## 1. Introduction

Certain solid solutions of perovskite-type ferroelectrics show excellent properties such as giant dielectric response and high electromechanical coupling constant in the vicinity of the morphotropic phase boundary (MPB). These materials are of importance to applications such as electrostrictive actuators and sensors, because of the large dielectric and piezoelectric constants (Jaffe et al., 1971; Sawaguchi, 1953; Kuwata et al., 1982; Newnham, 1997). The term “morphotropic” was originally used to refer to refer to phase transitions due to changes in composition (Ahart et al., 2008). Nowadays, the term ‘morphotropic phase boundaries’ (MPB) is used to refer to the phase transition between the tetragonal and the rhombohedral ferroelectric phases as a result of varying the composition or as a result of mechanical pressure (Jaffe et al., 1954; Yamashita, 1994; Yamamoto & Ohashi, 1994; Cao & Cross, 1993; Amin et al., 1986; Ahart et al., 2008). In the vicinity of the MPB, the crystal structure changes abruptly and the dielectric properties in ferroelectric (FE) materials and the electromechanical properties in piezoelectric materials become maximum.

The common ferroelectric materials used for MPB applications is usually complex-structured solid solutions such as lead zirconate titanate - PbZr_{1−x
}Ti_{
x
}O_{3} (PZT) and Lead Magnesium niobate-lead titanate (1-x)PbMg_{1/3}Nb_{2/3}O_{3}-xPbTiO_{3}), shortly known as PMN-PT. For example, PZT is a perovskite ferroelectrics which has a MPB between the tetragonal and rhombohedral FE phases in the temperature-composition phase diagram. However, these materials are complex-structured and require a complicated and costly process to prepare its solid solutions. Furthermore, the study of the microscopic origin of its properties is very complicated.

Recently, scientists started to pay attention to the MPB in simple-structured pure compound ferroelectric materials such as ferroelectric oxides. For example, a recent experimental study on lead titanate proved that PbTiO_{3} can display a large MPB under pressure (Ahart et al., 2008). These experimental results even showed richer phase diagrams than those predicted by first-principle calculations. Therefore, it is of particular importance to study the fundamental theory of dielectric as well as piezoelectric properties of such materials in the vicinity of the MPB. Such knowledge helps engineering specific simple-structured nonlinear (NL) materials with highly nonlinear dielectric and piezoelectric properties.

Apart from first principle calculations, an alternative way to investigate the dielectric or the piezoelectric properties of these materials is to use the free energy formalism. In this chapter, we investigate the behavior of both the dynamic and the static dielectric susceptibilities in ferroelectrics in the vicinity of the MPB based on the free energy formalism. The origin of the large values of the linear and the nonlinear dielectric susceptibility tensor components is investigated using semi-analytic arguments derived from both Landau-Devonshire (LD) free energy and the Landau-Khalatnikov (LK) dynamical equation. We show that, not only the static linear dielectric constant is enhanced in the vicinity of the MPB but also the second and the third-order static nonlinear susceptibilities as well. Furthermore, the behavior of the dynamic nonlinear dielectric susceptibility as a function of the free energy parameters is also investigated for various operating frequencies. This formalism enables us to understand the enhancement of the dielectric susceptibility tensors within the concept of ferroelectric soft-modes. The input parameters used to generate the results is taken from an available experimental data of barium titanate BaTiO_{3} (A common simple-structured ferroelectric oxide). The effect of operating frequency, and temperature, on the dynamic dielectric susceptibility is also investigated. The enhancement of various elements of particular nonlinear optical NLO process such as second-harmonic generation (SHG) and third-harmonic generation (THG) is investigated. The enhancement of these linear and nonlinear optical processes is compared with typical values for dielectrics and ferroelectrics.

The importance of this calculation lies in the idea that the free energy material parameters
_{3} or PbTiO_{3}. Such pure compounds with simple structure can be used for technological applications rather than material with complicated structure.

