we show that the latter can also be directly derived from elasticity…
There are three kinds of solid states of matter that can exist in physical space: quasicrystalline (quasiperiodic), crystalline (periodic) and amorphous (aperiodic). Herein, we consider the degree of orientational order that develops upon the formation of a solid state to be characterized by the application of quaternion numbers. The formation of icosahedral quasicrystalline solids is considered alongside the development of bulk superfluidity, characterized by a complex order parameter, that occurs by spontaneous symmetry breaking in three-dimensions. Crystalline solid states are viewed as hig…
Amorphous solids display numerous universal features in their mechanics, structure, and response. Current models assume heterogeneity in mesoscale elastic properties, but require fine-tuning in order to quantitatively explain vibrational properties. A complete model should derive the magnitude and character of elastic heterogeneity from an initially structureless medium, through a model of the quenching process during which the temperature is rapidly lowered and the solid is formed. Here we propose a field-theoretic model of a quench, and compute structural, mechanical, and vibrational observa…
We identify the possibility of realizing higher order topological (HOT) phases in noncrystalline or amorphous materials. Starting from two and three dimensional crystalline HOT insulators, accommodating topological corner states, we gradually enhance structural randomness in the system. Within a parameter regime, as long as amorphousness is confined by outer crystalline boundary, the system continues to host corner states, yielding amorphous HOT insulators. However, as structural disorder percolates to the edges, corner states start to dissolve into amorphous bulk, and ultimately the system be…
Glasses have a large excess of low-frequency vibrational modes in comparison with most crystalline solids. We show that such a feature is a necessary consequence of the weak connectivity of the solid, and that the frequency of modes in excess is very sensitive to the pressure. We analyze in particular two systems whose density D(w) of vibrational modes of angular frequency w display scaling behaviors with the packing fraction: (i) simulations of jammed packings of particles interacting through finite-range, purely repulsive potentials, comprised of weakly compressed spheres at zero temperature…
One of the long-standing issues concerning the thermal properties of amorphous solids is the complex pattern of phonon transport. Recent advances in experiments and computer simulations have indicated a crossover from Rayleigh scattering to viscous damping. A number of theories have been proposed, yet critical tests are missing and the validity of these theories is unclear. In particular, the precise location of the crossover frequency remains controversial, and more critically, even the validity of the Rayleigh scattering itself has been seriously questioned. The present work settles these is…
Finding new ionic conductors that enable significant advancements in the development of energy-storage devices is a challenging goal of current material science. Aside of material classes as ionic liquids or amorphous ion conductors, the so-called plastic crystals (PCs) have been shown to be good candidates combining high conductivity and favourable mechanical properties. PCs are formed by molecules whose orientational degrees of freedom still fluctuate despite the material exhibits a well-defined crystalline lattice. Here we show that the conductivity of Li+ ions in succinonitrile, the most p…
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 $D(ω)$ 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 logarithmic enhancement of (Rayleigh) phonon attenuation. A logarithmic correction of the form $\sim -ω^{2}\lnω$ is predicted …
We develop a new method to isolate localized defects from extended vibrational modes in disordered solids. This method augments particle interactions with an artificial potential that acts as a high-pass filter: it preserves small-scale structures while pushing extended vibrational modes to higher frequencies. The low-frequency modes that remain are "bare" defects; they are exponentially localized without the quadrupolar tails associated with elastic interactions. We demonstrate that these localized excitations are excellent predictors of plastic rearrangements in the solid. We characterize se…
The low-temperature thermal and transport properties of an unusual kind of crystal exhibiting minimal molecular positional and tilting disorder have been measured. The material, namely, low-dimensional, highly anisotropic pentachloronitrobenzene (PCNB) has a layered structure of rhombohedral parallel planes in which the molecules execute large-amplitude in-plane as well as concurrent out-of-plane librational motions. Our study reveals that low-temperature glassy anomalies can be found in a system with minimal disorder due to the freezing of (mostly in-plane) reorientational jumps of molecules …
