**6.8 First order phase transitions (***ξ***<sup>1</sup> < 0)**

As explained above the condition for the occurrence of spontaneous polarization (*Ps*) is that *β* should be negative while *ξ*<sup>1</sup> should be positive and there is secondorder transition, for first-order transition, the coefficient *β* is negative and also *ξ*<sup>1</sup> is negative as temperature is lowered. The Gibbs free energy curves with function of polarization (*P*) at different temperatures for this transition are shown in **Figure 11a**. It is obvious that the polarization state (*P* 6¼ 0) is stable at the temperature *Tc*1ð<*Tc*) in the **Figure 11b**. The P-E curves at various temperatures are plotted in **Figure 11b** from the hysteresis loop occurs at *Tc*1ð< *Tc*). For *E* ¼ 0, the spontaneous polarization (*Ps*) satisfies the Eq. (28) and from the Eq. (22), using the condition *G*<sup>1</sup> ¼ *G*<sup>10</sup> we get the following relations given by.

*General Introduction to Ferroelectrics DOI: http://dx.doi.org/10.5772/intechopen.97720*

**Figure 11.**

*<sup>E</sup>* <sup>¼</sup> *<sup>β</sup><sup>P</sup>* <sup>þ</sup> *<sup>ξ</sup>*1*P*<sup>3</sup> <sup>þ</sup> *<sup>ξ</sup>*2*P*<sup>5</sup> (32)

*<sup>s</sup>* ¼ � *<sup>β</sup> ξ*1

*<sup>ξ</sup>*1*<sup>C</sup>* (33)

, (*β* < 0, *ξ*<sup>1</sup> >0Þ, we get

We get (*P* � *E*) curve at different temperatures corresponding to different *β* values are displayed in **Figure 10c**. In fact, at temperature *Tc*1ð Þ <*Tc* , the segment in the curve from point A to point C corresponds to an unstable state, since the slope in this segment corresponds to *β* > 0. The experimental curves always jump directly from the state A to the state B and also directly from C to D. Thus, the observed

*Functional relations of (a)* <sup>χ</sup>�<sup>1</sup> *versus* <sup>T</sup>*; (b)* G1 � G10 *versus* <sup>P</sup>*; (c)* <sup>P</sup> *versus* <sup>E</sup> *and (d)* <sup>P</sup> *versus* <sup>T</sup> *near the*

results are hysteresis loop.

*second-order phase transition [40].*

*Multifunctional Ferroelectric Materials*

**Figure 10.**

**22**

**6.7 Spontaneous polarization**

When *<sup>E</sup>* <sup>¼</sup> 0, we putting Eq. (27) into relation *<sup>P</sup>*<sup>2</sup>

**6.8 First order phase transitions (***ξ***<sup>1</sup> < 0)**

*P*2

polarization (*P*) at different temperatures for this transition are shown in

condition *G*<sup>1</sup> ¼ *G*<sup>10</sup> we get the following relations given by.

in ferroelectric crystals exhibiting second-order phase transitions.

*<sup>s</sup>* <sup>¼</sup> ð Þ *Tc* � *<sup>T</sup>*

The function *Ps* changes continuously with temperature and becomes zero at *Tc* as shown in **Figure 10d**. This theoretical curve agrees with the experimental results

As explained above the condition for the occurrence of spontaneous polarization (*Ps*) is that *β* should be negative while *ξ*<sup>1</sup> should be positive and there is secondorder transition, for first-order transition, the coefficient *β* is negative and also *ξ*<sup>1</sup> is negative as temperature is lowered. The Gibbs free energy curves with function of

**Figure 11a**. It is obvious that the polarization state (*P* 6¼ 0) is stable at the temperature *Tc*1ð<*Tc*) in the **Figure 11b**. The P-E curves at various temperatures are plotted in **Figure 11b** from the hysteresis loop occurs at *Tc*1ð< *Tc*). For *E* ¼ 0, the spontaneous polarization (*Ps*) satisfies the Eq. (28) and from the Eq. (22), using the

