**5. Classification of ferroelectrics**

Ferroelectric crystals have been classified into the following types [1–6].

	- a. Hydrogen bonded and its isomorphs such as KH2PO4 (KDP), triglycine sulphate (TGS), Rochelle salt (RS) and lead hydrogen phosphate (LHP) etc.
	- b. Double oxides such as BaTiO3, potassium niobite (KNbO3) and lithium niobate (LiNbO3) etc.
	- a. Single-axis of polarization such as Rochelle salt, KDP etc.
	- b. Several-axes of polarization such as BaTiO3 etc.
	- a. Non-centre of symmetrical non-polar phase such as KDP and Rochelle salt.
	- b. Centre of symmetrical non-polar phase such as BaTiO3 and TGS crystals, etc.

4.According to the nature of the phase change.


In the order–disorder group of ferroelectrics, the ferroelectric phase transition is associated with an individual ordering of ions. These are the crystals that contain H-bonds and in which the motion of protons is related to the ferroelectric properties. The examples are KH2PO4, RS, TGS, CsH2PO4, PbHPO4 and RbH2PO4, etc. The displacive group of ferroelectrics is the one in which the ferroelectric phase transition is associated with the displacement of a whole sublattice of ions of one type relative to a sublattice of another type. The displacive type ferroelectrics possess perovskite ABO3 type structures. Examples are BaTiO3, LiNbO3 and KNbO3, etc. Consider the case of BaTiO3 crystal, as shown in **Figure 8**. The unit cell is cubic with *Ba*<sup>2</sup><sup>þ</sup> ions occupying at the corners, *O*<sup>2</sup>� ions occupying the face centres and *Ti*<sup>4</sup><sup>þ</sup> ion occupying the body centre of the cube. Thus, each *Ti*<sup>4</sup><sup>þ</sup> ion is surrounded by six *<sup>O</sup>*<sup>2</sup>� ions in an octahedral configuration. Above the Curie temperature (*<sup>T</sup>* <sup>&</sup>gt;*Tc*Þ, the prototype crystal structure is cubic, the centres of gravity of positive and negative charges exactly coincide with each other to produce a net dipole moment is zero. Below the Curie temperature (*<sup>T</sup>* <sup>&</sup>lt; *Tc*Þ, the structure is slightly deformed with *Ti*<sup>4</sup><sup>þ</sup> at the body centre while *Ba*<sup>2</sup><sup>þ</sup> ions at cube corners slightly move upwards, and the structure becomes tetragonal with centres of the positive (þ) and negative (�Þ charges not coinciding with each other.

If the relative displacement of the positive and negative ions is *d* and the charge on each ion is *q*, then the dipole moment per molecule is *p* ¼ *qd* and

*Pion* ¼ *Nqd* (17)

<sup>3</sup>*kT* (18)

the ionic polarization becomes

*Multifunctional Ferroelectric Materials*

*Frequency dependence of the various contributions to the polarizability [6].*

**Figure 7.**

**16**

where *N* is the number of atoms per unit volume.

iii. Dipolar polarizability (αdÞ: The dipolar polarizability, also called

*<sup>α</sup><sup>d</sup>* <sup>¼</sup> *<sup>p</sup>*

polarizability (*αe*Þ and ionic polarizability (*αi*Þ is called distortion polarization. Since *α<sup>e</sup>* and *α<sup>i</sup>* are temperature independent, the part of dielectric constant depending on them is essentially independent of the temperature. The contribution to the polarization made by dipolar polarizability (*αd*Þ which is a function of temperature in accordance with Eq. (18). The contributions to the total polarizability (polarizability versus

We find that in the optical frequency range, the dielectric constant (*ε*0) arises entirely due to the electronic polarizability. The ionic and dipolar contributions are

small at high frequencies because of the inertia of the ions and molecules.

frequency curve) are shown in **Figure 7** [2–17].

