**2. Relaxor ferroelectrics**

*Multifunctional Ferroelectric Materials*

ferroelectric domains [6].

etc.) [3]. The presence of spontaneous polarization (*P*) in ferroelectric material exhibits unique behavior in the presence of external stimuli such as electric field (*E*), temperature (*T*), and stress (*σ*). Nowadays, large numbers of reports are available on the ferroelectric properties of various kinds of materials and their possible applications in modern technologies [3]. Out of the several compounds, Lead Zirconate Titanate (PbZr0.48Ti0.52O3/PZT) is one of the most well-known ferroelectric materials with superior ferroelectric, dielectric, and piezoelectric properties [4]. For a layman's understanding, ferroelectric materials are those which exhibit a high dielectric constant. Furthermore, normal ferroelectrics are characterized by the temperaturedependent maximum dielectric constant near ferroelectric to paraelectric phase transition temperature (*T*c) and, also known as Curie's temperature [4]. Some of the basic characteristics of normal ferroelectric are the non-dispersive nature of transition temperature and follow the first-order phase transition. The temperaturedependent dielectric constant of a typical regular ferroelectric ceramic is shown in **Figure 1** [5]. Also, it is well known that the fundamental physics behind the normal ferroelectric lies in the presence of long-range order parameters (electric dipoles) in

As per the literature, PZT exhibits the normal ferroelectric in nature with the above-mentioned properties and, well understood experimentally as well as theoretically. However, the spatial substitution of a foreign element such as lanthanum (La3+) in PZT shows the intriguing behavior in terms of dielectric, ferroelectric and piezoelectric properties with respect to frequency, temperature, and electric field, which are different from the normal ferroelectric [7, 8]. The extraordinary properties of modified PZT are related to a special group of ferroelectric materials, which is known as relaxor ferroelectric after the name by the scientist Cross in 1987 [9]. He proposed few characteristic properties for the material to be relaxor ferroelectric, as discussed in the later section. Currently, large numbers of materials with relaxor ferroelectric behavior are available in various forms of crystal structures such as perovskite, layer perovskite, tungsten bronze structure, etc. However, the exact origin of extraordinary properties of relaxor ferroelectrics is still a matter of investigation. A series of explanations have been reported to explain the origin of relaxor behavior by using various models such as

*Temperature-dependent dielectric constant and loss of a typical normal ferroelectric ceramic (PZT ceramic). It* 

**50**

**Figure 1.**

*is adapted from Ref. [5] (open access).*

From the earlier discussion (introduction section), the relaxor ferroelectrics show the abnormal behavior as compared to normal ferroelectric in terms of dielectric, piezoelectric, ferroelectric properties and, subsequently, received much attention by the scientific community. To distinguish a relaxor ferroelectric, Cross assigned few basic properties as follows [9].


The temperature-dependent dielectric permittivity of a typical relaxor ferroelectric has been shown in **Figure 2** to visualize the different characteristic temperatures. As per the earlier reports, the interesting properties of relaxor ferroelectric are basically due to the presence of unique polar structure in nanometer size (polar nanoregions: PNRs) along with their response towards the external stimuli [11]. Therefore, it is necessary to understand the origin of PNRs and their effects on the physical properties, as discussed below.

#### **Figure 2.**

*Various characteristic temperatures are associated with a temperature-dependent dielectric constant of relaxor ferroelectric. Adapted from Ref. [11] and reproduced with permission (self-citation © 2021 IOP).*

#### **2.1 Polar nanoregions (PNRs)**

The concept of polar nanoregions in relaxor ferroelectric is quite impressive in terms of fundamental understanding of various peculiar physical properties. Also, it has been reported that the types as well as size of the ferroelectric domains greatly influence the external factors (i.e., electric field, temperature, stress, etc.) and, consequently, affect the different physical properties [12]. Initially, several models have been proposed to describe dielectric anomalies in relaxor ferroelectrics, which lead to formulate the concept of the origin of dynamic and formation of polar nanoregions (PNRs) [10]. The conclusions of various models have been described here to correlate the abnormal behavior of relaxor ferroelectric through the presence of nanosize polar domains. Smolenskii proposed the presence of compositional fluctuation in nanometer-scale within the crystal structure by considering a statistical distribution for the phase transition temperature [13].

