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

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Handbook of dielectric, piezoelectric and ferroelectric materials370

13.3.2 <110> Poled single crystals

A rhombohedral to orthorhombic (R-O) phase transformation can occur in

either <001> or <110> PZN–xPT and PMN–xPT single crystals under stress

and electric field loading (Lu et al. 2001, 2002; Viehland and Li, 2002; Feng

et al., 2003; Liu and Lynch 2003; McLaughlin et al. 2004, 2005). Crystal

variants present in a <110> poled single crystal and the rhombohedral–

orthorhombic (R-O) phase transition under electric field E

and stress σ

are illustrated in Fig. 13.2. This phase change is effectively a <111> to

<110> polarization rotation. Both the positive <110> electric field and the

<001> compressive stress drive the R-O phase transition and stabilize the O

phase.

Figure 13.3 shows the electric displacement and strain curves for single

crystal PZN–4.5%PT (Liu and Lynch, 2003) during an electric field-driven

R-O phase transformation at room temperature and zero stress. There is a

jump in electric displacement as well as strain during the phase transformation.

This is a discontinuous (first-order) phase transformation with hysteresis.

This phase transformation induced by combined stress and electric field

was mapped out in a series of experiments performed on relaxor PMN–

32%PT single crystals (McLaughlin et al., 2004), the electric field-induced

strain for a series of pre-stress levels and the stress-induced strain for a series

of bias electric field levels on <110> oriented PMN–32%PT single crystals

were measured at three different temperatures. These data were used to

generate plots of the electric displacement and strain versus the applied

electric field and stress. The plots at 40°C are shown in Fig. 13.4. There are

two distinct planar regions in these three-dimensional surfaces. These

correspond to the rhombohedral (R) region at low field level and the

orthorhombic (O) region at high field level. Between them is an R-O phase

transition region showing sharp changes of strain and electric displacement.

The phase transition is associated with a change of internal energy. The

question arises as to whether the onset and completion of the phase transition

can be identified by a scalar internal energy density level and whether the

(a) (b)

<001>

<110>

13.2

Crystal variants of a <110> poled single crystal and the

rhombohedral–orthorhombic phase change under electric field and

stress loadings: (a) rhombohedral phase polarized into a two-variant

state; (b) orthorhombic phase polarized into a single-variant state.

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Energy analysis of field-induced phase transitions 371

progression of the transition can be indicated by the scalar Gibbs free energy.

This is explored through numerical integration of the three-dimensional coupled

surfaces along various paths.

Compared with the phase change of PZN–4.5%PT in Fig. 13.3, the PMN–

32%PT single crystal shows a gradual transformation with little hysteresis.

The transformation takes place over a range of field levels and there is no

distinct phase transition point. (Note that the apparent hysteresis in the D-E

curves in Fig. 13.4b was attributed to difficulties with the polarization

measurements.)

The <110> electric field E

and the <001> compressive stress σ

each

provide a positive driving force for the R-O phase transition in the <110> cut

(MV/m)

(b)

2.52.01.51.00.50

PZN–4.5%PT, <110>, 20°C

(%)

0.4

0.3

0.2

0.1

(MV/m)

(b)

PZN–4.5%PT, <110>, 20°C

2.52.01.51.00.50

(C/m

)

0.40

0.42

0.44

0.38

0.36

0.34

0.32

0.30

13.3

Electric field driven R-O phase transition in a <110> oriented

relaxor PZN–4.5%PT single crystal at room temperature (Liu and

Lynch, 2003): (a) electric displacement vs. electric field; (b) strain vs.

electric field.

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Handbook of dielectric, piezoelectric and ferroelectric materials372

and poled crystals. This is apparent when considering the change in polarization

and strain that take place during the transformation. The strain change is in

the direction of the applied stress and the polarization change is in the

direction of the applied field. The driving force for this transformation decreases

with increasing temperature. As shown in Fig. 13.5, when plotted in the

electric field–stress–temperature (E

-σ

-T) coordinate system, the starting

points (E

, σ

, T) for the phase transformation lie in one plane and the

finishing points in another (McLaughlin et al. 2004). By least square fitting

of the data, the equation for the start plane is T = 91.1 – 41.3E

+ 2.12σ

and

the equation for the finish plane is T = 105 – 40.6E

+ 2.12σ

. These two

= 40°C

0.30

0.32

0.34

0.36

(c/m

)

0.00

0.25

0.50

0.75

1.00

(MV/m)

–28

–21

–14

–7

(MPa)

(b)

(a)

= 40°C

–0.3

–0.2

–0.1

0.0

(%)

0.00

0.25

0.50

0.75

1.00

(MV/m)

–28

–21

–14

–7

(MPa)

13.4

Strain (ε22) and electric displacement (

) response of PMN–

32%PT single crystals under combined electric field (

) and stress

(σ

) loadings at 40°C (McLaughlin

et al.

