Lallart M. (ed.) Ferroelectrics

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Morphotropic Phase Boundary in Ferroelectric Materials

(

)

(2) (2) (2)

3342

SHG SHG SHG

zxx xzx s TO LO

ω= =χ =χ =β ε ω ω (33)

The element

(2)SHG

zyy

χ in Eq. (32) contains one longitudinal optical mode

ω=αε

corresponds to the output wave of frequency 2

and one TO mode with frequency

()

1201

Mω=αβ−β ε β corresponds to the input frequency

. Therefore, this element is

expected to assume one peak at certain value of

. This expectation is assured upon

plotting

(2)SHG

zyy

χ versus

for different operating frequencies as shown in Fig. 6(a). The main

feature of those curves is that they all display a single pole at certain value of

β . In the

static case (curve i), the value of

(

)

(2)

SHG

zyy

χω→ increases systematically then diverges at the

MPB between the tetragonal and rhombohedral phase where

= . As seen from Eq. (33),

the static value of

(

)

(2)

SHG

zyy

χω→only depends on

and

. Since

is not a function of

β , thus the enhancement in the values of

(

)

(2)

SHG

zyy

χω→ is mainly due to

. The value of

()

1201

Mω=αβ−β ε βdecreases “softens” systematically when

approaches

β which

lead to a direct increment in the value of

(

)

(2)

SHG

zyy

χω→till the divergence occur

β→βas 0

ω→ at the MPB. It should be noted that, the association of a high static

dielectric constant with a specific low frequency optical mode is experimentally observed on

FE materials such as strontium titanate, SrTiO

(Kittel 1995; Sirenko et al., 2000; Katayama et

al., 2007).

Fig. 6. (a) The nonlinear dielectric susceptibility

()

(

)

2;,

SHG

zyy

−ωωω

versus

=β β in

tetragonal phase at room temperature for different operating frequencies

ωω shows

the improvement of the MPB upon decreasing the operating frequency.

The other curves in Fig. 6(a) show that both the dynamic value of

SHG

zyy

χ around the MPB and

the corresponding value of

are functions of the operating frequency. Particularly, at

0.3f = (curve ii), the maximum value of

SHG

zyy

χ is found to be around

3-1

4.8 10 mv

−

× at

Ferroelectrics – Physical Effects

1.2β  . At 0.5f = (curve iii), the maximum value of

SHG

zyy

χ was found to be around

3-1

3.5 10 mv

−

× at

1.5β  . At 0.6f

(curve iv), the maximum value of

SHG

zyy

χ was found to be

around

3-1

1.5 10 mv

−

× at

1.7β  . At 0.7f

(curve v), the maximum value of

SHG

zyy

χ was

found to be around

3-1

0.4 10 mv

−

× at

2β  . Therefore, a systematic increase of the operating

frequency

ω= ω results in a systematic decrease of the maximum value of

SHG

zyy

χ and a

systematic increase of the value

too. Thus the dynamic

SHG

zyy

χ assumes higher values at

lower frequencies. The peaks exhibited by

SHG

zyy

χ can be viewed clearly upon plotting both

the real and imaginary parts of the complex

SHG

at single operating frequency

0.1f =ωω = as shown in Fig. 6(b). The graph shows that the single pole exhibited by

SHG

zyy

χ nearly at

1β≈ results from the resonance-like response exhibited by both real and

imaginary parts of the complex

SHG

. The values obtained in Fig. 6(a) are typical for SHG in

BaTiO

(Murgan et al., 2004). However, the maximum value of

SHG

obtained in Fig. 6(b) is

dramatically increased (

2-1

2.5 10 mV

−

× ) at such low frequency. This value is higher than the

typical values of BaTiO

by about two to three order of magnitude. Needless to say that the

SHG value in ferroelectrics is initially very large (typically

(2)

11 -1

10 mV

−

χ  ) compared with

the order of magnitude for typical dielectric or semiconductors in the nonresonant region.

This is not surprising for resonant effects in highly polarizable materials such as

ferroelectrics. The TO mode softening may be explained as a result of the well-known

phenomenon of LO-TO splitting, that is, the shift in frequency between longitudinal optical

and transverse optical phonons at the Brillouin zone centre (Waser et al., 2006). In this case,

the softening of the TO modes is caused by a partial compensation of the long-range electric

fields on one hand and the short-range lattice elastic forces on the other hand.

Fig. 6. (b) The real and imaginary part of the nonlinear dynamic dielectric susceptibility

()

(

)

2;,

SHG

zyy

χ−ωωω versus

=β β in tetragonal phase at room temperature and at operating

frequency

f =ω ω =0.1.