Ishibashi & Iwata (1998) were the first to propose a physical explanation of the MPB on the basis of a Landau–Devonshire-type of free energy with terms up to the fourth order in the polarization by adopting a “golden rule” and obtaining the Hessian matrix. They expressed the static dielectric susceptibility
*F*. They explained the large dielectric and piezoelectric constants in the MPB region as a result of transverse instability of the order parameter (Ishibashi & Iwata, 1999^{a}
^{,b,c}; Ishibashi, 2001; Iwata et al., 2002^{a}
^{,b}). Such transverse instability is perpendicular to the radial direction in the order-parameter space near the MPB (Iwata et al., 2005). However, the work by Ishibashi et al. was limited to the study of the MPB for the static linear dielectric constant only and never extended to include the nonlinear dielectric susceptibility. Perhaps, this is because the expressions of the nonlinear dielectric susceptibility tensor components in terms of the free energy parameters were not yet formulated.

In earlier work by Osman et al. (1998^{a}
^{,b}), the authors started to derive expressions for the nonlinear optical (NLO) susceptibilities of ferroelectric (FE) in the far infrared (FIR) spectral region based on the free energy formulation and Landau-Khalatnikov equation. The core part of this formulation is that the NLO susceptibilities are evaluated as a product of linear response functions. However, the work by Osman et al. was obtained under the approximation of a scalar polarization which only allows them to obtain specific nonlinear susceptibility elements. Soon after that, Murgan et al. (2002), presented a more general formalism for calculating all the second and third-order nonlinear susceptibility coefficients based on the Landau-Devonshire (LD) free energy expansion and the Landau-Khalatnikov (LK) dynamical equation. In their work they provided detailed results for all the nonvanishing tensor elements of the second and third –order nonlinear optical coefficients in the paraelectric, tetragonal and rhombohedral phase under single frequency approximation and second-order phase transitions.

Our aim here is then to utilize the expressions for the NLO susceptibility tensor components derived by Murgan et al (2002) to extend the study of the MPB to the second and third-order nonlinear susceptibility. Further, both the dynamic and static case is considered and an explanation based on the FE soft modes is provided. Because the expressions for the dielectric susceptibility given by Murgan et al. (2002) do not immediately relate to the MPB, we will first transform them into an alternative form that shows the explicit dependence on the transverse optical (TO) phonon mode and the longitudinal optical (LO) modes. The enhancement of the dynamic nonlinear susceptibility tensors is then investigated within the concept of the ferroelectric soft-mode with normal frequency

## 2. Background on morphotropic phase boundary (MPB)

Most studies on MPB is performed on a complex structured ferroelectric or piezoelectric materials such as PZT or PZN-PT and only recently studies on simple structure pure ferroelectric materials such as BaTiO3 or PbTiO3 took place. In this section we will shortly review both theoretical and experimental results on the most common MPB materials and its main findings. Early experimental work on MPB focused mainly on the behavior of piezoelectric constant. This is because most of the measurements were based on diffraction which measure distortion of a unit cell. For example, Shirane & Suzuki (1952) and Sawgushi (1953) found that PZT solid solutions have a very large piezoelectric response near the MPB region. Results of this kind are reviewed by Jaffe et al (1971) who first introduced the phrase “morphotropic phase diagram”. A typical temperature-composition phase diagram for PZT is shown in Fig.1. The graph is after Noheda et al. (2000^{a}
). As shown in Fig. 1, the MPB is the boundary between the tetragonal and the rhombohedral phases and it occurs at the molar fraction compositions close to *x* = 0.47. In addition, the MPB boundary is nearly vertical in temperature scale. Above the transition temperature, PZT is cubic with the perovskite structure. At lower temperature the material becomes ferroelectric, with the symmetry of the ferroelectric phase being tetragonal (F_{T} ) for Ti-rich compositions and rhombohedral (F_{R}) for Zr-rich compositions. Experimentally, the maximum values of the dielectric permittivity, piezoelectric coefficients and the electromechanical coupling factors of PZT at room temperature occur at this MPB (Jaffe et al., 1971). However, the maximum value of the remanent polarization is shifted to smaller Ti contents.

