Ferroelectric Devices (Materials Engineering, 16)

Ceramic n Engineering: Properties, Processing, and Use in Design. d Edition, Revised and Expanded, DavidW. rich er so^ 2. ~ntro~uction to Engineering Materials: Behavior, Properties, and Selection, G. 1.

olidifiedAlloys:Processes

*

~ t r u c t ~ r e sApplications, e d ~ e dby 0

inforced Ceramics for Structural Applications,

~avjd

5. Thermal Analysis of Ceramics, robe^ F. S ~ e y e r tion and Wear of Ceramics, e d ~ e dby Said~ a h a n ~ j r hanical Properties of Metallic Composites, edjfed Sby~ o ~ jOchjaj ro 8. Chemical Processing of Ceramics,e d ~ e d by B ~ ~ r a 1. n Lee d and ~ ~ ~J. a A. r d Pope 9. HandbookofAdvancedMaterialsTesting, e d ~ e dby ~ i c ~ o / a P,s C ~ e r e ~ j s j n o and ~ Paul~.C h e r e ~ j s i n o ~ I O . Ceramic Processing and Sintering,M. N. R a h ~ ~ a n 11. Composites Engineering Handbook,~ d ~ by e P. d K. 12. Porosity of Ceramics, RoyW. Rice 13. Intermetallic and Ceramic Coatings, e d ~ e dby ~afendraB. aho of re and 7: S. on Techniques: Technological Applications, e d ~ e dby K. 6. eering Materials: Impact, Reliabili~,and Control, e d ~ e dby

International Centerfor Actuat~rsan^ Trans~ucers~ I C T ~ Pennsylvania e State ~niversity ~ , ylvani~ ~niversityP ~ r Penns

M A R C E L

D E K K E R

~

~

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Ferroelectrics can be utilized in various devices such~ as g h - p e ~ ~ i vdielectrics, ity pyroelectric sensors, piezoelectric devices, electrooptic devices and PTC components.Theindustriesareproducinglargeamountsofsimpledevices,e.g. ceramic capacitors, piezoelectric igniters, buzzers and PTC t h e ~ s t o r continuously. s But until now ferroelectric devices have failed to reach co~ercializationin more functionalcases.Inthelightsensor,forexample,semiconductivematerialsare superior to ferroelectrics in response speed and sensitivity. Magnetic devices are much more popular in the memory field, and liquid crystals are typically used for optical displays. Ferroelectric devices often fail to be developed in the cases where competitive materials exist. This is mainly due to a lack of systematic a c c ~ u l a t i o n of hndamental knowledge of the materials and developmental experiences on the devices. Duringmy12-yearteaching periodon"FerroelectricDevices," I foundthatno suitable textbook is available in this particular field, except some professional books I decidedtowriteasingle-authored likemulti-authorpapercollections.Hence, textbook based on my lecture notes, including my device development philos~phy. Thistextbookintroducesthetheoreticalbackgroundofferroelectricdevices, practical materials, device designs, drivelcontrol techniques and typical applications, andlooksforwardtothe hture progressinthisfield.Thoughthediscoveryof ferroelectricity i s relativelyold,sincethedevicedevelopmentisreallynewand interdisciplinary, it is probably impossible to cover all the recent studies in a limitedpage book. Therefore, I selected only important and basic ideas to understand how todesignanddeveloptheferroelectricdevices,puttingaparticularfocuson thidthick film devices. Letme introducethecontents.Chapter1introducestheoverallbackground, "General view of ferroelectrics," followed by the theoretical background in Chapter 2, "Mat~ematicaltreatmentofferroelectrics."Chapter 3, "Devicedesigningand fabricationprocesses,"providespracticaldesigningandmanufacturingofthe devices.CapacitorapplicationsaredescribedinChapter 4, "Highpermittivity devices,"Chapters 5 and 6 treat thidthick filmapplications,i.e."Ferroelectric memory devices" and "Pyroelectric devices," respectively. Chapter 7, "Piezoelectric devices" deals with piezoelectric actuators and ultrasonic motors as well as acoustic transducers and piezoelectric sensors. Optical devices such as light valves, displays, wave guides and bulk photovoltaic devices are described in Chapter 8, "Electrooptic devices." In Chapters 9 and 10, we learn basic concepts of "PTC materials" and

...

111

iv

Preface

"Compositematerials,"andtheirdeviceapplications.FinallyinChapter11we discuss "Future of ferroelectric devices," in which the rnarket size is estimated, and the author's strategy for developing bestseller devices is introduced. This textbook was written for graduate students and industry engineers s ~ d or~ g working in the fields of electronic materials, optical materials and c o ~ ~ c a t i o n s , this text is designed for a course with precision machinery and robotics. Though thirty 75-~ninutelectures, the reader can learn the content by himselflherself aided by the availability of examples and problems. Critical review and content corrections on this book are highly appreciated. Send the i ~ f o ~ t i directed on to Kenji Uchino at 134 Materials Research Laboratory, The Pennsylva~aStateUniversity,UniversityPark,PA16802-4800.Fax:814-8652326, E-mail: KenjiUchino~PSU.EDU For the reader who needs detailed i n f o ~ t i o non smart piezoelectric actuators and sensors, "Piezoelectric Actuators and Ultrasonic Motors" (349 pages) authored byK. Uchino, (Kluwer Adademic Publishers 1997) recomended. is EventhoughIamthesoleauthorofthisbook,itneverthelessincludesthe contributions ofmany others. I express my gratitude tomy ICAT center faculty who have generously given me their advice and help during the writing, particularly to Dr. U m a ~ ~ e l e g ~who d u ,worked out all the problems. Dr. Yulcio Ito (now in Rutgers University)allowedmetousesomeparagraphsandfiguresfromourcoauthored papers. Specific acknowledgement is given to Professor J a p e Giniewicz, Indiana Universi~of Pennsylvania, who reviewed and criticized the entire manuscript and provided linguistic corrections. Kenj i Uchino

iii vii

viii

ix

1.2 1.3 1.4 1 .S 1.6

Origin of ~pontaneousPolarization Origin of Field ~nducedStrain Electrooptic EBect Example of Ferroelectrics 18 Applications of Fe~oelectrics

L

2.1 Tensor Representation of Physical Properties ~ h e n o2~. 2e n o l o ~ y of Ferroelectricity

Resigning3.1 3.2 Resigning 3.3 3.4 3.5 105

Capacitors 4.1 Capacitors 4.2 4.3 4.4

Material Fa~ricationProcesses of Ceramics Device Size Grain Effect on Ferroelec~icity Ferroelectric Domain ~ontributions Ceramic Chip Hybrid Substrate Relaxor F e ~ o e l ~ t r i c s

1 2 4 9 13

20 23 38

57 67 73 84 89

106 108 108 119 126 V

Contents

.2

6.3

T e m p e r a t ~ e ~ n ~Light a r e ~Sensors Infrared Image Sensors

131 138 139 145

iezoelectric Vib

158

161 174 176 180 197

221 222

230 239 243 248 250

10.2

CompositeEffects

255 257 260 269 275 276 279 283

3~~

ion emanent ~ o l ~ z a t i o n yroelectric coefficient Lorentz factor elative ~ e ~ i t t i v i tdielectric y, constant tran§ition t~mPerature) Strain Spontaneous strain Stress

Electro§tric~vecoefficients mechanical co~plingfactor tran§mi§§ion coefficient ive index lmary electrooptic coefficient Secondary electrooptic coefficient hase ret~dation

Q. 1.

2.

3.

4. 5.

6, 7.

8. 9. 1Q. 11.

viii

Course Explanation & Prerequisite Knowledge Check General Viewof Ferroelectrics ~ a ~ e m a t i cTreatment al of Ferroelectrics Device Designing and Fabrication Processes High ~ e ~ i t t i v iDielectrics ty Ferroelectric Memory Devices Pyroelectric Devices Piezoelectric Devices Electrooptic Devices PTC Materials Composite Materials Future of Ferroelectric Devices ~ e v i e w l ~ ~ ~

1 Time 4 Times 4 Times 3 Times 2 Times 1 Time 1 Time 7 Times 2 Times 1 Time: 2 Times 1 Time 1 Time

In order to understand ferroelectric devices, some prerequisite knowledge is expected. to solve the following questions without seeing the answers on the next page. Q1 Q2

Q3 Q4 Q5

6 Q7 Q8

escribe the definitions of elastic stifness c and c o ~ ~ l i u n S, c eusing a stress X - strain x relation. Indicate a shear stress onthefollowingsquare.

~ a material with mass density p and elastic ~escribethe s~~~ v ~ l o cvi in compliance SE. Calculatethe capaci~nceC of a capacitor with area S and electrode gapt filled with a material of relutive ~ e r ~ ~ t Et.i v i ~ Calculatethe ~ ~ l u r i z a t i o nof a material with dipoledensity dipole momentqu (Gm). escribe the C ~ r i e - ~ law e ~ of ~ srelative p e ~ i t t i v i t yE, using a C ~ r i e Weiss temperature To and a Cu~e-Weissconstant C. Describethelightvelocityin a material with a refractive index n (c: light velocity in vacuum). Indicate the work function in the following energy band of a metal. eve1

In

of

Q9 Q10

id0

There is a voltagesupplywith an internal impedance 20.Indicate the external impedance21to obtain the maximum output power. Calculate the induced polarization P under an external stress X in a ~iezoelect~ with c a piezoelectric constantd.

ix

Prerequisite Knowledge

X

(Correct rate moreth 1

2

l

,~ e ~ o e ~ e care ~ isaid c s to be very

T

riC

C

1

Chapter 1

2

Ferroel~tricsareutilizedinvariousdevicessuchas h i g h - p e ~ i t t i v idielectrics, ~ pyroelectricsensors,piezoelectricdevices,electroopticdevices, and PTC (positive ~ m p e r a t ~ ec o e ~ c i e n toresistivity) f components.owever,ferroelectricdevices often fail to be c o ~ e r cini areas ~ ~of applicatio competitive materials exist.Lightsensors,€orexample,typically are manufac~edfrom se~conductive materialswhich are superiortoferroelectricsinresponse agnetic devices are much more popular formemo^ applicati are typicallyusedforopticaldisplays.Onereasonforthis is due tothe lack of systematic andcomprehensive com~ilationof knowledgeonferroelectricmaterials. In this chapter, we will learn~ n d ~ eknowledge n t ~ on f e r r o e l e c ~ c i ~ '

lectric materials, the constituent atoms are considered to be ionized In such i andareeitherpositivelyornegativelycharged. electricfield is applied,cationsareattractedtothecathode anions to the anode due to eiectros~ticinteraction. The electron clouds also deform, causingelectricdipoles. This phenomenon is own as electric po~rizationof the electric, andthepolarization is expressed ~ u ~ t i ~ t i vasethe l ysum of the electric 1.1 showsschematicallytheorigin of the lpolesperunitvolume[C/m2].Figure are threeprimary con~ibutions:electronic, ionic and electricpolarization.There ole r e o r i e n ~ ~ o n - ~The ~degree ~ t eto which ~ . eac 'sm c o n ~ i b ~ tto ethe s overall ~olarizationof thematerial d e ~ on nthe ~ ~lectronicpolarization can follow alte~atingfields W ond,higherthan visiblelight wave)and ionic polarization z (109-101~ cyclelsec, microwave region). Thus, you should relation between the relativepe~ittivityE and the refractive =O

l l

ti0

+

+

:

c

l

l

4 I)

l

f

l

l

origins of the electric polarization.

+

eneral View of Ferroel~~tflcs

3

is valid only when the applied electric field has a higher. ~ermanentdipole reo~en~tion can follow cyclelsec). "his is why fe~oelectricmaterials with permane for ~ c r o w a v edielectricmaterials;theirpermittivitiesaretypicallyhighatlow signi~cantlywithincreasingappliedelectricfield frequency. Compared with air-filled capacitors, dielectric capacitors can store more electric charge duetothedielectric pol~zation as shown in Fig. 1.2. Thephysical ~uantity comspo o the stored electric charge t called area isthe elec~*c displace ,and related is to the electric fi the following expression: *

Here, ;EO is the vacuum permittivity (= 8 . 8 5 4 ~ 1 0F/m), ' ~ ~ E is the material's reiiztive p e r ~ i t t i v(also i ~ simply called permittivityor d~electric consta~t, and in general is a tensor property).

Dependingonthe crystal structure, the centers of the positive and negative charges maynotcoincideevenwithouttheapplicationofanexternalelectric field.Such o n . the spon~neous crystalsare said to possessa s p o n ~ n e o ~ p o l a ~ z a ~When an electricfield, it is d e d polarization of thedielectriccanbereversedby

ferroelectric.

ob: Bound charge ot: True charge

Charge accumulation in a dielectric capacitor.

mom~ntsresult from th

(electric c h ~ 4) ~ relative e tothecrystal lattice. ~onsiderthecaseinwhichthe ~ o l ~ ~ a tisi caused o n by all the ions being displace^ equally in a lattice. ugh lattice vibrations at a finite eigen lattice vibrationsina

becomes zero. tfollowsthat,atany

indivi~ual ion site, there exists alocalfieldfrom ,even if there is no external field. own schematically in Fig. 1.4. It can be shown that:

(c) ossible eigen lattice vibration modes in a ~erovskitecrystal.

6

Chapter l

\ \

External Field Eo

Dielectric material

of the local field. Eloc is givenby loC= EO + X [3(pi*ri)ri $pi] / &Eo ri5.

-

i

This local field is thedrivingforcefortheionshift.Here factor. For an isotropiccubicsystem,it is known that

y is calledthe brentz y = 1 . l ) EO is the

pe~ittivity of vacuumand is equal to 8 . 8 5 4 ~ 1 0 F/m. " ~ ~ If the ionic ~oZarizabiZi~ of ion A is a,then the dipole momentof the unit cell of this crystal is:

The energyof this dipole moment (dipole-dipole coupling)is

Defining N to be the numberof atoms per unit volume:

F u ~ e ~ o rwhen e , the A ions are displacedfromtheirnonpolarequilibrium If thedisplacement is U, andthe fm positions, the elastic energy also increases. constants k and k', thentheincrease of the elastic energyperunitvolumecan be expressed as:

Here, k' (> 0) is thehigher-orderformconstant. It shouldbenotedthatin pyroelectrics, k' plays an important role in determining the magnitude of the dipole moment. B y rewriting Eq. (1.7) using:

General View of Ferroelectrics

7

where q is the electric charge, and combining withl3q. (1.6), the total energy can be expressed as follows (see Fig.1.5):

From this, one can see that if the coefficient of the, harmonic term of the elastic energy is equal to or greater than the coefficient of the dipole-dipole coupling, then = 0; the A ions are stable andremainat the -polar e~uilibriumpositions. = [(2Nay2/9&o2) oNqIL)]/ Otherwise, a shift fromtheequilibriumposition

-

[k'/N3q4]) is stable. Spontaneous polarization can occur more easily in perovskite type crystal structure (e.g. barium titanate) duea higher to value of Lorenz factory (= thanfoundforother crystalstructures.Notealsothatthepolarizability is sensitive to temperat~e,leadingtothephasetransition.Supposethattheionic polarizability of ion A,a, increases with decreasing temperature, evenif [(W2Nq2) (Nay2/9q2)] > 0 ~ ~ l e c t r i c at ! ) a hightemperature,thisvaluemaybecome negativewithdecreasingtemperature,leadingto a ferroelectricphase transition. Considering a first approximation, a linear relation of the a with temperature, the urie-Weiss law is derived, which will be discussed in detail in Section 2.2( 1).

