Ferroelectric phase transition and soft-mode behavior in Ba_xSr_1−xTiO₃: a refined treatment of a quasi-harmonic model

The composition and temperature dependences of the soft-mode and ferroelectric phase transition in Ba_xSr_1−xTiO₃ (BST) are presented and analyzed within the framework of a refined treatment of a quasi-harmonic model (QHM) for quantum particles. The QHM involves generalized simple microscopic quantum particles with anharmonic on-site double-Morse-type potential and harmonic interactions. The theory uses the variational principle scheme at finite temperature in the mean-field approximation while taking into account the predominant mass effect, the cell volume effect as well as ferroelectric distortion. The soft-mode frequency and the Ba concentration (x) dependence of the ferroelectric phase transition temperature are analyzed and show good agreement with experimental results throughout the full x range (0 ⩽ x ⩽ 1). It is found that BST is paraelectric below the critical impurity concentration x_c = 0.01 with a softening of the soft mode. It becomes ferroelectric above this critical concentration (x ⩾ x_c), and as the impurity concentration increases the predominant displacive soft-mode behavior stiffens in the ferroelectric phase with much less softening in crystals indicating a possible phase change at high x. Around x′_c = 0.5, T_c deviates gradually from linear behavior with a rather slight round stagnation which might characterize an equilibrium 'region' where several ferroelectric phases coexist. The proposed x dependence of the soft-mode behavior indicates that the ferroelectric phase transition is of first order in BST with x > x′_c, and of second order for low Ba concentrations (x_c ⩽ x < x′_c), throughout the tricritical point at x = x′_c. Finally, the quantum dynamics of the ferroelectric phase transition and the mode softening are also addressed and discussed, suggesting a net increase of quantum effects with a decrease of x for x ⩽ 0.1, and a dominant Curie–Weiss law for x > 0.1. (paper)