Theory of Ferroelectric Phase Transition

Yacoby, Y. and Girshberg, Y., Theory of ferroelectric phase transitions in pure and mixed perovskites. Mat. Res. Soc. Symp. Proc., Vol. 718, Materials Research Society, Warrendale, Pennsylvania, 2002. 

Fig. 3.26 The dependence of ferroelectric phase transition temperature on inverse mean grain size for nanogranular BaTiOs ceramics. Solid line is theory, the open squares are experimental points [95]...

One of the consequences of the suppression of the phase transition is the presence of a special critical point, Tc = 0 K. This point, called the quantum displacive limit, is characterized by special critical exponents. Its presence gives rise to classical quantum crossover phenomena. Quantum suppression and the response at and near this limit, Tc = 0 K, have been extensively studied on the basis of lattice dynamic models solved within the framework of both classical and quantum statistical mechanics. Figure 8 is a log-log plot of the 6 T) results for ST018 [15]. The expectation from theory is that in the quantum regime, y = 2 at 0.7 kbar, after which y should decrease. The results in Fig. 8 quantitatively show the expected behavior however, y is < 2 at 0.70 kbar. Despite the difference in the methods to suppress Tc in ST018, the results in Fig. 4a and Fig. 8 are quite similar. As shown in the results in Fig. 3b, uniaxial pressure also can be a critical parameter S for the evolution of ferroelectricity in STO. 

ORDER-DISORDER THEORY AND APPLICATIONS. Phase transitions in binary liquid solutions, gas condensations, order-disorder transitions in alloys, ferromagnetism, antiferromagnetism, ferroelectncity, anti-ferroelectricity, localized absorptions, helix-coil transitions in biological polymers and the one-dimensional growth of linear colloidal aggregates are all examples of transitions between an ordered and a disordered state. 

Determination of the critical temperature after equation (1), T0 = En/Syl, is considered in the cooperative JT and PJT effects [2,3,8,9]. In particular, the origin of structural ferroelectric phase transitions as due to the PJT effect (the vibronic theory of ferroelectricity) was suggested first in the sixties [10] (see also Ref. [9]). JT structural phase transitions are reviewed in Ref. [8]. 

A primary focus of our work has been to understand the ferroelectric phase transition in thin epitaxial films of PbTiOs. It is expected that epitaxial strain effects are important in such films because of the large, anisotropic strain associated with the phase transition. Figure 8.3 shows the phase diagram for PbTiOs as a function of epitaxial strain and temperature calculated using Landau-Ginzburg-Devonshire (lgd) theory [9], Here epitaxial strain is defined as the in-plane strain imposed by the substrate, experienced by the cubic (paraelectric) phase of PbTiOs. The dashed line shows that a coherent PbTiOs film on a SrTiOs substrate experiences somewhat more than 1 % compressive epitaxial strain. Such compressive strain favors the ferroelectric PbTiOs phase having the c domain orientation, i.e. with the c (polar) axis normal to the film. From Figure 8.3 one can see that the paraelectric-ferroelectric transition temperature Tc for coherently-strained PbTiOs films on SrTiOs is predicted to be elevated by 260°C above that of... 

The ferroelectric phase transition of second-order in tgs at 0C = 49° C can be described in the framework of the Landau-theory (e.g. [4]) by the thermodynamical potential... 

Thus, one may summarize the physical picture of the relaxation dynamics in KTN crystal-doped with Cu+ ions in the following way In the paraelectric phase, as the ferroelectric phase transition is approached, the Nb5+ ions form dipolar clusters around the randomly distributed Cu+ impurity ions. The interaction between these clusters gives rise to a cooperative behavior according to the AG theory of glass-forming liquids. At the ferroelectric phase transition the cooperative relaxation of the Cu+ ions is effectively frozen. ... 

Of central importance for understanding the fundamental properties of ferroelec-trics is dynamics of the crystal lattice, which is closely related to the phenomenon of ferroelectricity [1]. The soft-mode theory of displacive ferroelectrics [65] has established the relationship between the polar optical vibrational modes and the spontaneous polarization. The lowest-frequency transverse optical phonon, called the soft mode, involves the same atomic displacements as those responsible for the appearance of spontaneous polarization, and the soft mode instability at Curie temperature causes the ferroelectric phase transition. The soft-mode behavior is also related to such properties of ferroelectric materials as high dielectric constant, large piezoelectric coefficients, and dielectric nonlinearity, which are extremely important for technological applications. The Lyddane-Sachs-Teller (LST) relation connects the macroscopic dielectric constants of a material with its microscopic properties - optical phonon frequencies ... 