Morphotropic Phase Boundary in Ferroelectric Materials

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

(

)

nσω

and

(

)

. The integer number 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

THG

in Eq. (17) may be written as;

()

() ()

() () () () ()

34 2 2

222

22 22 2

2 3

THG

zyyz

LO LO

TO TO LO TO TO

MM MM M

−β

χ=

⎡Θ ω ⎤⎡Θ ω ⎤

ε−ω−ω

⎢⎥⎢⎥

⎣⎦⎣⎦

⎧ ⎫

⎪ ⎪

βω ω

⎪ ⎪

×++

⎨ ⎬

⎡Θ ω ⎤⎡Θ ω ⎤ ⎡Θ ω ⎤⎡Θ ω ⎤ ⎡Θ ω ⎤

⎪ ⎪

β+ω+ω −ω+ω +ω

⎢ ⎥⎢⎥⎢ ⎥⎢⎥⎢⎥

⎪ ⎪

⎣ ⎦⎣⎦⎣ ⎦⎣⎦⎣⎦

⎩ ⎭

(34)

The linear response functions for the second-harmonics

(

)

(

)

exists in Eq. (17) is

responsible for the appearance of the function argument

(

)

in Eq. (34) where

(

)

24 2MiΘω=− ω−Γω

. The function argument

(

)

is intrinsic part of the

(3)

χ response

and they do not result from cascade processes in which two second-order process follow

one another in time (Murgan et al., 2002). In nonlinear optics, a cascaded THG could arise

from SHG followed by sum-frequency generation, or symbolically 2

+ω→ ω, then

23ω+ ω→ ω. The element

THG

in Eq. (34) (apart from the prefactors and the LO modes)

include two terms related to the TO modes. The first term include a TO mode propagating

at the input frequency

while the second term include a TO mode propagating at 2ω .

Therefore, the element

THG

is expected to assume a double peak behavior which may lead

to a complicated enhancement effects at two different values of

. In the static limit, Eq. (34)

reduces to the following form;

()

3444 2

THG

zyyz

TO LO TO

⎡

⎤

−β βω

=−

⎢

⎥

εωωβω

⎣

⎦

(35)

The former equation clearly indicates that the value of

()

THG

zyyz

→∞ when the value of

ω→ at the MPB.

A plot of

THG

χ versus

is shown in Fig. 7(a). In the dynamic case, both the real (The -●- curve)

and the imaginary parts (The -

o- curve ) of

THG

χ assume a double peak behavior at two

different values of

as predicted by Eq. (34). In Fig. 7(b), the value

THG

χ is plotted versus

β for different operating frequency. The main feature of these curves is that they all display a

double peak response at certain value of

as predicted by Eq. (34). The value of

THG

χ is

generally increased upon decreasing the operating frequency f. As in the second-order case,

this is due to the softening of the TO mode which results in a direct increment of

THG

χ .

Ferroelectrics – Physical Effects

Fig. 7. (a) The third-order nonlinear susceptibility tensor element

THG

versus

=β β in

tetragonal phase at room temperature for operating frequencies f = 1.5 and for the static

case at f = 0.

Fig. 7. (b) The absolute value nonlinear dielectric susceptibility

THG

χ versus

=β β in

tetragonal phase at room temperature for different operating frequencies

ωω .

Morphotropic Phase Boundary in Ferroelectric Materials

For example, at 1.3f

(thin solid curve), the maximum value of

THG

χ is found to be

around

12 2 -2

1.15 10 m v

−

× at

4.5β  . At 1.1f

(the dashed curve) the maximum value of

THG

χ is

12 2 -2

4.8 10 m v

−

× at

3.5β  . At 1.01f = (solid curve), the maximum value of

THG

increases to

12 2 -2

9.1 10 m v

−

× at

3β  . At 0.9f

(the -●- curve), the maximum value of

THG

χ further increases to

11 2 -2

1.61 10 m v

−

× at

2.7β  . In the static case with 0f = (the –o-

curve), the value of

SHG

zyy

χ diverges at

= as predicted by Eq. (35). From the numerically

generated data, the typical value of the dynamic

(3)

in BaTiO

is found to lie within the

range of

−

depending on the operating frequency. Similar values of

THG

χ for

BaTiO

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

THG

for GaAs is

about

19 2 -2

6.7 10 m v

−

× while it is in the range of

22 20

10 10

−−

− for Glass (Eaton, 1991). From our

numerically generated data, the value of

(3)

in BaTiO

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

and

may be assumed as

function of the molar composition. Using both the free energy and Landau-Khalatnikov

equation, the nonlinear dielectric susceptibility is written as a product of the linear

response functions. This form of dielectric susceptibility is transformed into an alternative

form that shows the explicit dependence on the transverse-optical (TO) and longitudinal-

optical (LO) modes. The dielectric susceptibility is then investigated within the concept of

the ferroelectric soft-mode with normal frequency

where the material parameters

and

β enters on the same footing as temperature. The divergence of the static dielectric

susceptibility at the MPB occurs when the ferroelectric soft-mode becomes zero as a result