*Functional relations of (a) G*<sup>1</sup> � *<sup>G</sup>*<sup>10</sup> *versus P; (b) P versus E; (c) <sup>χ</sup>*�<sup>1</sup> *versus T and (d) P versus T near the first-order phase transition [40].*

$$P\_s^2(T\_\varepsilon) = \frac{3|\xi\_1|}{4\xi\_2}, \beta = \frac{3\xi\_1^2}{16\xi\_2^5}, P\_s^4(T\_\varepsilon) = \frac{3\beta}{\xi\_2} \tag{34}$$

Combing the Eq. (30) with Eq. (34) and keeping in the mind that *ξ*<sup>1</sup> <0 in the second term of Eq. (30) we get.

$$\chi^{-1} = 4\beta = 4\frac{(T - T\_c)}{C} \tag{35}$$

At the transition temperature ð Þ *Tc* , the *<sup>χ</sup>*�<sup>1</sup> in a first order transition is not zero but it is positive quantity as can be seen from Eq. (35). The variation of temperature dependence of inverse (*χ*�1) above and below the transition temperature ð Þ *Tc* is shown in **Figure 11c**. The *χ*�<sup>1</sup> at a temperature just below *Tc* is four times that at a temperature above *Tc*. The curve for the polarization (*P*) with temperature (*T*) is shown in **Figure 11d**.

#### **7. Model theories of ferroelectricity**

Various physicists have developed model theories of ferroelectricity. An introductory idea of model theories that have been developed to explain the phenomenon of ferroelectricity is given below. Many experimental and theoretical attempts were made to explain the phenomenon of ferroelectricity in single and polycrystals and proposed a number of theories. The first theoretical explanation of the ferroelectric properties of Rochelle salt was proposed by Kurchatov [41]. Slater [22] put forward the first molecular theory of ferroelectricity and suggested that the ferroelectric behaviour in KDP and Rochelle salt is principally due to the ordering of Hbonds. A general theory of ferroelectricity established at that time by Cochran's lattice dynamic theory [33] and Lines statistical theory [42] have provided for major understanding of the ferroelectric phenomena of ferroelectricity.

A simple order–disorder model Hamiltonian was proposed by Mason [43]. In this model, a proton motion along an H-bond. A proton may transfer from one-well to another, and vice versa, stochastically over a potential barrier, and this model

does not explain the isotope effect. Blinc [44] introduced the concept of proton tunneling motion between the two equilibrium sites in the double-well minimum O-H–O bond potential. The Blinc's [44] concept was quickly put into the simple formalism of the pseudospin-model by De Gennes [45] and independently by Matsubara [46]. De Gennes [45] showed that the proton in a double minimum type potential has described by a one-half pseudospin. The pseudospin model Hamiltonian developed to explain the proton system and the triggering of phase transition can be ascribed to the ordering of the proton in the double-well potential. The dipoles formed by the proton ordering make a small contribution to the spontaneous polarization ð Þ *Ps* which is the result of heavy atoms displacements along the ferroelectric axis. The major contribution to the spontaneous polarization ð Þ *Ps* is the displacement of heavy atoms projected on c-axis. To account for the displacement of the heavy atoms, Kobayashi [47] included pseudospin lattice interaction into the pseudospin model. The net result of this approach was to enhance the effective dipolar proton-proton interaction.

and the thermodynamic properties of the crystals. For *k* ¼ 0 modes of diatomic crystal, Lyddane et al. [53] relation gives the ratio of the static dielectric constant ð Þ *ε<sup>s</sup>* of the crystal to the high-frequency dielectric constant ð Þ *ε<sup>e</sup>* in terms of the frequencies of the longitudinal optical ð Þ *ω<sup>L</sup>* and transverse optical ð Þ *ω<sup>T</sup>* of infinite wavelength

> *εs εe* <sup>¼</sup> *<sup>ω</sup><sup>L</sup>* 2

*εs εe* <sup>¼</sup> <sup>Y</sup>*<sup>n</sup> j*

which there are *n* atoms in the elementary cell as

necessary to investigate the temperature of *ω<sup>T</sup>*

**8. Applications of ferroelectrics**

O3 are some examples of these applications.

