Introduction

Dielectrics are a special kind of insulating materials. As they are insulators, they possess virtually no free charge carriers. When dielectric materials are placed in an external electric field they undergo polarization. Polarization is the process of displacement of oppositely charged particles in opposite directions due to an externally applied field, which results in the formation of a permanent dipole moment. The strength of the dielectric properties of materials is measured in terms of the dielectric constant (ε) or relative permittivity (ε_r ). The dielectric constant (ε) or relative permittivity (ε_r ) is the measure of the ability of the material to be polarized in response to an applied electric field. Hence the greater the dielectric constant of a material, the greater is the polarization developed in a material for a particular applied field strength.

The energy stored in a capacitor is

$$\begin{eqnarray}&&E=\displaystyle \frac{1}{2}{C}{V}^{2}.\end{eqnarray} \tag{ 1.1 }$$

The total capacitance (C) of a capacitor is given by

$$\begin{eqnarray}&&C=\displaystyle \frac{{\varepsilon }_{0}{\varepsilon }_{r}A}{d}\end{eqnarray} \tag{ 1.2 }$$

$$\begin{eqnarray}&&{C}_{o}={\epsilon }_{o}\displaystyle \frac{A}{d}\end{eqnarray} \tag{ 1.3 }$$

$$\begin{eqnarray}&&C=\,{\varepsilon }_{r}{C}_{o},\end{eqnarray} \tag{ 1.4 }$$

where, ε₀ is the permittivity of the free space, ε is the permittivity or dielectric constant of the material which is equal to ε_o *ε_r , d is the distance between the parallel plates, and A is the area of the plates:

$$\begin{eqnarray}&&{\varepsilon }_{r}* ={\varepsilon }_{r}^{\prime} -j{\varepsilon }_{r}^{\prime\prime} \end{eqnarray} \tag{ 1.5 }$$

ε_r * represents the complex permittivity, the real part of permittivity (ε_r ') is a measure of the energy stored in a material. The imaginary part of the permittivity (ε_r '') is a measure of the energy lost in the material when an external electric field is applied. It is always greater than zero and is usually much smaller than the real part of permittivity (ε_r '). When the complex permittivity is plotted as a simple vector diagram (figure 1.3), the real and imaginary components are 90° out-of-phase. The vector sum forms an angle δ with the real axis (ε_r '). The relative 'lossiness' of a material is the ratio of the energy lost to the energy stored. The loss tangent or tanδ is defined as the ratio of the imaginary part of the dielectric constant to its real part. D denotes the dissipation factor and Q is the quality factor. The loss tangent tanδ is called the tangent loss or dissipation factor. Sometimes the term 'quality factor or Q-factor' is used with respect to an electronic microwave material, which is the reciprocal of the loss tangent. For very low-loss materials, since tanδ ≈ δ, the loss tangent can be expressed in angle units, milliradians or microradians:

$$\begin{eqnarray}&&\tan \,\delta =\,\frac{{\varepsilon }_{r}^{^{\prime\prime} }}{{\varepsilon }_{r}^{^{\prime} }},\end{eqnarray} \tag{ 1.6 }$$

$$\begin{eqnarray}&&D=\displaystyle \frac{1}{Q}=\displaystyle \frac{{\rm{Energy\; lost\; per\; cycle}}}{{\rm{Energy\; stored\; per\; cycle}}}.\end{eqnarray} \tag{ 1.7 }$$

The dielectric susceptibility χ of the material is the parameter that relates the applied external electric field (E) and electrical polarization (P) by the relation

$$\begin{eqnarray}&&P=\chi {\varepsilon }_{0}E.\end{eqnarray} \tag{ 1.8 }$$

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Also, ε_r of an isotropic medium is defined by the relation

$$\begin{eqnarray}&&{\varepsilon }_{r}=\frac{({\varepsilon }_{o}E+\,P\,)}{{\varepsilon }_{o}E}=1+\chi ,\end{eqnarray} \tag{ 1.9 }$$

where ε₀ E + P = D is the electric displacement field.

Therefore, the energy stored in a dielectric material is directly proportional to the dielectric constant of the material. The total polarization in a dielectric material is a combinational contribution of four different polarization mechanisms which depend on the frequency of the externally applied electric field.

The four mechanisms of polarization are the following.

