Zuo-Guang. Ye Advanced Dielectric Piezoelectric and Ferroelectric Materials: Synthesis, Characterisation and Applications

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Handbook of dielectric, piezoelectric and ferroelectric materials360

26 discrete directions. This was probably enough to embolden Pirc and Blinc

(1999) to propose the spherical model, within which polar clusters can have

P pointing in all possible directions. Yet besides these 26 discrete directions,

P for the monoclinic C (M

) can point in any direction in each of the three

<100>-type planes. For the M

and M

monoclinic phases, P can point in

any direction in the six <110>-type planes, and in fact the phase designation

or M

) switches as P rotates in one of these planes past a {111}-type

direction. These nine planes (which include the 26 points for the T, R, and O

domain polarizations) divide the solid angle sphere into 48 regions. Only for

Tri domains can P point in the interior of such a region. Such domains have

not yet been reported, and indeed it would require a careful measurement to

distinguish such a polarization direction from a nearby T, R, O, or M domain

polarization direction.

How can such a multiplicity of observed phases be explained? Vanderbilt

and Cohen (2001) shed much light on this question with a Landau-type free

energy analysis. This analysis involves expansion of the free energy F in

terms of even powers of P

, P

and P

and products of such factors. The

coefficients of the various terms must be consistent with the symmetry of the

parent C phase. To fourth order we have

FAPBPPP BPPPPPP

xyz xyyzzx

= + ( + + ) + ( + + + + )

444

222222

12.1

The coefficients such as A

may be functions of temperature T and hydrostatic

pressure p, but not of variables such as uniaxial pressure or electric field that

break the cubic symmetry. Then, examining Eq. (12.1), we see that if we

truncate the expansion at second order, we could have a T- or p-induced

ferroelectric transition. However, it would have the unphysical property of

going from zero polarization to infinite polarization in arbitrary direction as

changes sign to negative. Obviously, we must go to fourth order to begin

obtaining physically meaningful results.

Going to fourth order by using all terms in Eq. (12.1), we find that (as also

for uniaxial ferroelectrics) there can only be a second-order ferroelectric

transition, which again occurs when A

becomes negative. The requirements

that ensure that P remains finite are that B

> 0 and that B

> –B

. The first

requirement ensures that P remains finite in any {100} direction, while the

second ensures that P remains finite also in any {111} direction.

To find the P values, we first differentiate Eq. (12.1) to obtain

∂∂FP AP BP BPP P

xxx xyz

/ = 2 + 4 + 3 ( + ) = 0

12.2

and two analogous equations for the P

and P

derivatives. The three equations

together determine where maxima, minima, and saddle points occur. The P

= 0 solution of the three Eq. (12.2) expressions gives minimum F only for A

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Electric field-induced domain structures and phase transitions 361

> 0, corresponding to the paraelectric C phase. This solution gives a maximum

F for the ferroelectric phases that occur for A < 0.

Another class of solutions, for the six T domains, exemplified by polarization

along the +P

axis, is P

100

= [(–A

/2B

)

1/2

,0,0], and plugging this result into

Eq. (12.1) gives

FAB

= – /4

12.3

This solution indeed gives a minimum F only if all three second derivatives

obey three relations, typified by

∂∂

FP A BP BP P

xxyz

/ = 2 + 12 + 2 ( + ) = 0

12.4

The second P

derivative, found by plugging in P

100

, is –4A

which is positive.

The second P

and P

derivatives are 2A

–A

which is positive only if

> 2B

. Accordingly, we have a stable or metastable T phase only in the

wedge-shaped volume in A

, B

parameter space for which A

< 0, B

0, B

> 2B

. Note that for B

= 2B

the 2nd and 3rd terms in F in Eq. (12.1)

just add to B

so that all polarization directions have the same F in this

case.

Still another class of solutions, for the eight R domains, exemplified by

polarization in the [1,1,1] direction, is P

111

= [–A

/2(B

+ B

)]

1/2

[1,1,1], for

which (noting that [1,1,1] has magnitude

)

= –

/4(B

+ B

) 12.5

The second derivative of F

with respect to P

(or P

or P

), evaluated where

the first derivative vanishes, is –4A

/(1 + B

) which is positive if B

>–

, the condition already required to keep P finite. Accordingly, we have a

stable or metastable R phase in the wedge-shaped parameter space volume

for which A

< 0, B

> 0, B

> –B

. This volume is greater than that for the

T phase and includes all of the T phase volume.