<sup>2</sup> <sup>¼</sup> 0, we get *<sup>ε</sup><sup>s</sup>* <sup>¼</sup> <sup>∞</sup>. Cochran [32] developed a more general case in

*ωj* 2 � � *L ωj* 2 � � *T*

This Eq. (36) produces one essential anomaly needed to explain a ferroelectric transition. In order to complete understanding the ferroelectric behaviour, it is

static dielectric constant (*εs*) obeys Curie–Weiss law above *Tc*ð Þ *T* >*Tc* . Eq. (37) was derived for harmonic forces. In order to derive the temperature dependence of *ω<sup>T</sup>*

it will be necessary to introduce anharmonic interaction, which shows little effect on any mode other than*k* ¼ 0 mode, and only that mode exhibits any anomalous behaviour. Thus Eq. (37) through (36) implies that the transverse optical mode *ω<sup>T</sup>*

have anomalous temperature dependence given by the Curie–Weiss law by a rela-

ture dependence dielectric constant *ε*ð Þ *T* through the Lyddane et al. [53] relation.

<sup>2</sup> <sup>¼</sup> *K T*ð Þ � *Tc* , where the coefficient *<sup>K</sup>* is constant related to the tempera-

Ferroelectric materials have been extensive applications [3, 54] in a large number of areas due to their peculiar and interesting properties such as high permittivity capacitors (BaTiO3), ferroelectric non-volatile FeRAM memories (due to bi-stable polarization in modulation and deflector), pyroelectric sensors, piezoelectric and electrostrictive transducers (TGS crystal), electrooptic and optoelectronic devices (due to their non-linear polarizability), thermistors, storage and laser devices, sensors, resonators and actuators which have revolutionized consumer electronics, automobile industry, biomedical diagnosis, underwater acoustic technology, defense-related sectors, gas sensing devices and surface acoustic wave technology, etc. The major areas of applications [3, 54] of ferroelectrics have received a great deal of attention amongst all the above capacitors, ferroelectric memories, pyroelectric sensors, piezoelectric, electrostrictive transducers, electrooptic devices and thermistors. The basic specifications required for capacitors are small size, large capacitance (materials with a large dielectric constant are desired). High frequency characteristics (ferroelectrics with a high dielectric constant are sometimes associated with dielectric dispersions, which must be taken into account for practical applications). Ferroelectric relaxors such as Pb(Mg1/3Nb2/3)O3 and Pb(Zn1/3Nb2/3)

The bi-stable polarization of ferroelectrics makes them useful for binary memory systems. There are volatile and non-volatile memory devices in erasable semiconductor memories. Non-volatile memory does not require a holding voltage. Dynamic random-access memory (DRAM), which is widely used because of its high integration capability, is an example of volatile memory. Data stored in these

When *ω<sup>T</sup>*

*General Introduction to Ferroelectrics*

*DOI: http://dx.doi.org/10.5772/intechopen.97720*

tion *ω<sup>T</sup>*

**25**

*<sup>ω</sup>T*<sup>2</sup> (36)

2. In the ferroelectric crystal, the

(37)