When an atom is placed in an external electric field, the centre of mass of a spherical electron cloud of uniform density is displaced slightly with respect to the centre of mass of the nucleus with the charge +Ze due to the electrostatic force. This causes an induced dipole moment which results in electronic polarization (P_e ).

When the sharing of electrons among the atoms of a molecule is not symmetrical then in such cases ions acquire partial opposite charge. When such a molecule is placed in an external electric field, the net charge tends to change the equilibrium position of the ions themselves. Thus when an external electric field is applied, the positive and negative sub-lattices are displaced in opposite directions. This gives rise to an induced dipole moment resulting in ionic polarization (P_i ).

In some molecules, there is a permanent dipole moment even in the absence of an external electric field. This permanent dipole moment experiences a torque in the presence of an external applied electric field. This mechanism gives rise to orientational polarization (P_o ). For orientational polarization the molecule must possess a permanent dipole moment.

In addition to these three polarization mechanisms, one more polarization mechanism, known as the space charge polarization mechanism, also exists. The polarization which occurs due to the accumulation of charges at the electrodes or at the interfaces is known as space charge polarization (P_s ) or interfacial polarization. The main cause of space charge polarization is diffusion, in which the charges migrate over a limited distance under the influence of an applied field and forms positive and negative space charges at the grain boundaries or at the electrode–material interface. Except for P_o , all other kinds of polarization, that is P_e , P_i , and P_s , are independent of temperature. The total polarization (P_T ) of a material is given by the contribution due to all types of polarizations:

$$\begin{eqnarray}&&{P}_{T}={P}_{e}+{P}_{i}+{P}_{o}+{P}_{s}.\end{eqnarray} \tag{ 1.10 }$$

In the lower frequency region all types of polarizations (electronic, ionic, orientational, and interfacial) contribute to total polarization which gives rise to a high value of the dielectric constant. However, as the frequency increases, it becomes difficult for the dipoles to orient themselves in the direction of the applied electric field during each alternation of the electric field. Hence, a low value of dielectric constant at high frequency is obtained. The dependence of the polarization on frequency also indicates the time for which a particular mechanics of polarization remains. Electronic polarization (P_e ) is the fastest and occurs completely at the instant the voltage is applied. It persists around 10¹³–10¹⁵ Hz and occurs during every cycle of applied voltage. Ionic polarization (P_i ) is slower compared to P_e as ions are heavier than atoms, so the time taken for displacement is greater and it occurs at frequencies between 10⁹–10¹³ Hz. Orientational polarization occurs around 10⁹ Hz and is slower than P_i . The slowest is the space charge polarization occurring between 50–60 Hz since it involves the diffusion of ions over several interatomic distances. Figure 1.4 represents the dependence of each polarization mechanism as a function of frequency.

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Complex impedance spectroscopy is a powerful tool used for studying the electrical properties of polycrystalline materials. Most of the electrical properties are affected strongly by the microstructure, and impedance spectra usually contain features that are related directly to the microstructure (the grain and grain boundaries of differing phase compositions, the suspension of one phase within another, the porosity, electrode interface effects, etc). It also enables us to evaluate the relaxation frequency ω_max of the material. The relaxation frequency of the material at a given temperature is an intrinsic property of the material independent of the geometrical factors of the sample. The contribution of various processes such as grain, grain boundary, and electrode–sample interface effects, etc, can be resolved in the frequency domain. Impedance is the sum of resistance and reactance in which the latter is a function of frequency. Any material can be considered as a combination of resistance and reactance either in series or in a parallel combination of both. In the parallel mode admittance (Y) is dominant and in the series mode impedance (Z) plays a vital role. The parallel mode is more appropriate for inductive samples and the series mode for capacitive samples. Electrical insulators can be characterized using ac impedance data. There are various formalisms by which one can exploit the wealth of data. The effects which do not show up in one of the formalisms show up in another. All the four formalisms are interrelated. The different formalisms are electric modulus (M), admittance (Y), permittivity (ε), and impedance (Z), after Sinclair et al [10]. They are

$$\begin{eqnarray}&&{M}^{\ast }=\,\displaystyle \frac{1}{{\varepsilon }^{\ast }}=\,M^{\prime} +iM^{\prime\prime} \end{eqnarray} \tag{ 1.11 }$$

$$\begin{eqnarray}&&{Z}^{\ast }=\,\displaystyle \frac{1}{j\omega {C}_{o}{\varepsilon }^{\ast }}\end{eqnarray} \tag{ 1.12 }$$

$$\begin{eqnarray}&&{Y}^{\ast }=\,\displaystyle \frac{1}{{Z}^{\ast }}=\,\displaystyle \frac{j\omega {C}_{o}}{{M}^{\ast }}.\end{eqnarray} \tag{ 1.13 }$$