orientational polarizability, is important only in materials that contain complex ions having permanent dipole moment. In the absence of an external electric field, the dipoles have random orientations, and there is no net polarization. However, when the electric field is applied, the dipoles orient themselves along the direction of the field and produce dipolar or orientational polarization, as shown in **Figure 6c**. Such an orientation is opposed by the thermal agitation which tends. According to Debye's quantum theory, dipolar polarizability (*αd*Þ per dipole is given by

*<sup>E</sup>* <sup>¼</sup> *<sup>p</sup>*<sup>2</sup>

where *k* is the Boltzmann's constant, *T* is the absolute temperature, and *p* is the dipole moment of the atom. The polarization contributed by electronic

ferroelectricity in Rochelle salt. Thereafter, for about twenty years, the mechanism of ferroelectricity remained a mystery. In the period between 1935 and 1938, ferroelectricity in KDP crystal and its isostructural crystals was observed [21]. In order to explain ferroelectricity in KDP, a microscopic model Hamiltonian of disordered proton compositions was proposed, based on which a pseudospin model was developed by Slater [22] and Takagi [23] later independently. Ferroelectricity in Barium titanate (BaTiO3) was reported in 1945–1946 [24, 25], and a microscopic model of *Ti*4<sup>þ</sup> ion displacement was proposed by Slater in 1950 [26]. Devonshire [27] developed a phenomenological approach based on Landu-Ginburg phase transition theories [28, 29] to explain ferroelectric phase transitions. Later on, the thermodynamic theories of piezoelectricity were summarized by Cady [30] and Mason [31] independently. The most important concept in the theory of solid-state phase transitions is the concept of a "soft mode", which was developed on the basis of lattice dynamics by Cochran [32, 33] and Anderson [34]. According to the concept of a "soft mode", ferroelectric order stems from the instability of a transverse vibrational mode or a ferroelectric mode. Detailed lattice dynamic calculations for ferroelectric crystals and more rigorous mathematical treatments of the soft mode in ferroelectrics and anti-ferroelectrics have been made by Blinc and Zeks [35] and others [36]. Later on, ten years ago, it was believed that there were two different types of ferroelectric phase transition mechanism: displacement type and order– disorder type. However, several ferroelectric phenomena discovered that could be explained neither by a displacement type mechanism nor by an order–disorder type mechanism unequivocally. Therefore, several unified models based on a combination of both mechanisms have been proposed [37], such as in the general model developed by Stamenkovic et al. [38], two basic ordering parameters associated with the motion of active atoms etc. Based on the theory of phase transition of Landau-Ginzburg [28, 29] and Devonshire [27], developed the phenomenological theory of ferroelectricity by choosing the polarization as an order parameter. The most convenient treatment of the ferroelectric phase transition by using the elastic Gibbs function *G*<sup>1</sup> as a state function of the ferroelectric system and the temperature (*T*Þ, stress (*Λ*Þ and polarization (*P*) as independent variables [39, 40]. The

Gibb's free energy function is expressed as

*General Introduction to Ferroelectrics*

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

differential form of elastic Gibbs function

*i*, *j*

entropy, *Si* and *Λ <sup>j</sup>* are the *i*

*dG*<sup>1</sup> ¼ �*σdT* <sup>þ</sup><sup>X</sup>

*T*,*P*

*Si* ¼ � *dG*<sup>1</sup> *dΛ<sup>i</sup>* � �

**19**

*<sup>G</sup>*<sup>1</sup> <sup>¼</sup> *<sup>U</sup>* � *<sup>T</sup><sup>σ</sup>* <sup>þ</sup><sup>X</sup>

*i*, *j*

where *En* is the components of the electric field. Therefore, we can write the

*th* and *j*

*dU* <sup>¼</sup> *Td<sup>σ</sup>* �<sup>X</sup>

Making the use of a differential form of the internal energy

*Sid<sup>Λ</sup> <sup>j</sup>* <sup>þ</sup><sup>X</sup>

*n*, *m*

where *En* is the components of the electric field, we have *En* <sup>¼</sup> *dG*<sup>1</sup>

where *U* is the internal energy of the system, *T* is the temperature and *σ* is the

*i*, *j*

*<sup>Λ</sup> jdSi* <sup>þ</sup><sup>X</sup>

*n*, *m*

. Since the Gibbs free energy density *G* ¼ *G*<sup>1</sup> � *EnPm*, the stable state

*SiΛ <sup>j</sup>* (19)

*EndPm* (20)

*dPn* � �

*T*,*Λ* and

*th* component of mechanical strain and stress.