Furthermore, Cross extended the Smolenskii theory to superparaelectric model, which is associated with the relaxor ferroelectric behavior with thermally activated superparaelectric clusters [9]. Viehland *et al*. have reported the presence of cooperative interaction between superparaelectric clusters and forms the glasslike freezing behavior similar to the spin-glass system [14]. In later study, Qian and Bursil reported the role of random electric field on the formation and dynamic of polar clusters originated from the nanoscale chemical inhomogeneity and defects [15]. According to their theory, the dispersive nature of relaxor ferroelectric is associated with the variation of cluster size and correlation length (the distance above which the polar clusters become non-interactive) as a function of temperature. Although various models and theories have been explained the relaxor ferroelectric behavior, more additional theories and experiments need for its detailed description. Therefore, the dynamic of polar nanoregions (PNRs) with respect to an electric field as well as the temperature has been mentioned here to understand the clear picture (indirectly) of PNRs within relaxor ferroelectric in terms of dielectric and ferroelectric properties.

**53**

*Relaxor Ferroelectric Oxides: Concept to Applications DOI: http://dx.doi.org/10.5772/intechopen.96185*

In the normal unpoled ferroelectric, the large-size domains with specific dipole moments are randomly distributed throughout the material. However, the domains are switched and aligned in the field direction as far as possible with the application of the threshold electric field (coercive field). This process leads to the formation of macroscopic polarization. The polarization is further enhanced with the electric field and reached to the maximum polarization (*P*max). Usually, the normal ferroelectric exhibits large values of characteristic parameters (*P*sat, *P*r, *E*c*, and* hysteresis loss) [16]. However, it is interestingly observed that the ions (cations) in parent compounds replaced by the small number of foreign ions (in terms of different ionic charge and radius) lead to break the long range ferroelectric polar order domains and, converted into a large number of polar islands which are known as polar nanoregions (PNRs) [17, 18]. In the present context, the fundamental factor related to the formation of polar nanoregions has been related to the appearance of intrinsic inhomogeneity in the material due to the compositional fluctuation at the crystallographic sites and structural modification of the unit cell. N. Qu *et al*. reported the evolution of ferroelectric domains with PNRs in the ferroelectric ceramic (refer **Figure 1** [19]). It is clearly observed that the ferroelectric hysteresis loop becomes slim (having low hysteresis loss) in nature with the increase of the percentage of nanodomains. It occurs due to the reduction of correlation length between order parameters (dipole moments). As compared to the normal ferroelectrics, it is difficult to realize the *P*max at the moderated electric field. It is mainly due to the absence of inducing the long-

range polar ordering as well as an increase in the local random field. Furthermore, the temperature-dependent dielectric constant (

ε

Curie–Weiss law deviates from

ferroelectric provides detailed information about the dynamic and formation of polar nanoregions (PNRs). The temperature evolution of PNRs in (Bi0.5Na0.5TiO3)- SrTiO3-BaTiO3 based relaxor ferroelectric has been explained along with supported by the impedance (Z"/Z"max ~ *f*) as well as electric modulus (M"/M"max ~ *f*) analysis (refer Figure 11 [20]). In general, the relaxor ferroelectric exhibits the paraelectric phase at high temperature with zero net dipole moment (randomly oriented dipole moments) and well follows the Curie–Weiss law. However, the nucleation of polar nanoregions (PNRs) initiates during the cooling at the particular temperature known as Burn's temperature (*T*B). This is the temperature limit below which

