Moulson A.J., Herbert J.M. Electroceramics: Materials, Properties, Applications

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optical indicatrix. These can be speciﬁed by changes in the coeﬃcients of the

indicatrix DB

which are assumed to depend on E (and P) as follows:

¼ r

ijk

þ R

ijkl

ð8:18Þ

or, in terms of polarization P,

¼ f

ijk

þ g

ijkl

ð8:19Þ

The constants r

ijk

and f

ijk

are the Pockels electro-optic coeﬃcients, and R

ijkl

and

ijkl

the Kerr coeﬃcients. They are related as follows:

ijk

 e

ð8:20Þ

and

ijkl

ðe

 e

Þðe

 e

ð8:21Þ

where e

and e

are the permittivity values measured with E directed along k or l.

The subscript on E

has now been dropped because the distinction between a bias

ﬁeld and the electric ﬁeld component of an electromagnetic wave should be

apparent from the context.

Whether or not the dependence is expressed in terms of E or P is a matter of

choice; it seems customary in the literature relating to single crystals to use the r

coeﬃcient for the linear Pockels eﬀect and g for the quadratic Kerr eﬀect. In the

case of electro-optic ceramics r and R are most commonly used.

The r and f are third-rank tensors and the R and g are fourth-rank tensors,

and since they are both symmetrical the subscripts i and j can be interchanged,

as also can k and l. Under these circumstances it is helpful to use a ‘contracted’

or ‘reduced’ notation such that r

ijk

! r

and R

ijkl

! R

, where m and n run

from 1 to 6. The ij ¼ m and kl ¼ n contractions are as follows:

11 ! 1, 22 ! 2, 33 ! 3, 23 ! 4, 13 ! 5 and 12 ! 6.

A complete description of the electro-optic eﬀect for single crystals necessitates

full account being taken of the tensorial character of the electro-optic

coeﬃcients. The complexity is reduced with increasing symmetry of the crystal

structure when an increasing number of tensor components are zero and others

are simply interrelated. The main interest here is conﬁned to polycrystalline

ceramics with a bias ﬁeld applied, when the symmetry is high and equivalent to

1mm (6 mm) and so the number of tensor components is a minimum. However,

the approach to the description of their electro-optic properties is formally

identical with that for the more complex lower-symmetry crystals where up to a

maximum of 36 independent tensor components may be required to describe

their electro-optic properties fully. The methods are illustrated below with

reference to single-crystal BaTiO

and a polycrystalline electro-optic ceramic.

442 ELECTRO-OPTIC CERAMICS

TEAMFLY

Team-Fly

The Pockels electro-optic effect

Single-crystal BaTiO

(T5T

)

At temperatures below T

BaTiO

belongs to the tetragonal crystal class

(symmetry group 4mm); it is optically uniaxial, and the optic axis is the x

axis

¼ 2.416, n

¼ 2.364). When an electric ﬁeld is applied in an arbitrary

direction the representation quadric for the relative impermeability is perturbed

ðB

þ DB

Þx

¼ 1 ð8:22Þ

where DB

¼ r

ijk

. Employing the contracted notation and referring to the

principal axes of the unperturbed ellipsoid (Eq. (8.12)) we can expand Eq. (8.22)

þ r

þ 2x

¼ 1

ð8:23Þ

in which k takes the values 1, :::, 3. From considerations of crystal symmetry it

