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

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threshold, given that a change in voltage of 1 nV on the pyroelectric element can be

detected. [Answer:  2.8 mWm

]

5. (a) Describe and justify a suitable route for preparing a ceramic for use as a

pyroelectric detector. The description should include co-precipitation, hot pressing,

control of porosity and grain size. (b) Describe a route for the construction of a thin

ﬁlm PST-based pyroelectric array for thermal imaging.

6. Explain concisely the underlying physics of the l/4 radiation absorbing structure

shown in Fig. 7.13.

7. A ceramic pyroelectric material has the formula

PbZr

0.8

0.1

0.02

0.0021

What do each of these cations contribute?

8. What is the objection to using PZT compositions of the type developed for

piezoelectric applications in a pyroelectric infrared detector? What are the properties

needed and how have they been achieved?

9. A pyroelectric ceramic has a tan d of 0.005 arising solely due to dielectric losses

and e

¼ 250 at 100 Hz. What is the minimum resistivity that the ceramic can have

without incurring a 20% increase in tan d. What will be the consequent eﬀect on

the noise-related ﬁgure of merit of a 20% increase? [Answers: 7.2610

O m; 10%

decrease]

Bibliography

1. Whatmore, R.W. (1986) Pyroelectric devices and materials, Rep. Prog. Phys., 49, 1335–

86.

2. Whatmore, R.W. and Watton, R. (2001) Pyroelectric Materials and Devices. In:

Infrared Detectors and Emitters: Materials and Devices , Kluwer Academic, Chapter 5,

pp. 99–148.

3. Smith, R.A., Jones, F.E. and Chasmar, R.P. (1957) The Detection and Measurement of

Infra-red Radiation, Clarendon Press, Oxford.

432 PYROELECTRIC MATERIALS

TEAMFLY

Team-Fly

ELECTRO-OPTIC CERAMICS

The last decade has seen an intensiﬁcation of research and development eﬀort

into ‘photonics’, a term embracing the many related optical technologies

harnessed principally for communications and the processing of information.

Speciﬁcally, it is concerned with generating, amplifying, detecting, guiding,

modulating, or modifying optical radiation and covers a range of interests

including lasers, opto-electronics, ﬁbre-optics, non-linear optics, acousto-optics,

optical data storage and electro-optics. The focus of attention here is mainly on

the electro-optical eﬀects in polar ceramics. However, the last decade has also

seen a strong interest grow in the non-linear properties of poled glasses and the

potential they have for modifying and controlling optical signals. The discussion

includes a brief summary of this development.

The article by G.H. Haertling [1], a pioneer in the ﬁeld of electro-optical

ceramics, and the monographs by J.M. Herbert [2] and F. Aguillo

-Lo

pez et al.

[3], which cover much of the same ground as the present text but in greater detail,

are recommended to supplement the discussion. The classical work by M.E.

Lines and A.M. Glass [4] and the monograph by J.C. Burfoot and G.W. Taylor

[5] are invaluable. The text by R.M. Wood [6] and the review of ‘poled glasses’ by

W. Margulis et al. [7] are especially suited to the ‘materials’ student.

There are many books covering the background optics and that by A. Yariv

and P. Yeh [8] can be recommended.

8.1 Background Optics

To appreciate properly how electro-optic ceramics function, it is ﬁrst necessary

to consider the nature of light and its interaction with dielectrics.

James Clerk Maxwell (1831–1879), against a background of experimental and

theoretical work by Andre

Ampe

re (1775–1836). Karl Gauss (1777–1855) and

Michael Faraday (1791–1867), developed the electromagnetic wave theory of

Electroceramics: Materials, Properties, Applications. 2nd Edition. Edited by A. J. Moulson and J. M. Herbert.

& 2003 John Wiley & Sons, Ltd: ISBN 0 471 49747 9 (hardback) 0 471 49748 7 (paperback)

light. Maxwell’s equations describe how an electromagnetic wave originates from

an accelerating charge and propagates in free space with a speed of

2.998610

. An electromagnetic wave in the visible part of the spectrum

may be emitted when an electron changes its position relative to the rest of an

atom, involving a change in dipole moment. Light can also be emitted from a

single charge moving at high speed under the inﬂuence of a magnetic ﬁeld:

because the charge follows a curved trajectory it is accelerating and, as a

consequence, radiating. The radiation, termed ‘synchrotron radiation’, is emitted

naturally from regions of the Universe, e.g. the Crab Nebula. An electromagnetic

wave in free space comprises an electric ﬁeld E and a magnetic induction ﬁeld B

which vibrate in mutually perpendicular directions in a plane normal to the wave

propagation direction. The E vector of a single sinusoidal plane-polarized wave

propagating in the z direction is shown in Fig. 8.1. The wave is polarized in the

y–z plane, and the convention is to describe the plane to which E is conﬁned as

the plane of polarization.