[20]. Also, *T*B represents the manifestation of phase transition from paraelectric to ergodic relaxor with the formation of PNRs. This observation is well supported by the occurrence of normalized impedance and electric modulus around the same frequency above the Burn's temperature. This happens due to the higher flipping frequency of the local dipoles at the high-temperature region. Further reduction of temperature (< *T*B) leads to enhance formation of polar nanoregions (PNRs) and, subsequently, maximized in volume near dielectric maxima temperature (*T*M) with broad dispersive peak, which is supported by the observation of separated normalized impedance and electric modulus peaks. The separation in peak position

between Z"/Z"max Vs M"/M"max indirectly confirmed the maximum number of PNRs

(*T*m ~ 150 °C), the polar nanoregions associate with maximum number relaxation times as indicated by the broad dielectric maxima. Also, the dipole moments of PNRs are randomly distributed with approximately zero remanent polarization. The small polarized entities extend throughout the grains with different correlation lengths. Its dynamic behavior with respect to external stimuli has been observed successfully through various experiments such as in situ transmission electron microscope, neutron diffusion scattering, and piezoelectric force microscopy, and so on [21, 22]. Furthermore, the interaction between the polar nanoregions increases continuously with the reduction of temperature (< *T*m ~ 150 °C), which leads to

with the distribution of localized dipolar relaxation. At this temperature

ε

′ ~ *T* curve as shown in **Figure 4** with *T*B = 286 °C

′ ~ ) *T* of relaxor

#### *Relaxor Ferroelectric Oxides: Concept to Applications DOI: http://dx.doi.org/10.5772/intechopen.96185*

*Multifunctional Ferroelectric Materials*

**2.1 Polar nanoregions (PNRs)**

**Figure 2.**

The concept of polar nanoregions in relaxor ferroelectric is quite impressive in terms of fundamental understanding of various peculiar physical properties. Also, it has been reported that the types as well as size of the ferroelectric domains greatly influence the external factors (i.e., electric field, temperature, stress, etc.) and, consequently, affect the different physical properties [12]. Initially, several models have been proposed to describe dielectric anomalies in relaxor ferroelectrics, which lead to formulate the concept of the origin of dynamic and formation of polar nanoregions (PNRs) [10]. The conclusions of various models have been described here to correlate the abnormal behavior of relaxor ferroelectric through the presence of nanosize polar domains. Smolenskii proposed the presence of compositional fluctuation in nanometer-scale within the crystal structure by considering a

*Various characteristic temperatures are associated with a temperature-dependent dielectric constant of relaxor* 

*ferroelectric. Adapted from Ref. [11] and reproduced with permission (self-citation © 2021 IOP).*

statistical distribution for the phase transition temperature [13].

Furthermore, Cross extended the Smolenskii theory to superparaelectric model, which is associated with the relaxor ferroelectric behavior with thermally activated superparaelectric clusters [9]. Viehland *et al*. have reported the presence of cooperative interaction between superparaelectric clusters and forms the glasslike freezing behavior similar to the spin-glass system [14]. In later study, Qian and Bursil reported the role of random electric field on the formation and dynamic of polar clusters originated from the nanoscale chemical inhomogeneity and defects [15]. According to their theory, the dispersive nature of relaxor ferroelectric is associated with the variation of cluster size and correlation length (the distance above which the polar clusters become non-interactive) as a function of temperature. Although various models and theories have been explained the relaxor ferroelectric behavior, more additional theories and experiments need for its detailed description. Therefore, the dynamic of polar nanoregions (PNRs) with respect to an electric field as well as the temperature has been mentioned here to understand the clear picture (indirectly) of PNRs within relaxor ferroelectric in terms of dielectric

**52**

and ferroelectric properties.

In the normal unpoled ferroelectric, the large-size domains with specific dipole moments are randomly distributed throughout the material. However, the domains are switched and aligned in the field direction as far as possible with the application of the threshold electric field (coercive field). This process leads to the formation of macroscopic polarization. The polarization is further enhanced with the electric field and reached to the maximum polarization (*P*max). Usually, the normal ferroelectric exhibits large values of characteristic parameters (*P*sat, *P*r, *E*c*, and* hysteresis loss) [16]. However, it is interestingly observed that the ions (cations) in parent compounds replaced by the small number of foreign ions (in terms of different ionic charge and radius) lead to break the long range ferroelectric polar order domains and, converted into a large number of polar islands which are known as polar nanoregions (PNRs) [17, 18]. In the present context, the fundamental factor related to the formation of polar nanoregions has been related to the appearance of intrinsic inhomogeneity in the material due to the compositional fluctuation at the crystallographic sites and structural modification of the unit cell. N. Qu *et al*. reported the evolution of ferroelectric domains with PNRs in the ferroelectric ceramic (refer **Figure 1** [19]). It is clearly observed that the ferroelectric hysteresis loop becomes slim (having low hysteresis loss) in nature with the increase of the percentage of nanodomains. It occurs due to the reduction of correlation length between order parameters (dipole moments). As compared to the normal ferroelectrics, it is difficult to realize the *P*max at the moderated electric field. It is mainly due to the absence of inducing the longrange polar ordering as well as an increase in the local random field.