2004): (a) strain; (b) electric

displacement.

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Energy analysis of field-induced phase transitions 373

planes are nearly parallel, giving an estimate of the R-O phase change

temperature range T

= 90 ~ 105°C at zero electric field and stress.

It is of interest to compare the R-O phase transitions in <110> and <001>

cuts. In the <110> oriented crystals, with the electric field in <110>, one of

the spontaneous polarization directions of the orthorhombic phase, the

transformed orthorhombic crystals are poled into a single-domain state (Fig.

13.2b); in the <001> oriented crystals, under <001> compressive stress, the

transformed orthorhombic phase is in a multi-domain depolarized state with

spontaneous polarizations perpendicular to the poling direction (Fig. 13.1b).

13.4 Energy analysis of phase transitions

13.4.1 Energy-based modeling

A number of analytical and computational approaches have been used to

describe and predict phase transformation behavior in relaxor single crystals.

Theoretical investigations have been carried out to study structure–property

relations from the atomic level (Allen, 1976; Cohen, 1992; Rabe and Waghmare,

1996; Fu and Cohen, 2000; George et al., 2001; Tadmor et al., 2002) to the

continuum level (Chen and Lynch, 1998a,b; Lynch, 1998; Kamlah and

Tsakmakis, 1999; Mauck, 2002; McMeeking and Landis, 2002; Landis et

al., 2003; Glinchuk, 2004). Landau-Devonshire phenomenological theory

60°C

40°C

20°C

100

(°C)

(MV/m)

(MPa)

–60

–40

–20

13.5

R-O phase change plane for PMN–32%PT single crystal plotted

in electric field–stress–temperature (

-σ

) coordinate system

(McLaughlin

et al.

, 2004). Electric field

is in <110> direction, and

stress σ

is in <001> direction. The bottom (small) triangle refers to

the onset plane of the transition, and the top (big) triangle is in the

plane for the end of the transition.

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Handbook of dielectric, piezoelectric and ferroelectric materials374

has been widely used to explain observed phases and phase transition

phenomena (Su, et al., 2001; Vanderbilt and Cohen, 2001; Li et al., 2003;

Wang et al., 2004). An energy-based approach is particularly useful for

multi-physics coupled materials such as ferroelectrics, ferromagnetics

(Asamitsu et al., 1995), and ferroelectromagnets (Fiebig et al., 2002) because

it provides a scalar phase transformation criterion that can be measured.

Such an energy criterion can be applied to multi-axial multi-field loading.

Energy minimization has been used to analyze the engineered domain

configurations of ferroelectric single crystals (Shu and Bhattacharya, 2001;

Jiang and Dan, 2004). By considering the contributions to the free energy,

numerical simulations of the formation of domain structures can be performed

(Yang and Chen, 1995; Hu and Chen, 1997; Hu and Chen, 1998; Khachaturyan,

2000; Li et al., 2001). Energy-based polarization switching and phase transition

criteria have also been used to simulate the behavior of ferroelectric ceramic

PZT (Hwang et al., 1995; McMeeking and Hwang, 1997; Chen and Lynch,

1998a; Huber et al., 1999), electrostrictive PMN–xPT (Hom and Shankar,

1996), and antiferroelectric PLSnZT (Chen and Lynch 1998b; Essig et al.,

1999).

The R-O phase transitions observed in the <110> oriented PZN–4.5%PT

(Liu and Lynch, 2003) and PMN–32%PT crystals (McLaughlin et al., 2004)

is the subject of much of the following discussion. Work done by the electric

field and stress is computed by path integration of recorded data and the

energy balance during the phase transitions is analyzed. The path integrals

needed to analyze the data are obtained through a continuum mechanics

derivation that follows the details presented by Malvern (1969) plus an

expression for the work done by electrical loading.