β→β or

TT→ . Most dielectric susceptibility elements are systematically enhanced

upon decreasing the operating frequencies. In the vicinity of the MPB, the low-frequency

limit linear dielectric susceptibility is increased by one to two orders of magnitude in

comparison with the non-MPB value of the static dielectric susceptibility. For second-

order NL elements, the dynamic NL susceptibility may assume single or double pole

response. This actually depends on whether the dielectric susceptibility include a soft-

mode corresponding to the input frequency

or two soft-modes corresponds to both

ω and 2ω . We found that the second-order NL process is enhanced by two orders of

magnitude in the vicinity of the MPB in comparison with the non-MPB values. We have

also demonstrated the systematic enhancement of the dynamic nonlinear susceptibility

upon increasing the thermodynamic temperature of the material towards the Curie

temperature. Within the third-order process, we have investigated both the THG and the

intensity-dependant (IP) susceptibility. We have found that certain elements within the

THG process assumes a triple-response at three different values of

. This phenomena

occurs if the NL susceptibility element includes three different soft-modes frequencies

Ferroelectrics – Physical Effects

corresponds to

, 2

and 3

. Other elements within the third-order NL process may

assume single or double pole in their dynamic response. As in the second-order process,

the third order process is enhanced at the MPB by two to three orders of magnitude. All

nonlinear elements are found to diverge in the static limit upon approaching

1β= . It

should be noted that we have investigated the MPB within the tetragonal phase

where

0β>β> , it will be very interesting to study the MPB for the rhombohedral phase

within the free energy formalism. Finally we note that this calculation is important for

designing a new material with higher nonlinearity for technological applications. This is

achieved by investigating the MPB where the dielectric response is maximized.

10. Acknowledgement

The authors wish to express their sincere gratitude to Prof Y. Ishibashi and Prof D. R. Tilley

for their fruitful discussion and support especially during their collaboration period with

Universiti Sains Malaysia as visiting scientists.

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Morphotropic Phase Boundary in Ferroelectric Materials

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Ferroelectrics – Physical Effects

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Relaxor-ferroelectric PMN–PT Thick Films

Hana Uršič and Marija Kosec

Electronic Ceramics Department, Jožef Stefan Institute

Slovenia

1. Introduction

Electroceramic thick films are 2D (planar) structures that, in their simplest form, consist of a

substrate, a bottom electrode, a ceramic film and a top electrode, with the thickness of an

individual layer being typically between 1 and 100 µm. The processing of thick films is

similar to the processing of bulk ceramics, i.e., it involves powder synthesis, shaping and

sintering. The major difference, however, is in the clamping of the film onto the substrate.

The consequences of this are a constrained sintering, a possible reaction with the electrode

or the substrate during firing, and the thermal stresses that are generated during the

cooling. To minimize the chemical interaction of the film and the substrate the sintering

temperature must be kept relatively low in comparison to that for the bulk. This requires a

fine particle size for the powder and additives that form a liquid phase at the sintering

temperature. This liquid phase must also be present to enable the densification under the

constrained conditions, as was predicted and demonstrated in Kosec et al. (1999).

The effect of clamping the active film to the relatively stiff substrate results in smaller

displacements in comparison to substrate-free structures. Due to the mismatch of the

thermal expansion of the thick-film and substrate materials during the processing the

properties of the thick films differ from those of the respective bulk ceramics. Furthermore,

an additional drawback can be the deterioration of the thick film’s properties due to its

chemical interaction with the substrate. To ensure a sufficient displacement required for a

certain application, different solutions have been proposed, such as a reduction in the

thickness of the substrate, the processing of thick-film multilayer structures or the

preparation of “substrate-free” structures.

In the world of technology there is a continuous, global increase of interest in the

miniaturisation of devices, materials and system integration. Thick-film structures are a

good example of the opportunities offered for the miniaturization of electromechanical

systems by the successful implementation of new functional materials and technologies.

