Softening behavior of the ferroelectric A(1)(TO) phonon near the Curie temperature

It is well known that the A(1)(TO) phonon with its eigenvector parallel to the unique c axis of the 4mm symmetry is primarily responsible for the manifestation of displacive ferroelectricity in ABO(3)-type perovskites. We have theoretically examined the softening behavior of this A(1)(TO) phonon, particularly paying attention to its mode frequency around the Curie temperature (T-o). Our unusual approach is that we examine the microscopic Landau potential in terms of the displacement from the double-well minimum, rather than the displacement from the symmetry point. Adopting PbTiO3 as an ideal underdamped phonon system to test our formalisms, we have shown that the mode frequency of the "soft" A(1)(TO) phonon does not converge to zero even at the phase transition temperature, T-c, and that the computed mode frequency accurately reproduces the experimentally observed frequency over a wide range of temperature below T-c (763 K).