Vibrational dynamics of filled skutterudites $M_{1-x}$Fe$_4$Sb$_{12}$ ($M$ = Ca, Sr, Ba, and Yb)

First-principles density-functional theory and lattice-dynamics calculations were performed to study the vibrational dynamics and related observables of the ternary compounds $M_{1-x}$Fe$_4$Sb$_{12}$ ($A$ = Ca, Sr, Ba, Yb). The calculation results are supported by experimental data, which were obtained from neutron inelastic scattering, neutron-diffraction, and heat-capacity measurements. Within the calculation approach based on the theory of harmonic solids all observables are linked to the phonon density of states $Z(\omega)$. The good agreement with experimental data shows that the vibrational dynamics of the ternary skutterudite structures can be described by a set of normal modes. Features in the experimentally obtained density of states $G(\omega)$ reflecting the variation in properties (mass, ionic radius) of the cations Ca, Sr, Ba, and Yb are reproduced by the calculations.