Monoclinic and triclinic phases in higher-order Devonshire theory

Devonshire theory provides a successful phenomenological description of many cubic perovskite ferroelectrics such as ${\mathrm{BaTiO}}_{3}$ via a sixth-order expansion of the free energy in the polar order parameter. However, the recent discovery of a novel monoclinic ferroelectric phase in the PZT system by Noheda et al. [Appl. Phys. Lett. 74, 2059 (1999)] poses a challenge to this theory. Here, we confirm that the sixth-order Devonshire theory cannot support a monoclinic phase, and consider extensions of the theory to higher orders. We show that an eighth-order theory allows for three kinds of equilibrium phases in which the polarization is confined not to a symmetry axis but to a symmetry plane. One of these phases provides a natural description of the newly observed monoclinic phase. Moreover, the theory makes testable predictions about the nature of the phase boundaries between monoclinic, tetragonal, and rhombohedral phases. A ferroelectric phase of the lowest (triclinic) symmetry type, in which the polarization is not constrained by symmetry, does not emerge until the Devonshire theory is carried to twelfth order. A topological analysis of the critical points of the free-energy surface facilitates the discussion of the phase transition sequences.