Different graphical analyses can be performed using these three equations. The general forms of data presentation are complex impedance plots (Z'' versus Z'), loss spectra (ε'' or M'' versus frequency), and the plots of ε', σ', or σ'' versus frequency. Usually, a peak is observed when a graph with these dielectric parameters is plotted with frequency. These peaks within a particular frequency range depend on the strength of relaxation. The data over a wide range of frequencies give additional information compared to fixed frequency dielectric data. Particularly, polycrystalline materials exhibit a wide variety of properties depending on the frequency. Such studies would help in understanding the dielectric nature, conduction processes, and possible relaxation phenomena. Any intrinsic property that influences the conductivity of an electrode–material system or an external stimulus can be studied by impedance measurement. The parameters derived from such a study fall into two categories (i) those relevant only to the material itself, such as conductivity, dielectric constant, etc, and (ii) those pertinent to an electrode–material interface such as the adsorption–reaction rate, capacitance of interface regions, diffusion coefficients, etc [9]. All ferroelectric materials have coupled electrical and mechanical properties. The impedance data, generally compared to an equivalent circuit, are obtained, which represent physical processes taking place in the system under investigation. An equivalent circuit is made up of ideal resistors, capacitors, inductance, and possibly various distributed circuit elements. In such a circuit, the resistance represents a conductive path and a given resistor in the circuit might account for the bulk conductivity of the material. Similarly, capacitances and inductances will be generally associated with various polarization processes. The total impedance of a polycrystalline material is the sum of the grain, grain boundary, and electrode effects [10]. By analysing the results on impedance data in more than one of the formalisms, it may be possible to separate these effects effectively.

Piezoelectricity was first discovered by Jacques and Pierre Curie [11]. Piezoelectricity is defined as 'electrical polarization produced by applying mechanical strain to a crystal'. The polarization produced is directly proportional to the mechanical strain applied to the crystal. A basic requirement for a material to show piezoelectricity is that it must lack a centre of symmetry of the unit cell. This non-centrosymmetry of the unit cell creates an asymmetric charge distribution, which results in the formation of an electric dipole in the crystal as a whole. The converse is also true, i.e. when voltage is applied across the surface of the crystal the strain is produced across the surface. This strain produced in a crystal is directly proportional to the applied electric field.

The direct and converse piezoelectric effects are shown in figure 1.5 and can be expressed in tensor notation as

$$\begin{eqnarray}&&{D}_{i}={d}_{{ijk}}{\sigma }_{{jk}}\,(\text{direct effect})\end{eqnarray} \tag{ 1.14 }$$

$$\begin{eqnarray}&&{S}_{{ij}}={d}_{{kij}}{E}_{k}\,(\text{converse effect}),\end{eqnarray} \tag{ 1.15 }$$

where D_i is the polarization generated along the k-axis in response to the applied stress σ_jk . For the converse effect S_ij is the strain generated in a particular orientation of the crystal by the application of the electric field E_k along the k-axis [7, 12] and d_ijk and d_kij are the piezoelectric coefficients for the direct and converse effects, respectively.

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Pyroelectric materials belong to a category of non-centrosymmetric polar crystals that possess coupling between the electrical polarization P and temperature T, such that a change in temperature results in a change in the electric dipole moment. In pyroelectric materials, the spontaneous polarization is not necessarily switchable using an applied electric field. These materials can be used for converting thermal energy to electric energy. Pyroelectricity is the ability of the materials to generate a temporary voltage when they are heated or cooled. As a result of the temperature difference generated across the crystal, positive and negative charge displacement takes place. Due to this relative displacement of charges, the material becomes polarized (as a molecule possesses a permanent dipole moment) and hence a potential difference is generated.

The pyroelectric coefficient (p_i ) is described as the change in the spontaneous polarization with temperature [13] as

$$\begin{eqnarray}&&{p}_{i}=\displaystyle \frac{\partial {Ps}}{\partial T},\end{eqnarray} \tag{ 1.16 }$$

where (Cm⁻² K⁻¹) is the unit of the pyroelectric coefficient.

