Energy Renormalization and Damping of Long-Wavelength Phonons in Amorphous Solids

Abstract

We consider the energy renormalization and damping of long-wavelength phonons in monatomic amorphous solids within the harmonic approximation due to the influence of long-range structure fluctuations. By employing a formalism analogous to the Matsubara-Kaneyoshi fomalism (MKF) established for amorphous ferromagnets we derive the structure averaged Green's function 〈Gqq′(E)〉 which contains the phonon self-energy in “improved quasi-crystalline approximation” as a start approximation in the equation of motion for the Green's function, thus containing important structural information already. The typical static structure factor in the small-angle region is approximated by convenient analytical model functions. We then derive the damping and the sound velocity renormalization for long-wavelength longitudinal and transverse acoustic phonons, where the former is found to show a characteristic q4 behavior in the small-q range (phonon Rayleigh scattering) due to the density fluctuations acting as “point defects” for extremely long-wavelength phonons, while in the larger-q range a q2 law is found. The renormalization R(q) shows generally uniform behavior for all model functions used for the static structure factor. Extrapolation of the renormalized energy from the larger-q region provides an apparent gap energy induced by the amorphous structure, which can be explained by the nascency of “compensational phonons” which accompany the phonon compensating the inhomogeneities, thus forming a “phonon-dressed phonon”. The mentioned features of long-wavelength phonons in amorphous solids show wide similarities to long-wavelength magnons in amorphous ferromagnets.