Now we investigate which phases are stable, and which metastable, over

this parameter space, by comparing the free energies in Eqs. (12.3) and

(12.5). We already found that F

= F

for B

= 2B

because all P directions

give equal F, namely F = F

from Eq. (12.3), for this condition. This B

condition was also shown above to be the limit for a stable T phase, so

the T phase is stable and not metastable over the whole parameter space for

which T-phase minima exist. The R phase, on the other hand, is metastable

over the portion of parameter space for which both T-phase and R-phase

minima exist, and is stable over the portion of the space for which only R-

phase minima exist.

Vanderbilt and Cohen (2001) are correct in saying that the T↔R transition

at B

= 2B

is unphysical because all polarization directions have equal free

energy there, and thus are equally likely. However, this free energy isotropy

can be considered as the first clue (as we go to higher and higher powers of

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Handbook of dielectric, piezoelectric and ferroelectric materials362

polarization in the Landau expansion) that it is easy for other phases, such as

O, M, and possibly Tri, to exist near this (A

< 0, B

> 0, B

= 2B

) quarter-

plane, which could be called the morphotropic phase boundary (MPB) plane

in this parameter space. To summarize, going only to fourth order in the

Landau free energy expansion has already shown us why the T and R phases

are the most prevalent in ferroelectric perovskites, and why O and M phases

are often found near the MPB.

To obtain results more applicable to PMN–PT, we need to go to higher

order. The sixth order will allow first-order FE transitions (as for uniaxial

ferroelectrics), and going successively to higher orders allows more phases

to become metastable or stable, as was shown by Vanderbilt and Cohen

(2001). Specifically, they found that the sixth-order polarization expansion

evidences the O phase along with the T and R phases, whereas going to

eighth order brings in the various M phases. They proved that expansion to

twelfth order is required to bring in the Tri phase. No one has yet had the

courage to tackle the details of the twelth-order expansion.

12.8 Conclusions

PMN–PT crystals are well recognized for their superior piezoelectric properties

for actuator and sensor applications, and their optical properties are attracting

increasing interest. For basic science their multiplicity of phases and their

combination of relaxor and ordinary ferroelectric behavior offer challenges

for experimentalists and theorists to understand better both their static and

dynamic behavior.

For experimentalists, it is vital to use as many techniques as possible,

particularly in view of the complicated behavior. Measurements over an

extremely wide frequency range are needed because of the combination of

glassy behavior, nanoscale clusters, and ordinary domains. The closely balanced

tendencies for nanoclustering vs. domain formation, and for appearance of

one crystal phase vs. another, make it especially important to observe hysteretic

effects of both temperature cycling and electric field history. We have

particularly emphasized in this chapter the usefulness of the polarizing

microscopy technique.

Much attention has been paid to the various types of phases that can

appear, but more work is needed in developing phase diagrams that

systematically take electric field (both magnitude and direction) into account.

Not much is known about the nature of pseudo-spin glass, nanocluster, and

ordinary domain coexistence, and the nature and rate of transition of material

among these three ordering forms. The field-induced cracking phenomena

observed by many workers remain an obstacle for applications, and show the

importance of mechanical stress in governing which phases will appear, but

little is understood about the mechanisms for such behavior. For instance,

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Electric field-induced domain structures and phase transitions 363

when one observes by polarizing microscopy what appear to be monoclinic

regions with varying extinction angles, is this evidence that an unstressed

crystal at zero field would favor a monoclinic phase, or are the monoclinic-

type extinction angles merely evidence that a higher-symmetry phase has

been distorted by mechanical stress or applied electric field?

Close collaboration between theorists and experimentalists, and among

various research groups and crystal growers around the world, is needed to

speed progress on these challenging and technologically vital questions.

12.9 References and further reading

Blinc R, Laguta V V and Zalar B (2003), ‘Field cooled and zero field cooled

207

Pb NMR

and the local structure of relaxor PbMg

1/3

2/3

’, Physical Review Letters, 91, 247601/

1–4.

Bokov A A and Ye Z G (2004), ‘Domain structure in the monoclinic Pm phase of Pb(Mg

1/3

2/3

–PbTiO

single crystals’, Journal of Applied Physics, 87, 6347–6359.

Chien R R, Schmidt V H, Tu C-S, Hung L-W and Luo H (2004), ‘Field-induced polarization

rotation in (001)-cut Pb(Mg

1/3

2/3

)

0.76

0.24

’, Physical Review B, 69, 172101/1–4.