-

Dipole (a) interaction

(b) Elastic energy

)W

+ (~/4N3q4)P4

Welas =:(~2Nq2)P2

(c) Total energy wtot

=wdip + welas

Fig. 1.5 Energy explanation of the origin of spontaneous polarization.

i 0 3 exhibits ionic ~ s p l a c e m e nas~ i l l u s ~ ain t ~Fig. 1.6atroom culate the magnitudeof the spontaneouspol~zation. = 4.036 A and a = 3.992 A.

C

a

:number of the dipole

c ~ c u l by a ~taking e product of the c ~ ~ magnitude g e total dipolemomentin a unit cell is c ~ c u ~by a tsumming ~ dipoles (notice the ~actionalcontri i. e, l/$for corner atoms and 1/2 for face-centered atoms); S

)(0.061x10-10m~+ 4e(0.12~10-~

e unit cell volume is given

9

v = a2c = (3.992)2(~.036)x

m3

(P1 1.2) *

e spon~neousp o l ~ z a ~ is o ndefined as the pol~zation(total dipole moment) unit volume:

lomz9Cm(64.3 x

m3

(P1.1.3)

The e x ~ e r i m evalue ~ ~ l of PS is about 0.25 C/m2.

"eZectro~t~ctio~" is usedin a gener~sense to train, and hence ~ ~ u e nalso implies ~ y the "converse o ~ e v e rin , solid state theory, the converse p i e z ~ l e c ~ c e E i t 'cal coupling effect, that e l ~ ~ o s ~ c tisi ao nsec

material is a mono-

electric fieldand the anions in the opposite direction, leading to the relative change in of theelectricfield,the soft theinter-ionicdistance.Dependingonthedirection springexpandsorcontractsmorethanthecontractionorexpansion of the had spring, causing a strain x (a unit cell length change) in propo~onto the electric field eflect. When expressed as E. "his is the converse ~iezoezecEr~c X

= dE,

(1 .lo)

the propo~onalityconstant d is called thepiezoelec~cconstant. Ontheotherhand,in Fig.l.7(b), theamounts of extensionand contractio~of the springarenearlythesame,andthedistancebetweenthetwocations(lattice parameter)remainsalmostthesame,hence,there is nostrain. precisely, ions are not connected by such idealized springs (those are cded harmonic springs, in which force (F) = spring constant (k) x displacement (A) holds). In most u ~ ~ ~ ~(F o= k1A ~ i c k2A2>, i ~ that is, they m cases, thespringspossess somewhateasytoextendbuthardtocontract.Suchsubtle ~ e ~ n c in ethes displace~entcauses a changeinthe lattice parameter,producing a strainwhich is inde~ndentof the direction of the applied electric field (+E or -E), andhence is an even-function of the electric field. This is called the ezectrosEric~veefect, and can be expressed as

-

(1.11)

is the electrostrictive constant.

E

+

(a)P I ~ % ~ i ~Strain ctric

l

,-

C

(b) E l ~ t r o s t ~ ~ t i o n

.7 ~icroscopicexplanation of the piezostriction and electrostriction.

11

in Fig. 1.7(a) also possesses a s p o n ~ e ~ bias us dipolemoment.Thetotaldipolemoment puza~~atiun. When a large reverse bias elec s p o . ~ ~ e o u s p o l ~in ~ tai particular on polar mother stable crystal state in which are re (In terms of an u n ~ n n e dsingle the ions 180" aboutan axis ~ ~ nto ~ ntial double minima in Fig. 1.S.) ,also causesa remarkable change in ' c smen~oned , to a s ~ e ~ u e Z e c ~ as in Section 1.1. Generally,whatisactuallyobserved .as a field-indu~strainis a com~licatedcombination of the three basic effects just descri~. Figure 1.8 shows typical strain curves for a piezoelectric lead zirconate titanate ( strictive lead magnesium niobated (PMN) based c ~ ~ c . ~ ) in PZT becomes distorted md shows large hys~resis field level, which is duetothe p o l ~ ~ t i oreorientation. n doesnot exhibithysteresis under an electricfieldcycle. relation (E2) at a high e field level. theconverse

iez~lectriceffectabove.Then,what is thenormalor phenomenonwherebycharge (Coulo~b alstress(forceperunitarea).Note the same piezoelectric coefficientd is used as used in Eq.(1.lo), in the relation (1.12)

P=dX,

Ei6ctric field

(kV/cm)

Electric fieM (kV/cm)

Typical strain curves for a piezoelectric zirconate titanate (W based (a) and an el~~ostrictive lead ma~nesiumniobate (P"?) based ceramic (b).

c

~

l

1

ne of the lead zirconate tita~ate( 590 X C/N with adielectriccon§tant e3 = 3400 andan elastic com~liances33

)

calculateindu the

will in~oducethetensor ~ u ~ ectin~the su~§cript§ at

~

~

e

s

x=dE =d

~ n d e ar completely clamp 3 = x3/ s33 = 5.9 x

/2QX

= 3.0 x 1Q7N/m2

3 = P3/WE

= (1.77 X

= 5.9 x 105 Vlm

m2

c/m2)/(3400X 8.854 X I

13

en electric energy is supplied toa piezoelectric sample and some e l e c t r o ~ e c ~ ~ iccoaul ~ l ~ ~ ~ ~ a c t o r ener~y)/(~nput electrical energy).

= (1/2)(x2/ S)/( 1/2)( = d2 / S € 0 ~

,hen mechanical energy is suppli put mechanical energy).

0 x 10-l2 m 2 ~ ) ( 3 4 0 0x 8.854 x

F/m) (P1 -2.10)

(b) is about k2 of the 10 x 105 ~ansd~ction ratio accomp

(1.13)

ion isp placement is entlythe r e ~ ~ c ~ v e

14 Generally, the refractive indexis treated as a symme~icals e c o n d - r ~tensor quantity and is represented geometrically by the opticali ~ i c a which t ~ is described by

(1.14) where "1, n2 and n3 are theprincipalrefractiveindices.ththe appli~ationof an electric field,the change in refractive indexis given by an expansion expression:

ere n(E) and n(0) (no) are the refractive indices atE and zero rijk is the p r i electrooptic ~ ~ c o e ~ c i e n(t ~ o c ~ ee#e~t) l s and c ~ ~ c i e(Kerr n t e#&). ons side ring the paraelectric phase of a perovskite crystal (m3m) as an example, the

rr coefficients are represented in the following matrix: 1 1 R12 R12 0 0

0 12 R11 R12 0 0 0

0 0

0 0

0 0 0 0

0

so that the refractive indicatrix under an electric field applied along the z direction expressed as: x2 + y2 22 + = 1 . (1.16) no2(1 ( n 0 ~ / 2 ) R l 2 E ~ ~no2( ) ~ 1 (no2/2)Rl1E22)2

-

is

-

The refkactive index change under an external electric field is explained intuitively as en an electric field Ez is applied to a cubic perovskitecrystal, the crystal is elongated along the z-axis and contracted along both ~onse~uently, thematerial'sdensity or compac~esswill be axis and densified along the x and y axes, leading to a decrease nz andanincrease of theindices nx and ny . (Note thattherefkactiveindex is or ion compactness the polarized light propo~ionalto the electron density electricfielddirectionwhich is pe~endiculartothelightpron ~rection.) en light is ~ansmittedalong the y direction, the phase re~dationrYbetween the o r ~ andi e ~~ r~ a o r waves ~ i ~is given ~ by

i

15

+l

Optical phase retardation through an electrooptic crystal. Notice the crossed polarizerc o n ~ g ~ a t i o n .

where d is the electrode gap and L is the optical path length (See Fig. 1.9). Placing the crystal between crossed polarizers m g e d at a 45O angle with respect to the zaxis,theoutputlightintensityis modulatedas afunction ofappliedvoltage according to:

This is the p~ncipleb e ~ n dthe operation of a light shutterlvalve, and the voltage required for the first i~tensitymaximum (i.e., ry= E ) is an impo~ant ch~acte~stic called theha~-wave vo~ta~e.

ZT) samplewitha

r e c ~ g u lshape ~ (optical at 45" with respect to to the samplewith an dentintensity:IQ) is lightintensity I F z ) by ne~lectin are listed below:

16 refractive index at E = 0 :no electrooptic Ken coefficients : phase retardation:ry reflectance at the crystal surfac int into an ellipsoidalone under an

The initial sphericalindicatrixwillbedeformed applied electric field EZ: x2

+ y2

-

22

+

-

= 1. (P1.3.1)

no2( 1 ( n 0 ~ / 2 ) R l 2 E ~ ~no2( ) ~ 1 (no2/2)Rl 1Ez2)2 The output light intensity is reduced twice, once at the inlet and once again at the outlet crystal surfaces by a factor of (1 -Re)2, Also you shouldnoticethattheincident light (afterpassing ~ icomponen~ ~ ~ of qual mag~tudes. polarizer) has or^^^^ and e x ~ u o rlight

In a cubic s ~ c t u r ethe , refractive index change under an external electric field along z-axis is expressed by the following two equations:

l/nx2(Ez) - Uno2 =

l/nZ2(Ez) Uno2 = R1 1Ez2,

(P1.3.2)

-

Taking into account the relation, d(l/n2) = (2/n3) dn, (P 1.3.4) (P1.3.5) 11 >0 and R12 c0 in most cases. Sincethewavelengths

ofthe

e x ~ a o r ~@oldzed n ~ alongz-direction)andthe

or din^ (polarized dong x) waves are describedas

(P1.3.6) (P1.3.7) where

is thevacuum wavelengt~oftheincident light, and numbersofwaves in the crystal with an optical ~ a t h ~ n g tof h L L& and Llhx, ( ~ e ~ rY) u is~~ i v~e n by ~ ~ o ~ respectively, the phase difference between these waves

exist in^

the linearly polarized light incident on the electric field vectoras

PLZT in terns of its

(P1.3.9)

the output light fiom the

ZT can be described as in[(2n/?q))y -cot + (p]

-

sin[(2n/?q))y cot + (pat the ex 142 . .eZ142 = (1

, (P1.3.10)

- 45O orientation, the electric field

-

-

-

[sin[@ /ho)y.cot + (p] sin[(2 Iho)y a t + (p

comp~nent

-

lntensi~

A ~ ~ l i Voltage ed

voltage.

ariat ti on inthelightintensity

of an electroopticshutterwithapp

Chapter 1

18 Thus, the output intensity through the 2nd polarizer is obtained:

-

I = (112) (1 -Re)2 (Io 12) [(1 cosry)2 e (sinr'y)2] 4 1 2 ) IO(1 -cosry)

(P1.3.12)

Figure l. 10 shows the output intensity I as a function of applied voltage Vz. The ~ a ~ - w a voltage, ve which is &findas the minimum voltage required to produce the first m ~ i m u min the transmitted light intensity, is given by

A typical ceramic ferroelectric is barium titanate, which is used here as an example to illustrate somepropertiesofferroelectrics. As showninFig.1.11,BaTiQ3has a polar perovskite crystal structure. In the high temperature paraelectric phase (non phase) there is no spontaneous polarization (the symmetry is Q, -m3m). Below the

-

ans sit ion temperature TC called the Curie t e ~ p e r ~ (about ~ r e 13OoC), spontaneous polarizationoccurs, and the crystal s ~ c becomes ~ e slightlyelongated, that is, tetragonal (C4v - 4mm). Figure 1.12 shows schematically the tem~rature PSand pe~ittivityE. PS decreases with dependence of the spon~neous pol~zation increasing temperature and vanishes at the Curie temperature, while E tends to diverge 1 1known ~ to be linear with near Tc. Also, the reciprocal (relative) p e ~ t t i v i t y is Curierespect to the temperature over a wide range in the paraelectric phase (so-called Weiss law), E=C/(T-To),

(1.19)

where C is the C u r i e - ~ e i s s ~ ~ n s and t a n TO t is the C u r i e - ~ e i s s t e ~ p e r a ~TO r e .is slightly lower than the exact transition temperature Tc. It is also known that the spontaneous pol~zationPS and the spontaneous strain xs follow the relationship

and xs decreases almost linearly with increasing temperature. In the case of it exhibits the piezoelectriceffect in the ferroelectric phase, while in the p ~ e l ~ t r i c phase,it is non-piezoel~tricand exhibits onlythe el~trostrictive effect.With d ~ r e a s i n g ~ m p e rfrom a t ~ eroom tempera~e,however, barium titanate undergoes a series of complicatedphasetransitions.Figure1.13illustratesthesesuccessive phase ~ansitions.

General View of Ferroelectrics

A

TC :Curie temperature

Crystal structures of BaTi03.

(a) Capacitor

Te~per~ture

(d) Piezoelectric ~ ~ s d u c e rElectrostrictor (e) (f) Electrooptic device

Temperat~ede~ndence of the spontaneous polarization pe~ittivityin a ferroelectricmaterial.(a) (f) indicatethetemperaturerangesfor eachapplication. In otherwords, if wecanshiftsuchtemperaturerangecloserto room temperature,a practical materialis obtained.

-

Rhombohe~~

0 150

"

-100

-so

0

50

100

Tempe~ture (5)

V ~ i o phase ~ s transitions in b

~ titanate. i ~

~

150

1

1.1

1.

22

= Q I ~ P and ~ refkactive ~ , index changes An3 = - (112) no3gl 1P32 andAn 1= - (112) no3g 1 2 P 3 ~ . ~ x p e ~ m values e n ~of these are: Q1 1 = andx1

0.090 m 4 C 2 , 4 1 2

= - 0.035 m 4 C 2 ;

g1 1

= 0.136 m 4 C 2 ,

g12

= - 0.038

m4C2. Co~paringthe absolute valuesbetween Q and g and the ratios 12 and g1 1412, discuss s i ~ l ~ t i in e s terns of the crystal lattice

compac~essalong and pe~endicularto the electric field.

1)

2) 3) 4)

C. Kittel: In~oductiontoSolidStatePhysics6thedition,

Chap.13, John Wiley Sons, New York (1986) .Kinase, U. Ukmura and M.Kikuchi: J. Phys. .Uchino and S. Nomura: Bull. Jpn. Appl. Phy omura, L.E. Cross, R. E. Newnham and S. J. Perovskites and ItsTransducerApplications, J. Mater. Sci., 16, 569 (1981). trostrictive Actuators: Materials and Applications, Bull. Amer. No.4, 647 (1986).

Physicists usually treat a natural p~enomenonusing .a simple mathematical form: one is a linear approximation and anotheris a non-linear expansion theory. Hooke's law, the stress- strain relationand Ohm's law, the voltage current relation a m two of the most famous linear laws in physics. These linear relations are extended into matrix or tensor relations in linear algebra. Onthe other hand, the Maclaurin or Taylor series are popularly used to calculate slightly perturbed physical quanti~es aroundan equilibrium state inclu~ngnon-lineareffects.Inthischapter,wewill considerthe tensorrepresentation ofphysicalproperties(linear relation) pheno~enologyof ferr~lectricity(non-linear relation).