For ferroelectrics with rhombohedral and tetragonal symmetries on the two sides of the MPB, the polar axes are (1,1,1) and (0,0,1) (Noheda et al., 1999). The space groups of the tetragonal and rhombohedral phases (P4mm and R3m, respectively) are not symmetry-related, so a first order phase transition is expected at the MPB. However, this has never been observed and, only composition dependence studies are available in the literature. Because of the steepness of the phase boundary, any small compositional inhomogeneity leads to a region of phase coexistence (Kakegawa et al., 1995; Mishra & Pandey, 1996; Zhang et al., 1997; Wilkinson et al., 1998) that conceals the tetragonal-to-rhombohedral phase transition. The width of the coexistence region has been also connected to the particle size (Cao & Cross, 1993) and depends on the processing conditions, so a meaningful comparison of available data in this region is often not possible.

Various studies (Noheda et al., 1999; Noheda et al., 2000
^{a}; Noheda et al., 2000
^{b}; Guo et al., 2000; Cox et al., 2001) have revealed further features of the MPB. High resolution x-ray powder diffraction measurements on homogeneous sample of PZT of excellent quality have shown that in a narrow composition range there is a monoclinic phase exists between the well known tetragonal and rhombohedral phases. They pointed out that the monoclinic structure can be pictured as providing a “bridge” between the tetragonal and rhombohedral structures. The discovery of this monoclinic phase led Vanderbilt & Cohen (2001) to carry out a topological study of the possible extrema in the Landau-type expansions continued up to the twelfth power of the polarization. They conclude that to account for a monoclinic phase it is necessary to carry out the expansion to at least eight orders. It should be noted that the free energy used to produce our results for the MPB means that our results apply only to the tetragonal and rhombohedral phases, however, since these occupy most of the

As mentioned above, the common understanding of continuous-phase transitions through the MPB region from tetragonal to rhombohedral, are mediated by intermediate phases of monoclinic symmetry, and that the high electromechanical response in this region is related to this phase transition. High resolution x-ray powder diffraction measurements on poled PbZr_{1-x
}Ti_{
x
}O_{3} (PZT) ceramic samples close to the MPB have shown that for both rhombohedral and tetragonal compositions the piezoelectric elongation of the unit cell does not occur along the polar directions but along those directions associated with the monoclinic distortion (Guo et al., 2000). A complete thermodynamic phenomenological theory was developed by Haun et al., (1989) to model the phase transitions and single-domain properties of the PZT system. The thermal, elastic, dielectric and piezoelectric parameters of ferroelectric single crystal states were calculated. A free energy analysis was used by Cao & Cross (1993) to model the width of the MPB region. The first principles calculations on PZT have succeeded in reproducing many of the physical properties of PZT (Saghi-Szabo et al., 1999; Bellaiche & Vanderbilt, 1999). However, these calculations have not yet accounted for the remarkable increment of the piezoelectric response observed when the material approaches its MPB. A complicating feature of the MPB is that its width is not well defined because of compositional homogeneity and sample processing conditions (Kakegawa et al., 1995).

Another system that has been extensively studied is the
^{b}; Cross, 1987; Cross, 1994). The giant dielectric response in relaxors and related materials is the most important properties for applications. This is because the large dielectric response means a large dielectric constant and high electromechanical coupling constant.

Iwata et al. (2002
^{b}; 2005) have theoretically discussed the phase diagram, dielectric constants, elastic constants, piezoelectricity and polarization reversal in the vicinity of the MPB in perovskite-type ferroelectrics and rare-earth–Fe_{2} compounds based on a Landau-type free energy function. They clarified that the instability of the order parameter perpendicular to the radial direction in the order-parameter space near the MPB. Such instability is induced by the isotropy or small anisotropy of the free-energy function. In addition, the transverse instability is a common phenomenon, appearing not only in the perovskite-type ferroelectric oxides, but also in magnetostrictive alloys consisting of rare-earth–Fe_{2} compound (Ishibashi & Iwata, 1999
^{c}), in the low-temperature phase of hexagonal BaTiO_{3} (Ishibashi, 2001) and in shape memory alloys (Ishibashi &Iwata, 2003; Iwata & Ishibashi, 2003). They also noted that the origins of the enhancement of the responses near the MPB both in the perovskite-type ferroelectrics and the rare-earth–Fe_{2} compounds are the same. Even more, Iwata & Ishibashi (2005) have also pointed out that the appearance of the monoclinic phase and the giant piezoelectric response can be explained as a consequence of the transverse instability as well.