Furthermore, the temperature-dependent dielectric constant ( ε ′ ~ ) *T* of relaxor ferroelectric provides detailed information about the dynamic and formation of polar nanoregions (PNRs). The temperature evolution of PNRs in (Bi0.5Na0.5TiO3)- SrTiO3-BaTiO3 based relaxor ferroelectric has been explained along with supported by the impedance (Z"/Z"max ~ *f*) as well as electric modulus (M"/M"max ~ *f*) analysis (refer Figure 11 [20]). In general, the relaxor ferroelectric exhibits the paraelectric phase at high temperature with zero net dipole moment (randomly oriented dipole moments) and well follows the Curie–Weiss law. However, the nucleation of polar nanoregions (PNRs) initiates during the cooling at the particular temperature known as Burn's temperature (*T*B). This is the temperature limit below which Curie–Weiss law deviates from ε ′ ~ *T* curve as shown in **Figure 4** with *T*B = 286 °C [20]. Also, *T*B represents the manifestation of phase transition from paraelectric to ergodic relaxor with the formation of PNRs. This observation is well supported by the occurrence of normalized impedance and electric modulus around the same frequency above the Burn's temperature. This happens due to the higher flipping frequency of the local dipoles at the high-temperature region. Further reduction of temperature (< *T*B) leads to enhance formation of polar nanoregions (PNRs) and, subsequently, maximized in volume near dielectric maxima temperature (*T*M) with broad dispersive peak, which is supported by the observation of separated normalized impedance and electric modulus peaks. The separation in peak position between Z"/Z"max Vs M"/M"max indirectly confirmed the maximum number of PNRs with the distribution of localized dipolar relaxation. At this temperature (*T*m ~ 150 °C), the polar nanoregions associate with maximum number relaxation times as indicated by the broad dielectric maxima. Also, the dipole moments of PNRs are randomly distributed with approximately zero remanent polarization. The small polarized entities extend throughout the grains with different correlation lengths. Its dynamic behavior with respect to external stimuli has been observed successfully through various experiments such as in situ transmission electron microscope, neutron diffusion scattering, and piezoelectric force microscopy, and so on [21, 22]. Furthermore, the interaction between the polar nanoregions increases continuously with the reduction of temperature (< *T*m ~ 150 °C), which leads to

initiate the freezing process of PNRs with the phase transition from ergodic to nonergodic state (ferroelectric). After that, the re-oriental flipping of polar nanoregions get a freeze at a particular temperature, known as freezing temperature (*T*f) and, represented by the nonergodic state (as shown in **Figure 4** with *T*f ~ 76 °C).

#### **2.2 Theory of relaxor ferroelectric**

As per the literature, the formation of PNRs is related to compositional fluctuation at the lattice sites of the crystal structure. To understand the fundamentals behind the formation of PNRs in the relaxor ferroelectric, ABO3 type general perovskite compound with different charge states in A & B-site (i.e. A'xA"1-xB'yB″1 yO3) can be considered. The randomness of A (A', A") and B (B′, B") sites depends upon the ionic sizes, a charge of cations, and distribution of cations in the sublattice. Depending upon the randomness, the distribution of domains is classified into two categories such as LRO (long-range order) and SRO (short-range order). SRO can be represented by the continuous with order distribution of cations on the neighboring sites, and the size of the order domains are extended in the range from 20 to 800 Å in diameter, whereas LRO is extended above 1000 Å diameter [23]. Hence, the diffuse phase transition behavior is the characteristic feature of a disordered system in which the random lattice disorder-induced the dipole impurities and defects at the crystal sites. The correlation length (*r*c) in a highly ordered ferroelectric compound is defined as the extended length up to which dipole entities are correlated with each other. It is basically larger than the lattice constant (*a*) in normal ferroelectric and strongly dependent upon the temperature. With reducing the temperature, the correlation length increases gradually and promotes the growth of polar order in a corporative manner with long-range order in FE (ferroelectric) at *T* < *T*c [23]. However, the correlation length (*r*c) in relaxor ferroelectric significantly reduced, which forms the polar nanoregions by frustrating the longrange ordering in FE similar to the dipolar glass-like behavior.