-

Let us fiist consider the tensor for electric conductivity. The conductivity is &fined so astocorrelateanappliedelectricfieldandtheinducedcurrentdensity follows:

Sinceboththeelectricfieldthecurrentdensityare fiist ranktensor(that is, vector) ~ u a n ~ t i ethe s , conductivity should have a second rank tensor representation (that is, with two suffixes); this is described as

(2.3)

e x e ~ ~ l ibyf i~iezoelectric ~ coe~cients, providin~ a relatio~ ~etweenthe applie~field and the induced strain 23

are f~st-rankand second-r& tensors, respectively, the d should have a~ d - r a n tensor k form represent^ as Ei

Xjk

= L= dijk

(2.5)

i

The d tensor is composed of three layersof the symme~icalmatrices.

layer1st

dl11 dl12 dl21 dl23 dl22

(i = 1)

2nd layer (i= 2)

3rd layer (i= 3)

131

dl32

211

d221 d231

d21 d212 d222 d232

311 d321 d331

d312 d322 d332

dl1 d13

d223 d23 d31 d32 d33

Generally speaking, if two physical properties are represented using tensors of prank and q-rank, the quantity which combines the two properties linear in a relation is also c 4)-rank. represented by a tensor of (p

ryst

etry

A physical property me as^^ along two different dir~tionsmust be equal if these ally is consideration sometimes two directions are c ~ s ~ l l o ~ a p ~ cequivalent. reduces the number of i n d e ~ ~ d e ntensor t componentsrepresenting the above property. - r ~ If the Let us again take electric conductivity as an example of a s ~ o n ~tensor. as in an (x,y,z) coordinate system is described in an (x',y',z') system J', Jand J' are related using a unitarym a t r i ~ as~follows:

e electric field is~ ~ s € oin~thee same d way:

3

at~~~atical T r ~ a t o~f ~ ~ t

or

Then, we canc~culatethe co~espondinga' tensor defined by

(2.10)

#A

13

a3

012

012

01

21

a22

023

a31

032

03

all a12 a13

unit^ matrix without ani m a g i n part ~ has

a32

a21 a22 a23

the following relation:

31

For c e n ~ o - s y ~the e ~ a, n s f o ~ a t i matrix on is written as 0 -1 Q

0

Q

-1

and for rotation about a principal axis,

a31 a32 a33

26

Chapter 2

or 0'"

-

aikajl a k l

(2.12)

U

When the crystal has a 2-fold axis along the zLaxis, the electric conductivity should have the same tensor form in terms of the ans sf or mat ion:

0 -1 0

0 0 1

From the condition 0 -1 0

0 0 1

0 -1 0

0 0 1

(2.13)

the following equivalencies can be derived: 031

= 0 1 3 = G32 = 0 2 3 = 0

0229 0 3 3 0 0 1 2 = 021

(2.14)

1 1 9

It is very impo~anttonotethatmost physic^ c o n s t ~ t s form. [The proof involves t h e r m o d y n ~ cconsiderations ~

a s y ~ e t r i ctensor beyond the sco

tric tensor, the ~ a n s f o ~ a t due i o ~to a

(2.15)

~ t ~ e r n ~Tre~trnent t i c ~ ~ of Ferro~lectrics W e n thecrystalhasa4-foldaxisalongz-axis, matrix is given by 1 0 0

27 for example,the t r a n s f o ~ a ~ o n

0 0 1

Conside~ngthe tensor s y m m e ~with m and n such that dl23 = dl32 and d213 = d231 (each matrixof the ith layer of the d tensoris symmetrical), we can obtain: dl 11 = d222 = d l 12 = dl21 = d211= d221=d212 = dl22 = d33 1= d313= d l 33 = d332 = d323 = d233 = d312 = d321= 0 d333 = 0 d311 =d322 d113=d131=d223=d232 dl23 =dl32= -d213 = -d231

(2.16)

Then we getthe d tensor as follows: 1st lay

131

0 0

dl 3 dl2

dl23

0

2nd la er

0 0 -dl23

(2.17)

dl31

0

0 d311 0

0 0 d33

3rd layer 311

A. general ~ d - r tensor a ~ has 33 = 27 inde~ndentcomponents. Since dijk is s y m m e ~ i cin~ j and k some ofthecoefficientscanbeeliminated,leaving18 independent dijk coefficients; this facilitates the use of matrix notation.

28 the num~erof suffixes as

,for instance, d21 = by a single suffix 1 to 6 in matrix notation, as follows:

S

of these new symbols themay (2.6) is rewritten as:

e last twosuffixesinthetensornotation c o ~ e s ~ o ntod ~ o ~ ~ o n e n eref t s ; fore, forconsistency,wemake ~otationfor the s ~ com~onents. ~ n

i (i =: I, 2, 3; j

or

= 1, 2, ...,6)

nk

electrics

(2.21)

onents, the(1/2)s are ~ n n e c e s s ~ . 1

(2.22)

6 5

Themarixnotationhasvantageofcompactnessover makes it easytodispl c ~ ~ c i on eaplane n ~ di remem~~red thatin sp irform,thedij'sdonot of a second-rank tensor. An example of a piezoelectric matrix for the point group 4 is written as Q Q

d33

d31

d31

0 Q

15

0

-dl4

0

0

0

theoretical ~ e a ~ e ofn tthe pheno~enonof strain xkl is expressed in termsof the electric

(2.23)

'

(2.24)

coe and

ere, diH and giM are called the piezo ectric electrostrictive respectively, d

coef~cients,and and

ijkl the

tensors,

Using a similar reduction of the notation for the elec~ostrictivec ~ f ~ ~ i e n ~ we get the followingequatio~ ~o~esponding to Eq.(2.24):

30

hapter

11

21

M24 M14

Tables2.1and2.2summarizethematrices groups.1)

d andforall

c r y s ~ l o ~ a ppoint ~c

Suppose thata shear stressis applied to a square crystal and the crystal is deformed as illus~atedin Fig. 2.1. Calculate the induced strainx5 (= 2x3 1).

F

F

.2.1

Shear stress andstrain c o n ~ ~ u r a ~ o n .

Since x5 = 2x31 = tan 8 = 8 and 1' = IC /l80 rad., x5 = 0.017.

31

at~~matica~ Tr~atm0ntof F 0 r r o 0 ~ ~ ~ c s

T~c~ic

wit p u p 1

.. . .

...

3

(~ontinued)~ i e z o e l e c ~ c c o e f ~ c ime n t

. . . .

* . . .

Electro~t~ctive coe~ficie~t

t

b

. . . . . .

X.

*

34

Chapter 2 continue^) Electrostrictive coefficient m a ~ x . *

.

.

Point group 23, m3

I

.

*

I

.

)

.

.

Point group 43m,432, m 3 ~

. . . *

.

I

I

.

.

a t ~ e ~ ~ t iTreat~ent cal of Ferroelectrics

35

For acube-sha~d specimen, tensile s a s s X and compressive stresssimultaneously along the (1 0 1) and (1 0 1) axes, respectively (Fig. 2.2). When we take the p~me-coordinatesas illustrated in Fig. 2.2, the stress tensoris represented as 0 0 0

Using the transformation matrixA

calculate Ax-A"~,and verify that the above-stressis equivalent toa pure shear stress in the original (non-prime) coordinates.

Application of a pair of stresses X and . X to a cube of material Solution

-

Using 8 = 45O, we can obtain the transformed stress r~presen~tion: A.x.A-~=

0 0 X $

0 0 0

X 0 0

(P2.2.l)

The off-diagonal componentsX13 and X31 have the same magnitudeX, and represent a pureshearstress.Notethat a shearstress is equivalentto a combination of

3

extensional and c o n ~ a c t i o nstresses. ~

an extensio~alastress

ntly s i ~ i diagonal l ~ e ng, withoutthe co~trac~on along the 3' d ~ e c ~ o n , an

0 0

d3 l

0 0 31

is t r a ~ s ~ o into ~ed

0 0 d33

0

dl5

0

0 0

0

0 0

0

dl5

0 0

0

15

0 0 0

0 0 0

is inducedunderan

ate exhibits a cubic crystal symne does not show ~ i e ~ o e l ~ t r i c iHo ty. a ~ ~ l i eelectric d field.Therelation

1 2 0 1 1 0

0 0 0

0 0

0

1 2 0

0 0

0 0 0

0 0 0

alculatetheinduced

§olution

0

0 0

S

n underanelectricfie1

38

Chapter 2

Thedistortion is illustratedin Fig. 2.3(b). Thestrain x indud alonganarbitrary direction is given by

x = I:X"1J I*1 I*J

(P2.4.3)

where li is a direction cosine with respect to the i axis. "herefore, the strain induced is given by along the [11 l] direction, ~~111111, X[I 1 l]//= Z; xij (1/43)(1/43)

= [X 1 + x2 + x3 + 2 ( ~ 4 +2 x512 + xd2)]/? = (M11 2M12 + E[111I2/3* M441

(P2.4.4)

On the other hand, the strain induced perpendicular to the[1 1 l] direction, x[ 1 1 1 U, is calculated in a similar fashion as

Figure 2.1 l(b) shows the distortion schematically. volume~cstrain (AVfV)given by

X[lll]//+ 2 X[lll]l= (M11

It is importanttonotethatthe

+ 2M12) E[111I2

(P2.4.6)

is independentof the applied field direction.

(1)

~ a n ~ Theory a u of thePhaseTransition

A t h e r m o d y n ~ theory c ~ explaining the behavior of a ferroelectric crystal can be

obtained by considering the formof the expansion of the free energy as a ~ n c t i o nof thepolarization P. Weassumethatthe Landau frwj energy F inonedimension is represented formally as: F(P,T)

= (112)a P2 + (114)p P4.+ ( 1 1 6 ) ~ P6 +

(2.26)

The coefficients a,p,y depend, in general, on the temperature. Note that the series does not contain terms in odd powers of P because the fiee energy of the crystal will not change with polarization reversal (P -P). The phenomenological fo~ulationshouldbeappliedforthewholetemperaturerangeoverwhichthe material is in the paraelectric and ferroelectric states. m->

~at~ematical Treatment of Ferroelectrics

39

The equilibrium polarization in an electric fieldE satisfies the condition: (2.27) To obtaintheferroelectricstate,the coescient of thetermmustbenegativefor thepolarizedstatetobe stable, whileintheparaelectricstate it mustbepositive passing through zero at some temperature To (Curie-Weiss temperature):

cx = (T -TO)/@C

(2.28)

where C is taken as a positive constant called the Curie-Weiss constant and To is equal to or lower than the actual transition temperature Tc (Curie tempera~e). The temperaturedependence of a is relatedon amicroscopicleveltothetemperature dependence of theionicpolarizabilitycoupledwiththermalexpansion andother effects of a n h ~ o n ilattice c interactions. Refer to the discussion in Section 1.2.

When p ispositive, y is oftenneglectedbecausenothingspecial term. The polarization for zero applied field is obtained from

is

m.(2.27) as

[(T -To)/&() C]PS + p Ps3 = 0

bythis

(2.29)

-

so that either PS = 0 or Ps2 = (To T)/p EO C.

For T >To, theuniquesolution PS = 0 is obtained. For T the Landau free energy is obtained at:

PS L= ZJ(T0 -T)/(p Q C).

C

To theminimum of (2.30)

The phase transition occurs at Tc = To and thepolarizationgoescontinuouslyto zero atthis temperature; this is called asecond-ordertra~si~ion. The relative permittivityE is calculated as: )= &()(a + 3p P2)

(2.3 1)

Then, (2.32) Figure 2.4(a) showsthevariations of PS and E withtemperature. It is notable that thepermittivitybecomesinfiniteatthetransitiontemperature.Triglycinesulphate is an example of a ferroelectric exhibiting the second-order transition.

40 Pe~ittivityE

P e ~ t t i v iEt ~

Tc

(Curie Temp.)

Te~perature

'c

(Curie Temp.)

Te~~eratur~

0)

(a)

hase transitions in a ferroelectric: (a) second-order and (b) first-order. First-order ~ a n s i ~ o n

p is negative in Eq. (2.26) and y is

. The equilib~umconditionfor E =

(2.34).

ositive, the transition becomes first

.(2.33) leads to either PS = 0 or Eiq.

(2.34) e transition temperat~eTc is obtained from thecondi~onthat the the paraelectric and ferroelectric phases are equal: i.e., F = 0, or:

-TO)/&OC] + (112) p

(2.35)

fore:

TC = To + (3/16)(p2 Q C1 y)

(~.36)

ote that the Curie temperature TC is lightly higher than the C te~peratureTo,andthat a discrete jump of appearsat Tc. Also, the p exhibits a finite maximum at TC for a ~ ~ s t -t ro ~r ~~s ~i t[Fig. i o ~2.4(b) tana ate is an example of a ferroelectricthat undergoes a ~rst-orderphase ~ a n s i t i o ~ . es are plotted for the second- and firs resin Fig. 2.5. In thecase of p > 0,

shows a maximum and a discon~nuityof the

41

Free Energy

P

Freeenergycurvesplotted€orthesecond-(a)andfirst-order ans sit ions at various temperatures.

Veri@thedifferencebetweentheCurie by:

TC = TQ+ (3/16)(p2

(b) phase

and C ~ i e - ~ e itempera~res ss asexpressed

EO

€or a €iist-order phase~ ~ s i ~where o n the , Landau free energyis expanded as

a = (T - TQ)/EQ C.

The potential ~ i n i m are a obt~ne

(P2.5.1) There are generally three minima including P = 0 (F = 0). At the Curie temperature, the free energy at the non-zero pol~zationmust be to zero (F = 0). Thus we o b t ~ nanother condition: F = (112)~~ P2 + (1/4)p

+ (1/6)y P6 = 0 .

42

Chapter 2

Equations (P251) and (P2.5.2) are reduced for non-zero polarizations to a+pp2+yP4=0,

(P2.5.3)

a + (1/2) p P2 + (113) y $= 0

(P2.5.4)

g

.(P2.5.3) is validforalltemperaturesbelow Tc, but Q. (P2.5.4) is only valid at T = Tc. Eliminating the P terms from these two equations, we obtain

TC = To + (3/16)(p2

(P2.5.6)

C/ y) ,

~ e n o m e n o ~ o of ~ y E~ectrostrictio In a ferroelectricwhoseprototypephase(high temperat~eparaelectricphase) is centrosy~etricandnon-piezoelectric,thepi ectric couplingterm and only the electrostrictive coupling term is introduced. electrostrictioninferroelectricswereformulatedinthe 1950s byDevonshire2) and aye3) Let usassumethatthe elastic Gibbs energyshould be expandedin a onedimensional form:

X,

(a= (T-TO)/EO C)

(2.37)

,X, T are the polarization, stress and temperat~e,respectively, and S and

are calledthe

elastic compliance andthe

This leads to Eqs. (2.38) and (2.39).

electrostrictivecoefficient,respectively. (2.38)

X

-(aGl/aX)= SX+ Q

(2.39)

en the external stress is zero, the following equations are derived: E=aP+pP3+yP5 X = QP2 l/EO&=a+3pP2+5yP4

(2.40) (2.41) (2.42)

~ a t h e ~ a t i cTreat~ent a~ of Ferroelectrics

43

(E = 0), two different states are derived; If the external electric field is equal to zero

P = 0 and P2= (4 p2-4ay - p)/2y. (i) Paraelectric phase:PS = 0 or P = Q E E (under small E) Permittivity:

E = C/(T -To) (Curie-Weiss law) (2.43)

Electrostriction: x = Q &02e2E2

(2.44)

The previously mentioned electrostrictive coefficient M in Eq.(2.24) is related to the electrostrictive Q coefficient through

M = Q &02e2 (ii) Ferroelectric phase: Ps2= (d X

(2.45)

p2-4ay - p)/2y or P = PS + EO& E (under small E)

= Q(Ps + EO E E)2 = QPs2 + 2 EO E QPsE + Q &02e2E2 (2.46)

where we define the spontaneous strainxs and the piezoelectric constantd as: Spontaneous strain:

xs = QPs2

(2.47)

Piezoelectric constant:

d = 2 EO E QPs

(2.48)

Eq. (2.48)thatpiezoelectricityisequivalenttotheelectrostrictive Weseeby phenomenon biasedby the spontaneous polarization. The temperature dependences of are plotted in Fig. 2.6. the spontaneous strain and the piezoelectric constant

-

When a hydrostatic pressurep (X = p) is applied, the inverse permittivity is changed in proportion to p:

= a + 3 p p2 + 5 y Pc + 2 ~ p (Ferroelectricstate) a + 2Qp = (T - To + 2Q~Cp)/(&oC)(Paraelectric state) (2.49)

I/EOE

Therefore,thepressuredependenceoftheCurie-Weisstemperature transition temperature T c is derived as follows:

To orthe

In general, the ferroelectric Curie temperature is deaeaxd with increasing hydros~tic pressure(i.e. Q h 0).

