Effect of ${\mathrm{Ba}}^{2+}$ substitution on the stability of the antiferroelectric and ferroelectric phases in $\left({\mathrm{Pb}}_{1-x}{\mathrm{Ba}}_x\right){\mathrm{ZrO}}_3:$ Phenomenological theory considerations

The relative stability of the ferroelectric rhombohedral ${(F}_R)$ and antiferroelectric orthorhombic ${(A}_O)$ phases of $\left({\mathrm{Pb}}_{1-x}{\mathrm{Ba}}_x\right){\mathrm{ZrO}}_3$ has been investigated as a function of ${\mathrm{Ba}}^{2+}$ concentration (x) using phenomenological theory considerations. It is shown that all the unknown parameters in the free-energy expressions for the $F_R$ and $A_O$ phases can be evaluated using the dielectric and x-ray-diffraction data as a function of temperature. We show that the differences in the energies of antiferroelectric and ferroelectric phases decrease with increasing ${\mathrm{Ba}}^{2+}$ concentration, indicating the gradual destabilization of the antiferroelectric phase. The theoretically predicted critical composition $x=0.17\mathrm{}\pm 0.0025,$ above which no antiferroelectric phase can exist, is found to be in excellent agreement with the experimental observations.