According to Landau-Devonshire theory of free energy (*F*) of a uniaxial ferroelectric material can be expressed in terms of electric field and polarization by ignoring the stress field as follow; [24]

$$F = \frac{1}{2}aP^2 + \frac{1}{4}bP^4 + \frac{1}{6}cP^6 + \dots - EP \tag{1}$$

**55**

Here, *<sup>r</sup>* ε

follow. [31, 32]

*Relaxor Ferroelectric Oxides: Concept to Applications DOI: http://dx.doi.org/10.5772/intechopen.96185*

dielectric permittivity

*Adapted from ref. [23] (open access).*

*,T ,P E* ′ and

**Figure 3.**

*(Tm), <sup>r</sup>* ε

shown in **Figure 4** [29].

ε ω

(diffuseness) for RFEs by the following relation [30].

dielectric maxima temperature (*T*m) and becomes independent at the higher temperature region. Also, the *T*m shifts towards the higher temperature with an increase in frequency, which is clearly shown in **Figure 3**. Also, the reduction in relaxation processes below *T*m in RFEs suggests the onset of relaxor freezing. Therefore, the characteristic relaxation time diverges near freezing temperature

*Confirmation of the presence of polar nanoregions at above the maximum dielectric temperature* 

 *are the permittivity, temperature, electrical polarization, and electric field, respectively.* 

( ) <sup>0</sup> exp *<sup>a</sup> Bm f*

*<sup>E</sup> f f KT T* <sup>−</sup> <sup>=</sup> <sup>−</sup>

Here, *f* is the measuring frequency, and *Ea* , *Tf* an 0*f* are the parameters. In general, Tf is believed to be the temperature corresponds to freezing of the dynamic of PNRs due to the increase in the cooperative interaction between them. A typical VF relation for BiAlO3-(x) BaTiO3 solid solution with x = 0.05 and 0.1 has been

Furthermore, the Curie–Weiss (C-W) law is well known to describe the ferroelectric to paraelectric phase transition for normal ferroelectric and, well fitted above the phase transition temperature (Curie temperature: *T*c). However, it deviates for relaxor ferroelectrics due to the presence of diffuse phase transition. Therefore, reciprocal of *C-W* law can be used to estimate the degree of deviation

> 1 *<sup>c</sup> r*

ε *C* <sup>−</sup> <sup>=</sup>

explained that, the C-W law is well fitted at above the Burn's temperature (*T*B). Hence, the degree of diffuseness ( ∆*T* ) in term of temperature can be calculated as

*T T*

′ is the dielectric permittivity, *C* is the Curie temperature. It has been

′ (3)

(*T*f) as per the following Volgel-Fulcher (VF) law relation [27, 28].

′(*T*, ) with applied frequency observes at the lower side of

(2)

Where *E* is the electric field, *P* is electrical polarization, *a*, *b*, and *c* are the unknown coefficient with temperature-dependent. Here, the even powers of *P* are considered because the energy should be the same for ±*Ps* . *Ps* is the saturation

polarization. The equilibrium configuration can be estimated by finding the minima value of *F* [24]. The phenomenological description of ferroelectrics states that the spontaneous polarization varies continuously with temperature and attains zero value at the Curie's temperature (*T*c: dielectric maxima temperature) in normal ferroelectric. However, the properties of relaxor ferroelectrics, which depend upon polarization in the order of *P*<sup>2</sup> have been measured experimentally at above the maximum dielectric temperature (shown in **Figure 3**), which supported the presence of polar nanoregions (PNRs) [25, 26]. Furthermore, different formulations have been extensively used to estimate the various characteristic parameters and the degree of relaxor ferroelectric behavior (diffuseness) of materials as described below.

