**4. Dielectric properties**

## **4.1 Local field**

occurs under the influence of an internal process in a dielectric, without the effect of external factors. A ferroelectric crystal generally consists of regions called domains of homogeneous polarization, within each of which the polarization is in the same direction, but in the adjacent domains, the polarization is in different directions so that the net polarization of the specimen is equal to zero in the beginning when no electric field (*E* ¼ 0) is applied. The polarization varies in a non-linear configuration with an electric field (*E*). This non-linear relation exhibits the closed curve called the hysteresis loop in the polarization when an electric field is applied, as shown in **Figure 5** [2–6]. When the electric field (*E* ¼ 0) is zero, the spontaneous polarization (*Ps*Þ in a single domain specimen is either positive or negative in sign. As an applied electric field strength gradually increases in the direction of spontaneous polarization (*Ps*Þ, the polarization (*P*) increases due to induced polarization such as electronic, ionic and dipolar types. As the electric field is increased further, more and more domains rotate along the direction of the electric field (*E*) until the polarization

At this stage, the whole specimen represents a single domain. This is usually accompanied by a distortion in the crystal along the polarization direction. The extrapolation of the saturation value to zero field gives the magnitude of the spontaneous polarization (*Ps*Þ. This value of *Ps* is the same as possessed by each domain before the application of the electric field. However, if the applied electric field decreases, the polarization also decreases but follows another path and does not become zero for zero electric field. The remaining polarization at this stage is called remnant polarization ð Þ �*Pr* and the intercept on the E-axis, where *Pr* refers to the whole crystal block. In order to destroy the remnant polarization (*Pr*Þ, the polarization of nearly half of the crystal is to be reversed by reversing the direction of the field, the electric field required to make the polarization zero is called the coercive field (*Ec*Þ. Furthermore, an increase in the reverse field results in the saturation of polarization in the reverse direction. Reversing the electric field again, the hysteresis curve will be obtained. The relation between polarization (*P*) and applied electric field ð Þ *E* is thus represented by a hysteresis loop (BDFGHB) which is the most important characteristic of the ferroelectric crystals. The most important feature of a ferroelectric is thus not the fact that it has a spontaneous polarization (*Ps*Þ but rather the fact that this spontaneous polarization can be reversed by means of an

reaches a maximum value called the saturation value.

*Multifunctional Ferroelectric Materials*

electric field [2–14].

**Figure 5.**

**12**

*Ferroelectric (P-E) hysteresis loop [13].*

The electric field acting at the site of atom or molecule is, in general, significantly different from the macroscopic filed *E* and is known as the local field ð Þ *Eloc* . This field is responsible for the polarization of each atom or molecule of a material [5, 6]. For an atomic site with cubic crystal symmetry, the local field ð Þ *Eloc* is expressed by the Lorentz-relation as

$$E\_{\rm loc} = E\_0 + E\_P + \frac{P}{\mathfrak{B}\varepsilon\_0} = E + \frac{P}{\mathfrak{B}\varepsilon\_0} \tag{2}$$

where *E* ¼ *E*<sup>0</sup> þ *EP*. The field *EP* is called the polarization field as it tends to oppose the external applied field *E*0. Thus, apart from the macroscopic field Eð Þ, the local field also contains a term called Lorentz field ð Þ *Eloc* . The difference between the macroscopic field ð Þ *E* and the Lorentz field ð Þ *Eloc* may be understood as follows. The macroscopic field is macroscopic in nature, is an average value and is constant throughout the medium. On the other hand, the Lorentz field ð Þ *Eloc* is a microscopic field and is periodic in nature. This is quite large at molecular sites representing that the molecules are more effectively polarized than they are under the average field.

#### **4.2 Polarization and dielectric susceptibility**

Generally, at ordinary electric fields, the magnitude of polarization (*P*) is directly proportional to the macroscopic electric field Eð Þ at a given point of a dielectric [5–15]. It is expressed as

$$P \propto E \Rightarrow P = \varepsilon\_0 \chi\_e E \tag{3}$$

where *ε*<sup>0</sup> is the permittivity of free space and *χ<sup>e</sup>* is the dielectric susceptibility. Thus, except for a constant factor ð Þ *ε*<sup>0</sup> , the dielectric susceptibility is a measure of the polarization produced in the material per unit electric field. If the dielectric material slab is placed in a uniform electric field (*E*) with its normal parallel to the field. The dielectric displacement vector (*D*) for an isotropic or cubic medium relative to vacuum is defined in terms of the macroscopic field ð Þ *E* as