Indenbom, V.L. Phase transitions without atoms number change in elementary cell of crystals. Crystallography 5(1), 115-125 (1960) On thermodynamic theory of ferroelectricity, Izvestiya AN SSSR, ser. Phys. 24(10), 1180-1183 (1960) (in Russian)... 

However, the molecular interaction that is responsible to the occurrence of a particular transition is rarely known. Ferroelectric, antiferro-electric and other phase transitions in dielectric and molecular crystals are among the most complicated in their microscopic mechanism [2]. The clathrate compounds offer possibility of studying phase transitions in the substances in which molecular interactions are relatively well-specified. It should be mentioned parenthetically that guest-guest interactions play an essential role in a theory of clathrate formation [3]. The effects considered there are those of more drastic change (presence or otherwise of the guest molecules in the neighbouring cavities) than discussed in this paper. 

The fundamental problem of the theory of ferroelectricity is the origin of stmc-tural phase transition, when the spontaneous polarization appears or disappears. At first, we will describe such phase transition from the thermodynamic point of view. Phenomenological ferroelectricity theory is based on the works of Landau and Lifshitz (1974) and Devonshire (1949,1951). Starting point of the theory is the elastic Gibbs potential G, where we would select polarization P (except of electric displacement D) as an independent variable. Therefore the Gibbs potential is... 

Althoiigh, experimentally, SrTiOj 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 fluctuations and makes it ferroelectric. Hence, SrTiOs 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 cristobaUte (Schmahl et al., 1992). 

In conclusion, the Landau theory provides a welcome quantitative description of the thermodynamics and kinetics of phase transitions in minerals, including ferroelectric ceramics, by using macroscopic order parameters and their relationship to physical properhes and symmetry, as shown below. For an excellent review of the apphcation of Landau theory to displacive phase transitions in minerals, see Dove (1997). 

Kumar, A., and Waghmare, U. V. 2010. First-principles free energies and Ginzburg-Landau theory of domains and ferroelectric phase transitions in BaTiOj. Phys. Rev. B 82 05411. 

The dynamics of atoms in solids is responsible for many phenomena which cannot be explained within the static lattice model. Examples are the specific heat of crystals, thermal expansion, thermal conductivity, displacive ferroelectric phase transitions, piezoelectricity, melting, transmission of sound, certain optical and dielectric properties and certain aspects of the interaction of radiation such as X-rays and neutrons with crystals. The theory of lattice vibrations, often called lattice dynamiosy and its implications for many of the above mentioned phenomena is the subject of this two-volume book. 

Whereas the first microscopic theory of BaTiOs [1,2] was based on order-disorder behavior, later on BaTiOs was considered as a classical example of displacive soft-mode transitions [3,4] which can be described by anharmonic lattice dynamics [5] (Fig. 1). BaTiOs shows three transitions at around 408 K it undergoes a paraelectric to ferroelectric transition from the cubic Pm3m to the tetragonal P4mm structure at 278 K it becomes orthorhombic, C2mm and at 183 K a transition into the rhombohedral low-temperature Rm3 phase occurs. 

Recently, perfect softening of the ferroelectric TO mode was confirmed by Takesada et al., so ST018 is considered to be a displacive-type ferroelectric [11]. Kvyakovskii [ 12] gave an explanation for the phase transition mechanism. Part of the theory is summarized below. 

Ferroelectric-paraelectric transitions can be understood on the basis of the Landau-Devonshire theory using polarization as an order parameter (Rao Rao, 1978). Xhe ordered ferroelectric phase has a lower symmetry, belonging to one of the subgroups of the high-symmetry disordered paraelectric phase. Xhe exact structure to which the paraelectric phase transforms is, however, determined by energy considerations. 

In order to compare calculated and experimentally observed phase portraits it is necessary to know very exactly all the coefficients of the describing nonlinear differential Equation 14.3. Therefore, different methods of determination of the nonlinear coefficient in the Duffing equation have been compared. In the paraelectric phase the value of the nonlinear dielectric coefficient B is determined by measuring the shift of the resonance frequency in dependence on the amplitude of the excitation ( [1], [5]). In the ferroelectric phase three different methods are used in order to determine B. Firstly, the coefficient B is calculated in the framework of the Landau theory from the coefficient of the high temperature phase (e.g. [4]). This means B = const, and B has the same values above and below the phase transition. Secondly, the shift of the resonance frequency of the resonator in the ferroelectric phase as a function of the driving field is used in order to determine the coefficient B. The amplitude of the exciting field is smaller than the coercive field and does not produce polarization reversal during the measurements of the shift of the resonance frequency. In the third method the coefficient B was determined by the values of the spontaneous polarization... 

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