13.4.2 Equations of energy equilibrium

Energy balance of a system can be expressed in rate form as

˙˙˙ ˙

KEW W Q + = +

13.1

where the terms are rate of change of kinetic energy

and rate of change

of internal energy

, mechanical work rate

, and electrical work rate

(superscripts m and e refer to mechanical and electrical components),

and rate of heat addition

With the restriction to stationary bodies and an absence of body forces

and body charges, a local expression for the rate of change of internal energy

ue EDq

ij ij j j

= = + + ρσε

13.2

where

is the rate of heat added per unit volume from external sources, ρ

is the mass density, and

is the rate of change of internal energy per unit mass.

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Energy analysis of field-induced phase transitions 375

For a single crystal ferroelectric with rate-independent dissipation, the

internal energy can be written in incremental form

du = σ

dε

+ E

+ dq 13.3

where dq is the thermal energy per unit volume added from external sources.

Integrating each term of Equation (13.3) along a loading path gives

∆uu

= d

∫

13.4

ij ij

= d

∫

σε

13.5

wED

= d

∫

13.6

∆qq

= d

∫

13.7

Combining equations (13.4–13.7) gives the following form of (13.3):

∆u = w

+ w

+ ∆q 13.8

Equations (13.5) and (13.6) can be applied directly to measured stress–strain

and electric field–electric displacement data. Under adiabatic loading (∆q =

0) the temperature will increase after a cycle due to internal dissipation.

Under isothermal loading through a closed cycle (∆u = 0), Equation (13.7)

represents the amount of heat leaving the body that is equal to the heat

generated by the dissipative terms.

Assume the final state is the same as the initial state after a loading cycle,

i.e. ∆u = 0

( + ) = (– )

ww q

∫∫

∆

13.9

This states that the irreversible work done by the stress and the electric field

will be converted into heat and dissipate into the environment to maintain a

constant temperature. Equation (13.9) can be used to obtain the relative

internal energy density levels before and after a phase transition.

The Gibbs free energy gives a measure of the driving force for the

transformations. This can also be determined by direct integration of the

experimental data. Performing a Legendre transformation on the expressions

for electrical and mechanical work leads to the equation for Gibbs free

energy

∆∆σε εσ ∆gu ED DE q

ij ij j j

ij ij

= – – = – d + – d +

∫∫

13.10

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Handbook of dielectric, piezoelectric and ferroelectric materials376

13.4.3 Work–energy analysis

Details of the work in this section can be found in Liu et al. (2006). Path

integrals were conducted to compute the work done by the electric field and

stress. The external work for the loading scheme shown in Fig. 13.2 includes

mechanical work done by the stress σ

and electrical work done by the

electric field E

. For the PZN–4.5%PT single crystal, only electrical work is

done during the phase transition due to the stress-free condition. Equation

13.6 is applied to compute the electrical work and Fig. 13.6(a) shows the

result of the numerical integration of the PZN–4.5%PT data. Note that there

is an offset (non-zero external work) over a loading cycle. This is the irreversible

work converted into heat in a cycle. Assuming that half of the heat is generated

during the forward transformation and half during the reverse transformation,

the results after subtracting the heat are shown in Fig. 13.6(b). This is the

internal energy before and after the phase transition.

With the assumption that the material is linear piezoelectric both before

and after the transformation, the energy functions can be written in quadratic

form. The general ferroelectric constitutive relations can be expressed in

incremental form as:

∆ε ε ∆σ ∆( – ) = +

ijkl

nij n

sdE

13.11

∆∆σ∆

( – ) = +

DD d kE

mkl kl

mn n

13.12

The remnant strain

and remnant electric displacement

are associated

with the non-linearity and the irreversibility. In a linear region under the

prescribed loads σ

, E

(either in the R phase or O phase), the constitutive

equations reduce to:

εε σ

2222

22 322 3

– = + sdE

13.13

DDd kE

322 22

– = + σ

13.14

For the linear behavior of a single phase at constant temperature, the quadratic

form of the internal energy function for the single crystal is then given by

uu s d E kE = +

+ +

2222

322 22 3

33 3

σσ

13.15

The quadratic form of the Gibbs free energy is given by

sdEkE

= –

+ +

2222

322 22 3

33 3

σσ









13.16

Where u

and g

are the reference internal energy and Gibbs free energy at

zero electric field and stress. The experimental data give direct measures of

the

2222

, d

322

and

coefficients for both the R phase and the O phase

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Energy analysis of field-induced phase transitions 377

(McLaughlin et al., 2004). With these coefficients, quadratics given by

equations (13.15) and (13.16) are obtained and overlain on the data in Fig.

13.6.