*EndPm*ð Þ *i*, *j* ¼ 1, 2, … 6; *m*, *n* ¼ 1, 2, 3 (21)

**Figure 8.** *Structures of BaTiO3: T* >*Tc left and T* <*Tc right [18].*

**Figure 9.** *Single cell potential for (a) displacive (b) order–disorder ferroelectrics [19].*

This situation leads to net dipole moment and hence produces the spontaneous polarization ð Þ *Ps* of the crystal [6–18]. **Figure 9** shows that in displacive ferroelectrics active atom has a single potential, while in the order–disorder ferroelectrics, the active atom has a double-well potential [19]. In the order–disorder systems, the proton can tunnel through the barrier, which separates the two minima of the potential energy in the hydrogen bond, and the ground state of the system splits into two levels separated by an energy Ω. The magnitude of Ω depends on the overlap of the wave functions appropriate to a proton located in each of the two separated minima.

### **6. Thermodynamics of ferroelectricity**

Many of the experimental results on the macroscopic properties of ferroelectrics such as polarization and dielectric constant as well as their temperature, electric field and pressure dependence, etc. In 1920, Valasek [20] was discovered

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

ferroelectricity in Rochelle salt. Thereafter, for about twenty years, the mechanism of ferroelectricity remained a mystery. In the period between 1935 and 1938, ferroelectricity in KDP crystal and its isostructural crystals was observed [21]. In order to explain ferroelectricity in KDP, a microscopic model Hamiltonian of disordered proton compositions was proposed, based on which a pseudospin model was developed by Slater [22] and Takagi [23] later independently. Ferroelectricity in Barium titanate (BaTiO3) was reported in 1945–1946 [24, 25], and a microscopic model of *Ti*4<sup>þ</sup> ion displacement was proposed by Slater in 1950 [26]. Devonshire [27] developed a phenomenological approach based on Landu-Ginburg phase transition theories [28, 29] to explain ferroelectric phase transitions. Later on, the thermodynamic theories of piezoelectricity were summarized by Cady [30] and Mason [31] independently. The most important concept in the theory of solid-state phase transitions is the concept of a "soft mode", which was developed on the basis of lattice dynamics by Cochran [32, 33] and Anderson [34]. According to the concept of a "soft mode", ferroelectric order stems from the instability of a transverse vibrational mode or a ferroelectric mode. Detailed lattice dynamic calculations for ferroelectric crystals and more rigorous mathematical treatments of the soft mode in ferroelectrics and anti-ferroelectrics have been made by Blinc and Zeks [35] and others [36]. Later on, ten years ago, it was believed that there were two different types of ferroelectric phase transition mechanism: displacement type and order– disorder type. However, several ferroelectric phenomena discovered that could be explained neither by a displacement type mechanism nor by an order–disorder type mechanism unequivocally. Therefore, several unified models based on a combination of both mechanisms have been proposed [37], such as in the general model developed by Stamenkovic et al. [38], two basic ordering parameters associated with the motion of active atoms etc. Based on the theory of phase transition of Landau-Ginzburg [28, 29] and Devonshire [27], developed the phenomenological theory of ferroelectricity by choosing the polarization as an order parameter. The most convenient treatment of the ferroelectric phase transition by using the elastic Gibbs function *G*<sup>1</sup> as a state function of the ferroelectric system and the temperature (*T*Þ, stress (*Λ*Þ and polarization (*P*) as independent variables [39, 40]. The Gibb's free energy function is expressed as