44

Temperat~edependence of the spontaneous strain and the p i e ~ o e l ~ t r i c constant.

ariumtitanatehasd33 = 320 x C N , (=€3) = 800 and33 at room te~perature.Estimate the spontaneousp o l ~ z a t i o n

= 0.1 1 m4c-2

Let us use the relation: (P2.6.1) S

= d33Q EO €3 Q33

= 3 2 0 ~ 1 0 - ~ ~ [ C N ]x/ {82. 8 ~ 4 ~ 1 0[- l ~ = 0.21 [c/mZ]

In the case of a second-order phase transition, the elasticGibbs energy is expanded in a one-di~ensionalform as follows: G1 (P,X,T)= (1/2)a P2 + (1/4)p - (112)s x2 Q P2 x ,

(P2.7.1)

*

-

where only the coefficient a is dependent on temperature, a = (T To)/&oC. Obtain the dielectric constant,spont~eous pol~ization, spontaneous strain and piezoelectric constant as a function of te~perature.

4

b e ech~acte~stic e~ua~ons:

et tin^ E = 0 i~itially,we ob n the follow in^ two stable states:Ps2 = 0 or ~ a e l ~ c t rphase i c --T >To --

(P2.7.9)

iezoel~tric cons~nt is obtained as

o far wehavediscussthe electric field ~ d u strains, c ~ i.e. piezoelec~c strain rse ~ ~ e ~ ~~ ~~ e~cx et=*dc t ~ c . Let us consider here the CO

46

A( 1k0 E)= 2QX

(2.52)

This is the co~verse elecFrost~c~ve e+ecF. The converse ~:lectrostrictive effect, the stress ~ ~ of the npe~ittivity, ~ stress ini n ~ sensor^.^) A b i m o ~ h s ~ c t u r ewhich subtracts the static capac of two electric provide superior stress sensitivity andtem~ratures ~ b i l i The ~. c of the top and bottom plates have opposite signs for uniaxial S urechange.Theresponse toabout 1 inthe low p i e z ~ l e c ~will c s be discuss^ in Section 7.2 of apter er ?.

Severalexpressions for theelectrostrictive coe~cient havebeengiven From the data obtained by independent experimen~methods such as

so far.

1)electric ~ e l d - i n d ustrain c ~ in thep~aelectricphase,

eous pol~zationand s p o n ~ n ~ strain ~ u s (x-ray ~ ~ c t i oinn the ) ferroelectric phase, 3)d c o n s ~ t &om s the ~eld-inducedstrain in the ferroelectric phase or fkom piezoelectric resonance, 4) ~ressuredependence of ~ e ~ i tin~thevparaelectric i ~ phase,

T r\

l x lo-2

( U

U

\

n

N

"E

0 \

"E W

1

-X50

-100

50

0

~ e r n ~ r a ~("c) re

50

100

.7 Temperature dependence of the electrostrictivecon st^^ Q33 and

athe~aticalT r ~ a t ~ e of n tFerroelectrics

47

nearly equal values ofQ were obtained. Figure 2.7 shows the temperature dependence of the electrostrictivecoefflcients Q33 and Q31 forthecomplexperovskite ~ b ( ~ g i / 3 ~ b 2 / 3 )whose 0 3 , Curie tempera~reis near O * C ~ ) It is seen that there is no significant anomaly in the electrostrictive coefiicient Q through the tem range in which the paraelectric to ferroelectric phase transition occurs piezoelectricity a p ~ a r s .Q is almost temperatureinde~ndent.

(1)

~ntife~roe~ectrics

The previous sections dealt with the case in which the directions of the spontaneous dipoles are parallel to each other in a crystal (polar crystal). There are cases in which antiparallel orientation lowers the dipole-dipole interaction energy. Such crystals are calledanti-polarcrystals.Figure 2.8 showstheorientation of the spontaneo~s electric dipoles in an anti-polar state in comparison with a non-polar and a polar ikee energy of an antipolar state does not state. In an anti-polar crystal, where the differ appreciatively from that of a polar state, the application of an external electric field or mechanical stressmay cause a transitionof the dipole orientation to a parallel u~?~e~~oe~ec?~c~. state. Such crystals are called Figure 2.9 showstherelationshipbetween(appliedelectricfield)and in paraelectric, ferroelectricand antiferroelectric phases. In a paraelectric E relation is linear; in a ferroelectric phase there appears a hys ans sit ion of thespontaneouspolarizationbetweenthepositive negative directions; anti an in ase, low atelectric field, the induced polarization is propo~onalt crystalbecomesferroelectric pol~zation hysteresis shows removal electric of field, the the crystal returns to its anti-polar state, and hence, no spontaneous polarization can ~ ~curve. ~ ~ e be observed as a whole. This is called double a

stripe type checker

board typ

Schematic ~ a n g e m e nof t the spontaneous dipoles in non-polar, polar and antipolar materials.

~

e

~

48

Polarization

Polarization

(a) Paraelectric (b) Ferroelectric

'

field

(c)Antiferroelectric

Polarizationvs.electricfieldhysteresiscurvesinparaelectric,ferroelectric and antiferroelectric materials.

We willdiscussheretheintroductionofelectrosctivecouplingin energy expression for ~tife~oelec~ics.6,7)esimplestmodel for ~ t i f e ~ ~ l ~ t r i c s is the"one-dimensionaltwo-sublatticemodel." It treatsthe coordinat~as one~mensional, and asuperlattice(twicetheunitlattice) is formedfromtwo neighbor in^ sublattices each having a sublattice polarization Pa and Q,. The state Pa = Pb represents the ferroelectric phase, while Pa = - Pb, the antiferroelectric phase. For the electrostrictive effect, ignoring the coupling between the two sublattices, the strains from the two sublattices are QPa2 and QPb2, respectively (assuming equal electrostrictiveconstants Q forbothsublattices).Thetotalstrain of thecrystal becomes (2.53)

owever, since ~tiferroelec~cityoriginates from s~blattices,it is appropriatetoconsiderthesublattice

the coupling between the coupli~g also forthe

at~emati~al Tr~atm~ of~ t

49

electrostrictive effect. The coupling term for the elec~ostriction the following form:

which in hy~ostaticpressure p is employed, and XT is the i s o t h e ~ a l compressibility, Qh and are the electrostrictive constants. Introducing the the followin transfo~ationsPF = (Pa Pb)/2 and PA = (Pa Pb)/2leadsto expression:

-

The dielectric and elastic ~ u a t i o n of s state followas (l+~)p+pPF2+3pPA2+y

+ l@' PF2PA2 + 5" PA4]

(2.56)

Hence,theinducedvolumechangeintheparaelectricphasecanberelated induced ferroelectricpol~zationby the following formula:

to the

Below the phaseans sit ion temperature (this temperature forantife~~lectrics is called ~ e e~ Z e ~ ~ e the r ~ spontaneous ~ ~ r e ) volume strain and the s p o n ~ e o uantiferr~lectric s pol~zationare relatedas (2.60) Even if the perovskite cystal shows Qh 0, the spontaneous volume strain can depending on the value of > l), that is, the if positive or negative inter-sublattice coupling is s~ongerthan the coupling, volume a contraction is observed at the Nee1 point. This is quite different from f e ~ ~ l ~ t r i c s , C t. Figure 2.10 illustrates whichalwaysshowavolumeexpansionatthe thespontaneous strai~sinacrystal scheme tic all^ 0. en PaandPbare in theparallel con~~uration (ferroelectricphase), the actstoincrease the strain

Chapter 2

50

xs, whenthey are inthe anti-par~lelconfig~ation(antife~oelctricphase),the term acts to decrease the strain.

a-

This phenomenological theory explains well the experimental results for the ~tiferroelectricperovskitecrystal PbZr03 and others.8)Figure 2.11 showsthe strain in the antiferroelectric ceramic Pbo,ggNbo,o2[(Zro,6sno.4)o.g4Tio.o6]0.g8o3 as a function of an applied electric field91 The large change in the strain associated with the field-induced transition from themtiferrwlectric to ferroelectric phase canbe estimated to be

Here, we assume that the magnitudes of Pa and pb do not change drastically through the phase transition. (a) Ferroelectric Arran~e~ent X = Q (I+Q) (Pa + Pb)2/4

x =QPa2

Intuitive e~planationof thesublatticecouplingwithrespectto electros~ction(for S2 >0).

51

at~e~atical Treat~ent of Ferroelectrics

Antiferroelectricphase I

(kV /m)

1 Field induced strain in a P b ( ~ , S n ) ~ based 3 antife~oelec~c.

1. Tensor representation: when two physical properties are represented using tensors

of p-rank and q-rank, the quanti~which combines the two properties in a linear relation is also represented by a tensorof (p + 9)-rank.

2. A physicalpropertymeasuredalongtwo

diffe~ntdirectionsmustbeequal if are c~s~lographically equivalent. This consideration these two directions reduces the numberof the independent tensor components representing the above property.

3.

Shear strain: x5 = 2 x31 = 2 Qb,taken as positive for smaller angle.

4. Phenomenology: (M) >Q --> second-orderphasetransition e Q --> first-order phase~ n s i t i o n

x = Q PS2 spontaneous strain

+ 2 Q &Q& PS E

+ Q Y )2 &2 E2

piezos~ction electros~ction

constant is insensi

7 . In ~ t i f e ~ o e l e c ~consid~ratio~ cs, of ~ublat~ce the stable sublattice ~olariza~on con jump in strain associate^ i n d ~ by c ~an external el

'1

e room tem~eratureform of ~u~~belon at the ~ i e z o ~ l em c~c 11

-dl1 0

0 0

0

0

atthe ~ i e z ~ l ~ ttensor r i c must be in ~ o u nthe %axis ~ a d for a 1~0'rotation transformation matrices are

0 -1 0

.3 2.3

0

0

0 0 '

0

0

0

0

3

int

t

212

31 331

54

Chapter Next, a 120° rotation is considered such that a11 = -112, a12 = 1/3/2,a21 = . 1/3/2,a22 = -112, a33 = 1 :

Continuingthe c~culationsfor d123, d212, d23 1, d312, d331, we can obtain all the necessary e~uationsfor deriving the final matrix form. 2.2

In the case of a first-order phase transition, the h d a u free energy is expanded as in Example Problem 2.5. Calculate the inverse pe~ittivityin the vicinity of the Curie tem~rature,and verify that the slope ((l/e)/ilT) just below Tc is 8 times larger than the slopejust above Tc.

In a fmt-orderphasetransition, PS satisfies the following ~ u a t i o ninthe temperature rangeof T p i ~ in-~ -"hard" > piezoelectric Donor ---Pb > deficiencycom~nsation---"soft" > piezoelectric aration of ceramic powders: o x i d e - ~ x i ntechnique ~ co~r~ipitation alkoxide hydrolysis

3. Devicedesigns: Single disk ultilayer ~ n i m o ~ ~ i m o ~ h ~oonie/cymbal Flexible composite Thin/thick film 4.

omp par is on betweenmultilayersandbimorphs: 1. The multilayer type does not exhibit large displacements, but hasadvanta~es in generative force, response speed, life time and elec~om~hanical coupling k33 2. The bimorph type exhibits large displace men^, butshows disadvantag~in generative force, response ,life time and the elec~om~h~c coupling bff.

5. Tipdisplacementin

a b i m o ~ hunder a one-endclampcondition (cantilever): 6 = (312) d31 ( ~ 2t2) / V or

6 = 3 d31 (L2/t2) V

(according to the structure)

a m e n ~ l~esonancefre

ialceases to be ~ e ~ o e l(i.e. e ~ t ~ ~

7.

..to12

or

3.

102

q

(a)

+

q

q

+

q

0

-a -2a-3a

q

+

q

q

4

+3a 0 +2a . a

l- dimension^ finite chain of two kindsof ions +q and -4. 3.5

Bariumtitanateexhibits a tetragonalcrystalsymmetryatroom tem~ra~e and the distortion firom the cubic structure is not very large (cla = 1.01). Calculate all the possible angles between the two non-180' domain walls.

3.6

In calculating Eqs. (3.10) and (3.1 l), the volume element dv is given by Chr = 2nr2 d r sine de. Usingthisdv,calculatedv,cos0 dv and cos2@dv, when the polarization is uniformly ~ s ~ i b u t with e d respect to

e.

9)

B. Jaffe, W. R. CookandH.Jaffe:PiezoelectricCeramics,p.142,Academic Press, NY (1971). K. Uchino and S, Nomura: Jpn. J. Appl, Phys. It K. Abe, 0. Furukawa and H. Inagawa: Ferroelectrics 87,55 (1988). A. Hagimura and K. Uchino: Ferroelectrics, 93, 373 (1989). K.Uchino,H,NegishiandT.Hirose:Jpn.J.Appl,Phys., 28, Suppl. 28-2, 47 (1989). S. Hirose, Y. Yamayoshi, M. TagaandH.Shimizu:Jpn.J.Appl.Phys., 30, Suppl.30-1,1117(1991). S. Takahashi and S. Hirose: Jpn. J. Appl. Phys., 32, Pt. l , No.SB, 2422 (1993). K. Uchino, J. Zheng, A. Joshi, Y. H. Chen, S. Yoshikawa, S. Hirose, S. Takahashi and J. W. C. de Vries: J. Electroceramics, 2, 33 (1998). S. Hirose, N. Aoyagi, Y. Tomikawa, S. T~ahashiand K. Uchino: Proc. Ultrasonics Int'l. '95, Minburgh, p.184 (1995).

