Weak localization and anharmonicity of phonons

Abstract

The problem of localization of phonons in disordered materials is studied in the framework of the weak-localization theory. Quantum correction to phonon diffusion is calculated by the resummation technique of maximally crossed diagrams in two and three dimensions. The strong energy dependence of the elastic mean free path—the Rayleigh-Klemens scattering—is responsible for the existence of a threshold frequency ${\omega }_3^{\mathrm{*}}$ where the diffusion constant vanishes. The value of ${\omega }_3^{\mathrm{*}}$ depends only on the local fluctuation of masses and on the Debye frequency in three dimensions. This threshold describes the phenomenon of localization of phonon density fluctuations or second sound. The self-energies of the phonons are strongly affected by this quantum correction via the anharmonic interactions. The two basic anharmonic couplings contribute to the one-phonon renormalization and provide shortening of the mean lifetime as well as excess of spectral density in the vicinity of the threshold. In two dimensions, as for the electrons, the dynamical quantum correction diverges logarithmically when the frequency goes to zero. A procedure of convergence is used by cutting off the low-frequency contributions at the inelastic relaxation rate. Renormalization of phonons are obtained in a self-consistent way. Finally a tentative application of the previous results to the low-temperature properties of glasses is discussed. In particular the existence of a plateau in thermal conductivity accompanied by excess specific heat in all the glasses measured so far could be understood as the manifestation of localization of acoustic-phonon density at the critical threshold ${\omega }_3^{\mathrm{*}}$. .AE