$$G\_1 = U - T\sigma + \sum\_{i,j} \mathbb{S}\_i \Lambda\_j \tag{19}$$

where *U* is the internal energy of the system, *T* is the temperature and *σ* is the entropy, *Si* and *Λ <sup>j</sup>* are the *i th* and *j th* component of mechanical strain and stress. Making the use of a differential form of the internal energy

$$d\mathbf{U} = T d\sigma - \sum\_{i,j} \Lambda\_j d\mathbf{S}\_i + \sum\_{n,m} E\_n dP\_m \tag{20}$$

where *En* is the components of the electric field. Therefore, we can write the differential form of elastic Gibbs function

$$dG\_1 = -\sigma dT + \sum\_{i,j} \mathbf{S}\_i d\Lambda\_j + \sum\_{n,m} E\_n dP\_m(i,j=1,2,\ldots \mathbf{6}; m, n=1,2,3) \tag{21}$$

where *En* is the components of the electric field, we have *En* <sup>¼</sup> *dG*<sup>1</sup> *dPn* � � *T*,*Λ* and *Si* ¼ � *dG*<sup>1</sup> *dΛ<sup>i</sup>* � � *T*,*P* . Since the Gibbs free energy density *G* ¼ *G*<sup>1</sup> � *EnPm*, the stable state

This situation leads to net dipole moment and hence produces the spontaneous polarization ð Þ *Ps* of the crystal [6–18]. **Figure 9** shows that in displacive ferroelectrics active atom has a single potential, while in the order–disorder ferroelectrics, the active atom has a double-well potential [19]. In the order–disorder systems, the proton can tunnel through the barrier, which separates the two minima of the potential energy in the hydrogen bond, and the ground state of the system splits into two levels separated by an energy Ω. The magnitude of Ω depends on the overlap of the wave functions appropriate to a proton located in each of the two

Many of the experimental results on the macroscopic properties of ferroelectrics such as polarization and dielectric constant as well as their temperature, electric field and pressure dependence, etc. In 1920, Valasek [20] was discovered

separated minima.

**18**

**Figure 8.**

**Figure 9.**

*Structures of BaTiO3: T* >*Tc left and T* <*Tc right [18].*

*Multifunctional Ferroelectric Materials*

**6. Thermodynamics of ferroelectricity**

*Single cell potential for (a) displacive (b) order–disorder ferroelectrics [19].*

of the system can be determined by the minimum of Gibb's free energy (*G*). If *T* and *Λ<sup>i</sup>* are constants, *G*<sup>1</sup> is a function of the polarization *P* (if *G*<sup>1</sup> and *Pn* are known, then *En* are entirely determined).

By combining Eqs. (26) and (27) we get

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

*General Introduction to Ferroelectrics*

**6.3 Second-order phase transitions**

*Ps* 2 ¼

One of the two roots of *Ps*

**6.4 Susceptibility**

dielectric susceptibility

The term of *ξ*2*Ps*

Eq. (30) we have.

**6.5 Free energy**

**21**

**6.6 Ferroelectric hysteresis loop**

Using the value of *<sup>β</sup>* <sup>¼</sup> ð Þ *<sup>T</sup>*�*Tc*

Consider first case *ξ*<sup>1</sup> >0. The roots of Eq. (24) for *Ps*

�*ξ*<sup>1</sup> � *ξ*<sup>1</sup>

value of *Ps*. When *β* <0, we may get a positive root of *Ps*

*<sup>χ</sup>*�<sup>1</sup> <sup>¼</sup> *<sup>∂</sup><sup>E</sup> ∂P* � �

because then *Ps* is very small and using the relation *Ps*

*<sup>χ</sup>*�<sup>1</sup> ¼ �2*<sup>β</sup>* ¼ �<sup>2</sup>

real value of *Ps*. However, for a real ferroelectric crystal j j *β ξ*<sup>2</sup> ≪ *ξ*<sup>1</sup>

*<sup>χ</sup>* <sup>¼</sup> *<sup>C</sup>* ð Þ *T* � *T*<sup>0</sup>

<sup>2</sup> � <sup>4</sup>*βξ*<sup>2</sup> � �<sup>1</sup>*=*<sup>2</sup> h i

When the temperature is below *Tc*. From Eq. (26), we get reciprocal of the

¼ *β* þ 3*ξ*1*Ps*

ð Þ *T* � *Tc*

*P*¼*Ps*

We plot the reciprocal of the susceptibility as a function of temperature in **Figure 10a**. This theoretical plot agrees well with the experimental data; the slope of the *χ*�<sup>1</sup> curve in the ferroelectric phase is twice that of the *χ*�<sup>1</sup> curve in the paraelectric phase.