evice ~ e s i ~ nand i n ~F~~rication Processes

103

Kato:FineCeramicsTechnology,Vo1.3FabricationTechnology of Ceramic Powder and Its Future, p.166, Industry Research Center, Japan (1983). M, Lejeune and J. P. Boilot: Ferroelectrics 54, 191 (1984). S. L. SW-,T.R.Shrout, W. A. Schulze and L. E. Cross: J. Amer. Ceram, Soc. 67, 311(1984). Tanada, Yamam~a,Shirasaki: Abstract 22nd Jpn. Ceram. Soc. Fundamental Div. 3B5,p.81(1984). Ozaki: Electronic Ceramics 13, Summer, p.26 (1982). Kakegawa,Mohri,Imai,ShirasakiandTekahashi:Abstract21stJpn.Ceram. Soc. Fundamental Div. 2C6, p.100 (1983). H.Abe: ~ec~stuZZizution, Mater. Sci. Series 2, ,Kyoritsu Pub., Tokyo (1969). K. Uchino and T. Takasu: Inspec. 10, 29 (1986). A. Yamaji, Y. Enomoto, E. Kinoshita and T. Tanaka: Proc, 1st Mtg. Ferroelectric Mater. &c Appl. p.269, Kyoto (1977). K,Nakamura, H. Ando and H. Shimizu: Jpn. J. Appl. Phys. 26, Supp1.26-2, 198 (1987). J. Kuwata, K. Uchino and S. Nomura: Ferroelectrics 37, 579 (1981). J. Kuwata, K. Uchino and S. Nomura: Jpn. J. Appl. Phys. 21(9), 1298 (1982). J. Zheng, S. Takahashi, S. Yoshikawa, K. Uchino and J. W. C. de Vries: J, Amer. Ceram. Soc. 79, 3193 (1996). K, Nagai and T. Konno Edit.: Electromechanical Vibrators and Their Applications, Corona Pub. (1 974). K. Uchino:PiezoelectricActuatorsandUltrasonicMotors,KluwerAcademic Publishers,MA,p.241(1997). K, Abe, K. Uchino and S. Nomura: Jpn. J. Appl, Phys. 21, L408 (1982). Y. Sugawara, K. Onitsuka, S. Yoshikawa, Q. C.Xu, R. E.Newnhamand K. Uchino: J. Amer. Ceram. Soc. 75, 996 (1992). H, Goto, K. Imanaka and K. Uchino: Ultrasonic Techno 5,48 (1992). A. Dogan: Ph. D. Thesis, Penn State University (1994). Kitayama:Ceramics 14, 209(1979). M. Ishida et al.: Appl. Phys. Lett. 31, 433 (1977). M. Okuyama et al.: Ferroelectrics 33, 235 (1981). S. K. Dey and R. Zuleeg: Ferroelectrics 10 A. Yamaji, Y. Enomoto, K. Kinoshita Ferroelectric Mater. 8t Appl., Kyoto, p.269 (1977). K.Uchino and T. Takasu: Inspec. 10, 29 (1986). .Uchino, E. Sadanaga and T. Hirose: J. Amer. Ceram, Soc. 72, 1555 (1989). T.Yamakawaand K.Uchino: Proc. Int'l. Symp. Appl. Ferroelectrics '90, p.610 (1991). K.Saegusa et al.: Amer. Ceram. Soc., 91th Ann. Mtg. (1989). G.A.. Samara: Ferroelectrics, 2, 277 (1971). K. Uchino,E,Sadanaga, K. Oonishiand H. Yamam~a:CeramicTrans. Ceramic Dielectrics, 107 (1990). N. Uchida and T. Ikeda: Jpn. J. Appl. Phys. 6, 1079 (1967). N. Uchida: Rev. Elect. Commun. Lab. 16, 4 N. Uchida and T. Ikeda: Jpn. J. Appl. Phys. N. A. Schmidt: Ferroelectrics 31 ,105 (1981). J. Kuwata, K. Uchino and S. Nomura: Jpn. J. Appl. Phys. l P. Gerthsen and G, Kmger: Ferroelectrics 11,489 (1976).

hould dis~nguishthedevice te~inology: monomo~h, unimo~h, bimo ultimo~h.All are bending devices, however, the d e ~ n i ~ o n are: s

b i m o ~ h. . h

-

singleactuator c e r plate ~ ~ single actuator plateBr; an elastic shim double ac~atorplatesbondedtogetherwith or without an elastic shim multipleactuatorplatesbondedtogether with orwithout multiple elastic shims

emajorapplication of ferro the Curie

con st^^ around

CS is

forcapacitors,utilizingtheirhigh

two classes ofitors: cone rcuits, using a Ti

is for thermal compensation of is a high permittivity

dl and the other

low-dielec~icconstan

1 ~ , ~ 0 .

Figure 4.1 s ~ ~ ~thez v e~ osu scapacitortypes,highlightingtheirsizes ranges.1) ~ e r ~ i c c a p with ~ iatsingle o~p opular, while multilayer ceramic capacitors parallel plate type. Se~conductorcapacitors e x ~ b i tl capacitance using very thi tric layers in a se~conductorbased ceramic (see Chapter 9, Section 9.3). capacitors ip are ul~a-smallcapacitors for high frequency applications. basic s p ~ i ~ c a ~reo n s (a) Small size, large capacitance

arge dielectric constant are des th a high dielectric constant are sometimes

(c) Te

tricdispersion,whichmustbe account forprac~calapplications. mate~dls stabilize to the

temp

105

hapter 4

106

Satellite Commun. Automobile Commun.

mTv mTv

lultilayer :eramic lapacitor

FM Radio

ledonductor lapacitor

AM Radio

Various capacitor types classified according to their sizes and operating frequency ranges.

Calculate the wavelength in air (E = 1) and in a dielectric material with E = 30 for electromagnetic wave at 10 GHz.

_ .

Taking accountof c = 3.0 x lo8 m/s in air and v t:c / de in the dielectric,

=3x

-

/d30 [m] = 5.5 [mm]

~ ~ Z ? istructures ~ e r havebeen developed as part of capacitorm ~ u f a c 4.2 schematically shows a the inte~ationof electrical circuit components. Figure multilayer capacitor chip. "hin sheets made by the tape casting tec~ique,starting from a slurry of the dielectric powder and organic solvents, are coated with Ag-Pd, Ag, or cheaper Ni or Cu paste is used to form the electrodes, then several tens of sheets are stacked together and sintered. Finally, external electrodes,used to connect the chip with the circuit, are painted on. See Chapter 3, Section 3.3(2) for the details of the manufact~ingprocess.

107 Thelayerthickness of multilayercapacitorshasbeen d u d remarkably,with thicknesscurrentlyontheorder of 7 10 pm. "he electrostaticcapacitance of a multilayer capacitor is given by the following formula:

-

where E is the relative permittivityof the dielectric material,n the number of layers, S theelectrodearea,and L thetotalthickness of thecapaciator.Notethatthe c a p a c i ~ c increases e in propo~onto the square of the number of layers, whenthe 4.1 s u ~ ~ z specifications e s forseveralmultilayer totalsize is fixed.Table capacitors.2)Theconventionalcapacitor of 10 witha 30 pmlayerthicknesshas a volume of 70mm3. B y decreasing the layer thickness down to 10 pm, the device volumecan be reduced to7.7mm3.Notethatbyreducingthelayerthicknessby h , the total volume is reducedbyafactor of ( to sustain the same capacitance.

layers n Totalthickness L ctric field direction Internal electrode "

Structure of a multilayer capacitor.

.l ~imensions ofthemultilayerceramiccapacitors. "

~ . . ~ . . .

Capacitance Dimensions VolumeRelativeGreenSheet at Room (mm3) Volume Thickness (-1 Temp. (PI?) L W T (W)

Present Ceram.Cap.

1 10

2.0 1.3 0.8 3.2 1.6 1.5

Conventional 7.0 Ceram.Cap.

1 10

3.3 1.7 1.2 4.2 2.4

-~.

1 Tantalum Electrolytic Cap. 2.64.7 10

1.6 3.2 1.6

2.1

2.110 7.7'

(100)

100

10

6.7 70.0 30

(319) 909

25

8.2

(390)

---

334 25.7

" .

S @ C

6C 6C 6C

6C

e l ~ ofe~roelectrics r such as been utiliz~for very comp ~ e r o v § ~ thave e § been investi

h

ivi

ctri

ir very high ~ e ~ i t ~and v i ~ , - in§ensi~vec h ~ a c t ~ ~ § t(i.e., ics a1 f ~ ~ o e l e c ~ c ~ e Sr o v s ~ t e

ions.

110

act reason why the phase an sit ion is & h e in the relaxor ferroelectrics has been clarified, We i c composi~~n fluc~a~on" which is one of the most widely models for the relaxor

Fi ure 4.5 shows a comput~rsimulati hasreportedthesh electron mi~roscopy.8) Thehighresolutionimage in Fig. 4. ordered islands in the range of 2 -5 nm, each of which may hav tr~sitiontemperature. v)

0.3 0.4 0.50.6 0.7

Fraction of

0

20

0.3 0.4 0.5 0.6 0.7 Fraction of 0 e 0 0 0 IO. e e

M

e 0 0 0 e o o e l.

to. e o toeoa.e*s 0 0 . eo l.000000. 100. e roe e 0 0 0 . . l.0.0.... 'O... .OO. a 10..00..0 eo 0 . ..oe 000 0 100 0. ~ O . O . . O O O a lOOO..OO. . 0 0 . l.00. oeoo ~ 0 0 0 0~ 0 0 . 0 . 0 . 0 0 .eo. ..e

0.0.

k..00.

e00

v0.0

1.0.0

000.

t e e 0 0 kO.O.O..O

eo00 100.0 lO.OO0O.O

e eo0 ) . O O . tO.O.OO.. .eo. t00.0 teoveoooa 0 0 0 0 ~ o * o e~ . 0 0 . 0 0 0 0

*e 0 0 00.0

l00.0 POOO.

o.00e PO... ..v ).a00 0.00

)O...

om~uter simulation calcul crystaltype zig region size: 4 x 4).

-

~0.0.00.0 lO.OO..O. ).O.r000.

B.

1.000....

IO

rooooeooe

0.3 0.4 0.5 0.6 0.7 Fraction of 0

uctua~on of ionic

h P e ~ i ~ i~ieiectrics v i ~

111

High resolution e l e c ~ o n - ~ c r o s cimage o ~ of a Pb single crystal(110). Note ion-ordered islands in the rangeof 2 -5 nm. In the case of the pe~ittivity,for example, by rimpos posing the Curiewith a ~ ~ e Curie ~ n t etm ~ e r a ~ r ewe , obtain a rather which provides more stable temperature change. Thus, some is specifiedratherthanthe " ~ u r i epoint."The p e ~ i t t i v i t yof relaxor f e ~ ~ l e c in~ the c sp ~ a e l e c region ~c obeys the followingquadra~crelation:

rather t ~ a nthe n o ~ alaw l

ve the tem~eraturec~fficientof pe d i ~ ~ s phase e d sitio ion, the followi~

112

provement of the t e r n ~ e ~ ~t ~ o ee~ c i e noft ~ ~ i t ~ v i t in tQ

type9

45, (4) 450, (5)

1s

h

113 (a)S

~ ~ ..type a ~relaxor i

(b)~ ~ ~ orelaxor e l ~ t ~ ~

ulti-potential-well model for (a) the ~ k ~ a v i - t y fe~oelectricrelaxors: Note the differencein the coo~rative ~heno~enon. 0

near O°C shifts towards higher

di

perovs~tecellduetothe

pol~zationappears.

Consider an order-disorder ty electric with potential with a relatively lo quasi-dc field, the ion follo tric field negative potentials. wever, with increasing drive e ~ h i bai delay ~ with is an intuitive explanation for the d

an ion

d in a double-mini~um a (Pig. 4.9). Under a alte~atin between the positive

(1) sing a mathematical represen~tion,derive the ~ o n o ~ s p e r s i vcase: e

ion rela~onfor a

+

E(@) = eS f (1 j WT)

(2) Also discuss how

the above dispersion obeys so-called the (i.e., the real and i ~ a ~ parts i nof~pe~ittivitytrace a half ci pe~ittivityplane).

n X

Chapter 4

114

F Ion in a double-minimum potential.

F in the crystal is described When an external electric fieldE is applied, the local field by F=E+yP. The transition probability for an ion from the

(P4.2.2)

-

to the opposite transition probabilitya , , are expressedas

r exp[- (AU -p)/k'I'l, a..= r exp[- (AU + pF)/k'I'I .

a+=

ere, AU is thebarrierheightbetweenthetwo moment, and r is a constant. we in~oducethe numb~rof + (or -) e total dipole number is given by volume) is re~resentedas

e ~ e ~ n d e n will c e be expres

Then,

+ in Fig. 4.9, a+, and the (P4.2.3) (P4.2.4)

potentialminima, p thedipole

115

(P4.2.8)

N+ = (112) (P4.2.9) (N + P/p), (P4.2.10) Suppose that the external electric field E = Eo ,jot is small and that thep o l ~ ~ ~ o n is given by P=Ps+&o&Eoejot

,

(P4.2.11)

From Eq.(P4.2.$),

Consequently, we obtain

= Es / (1 + j

0%),

( P 4 2 13)

TO = 1 I2r exp(- AUkT)

(P4.2.15)

E(O)

where

The subsc~ptS stands for a static value(o= 0), and in theparaelectric phase

-

Es =Z C(P4.2.16) / (T Tc).

11

Cole-Coleplot for a ~ouble-minim~m .2.13) can be r e ~ ~ t as te~

+ j €"(~),

€(~) = €'(~)

ivi

0.3

6 lo' S IO'

4

8.z B 8

0.24

410'

0.18

3 IO'

0.12

c.

2 10'

0.06

1 10'

0

0 20

40

60

80

100

120

140

160

I

118 3.

~haracte~stics relaxor of ferroelectrics: (a) high pe~ittivity (i.e., diffuse phasetr~sition) re -insensitive ch~acte~stics (c) dielectric relaxation

to the .presence of ielectric rel~ationin some relaxor ferroelec CS is a ~ b u t e d ~ c r ~ o m ~Once n s .macrodom~nsare induced by an external electric field, the dielectric dispe~ion disap~ars and the loss becomes very small.

4.1

A multilayercapacitor (50 layers) is madefiom a 10 dielectric mate~alE = 3000. ~ s s u m i na ~90% ratio

areaoverthechipsurfacearea,calculatethechipareatoobtainatotal capaci~nce of 10 p.

4.2

relaxation e time as

& ( ~= ) &S / (1

is ~stributed,the pe~ittivitydispersion follovvs

+ (j~~)

iscuss the Cole-Cole plotchangein

Murata Catalog: Miracle Stones. K. Utsumi:Privatecommunicationat 3)

~e~oelectricity, ~ijmegen (1995).

c o m ~ ~ s owith n the

4th U-JapanSeminaronDielectrics

h

Recently, very large scale semiconductor memories using ferroelectric y. Sincetheconventional Si micromachining films have been investigat~ technology coupled with silicon oxide or nitride, and metal, i s limited in its ability to produce fine-scale capacitors,u ~ l i z a ~ oofnferroelectiics with high ~ e ~ i t t i v i tory polarization hysteresis has been considered as a possible solution to the problem.

e devices ~ inerasable o semicond ~ Thereare voZutiZe and ~n-voZatiZe~ memories. ~~A~ m Access memo^), which is widely because of its high ty, in is a volatile memory. Data stored i memory are lost when the electric power is shut off. On the contrary, non-volatile memories include a circuit-latch mu1 s ~ a c e - ~ o t e ncontrol ~al both types,in general, h

Figure 5.1 shows the~ n d ~ S e n ~ a capacitor; a Si02 film capacitoris connected to the sourceof a 5.2 showsthestructure ofthe g; i chosen by x-y ~ ~ e s s i nthat electrodes simultaneously,thus ( ~ e ~ o r i z i n g Since ), the ~ c u m u l a charge t~ leaks, the capacitor must be repeatedly (re~es~ing). ord Line

. capacitor. 119

p-type Si

e structure of a D The el~c~on-hole pair genera~onaroundthe radia~onchanges the ~ o u nof t charge on memo^ (SOJ? error). In order retain memo^, the c ~ a c ofi the ~ memo^ ~ capacitormustbehigherthan 30 (remem~rf = 10-15).

E x p l ~ nthe genera~onprocess of the ~ e p l e ~ oand n inversionlayersin p-type Si) using a simple energy b voltage is appliedonthemetal. the hole and electronconcentra~on band model. For simplicity, you can use th close to zero.

