Martienssen W., Warlimont H. (Eds.). Handbook of Condensed Matter and Materials Data

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912 Part 4

spontaneous polarization in the crystalline regions. The

electric-ﬁeld distribution is expected to be complex in

the thin ﬁlm because of the interposition of the amor-

phous regions. A ferroelectric hysteresis loop can be

observed when the ratio of the volume of the crys-

talline regions to the total volume is relatively large

(e.g. more than 50%). The coercive ﬁeld is larger (e.g.

> 50 MV/m) than that of solid ferroelectrics (usually

afewMV/m or smaller). The ferroelectric properties

depend sensitively upon the details of sample prepa-

ration, for example the use of melt quenching or melt

extrusion, the annealing temperature, or the details of

the poling procedure. Polymer ferroelectrics are useful

for soft transducers.

4.5.4 Physical Properties of 43 Representative Ferroelectrics

This section surveys the characteristic properties of

43 representative ferroelectrics with the aim of demon-

strating the wide variety in the behavior of ferroelectrics.

The 43 representative ferroelectrics are selected from

29 of the above-mentioned families. The presentation

is mainly graphical. Most of the ﬁgures are repro-

duced from LB III/36, where the relevant references

can be found. Table 4.5-2 summarizes the meaning

of the symbols frequently used in the ﬁgure cap-

tions and indicates the units in which the data are

given.

To facilitate to get more information on each repre-

sentative substance in LB II/36, the number assigned to

the substance in LB III/36 is given in parenthesis fol-

Table 4.5-2 Symbols and units frequently used in the ﬁgure captions

a, b, c unit cell vector, units Å

∗

, b

∗

, c

∗

unit cell vector in reciprocal space, units Å

−1

coercive ﬁeld, units V/m

f frequency, units Hz = 1/s

spontaneous polarization, units C/m

T temperature, units K or

◦

κ dielectric constant (or relative permittivity) = ε/ε

, where ε is the permittivity of the

material and ε

is the permittivity of a vacuum. κ is a dimensionless number



,κ



real and imaginary parts of the complex dielectric constant κ

∗

= κ



+iκ





and κ



are both dimensionless numbers

component of dielectric-constant tensor; dimensionless numbers

,κ

κ measured along a, b, c axes; dimensionless numbers

(hkl)

κ measured perpendicular to the (hkl) plane; dimensionless number

[uvw]

κ measured parallel to the [uvw] direction; dimensionless number

κ of free crystal, i. e. κ measured at constant stress T; dimensionless number

κ of clamped crystal, i. e. κ measured at constant strain S; dimensionless number

ferroelectric transition temperature, units K or

◦

lowing its chemical formula, e.g., KNbO

(LB number

1A-2).

4.5.4.1 Inorganic Crystals Oxides [5.1, 2]

Perovskite-Type Family

KNbO

(LB Number 1A-2). This crystal is ferroelec-

tric below about 418

◦

C. Further phase transitions

take place at about 225

◦

C and about −10

◦

C, re-

taining ferroelectric activity. The crystal has large

electromechanical coupling constants and is useful

in lead-free piezoelectric elements and SAW (surface

acoustic wave) ﬁlters in communications technology

(Fig. 4.5-13,4.5-14).

Part 4 5.4

Ferroelectrics and Antiferroelectrics 5.4 Physical Properties of 43 Representative Ferroelectrics 913

5000

4000

3000

2000

1000

Dielectric constant



–200 –100 0 100 200 300 400 500

Temperature T (°C)

f = 10 kHz

E = 500 Vm

–1

Fig. 4.5-12 KNbO

. Dielectric constant κ versus tempera-

ture T

KTaO

(LB Number 1A-5). KTaO

is cubic at all tem-

peratures, and its dielectric constant becomes very

large at low temperatures without a phase transition

(Fig. 4.5-14). It is generally believed that this behavior is

related to the zero-point lattice vibrations. Replacement

of Nb by Ta generally lowers drastically the ferroelec-

tric Curie temperature , as seen by comparing Fig. 4.5-14

with Fig. 4.5-12. (This effect is well demonstrated later

in Figs. 4.5-39 and 4.5-40).