The relaxor ferroelectrics exhibit the broad diffuse maxima in the temperature dependence dielectric permittivity. For a typical RFE, strong dispersion behavior of

#### **Figure 3.**

*Multifunctional Ferroelectric Materials*

**2.2 Theory of relaxor ferroelectric**

initiate the freezing process of PNRs with the phase transition from ergodic to nonergodic state (ferroelectric). After that, the re-oriental flipping of polar nanoregions get a freeze at a particular temperature, known as freezing temperature (*T*f) and, represented by the nonergodic state (as shown in **Figure 4** with *T*f ~ 76 °C).

range ordering in FE similar to the dipolar glass-like behavior.

ignoring the stress field as follow; [24]

polarization in the order of *P*<sup>2</sup>

described below.

According to Landau-Devonshire theory of free energy (*F*) of a uniaxial ferroelectric material can be expressed in terms of electric field and polarization by

> 111 2 46 246

Where *E* is the electric field, *P* is electrical polarization, *a*, *b*, and *c* are the unknown coefficient with temperature-dependent. Here, the even powers of *P* are considered because the energy should be the same for ±*Ps* . *Ps* is the saturation polarization. The equilibrium configuration can be estimated by finding the minima value of *F* [24]. The phenomenological description of ferroelectrics states that the spontaneous polarization varies continuously with temperature and attains zero value at the Curie's temperature (*T*c: dielectric maxima temperature) in normal ferroelectric. However, the properties of relaxor ferroelectrics, which depend upon

maximum dielectric temperature (shown in **Figure 3**), which supported the presence of polar nanoregions (PNRs) [25, 26]. Furthermore, different formulations have been extensively used to estimate the various characteristic parameters and the degree of relaxor ferroelectric behavior (diffuseness) of materials as

The relaxor ferroelectrics exhibit the broad diffuse maxima in the temperature dependence dielectric permittivity. For a typical RFE, strong dispersion behavior of

*F aP bP cP EP* = + + +…− (1)

have been measured experimentally at above the

As per the literature, the formation of PNRs is related to compositional fluctuation at the lattice sites of the crystal structure. To understand the fundamentals behind the formation of PNRs in the relaxor ferroelectric, ABO3 type general perovskite compound with different charge states in A & B-site (i.e. A'xA"1-xB'yB″1 yO3) can be considered. The randomness of A (A', A") and B (B′, B") sites depends upon the ionic sizes, a charge of cations, and distribution of cations in the sublattice. Depending upon the randomness, the distribution of domains is classified into two categories such as LRO (long-range order) and SRO (short-range order). SRO can be represented by the continuous with order distribution of cations on the neighboring sites, and the size of the order domains are extended in the range from 20 to 800 Å in diameter, whereas LRO is extended above 1000 Å diameter [23]. Hence, the diffuse phase transition behavior is the characteristic feature of a disordered system in which the random lattice disorder-induced the dipole impurities and defects at the crystal sites. The correlation length (*r*c) in a highly ordered ferroelectric compound is defined as the extended length up to which dipole entities are correlated with each other. It is basically larger than the lattice constant (*a*) in normal ferroelectric and strongly dependent upon the temperature. With reducing the temperature, the correlation length increases gradually and promotes the growth of polar order in a corporative manner with long-range order in FE (ferroelectric) at *T* < *T*c [23]. However, the correlation length (*r*c) in relaxor ferroelectric significantly reduced, which forms the polar nanoregions by frustrating the long-

**54**

*Confirmation of the presence of polar nanoregions at above the maximum dielectric temperature (Tm), <sup>r</sup>* ε *,T ,P E* ′ and  *are the permittivity, temperature, electrical polarization, and electric field, respectively. Adapted from ref. [23] (open access).*

dielectric permittivity ε ω ′(*T*, ) with applied frequency observes at the lower side of dielectric maxima temperature (*T*m) and becomes independent at the higher temperature region. Also, the *T*m shifts towards the higher temperature with an increase in frequency, which is clearly shown in **Figure 3**. Also, the reduction in relaxation processes below *T*m in RFEs suggests the onset of relaxor freezing. Therefore, the characteristic relaxation time diverges near freezing temperature (*T*f) as per the following Volgel-Fulcher (VF) law relation [27, 28].