Radiation from a single atom, say in an electric light bulb ﬁlament, only

persists in phase and polarization for a time of the order of 10

s, so that light

from such sources contains a random mixture of polarizations and phases as well

as a wide range of wavelengths. Such radiation is said to be unpolarized,

incoherent and white, but by various means (particularly the use of laser sources)

light can be obtained with a speciﬁed polarization, a coherence persisting for

times in the neighbourhood of 10

s and a linewidth of less than 10 Hz at a

frequency of 10

Hz. For many purposes the polarization is the simplest feature

to control optically and to alter electrically.

8.1.1 Polarized light

The various forms of polarization can be understood by considering a plane-

polarized wave travelling in the z direction. It can be resolved into two

434 ELECTRO-OPTIC CERAMICS

Fig. 8.1 E vector of a wave polarized in the y–z plane and propagating in the z direction

towards the observer.

components, one polarized in the x–z plane and the other in the y–z plane, as

shown in Fig. 8.2. A phase diﬀerence can be established between the two

components, for instance by passage through a medium with an anisotropic

refractive index such that the velocity of the y–z wave is greater than that of the

x–z wave. This will cause the y–z wave to lead the x–z wave by a distance, say,

Dz, i.e. by a phase angle d ¼ 2pDz=l. d will remain constant when the light

emerges into an isotropic medium such as air.

Conﬁning attention to the electric ﬁelds, we have for the x wave,

¼ E

sinðkz  otÞð8:1Þ

and for the y wave

¼ E

sinðkz  ot þ dÞð8:2Þ

in which E

and E

are the amplitudes, k is the propagation number ðk ¼ 2p=lÞ

and o is the angular frequency. The locus of the tip of the resultant E can now be

determined as the waves pass a plane z ¼ constant. If, for simplicity, we put z ¼ 0

in Eqs (8.1) and (8.2) we obtain

¼E

sinðotÞ¼E

sinðotÞð8:3Þ

and

¼ E

sinðot þ dÞ¼E

sinðot  dÞ

Then

¼E

fsinðotÞcos d  cosðotÞsin dgð8:4Þ

Substituting sin(otÞ¼E

from Eq. (8.3) into (8.4) and rearranging yields

BACKGROUND OPTICS 435

Fig. 8.2 The two components of a right elliptically polarized wave.

 2

cos d ¼ sin

d ð8:5Þ

which represents an ellipse with semiminor and semimajor axes E

and E

respectively. The major axis of the ellipse is inclined to the x axis at an angle a,

where

tanð2aÞ¼

cos d

 E

ð8:6Þ

The above treatment is general, but the following special cases are important.

1. When d ¼ p=2

¼ 1 ð8:7Þ

and the tip of E describes an ellipse whose major and minor axes coincide with

the x and y axes. When E

¼ E

the ellipse becomes a circle.

2. When d ¼ 0

 2

¼ 0

i.e.



¼ 0

ð8:8Þ

The tip of E describes a straight line passing through the origin with slope

The paths described by the tip of E for the various values of d are summarized

in Fig. 8.3. It is apparent that plane-polarized light is a special case of elliptically

polarized light, as indeed is circularly polarized light. The sense of the

436 ELECTRO-OPTIC CERAMICS

polarization is taken to be the sense of rotation of the E vector for oncoming

light. For example (b) is described as ‘right elliptically polarized light’, (f) as

‘left elliptically polarized light’, and (a) and (e) simply as ‘plane-polarized

light’. This seems to be the most commonly adopted convention, although the

opposite one might be encountered which is, in fact, more consistent with

the deﬁnitions regarding the polarization and angular momentum of a

photon.