SrTiO

(LB Number 1A-8). This crystal is cubic at room

temperature and slightly tetragonal below 105 K. The

phase transition at 105 K is caused by softening of

the lattice vibration mode at the (1/2, 1/2, 1/2) Bril-

2000

1000

100 150 200 250 300 350 400

(10

–2

) E

(10

–1

)

T (°C)

Fig. 4.5-13 KNbO

. P

and E

versus T . Measurements

were made by applying the electric ﬁeld parallel to the

pseudocubic [100] direction

3850

3845

3840

3835

3830

07123456

T (K)

Fig. 4.5-14 KTaO

. Real part of dielectric constant κ



ver-

sus T. f = 1kHz

(10

Hz)

T (K)

100 240

1.50

120 140 160

1.25

1.00

0.75

0.50

0.25

180 200 220

Fig. 4.5-15 SrTiO

. ν

versus T. ν

is the frequency of the

(Γ

) optical phonon

louin zone corner (Fig. 4.5-15) without an appreciable

dielectric anomaly. At very low temperatures, the di-

electric constants become extraordinarily high without

a phase transition (Fig. 4.5-16), owing to softening of

the optical phonon at the origin of the Brillouin zone

(Fig. 4.5-17). It is generally believed that the absence of

a low-temperature transition is related to zero-point lat-

Part 4 5.4

914 Part 4

T (K)

2.25

2.00

1.75

1.50

1.25

1.00

0.75

0.50

0.25

(10

)

100 150 200 250 300

(111)

(110)

(100)

50 kHz

(10

Hz)

350

T (K)

100 150 200 250 300

Fig. 4.5-17 SrTiO

. ν

versus T . ν

is the frequency of

the soft phonon at q = 0. Open circles: inelastic neutron

scattering. Filled circles: Raman scattering. The solid curve

is 194.4 κ

1/2

and the dashed curve is 0.677(T −T

)

1/2

, with

= 38 K

Fig. 4.5-16 SrTiO

. κ

(111)

,κ

(110)

,andκ

(100)

versus T at

f = 50 kHz. κ

(111)

,κ

(110)

,andκ

(100)

are slightly different

from each other in the tetragonal phase

tice vibrations as in KTaO

. Solid solutions of SrTiO

are useful as ceramic thin ﬁlms in memory devices.

(10

)

T (°C)

0 120

–160 –120 –80 –40 0 40 80

Fig. 4.5-18 BaTiO

. κ

and κ

versus T . κ

and κ

are

the values of κ along the a and b axes, respectively, of the

tetragonal phase

0.30

0.25

0.20

0.15

0.10

0.05

200 40 60 80 100 120 140

T(°C)

(C m

–2

)

Fig. 4.5-19 BaTiO

. P

versus T in the tetragonal phase.

is parallel to the c axis

Part 4 5.4

Ferroelectrics and Antiferroelectrics 5.4 Physical Properties of 43 Representative Ferroelectrics 915

BaTiO

(LB Number 1A-10). BaTiO

is the most exten-

sively studied ferroelectric crystal. It is ferroelectric

below about 123

◦

C, where the crystal symmetry

changes from cubic to tetragonal. Further phase transi-

tions take place from tetragonal to orthorhombic at about

◦

C and to rhombohedral at about −90

◦

C (Figs. 4.5-18

to 4.5-20). It is believed that the ferroelectric transition

–140 –120 –100 –80 –60 –40 –20 0 20

T (°C)

(10

–2

C m

–2

)

Fig. 4.5-20 BaTiO

. P

s[001]

versus T below 20

◦

C, where

s[001]

is the component of the spontaneous polarization

parallel to the c axis in the tetragonal phase

'' (arb. units)

–1

–2

ν/c (cm

–1

)

0 10 20 20 40 60 80 100

120 140

0 3 6 6 12 18 24 30 36 42

ν (10

Hz)

T = 408 K

T = 423 K

T = 473 K

T = 553 K

Fig. 4.5-21 BaTiO

. κ



versus ν. Parameter: T . κ



the imaginary part of κ obtained from the hyper-Raman

spectrum. Curves: calculations based upon the classical

dispersion oscillator model. c: light velocity

4.840

3.872

2.904

1.936

0.968

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

ν (10

Hz)

ζ (Å

–1

)

[ζ 00]