Alternatively, p_i is calculated using the relation

$$\begin{eqnarray}&&{p}_{i}=\displaystyle \frac{I}{A\displaystyle \frac{{dT}}{{dt}}},\end{eqnarray} \tag{ 1.17 }$$

where I is the pyroelectric current measured during the heating cycle, A is the area of the electrode, and $\frac{{dT}}{{dt}}$ is the rate of heating. The spontaneous polarization disappears above the Curie temperature (T_c ), the temperature above which a pyroelectric unit cell transforms into the centrosymmetric paraelectric phase. In the centrosymmetric paraelectric phase no dipole moment exists within the unit cell and thus it exhibits a reduced piezoelectric effect [5].

The subclass of dielectric materials in which spontaneous polarization exists even in the absence of an electric field and the polarization can be reversed with the direction of the applied external field are called ferroelectric materials. Ferroelectrics are polar dielectric materials that possess at least two equilibrium orientations of the spontaneous polarization vector in the absence of an external electric field, and in which the polarization may be switched between those orientations by means of an electric field. The magnitude of the external electric field applied should be less than the dielectric breakdown of the material itself. Therefore, for the material to show ferroelectricity it must possess the following properties:

• 1.  
  Spontaneous polarization.
• 2.  
  The direction of polarization can be reversed by an externally applied electric field.
• 3.  
  A ferroelectric hysteresis loop.
• 4.  
  A ferroelectric transition temperature.

Ferroelectric materials must possess non-centrosymmetric structure. As the material is non-centrosymmetric, ferroelectricity arises due to the change in the positions of the cations and anions within the unit cell, resulting in reversible spontaneous dipole moments. This gives rise to a dipole moment equal to qd/V, where q is the electric charge on a displaced ion, d is the relative displacement, and v is the volume of the unit cell.

The dipole moment is related to the electric displacement as

$$\begin{eqnarray}&&D=\chi \,{\chi }^{0}E={\chi }^{0}E+P,\end{eqnarray} \tag{ 1.18 }$$

where χ⁰ and χ are the free space and relative susceptibilities, respectively.

Ferroelectric materials show the phase transition phenomenon. These materials undergo phase transition from ferroelectric to paraelectric above a certain temperature. During this phase transition they undergo a change in symmetry. The temperature at which the phase transition occurs is called the Curie temperature (T_c ). At this temperature the symmetry of the material changes from the non-centrosymmetric to symmetric phase. Due to a change in the symmetry above the Curie temperature, spontaneous polarization will be lost and the ferroelectric material becomes paraelectric. Hence, at the Curie temperature the hysteresis curve becomes linear. In addition to the change in spontaneous polarization, temperature affects the dielectric constant of the materials.

The spontaneous polarization (P_s ) decreases with an increase in temperature and vanishes at a transition temperature (T_c ), while the temperature dependence of the dielectric constant (ε) shows a sharp peak at T_c as shown in figure 1.6. The Curie–Weiss temperature (T_o ) coincides with the true transition temperature T_c for a ferroelectric material, which undergoes a second-order phase transition. However, in the case of a first order phase transition, the Curie–Weiss temperature T_o is several degrees lower than the transition temperature (T_c ).

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Some ferroelectric materials do not show a sharp peak in the temperature dependence of the dielectric constant at the transition temperature. Such materials show a broad peak at the transition temperature instead of a sharp peak. This type of dielectric behaviour is called a diffuse phase transition (DPT). It is well known that the dielectric constant/permittivity of a classical ferroelectric material above the transition temperature follows the Curie–Weiss law. However, the dependence of the dielectric permittivity of DPT ferroelectrics on a temperature above the transition temperature differs from the Curie–Weiss law over a wide temperature range. Smolenskii et al [14] predicted the following quadratic dependence of the dielectric permittivity on temperature:

$$\begin{eqnarray}&&\displaystyle \frac{1}{\varepsilon }-\displaystyle \frac{1}{{\varepsilon }_{m}}=\displaystyle \frac{{\left(T-{T}_{m}\right)}^{2}}{2{\varepsilon }_{m}{\delta }^{2}},\end{eqnarray} \tag{ 1.19 }$$