Recently, progress has been made in the understanding of anomalous vibrational excitations in amorphous solids. In the lowest-frequency region, the vibrational spectrum follows a non-Debye quartic law, which persists up to zero frequency without any frequency gap. This gapless vibrational density of states (vDOS) suggests that glasses are on the verge of instability. This feature of marginal stability is now highlighted as a key concept in the theories of glasses. In particular, the elasticity theory based on marginal stability predicts the gapless vDOS. However, this theory yields a quadratic…
The low-temperature properties of glasses present important differences with respect to crystalline matter. In particular, models such as the Debye model of solids, which assume the existence of an underlying regular lattice, predict that the specific heat of solids varies with the cube of temperature at low temperatures. Since the 1970s' at least, it is a well established experimental fact that the specific heat of glasses is instead just linear in $T$ at $T \sim 1K$, and presents a pronounced peak when normalized by $T^{3}$, known as the boson peak. Here we present a new approach which sugge…
We study the phase ordering dynamics of a two dimensional model colloidal solid using molecular dynamics simulations. The colloid particles interact with each other with a Hamaker potential modified by the presence of equatorial "patches" of attractive and negative regions. The total interaction potential between two such colloids is, therefore, strongly directional and has three-fold symmetry. Working in the canonical ensemble, we determine the tentative phase diagram in the density-temperature plane which features three distinct crystalline ground states viz, a low density honeycomb solid fo…
As molecular dynamics is increasingly used to characterize non-crystalline materials, it is crucial to verify that the numerical model is accurate enough, consistent with experimental data and can be used to extract various characteristics of disordered systems. In most cases the only derived property used to test the "realism" of the models has been the radial distribution function. We report extensive ab-initio simulation of hydrogenated amorphous silicon that demonstrates that although agreement with the RDF is a necessary requirement, this protocol is insufficient for the validation of a m…
The vibrational spectra of solids, both ordered and amorphous, in the low-energy regime, control the thermal and transport properties of materials, from heat capacity to heat conduction, electron-phonon couplings, conventional superconductivity etc. The old Debye model of vibrational spectra at low energy gives the vibrational density of states (VDOS) as proportional to the frequency squared, but in many materials the spectrum departs from this law which results in a peak upon normalizing the VDOS by frequency squared, which is known as the "boson peak". A description of the VDOS of solids (bo…
We calculate the energy and heat capacity of a liquid on the basis of its elastic properties and vibrational states. The experimental decrease of liquid heat capacity with temperature is attributed to the increasing loss of two transverse modes with frequency $ω<1/τ$, where $τ$ is liquid relaxation time. In a simple model, liquid heat capacity is related to viscosity and is compared with the experimental data of mercury. We also calculate the vibrational energy of a quantum liquid, and show that transverse phonons can not be excited in the low-temperature limit. Finally, we discuss the implica…
Many systems, including polymers and molecular liquids, when adequately cooled and/or compressed, solidify into a disordered solid, i.e., a glass. The~transition is not abrupt, featuring progressive decrease of the microscopic mobility and huge slowing down of the relaxation.} A~distinctive aspect of glass-forming materials is the microscopic dynamical heterogeneity (DH), i.e., the presence of regions with almost immobile particles coexisting with others where highly mobile ones are located. Following the first compelling evidence of a strong correlation between vibrational dynamics and ultras…
The phonon spectra of solids, described through the measurable vibrational density of states (VDOS), provide a wealth of information about the underlying atomic structure and bonding, and they determine fundamental macroscopic properties such as specific heat, thermal conductivity and superconductivity. In amorphous solids, structural disorder generates a number of poorly understood phenomena that are absent in crystals. Starting from an effective field theory description which accounts for disorder in terms of momentum diffusion, we provide a unified theoretical description of the VDOS of amo…