For *ξ*<sup>1</sup> >0, using Eq. (22), *G*<sup>1</sup> � *G*<sup>10</sup> is plotted as a function of the polarization (*P*) at several temperatures (*Tc*<sup>1</sup> <*Tc* < *Tc*2Þ in **Figure 10b**. Since the sign of *β*, positive at *T* ¼ *Tc* and at *Tc*2, turns negative at *Tc*1, the curve representing

(*G*<sup>1</sup> � *G*10) changes from a minimum at *Tc*<sup>2</sup> to maximum at *Tc*<sup>1</sup> at *P* ¼ 0. At *Tc*1, the two minima of the free energy (*P* 6¼ 0) corresponds to stable ferroelectric states.

> *∂Pi* � �

*T*,*Λ*

impiles

*<sup>C</sup>* and *Ei* <sup>¼</sup> *<sup>∂</sup>G*<sup>1</sup>

2*ξ*<sup>2</sup>

where *C* is the Curie–Weiss constant. This is the Curie–Weiss law, which applies to the dielectric susceptibility in a paraelectric phase. In the case of spontaneous polarization *Ps* ¼6 0, we will see that the one result corresponds to a second-order phase transition when *ξ*<sup>1</sup> >0, and that the other result to a first-order phase transition when *ξ*<sup>1</sup> <0.

<sup>2</sup> 6¼ 0 are

<sup>2</sup> is always negative and corresponding to an imaginary

<sup>2</sup> <sup>þ</sup> <sup>5</sup>*ξ*2*Ps*

<sup>2</sup> ¼ � *<sup>β</sup> ξ*1

*<sup>C</sup>* ,ð Þ *<sup>T</sup>* <sup>&</sup>lt;*Tc* (31)

<sup>4</sup> can be neglected when the temperature is below and near *Tc*

*ξ*<sup>1</sup> ð Þ , *ξ*<sup>2</sup> >0, *β* <0 (29)

<sup>2</sup> and corresponding to a

<sup>4</sup> (30)

,ð Þ *β* <0, *ξ*<sup>1</sup> > 0 into

2 . (28)

### **6.1 Equation of state**

We consider a ferroelectric crystal having an intrinsic spontaneous polarization (*Ps*) along a specific-axis in the space coordinate system and assume that the external pressure is constant (say one atmosphere). As *G*<sup>1</sup> of the system is not changed by reversing the direction of the axes of space coordinate system, *G*<sup>1</sup> is independent of the direction of polarization (*P*). Thus, *G*<sup>1</sup> is an even function of *P*. Therefore, we can expand *G*<sup>1</sup> as a power series, in even powers of polarization *P* and neglecting the odd powers of *P* for symmetry reasons

$$\mathcal{G}\_1(T, P) = \mathcal{G}\_{10}(T) + \frac{1}{2}\beta(T)P^2 + \frac{1}{4}\xi\_1(T)P^4 + \frac{1}{6}\xi\_2(T)P^6 + \dots \tag{22}$$

where *G*10ð Þ *T* is the value of elastic Gibb's free energy (*G*1Þ of the system at *P* ¼ 0 and in general, the coefficients *G*10, *β*, *ξ*<sup>1</sup> and *ξ*<sup>2</sup> … ., are the functions of temperature (*T*). A stable state of a thermodynamic system is characterized by a minimum value of the Gibbs free energy *G*. We have *G*<sup>1</sup> ¼ *G* when *E* ¼ 0, *G* can be replaced by *G*1. When a crystal exhibits a stable spontaneous polarization (*Ps*) at a certain temperature, the conditions for a minimum of *G*<sup>1</sup> are

$$\left(\frac{\partial G\_1}{\partial P}\right)\_{P\_\iota} = 0, \left(\frac{\partial^2 G\_1}{\partial P^2}\right)\_{P\_\iota} > 0 \text{ or } \left(\frac{\partial E}{\partial P}\right)\_{P\_\iota} = \chi^{-1} > 0 \tag{23}$$