2

2

Later on, Arefev et al. [48] and Brout et al. [49] suggested that essentially the same concept could also be applied such that KDP etc. In this case, where the permanent electric dipoles move-in a potential with more than one equilibrium position and the soft mode collective excitations are not phonons but rather the unstable pseudospin phonon coupled-wave. Kaminow [50] experimentally confirmed the existence of soft mode in KDP crystal, and the other investigators also confirmed the soft mode in other ferroelectric crystals. It is now well recognized that the several interesting properties of ferroelectrics are associated with the hightemperature dependence of the soft mode. Cowely [51] has given a microscopic theory of ferroelectricity in which the temperature dependence of the normal mode (soft mode) arises from anharmonic interactions between normal modes. These anharmonic interactions in a crystal are quite small, at least at low temperatures, so that an anharmonic crystal provides an example of the many-body system in which interactions between the elementary excitations are both small and non-singular. In the harmonic approximation, the equations of motion of the shell model can be obtained from a quadratic function of the displacements of the ions and of the electronic dipoles produced on the ions during the lattice vibrations. The anharmonic interactions arise from the cubic and higher terms in potential function, and in general, there will be anharmonic interactions between all the displace-

ments and all the dipoles. In the ferroelectrics, the root at wave vector *k* ! ¼ 0 is imaginary in the harmonic approximation showing the instability of the lattice [51]. This indicates that the harmonic forces alone are not sufficient to stabilize the system at any temperature. The stabilization of the mode can only be brought about by a consideration of anharmonic interactions terms. The anharmonic interactions thus play a fundamental role as regards the stability of the crystal system. A number of physical properties of solids could be well explained by considering the effects of phonon anharmonic interactions.

A very successful attempt has been made to give a microscopic theory of ferroelectric crystals given by Cochran [33, 52]. This lattice dynamical theory is based on the hypothesis that the ferroelectric transitions are the results of the instability of the crystal lattice with respect to one of the homogeneous (wave number *<sup>k</sup>* <sup>¼</sup> <sup>2</sup>*<sup>π</sup> <sup>λ</sup>* ¼ 0) transverse optic mode. If a crystal is fully or partly ionic, lattice vibrations are accompanied by polarization oscillations having an equal frequency which provide a Lorentz field called local field interacting with the ions through long-range Coulomb forces. The crystal becomes unstable for one particular mode of vibration at which the long-range forces are equal and opposite to the short-range forces. A relation given by Lyddane et al. [53] explains the relation between the ferroelectric properties

#### *General Introduction to Ferroelectrics DOI: http://dx.doi.org/10.5772/intechopen.97720*

does not explain the isotope effect. Blinc [44] introduced the concept of proton tunneling motion between the two equilibrium sites in the double-well minimum O-H–O bond potential. The Blinc's [44] concept was quickly put into the simple formalism of the pseudospin-model by De Gennes [45] and independently by Matsubara [46]. De Gennes [45] showed that the proton in a double minimum type potential has described by a one-half pseudospin. The pseudospin model Hamiltonian developed to explain the proton system and the triggering of phase transition can be ascribed to the ordering of the proton in the double-well potential. The dipoles formed by the proton ordering make a small contribution to the spontaneous polarization ð Þ *Ps* which is the result of heavy atoms displacements along the ferroelectric axis. The major contribution to the spontaneous polarization ð Þ *Ps* is the displacement of heavy atoms projected on c-axis. To account for the displacement of the heavy atoms, Kobayashi [47] included pseudospin lattice interaction into the pseudospin model. The net result of this approach was to enhance the effective

Later on, Arefev et al. [48] and Brout et al. [49] suggested that essentially the same concept could also be applied such that KDP etc. In this case, where the permanent electric dipoles move-in a potential with more than one equilibrium position and the soft mode collective excitations are not phonons but rather the unstable pseudospin phonon coupled-wave. Kaminow [50] experimentally confirmed the existence of soft mode in KDP crystal, and the other investigators also confirmed the soft mode in other ferroelectric crystals. It is now well recognized that the several interesting properties of ferroelectrics are associated with the hightemperature dependence of the soft mode. Cowely [51] has given a microscopic theory of ferroelectricity in which the temperature dependence of the normal mode (soft mode) arises from anharmonic interactions between normal modes. These anharmonic interactions in a crystal are quite small, at least at low temperatures, so that an anharmonic crystal provides an example of the many-body system in which interactions between the elementary excitations are both small and non-singular. In the harmonic approximation, the equations of motion of the shell model can be obtained from a quadratic function of the displacements of the ions and of the electronic dipoles produced on the ions during the lattice vibrations. The