There are materials, for example in the form of certain specially prepared

polymer ﬁlms, which, for light incident normal to the ﬁlm, absorb to an extent

dependent on the inclination of the plane of polarization to a unique axis in the

plane of the ﬁlm. Devices made from such ﬁlms are termed ‘polarizers’:

approximately 60% of the incident unpolarized light is absorbed, and that part

transmitted is plane polarized. The E vectors for the transmitted light are

perpendicular to the high-absorbance direction. If the incident light is plane

polarized, the intensity transmitted depends on the orientation of the polarizer

axis with respect to the plane of polarization of the light. A device used in this

mode is usually referred to as an ‘analyser’.

8.1.2 Double refraction

In the discussion of the piezoelectric eﬀect in Chapter 6 the tensor character of

the permittivity of a dielectric was recognized although attention was focused on

the piezoelectric coeﬃcients. Because the optical and electro-optical properties of

dielectrics are determined by their refractive indices or, equivalently, by their

permittivities (see Eq. (2.120)), it is now necessary to consider these parameters in

some detail.

In an isotropic dielectric such as a normal glass, the induced electrical

polarization is always parallel to the applied electric ﬁeld and therefore the

susceptibility is a scalar. In general this is not the case in anisotropic dielectrics

when the polarization depends on both the direction and magnitude of the

applied ﬁeld. The three components of the polarization are written

BACKGROUND OPTICS 437

Fig. 8.3 How elliptically polarized light depends on the phase diﬀerence d between plane-

polarized components.

¼ e

ðw

þ w

¼ e

ðw

þ w

¼ e

ðw

þ w

ð8:9Þ

¼ e

þ e

¼ e

þ e

¼ e

þ e

ð8:10Þ

which can be abbreviated to

¼ e

ð8:11Þ

with summation over repeated indices implied.

In an anisotropic dielectric the phase velocity of an electromagnetic wave

generally depends on both its polarization and its direction of propagation. The

solutions to Maxwell’s electromagnetic wave equations for a plane wave show

that it is the vectors D and H which are perpendicular to the wave propagation

direction and that, in general, the direction of energy ﬂow does not coincide with

this.

The classical example of an anisotropic crystal is calcite (CaCO

). This is the

crystal with which the ﬁrst recorded observation of ‘double refraction’ was made

by Bartolinius in 1669. Because of the particular arrangement of atoms (and so

of electric charges) in calcite, light generally propagates at a speed depending on

the orientation of its plane of polarization relative to the crystal structure. For

one particular direction, the optic axis, the speed of propagation is independent

of the orientation of the plane of polarization. Because in calcite there is only one

such axis the crystal is termed uniaxial.

Crystals may have two optic axes, not necessarily perpendicular, in which case

they are termed biaxial. Orthorhombic, monoclinic and triclinic crystals are

biaxial; hexagonal, tetragonal and trigonal crystals are uniaxial; cubic crystals

are isotropic. In the following discussion attention is conﬁned to uniaxial

crystals.

The situation can be analysed more closely by considering a source of

monochromatic light located at S in Fig. 8.4 from which two wavefronts

propagate, one with a spherical surface and the other with an ellipsoidal surface.

A wavefront is the locus of points of equal phase, i.e. the radii (e.g. SO and SE),

which are proportional to the ray speeds and inversely proportional to the

refractive indices. Figure 8.4 is a principal section of the wavefront surfaces

because it contains the optic axis SY.

Those rays, e.g. SO, for which the electric displacement component of the

wave vibrates at right angles to the principal section (indicated by the dots) travel

at a constant speed irrespective of direction; they are the ordinary or o rays.

438 ELECTRO-OPTIC CERAMICS

Those rays, e.g. SE, for which the electric displacement component lies in the

principal section travel at a speed which depends on direction; they are the

extraordinary or e rays. The refractive index n

for an e ray propagating along

SX is one of the two principal refractive indices of a uniaxial crystal; the other,

, refers to the o rays.

In Fig. 8.4 the velocity of the o ray is less than that of the e ray, except for

propagation along the optic axis. It follows that n

 n

50, a situation deﬁned as

negative birefringence. Cases of both positive and negative birefringence occur:

calcite and rutile (TiO

) have values of 70.18 and +0.29 respectively, and both

crystals are very birefringent. Values for the common ferroelectrics lie in the

range 70.01 to 70.1.