Using Eq. (22) into (23), we obtained the equation of state for the ferroelectric system of the form

$$P\_s \left( \beta + \xi\_1 P\_s^{\;2} + \xi\_2 P\_s^{\;4} \right) = \mathbf{0} \tag{24}$$

$$\chi^{-1} = \left(\beta + \mathfrak{Z}\xi\_1 P\_s^{\ 2} + \mathfrak{Z}\xi\_2 P\_s^{\ 4}\right) > 0\tag{25}$$

Eq. (24) has two roots: (i) the first root *Ps* ¼ 0 corresponds to a paraelectric phase and (ii) the second root *Ps* 6¼ 0 corresponds to a ferroelectric phase.

#### **6.2 Paraelectric phase**

Suppose spontaneous polarization *Ps* ¼ 0; from Eq. (25), the reciprocal of the dielectric susceptibility can be explained as.

$$\chi^{-1} = \beta(T) > 0 \tag{26}$$

It is obvious that the value of *β* have a positive value when a stable state of the crystal is a paraelectric phase. Therefore, the boundary conditions at the critical temperature is ð Þ *β*ð Þ *T <sup>T</sup>*<sup>0</sup> ≥0. Expanding *β*ð Þ *T* as a Taylor's series in ð Þ *T* � *T*<sup>0</sup> and taking into account only the first-order term in ð Þ *T* � *T*<sup>0</sup> , we have

$$\beta(T) = \frac{(T - T\_0)}{C} \tag{27}$$

of the system can be determined by the minimum of Gibb's free energy (*G*). If *T* and *Λ<sup>i</sup>* are constants, *G*<sup>1</sup> is a function of the polarization *P* (if *G*<sup>1</sup> and *Pn* are known,

We consider a ferroelectric crystal having an intrinsic spontaneous polarization

1

where *G*10ð Þ *T* is the value of elastic Gibb's free energy (*G*1Þ of the system at *P* ¼

<sup>4</sup> *<sup>ξ</sup>*1ð Þ *<sup>T</sup> <sup>P</sup>*<sup>4</sup> <sup>þ</sup>

1

<sup>6</sup> *<sup>ξ</sup>*2ð Þ *<sup>T</sup> <sup>P</sup>*<sup>6</sup> <sup>þ</sup> … (22)

<sup>¼</sup> *<sup>χ</sup>*�<sup>1</sup> <sup>&</sup>gt;0 (23)

(*Ps*) along a specific-axis in the space coordinate system and assume that the external pressure is constant (say one atmosphere). As *G*<sup>1</sup> of the system is not changed by reversing the direction of the axes of space coordinate system, *G*<sup>1</sup> is independent of the direction of polarization (*P*). Thus, *G*<sup>1</sup> is an even function of *P*. Therefore, we can expand *G*<sup>1</sup> as a power series, in even powers of polarization *P* and

neglecting the odd powers of *P* for symmetry reasons

2

certain temperature, the conditions for a minimum of *G*<sup>1</sup> are

<sup>¼</sup> 0, *<sup>∂</sup>*<sup>2</sup>

*<sup>β</sup>*ð Þ *<sup>T</sup> <sup>P</sup>*<sup>2</sup> <sup>þ</sup>

0 and in general, the coefficients *G*10, *β*, *ξ*<sup>1</sup> and *ξ*<sup>2</sup> … ., are the functions of temperature (*T*). A stable state of a thermodynamic system is characterized by a minimum value of the Gibbs free energy *G*. We have *G*<sup>1</sup> ¼ *G* when *E* ¼ 0, *G* can be replaced by *G*1. When a crystal exhibits a stable spontaneous polarization (*Ps*) at a