Chien R R, Schmidt V H, Hung L-W and Tu C-S (2005a), ‘Temperature- and electric-

field-dependent domain structures and phase transformations in (001)-cut tetragonal

Pb(Mg

1/3

2/3

)

1–x

(x = 0.40) single crystal’, Journal of Applied Physics, 97,

114112/1–4.

Chien R R, Schmidt V H, Tu C-S and Wang F-T (2005b), ‘Prior poling effect on thermal

phase stability in (110)-cut Pb(Mg

1/3

2/3

)

0.74

0.26

single crystal’, Journal of Applied

Physics, 98, 114106/1–5.

Chien R R, Tu C-S, Schmidt V H and Wang F-T (2006), ‘Electric-field- and temperature-

induced phase transitions in high-strain ferroelectric Pb(Mg

1/3

2/3

)

0.67

0.33

single

crystal’, Journal of Physics: Condensed Matter, 18, 8337–8344.

Colla E V and Weissman M B (2005), ‘Two-step phase changes in cubic relaxor

ferroelectrics’, Physical Review B, 72, 104106/1–7.

Cross L E (1987), ‘Relaxor ferroelectrics’, Ferroelectrics, 76, 241–267.

Davis M, Damjanovic D and Setter N (2006), ‘Electric-field-, temperature-, and stress-

induced phase transition in relaxor ferroelectric single crystals’, Physical Review B,

73, 014115/1–16.

Feng Z, Luo H, Guo Y, He T and Xu H (2003), ‘Dependence of high electric-field-

induced strain on the composition and orientation of Pb(Mg

1/3

2/3

–PbTiO

crystals’,

Solid State Communications, 126, 347–351.

Feng Z, Zhao X and Luo H (2006), ‘Composition

and orientation dependence of dielectric

and piezoelectric properties in poled

Pb(Mg

1/3

2/3

–PbTiO

crystals’, Journal of

Applied Physics, 100, 024104/1–5.

Fu H and Cohen R E (2000), ‘Polarization rotation mechanism for ultrahigh

electromechanical response in single-crystal piezoelectrics’, Nature, 403, 281–283.

Guo Y, Luo H, He T, Pan X and Yin Z (2003), ‘Electric-field-induced strain and piezoelectric

properties of a high Curie temperature Pb(In

1/2

–PbTiO

single crystal’, Materials

Research Bulletin, 38, 857–864.

Han J and Cao W (2003), ‘Electric field effects on the phase transitions in [001] oriented

(1–x) Pb(Mg

1/3

2/3

–xPbTiO

single crystals with compositions near the

morphotropic phase boundary’, Physical Review B, 68, 134102/1–6.

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Handbook of dielectric, piezoelectric and ferroelectric materials364

Hartshorne N H and Stuart A (1970), Crystals and the Polarizing Microscope, London,

E. Arnold Ltd.

Jona F and Shirane G (1962), Ferroelectric Crystals, New York, Pergamon Press, 254–

255.

Noheda B, Cox D E, Shirane G, Gao J and Ye Z–G (2002), ‘Phase diagram of the

ferroelectric relaxor (1–x)Pb(Mg

1/3

2/3

–xPbTiO

’, Physical Review B, 66, 054104/

1–10.

Pirc R and Blinc R (1999), ‘Spherical random-bond-random-field model of relaxor

ferroelectrics’, Physical Review B, 60, 13470–13478.

Shebanov L A, Kaspostins P and Zvirgzds J (1984), ‘The structure change of PMN in the

diffuse phase transition region’, Ferroelectrics, 56, 53–56.

Shrout T R, Chang Z P, Kim N and Markgraf S (1990), ‘Dielectric behavior of single

crystals near the (1–x)Pb(Mg

1/3

2/3

–(x)PbTiO

morphotropic phase boundary’,

Ferroelectrics Letters Section, 12, 63–69.

Sommerfeld A (1964), Optics, New York, Academic Press, 129–139.

Tu C-S, Schmidt V H, Shih I-C and Chien R (2003a), ‘Phase transformation via a monoclinic

phase in relaxor-based ferroelectric crystal (PbMg

1/3

2/3

)

1–x

(PbTiO

)

’, Physical

Review B(R), 67, 020102/1–4.

