Theory of phonon glass behaviour in host-guest thermoelectrics with avoided crossing

An analytical model is developed to describe the phonon dispersion relations of thermoelectrics with the presence of heavy guest atoms (rattlers). The model also accounts for anharmonic effects in phonon damping. The spectrum of low energy states contains acoustic-like and (soft) optical-like modes, which display the typical avoided crossing, and which can be derived analytically by considering the dynamical coupling between host lattice and guest rattlers. Inclusion of anharmonic damping in the model allows us, for the first time, to compute the vibrational density of states (VDOS) and the specific heat, unveiling the anomalous boson peak (BP) relating to the glassy behaviour of phonons in the otherwise crystalline material. We discuss the dynamics of the BP as a function of the strength of the interaction between the soft modes and the anharmonic lattice, and of the energy gap between the two avoided-crossing branches. Moreover, we find a robust linear correlation between the BP frequency and the energy of the soft optical-like modes.