Zone

boundary

T =

230 °C

430 °C

Fig. 4.5-22 BaTiO

. Phonon dispersion relation deter-

mined by neutron scattering along the [100] direction in

the cubic phase. ν is the phonon frequency. LA, longitu-

dinal acoustic branch; TA, transverse acoustic branch; TO,

transverse optical branch. The frequency of the TO branch

is lower (softer) at 230

◦

C than at 430

◦

C, indicating mode

softening

is caused by softening of an optical mode at the center of

the Brillouin zone. Hyper-Raman scattering studies sup-

port this model (Figs. 4.5-21 and 4.5-22). The results of

neutron scattering studies favor this model (Fig. 4.5-23),

but the measurement was not easy owing to the intense

elastic peaks (Fig. 4.5-24a). These elastic peaks indicate

marked cluster formation in the vicinity of the Curie

point. As discussed in Sect. 4.5.3, there seem to be two

components contributing to the dielectric constant meas-

ured in the vicinity of the Curie point: a component due

to the soft mode and another one due to the motion

of the cluster boundaries. Presumably the two compo-

nents suggested by infrared and hyper-Raman scattering

data correspond to these two components. For theoreti-

cal studies of the phase transitions, readers should refer

to [5.7]. Solid solutions of BaTiO

are the most useful

ferroelectrics in ceramic condensers and as thin ﬁlms in

memory devices.

When the temperature is raised above 1460

◦

C, cubic

BaTiO

performs another phase transition to a hexa-

gonal structure. This hexagonal phase can be quenched

Part 4 5.4

916 Part 4

T (K)

300

250

200

150

100

400 500 600 700 800

∆ν

, Γ (10

Hz) ∆ν

/c (cm

–1

)

∆ν

3.6

3.2

2.8

2.4

2.0

1.6

1.2

0.8

0.4

0.50

0.25

3.7 3.8 3.9 4.0 4.1 4.2 4.3

I (10

counts)

Cubic BaTiO

150 °C

Q = (H, K, O)a*

K = 0.1

K = 0.2

K = 0.3

ν = 0

ν = 0.97 ×10

Fig. 4.5-23 BaTiO

. ∆ν

and Γ versus T , obtained from

hyper-Raman scattering in the cubic phase. ∆ν

and Γ

are the optical mode frequency and damping constant, re-

spectively. The different symbols (brown and gray)show

results from different authors. ∆ν

decreases as the tem-

perature decreases to the Curie point, showing the presence

of mode softening. c: ligth velocity

10 000

8000

6000

4000

2000

0 100 200 300 400 500 600

T (°C)

Fig. 4.5-25 PbTiO

. κ versus T

0 100 200 300 400

500

(10

–2

C m

–2

)

T (°C)

Fig. 4.5-26 PbTiO

. P

versus T

Fig. 4.5-24a,b BaTiO

. Triple-axis neutron spectrometer

scans at constant frequency across the sheet of diffuse scat-

tering at 150

◦

C. The path of the scans is shown in the

inset.

(a) shows the elastic scan (ν = 0), where the high

background level is due to nuclear incoherent scattering.

(b) Ineleastic scan (ν = 0.97 ×10

Hz)

Part 4 5.4

Ferroelectrics and Antiferroelectrics 5.4 Physical Properties of 43 Representative Ferroelectrics 917

4000

3500

3000

2500

2000

1500

1000

500

175 200 225 250 275 300 325

T (°C)

f = 1 MHz

–1

(10

–4

)

–1

Fig. 4.5-27 PbZrO

(single crystal). κ

and κ

−1

versus T

to room temperature by relatively rapid cooling. This

hexagonal BaTiO

also shows ferroelectric activity be-

low −199

◦

PbTiO

(LB Number 1A-11). This crystal is ferroelec-

tric below about 500

◦

C (Figs. 4.5-25, 4.5-26). The

spontaneous polarization is large (Fig. 4.5-26) and

thus the pyroelectric coefﬁcient is large, which makes

crit

(10

V m

–1

)

200 205 210 215 220 225 230

crit

T (°C)

Fig. 4.5-28 PbZrO

. E

crit

versus T. E

crit

is the critical ﬁeld

of the antiferroelectric hysteresis loop

the crystal useful in infrared sensors. Solid solutions

with other perovskite-type oxides provide good dielec-

tric and piezoelectric materials (see PZT, PLZT, and

(PbTiO

)

(Pb(Sc

1/2

)

1−x

below).

PbZrO

(LB Number 1A-15). This crystal is antiferro-

electric below about 230

◦

C (Fig. 4.5-27), exhibiting

a typical antiferroelectric hysteresis loop (Fig. 4.5-28).

Solid solutions of this substance are very important in

technological applications (see PZT and PLZT below).