Driven by the versatility of conventional thick-film technology, the processing of functional

structures with thick films on different substrates is possible, along with many design

possibilities.

In this chapter the progress in relaxor-ferroelectric Pb(Mg

1/3

2/3

–PbTiO

(PMN–PT)

thick films is reported. The preparation procedures, the processing of PMN–PT thick films

on different substrates, the structural and electrical characterization as well as PMN–PT

thick-film devices are discussed. The phase composition and functional properties of PMN–

PT thick films are discussed in terms of the thermal stresses generated in the films during

Ferroelectrics – Physical Effects

the thermal treatment. Due to these process-induced stresses the electrical and structural

properties of the films can be changed dramatically in comparison to the unstressed films.

Some representative examples of PMN–PT thick-film applications are described, including a

novel approach to the manufacture of large-displacement “substrate-free” PMN–PT/Pt

actuators using thick-film technology.

2. Relaxor-ferroelectric PMN–PT material

Ferroelectric and piezoelectric films are mostly based on lead oxide compounds, mainly

Pb(Zr,Ti)O

(PZT) solid solutions. An alternative to PZT are the relaxor-based systems,

where the term relaxor-ferroelectrics is used for solid solutions between relaxors such as

Pb(Mg

1/3

2/3

(PMN) and ferroelectrics such as PbTiO

(PT) (Cross, 1987; Damjanovic,

2008). One of the most widely studied relaxor-ferroelectrics is Pb(Mg

1/3

2/3

–PbTiO

(PMN–PT). These PMN–PT-based materials are characterized by a high dielectric

permittivity, high piezoelectric properties, a high electrostriction, and are suitable for

applications in multilayer capacitors, actuators, sensors and electro-optical devices (Park &

Shrout, 1997a, 1997b).

The morphotropic phase boundary (MPB) in the (1-x)Pb(Mg

1/3

2/3

–xPbTiO

system is

located close to the x=0.35 composition. The strong piezoelectric properties are related to the

“polarization rotation” between the adjacent rhombohedral phase with the space group

R3m (subsequently referred to as the rhombohedral R3m phase in the text) and tetragonal

phase with the space group P4mm (subsequently referred to as the tetragonal P4mm phase)

through one (or more) intermediate phase(s) of low symmetry, i.e., a monoclinic

(orthorhombic or triclinic) phase (Davis et al., 2006; Noheda et al., 2001). As a consequence,

the observation of a low-symmetry phase, e. g., a monoclinic phase, may suggest strong

electromechanical responses.

It was shown (Singh & Pandey, 2003) that the MPB region of the PMN–PT ceramic material

contains two monoclinic phases with the space groups Cm (subsequently referred to as the

monoclinic Cm phase) and Pm (subsequently referred to as the monoclinic Pm phase) stable

in the composition ranges 0.27 < x < 0.30 and 0.31 < x < 0.34, respectively. For ceramics of the

composition x = 0.35, the coexistence of the monoclinic Pm and the tetragonal P4mm phases

was reported (Alguero et al., 2006, 2007; Carreaud et al., 2005; Singh & Pandey, 2003; Uršič et

al., 2011a), stressing that the phase composition ratio between the monoclinic and the

tetragonal phases depends on different parameters, such as the grain-size effect (Alguero et al.,

2006, 2007; Carreaud et al., 2005) and the poling field (Uršič et al., 2011a) of the ceramics.

PMN–PT ceramics and a single crystal with its composition on the MPB can have the

piezoelectric coefficients d

as high as 700 pC/N (Kelly et al., 1997; Xia et al. 2007) and 1500–

2800 pC/N (Park & Shrout, 1997a; Shrout et al., 1990; Zhang et al., 2001), respectively. The

commonly used poling electric fields for the PMN–PT material vary from 2 to 3.5 kV/mm

(Alguero et al., 2006, 2007; Leite et al. 2002; Kelly et al., 1997; Xia et al. 2007). On the other

hand, the PMN–PT material with compositions x=0–0.1 shows relaxor behaviour (Cross et

al., 1987; Swartz et al., 1982; Swartz et al., 1984) and are known as good electrostrictive

materials (Kighelman et al., 2001; Uchino et al., 1890; Vikhnin et al., 2003; Zhao et al., 1995 ).

Likewise, the electrostrictive effect of the MPB compositions was also reported to be

relatively high (Bokov & Ye, 2002). Furthermore, due to its large responses to an applied

electric field, the PMN–PT material was investigated as a promising material for actuator

applications (Uršič et al., 2011).

Lallart M. (ed.) Ferroelectrics - Physical Effects

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