Tu C-S, Shih I-C, Schmidt V H and Chien R (2003b), ‘E-field-induced polarization

rotation in (PbMg

1/3

2/3

)

1–x

(PbTiO

)

crystal’, Applied Physics Letters, 83, 1833–

1835.

Tu C-S, Chien R R, Wang F-T, Schmidt V H and Han P (2004), ‘Phase stability after an

electric-field poling in Pb(Mg

1/3

2/3

)

1–x

crystals’, Physical Review B(R), 70,

220103/1–4.

Tu C-S, Wang F-T, Chien R R, Schmidt V H and Tuthill G F (2005), ‘Electric–field

effects of dielectric and optical properties in Pb(Mg

1/3

2/3

)

0.65

0.35

crystal’, Journal

of Applied Physics, 97, 064112/1–5.

Tu C-S, Wang F-T, Chien R R, Schmidt V H, Hung T-M and Tseng C-T (2006a), ‘Dielectric

and photo–voltaic phenomena in tungsten doped Pb(Mg

1/3

2/3

)

1–x

crystal’, Applied

Physics Letters, 88, 032902/1–3.

Tu C-S, Hung C-M, Wang F-T, Chien R R and Yang S-W (2006b), ‘Dielectric and optical

behaviors in relaxor ferroelectric Pb(In

1/2

)

1–x

crystal’, Solid State

Communications, 138 (4), 190–193.

Tu C-S, Chien R R, Schmidt V H, Wang F-T and Lim L C (2006c), ‘Electric-induced

dielectric anomalies and optical birefringence in Pb(Zn

1/3

2/3

)

1–x

’, Journal of

Applied Physics, 100, 074105/1–6.

Tu C-S, Wang F-T, Hung C-M, Chien R.R and Luo H (2006d), ‘Dielectric, domain, and

optical studies in high-Curie-temperature Pb(In

1/2

)

1–x

(x=0.40) single crystal’,

Journal of Applied Physics, 100, 104104/1–5.

Vanderbilt D and Cohen M H (2001), ‘Monoclinic and triclinic phases in higher-order

Devonshire theory’, Physical Review B, 63, 09418/1–9.

Viehland D and Li J F (2002), ‘A hysteretic field-induced rhombohedral to orthorhombic

transformation in <110>–oriented 0.7Pb(Mg

1/3

2/3

–0.3PbTiO

crystals’, Journal

of Applied Physics, 92, 7690–7692.

Viehland D, Wuttig M and Cross L E (1991), ‘The glassy behavior of relaxor ferroelectrics’,

Ferroelectrics, 120, 71–77.

Wan X, Xu H, He T, Lin D and Luo H (2003), ‘Optical properties of tetragonal

Pb(Mg

1/3

2/3

)

0.62

0.38

single crystal’, Journal of Applied Physics, 93, 4766–4768.

Westphal V, Kleemann W and Glinchuk M D (1993), ‘Diffuse phase transitions and

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Electric field-induced domain structures and phase transitions 365

random-field-induced domain states of the ‘‘relaxor’’ ferroelectric PbMg

1/3

2/3

’,

Physical Review Letters, 68, 847–850.

Xu G, Luo H, Xu H and Yin Z (2002), ‘Third ferroelectric phase in PMNT single crystals

near the morphotropic phase boundary composition’, Physical Review B, 64, 020102/

1–3.

Yamashita Y, Hosono Y, Harada K and Yasuda N (2002), ‘Present and future of piezoelectric

single crystals and the importance of B-site cations for high piezoelectric response’,

IEEE on Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 49,

184–192.

Yasuda N, Sakaguchi M, Itoh Y, Ohwa H and Yamashita Y (2003),’ Effect of electric

fields on domain structure and dielectric properties of Pb(In

1/2

–PbTiO

near

morphotropic phase boundary’, Japanese Journal of Applied Physics, 42 part 1 (9B),

6205–6208.

Ye Z-G (1998), ‘Relaxor ferroelectric complex perovskites: structure, properties and

phase transitions’, Key Engineering Materials, 155–156, 81–122.

Ye Z-G and Dong M (2000), ‘Morphotropic domain structures and phase transitions in

relaxor-based piezo-/ferroelectric (1–x)Pb(Mg

1/3

2/3

–xPbTiO

single crystals’,

Journal of Applied Physics, 87, 2312–2319.

Ye Z-G, Noheda B, Dong M, Cox D and Shirane G (2001), ‘Monoclinic phase in the

relaxor-based piezoelectric/ferroelectric Pb(Mg

1/3

2/3

–PbTiO

’, Physical Review

B, 64, 184114/1–5.