Pb(Mg

1/3

2/3

(LB Number 1B-d4). This crystal

exhibits a broad ferroelectric phase transition with

an average transition temperature of −8

◦

C deter-

mined by the maximum of the low-frequency dielectric

constant. A marked frequency dispersion of the

0.16

0.08

–150 –100 –50 0 50 100

T (°C)

100

600

kHz

MHz

δtan

(10

–3

)

tan

Fig. 4.5-29 Pb(Mg

1/3

2/3

(ceramic). κ and tan δ ver-

sus T. Parameter: f

(10

–2

)

100 125 150 175 200 225 250

T (°C)

x= 0.05

0.04

0.03

0.02

0.005

Fig. 4.5-30 Pb(Zr

1−x

(ceramic). P

versus T .Pa-

rameter: x

Part 4 5.4

918 Part 4

dielectric constant occurs around the transition tem-

perature. These is a typical behavior of a relaxor

(Fig. 4.5-29).

Pb(Zr,Ti)O

(LB Number 1C-a62). PbZrO

is antiferro-

electric, as described above, while a small addition of

PbTiO

induces a ferroelectric phase (Fig. 4.5-30).

PZT (LB Number 1C-a63). Solid solutions

Pb(Ti

1−x

with x = 0.5–0.6 are commonly called

PZT. They have very large electromechanical coupling

constants and are widely utilized as piezoelectric elem-

ents. Thin ceramic ﬁlms are useful in memory devices

(Fig. 4.5-31).

1000

800

600

400

200

(10

–2

C/m

)

(10

–4

C/(K m

))

– d

(10

–12

C/N)

x (PbZrO

)

100

0.3 0.4 0.5 0.6 0.7 0.8

Fig. 4.5-31 Pb(Ti

1−x

2% Zr(Mn

1/3

2/3

as an

additive (ceramic). κ, p

, d

,andP

versus x. p

is the py-

roelectric coefﬁcient, d

is thepiezoelectric strain constant,

and P

is the remanent polarization (see Fig. 4.5-1)

(10

–2

C m

–2

)

–25 25 50 75 100 125125 150

T (°C)

Fig. 4.5-32 (Pb

0.92

0.08

)(Zr

0.40

0.60

)

0.98

(PLZT). P

versus T

PLZT (LB Number 1C-c66).

The acronym PLZT means

La-modiﬁed PZT. Hot-pressed ceramic PLZT is trans-

parent and useful for opticalswitches and similar devices

(Fig. 4.5-32).

(PbTiO

)

(Pb(Sc

1/2

)

1−x

(LB Number 1C-b34).

These solid solutions have very large electromechan-

x (PbTiO

)

0.2 0.3 0.4 0.5 0.6 0.70.10

x = 0 corrsponds to Pb(Sc

T = 25 °C

0.5

0.4

0.3

0.2

0.1

Fig. 4.5-33 (PbTiO

)

(Pb(Sc

1/2

)

1−x

(ceramic).

versus x. k

is the planar electromechanical coupling

factor

Part 4 5.4

Ferroelectrics and Antiferroelectrics 5.4 Physical Properties of 43 Representative Ferroelectrics 919

0 100 200 300 400 500 600 700 800

T (°C)

f = 10kHz

Fig. 4.5-34 LiTaO

. κ

and κ

versus T . f = 10 kHz

–100

1.0

0.8

0.6

0.4

0.2

0 100 200 300 400 500 600 700

T (°C)

(0)

Fig. 4.5-35 LiTaO

. P

(0) versus T. P

(0) is the value

of P

at 0 K (about 0.5Cm

−2

)

ical coupling factors, and are utilized for piezoelectric

actuators and similar devices (Fig. 4.5-33).

LiNbO

Family

LiTaO

(LB Number 2A-2). This crystal is ferroelectric

below 620

◦

C. The coercive ﬁeld is large. The crys-

tal is useful for piezoelectric elements, for linear and

nonlinear optical elements, and for SAW ﬁlters in com-

munications technology (Figs. 4.5-34 and 4.5-35).

SrTeO

Family

SrTeO

(LB Number 4A-1). This crystal is ferroelectric

between 312 and 485

◦

C (Figs. 4.5-36 and 4.5-37).