$$f = f\_0 \exp\left(\frac{-E\_a}{K\_B \left(T\_m - T\_f\right)}\right) \tag{2}$$

Here, *f* is the measuring frequency, and *Ea* , *Tf* an 0*f* are the parameters. In general, Tf is believed to be the temperature corresponds to freezing of the dynamic of PNRs due to the increase in the cooperative interaction between them. A typical VF relation for BiAlO3-(x) BaTiO3 solid solution with x = 0.05 and 0.1 has been shown in **Figure 4** [29].

Furthermore, the Curie–Weiss (C-W) law is well known to describe the ferroelectric to paraelectric phase transition for normal ferroelectric and, well fitted above the phase transition temperature (Curie temperature: *T*c). However, it deviates for relaxor ferroelectrics due to the presence of diffuse phase transition. Therefore, reciprocal of *C-W* law can be used to estimate the degree of deviation (diffuseness) for RFEs by the following relation [30].

$$\bigvee\_{\mathcal{E}\_r} \mathcal{E}\_r = \frac{T - T\_c}{C} \tag{3}$$

Here, *<sup>r</sup>* ε ′ is the dielectric permittivity, *C* is the Curie temperature. It has been explained that, the C-W law is well fitted at above the Burn's temperature (*T*B). Hence, the degree of diffuseness ( ∆*T* ) in term of temperature can be calculated as follow. [31, 32]

$$
\Delta T = T\_{\text{B}} - T\_{\text{m}} \tag{4}
$$

To visualize the above relations for the experimental observations, the fitted curve has been shown in **Figure 5** [11]. The separation between maximum dielectric temperature and Burn temperature represents the characteristic of diffuse phase transition.

Furthermore, Uchino and Nomura quantified the dielectric material's relaxor behavior from the ferroelectric to the paraelectric phase transition. They have estimated the diffuse behavior by employing the modified Curie–Weiss (MCW) law. According to this law, the reciprocal of the permittivity can be expressed as; [33].

$$\frac{1}{\epsilon^{'}} - \frac{1}{\left(\epsilon\_{\max}\right)^{'}} = \mathbf{C} \left(T - T\_m\right)^{'} \mathbf{1} \le \gamma \le 2\tag{5}$$

Here γ and *C* are the constants. γ = 1 for the normal ferroelectric material. 1 2 ≤ ≤ γ represents the relaxor ferroelectric behavior and, γ = 2 is known as the completely diffuse phase. The MCW law is well fitted for (Lax(Bi0.5Na0.5)1–1.5x) 0.97Ba0.03TiO3 based relaxor ferroelectric with different mole fraction of La (refer **Figure 3** [34]).

According to Smolenskii's work, the temperature dependence of dielectric permittivity above *T*m is extensively studied to explain the DPT (diffuse phase transition) of relaxor ferroelectrics. Besides this, there are other models reported to describe DPT behavior quantitatively at above the maximum dielectric temperature (*T > T*m). One of the most applied models is based on Lorentz-type empirical relation as follow [35, 36].

( ) 2 <sup>2</sup> 1 2 *A A r A* ε *T T* ε δ <sup>−</sup> = + (6)

**57**

at *T*A. δ

**Figure 6.**

**Figure 5.** *Variation of* 

1 *r* ε

*(self-citation © 2021 IOP).*

*(© 2021 AIP).*

Where *TA* (*TA* ≠*T*m) and *<sup>A</sup>*

ε

temperature corresponding to the dielectric peak and *<sup>A</sup>*

are the fitting parameters. *TA* can be defined as the

is the dielectric constant

ε

*<sup>A</sup>* is the diffuseness parameters which is independent of temperature and

 *as a function of temperature, and the solid straight lines represent the C-W law at a* 

*particular frequency. TB is the burn temperature. Adapted from ref. [11] and reproduced with permission* 