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366

13.1 Introduction

Relaxor ferroelectric single crystals, when cut and poled in certain orientations,

display extraordinary properties that include very large piezoelectric

coefficients, high electromechanical coupling coefficients, and very low loss

tangents. These properties are found within a limited range of temperature,

electric field, and stress. The limitations on loading range are associated

with temperature and field-induced phase transitions (Zhao et al., 2002; Han

and Cao, 2003; Liu and Lynch, 2003), composition-temperature behavior

(Noheda et al., 2002), electric field–temperature behavior (Ren et al., 2002),

and the coexistence of multiple phases (Ye and Dong, 2000; Bertram et al.

2003). The stability of phases and field-induced phase transformations show

composition dependence. Recent studies include the effect of stress on phase

transformations (Samara, et al., 2000; Viehland and Powers, 2001a; Viehland

et al., 2003).

Phase transformations can be classified into two types, namely, discontinuous

and continuous. Discontinuous phase transformations, also called first-order

transformations, involve a discontinuous change in the entropy; continuous

phase transformations, also called second-order transformation, occur with

continuous changes of entropy. These two types of transformation have been

observed in relaxor ferroelectric single crystals (Liu and Lynch, 2003;

McLaughlin et al., 2004). Discontinuous phase transformations with large

hysteresis result in rapid degradation of the material due to cracking.

Phase transformations can be described within a thermodynamics

framework. Thermodynamics-based approaches have been used extensively

to elucidate the complex behavior of multi-physics coupled materials (Fiebig

et al., 2002). In this chapter domain engineering and multi-field-driven phase

transitions are reviewed and a thermodynamics-based analysis of measured

material behavior under combined stress, electric field, and temperature loading

is discussed (Liu, et al., 2006). This leads to a technique for the direct

measurement of internal energy and Gibbs free energy based on combined

Energy analysis of field-induced phase

transitions in relaxor-based piezo- and

ferroelectric crystals

T LIU and C S LYNCH, The Georgia Institute of

Technology, USA

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Energy analysis of field-induced phase transitions 367

electromechanical loading. The approach is applied to an electric field-driven

transition from rhombohedral to orthorhombic with associated hysteresis in

<110> loaded PZN–4.5%PT (Liu and Lynch, 2003), and to an electric field-

driven transition from rhombohedral to orthorhombic with rotation through

an intermediate monoclinic phase in <110> PMN–32%PT driven by both

electric field and stress (McLaughlin et al., 2004).

13.2 Background

13.2.1 Anisotropy and domain engineering

Ferroelectric single crystals exhibit anisotropic elastic, dielectric, and

piezoelectric properties. The macroscopic behavior of the crystals can be

described in terms of a volume average of the behavior at smaller length

scales within the crystals. This includes domain wall motion, nucleation and

growth of phases, phase boundary motion, diffusion of defects, and unit cell

behavior (intrinsic piezoelectricity). Actuators fabricated from ferroelectric

materials are generally pre-loaded under uniaxial stress (one non-zero stress

component) and under uniaxial electric field. The actuators are designed

such that the stress and electric field within the active material are uniform.

If the electric field is bipolar and exceeds the coercive field, polarization

reversal will occur and the typical electric displacement/electric field (D-E)

and the butterfly-shaped strain/electric field (S-E) hysteresis loops will be

observed. If the ferroelectric material is subjected to a unipolar field, minor

hysteresis loops can be observed. The area within the D-E loop is associated

with material loss and results in the generation of heat. If the stress is varying

during the cycle, the area within the stress/strain (T-S) hysteresis loop represents

additional loss. If the stress and electric field at a given temperature exceed

critical values, a phase transformation occurs and additional hysteresis takes

place with the associated generation of heat.