where ε_m is the dielectric permittivity maximum, T_m is the temperature at which ε_m occurs (transition temperature), and d is the deviation factor that describes the degree of the DPT. However, Uchino et al [15] found that for many relaxor ferroelectric materials, the diffuseness does not exactly obey Smoleskii's [16] quadratic law and they then proposed an empirical expression to describe the diffuseness of the phase transition:

$$\begin{eqnarray}&&\displaystyle \frac{1}{\varepsilon }-\displaystyle \frac{1}{{\varepsilon }_{m}}=\displaystyle \frac{{\left(T-{T}_{m}\right)}^{\gamma }}{2{\varepsilon }_{m}{\delta }^{2}}.\end{eqnarray} \tag{ 1.20 }$$

In this expression γ is the degree of diffuseness and ranges from 1 to 2 for normal to relaxor ferroelectrics. The relaxor behaviour in the sample is observed due to polar domain fluctuations in the neighbourhood of the maximum permittivity and is reflected in the values of γ [17].

1.3.5.1. Hysteresis loop

The ferroelectric properties of a material are studied using a hysteresis P–E loop, as the ferroelectric materials exhibit hysteresis analogous to ferromagnetic materials. For these materials polarization is not a unique function of applied electric field strength. However, in dielectric materials, in general, polarization is a linear function of applied field strength, whereas it not a linear function in ferroelectric materials.

The ferroelectric specimen that is not previously subjected to an applied external electric field is said to be a virgin ferroelectric specimen. If this virgin specimen of ferroelectric material is subjected to an applied external electric field and this field is gradually increased, the polarization P increases along the curve 'ABCD' as shown in the figure 1.7. When the field gradually decreases, it is found that even when the applied electric field becomes zero (E = 0) there is polarization still left in the specimen. This polarization is called remanent polarization (P_r ). In other words, the material is said to be 'spontaneously polarized'. The electric field is to be applied in the opposite direction in order to bring the polarization to zero. The required electric field to reduce the polarization to zero is called the coercive field (E_c ).

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The existence of a hysteresis loop in a material implies that the substance possesses spontaneous polarization, i.e. polarization when the applied field is zero. Extrapolation of the linear segment P_s which intersects on the P-axis at a value P_s is the maximum value of polarization attained by the specimen and is called the saturation polarization (P_s ) and specimen is said to be saturated. Spontaneous polarization of a specimen depends on:

• 1.  
  The dimensions of a specimen.
• 2.  
  Temperature.
• 3.  
  Humidity.
• 4.  
  The texture of the crystal.
• 5.  
  The previous history of the specimen.

1.3.5.2. Ferroelectric domain theory and domain walls

A quantitative explanation of hysteresis in ferroelectric materials is given by the domain theory of ferroelectrics. According to the domain theory, ferroelectric materials possess regions with uniform polarization called ferroelectric domains, i.e. within a domain all electric dipoles are aligned in the same direction (shown in figure 1.8).

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The direction of polarization is not the same throughout the specimen. In fact, the specimen may be considered to consist of a number of domains. These domains themselves become spontaneously polarized but the direction of polarization varies from one domain to another domain. Thus, the virgin specimen may have zero polarization as a whole, i.e. the resultant polarization vector of the domains may vanish.

When an external electric field is applied, the domains in which the polarization points along the direction of the applied electric field grow at the expense of other domains in which the polarization points in the other direction. This process corresponds to the curve 'ABC' in figure 1.7.

The domains in the specimen are separated by interfaces called domain wall. The walls between domains with opposite orientation and mutually perpendicular polarizations are called the 180° and 90° walls, respectively [18]. A single domain can be obtained by domain wall motion made possible by the application of an appropriate externally applied electric field. The specific mechanism of domain wall motion is simply the small shift of the ions' positions within the unit cell resulting in the net change of orientation of the tetragonal C-axis. Such domain wall motion results in spontaneous polarization (shown in figure 1.8).

Depending on the structure, the ferroelectric materials are classified into four major categories.