anharmonic interactions arise from the cubic and higher terms in potential function, and in general, there will be anharmonic interactions between all the displace-

imaginary in the harmonic approximation showing the instability of the lattice [51]. This indicates that the harmonic forces alone are not sufficient to stabilize the system at any temperature. The stabilization of the mode can only be brought about by a consideration of anharmonic interactions terms. The anharmonic interactions thus play a fundamental role as regards the stability of the crystal system. A number of physical properties of solids could be well explained by considering the effects of

A very successful attempt has been made to give a microscopic theory of ferroelectric crystals given by Cochran [33, 52]. This lattice dynamical theory is based on the hypothesis that the ferroelectric transitions are the results of the instability of the

crystal lattice with respect to one of the homogeneous (wave number *<sup>k</sup>* <sup>¼</sup> <sup>2</sup>*<sup>π</sup>*

transverse optic mode. If a crystal is fully or partly ionic, lattice vibrations are accompanied by polarization oscillations having an equal frequency which provide a Lorentz field called local field interacting with the ions through long-range Coulomb forces. The crystal becomes unstable for one particular mode of vibration at which the long-range forces are equal and opposite to the short-range forces. A relation given by Lyddane et al. [53] explains the relation between the ferroelectric properties

!

¼ 0 is

*<sup>λ</sup>* ¼ 0)

ments and all the dipoles. In the ferroelectrics, the root at wave vector *k*

dipolar proton-proton interaction.

*Multifunctional Ferroelectric Materials*

phonon anharmonic interactions.

**24**

and the thermodynamic properties of the crystals. For *k* ¼ 0 modes of diatomic crystal, Lyddane et al. [53] relation gives the ratio of the static dielectric constant ð Þ *ε<sup>s</sup>* of the crystal to the high-frequency dielectric constant ð Þ *ε<sup>e</sup>* in terms of the frequencies of the longitudinal optical ð Þ *ω<sup>L</sup>* and transverse optical ð Þ *ω<sup>T</sup>* of infinite wavelength

$$\frac{\varepsilon\_t}{\varepsilon\_t} = \frac{\alpha \nu\_L^{-2}}{\alpha \nu\_T^{-2}}\tag{36}$$

When *ω<sup>T</sup>* <sup>2</sup> <sup>¼</sup> 0, we get *<sup>ε</sup><sup>s</sup>* <sup>¼</sup> <sup>∞</sup>. Cochran [32] developed a more general case in which there are *n* atoms in the elementary cell as

$$\frac{\varepsilon\_{\varepsilon}}{\varepsilon\_{\varepsilon}} = \prod\_{j}^{n} \frac{(\alpha\_{j}^{2})\_{L}}{(\alpha\_{j}^{2})\_{T}} \tag{37}$$

This Eq. (36) produces one essential anomaly needed to explain a ferroelectric transition. In order to complete understanding the ferroelectric behaviour, it is necessary to investigate the temperature of *ω<sup>T</sup>* 2. In the ferroelectric crystal, the static dielectric constant (*εs*) obeys Curie–Weiss law above *Tc*ð Þ *T* >*Tc* . Eq. (37) was derived for harmonic forces. In order to derive the temperature dependence of *ω<sup>T</sup>* 2 it will be necessary to introduce anharmonic interaction, which shows little effect on any mode other than*k* ¼ 0 mode, and only that mode exhibits any anomalous behaviour. Thus Eq. (37) through (36) implies that the transverse optical mode *ω<sup>T</sup>* 2 have anomalous temperature dependence given by the Curie–Weiss law by a relation *ω<sup>T</sup>* <sup>2</sup> <sup>¼</sup> *K T*ð Þ � *Tc* , where the coefficient *<sup>K</sup>* is constant related to the temperature dependence dielectric constant *ε*ð Þ *T* through the Lyddane et al. [53] relation.
