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Register a new userA random matrix approach to the boson peak and Ioffe-Regel criterion in amorphous solids
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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 close to the Ioffe-Regel crossover. We also present a simple numerical random matrix model with finite interaction radius, which properties rapidly converges to the analytical results with increasing the interaction radius. For fine interaction radius, the numerical model demonstrates the presence of the quasilocalized vibrations with a power-law low-frequency density of states.
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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 vibrational density of states.
The vibrational properties of model amorphous materials are studied by combining complete analysis of the vibration modes, dynamical structure factor and energy diffusivity with exact diagonalization of the dynamical matrix and the Kernel Polynomial Method which allows a study of very large system sizes. Different materials are studied that differ only by the bending rigidity of the interactions in a Stillinger-Weber modelization used to describe amorphous silicon. The local bending rigidity can thus be used as a control parameter, to tune the sound velocity together with local bonds directionality. It is shown that for all the systems studied, the upper limit of the Boson peak corresponds to the Ioffe-Regel criterion for transverse waves, as well as to a minimum of the diffusivity. The Boson peak is followed by a diffusivitys increase supported by longitudinal phonons. The Ioffe-Regel criterion for transverse waves corresponds to a common characteristic mean-free path of 5-7 AA{} (which is slightly bigger for longitudinal phonons), while the fine structure of the vibrational density of states is shown to be sensitive to the local bending rigidity.
We consider diffusion of vibrations in 3d harmonic lattices with strong force-constant disorder. Above some frequency w_IR, corresponding to the Ioffe-Regel crossover, notion of phonons becomes ill defined. They cannot propagate through the system and transfer energy. Nevertheless most of the vibrations in this range are not localized. We show that they are similar to diffusons introduced by Allen, Feldman et al., Phil. Mag. B 79, 1715 (1999) to describe heat transport in glasses. The crossover frequency w_IR is close to the position of the boson peak. Changing strength of disorder we can vary w_IR from zero value (when rigidity is zero and there are no phonons in the lattice) up to a typical frequency in the system. Above w_IR the energy in the lattice is transferred by means of diffusion of vibrational excitations. We calculated the diffusivity of the modes D(w) using both the direct numerical solution of Newton equations and the formula of Edwards and Thouless. It is nearly a constant above w_IR and goes to zero at the localization threshold. We show that apart from the diffusion of energy, the diffusion of particle displacements in the lattice takes place as well. Above w_IR a displacement structure factor S(q,w) coincides well with a structure factor of random walk on the lattice. As a result the vibrational line width Gamma(q)=D_u q^2 where D_u is a diffusion coefficient of particle displacements. Our findings may have important consequence for the interpretation of experimental data on inelastic x-ray scattering and mechanisms of heat transfer in glasses.
We establish a phase diagram of a model in which scalar waves are scattered by resonant point scatterers pinned at random positions in the free three-dimensional (3D) space. A transition to Anderson localization takes place in a narrow frequency band near the resonance frequency provided that the number density of scatterers $rho$ exceeds a critical value $rho_c simeq 0.08 k_0^{3}$, where $k_0$ is the wave number in the free space. The localization condition $rho > rho_c$ can be rewritten as $k_0 ell_0 < 1$, where $ell_0$ is the on-resonance mean free path in the independent-scattering approximation. At mobility edges, the decay of the average amplitude of a monochromatic plane wave is not purely exponential and the growth of its phase is nonlinear with the propagation distance. This makes it impossible to define the mean free path $ell$ and the effective wave number $k$ in a usual way. If the latter are defined as an effective decay length of the intensity and an effective growth rate of the phase of the average wave field, the Ioffe-Regel parameter $(kell)_c$ at the mobility edges can be calculated and takes values from 0.3 to 1.2 depending on $rho$. Thus, the Ioffe-Regel criterion of localization $kell < (kell)_c = mathrm{const} sim 1$ is valid only qualitatively and cannot be used as a quantitative condition of Anderson localization in 3D.
The density of vibrational states $g(omega)$ of an amorphous system is studied by using the random-matrix theory. Taking into account the most important correlations between elements of the random matrix of the system, equations for the density of vibrational states $g(omega)$ are obtained. The analysis of these equations shows that in the low-frequency region the vibrational density of states has the Debye behavior $g(omega) sim omega^2$. In the higher frequency region, there is the boson peak as an additional contribution to the density of states. The obtained equations are in a good agreement with the numerical results and allow us to find an exact shape of the boson peak.
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