A first principles study was done by Fu & Cohen (2000) on the ferroelectric perovskite, BaTiO3, which is similar to single-crystal PZN-PT but is a simpler system to analyze. They suggested that a large piezoelectric response could be driven by polarization rotation induced by an external electric field rotation (Fu & Cohen, 2000; Cohen, 2006). Recently, these theoretical predictions of MPB on a single BaTiO_{3} crystal have been experimentally confirmed by Ahart et al. (2008) on a pure single crystal of PbTiO_{3} under pressure. These results on BaTiO_{3} and PbTiO_{3} open the door for the use of pure single crystals with simple structure instead of complex materials like PZT or PMN-PT (PbMg1/3Nb2/3O3-PbTiO3) that complicates their manufacturing as well as introducing complexity in the study of the microscopic origins of their properties (Ahart et al., 2008). Moreover, Ahart et al. (2008) results on the MPB of PbTiO_{3} shows a richer phase diagram than those predicted by first principle calculations. It displays electromechanical coupling at the transition that is larger than any known and proves that the complex microstructures or compositions are not necessary to obtain strong piezoelectricity. This opens the door to the possible discovery of high-performance, pure compound electromechanical materials, which could greatly decrease costs and expand the utility of piezoelectric materials. For the above mentioned reasons, we are motivated here to study the NL behavior of a pure single FE with simple crystal structure such as PbTiO_{3} or BaTiO_{3} at the MPB on the basis of the free-energy model.

## 3. The concept of morphotropic phase boundary (MPB) in the free energy

The first published paper on modeling the MPB using the Landau–Devonshire-type of free energy was made by Ishibashi and Iwata (1998). The authors basically used the free energy *F* as a function of the dielectric polarization in the following form;

The former expression for the free energy may simply be written in the form
*α* is a temperature dependent coefficient with
*a* is the inverse of the Curie constant, T is the thermodynamic temperature, and *T*
_{
c
} is the Curie temperature. The authors found that the static linear dielectric constant for both tetragonal and rhombohedral phases diverges at the MPB when
*P*. In the region of the

In Eq. (1), if
*xyz* frame of reference. In the tetragonal phase the free energy surface is elongated in the direction of the spontaneous polarization to assume the shape of an ellipse (Murgan et al., 2002^{a}
). For example, if the spontaneous polarization is taken along the z-direction, therefore, the ellipsoid is elongated along this axis as seen in Fig.3 which illustrates the uniaxial nature of the tetragonal symmetry. The intersection of the isotropic surface and the tetragonal surface occurs only at the

along the
^{a}
). In the previous theoretical

calculations using the free energy, Haun et al., (1989) had directly related the MPB to the composition in PbZrO_{3}:PbTiO_{3} solid solution family where the relation between the material parameters
^{a}
, 1999^{b}
) proposed that the material parameter
*x* passing through 1 at

## 4. Dielectric susceptibility from Landau-Devonshire free energy

Ishibashi & Orihara (1994) was the first to consider the Landau-Devonshire theory to give expressions for the nonlinear dynamic dielectric response by using the Landau-Khalatnikov (LK) equation. They evaluated the NLO coefficients and the third-order nonlinear (NL) susceptibility coefficients in the paraelectric (PE) phase above the Curie temperature *T*
_{
c
}. Subsequently, Osman et al. (1998^{a}
^{,b}) have extended the theory to evaluate the NLO coefficients in the FE phase. They have demonstrated that all second order
*P* Soon after that, Murgan et al. (2002), used a more general form of the free energy to*.* calculate the dielectric susceptibility elements. In their expression, they considered the free energy expansion to be a function of a vector polarization Q and additional terms were added to Eq. (1). They considered a free energy of the following form;

In the above expression,
*M* and

Expressions for the second-order nonlinear susceptibility tensor elements are shown in table 1 while expressions for third-order nonlinear susceptibility tensor components are shown in Table 2. In particular, table 1 shows the nonvanishing tensor elements for second-harmonic generation (SHG) and optical rectification (OR) while Table 2 shows the nonvanishing tensor elements for the third-harmonic generation (THG) and the intensity-dependent (IP) refractive index process. The expressions in both table 1 and Table 2 are all written in terms of the above linear response functions

In the above, the frequency-dependent term is
*n* is an integer number.