can be shown that r

¼ r

and r

¼ r

, and that, except for r

, the remaining

13 tensor components are zero. Furthermore if we assume that the electric ﬁeld E

is directed along the x

axis so that E

¼ E

¼ 0, Eq. (8.23) reduces to

þ r

¼ 1 ð8:24Þ

It follows that, in comparison with Eq. (8.22),

¼ r

E and D

¼ r

or, since Dn

 n

and Dn

 n

¼

ð8:25Þ

The induced birefringence is Dn, where

Dn ¼ Dðn

 n

Þ¼Dn

 Dn

¼



E ð8:26Þ

or, simply,

Dn ¼

where

BACKGROUND OPTICS 443

¼ r



ð8:27Þ

 r

 r

if n  n

 n

Polycrystalline ceramic

The form of the electro-optic tensor for 6mm symmetry is identical with that

for the 4mm symmetry dealt with above. Therefore the induced birefringence for

a ﬁeld directed along the x

axis is again

Dn ¼

E ð8:28Þ

where r

¼ r

 r

A useful ﬁgure of merit for linear electro-optic materials is the half-wave

voltage V

which is the voltage that must be applied to opposite faces of a unit

cube of material to induce a phase diﬀerence of p between the extraordinary and

ordinary rays. It is discussed further in Section 8.2.2 (see Eq. (8.44)). Values for

are included in Table 8.1.

The Kerr quadratic electro-optic effect

Single-crystal BaTiO

(T4T

)

At temperatures above 130 8C BaTiO

is cubic (symmetry group m3m); it is

optically isotropic with n ¼ 2.42. When an electric ﬁeld is applied in an arbitrary

direction the representation quadric is perturbed to

þ DB

¼ 1 ð8:29Þ

where DB

¼ R

ijkl

444 ELECTRO-OPTIC CERAMICS

Table 8.1 Properties of some electro-optic materials

Material r

/pm V

/V R/nm

 g

j=m

2

Single crystals

BaTiO

(room temperature) 80 500 – –

BaTiO

(4130 8C) – – – 0.13

LiNbO

17.5 3030 – –

LiTaO

22 2800 – –

SrTiO

– – – 0.14

KTa

0.65

0.35

– – – 0.17

Ceramics

PLZT(8/65/35)

520 400 – –

PLZT(8/40/60) 100 200 – –

PLZT(9/65/35) – – 380 0.018

For explanation of the notation see Section 8.2.1.

Because the material is isotropic, the electric ﬁeld can be directed along the x

axis without any loss in generality, and so E

¼ E

¼ 0. Symmetry requires

¼ R

and R

¼ R

, R

¼ R

, and the

remaining components to be zero. Therefore the indicatrix can be written



þ R



þ x



þ R



þ x



þ R



¼ 1 ð8:30Þ

Now

¼ n þ Dn

¼ n 

¼ n

and

¼ n þDn

¼ n 

The induced birefringence is given by

Dn ¼ n

 n

¼ n

 n

¼

ðR

 R

ÞE

ð8:31Þ

or, in terms of the polarization,

Dn ¼

ðR

 R

ÞP

ðe

 1Þ

¼

ðg

 g

ÞP

ð8:32Þ

Polycrystalline ceramic

In the case of polycrystalline ceramic (6mm) the form of the electro-optic tensor

is the same as that for m3m symmetry except that R

ðR

 R

Þ. Therefore,

when a ﬁeld is applied along the x

axis, the induced birefringence is again

Dn ¼

ðR

 R

ÞE

ð8:33Þ

or, in terms of the polarization,

Dn ¼

ðg

 g

ÞP

ð8:34Þ

The electro-optic properties of a few important materials are summarized in

Table 8.1.

8.1.4 Non-linear optics

Second harmonic generation (SHG)

In the previous section the eﬀects of a d.c. bias ﬁeld on optical properties were

discussed. The eﬀects arise because the bias ﬁeld is suﬃciently strong (10

1

)

to signiﬁcantly disturb the structure of an anisotropic crystal such as LiNbO

BACKGROUND OPTICS 445

disturbance described by changes in the optical indicatrix. The strength of the

electric vector (10

1

) of the light used in such experiments acts on the bound

electrons in an essentially linear way. We now turn attention to laser light for which

the electric vector can have magnitudes in the range 10

–10

1

. Under such

intense ﬁelds the interaction with the electrons is non-linear.

Field strengths as high as this, if applied statically or at low frequency, are

strong enough to detach electrons from atoms or ions and cause breakdown. As

a component of light the peak ﬁelds persist for only very short times (10

15

and so do not cause breakdown.