$$D = \varepsilon\_0 \varepsilon\_r E = \varepsilon E = \varepsilon\_0 E + P \tag{4}$$

where *ε<sup>r</sup>* is called the relative permittivity or dielectric constant of medium and *ε*<sup>0</sup> is the permittivity or dielectric constant of free space, and *P* is the polarization. It is a scalar quantity for an isotropic medium and is always dimensionless. The dielectric constant (also called as permittivity of medium) is a measure of the degree to which a medium can resist the flow of charge, defined as the ratio of the dielectric displacement *D* to the macroscopic field intensity ð Þ *E* as

$$
\varepsilon\_r = \frac{D}{\varepsilon\_0 E} = \frac{\varepsilon\_0 E + P}{\varepsilon\_0 E} = \mathbf{1} + \frac{P}{\varepsilon\_0 E} = \mathbf{1} + \chi\_e \tag{5}
$$

Eq. (5) gives the dielectric constant of an isotropic medium or cubic medium. This is represented by a scalar quantity. The dielectric susceptibility (*χe*Þis related to the dielectric constant defined as

$$\chi\_{\varepsilon} = \frac{P}{\varepsilon\_0 E} = e\_r - \mathbf{1} \tag{6}$$

Solving for P

polarization [2–17] given as

tively shown in **Figure 6**.

**Figure 6.**

**15**

*Atomic contributions to electric polarization [18].*

*j*

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

*General Introduction to Ferroelectrics*

*N <sup>j</sup>α <sup>j</sup>* we get

*<sup>n</sup>*<sup>2</sup> � <sup>1</sup> *<sup>n</sup>*<sup>2</sup> <sup>þ</sup> <sup>2</sup> <sup>¼</sup> <sup>1</sup>

constant (*εr*Þ by the relation

3*ε*<sup>0</sup> X *j*

where *n* being the refractive index which is related to the dielectric

ii. Ionic polarizability (*αi*Þ: The ionic polarizability (*αi*Þ arises due to the relative displacement of positive and negative ions from their equilibrium positions to a distance less than the distance between adjacent ions. The cations are displaced parallel to the Lorentz field, and the anions are displaced in the opposite direction, as shown in **Figure 6b** [5–18].

*ε<sup>r</sup>* � 1 *<sup>ε</sup><sup>r</sup>* <sup>þ</sup> <sup>2</sup> <sup>¼</sup> <sup>1</sup>

3*ε*<sup>0</sup> X *j*

This is known as the Clausius-Mossotti relation. It relates the dielectric constant to the atomic polarizability, but only for crystal structures for which the Lorentz field Eq. (2) obtains. The total polarization (*α*Þ can be expressed as the sum of three types of basic polarizability representing the most important contributions to the

where *αe*, *α<sup>i</sup>* and *α<sup>d</sup>* are the electronic, ionic and dipolar polarizabilities, respec-

i. Electronic polarizability (*αe*Þ: The electronic polarizability (*αe*Þ arises due to the displacement of electrons in an atom relative to the atomic nucleus in the external electric field, as shown in **Figure 6a**. The polarization, as well as the dielectric constant of a material at optical frequencies, results mainly from the electronic polarizability (*αe*Þ. The optical range Eq. (13) reduces as

*N <sup>j</sup>α <sup>j</sup>* (13)

*α* ¼ *α<sup>e</sup>* þ *α<sup>i</sup>* þ *α<sup>d</sup>* (14)

*N <sup>j</sup>α <sup>j</sup>*ð Þ electronic (15)

*<sup>n</sup>*<sup>2</sup> <sup>¼</sup> *<sup>ε</sup><sup>r</sup>* (16)

Thus, like susceptibility, the dielectric constant (*εr*Þ is also a measure of the polarization (*P*) of the material. The larger the polarization per unit resultant macroscopic field, the greater will be the dielectric constant of the dielectric medium. However, for anisotropic medium, the dielectric response (*εr*or *χe*Þ depends on the direction of the field and described by the components of the susceptibility tensor or of the dielectric tensor of the second rank

$$P\_{\mu} = \chi\_{\mu\nu}\varepsilon\_0 \varepsilon\_\nu; \varepsilon\_{\mu\nu} = \delta\_{\mu\nu} + \chi\_{\mu\nu} \tag{7}$$

