Effect of cation substitution on the lattice dynamics and ferroelectric instability of cubic BaTiO$_3$ and BaZrO$_3$ doped with Bi and La ions

The spectra of lattice vibration frequencies of solid solutions Ba$_{1-x}$Bi$_{2x/3}\Box_{x/3}$Ti(Zr)O$_3$ and Ba$_{1-x}$La$_x$Ti(Zr)$_{1-x/4}\Box_{x/4}$O$_3$ are calculated in terms of a generalized Gordon–Kim model with inclusion of the dipole and quadrupole polarizabilities. Over the entire concentration range, the calculated phonon spectra contain a ferroelectric soft mode. The effect of various interactions on the ferroelectric instability of these solid solutions is studied. It is shown that the character of ferroelectric instability is largely determined by the mechanism of charge compensation.