The non-linearity in the response can be expressed by

P ¼ e

ðw

1Þ

E þ w

2Þ

þ w

3Þ

þ :::Þ (8:35Þ

Because the susceptibilities are tensors, for the linear dependence we have

¼ e

1Þ

(8:36Þ

and for the quadratic

¼ e

2Þ

ijk

(8:37Þ

In the literature relating to second harmonic generation d-coeﬃcients are

sometimes used rather than ws when, employing a contracted notation

ðjk ¼ s ð1,2, :::,6ÞÞ

¼ w

2Þ

=2(8:38Þ

with ‘s’ running from 1 to 6, the values of the permutated ‘jk’.

If laser light of suﬃcient intensity is propagating through an optically non-

linear material, and the time dependence of the electric vector component of the

electromagnetic ﬁeld is given by E ¼ E

sinðotÞ, then from Eq. (8.35), the

polarization is given by

P ¼ e

1Þ

sinðotÞþw

2Þ

sin

ðotÞþw

3Þ

sin

ðotÞþ :::g

¼ e

1Þ

sinðotÞþ

2Þ

f1  cosð2otÞg

3Þ

f3 sinðotÞsinð3otÞg ð8:39Þ

The term w

1Þ

sinðotÞ represents the response expected from a linear dielectric.

The second term contains the components

2Þ

and 

2Þ

cosð2otÞ: the ﬁrst

of these represents a constant polarization which would produce a voltage across

the material, that is rectiﬁcation; the second corresponds to a variation in

polarization at twice the frequency of the primary wave. Thus the interaction of

laser light of a single frequency with a suitable non-linear material leads to both

frequency doubling (SHG) and rectiﬁcation.

Whether or not second harmonics can be generated in a material depends

upon the symmetry of its structure. For example, in the case of isotropic

446 ELECTRO-OPTIC CERAMICS

materials such as cubic crystals (e.g. calcite) and normal glasses, reversal of E

simply reverses P; therefore there can be no even-power terms in Eq. (8.35) so

that w

2Þ

etc. are zero and SHG is not possible. Specially modiﬁed glasses can

generate second harmonics, as discussed in Section 8.4.

A diﬃculty arises in the production of second harmonic radiation because of

‘dispersion’, that is the dependence of wave velocity on frequency. We consider

the primary wave (angular frequency o) interacting with a bound electron and

producing a second harmonic (2o) wave. At this point the two waves are in

phase. As the ‘o-wave’ propagates it generates new second harmonic, ‘2o-waves’

and, because of dispersion, the ‘o-wave’ will, in general, travel with a diﬀerent

phase velocity than that of the ‘2o-wave’. As a result a ‘new’ ‘2o-wave’ will

interfere with those generated earlier, and only constructively if they have the

required phase relationship.

For a plane wave travelling in a direction normal to the surfaces of a plate of

thickness L the following proportionality can be derived [9] for the intensity I of

the second harmonic wave leaving the plate:

I /fsin

2pðn

 n

ÞL=l

g=ðn

 n

(8:40Þ

where n

and n

are the refractive indices for angular frequencies o and 2o

respectively and l

(or 2pc/o), is the wavelength of the fundamental frequency

wave travelling in a vacuum.

From Eq. (8.40), because the intensity is inversely proportional to (n

n

)

, the

eﬃcient production of SHG radiation in a crystal of appreciable thickness

requires the matching of n

and n

. In certain birefringent materials it is possible

to choose a direction of propagation such that the refractive indices of the

ordinary and extraordinary rays are equal. This is possible in, for example, the

crystal potassium dihydrogen phosphate (KDP).