1.4.1.1. Perovskites

Perovskite is the family name of a group of materials with an archetypal ABO₃ type structure. Here the A-site is occupied by divalent cations such as Ba²⁺, Sr²⁺, Ca²⁺, and Pb²⁺, which are usually larger than the B-site cations that are tetravalent, such as Ti⁴⁺, Zr⁴⁺, and Sn⁴⁺, and O represents the oxygen atoms. The A-site cations are surrounded by twelve anions in a cube-octahedral coordination and the B-site cations are surrounded by six anions in an octahedral coordination. Many piezoelectric (including ferroelectric) ceramics such as barium titanate (BaTiO₃), lead titanate (PbTiO₃), lead zirconate titanate (PZT), lead lanthanum zirconate titanate (PLZT), lead magnesium niobate (PMN), and potassium niobate (KNbO₃) have a perovskite type structure [19].

A typical ABO₃ unit cell structure of BaTiO₃ is illustrated in figure 1.9 and can be viewed as a closely packed cubic cell with large cations (A) at the cube corners, small highly charged cations at the body centre, and oxygen ions at the face centres. The structure is a network of corner-linked (BO₆) octahedral holes and the large cations filling the dodecahedral holes [20]. The A-site cations are in 12-fold coordination with oxygen ions and the B-site cations are in six-fold coordination. The terms A- and B-sites are used frequently in describing perovskites and perovskite-like structures. Goldschmidt [21] determined the size of the cations that can be accommodated by the perovskite structure. The A-site can contain ions with ionic radii [22] between 1.34 Å (Ca²⁺) and 1.61 Å (Ba²⁺) while the B-site can accommodate ions from 0.535 Å (Al³⁺) to 0.87 Å (Ce⁴⁺). Figure 1.9 shows the cubic and tetragonal perovskite structures with dipole moments. Generally, perovskite structure materials are ionic in nature.

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1.4.1.2. Tungsten bronze

Tungsten bronze structure can be represented by nonstoichiometric compounds with the general formula M_x WO₃ (0 < x < 1) where M is an alkali metal. It is a highly coloured lustrous crystalline compound. The 'tetragonal tungsten bronze' (TTB) structure consists of interconnected corner-sharing oxygen octahedrals with three types of pseudo-symmetric open channels (three-, four-, and five-fold), which is a typical characteristic of the TTB crystal structure. It is related to the potassium tungstate (K_0.475WO₃)-like structure. The general formula for the TTB ferroelectric crystal structure is given by (A1)₂(A2)₄(C)₄(B1)₂(B2)₈O₃₀ [24]. The TTB structure is shown in figure 1.10.

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In TTB there are three open channels (the A1-, A2-, and C-sites) and the space occupied by the oxygen octahedron (B1 and B2) can accommodate a broad range of cations and anions. Hence, different properties can be tailored in TTB.

The first crystal of the tungsten bronze type structure to show useful ferroelectric properties was lead niobate (PbNb₂O₆) [3]. Various dopings in the TTB structure have led to the discovery of many new normal and relaxor-type ferroelectrics [25].

1.4.1.3. Pyrochlore

The general stoichiometric formula of pyrochlore structure ferroelectric material is A₂B₂O₇ [26], where A represents a trivalent or divalent cation and B represents a tetra- or pentavalent cation. Figure 1.11 shows the one octant part of the pyrochlore structure. Due to a broad compositional range of A- and B-site cations, pyrochlore compounds show many properties which are highly useful for anodes and cathodes for fuel cells and sensors, solid electrolyte catalysts and dielectrics [27]. Examples are Ce₂Ti₂O₇, Sr₂Nb₂O₇, La₂Ti₂O₇ (LTO), and Nd₂Ti₂O₇ (NTO).

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1.4.1.4. Layer structured ferroelectrics

Based on their physical and electrical properties, ferroelectric materials are divided broadly into two types: (i) normal ferroelectrics and (ii) relaxor ferroelectrics.

1.4.2.1. Normal ferroelectrics

1.4.2.2. Relaxor ferroelectrics

Relaxor ferroelectrics are characterized by the following characteristics:

• (i)  
  A broad or diffuse phase transition.
• (ii)  
  Significant dependence of the dielectric maximum temperature (T_m ) on frequency (figure 1.12).
• (iii)  
  The 1/ε' versus temperature curve follows the modified Curie–Weiss law (equation (1.7)).
• (iv)  
  Spontaneous polarization of the relaxor ferroelectric becomes zero at a temperature T_d (depolarization temperature) that is greater than T_m .
• (v)  
  Slim P–E loops with no sign of saturation near T_m .
• (vi)  
  Weak P_r (remanent polarization).
• (vii)  
  Exhibition of a considerable second-order piezoelectric effect by most of these materials.

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