Quantum theory of elastic strings and the thermal conductivity of glasses

Abstract

We study the thermal conductivity of amorphous solids by constructing a continuum model whose degrees of freedom are propagating vibrational modes (phonons) and extended Volterra dislocation line defects with their own vibrational degrees of freedom which do not propagate in space. Our working assumption is that these additional degrees of freedom account for the "boson peak" that is observed in glassy materials. This identification allows us to obtain the length distribution of dislocations from experimental data of the boson peak for each material, which we use as input to calculate the phonon self-energy in a quantum field theory framework using that the phonon-dislocation interaction is given by the Peach-Koehler force. The tail of the distribution for long dislocations is consistent with an $L^{-5}$ power law. Our results show that this power law yields a linear rise in the thermal conductivity, as observed in glasses at low temperatures. We then consider two approaches to describe thermal conductivity data quantitatively. In the simplest approach we only keep the low-frequency behavior of the phonon self-energy with one free parameter, plus an adjustable UV cutoff. In the more realistic approach we keep the full frequency dependence of the phonon self-energy as dictated by the phonon-dislocation interaction plus an additional contribution due to scattering with point defects, with a cutoff set by the typical interatomic spacing of the material. We obtain a satisfactory description of thermal conductivity data with both approaches. We conclude by discussing prospects to test the predictive power of this model.