Frequency mixing

It is also possible to ‘mix’ frequencies when light beams diﬀering in frequency are

made to follow the same path through a non-linear medium. For example, if two

waves with electric ﬁeld components E

sinðo

tÞ and E

sinðo

tÞ follow the same

path through a dielectric with the characteristic

P ¼ e

ðw

1Þ

E þ w

2Þ

Þð8:41Þ

then the second-order contribution is given by

2Þ

sin

ðo

tÞþE

sin

ðo

tÞþ2E

sinðo

tÞsinðo

tÞg ð8:42Þ

The ﬁrst two terms in braces describe SHG and the last term describes waves of

frequency o

þ o

and o

 o

. The output of a wave of frequency o

þ o

known as ‘up-conversion’ and has been used in infrared imaging. For example,

BACKGROUND OPTICS 447

as shown in Fig. 8.7, infrared laser light reﬂected from an object can be mixed

with a suitably chosen infrared reference beam to yield an up-converted

frequency in the visible spectrum. Also, the possibility of obtaining outputs

corresponding to the sums and diﬀerences of the frequencies of two signals opens

up the possibility of processing optical signals in similar ways to electronic

signals. This is the subject of intensive research eﬀort.

8.1.5 Transparent ceramics

For a ceramic to be useful as an electro-optic material it must, of course, be

transparent. Ceramic dielectrics are mostly white and opaque, though thin

sections are usually translucent. This is due to the scattering of incident light.

Scattering occurs because of discontinuities in refractive index which will usually

occur at phase boundaries (including porosity) and, if the major phase itself is

optically anisotropic, at grain boundaries. For transparency, therefore, a ceramic

should consist of a single-phase fully dense material which is cubic or

amorphous; glasses usually fulﬁl these requirements.

It is instructive to examine Rayleigh’s expression

1 þ cos







 n



ð8:43Þ

for the intensity I

of light scattered through an angle y by a dispersion of

particles of radius r and refractive index n

in a matrix of refractive index n

where n

n

is small. The incident intensity is I

, I

is measured at a distance x

from the particles and l is the wavelength of the scattered light (r5l).

It should be noted that the scattering is proportional to (r/l)

and so, for

example, if r/l is reduced from 10

to 10

, the scattering decreases by a factor

448 ELECTRO-OPTIC CERAMICS

Fig. 8.7 Up-conversion of infrared to the visible frequency range.

of 10

. This suggests that pores much smaller than the wavelength of the light will

have only a minor scattering eﬀect. Indeed, any change in refractive index that

only persists for a fraction of a wavelength can be expected to have little eﬀect.

Thus dense ceramics composed of ultraﬁne particles of six 0.1 mm or less are

expected to be transparent to visible light (l(yellow)0.5mm) even if the particles

are optically anisotropic.

Because the scattering is proportional to (n

n

it is expected to fall oﬀ

rapidly as the refractive indices of the two phases become closer or as the

birefringence of a single phase becomes less. However, if birefringence is

introduced into an isotropic ceramic by the Kerr eﬀect this may cause scattering

in an otherwise transparent material.

The position is complicated in ferroelectric materials by the presence of

domain walls which divide the crystals into small regions with diﬀering refractive

indices because of birefringence. Poling reduces the concentration of domain

walls and may thereby reduce scattering in anisotropic ceramics. Crystal size

within a ceramic aﬀects the birefringence. Small crystals (52 mm) are under

greater internal stress than larger crystals because they contain fewer domains

and are less able to adjust to the essentially isotropic cavities in which they are

embedded; consequently their optical anisotropies are reduced. The precise

contributions to scattering from crystal size and domain structure have yet to be

determined. However, it is found in practice that electrically controllable

birefringence can be obtained in ceramics consisting of crystals with sizes below

2 mm and low birefringence, whilst controlled scattering can be obtained in

ceramics with large crystals (45 mm).