Shown in Fig. 13.6, both u

and g

are higher for the O phase. At the phase

transition field level, the Gibbs free energies for the two phases become

(b)

2.52.01.51.00.50

(MV/m)

Driving force

PZN–4.5%PT, <110>, 20°C

–100

–80

–60

–40

–20

100

∆

, ∆

(kJ/m

)

(a)

2.52.01.51.00.50

(MV/m)

PZN–4.5%PT, <110>, 20°C

∆

(

–

–100

–80

–60

–40

–20

100

(kJ/m

)

13.6

Work–energy analysis of the PZN–4.5%PT single crystal under

electric field loading (Liu

et al.

, 2006). Internal energy and Gibbs free

energy values for the R phase and O phase based on linear

electromechanical response are superimposed: (a) work done by the

electric field; (b) change of internal energy and Gibbs free energy

density.

∫

–

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Handbook of dielectric, piezoelectric and ferroelectric materials378

equal. The difference of the Gibbs free energy between the two phases gives

the driving force for the phase transformation as the electric field increases.

The crystal will stay in the phase with lower Gibbs free energy at given

stress and electric field. There is a jump in the internal energy during the

phase transition corresponding to the differences in phase energies.

Two phases may coexist when they have the same Gibbs free energy. In

addition, considering inhomogeneous microscopic stress and electric field

present in the crystals, coexistence of two or more ferroelectric phases can

be expected when the phase energies are not too different in value.

Internal energy density and the Gibbs free energy surfaces for the <110>

PMN–32%PT single crystals were computed from experimental data (Fig.13.4)

through path integrals and plotted in Fig. 13.7. The quadratic energy functions

for the linear R and O phases are superimposed on these plots. Figure 13.7(b)

shows the Gibbs free energy of the crystal as a function of applied stress and

electric field at fixed temperature (40°C). At low stress and electric field, the

R phase has a lower Gibbs free energy compared with the O phase, therefore

it is the stable phase; as the applied stress and electric field increase, the

Gibbs free energy decreases, but the Gibbs free energy of the O phase decreases

faster than that of the R phase. When the Gibbs free energies of the two

phases are equal, the crystal reaches a critical point. A further increase of

stress and/or electric field leads to a lower Gibbs free energy of the O phase

and R-O phase transition occurs. The observed gradual phase transition behavior

with small hysteresis is believed to be the result of multi-phase coexistence

with intermediate monoclinic phases (Bai et al., 2004).

A similar analysis was performed on the data for each of the three

temperatures. The relative internal energy density at the onset of non-linearity

on the rhombohedral and the orthorhombic sides of the transition are plotted

as a function of temperature in Fig. 13.8. Note that the relative energies

converge very near the rhombohedral to cubic transition temperature seen in

the zero load phase diagram and that the energies of the phases are linear

with temperature in the range measured.

13.4.4 Irreversible work and hysteresis loss

Ferroelectric materials exhibit major hysteresis loops during field-induced

polarization switching. Hysteresis also occurs during a phase transition, which

is dependent on the material composition, temperature and other factors.

Minor hysteresis loops are displayed under unipolar loading, even though

the electric field and stress cycles do not induce apparent polarization switching

and phase transitions.

Domain and phase activities in ferroelectric materials contribute to non-

linear and hysteretic electromechanical response and other reliability issues

(Lines and Glass, 1977; Lynch, 1996; Damjanovic, 2001). Hysteresis below

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Energy analysis of field-induced phase transitions 379

the Curie temperature is mainly caused by domain wall movements (Hardtl,

1982). Hysteresis is associated with energy loss and heat generation in the

materials. Heat generation is a serious issue in piezoelectric devices working

at high frequencies such as ultrasonic motors. It is important to be able to

accurately model the hysteretic loss in piezoelectric actuators under dynamic

loading (Hall, 2001). The minor hysteresis loops under small variations of

stress and electric field may be expressed by a Rayleigh relationship in

which the coefficients are taken to be quadratic functions of applied stress

and electric field, with a different function for loading and unloading

(a)

∆

(kJ/m

)

1.00

0.75

0.50

0.25

0.00

(MV/m)

–28

–21

–14

–7

(MPa)

(b)

–20

–40

–60

–80

∆

(kJ/m

)

0.00

0.25

0.50

0.75

1.00

(MV/m)

–28

–21

–14

–7

(MPa)

13.7

Internal energy and Gibbs free energy of PMN–32%PT single

crystal as functions of stress and electric field (Liu

et al.

, 2006).

Values for the R phase and O phase based on linear

electromechanical response were overlain for comparison: (a) change

of internal energy; (b) change of Gibbs free energy.

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