#### **4.3 Dielectric constant and polarizability**

The polarizability ð Þ *α* of an atom is defined in terms of the local electric field (Lorentz field) at the atom. The induced dipoles of moments (*p*) are proportional to the local field (*Eloc*Þ can be expressed as

$$p = aE\_{loc} \tag{8}$$

where *α* is known as the polarizability of an atom. For a non-spherical atom or isotropic medium, *α* will be a tensor quantity [6–16]. Thus, polarizability is an atomic property, whereas dielectric constant is a macroscopic property that depends upon the arrangement of atoms within the crystal. If all the atoms have the same polarizability (*α*Þ and there are *N* number of atoms per unit volume, the total polarization of the crystal may be expressed as the product of the polarizabilities of the atoms times the local field

$$P = \sum\_{j} N\_{j} p\_{j} = \sum\_{j} N\_{j} a\_{j} E\_{\text{loc}}(j) \tag{9}$$

where the summation is over all the atoms or atomic sites. *N <sup>j</sup>* is the concentration and α<sup>j</sup> is the polarizability of atom *j* and *Eloc*ð Þ*j* is the local field at atom sites *j*. For an isotropic dielectric medium, the local field given by the Lorentz relation Eq. (2) inside the crystal is everywhere the same so that it can be taken out of the summation sign from Eq. (9). Substituting the value of the local field from Eq. (2), the Eq. (9) becomes

$$P = \left(E + \frac{P}{3\varepsilon\_0}\right) \sum\_{j} N\_j \alpha\_j \tag{10}$$

On rearranging the terms and making use of Eq. (5) gives

$$\chi\_{\varepsilon} = \frac{P}{\varepsilon\_{0}E} = \frac{\sum\_{j} N\_{j}a\_{j}}{\left(\varepsilon\_{0} - \frac{1}{3}\sum\_{j} N\_{j}a\_{j}\right)}\tag{11}$$

Using Eq. (6), we get

$$\varepsilon\_r = \mathbf{1} + \frac{\frac{\sum\_j N\_j a\_j}{c\_0}}{\mathbf{1} - \frac{1}{3c\_0} \sum\_j N\_j a\_j} = \frac{\mathbf{1} + \frac{2}{3c\_0} \sum\_j N\_j a\_j}{\mathbf{1} - \frac{1}{3c\_0} \sum\_j N\_j a\_j} \tag{12}$$

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

Solving for P *j N <sup>j</sup>α <sup>j</sup>* we get

*<sup>χ</sup><sup>e</sup>* <sup>¼</sup> *<sup>P</sup>*

susceptibility tensor or of the dielectric tensor of the second rank

*<sup>P</sup>* <sup>¼</sup> <sup>X</sup> *j*

*P* ¼ *E* þ

P *j N <sup>j</sup>α <sup>j</sup> ε*0

<sup>1</sup> � <sup>1</sup> 3*ε*<sup>0</sup> P *j N <sup>j</sup>α <sup>j</sup>*

On rearranging the terms and making use of Eq. (5) gives

*<sup>χ</sup><sup>e</sup>* <sup>¼</sup> *<sup>P</sup> <sup>ε</sup>*0*<sup>E</sup>* <sup>¼</sup>

*ε<sup>r</sup>* ¼ 1 þ

**4.3 Dielectric constant and polarizability**

*Multifunctional Ferroelectric Materials*

the local field (*Eloc*Þ can be expressed as

the atoms times the local field

the Eq. (9) becomes

Using Eq. (6), we get

**14**

Thus, like susceptibility, the dielectric constant (*εr*Þ is also a measure of the polarization (*P*) of the material. The larger the polarization per unit resultant macroscopic field, the greater will be the dielectric constant of the dielectric medium. However, for anisotropic medium, the dielectric response (*εr*or *χe*Þ depends on the direction of the field and described by the components of the

The polarizability ð Þ *α* of an atom is defined in terms of the local electric field (Lorentz field) at the atom. The induced dipoles of moments (*p*) are proportional to

where *α* is known as the polarizability of an atom. For a non-spherical atom or isotropic medium, *α* will be a tensor quantity [6–16]. Thus, polarizability is an atomic property, whereas dielectric constant is a macroscopic property that

depends upon the arrangement of atoms within the crystal. If all the atoms have the same polarizability (*α*Þ and there are *N* number of atoms per unit volume, the total polarization of the crystal may be expressed as the product of the polarizabilities of

*<sup>N</sup> jp <sup>j</sup>* <sup>¼</sup> <sup>X</sup>

*j*

where the summation is over all the atoms or atomic sites. *N <sup>j</sup>* is the concentration and α<sup>j</sup> is the polarizability of atom *j* and *Eloc*ð Þ*j* is the local field at atom sites *j*. For an isotropic dielectric medium, the local field given by the Lorentz relation Eq. (2) inside the crystal is everywhere the same so that it can be taken out of the summation sign from Eq. (9). Substituting the value of the local field from Eq. (2),