~ o n d u c ~ oband n

vel

Fermi le

EF etal

I

"

Oxide0

Energy band model for a

EF Valence band Se~conductor (p-type)

a

p-type s e ~ i c o n ~ ~ t o r

E3E2

E3 (c) Inversion State

Inversion layer

Let usconsider an n-channelenhancementmode MOS asillustratedin Fig. 5.5. A positive gate voltage induces the electron inversion layer, which then connects the n-typesourceandthen-typedrainregions.Discussthedraincurrentbehavioras a function of the drain/source voltage.1)

p-type se~conductor

with a p-type se~conductor ( n - c h ~ e l

ositive gate voltage induces the electron inversion layer, which then connects the n-typesourceanddrainregions.Thesourceterminal is thesource of carriersthat flow throughthechanneltothedrainterminal. In suchann-channel electrons travel from the source to the drain so that the c o n v e n ~ o nc~~ e n from the drain to the source. hich is analogo~stoa aninsulatingcoat(the be, wherethewater(the

n that the flat band

. Since forsmall

S

increase^ to the point where is equal to zero (~reciselysp versioncharge d e n s i ~is shownin Fig. 5.6(b). A

E

123

c o n d u c ~ c eatthedrainbecomeszero.The omes zero.

slope ofthe

ID versus E

en EDS becomes larger than the above value (Ea),the point in the channel at which the inve~ioncharge is just zero shifts toward the source t trons enter the channel at the source, travel and then, at the pinch-off point the electrons ion ( ~ e p l e ~ olayer) n where they are swept by the E-field to the contact. Ifwe assume that the c

l

m

I

/I

I

Gate

I

71nversioi layer Electron flow (n chmel) (a) Drain voltage EDSa Gate voltage

I

" Inversion lay Electron flow rain voltage EDS= Gate voltage EG

1

Gate

(c) Drain voltage EDS>Gate voltage

-channelwith the

dsource voltagefor an n-ch~nel

r ~ ~ ~ o Volta u r c ~

vers

300

10 k ~ z

I

0

a ferroelectricthin film witha large pol~zation-electricfield hysteresisis acitorthe in structure pictured Fig. in atile memo a voltage is applied to the gate and the the “on” state, a the drain generates a drain current on the nt rem~ent pol~zationstate. Let us assume a P-E hysteresis loop of the f e r r ~ l e c ~film c as i l l u s ~ ain~Fig. pol~zationstate is on A. the current flows according contrary, when the pol~zationstate is on C! first, thecurrent increases ~ ~ a t i c a l because ly thespontaneous pol~zationreversal is associated. Figure 5.10 shows the current responses to a series of pulses (two positive pulses 2 ~ o l l o wby~ two negative pulses) on a PZT film with 20 x 20 ~ m electrodes.~ en a positive pulseis applied just after the negative pulses,a large c ~ e nIposi t is ,whichincludesthe pol~zationreversal.However,thesecondpositive pulse generates only a small current Iup. Thus, the observed c ~ e n t ~ o for u nat positive pulse can indicate the initialp o l ~ z a ~ ostate; n that is, an on or off state, or 1 or 0 state. In this memory device, after reading the initial state by applying the positive voltage, the minimum pol~zationstate becomes A for all the times; that is,thereadingprocess is destructive. ,inordertoretain thememory state, a wri~ngprocesssimilar to thecase ofisrequiredeverytime. on a ferroelectric film at every reading process in pol~zationhysteresis ch~cteristicdegradeswithincreasing cycles.This is c ‘ ~ ~ t ~which g ~ ~is ,the ” most serious problem of a ferroelectric film to overcome for non-volatile memory applications. From a practical point of view, a lifetime (that is, the time until the ~ o l ~ ~ a tde~adation ion is observed) of more than10l5 cycles isrequ~ed.

, asdiscussedabove,the

Polarization versus electric field curvefor a ferroelectric film.

127

o a series of pulses (two positive pulses follo film with 20 x 20 pm2 electrodes. The possible origins for the fatigue are related to the generation of oxygen vacancies uch efforthas beenmade to remedy this proble andthe diffusion o proposed ideas can (l )improvement of the film fabrication process,

(2) search for new materials, (3) improvement of electrode materials.

@ ~r e@Z e c ? ~ c s . ecent new thin film mate~alsinclude ~ e r - s ~ c~ ~ material patented by S y m m e ~ xwhich , hasa basic compo ws superior an~-fatigue prope~ies. F i g ~ e5.1 1 shows for rew~tingthe remanent pol~zationin Y1 and Y1 evenaftertestingfor theremanent pol~zationdoesnotchange signific 1012cycles, an improvement as is CO wed to the lifetime of lo7 cycles for

New electrode materials RuO2 and Ir have been found to exhibit improvement in e , drive fatigue in c o m p ~ s o nwiththe convention^ Ptelectrode. F u ~ e ~ o r new modes such as a combination of the D M operation during the switch-on sta the memo^ mode during the switch-off stage have been proposed,

1

tin

0

wi

1.

130

3. ~inimummemory capacitance 4.

is an inversion current

is about 30 P. (f =

.

type of reading device.

FSET is achannel surface potential control typeof FET.

5.1

S ~ e y i n the recent literature, discuss and s u m ~ ~ zthe studies e on ferroelectric thin films from the following viewpoin~. (1) List thepapers(minimum 5) whichreport on epi~xiallygrown PZT

films. (2) Tabulatethe experimen~llyobtainedphysicalparameters of the PZT films and compare with the data for bulk ceramics. (3) Discuss the above deviation briefly with reference to the papers' results and conclusions. (4)Discuss the crystal orienta~on the PZT films byreferringtothe ate: theoretic^ E x p ~ ~ t i o for Thin nJpn. Phys., V01.36 [9A], 55$0-55$7,1997). 5.2

WelearnedinChap. 4 thatleadmagnesium ~ we , c exhibit ve high dielectric c o n s ~ n If N, itis applicable the to Discuss the fe~ibilityof this operation frequencyof the ~crocomputer.

1)

D, A. Neamen: Semiconductor Physics and evices, 2nd Edit., Irwin,

2)

3) 4) 5)

6) 7)

J. Appl.

(1 997). Okuyama: Ferroelectric emory, Bull. Ceram. Soc. Jpn., (1 995). Yam~ichi,T. Sakuma, ,2193(1991). .S~aemori,S. Ohno, H.Ito, T. Nishimura, T. and T. Namba: Nikkei Micro Mihara, H,~atanabe,C. A. Pas de Araujo,J. Cuchi~o,M.Scottand L. D, cMillan: Roc. 4th Int. Symp.onIntegratedFerroelectrics,Monterey,US,

H.Fujii, T. Ohtsuki, Y. Uemotoand K. Shimada:Jpn.Appl.Phys., Phys.Electronics,No.456,AP942235,p.32(1994). Matsui, H.Nakano, M. Okuyama, T. Nakagawa and Y, Yamakawa:Proc,2nd tg. Ferroelectric Mater. and Appl,, Kyoto, p239 (1979).

~yroezectriceflect incertainmaterialswas n=co a longtimeago, andsuch materials were referred as "electric stones." It was observed when such a stone was to gene^^ electric charges and a 'fc~c~ngff thrown in the fire, and it t e m ~ r a ~dependence e of the spon~neous sound. This is basicallyduetothe pol~zationof a polar material.

ct

Practical applications ofthepyroelectriceffect in temperat~esensors and light detectorshavebeenpromoted,enablingsomecommercialmarketing ferroelectric ceramics.

of

The merits of ~yrosensorsas compared to se~conducting inbed-sensormateri are summari~das follows: a) wide range of response frequency, b) use at room temperature, c) quick response in comparison with other temperature sensors, d) high quality (optical-grade homogeneity, etc.) materials for pyrosens~rs the are unnecess~. The principle on which the pyroelectric effect is based concerns thec ~ ~ generation g e associated with the spontaneous~ l ~ i z a t i change on witht e m ~ r a ~ e :

-

-

j = aPs/a t = (~Ps/aT)(~T/a t) = p(aT/a t).

(6.1)

Here p (= laPS/aTl) is denoted as the ~ y r o e Z e c t ~ c c o e ~ c iThe e ~ tphenomenon . is illustrated schematically in Fig. 6.1. Two typical electrode ~ ~ g e m e nfor t s pyrosensors are illustrated in Fig. 6.2: (a) face electrodes with the polarization direction irradiation, i and (b) edge electrodes with the polarization parallel to the direction ~ ~ n d i c u lto a rthe irradiation. The former type has higher efficiency, but requires a s o p ~ s t i c afabrication t~ process for applying uniform transparent electrodes for the inflared light. 131

13

S

in Qf

i. e., chop bY

Chapter 6

134

where q is the transmitt~ceof the incidentradiation, A a detectingarea, coefficient c o ~ e s p o n ~ ntog the loss ofheatperunitareaofthedetectorto s ~ o u n d i n g due s to its increase in temperature, and

y a its

where p is the density of the pyro-material, cp the specific heat andh is the thickness of the detector [refer to Fig. 6.2(a)]. The ~ ~ r r e n t r e s p u nri,s iis~defined i ~ , by ri = (IWA) (dq/dt)

.

Since the charge generatedby a temperature rise AT is given as q=pAAT, using Eq.(6.2), we obtain: ri = q p UOA( 9 ~ +2~ 0 2 ~ 2 ) - 1 / 2 . Introducing a thermal time constant

we obtain finally

When ozy) >> 1, ri = q p / p cp h. In order to increase ri, neglecting the size or surface effect, the value(p/ p cp) should be increased. Figure 6.4 shows an amplifier circuit for measuring a pyroelectric voltage signal. The resistance R is relatively high and is inserted to remove the charge after it is a ~ g h thermallyinducedon thepyroelectric (Cy)). Thetransistormusthave impedance (e.g., €ET).

Amplifier for a pyroelectric infrared detector.

Pyro@~ectric Devices

135

The voltage responsivity for suchan amplifier is expressed as: rv = (l/WA)(dV/dt) = ri lzl

(6.9)

where z is the impedanceof the detector-amplifier combination. AssumingRLCC R,

(6.10)

where TIE, = R (CD + CA),and CD and CA are the capacitances of the detector the amplifier. Therefore, Eq.(6.9)may be written as

(6.11) At a high frequency (>> l / z ~ME), , we obtain

(6.12)

rV=qp/pc~&Ao9

assuming that CE)> CA. In order to increase rv9neglecting again size or surface effects, the value(p/ p cp E) should increase. Note thatrv differs from ri by a factor of (l/e). The rv decreases with fiquency at high frequencies, but that is relatively independent of frequency between ~ / T(0.1 D 10 Hz) and l/w (0.01 Hz)2) Thus, in practice, the i ~ a ~ a t i chopping on frequency is chosen just between UTI) and l/%.

-

The pyroelectric sensor isa device for transducingoptic~thermalenergy to electrical energy, and its efficiency or figure of merit is evaluated in several ways; for example, in terms of p, p/cp or p/(cpe). Figures of merit for pyroelectric materials. Figure of Merit P’Cp P/(CpE) p’tcpae) thermal p/cp(e tan6)lI2

Application low impedance amplifier impedance amplifier high imaging device (vidicon) high impedance amplifier when the pyroelectric element is the main noise source

p: pyroelectric coefficient; cp: specific heat;E: relative permittivity a: thermal diffusivity

oom-temperat~eprop some "figures of merit" for their ateri

30 19

1

rature ~ e p e ~ ~ofe ~ cf ei ~ ~ofe m § e ~for t a

om, we can calc~late

138

P 1Cp& 800

300

600 200

400

100

200

18

20

22

(a)

that

Note

24

n

"0

5

10

ias Field (k:V/cm) cb)

Figure of merit ( p / c ~ &change ) with temperature (a) and bias field (b) for ~.67Sr0.33TiO3-based ce as the (a) voltage. s i g n i ~ c ~ t l y(b) . Maximum black b ST at a chopper fr~uencyof 40

Cushion ring Silicon window

I

A polymer-based (PVDF) pyroelectric infrared sensor.

139

infrared ray (input),

-type pyroelectric temperature sensor. Figure6.6shows a typical s t r u c t ~ eformer p ~ ~ l ~ tinr i c practicalusage, a pyrosensorrequiresanilight(thermalray the electrical signal canbe detected only at the~ ~ s i estage n t of light illuminationor as a light-chop shutoff. An elec~omagneticmotor is conventionallyused mechanism,butrecently a piez~lectricb i m o ~ hchopperhasbeendevelopedby ~ u w et~al.$) o which allows forminiatu~zationof the pyrosensors (Fig.6.7).

In Fig.6.8thevisualization of a thermal-dis~butionimage is exemplified by a pyro-vidicon tube.7) The light emitted from an object is filtered with a g e ~ a n i u m lens producingan infrared beam whichis focused onto the pyroelectric target ~ough an optical chopper. The ~ m ~ r a t u distribution re of the object is represented on the target as a voltage dist~bution.This is monitoredfrom the back surface of the target by elec~on-beamscannin~using a conventional TV tube. One of the ~ s ~ of the v p~o-vidicon ~ ~ is gthe degradation ~ of the image over a longperiodofusageduetothermaldiffusiononthetarget. et al. proposed a s e ~ e n t e dtargetdesign to solvethe d i ~ s i o nproblem.8)Figure 6.9 shows the sulphate, microscopic structure of a D-TGS [deuterated triglycine ( ~ D 2 ~ ~ 2 ~ O O ~ ) 3target, ~ 2 Sand O Fig. 4 ~ 6.10 is an example of a picture taken in darkness.

h0

cit ~

~

cof at~yro-vi~ico~ ~ r ~ tu

l

142

hapter

erits pyrosensors of se~conducto~: wide

a)

r

~

m to other p

~

sensor materials such

as

uency,

b) use at c) quick response in c o m p ~ s o nwi temperature d) highquality(optical-gmogeneity,etc.)materi p~osensorsis unnece

sensors,

. Figures of merit for p y r o e l ~ materials: ~c Figure Application of Merit P/Cp P/(CpE) pl(cpaa p/cP(s tan~)1/2

t h e ~ a laging device (vi~con) ~ g h i m~ p~l i f~i e cwhen r e the pyroelec~c element is the main noise source

coefficient; cp:speci~cheat; E: relative ~ e ~ t t i v i t y ;

a: thermal diffusivity ck film s ~ c t is~ essential e for quick rensivity, and a l i ~ h t - c h o p ~ r to mi~aturization. m ~ h a n i s m(e.g., p i e z o e l ~ ~ i c b i m o ~ish sthe )

6.1

as sum in^ the ~ ~ t - o r d ephase r an sit ion for the free energy, calculate thete~peraturedependence of the figures of merit fora pyroelectric detector: p, plcP and p/cp E.

6.2

There is a PLZT (6/80/20)ceramicdiskwith 1 cm2 in thickness electrically poled along the thickness with When the sample is illuminated 0.1 second, calculate the followi ~ ~ s p ~electrode, e n t and (c) theo ~ n - c i r c voltage ~t gene rat^.

143 Assume that all the light energy is absorbed by the sample, and thatno heat loss norelectric loss is taken into account.RefertoTable6.2forthe necessary data.

Total heat energy: 10 ( m ~ / c m 2x) 1 (cm2)x 0.1 (S) = 1 (mJ) Sample volume v: 1 (cm21 x 0.01 (cm) = 0.01 (cm31 Temperature riseAT: 1 (mJ)/ [2.57 (J/cm3K) x 0.01 (cm3)] = 0.039 (K) 6.3

Consider thee materials:sharpphasetransition,diffusephase an sit ion successivephasetransitionmaterials(a, b and c inthefigure)with spontaneouspolarionvs.temperaturerelationsasillustratedinthe followingfigures.cussthemerits anddemeritsofeach fiom a pyrodetectorapplicationviewpointwithrespect to thefollowing:(1)the m a g ~ ~ of d p, e (2) the relative pe~ittivity,(3) temperature stability (4) aging.

