Ioffe-Regel criterion and diffusion of vibrations in random lattices

Using a stable random matrix approach, we consider the diffusion of vibrations in 3d harmonic lattices with strong force-constant disorder. Above some frequency omega(IR), corresponding to the Ioffe-Regel crossover, the 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 et al. [P. B. Allen, J. L. Feldman, J. Fabian, and F. Wooten, Phil. Mag. B 79, 1715 (1999)] to describe heat transport in glasses. The crossover frequency omega(IR) is close to the position of the boson peak. By changing the strength of disorder we can vary omega(IR) from zero value (when the rigidity is zero and there are no phonons in the lattice) up to a typical frequency in the system. Above omega(IR), the energy in the lattice is transferred by means of diffusion of vibrational excitations. We calculated the diffusivity of the modes D(omega) using both the direct numerical solution of the Newton equations and the formula of Edwards and Thouless. It is nearly a constant above omega(IR) and goes to zero at the localization threshold. We show that apart from the diffusion of energy, a diffusion of particle displacements in the lattice takes place as well. Above omega(IR), a displacement structure factor S(q, omega) coincides well with a structure factor of random walk on the lattice. As a result, the vibrational line width is Gamma(q) = D(u)q(2), where D-u is a diffusion coefficient of particle displacements. These findings may be important for the interpretation of experimental data on inelastic x-ray scattering and mechanisms of heat transfer in glasses. We also show that the scaling with model parameters in our model maps directly onto the scaling observed in the jamming transition model. DOI: 10.1103/PhysRevB.87.134203