It is customary to develop the optical theory of crystals in terms of the relative

impermeability tensor B

¼ðe

1

, which is a second-rank tensor. It can be shown

that B

is symmetrical, a property that requires B

¼ B

. It can also be shown that

because B

is a symmetrical second-rank tensor it can be represented by

¼ 1 ð8:12Þ

which, when expanded, is a second-degree equation representing a surface. The

notation in Eq. (8.12) implies summation over values of j ¼ 1, :::, 3 for each

value of i ¼ 1, :::, 3. The equation is called the ‘representation quadric’ and

possesses principal axes. The representation quadric for the relative imperme-

ability tensor B

, with reference to its principal axes, reduces to

þ B

¼ 1 ð8:13Þ

in which B

etc. are the principal relative impermeabilities. Under the assumption

that the material is non-magnetic (m

¼ 1) (see Eq. 2.120)

¼ðe

1

etc. ð8:14Þ

BACKGROUND OPTICS 439

Fig. 8.4 A principal section of the wavefront surfaces for a uniaxial crystal; the dots on SO

represent the E (and D) vectors which are normal to the plane of the paper.

where n

is the refractive index for light whose electric displacement is parallel to

. Eq. (8.13) can be written

¼ 1 ð8:15Þ

and is known as the optical indicatrix. This is an ellipsoidal surface, as shown in

Fig. 8.5, and n

, n

and n

are the principal refractive indices of the crystal. For

biaxial crystals n

6¼ n

If in Fig. 8.5 OP is an arbitrary direction, the semiminor and semimajor axes

OR and OE of the shaded elliptical section normal to OP are the refractive

indices of the two waves propagated with fronts normal to OP. For each of the

two waves associated with a given wave normal, the electric displacement D

vibrates parallel to the corresponding axis of the elliptical section.

In the special case when Ox

is the propagation direction the two waves have

refractive indices n

and n

; similarly, the two waves propagated along Ox

have

refractive indices n

and n

, and those propagated along Ox

have refractive

indices n

and n

For a uniaxial crystal the indicatrix is symmetrical about the principal

symmetry axis of the crystal – the optic axis.Ifx

is the optic axis, the central

section of the ellipsoid is a circle of radius n

and the equation becomes

¼ 1 ð8:16Þ

8.1.3 The electro-optic e¡ect

In the earlier discussion on dielectrics (see Section 2.7.1) a linear relationship

between P and E was assumed. The justiﬁcation for this assumption rests on

440 ELECTRO-OPTIC CERAMICS

Fig. 8.5 The optical indicatrix.

experiment, and the underlying cause is that the largest ﬁeld strengths commonly

encountered in practice (about 10

) are small compared with those that

bind electrons in atoms (about 10

However, the linear response of a dielectric to an applied ﬁeld is an

approximation; the actual response is non-linear and is of the form indicated in

Fig. 8.6. The electro-optic eﬀect has its origins in this non-linearity, and the very

large electric ﬁelds associated with high-intensity laser light lead to the non-linear

optics technology discussed brieﬂy in Section 8.1.4. Clearly the permittivity

measured for small increments in ﬁeld depends on the biasing ﬁeld E

, from

which it follows that the refractive index also depends on E

. The dependence can

be expressed by the following polynomial:

n ¼ n

þ aE

þ bE

 ð8:17Þ

in which n

is the value measured under zero biasing ﬁeld and a and b are

constants. This dependence of n on E

is the electro-optic eﬀect.

It is evident that if the material has a centre of symmetry or a random

structure as in the case of a normal glass, reversal of E

will have no eﬀect on n.

This requires a to be zero, so that there is only a quadratic dependence of n on E

(and, possibly, a dependence on higher even powers). If, however, the crystal is

non-centrosymmetric, reversal of E

may well aﬀect n and so the linear term has

to be retained.

In 1875 John Kerr carried out experiments on glass and detected electric-ﬁeld-

induced optical anisotropy. A quadratic dependence of n on E

is now known as

the Kerr eﬀect. In 1883 both Wilhelm Ro

ntgen and August Kundt independently

reported a linear electro-optic eﬀect in quartz which was analysed by Pockels in

1893. The linear electro-optical eﬀect is termed the Pockels eﬀect.

The small changes in refractive index caused by the application of an electric

ﬁeld can be described by small changes in the shape, size and orientation of the

BACKGROUND OPTICS 441

Fig. 8.6 Non-linearity in the D versus E relationship.