I

:

(a) Sharp phase transition

(b)Diffuse phase

tr~sition

(c) Successive phase transition

1) 2)

3)

erbert: ~ e ~ o e l e cTt r~ ~c d ~ e d r s Sensors, p.267, Gordon & New York (1982). Use Its in I n ~ a ~ e d Towcester, NN12 7JN, :~rinciplesand Applic~tion~ Press, Oxford (1977).

144

~ e ~ a materials i n electric charges on their surfaces

ascons a

~iez~lectricity is extensively utilized in the fabrica~onof variousdevicessue t r ~ s d u c eactuators, ~, surface acoustic~ a v devices, e ~ ~ u ~control n c yand so on. used, and various potenti

Thereare five ~mpo~ant figures of meritin p i e z ~ ~ ~ ~ c s : constant g, the elwtromec ,and the acoustic impedan

emagnitude of theinducedstrainx by an external electricfield E is repres this figure of merit (an i m p o ~ nfigure t of merit for actuator applications):

Y

l c field E is related toan e x t e ~ astress voltage cons~ntg (an impo~antfigure of merit for sensor applications): 145

146

Chapter 7

T&ng into account the relation, P = d X, we obtain an impo~antrelation between g and d g = d / EOE

(E

:pe~ittivity)

(7.3)

Obtain the relations~pbetween the piezoelectric d and g constants, which indicates the strain per unitelectric field and theelectric field per unit stress.

From the f u n d ~ e piezoelectric n~ equations:

(P7.1.1)

(P% 1.2) the actuator figure of merit d (external X = 0) is given by Eq. ( the sensor figure of merit g (external E = 0) is given by E q . (P3.4.2): P = d X. The polarization P induced in a material with eo&xresults in an electric fieldof E=P/€()EX = (d X) / €()ex.

( W *1.3)

ng into account E = g X, g = d /E ~ E ~ .

(P7.1.4)

eterms, e l e c ~ o m ~ h ~coupling c a l factor, e ef~ciencyaresometimes C O ~ ~ S ~~ 1. ~1 ) electrical energy andmech~icalenergy, but

or

= (Storedmechanicalenergy

/I ut electricalenergy)

electrical e n e r /~

("7.4)

147 LetuscalculateEq.(7.4),whenanelectricfield E is applied to a piezoelectric material. Since the input electrical energy is (112) EO& E2 per unit volume and the stored mechanical energy per unit volume under zero external is stress given by (1/2) x2 1 S = (112) (d E)2 / S, k2 can be calculatedas k2 = [(1/2) (d E)2 1 S] / [(1/2) E2]

= d2 1 EOE*S. (b)m e energy

trans~ission &oe~&i~nt

Notallthestoredenergycan be actuallyused,andtheactualworkdoneon the mechanical load.~ i t zero h mechanical load ora complete clamp (no strain)m output workis done.

&ax = (~utput mech~ical energy 1 Input electrical energy)max (7.7) or

&ax = ( ~ u ~electrical u t energy

/ Input mechanical (7.8)

Let us consider the case where an electric field E is applied to a piezoelectric under constant external stressX (< 0, because a compressive stress is necessary to work to the outside). As shown in Fig. 7.1, the output work can be calculated as

while the input electrical energy is given by ~EdP=(~&E+dX)E. to choose a proper load to maximize the energy From the maximumc o n ~ ~ of on

(7.10) tr~smission c~fficient. (7.1 1)

we can obtain

& a x = [(Ilk)-II( lk2) ..1 12 = [(llk) + II( lnC2) - 1 1-2.

(7.12)

blem 7.1). Notice that k2/4 h (7.13)

~ c k212 ~ x value. For a small k, hmax = k2/4, and for a large k, hmax =

lc212.

~ h i c his close to the v theoretic~ly.

icalener~y) I( ~ o n s ~ r nelec~ical ener~y)(7.1

or

y ~a~srnission coe

150

~ ~ ~ 7t e r h = - ( S y2 + d y) / (d y + Q&).

(P7.2.3)

e maximum h can be obtained wheny satisfies W d y = [-(2sy +d)(dy +W)+ (sy2 +dy) dl / (dy

+w)~

= 0. en

yo2

(P7.2.4)

+ 2(&0e/d)yo + ( & ~ d s=) 0,

yo = (Qe/d)(- 1 + dl -k2).

(P7.2.5)

ere, k2 = d2 I S EO&. y putting y = yo into h(y), we can get the m ~ i m u mvalue of h:

= [d y0(21k2 - 1)+ EOE]/ (d yo + Q&) = [(-1+ dl -k2)(21k2

- 1)+ 11/ [(-1+ dl -k2) + l]

= [(llk) -d(l/k2) - 112.

e m e c ~ a ~ cquality al factor, e e l ~ ~ o m ~ h ~resonance i c a l spectrum. e resonance kquency 009 the 2 as : defined with respect to the full width at ~ m / d[2do]

M = 0 0 / 2do.

(7.17)

echanical loss (tan S,). de oftheresonant S litude at an off-reson~cef'requency (d E L, L: length of the s ~ p l e is ) ~ ~ ~ f i e d ~ c ~ For a long by a factor propo~ionalto QM atthe ~ s o n ikquency. vibration r e c ~ ~ plate u l ~t ~ o u g hd3 l , the maximum ~ s ~ l a c e m e nist

151

The acoustic is a p ~ e used ~ for evaluating r the acoustic transfer betweentwo materials. It is de~ned,in general, by

en

2 2 = ~ressure/volume (7.18) velocity). In a solid material,

=G,

(7.19)

where p is the density andc is the elastic stiffness of the material. ~iscussions,there m h e kinds of impedances; specific acoustic

acoustic of mech conceptually.

or mechan from one

matching other? E

e m ~ h a n i work c ~ one by one m a t e r i ~on the e a~pliedforce F an

Fi~ure7.2 shows a ~ o n c e p t u ~

c illustrating ~oon two extreme cases. If the materi

F=O

"0

echanical impedance m a t c ~ n

is section s u m m ~ z e sthecurrentstatus of piezoelec c mate~als:s i n ~ l e - c ~ s ~ l materia~s,piezocer~ics,piezopolymers,piezocomposi and ~ i e z o ~ l mTable s, 7.1 ~ ~ ~ e tofesome r s of the p i e z ~ l e c ~ c m a t e ~ ~ s . 8 ) roperties of representati~ep i e z o e l ~

d33 ( P C N

2.3 33 ( 1 0 ' 3 V ~ )57.8

5

593 190 12.6

289 26.1

1700

1300 175

105

500

120 193

328

19.7 3400 65

6 42 0.50 0.03 9 355

33 380 0.30 6 3 10

-

rial p r o p e ~ e depending s on thecut of the materialsand e wave propagation.

z is a ~ e l l - ~ o ~ n

Lithium niobate and lithium tantalate belong to an isomorph0 are composed of oxygen ~ t ~ ~ r The o nCurie . t e m ~ r a t ~of es are 1210 and 660°~,respectively. The crystal s y ~ of the e ~ these single crystals is 3mandthe pol~zationdirection is m ~ h a n i co~pling c~ c ~ ~ c i for e surface n ~ acoustic wave.

3 is one of the most

d o ~ a entering n~ onto with dopants such as phaseover a widertemper 0 3 solid sol~tions[ r p i e z ~ l prop ~ ~ c lutionsystem is d

e

~ by ~theZr n content, ~

154

Chapter 7

~tragonalferrmlec phaseofperovskite s ~ c t ~ e th , increasing a content, x, ~ e fromthe thetetragonaldistortion decmses andat x > OS2 the s ~ c t changes tetragonal 4mm phase to another ferrmlec~cphase of r h o m ~ h ~ a l o ~ i c The line dividing these two phases is called the ~ o ~ ~ t rphase boundary composition is considered to have both tetragonal and rhom coexistingtogether.Figure 7.4 showsthedependence of S cons tan^ on compo~itionnear the mo~hotropicphase boundary. have their highest values near the mo~hotropicphaseboundary. This e~ancement inpiezoelectriceffect is attributedtotheincreasedease of r ~ ~ e n ~ t of i o the n pol~zationunder an applied electric field. ~ o p i nthe ~ PZT material wi donor or acceptor ions changes

its p r o ~ ~ e s

5.t

~ ~ a t i c ~ Donor l y .doping w ionssuch as NborTa5"providessoft like PZT-5, because of the facility of domainmotionduetotheresultingPb3"

P

3.t

vacancies. On theotherhand,acceptordopingwith Fe or Sc leadsto PZTs, suchas PZT-8, becausethe oxy~envacancieswillpindomainwall ~ o t i o n . efer to ~hapter3, Section 3.1(3). in

PZTs

Subse~uently,PZT in ternary solid solution with another perov are: ositions investigat~ m intensiv solution with *

hich are patented by different companies. 500

400

300

00 I I

*

100

n

"

3

10

20

30

zirconate titanate (

3

155

800

600

A

N

2

400

_ . I

X

W

:S

“cl

48 60

50 58

52 56

54

ependence ,of several d c o n s ~ non~composition ne

phase

bound^ in the PZT system.

the m o ~ h o ~ o p i c

e endmemberof PZT, tanatehas a largecrystal dis ro with its te~agonality onal s ~ c t u r eat erature .~ e n s e l ysinter^ P b T i ~ 3 c e r cannot ~ c s be o b ~ n

31°) exhibits an extremely low ere, kt and kp are ~ c ~ e s s tors, respectively. Since these transducers can generate purely longitu~inalwaves ~ o u kt~ ~h s ~ i with a no t tran ~ k3 1, clear ul~asonicimaging is e a zero ~ m ~ r c a ~ e wave. ( acoustic S supe~orsubstrate device applica~ons.

Relaxor ferroelectrics can be either in polycrystalline form or as single crystals.Theydifferfromthelymentionednormalferroelectricsin th exhibit a broad phase ~ a n s i ~ ofrom n the p~electricto ferroelectric state, a fr~uency de~ndence of the dielectric c o n s ~ n t(i.e., dielectric relaxation) and a remanent p o l ~ i ~ a t i o n . relaxor materials complex have perovskite s ~ c t u r e s .

solid solutions.

Y

fer 7

158

One of the very basic applications of piezoelectric ceramics is a gas igniter. The very

high voltage generated in a piezoelectric ceramic under applied mechanical stress can cause s ~ ~ and ~ ignite n gthegas (Fig. 7.6). Therearetwomeanstoapplythe

mechanical force, either by continuous increase.

a rapid, pulsed

applica~onor by

a more

From the expe~mentaldata shown in Fig. 7.6(b), can you estimate the length the ~ i e z o c e rrod ~ c in Fig. 7,6(a)?

al,

L of

If youknowthe

relations~pbetweenthelength L and the mech~icalresonance uencyfr: fr a: 1 I L, andthat 10 mm roughlycorrespondsto 1 0 0 canestimatetherodlength,Fromtheoutputvoltageringing, od is roughly e s ~ m a to ~ dbe 30 pec,or a resonance ~equency of leading to a length L = 30 mm.

~iezoelectricceramicscanbeemployed as stresssensors and accelerationsensors, t~c Figure '7.7 shows a 3-D stresssensor because of the direct ~ ~ e z o e Z e ceffect. stler. By combining an approp~atenumber of quartzcrystalplates (extensionalandsheartypes),themultilayerdevicecandetect ~e-~mensional stresses.17)

output voltage

@W

(a) Gas igniter and (b) output voltage.

159

~oeiectricDevices

Z Y

1

stre§§ sensor (by ~ § t l e r ) .

c ceramic disk

ie

oltage of the piem-

sensor.

= Do sin cut provides the acceleration

iezo-disk is given by 8sS

ezoeiectric disk 0 sin at

Basic s ~ c t ~ of r ean accelerometer.

Base

160

U

rical ~ y r o s c o ~(by e

ctric

€3

e

Chapter 7

162

ese m called the ~iezoeZect~c e ~ ~ ~The i number o ~ ~of .inde~ndentparameters for the lowest s y ~ trigonal e ~crystal are 21 for Si*E, 18 for h i and 6 for h e number of independentparameters incre~ing crystallo~ap~c s y m m e ~ .C o n c e ~ i n g the polycrys oled axis is usually denoted as the z-axis and the ceramic is tothisz-axis(Curie groupC,,(mm)). The number of no inthiscase is 10 (S 1,'l

S1

zE, S 1 3E, ~33'~q4E, d3 1, d3

,Section 2.1.

Next let us in&

S.

(7.20) and (7.21) are applicable :

e S and E termsrepresentpurely m e c h ~ c a and l electricalenergies (U UEE), respectively, andthe d term denotes the energy ~ d &om el u mechanical energy or vice versa through the piezoelectric effect. k is defined by : (7.23) k valuevarieswiththevibrationalmode(eveninsameceramicsample), can have a positive or negative value (seeTable 7.2).

Note that this definitionis equivalent to thedefini~onprovided in Section 7.1( 1): k2 = (Stored mechanical energyIInput electrical energy)

or

k2 = (Stored electrical energy IInput mechanical energy).

~

163 le 7.2 Several shapes of the p i e z o e l ~ ~resonator ic and their e l ~ ~ o r n ~ h ~ c coupling factors. Elastic bundarv. conditions

kr

Resonator shaw

I

Thick mode

I

Width mode

I

Mnition

x1=x2+ 0 x3* 0 Q= 0 ,

XI#

t3+0

I

! /

' -

a1 length tension mode (//E) : nsion mode of the circul

Y

fol~owin~ ~ y n ~ c

1 volume

element in

: e ~ ~ a ~ o n

166 2

X

0

Longitudinal vibration through the transverse piezoelectric effect (d31) in a rectangular plate. Introducing Eq. (7.26) into Q. (7.24), and allowing for XI=U/ x and E,/ x=O to the equal potential on each electrode), leadsa to harmonic vibration equation:

- 0 2 p S1 p U = a2u/ax2 .

(7.27)

Here, o is the angular frequency of the drive field, and p is the density. ~ubstituting a general solution u=~l(x)eiO~+u,(x)e-j~~ into Eq. (7.26), andwiththeboundary condition X1 = 0 at x = 0 and L (sample length), the following solution can be obtained: adax = x1 = d31Ez [sino(L-x)/v + sin(ox/v) /sin(0Wv) . (7.28) Here, v is the sound velocity in the piezoceramic whichis given by v=l/.lps11E.

("729)

W e n thespecimen is utilized as an electricalcomponentsuch as a filter or a vibrator, the electrical impedance [(applied v o l ~ g ~ current) ~ d u ratio] ~ plays an important role. The current flow into the'specimen is described by the surface charge increment, aD3/a t, and the total currentis given by : L L d312/sllE)Ez + (d31/sllE)xl] dx . i = j a w D3 dx = j a w [(E# 0 0 (7.30)

-

c ethe mechanically free sample is calculated tobe: Using Q. (7.28), the a d ~ t ~for ( l a ) = (inr)= (rnzt)

= 00wWt) E33Lc[1 + (d312/ ~ 3 3 L c slE)(tan(0~/2v) l /(m~/2v>l, (7.31)

where W is the width, L the length, t the thickness of the sample, and V the applied voltage.E33LC is the ~ ~ t t i v i in t ya l o n g i ~ ~ n a l clamped ly sample, which is given by

Piezoelectric Devices

167

(7.32) The piezoelectric resonance is achieved where the admittance becomesinfinite or the impedance is zero. The resonance frequency fRis calculated from Q. (7.31), and the ~ n ~ e n tfrequency a l is given by fR

= v/2L = 1/(2L+ S1 1E .