> *P* 3*ε*<sup>0</sup> � �X

*j*

<sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup> 3*ε*<sup>0</sup> P *j N <sup>j</sup>α <sup>j</sup>*

<sup>1</sup> � <sup>1</sup> 3*ε*<sup>0</sup> P *j N <sup>j</sup>α <sup>j</sup>*

P *j N <sup>j</sup>α <sup>j</sup>*

*<sup>ε</sup>*<sup>0</sup> � <sup>1</sup> 3 P *j N <sup>j</sup>α <sup>j</sup>*

*<sup>ε</sup>*0*<sup>E</sup>* <sup>¼</sup> *<sup>ε</sup><sup>r</sup>* � <sup>1</sup> (6)

*P<sup>μ</sup>* ¼ *χμνε*0*Eν*; *εμν* ¼ *δμν* þ *χμν* (7)

*p* ¼ *αEloc* (8)

*N <sup>j</sup>α jEloc*ð Þ*j* (9)

*N <sup>j</sup>α <sup>j</sup>* (10)

(12)

� � (11)

$$\frac{\varepsilon\_r - 1}{\varepsilon\_r + 2} = \frac{1}{3\varepsilon\_0} \sum\_j N\_j a\_j \tag{13}$$

This is known as the Clausius-Mossotti relation. It relates the dielectric constant to the atomic polarizability, but only for crystal structures for which the Lorentz field Eq. (2) obtains. The total polarization (*α*Þ can be expressed as the sum of three types of basic polarizability representing the most important contributions to the polarization [2–17] given as

$$a = a\_{\epsilon} + a\_{i} + a\_{d} \tag{14}$$

where *αe*, *α<sup>i</sup>* and *α<sup>d</sup>* are the electronic, ionic and dipolar polarizabilities, respectively shown in **Figure 6**.

i. Electronic polarizability (*αe*Þ: The electronic polarizability (*αe*Þ arises due to the displacement of electrons in an atom relative to the atomic nucleus in the external electric field, as shown in **Figure 6a**. The polarization, as well as the dielectric constant of a material at optical frequencies, results mainly from the electronic polarizability (*αe*Þ. The optical range Eq. (13) reduces as

$$\frac{m^2 - 1}{m^2 + 2} = \frac{1}{3\epsilon\_0} \sum\_{j} N\_j a\_j (\text{electronic}) \tag{15}$$

where *n* being the refractive index which is related to the dielectric constant (*εr*Þ by the relation

$$m^2 = \varepsilon\_r \tag{16}$$

ii. Ionic polarizability (*αi*Þ: The ionic polarizability (*αi*Þ arises due to the relative displacement of positive and negative ions from their equilibrium positions to a distance less than the distance between adjacent ions. The cations are displaced parallel to the Lorentz field, and the anions are displaced in the opposite direction, as shown in **Figure 6b** [5–18].

**Figure 6.**

*Atomic contributions to electric polarization [18].*

**Figure 7.** *Frequency dependence of the various contributions to the polarizability [6].*

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 the ionic polarization becomes

$$P\_{\rm ion} = \mathcal{N}qd \tag{17}$$

**5. Classification of ferroelectrics**

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

*General Introduction to Ferroelectrics*

niobate (LiNbO3) etc.

etc.

of two types.

salt.

etc.

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)

b. Double oxides such as BaTiO3, potassium niobite (KNbO3) and lithium

2.Based on the number of directions allowed to the spontaneous polarization are

3.According to the existence or lack of centre of symmetry in non-polar phase.

a. Non-centre of symmetrical non-polar phase such as KDP and Rochelle

b. Centre of symmetrical non-polar phase such as BaTiO3 and TGS crystals,

a. Order–disorder type such as KDP, RS, TGS, LHP and CsH2PO4 (CDP) etc.

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

b. Displacive type such as BaTiO3, LiNbO3 and KNbO3 etc.

a. Single-axis of polarization such as Rochelle salt, KDP etc.

b. Several-axes of polarization such as BaTiO3 etc.

4.According to the nature of the phase change.

charges not coinciding with each other.

**17**

1.According to the chemical composition of the crystal.

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

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

$$a\_d = \frac{\bar{p}}{E} = \frac{p^2}{3kT} \tag{18}$$

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

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.