Rhombohedral phase relaxor ferroelectric single crystals with compositions

close to the morphotropic phase boundary (MPB) display large piezoelectric

coefficients and low loss when poled in a <001> or a <110> direction (Park

and Shrout, 1997; Liu et al., 1999; Liu and Lynch, 2003). When single

crystals are cut and poled in the <001> or <011> orientations, there is optimal

domain compatibility and minimized domain wall motion. This greatly reduces

the minor hysteresis loops and the corresponding heat generation. The

electromechanical properties of these crystals are directly related to their

engineered domain structures: an effective four-domain state for the <001>

poled crystals and a two-domain state for the <110> poled crystals. Such

engineered multi-domain states enable the calculation of piezoelectric

coefficients from that of a single domain (crystal variant). The technique is

called crystal variant-based modeling. Various calculations of piezoelectric

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Handbook of dielectric, piezoelectric and ferroelectric materials368

coefficients have been conducted for engineered multi-domain ferroelectrics

(Yin et al., 1999, 2000b; Damjanovic et al., 2003; Dammak et al., 2003; Liu

and Lynch, 2003; Zhang et al., 2003; Topolov, 2004; Davis et al., 2005). The

crystal variant model can be extended to describe the evolution of polarization

switching and phase changes by including the appropriate variant evolution

criteria.

13.2.2 Phase stability

The phases present in relaxor PZN–xPT and PMN–xPT single crystals depend

on their composition and temperature (Lu et al., 2001; La Orauttapong et al.,

2002; Ye, 2002). The most widely used compositions (PMN–28~32%PT)

are very near the MPB. High-resolution diffraction studies have revealed

that the morphotropic phase boundary compositions comprise the rhombohedral

(R), tetragonal (T), orthorhombic (O) and monoclinic (M) phases (Fujishiro

et al., 1998; Uesu et al., 1998; Ye et al., 1999, 2001; Zhang et al., 2000; Cox

et al., 2001; Lu et al. 2001; Noheda et al., 2001a,b). The coexistence of

rhombohedral and tetragonal domains in PZN–xPT and PMN–xPT crystals

have been observed at room temperature (Fujishiro et al. 1998; Uesu et al.

1998; Belegundu et al., 1999; Ye et al. 1999; Ye and Dong 2000). Recently

the extent of different phases (R, M/O, T) in PZN–xPT and the dependency

of their volume fractions on the PT content were characterized (Bertram

et al. 2003).

Multi-domain structures and multi-phase coexistence around the MPB

contribute to the enhanced electromechanical properties of relaxor single

crystals (Yin et al., 2000a; Zhang et al., 2001). At the same time the coexistence

of various phases leads to complex microscale behavior (Noheda et al.,

2002; Ye, 2002; Singh and Pandey, 2003). The stability of the phases depends

on composition, orientation, and temperature as well as the electrical and

mechanical boundary conditions (Viehland et al., 2001; Zhao et al., 2002).

Application of temperature, electric field, and stress leads to polarization

switching and phase transitions that dramatically alter the electromechanical

properties of these crystals and may induce cracking and fatigue that lead to

premature failure in applications.

13.3 Multi-field-induced phase transitions

Phase transformations can be driven by combinations of stress and electric

field in PZN–xPT and PMN–xPT single crystals (Samara et al. 2000; Shang

and Tan, 2001; Viehland and Powers, 2001a,b; Fang and Yang, 2002; Viehland

et al. 2003). This effect is most easily described in terms of the <001> and

<011> cut and poled crystals.

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Energy analysis of field-induced phase transitions 369

13.3.1 <001> poled single crystals

Transformation between the rhombohedral phase and tetragonal phase can

be induced by stress and electric field in <001> oriented relaxor single

crystals (Meeks and Timme, 1975; Liu et al. 1999; Viehland, 2000; Noheda

et al. 2001a; Chen et al., 2002; Park and Hackenberger, 2002; Ren et al.

2002). The <001> electric field provides a driving force to align the polarization

vector with the electric field vector. When the driving force is large enough,

the transformation takes place. Han and Cao (2003) found that application of

an electric field bias along <001> decreases the R-T phase transition

temperature. Ren et al. (2002) measured the R-T phase transition fields of

PZN–4.5%PT and PZN–8%PT crystals as a function of temperature and

found linearly decreasing field values relative to increase of temperature.

Schmidt et al. (2003) and Tu et al. (2003) suggested that the R-T phase

transition in PMN–PT single crystals progresses through a rotation of

polarizations of monoclinic (M) phases. Multiple phase transitions have

been observed in the <001> PMN–32%PT single crystal under combined

electric field and stress loading (both along <001>) at different temperatures

(McLaughlin et al., 2005). These field-induced phase transitions are depicted

in Fig. 13.1.

(b)

(a)

(c)

(d)

13.1

Crystal variants of a <001> poled single crystal and the

possible phase changes under electric field and stress loadings:

(a) rhombohedral phase polarized into a four-variant state;

(b) orthorhombic phase depolarized under large compressive stress;

phase under further increased electric field.

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