The linear dielectric susceptibility elements

where the linear dielectric susceptibility tensor

## 5. The input parameters

To plot the dielectric susceptibility, various input parameter is required. Input parameters such as
_{3} (Ibrahim et al 2007, 2008, 2010). For convenience, we may write the operating frequency

To estimate the value of

For ferroelectric material, the dielectric constant is approximated by
^{-2} at room temperature. Hence, a value of
_{3} may also be done by comparing the dielectric function in Eq. (4) and the equation
^{a}). This yields the relation;

which express
*M*,
_{3}. In fact, each oscillating mode in the crystal may assume different damping ratio in a real crystal and the stability of each mode depends on its damping ratio. The average damping parameter of all the relevant modes is usually obtained. However, for convenience, we have to fix the damping parameter at specific value within the range
^{-2},

## 6. Morphotropic phase boundary (MPB) in linear dielectric susceptibility

The divergence of the static dielectric susceptibility near the MPB for tetragonal and rhombohedral symmetry was first investigated by Ishibashi and Iwata (see for example Ishibashi & Iwata 1998). They have derived the static dielectric constant

In this section, we investigate the dynamic linear dielectric susceptibility

We note that in deriving equations (27) and (28), we have used the spontaneous polarization for tetragonal phase defined by

Since the stability region of the tetragonal phase lies at

Now the origin of the enhancement of the dielectric susceptibility is clear, when

In Eq. (29) and (30), the static linear dielectric constant shows that at the MPB,

To examine the effect of operating frequency, we plot the average value of the dynamic linear dielectric susceptibility versus
*i*) shows the linear susceptibility
*ii*) shows the linear susceptibility
*iii*)), the linear susceptibility
*iv*)),
*v*)),
*vi*)), the static limit of the linear susceptibility

## 7. Morphotropic phase boundary in second-order nonlinear susceptibility

In the free energy formalism, there is only one underlying dynamic equation and the NLO coefficients take the form of products of linear response functions. This formalism does not explicitly show the dependence of the NL susceptibility on the MPB or the ferroelectric soft mode. As shown in Table 1 and table 2, the susceptibility elements takes the form of a product of linear response functions,
*z*, and
*x* or *y* and the argument is the related frequency. In this case, it is convenient to transform the second-order NL susceptibility tensor elements to an alternative form that shows a direct dependence on the lattice-vibrational modes.

These are the transverse-optical modes (TO) with normal frequency
*T* approaches *T*
_{
c
} or

This is achieved by substituting the linear response functions

The element
*i*), the value of
_{3} (Kittel 1995; Sirenko et al., 2000; Katayama et al., 2007).

The other curves in Fig. 6(a) show that both the dynamic value of
*ii*), the maximum value of
*iii*), the maximum value of
*iv*), the maximum value of
*v*), the maximum value of
_{3} (Murgan et al., 2004). However, the maximum value of
_{3} by about two to three order of magnitude. Needless to say that the SHG value in ferroelectrics is initially very large (typically

## 8. Morphotropic phase boundary in third-order nonlinear susceptibility

The expressions for the third-order nonlinear susceptibility elements also consist of products of linear response functions. Table 2 shows the dynamic nonlinear susceptibility elements for the third-harmonic generation (THG) is also expressed in terms of the linear response functions
*n* may assume one, two or three. The elements in Table 2 may be written in terms of the TO and LO phonon frequencies following the same procedure describes in the previous chapter. For example, the third-harmonic element

The linear response functions for the second-harmonics

The former equation clearly indicates that the value of

A plot of
*f*. As in the second-order case, this is due to the softening of the TO mode which results in a direct increment of

For example, at
_{3} is found to lie within the range of
_{3} are found by Murgan et al., (2002). These values for THG in ferroelectrics are very large compared with typical semiconductors or dielectrics. For example, the
_{3} at its peak is increased by two or three orders of magnitude in comparison with the values far from its peak.

## 9. Conclusion

In this chapter we have examined the behavior of both linear and nonlinear dielectric susceptibility as a function the free energy parameters for different operating frequencies. Both dynamic and static dielectric susceptibility is examined. Within the free-energy formulation, the material-dependent nonlinear coefficients