> *G*1 *∂P*<sup>2</sup>

*Ps β* þ *ξ*1*Ps*

*<sup>χ</sup>*�<sup>1</sup> <sup>¼</sup> *<sup>β</sup>* <sup>þ</sup> <sup>3</sup>*ξ*1*Ps*

taking into account only the first-order term in ð Þ *T* � *T*<sup>0</sup> , we have

phase and (ii) the second root *Ps* 6¼ 0 corresponds to a ferroelectric phase.

*Ps*

Using Eq. (22) into (23), we obtained the equation of state for the ferroelectric

<sup>2</sup> <sup>þ</sup> *<sup>ξ</sup>*2*Ps*

Eq. (24) has two roots: (i) the first root *Ps* ¼ 0 corresponds to a paraelectric

Suppose spontaneous polarization *Ps* ¼ 0; from Eq. (25), the reciprocal of the

It is obvious that the value of *β* have a positive value when a stable state of the crystal is a paraelectric phase. Therefore, the boundary conditions at the critical temperature is ð Þ *β*ð Þ *T <sup>T</sup>*<sup>0</sup> ≥0. Expanding *β*ð Þ *T* as a Taylor's series in ð Þ *T* � *T*<sup>0</sup> and

*<sup>β</sup>*ð Þ¼ *<sup>T</sup>* ð Þ *<sup>T</sup>* � *<sup>T</sup>*<sup>0</sup>

<sup>2</sup> <sup>þ</sup> <sup>5</sup>*ξ*2*Ps*

>0 *or*

*∂E ∂P* 

*Ps*

<sup>4</sup> <sup>¼</sup> <sup>0</sup> (24)

<sup>4</sup> >0 (25)

*<sup>χ</sup>*�<sup>1</sup> <sup>¼</sup> *<sup>β</sup>*ð Þ *<sup>T</sup>* <sup>&</sup>gt;<sup>0</sup> (26)

*<sup>C</sup>* (27)

*<sup>G</sup>*1ð Þ¼ *<sup>T</sup>*, *<sup>P</sup> <sup>G</sup>*10ð Þþ *<sup>T</sup>* <sup>1</sup>

*∂G*<sup>1</sup> *∂P* 

dielectric susceptibility can be explained as.

system of the form

**6.2 Paraelectric phase**

**20**

*Ps*

then *En* are entirely determined).

*Multifunctional Ferroelectric Materials*

**6.1 Equation of state**

By combining Eqs. (26) and (27) we get

$$\chi = \frac{\mathcal{C}}{(T - T\_0)} \tag{28}$$

where *C* is the Curie–Weiss constant. This is the Curie–Weiss law, which applies to the dielectric susceptibility in a paraelectric phase. In the case of spontaneous polarization *Ps* ¼6 0, we will see that the one result corresponds to a second-order phase transition when *ξ*<sup>1</sup> >0, and that the other result to a first-order phase transition when *ξ*<sup>1</sup> <0.

#### **6.3 Second-order phase transitions**

Consider first case *ξ*<sup>1</sup> >0. The roots of Eq. (24) for *Ps* <sup>2</sup> 6¼ 0 are

$$P\_s^{-2} = \frac{\left[-\xi\_1 \pm \left(\xi\_1^{-2} - 4\beta\xi\_2\right)^{1/2}\right]}{2\xi\_2} (\xi\_1, \xi\_2 > 0, \beta < 0) \tag{29}$$

One of the two roots of *Ps* <sup>2</sup> is always negative and corresponding to an imaginary value of *Ps*. When *β* <0, we may get a positive root of *Ps* <sup>2</sup> and corresponding to a real value of *Ps*. However, for a real ferroelectric crystal j j *β ξ*<sup>2</sup> ≪ *ξ*<sup>1</sup> 2 .