*The temperature-dependent dielectric permittivity of (a) BZT25 (b) BZT30, (c) BZT35 and (d) PMN ceramic fitted with Lorentz relation (solid lines). Adapted from ref. [37] and reproduced with permission* 

*Relaxor Ferroelectric Oxides: Concept to Applications DOI: http://dx.doi.org/10.5772/intechopen.96185*

#### **Figure 4.**

*Volgel-Fulcher relation plot for BiAiO3-(x) BaTiO3 solid solutions with x = 0.05 & 0. Adapted from ref. [29] and reproduced with permission (© 2021 AIP).*

*Relaxor Ferroelectric Oxides: Concept to Applications DOI: http://dx.doi.org/10.5772/intechopen.96185*

*Multifunctional Ferroelectric Materials*

Here γ

**Figure 3** [34]).

relation as follow [35, 36].

1 2 ≤ ≤ γ

∆= − *TT T B m* (4)

To visualize the above relations for the experimental observations, the fitted curve has been shown in **Figure 5** [11]. The separation between maximum dielectric temperature and Burn temperature represents the characteristic of diffuse phase transition. Furthermore, Uchino and Nomura quantified the dielectric material's relaxor behavior from the ferroelectric to the paraelectric phase transition. They have estimated the diffuse behavior by employing the modified Curie–Weiss (MCW) law. According to this law, the reciprocal of the permittivity can be expressed as; [33].

> ( ) ( ) max 1 1 *CT Tm* 1 2

completely diffuse phase. The MCW law is well fitted for (Lax(Bi0.5Na0.5)1–1.5x) 0.97Ba0.03TiO3 based relaxor ferroelectric with different mole fraction of La (refer

According to Smolenskii's work, the temperature dependence of dielectric permittivity above *T*m is extensively studied to explain the DPT (diffuse phase transition) of relaxor ferroelectrics. Besides this, there are other models reported to describe DPT behavior quantitatively at above the maximum dielectric temperature (*T > T*m). One of the most applied models is based on Lorentz-type empirical

> ( ) 2

> > δ

<sup>2</sup> 1 2 *A A r A*

*T T*

*Volgel-Fulcher relation plot for BiAiO3-(x) BaTiO3 solid solutions with x = 0.05 & 0. Adapted from ref. [29]* 

γ

represents the relaxor ferroelectric behavior and,

ε

ε

∈

and *C* are the constants.

∈

γ

γ

= 1 for the normal ferroelectric material.

<sup>−</sup> = + (6)

γ

= 2 is known as the

′ ′ − = − ≤≤ (5)

**56**

**Figure 4.**

*and reproduced with permission (© 2021 AIP).*

**Figure 5.** *Variation of*  1 *r* ε *as a function of temperature, and the solid straight lines represent the C-W law at a particular frequency. TB is the burn temperature. Adapted from ref. [11] and reproduced with permission (self-citation © 2021 IOP).*

#### **Figure 6.**

*The temperature-dependent dielectric permittivity of (a) BZT25 (b) BZT30, (c) BZT35 and (d) PMN ceramic fitted with Lorentz relation (solid lines). Adapted from ref. [37] and reproduced with permission (© 2021 AIP).*

Where *TA* (*TA* ≠*T*m) and *<sup>A</sup>* ε are the fitting parameters. *TA* can be defined as the temperature corresponding to the dielectric peak and *<sup>A</sup>* ε is the dielectric constant at *T*A. δ*<sup>A</sup>* is the diffuseness parameters which is independent of temperature and

frequency. Hence, the DPT behavior of RFEs represents in terms of temperature (*T*m*-T*A) and dielectric constant ( ) *r A* ε ε − . Lei *et al*. have reported that the Lorentz formula is well fitted in both lower and higher temperature regions in ε *<sup>r</sup>* ~ *T* curve for Ba (Ti0.8Sn0.2)O3 relaxor ferroelectric [37]. Similarly, Lorentz type relationship in temperature-dependent dielectric permittivity in Ba (ZrxTi1-x)O3 solid solutions, PbMg1/3Nb2/3O3 relaxor with diffuse phase transition has been reported by S. Ke *et al*. A typical plot of Lorentz formula is shown in **Figure 6**.
