Appendix B: Thermodynamics of Displacive Phase Transitions

Abstract

Although, experimentally, SrTiO 3 is not ferroelectric even at low temperatures, it is very close to the ferroelectric threshold. The isotopic replacement of oxygen or partial cation substitution reduces quantum fl uctuations and makes it ferroelec-tric. Hence, SrTiO 3 will serve as a model material to apply the Landau theory to a quantitative description of the displacive phase transition. Similar descriptions of structural phase transitions have been provided for many minerals, including feldspars (e.g., Carpenter, 1988), garnets (Carpenter and Boffa Ballaran, 2001), quartz (Carpenter et al. , 1998), or cristobalite (Schmahl et al. , 1992). The soft optical phonon mode at the center of the Brillouin zone is a rigid unit mode (RUM), whereby the BO 6 octahedra of the perovskite structure or the SiO 4 tetrahedra in silicates are taken as rigid units that rotate by the angle Φ around their vertices to attain phase transition (Figure B.1 ; Dove et al ., 2000). The soften-ing of the transverse optical (TO) phonon mode causes the frequency of the associ-ated transverse acoustic (TA) branch to be depressed with increasing wave vector k to zero frequency close to k = 0 (see Figure 8.6). As a result, an incommensurate displacive phase transition occurs. The angle of rotation Φ of the RUM octahedra is the order parameter Q in the Landau – Ginzburg theory of phase transition (Landau and Lifshitz, 1980), that converges towards zero when the Curie tempera-ture has been reached (Carpenter, 1992). As indicated above, the transition from a low -temperature tetragonal antiferro-electric phase to a high -temperature cubic paraelectric phase in strontium titanate can be described by the rotation of rigid [TiO 6 ] octahedra in opposite directions around the [001] direction (Figure B.1 a) The angle of rotation, Θ , characterizes the average displacement of oxygen atoms in the crystal lattice, and hence may be taken as an order parameter Q in the Landau – Ginzburg thermodynamic theory of phase transition (as explained in the following paragraph). As evidenced from experimental data obtained by M ü ller and von Waldkirch (1973) , the angle Θ decreases with increasing temperature, and reaches zero once the transition tem-perature T C – and hence the cubic high -temperature phase – have been attained (Figure B.1 b). Appendix B 508 Appendix B Thermodynamics of Displacive Phase Transitions The rotation of the [TiO 6 ] octahedra causes a reduction of the unit cell parameter that is proportional to cos Θ ; for small values of Θ it may be expanded as [1 − (Θ 2 /2)], so that the lattice distortion varies with Q 2 . Landau Theory A new look at the thermodynamics of structural phase transitions has been pro-vided by the Landau theory (Landau and Lifshitz, 1980 ; Tol é dano and Tol é dano, 1987). Its principal element is a Gibbs free energy expansion in the form of a Taylor series in an order parameter Q . As the value of Q cannot be directly assessed, the coeffi cients of the power series G AQ BQ CQ DQ = + + + +