#### **6.4 Susceptibility**

When the temperature is below *Tc*. From Eq. (26), we get reciprocal of the dielectric susceptibility

$$\chi^{-1} = \left(\frac{\partial E}{\partial P}\right)\_{P=P\_r} = \beta + 3\xi\_1 P\_s^{-2} + 5\xi\_2 P\_s^{-4} \tag{30}$$

The term of *ξ*2*Ps* <sup>4</sup> can be neglected when the temperature is below and near *Tc* because then *Ps* is very small and using the relation *Ps* <sup>2</sup> ¼ � *<sup>β</sup> ξ*1 ,ð Þ *β* <0, *ξ*<sup>1</sup> > 0 into Eq. (30) we have.

$$
\chi^{-1} = -2\beta = -2\frac{(T - T\_c)}{\mathcal{C}}, (T < T\_c) \tag{31}
$$

We plot the reciprocal of the susceptibility as a function of temperature in **Figure 10a**. This theoretical plot agrees well with the experimental data; the slope of the *χ*�<sup>1</sup> curve in the ferroelectric phase is twice that of the *χ*�<sup>1</sup> curve in the paraelectric phase.

#### **6.5 Free energy**

For *ξ*<sup>1</sup> >0, using Eq. (22), *G*<sup>1</sup> � *G*<sup>10</sup> is plotted as a function of the polarization (*P*) at several temperatures (*Tc*<sup>1</sup> <*Tc* < *Tc*2Þ in **Figure 10b**. Since the sign of *β*, positive at *T* ¼ *Tc* and at *Tc*2, turns negative at *Tc*1, the curve representing (*G*<sup>1</sup> � *G*10) changes from a minimum at *Tc*<sup>2</sup> to maximum at *Tc*<sup>1</sup> at *P* ¼ 0. At *Tc*1, the two minima of the free energy (*P* 6¼ 0) corresponds to stable ferroelectric states.

#### **6.6 Ferroelectric hysteresis loop**

Using the value of *<sup>β</sup>* <sup>¼</sup> ð Þ *<sup>T</sup>*�*Tc <sup>C</sup>* and *Ei* <sup>¼</sup> *<sup>∂</sup>G*<sup>1</sup> *∂Pi* � � *T*,*Λ* impiles

**Figure 10.**

*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 second-order phase transition [40].*

$$E = \beta P + \xi\_1 P^3 + \xi\_2 P^5 \tag{32}$$

*P*2 *<sup>s</sup>*ð Þ¼ *Tc*

second term of Eq. (30) we get.

*first-order phase transition [40].*

*General Introduction to Ferroelectrics*

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

**Figure 11.**

shown in **Figure 11d**.

**23**

**7. Model theories of ferroelectricity**

3 *ξ*<sup>1</sup> j j 4*ξ*<sup>2</sup>

, *<sup>β</sup>* <sup>¼</sup> <sup>3</sup>*ξ*<sup>1</sup>

*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*

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

*<sup>χ</sup>*�<sup>1</sup> <sup>¼</sup> <sup>4</sup>*<sup>β</sup>* <sup>¼</sup> <sup>4</sup> ð Þ *<sup>T</sup>* � *Tc*

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

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

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

understanding of the ferroelectric phenomena of ferroelectricity.

2 16*ξ*<sup>2</sup> , *P*<sup>4</sup> *<sup>s</sup>* ð Þ¼ *Tc*

3*β ξ*2

*<sup>C</sup>* (35)

(34)

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 results are hysteresis loop.

#### **6.7 Spontaneous polarization**

When *<sup>E</sup>* <sup>¼</sup> 0, we putting Eq. (27) into relation *<sup>P</sup>*<sup>2</sup> *<sup>s</sup>* ¼ � *<sup>β</sup> ξ*1 , (*β* < 0, *ξ*<sup>1</sup> >0Þ, we get

$$P\_s^2 = \frac{(T\_c - T)}{\xi\_1 \mathcal{C}} \tag{33}$$

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 in ferroelectric crystals exhibiting second-order phase transitions.
