**3. Spontaneous polarization**

The intensity of polarization (*P*) is defined as the electric dipole moment per unit volume of the dielectric material. Spontaneous polarization (*Ps*Þ is a polarization that

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 reaches a maximum value called the saturation value.

**4. Dielectric properties**

*General Introduction to Ferroelectrics*

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

expressed by the Lorentz-relation as

**4.2 Polarization and dielectric susceptibility**

dielectric [5–15]. It is expressed as

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

> *P* 3*ε*<sup>0</sup>

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

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

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

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

*<sup>ε</sup>*0*<sup>E</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup>

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

*P*

*<sup>ε</sup>*0*<sup>E</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>χ</sup><sup>e</sup>* (5)

relative to vacuum is defined in terms of the macroscopic field ð Þ *E* as

dielectric displacement *D* to the macroscopic field intensity ð Þ *E* as

*<sup>ε</sup>*0*<sup>E</sup>* <sup>¼</sup> *<sup>ε</sup>*0*<sup>E</sup>* <sup>þ</sup> *<sup>P</sup>*

*<sup>ε</sup><sup>r</sup>* <sup>¼</sup> *<sup>D</sup>*

the dielectric constant defined as

**13**

¼ *E* þ

*P* 3*ε*<sup>0</sup>

*P*∝*E* ) *P* ¼ *ε*0*χeE* (3)

*D* ¼ *ε*0*εrE* ¼ *εE* ¼ *ε*0*E* þ *P* (4)

(2)

*Eloc* ¼ *E*<sup>0</sup> þ *EP* þ

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

**4.1 Local field**

the average field.

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 electric field [2–14].

**Figure 5.** *Ferroelectric (P-E) hysteresis loop [13].*