8.2 Lanthanum-substituted Lead Zirconate

Titanate

8.2.1 Structure and fabrication

Transparent single-crystal ferroelectrics, such as potassium dihydrogen phos-

phate (KDP), BaTiO

and Gd(MoO

)

, have long been recognized as useful

electro-optic materials. However, the use of single crystals is limited by available

size, cost and, in the case of KDP, susceptibility to moisture attack. In contrast,

electro-optic ceramics do not suﬀer the same limitations, but prior to about 1960

transparent forms were not available. The 1960s saw the development of

processing routes for the production of highly transparent ceramics in the

PbZrO

–PbTiO

–La

(PLZT) system which, as mentioned earlier (see Section

6.3.2), have the facility for changing their polar state under an applied ﬁeld and

have been exploited in a variety of successful electro-optic devices. The PLZT

phase diagram is shown in Fig. 8.8; compositions are deﬁned by y/z/(1z)in

which y is the percentage of Pb sites occupied by La and z/(1z) is the Zr/Ti

ratio.

LANTHANUM-SUBSTITUTED LEAD ZIRCONATE TITANATE 449

Optimum homogeneity is needed in electro-optic materials both to avoid

scattering due to local variations in composition and to ensure uniform electro-

optic properties. Adequate mixing of the more slowly diﬀusing ions such as Zr,

Ti and La is essential, and this calls for special chemical routes for the

preparation of starting powders. As an example, the starting materials can be

zirconium and titanium alkoxides mixed in predetermined ratios as liquids. An

aqueous solution of lanthanum acetate is added and the resulting hydrolysis

precipitates an intimate mixture of the three metal hydroxides. Lead oxide is

milled with the precipitated material and the mixture is calcined at 900 8C.

During the subsequent sintering stage the Pb

ions diﬀuse from the liquid phase

and are distributed uniformly without diﬃculty. The slower-diﬀusing Ti

,Zr

and La

ions are already intimately mixed and have short paths to their ﬁnal

lattice sites.

Pore-free PLZT can be obtained by sintering under atmospheric pressure in

oxygen or by applying a vacuum during the early stages of sintering. It is more

usual to prepare cylindrical blocks of material by hot-pressing in Si

or SiC

dies as shown in Fig. 8.9. In both the pressureless sintering and hot-pressing

routes the sample is surrounded by powders approximating in composition to

that of the sample; the correct design of the powder is particularly critical if good

quality material is to be made by the pressureless sintering route.

PLZT ceramic with a grain size less than 5 mm can be obtained by hot-pressing

at 1200 8C and 40 MPa and holding for about 6 h. The crystals can be increased

in size by a subsequent anneal in an atmosphere containing PbO vapour. Plates

of large area for wide aperture devices require very careful annealing in order to

eliminate variations in birefringence due to minor variations in density without

undue crystal growth. Plates of the required thickness are cut from the hot-

pressed pieces and the major surfaces are lapped and polished. If the plates are to

450 ELECTRO-OPTIC CERAMICS

Fig. 8.8 Phase diagram of Pb

13y/2

(Ti

1z

: A, memory; B, linear; C, quadratic.

be used exploiting a longitudinal electro-optic eﬀect, where the electric ﬁeld and

light are parallel, transparent electrodes of indium tin oxide (ITO) can be applied

to the major surfaces by sputtering and coated with blooming layers to reduce

the loss of light by reﬂection. Where the ﬁeld is to be at right angles to the

direction of light propagation, grooves can be formed in the major surfaces and

gold deposited in them to form an interdigitated electrode system. Blooming

layers are again essential because the refractive indices may be about 2.5, so that

approximately 40% of normally incident light would otherwise be reﬂected.

8.2.2 Measurement of electro-optic properties

In their ferroelectric state, the electro-optically useful PLZT compositions have

an almost cubic structure, with the polar c axis being typically only about 1%

longer than the a axes. Consequently the optical properties are almost isotropic

and this, in part, is why high transparency can be achieved in the ceramic form.

When an electric ﬁeld is applied to the ceramic, domain alignment, or a ﬁeld-

enforced transition to the ferroelectric state, leads to the development of

macroscopic polarization and so to uniaxial optical properties, i.e. the optic axis

LANTHANUM-SUBSTITUTED LEAD ZIRCONATE TITANATE 451

Fig. 8.9 Apparatus for hot-pressing PLZT.