Existence of a quadruple point in a binary ferroelectric phase diagram

In experimentally measured temperature-composition ferroelectric phase diagrams of ${\mathrm{BaTiO}}_3$-based binary systems, a quadruple point where cubic (C), tetragonal (T), orthorhombic (O), and rhombohedral (R) phases converge has been frequently reported in previous work. More interestingly, the quadruple points are experimentally found to behave as a critical point with large enhancement in properties. However, it has remained a fundamental question as to whether a quadruple point in a binary ferroelectric system defies the thermodynamic phase rule and whether such a point necessarily goes critical. In this study, it is demonstrated by Landau theory that a C-T-O-R quadruple point in a binary ferroelectric system can only exist in the form of a unique type of critical point at which two first-order transition lines and two second-order ones meet, and such critical quadruple points do not defy the thermodynamic phase rule. It is further shown that at such a critical C-T-O-R quadruple point, the system exhibits infinitely large piezoelectric coefficients, which agrees with the high piezoelectricity observed at the C-T-O-R quadruple point in a number of ${\mathrm{BaTiO}}_3$-based binary ferroelectric systems and also helps to explain the large piezoelectricity obtained at the morphotropic phase boundaries of these quadruple point based systems.

Figure 1

A binary ferroelectric phase diagram with a unique type of critical C-T-O-R quadruple point where two first-order and two second-order transition lines intersect. It is constructed by a Landau polynomial [Eq. (1)] with $\alpha ={\beta }_2=0$ and ${\beta }_1>0$ at the quadruple point. The solid lines represent first-order transition lines and the dashed lines represent second-order transition lines.

Figure 2

(a) $p-e$ loop and (b) $s-e$ loop at the unique type of critical quadruple point shown in Fig. 1. $p$, $e$, $s$ are reduced polarization, electric field, and strain, respectively. The slopes of both curves at $e=0$ approach infinity, which indicates infinitely large ${\varepsilon }_{33}$ and $d_{33}$.