100 200 300 400 500

600

T (°C)

Fig. 4.5-36 SrTeO

. κ versus T . f = 10 kHz

200

0.04

0.03

0.02

0.01

320 360 400 440 480 520

T (°C)

(C m

–2

)

Fig. 4.5-37 SrTeO

. P

versus T

Part 4 5.4

920 Part 4

(relative)

3.5

3.0

2.5

100

200

300

400

500

600

(10

–5

T (°C)

(relative)

Fig. 4.5-38 Ba

NaNb

. d

and l

versus T . d

is the

nonlinear optical susceptibility (relative value), and l

the coherence length

Θ (K)

0.0

1000

750

500

250

0.2 0.4 0.6 0.8 1.0

Fig. 4.5-39 Ba

Na(Nb

1−x

)

. Ferroelectric transi-

tion temperature Θ versus x. Brown circles and gray circles

represent data measured by different authors

Tungsten Bronze-Type Family

NaNb

(BNN) (LB Number 6B-a7). This crystal is

ferroelectric below about 580

◦

C. The crystal structure

is modulated below 300

◦

C. This material is utilized

for optical second-harmonic generation and in optical

parametric oscillators (Fig. 4.5-38).

Na(Nb

1−x

)

(BNNT) (LB Number 6C-b30). The

transition temperature varies over a wide temperature

range as x varies from 0 to 1.0 (Fig. 4.5-39).

Family

(Nb

1−x

)

(LB Number 8B-6). These solid solu-

tions can be made over the whole range of x = 0–1.0,

and the Curie point varies over a wide temperature range

from 1342

◦

Cto−107

◦

C (Fig. 4.5-40). The solid so-

lutions are very useful as high-temperature dielectric

materials, especially because they do not contain Pb,

which is volatile at high temperatures.

Family

(LB Number 13A-1). This crystal is ferroelectric

below 283.5 K. The sign of the spontaneous polarization

is reversed around 130 K (Figs. 4.5-41 and 4.5-42).

GMO (Gd

(MoO

)

) Family

(MoO

)

(GMO) (LB Number 17A-3). Three crys-

tal structures of GMO are known, α, β,andγ . The

β structure is stable above 850

◦

C but can be ob-

tained at room temperature as a metastable state by

Θ (°C)

0 1.0

1500

1000

500

0.2 0.4 0.6 0.8

–273

Fig. 4.5-40 Sr

(Nb

1−x

)

. Ferroelectric transition

temperature Θ

versus x.Thelower curve shows another

phase transition temperature

Part 4 5.4

Ferroelectrics and Antiferroelectrics 5.4 Physical Properties of 43 Representative Ferroelectrics 921

270 315

275 280 285 290 295 300 305 310

T (K)

f =10 kHz

Fig. 4.5-41 Li

. κ

versus T . f = 10 kHz

–25

–50

0 50 100 150 200 250 300

T (K)

(10

–5

C m

–2

)

bias

= 420 kV m

–1

bias

= – 420 kV m

–1

Fig. 4.5-42 Li

. P

versus T . P

was determined

by pyroelectric-charge measurement

rapid cooling. Ferroelectric activity takes place in this

metastable β-GMO below 159

◦

C. The phase transi-

tion is the indirect type III

discussed in Sect. 4.5.3.

The dielectric constant of the clamped crystal (κ

Fig. 4.5-6) shows no anomaly at the transition point.

The ferroelectric phase results from a phonon insta-

bility at the (1/2, 1/2, 0) Brillouin zone corner of the

parent tetragonal phase. Anharmonic coupling to the

T (K)

0 450

0.30

0.25

0.20

0.15

0.10

0.05

(10

–2

)

50 100 150 200 250 300 350 400

426

(10

–2

)

T (K)

0.08

0.06

0.04

0.02

428 430 432 434

Near the

transition

Fig. 4.5-43 Gd

(MoO

)

. P

versus T

resulting antiparallel displacements produces a spon-

taneous strain, which in turn causes a spontaneous

polarization through the normal piezoelectric coupling

(Fig. 4.5-43).

Boracite-Type Family

I (LB Number 18A-23). This crystal is

ferroelectric and ferromagnetic below about 60 K

(Figs. 4.5-44, 5-45, 5-46): a rare example where both fer-

roelectric and ferromagnetic spontaneous polarizations

20 40 60 80

100

T (K)

(10

–5

–2

)

Fig. 4.5-44 Ni

I. P

versus T . P

was measured

with the specimen parallel to the cubic (001) plane

Part 4 5.4