Ioffe-Regel criterion and diffusion of vibrations in random lattices

作者：Y. M. Beltukov, V. I. Kozub, Д. А. Паршин · 发表于：Physical Review B · 年份：2013 · DOI：10.1103/physrevb.87.134203 · 被引用次数：134 · 研究领域：Theoretical and Computational Physics、Advanced Mathematical Modeling in Engineering、Diffusion and Search Dynamics

Using a stable random matrix approach, we consider the diffusion of vibrations in $3d$ harmonic lattices with strong force-constant disorder. Above some frequency ${\ensuremath{\omega}}_{\mathrm{IR}}$, 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 ${\ensuremath{\omega}}_{\mathrm{IR}}$ is close to the position of the boson peak. By changing the strength of disorder we can vary ${\ensuremath{\omega}}_{\mathrm{IR}}$ 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 ${\ensuremath{\omega}}_{\mathrm{IR}}$, the energy in the lattice is transferred by means of diffusion of vibrational excitations. We calculated the diffusivity of the modes $D(\ensuremath{\omega})$ using both the direct numerical solution of the Newton equations and the formula of Edwards and Thouless. It is nearly a constant above ${\ensuremath{\omega}}_{\mathrm{IR}}$ 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 ${\ensuremat...