Effects of tricritical points and morphotropic phase boundaries on the piezoelectric properties of ferroelectrics

The longitudinal piezoelectric coefficient d*(33) of a PZT-like ferroelectric is computed in the full composition-temperature parameter space using sets of parameters that control the position of the tricritical points and the degree of tilting of the morphotropic phase boundary separating the ferroelectric rhombohedral phase from the ferroelectric tetragonal phase. The system is modeled using a Ginzburg-Landau expansion of the free energy in terms of the electric polarization up to sixth order, including all the symmetry-allowed terms. We obtain two regions of the phase diagram with a large piezoelectric response. In the polar direction, d*(33) is large in the vicinity of the paraelectric to ferroelectric line of phase transitions, whereas in a nonpolar direction d*(33) is large in the vicinity of the morphotropic phase boundary. We find that a given degree of tilting of the morphotropic phase boundary can be obtained from free energies with different degrees of anisotropy, and therefore the titling and anisotropy are not directly related. On the other hand, the piezoelectric response is larger when the two tricritical points of the phase diagram are farther apart from each other than when they collapse onto a single tricritical point.