(7.33)

On the other hand, the antiresonance stateis generated for zero admittance or infinite ce:

The finaltransfo~ationis providedby the definition, (7.35)

-

The resonance and antiresonance states are described by the following int~tivemodel. In a high electromechanic~coupling material withk almost equalto 1, the resonance or antiresonance states appear for t a n ( a ~ 2 v )= or 0 [i. e., aL/2v = (m-l/2) or m (m:integer)],respectively.Thestrainamplitudex1 dis~butionfor eachstate [calculated using Eq.(7.28)] is illustrated in Fig. 7.13. In the resonance state, large ~ o ~ j o nc~ Z~ i ~ a me ) strainamplitudesandlargecapacitancechanges(called induced,andthecurrentcan easilyflowintothe device.On theotherhand,at an~esonance,the strain induced in the device compensates completely, resulting in no capacitance change, and the current cannot flow easily into the sample. Thus, for a high k material the first antiresonance frequency fA should be twiceas large as the first resonance frequencyk.

Resonan~e m=1

e

Antiresonance

Lrtw coupling

High coupling

7.113 Strain generation in the resonant or antiresonant state.

168 h a typical case,where k31 = 0.3, the an mentioned mode and becomes closer to material exhibits c h ~ g is e c o m ~ n s a ct ~

a ~ ~ r othe a c resonance ~ frequency fR. e general processes for calculating the

e l ~ ~ o m e c h P~ i c ~

s11E, and~ 3 3are ~ )described below: e soundvelocity v inthespecimen is o (refer to Fig. 7.14), using

~ from~theresonance n ~

density p, the elastic com echanical coupling factor k31 isc couplingpiezoelectricmaterials,thefollowing available:

a~proximateeq~ationis

"

-

k312 f (1 k3I2) = (~2f4)(Af f

(7.36)

Figure 7.14 shows observedim~edancecurves for a typical k mate~al(PZT 5 = 0.70) and a high k mate~al(PZ~-PT single crystal, k33 = 0.90). Note and ~ t ~ ~ s o n a n c e

(7.37) (7.38) co~espondsto the mech~icalloss.

In con~ast,the equiv ent circuit for the ~ ~ ~ s o state n ~of c e shown in Fig. 7.15 (b), which has high im~e~~ce.

P

P

I

~ ~ i v a l ecircuit nt of a the an~resonancestates.

vice for (a) the

r e s o n ~ c eand (b)

170

Chapter 7 Elastic vibrator \

Piezoceramic U

U

Piezoelectric buzzer.

In the use of m e c h ~ c a vibration l devices such as filters or oscillators, the size shape of a device are very impo~ant,and both the vibrational mode and the w r m c f bending the mode in a material must considered. be The reson &om 100much to lower than that centimeter-size sample ranges ofthe thickness mode (100kHz).Forthatorapplicationsthe p i e z ~ e ~ c )ratherthanalargepiezoelectric shouldhaveahighmechanicalqualityfac coe~lcientd; that is, hard piezoelectric For speakers or buzzers, audible by humans, devices with a rather low ~sonance ~ ~ u e n are c y used (kHz range). Examples are a b i m o ~ hconsisting of two piezoceramic plates bonded together, and a piezoelectric fork consisting of a piezo-device and a metal fork. A. piezoelectricbuzzer is shown in Fig. 7.16, whichhas merits such as high electric power efficiency, compact size and long life.

Ultrasonicwaves are now used in various fields. The sound source is made &om piezoelectricceramics as well as magnetostictive materials. P i e z ~ e r ~ m cs generallysuperiorinefficiencyandinsize to magnetostrictivematerials. QM are preferable. A liquid p ~ c u l a r ,M piezoelectricmaterialswithahigh m e d i is ~ usually used forsoundenergytransfer.Ultrasonicwashers,ultrasonic microphones for short-distance remote control andu n d e ~ a t e detection, r such as sonar and fish finding, and nondes~ctivetesting are typicalapplications.Ultrasonic scannin~detectors are useful in medical electronics for clinical applications ranging from diagnosis to therapy and surgery. Oneofthemost importantapplications is basedon ultrasonicechofield.20y21) Ultrasonic transducers convert electrical energy into mechanical form when generating an acousticpulse andconvert m ~ h energy ~ c into~ an electricalsignal when detecting its echo.Thetransmittedwavespropagate into a bodyandechoes generated which travel back to be received by the same~ n s d u c e r .These echoes vary in intensity accor~ngto the type of tissue or body s~cture, there~y creati~g images. of the tissue, such as An ultrasonicimagerepresentsthemechanicalproperties

171 density and elasticity. We can recognize anatomical structures in an ultrasonic image are easilydiscerned.The sincetheorganboundariesandfluid-to-tissueinterfaces ultrasonic imaging process can also be done in real time. This means we can follow rapidlymovingstructuressuch as theheartwithoutmotiondistortion. In addition, ultrasound is one of thesafestdiagnosticimagingtechniques. It doesnotuse ionizingradiation like x-rays andthus is routinelyusedforfetalandobstetrical cardiac structures, the vascular imaging. Useful areas for ultrasonic imaging include systems,thefetus andabdominalorganssuch as liver andkidney. In brief,it is possible to see inside the human body without breaking the skin by usinga beam of ultrasound. Figure 7.17 showsthebasic ultrasoni~t r ~ d u c e rgeometry. The ~ ~ u c ise r mainly composed of matc~ng, piezoelectric material and backing layers.22) One or morematchinglayers are usedtoincreasesound tr~smissionsintotissues.The backing is added to the rear of the transducer in order to damp the acoustic backwave and toreducethepulseduration.Piezoelectricmaterialsareusedtogenerate detectultrasound. In eneral,broadbandtransducersshould be usedformedical ultrasonicimaging.Thebroad band wid^ responsecorrespondsto a shortpulse length, resulting in better axial resolution. Three factors are importantindesigning broad band wid^ transducers; acoustic impedance matching, a high e l ~ t r o m e c h ~ c a l couplingcoefficient of thetransducer,andelectricalimpedancematching.These of the piezoelectric pulse echo transducers operate based on thickness mode resonance thinplate.Further, a lowplanarmodecoupling coefficient, kp, isbeneficialfor A largedielectric limitingenergiesbeingexpendedinnonproductivelateralmode. constant is necessarytoenable a goodelectricalimpedancematchtothe system, especially with tiny piezoelectric sizes,

ig. 7.17 Basic transducer geometry for acoustic imaging applications.

(a)Vi~rator~ I ~ r n ~ ~ t

ay type ultrasonic are various typesof tran dis~eteelementsto be in& ic focus in^ in the S the use of phase del A lin (orsector). lrectlon, pro~ucin~ a rec is a modified linear

so

S

iezoelectric body vibrates resonant its at absorbs consid anatother ~ ~ u e n result c i ~in^ iezoelectric m a t e ~ ~ s

173

~ e ~ ~band e ~orcto yblo ~ i e ~ o emate~al l ~ ~ cis

e about 5.6 mm. m ~ e of s vi~ratio fits smaller size.

Chapter 7

174

A swji3ce a c o u s ~wave ~ (SAW),alsocalled a ~ a y l e i gwave, ~ is essentially a coupling between longitudinaland shear waves. The energy carried by the SA^ is confined near the surface. An associated electsostatic wave exists for a SAW on a oelectric substrate, which allows electroacoustic coupling via a transducer. ages of SA^ technology are:23,24) *

(1)Thewavecanbeelectroacoustically surfaceand its velocity is approxi electromagnetic wave. SA^ wavelength is on the same order of magnitudeas line dimensions produced by photolitho~aphy the lengths for both short and long delays are achievable on reasonably si There is a very broad range of commercial system applications which include frontte filters, CATV (Community Antenna Television) end and IF ( I n ~ ~ e d i aFrequency) and VCR (Video Cassette Recorder) components,synthesizers,analyzers navigators. In SA^ transducers, finger (i~Eer~igiEa~ electrodes provide the ability to sample or tap the wave and the electrode gap gives the relative delay. A SA^ filter is composed of a minimum of two transducers. A schematic of a simple SAW bi~ r e c t i o nfilter ~ is shown in Fig. 7.20. A bi-directional transducer radiates energy equally from each side of the transducer. Energy which is not associated with the received signal is absorbed to eliminate spurious reflection. Various materials are curren single-cry st^ SA^ material materials have different prop directionofpropagation.The material fora given device applications are SAW velocity, temperature c ~ f f l c i e nof~ delay(TCD), electromechanic~coupling .factor and propaga~on loss. Surface be generatedand by spatiallyperiodic, in~rdigial acousticwavescan S on the plane surface of a piez~lectricplate. A periodic electric field is when an RF source is connected to theelectrode,thus pe~itting source with a Wuency, piezoelectric coupling to a traveling surface wave.If an f, is applied to theelectrodehavingperiodicity, d, energyconversionfrom an electrical to mechanical form will be m ~ i m u mwhen ~~~

(7.39) where vs is the ~A~ velocity and fo is the center frequency of the device. The SAW velocity is an important parameter d e ~ ~ n i nthegcenter i m p o ~parameter t for many applications is t e m ~ r a sensitivity. ~e For example, the tempera~restability of the center frequency of SAW bandpass filters is a direct function of the tempera~recoefficient for the velocity and the delay for the materi~ used. The first-order temperaturec~fficientof delay is given by:

iezoe~ectric Devices

175

r-"

"l

I

l"

~ u n d ~ e n tstructure al of a surface acoustic wave device.

where z = L / vs is the elay time and L is the S

propagation length. The s d a c e

wave coupling factor,ks2 ,is defined in terms of the change in SA^ velocity which occurs when the wave passes across a surface coated with a thin massless conductor, so that the piezoelec~cfield associated with the wave is effectively short-circui The coupling factor,ks2 ,is expressed by : 2 (7.41) S =2(vf-vm)/vf e wavevelocity and vm the velocity onthe metdliz ks2 relates to the ma~imum plications,thevalueof bandwidth obt~nableand the ~ o u nof t signal loss between inputand output, which ~ e t e ~ n the e s ~ a c t i o ~band a l wid^ as a f~nctionof mini mu^ insertion -loss for a given material andfilter. Propagation loss is oneof the major factors thatd e t e ~ i n e s

insertion loss.

marizes some impo~ant

SA^ material properties.

ark Li N W 3

ST-X 12wY - X

0.16

5.

LiTQ Li2B49

0

3158

4.5

-74 -18

(110)-

0.8

0

1.o

10

26

c l

0.

-15

1.o

8.5 5

A delay line can be forrned from a slice of glass such as Pb glass inwhich the velocity ofsound is nearly inde~ndent ceramic transducers are soldered on two metalli inputtransducerconvertstheelectricalsignalto a shearwavewhichtravels through the slice. At the output transducer the waveis signal delayedby the length oftime takento travel arou are used in colorW sets to introducea delay of ap loyed in videotape recorders.

input and output terminals are f a b n c a ~on a

e is changedthrough thevibrationenergytr ~ ~ e z o e l e c t ~ c t r a n s f Piezoelectric o~er. transforrners of their com act size in c o m p ~ s o nwith the convention^ electroma~neticcoilt r ~ S f 0 ~ erious problems were found initially in the m~hanicalstre (collapse nodal thepoint!) heat and in ~ e n e r a ~ o nthe , development approach e as that used for fabricating ceramic actuators. Recent lap-top computers with a liquid crystal display require a very thin, no electromagnetic-noise transforrner to starttheglowof a ~uorescentb a c k - l ~ p . is application has recently accelerated the development of the piezo-transformer.

177

i e z o e l e c ~ c ~ a n s fproposed o ~ e r by theoriginal p i e ~ o - ~ ~ s fwas o proposed ~ e r by C. A. variety of such transfo~ers investi~ate~. Figure 7.21 wheretwo ~ e r e n ~ y - p o l eparts d coexist inone 1 s~ndingwave wi a wavelen~ the sample to wavelength existing on both the in~ut(L1) and output ratio r ( ~ t er ~ t -iis~~~iven ) ~ for the unloaded condi~onby : *

(7.42) e r ratio is incre~edwith an increase of (L2I t), where t is the e ~ ~ s f o (Fig. ~ e 7.22) r in to i n ~ e the ~ e voltage rise r ~ ~ o . z 6Usage ) of thethirdorder l~ngitudinalmode is anotheridea to ist tribute the stress concentration.

ultilayer type transfo~erby

178

Chapter 7

UsingMason'sequivalentcircuitsfortwolengthexpanderbars,surface and end electrded, as shown in Fig. 7.23, calculate the e q ~ v ~ e circuit n t for a Rosen type transfo~er.

A completeequivalentcircuitfor

a length expanderbar (top and bottom swfkce with electric field ~ ~ n ~ c utol the a rdirection of wave propagation is provided in Fig.7.23 (a), where

i

(P7.7.1)

/sinh (@L,/vbE),

22i =:

l:N

2

l" V I I

*

ent c i r c u i ~forlen

179 The ch~ctensticmechanicalimpedance 20and theclampedcapacitance provided by: ~ i = p w t v b ' = w t ( p / s l l E 112 ,

CO

(P7.7.3)

a lengthexpanderbar In asimilarfashion,the necessaryparametersfor electroded) with electric field parellel to ~rectionof wave propagation are givenby:

(P7.7.6) (P7.7.7) (P7.7.8) 0 = p W t V b ~ = W t ( p / s 3D 3 )112 ,

-

coo = wt &33T(1 k332)/ L,

(P7.7.9) (€37.7.10)

NO = wt d33 / L ~ 3 = (3w a~) (&33T/~33D)1/2 k33.(P7.7.11)

ircuits for (a) one-end five length expander bar (surhce length expander bar (end electmded), and (c) the Rosen

n one end of the piezoelectric element is free lication a

as s

must be replaced by (L/2)

voltage ratio for an ope^-circuit condition c

L2) = n / L2 (p /S33 ng into account the relation:

S

182

ter 7’ Piezostrictof

-20-10

Electric (kV/cm) field Electric

~iectrostric~or PMN-PT

BST

-5

-15-10

0 10 20 field (kV/cm) Electric

0 5 IO 15 field (kV/cm) (b)

(a) Phase-change material

PNZST

-30 -20 -10 0 (kV/cm) field Electric (kV/cm) field Electric

10

20

30

-30 -20 -10

in

ctive

0

10

20

30

18 MOVING P I E C E LEAF SPRING

CONT

La~hingrelayusing a shape memo^ ceramicunimoh. ires a 4 mspulsevoltage,not a continuousvoltage,whiches unimo~htip displacement.

a 150 pm

Two of the most popular actuator designs are the mu1tilayers3’) and b i m o ~ h s(S l 100 thin p i e z ~ l e c ~ c / e l ~ t r o s ~ c t i v e es of low driving voltage (1 force (10oO N), and hi e l ~ t r o m e c h ~ ccoupling. al Butthe dis~lacement,ontheorder of 10 pm, is n suf~cientforsomeapplications. This contrastswiththe characte~sticsof the b i m o ~ hwhichconsists of multiple p i e z ~ l e c ~ c tic platesbondedtogetherto gene^^ a large ~ n d i n gdisplacement of sev pm, but has relatively low response time (1 ms) and genera~veforce (1

Multilayer

I

!

U Single Plate

L li ./ i

i

m '

Moonie

/ Typical designs for c e r ~ i actuators: c mul~layer,moonie and bimo

Z-stack (1 0 layers) (extension)

X-stack (10 layers) (shear)

Y-stack (10 layers) (shear)

ing ~ u l t i l a ~actuat er

electric field d i r e c ~ o ~ .

instance, requires a very hardp i e z o e l ~ with ~ c a hig ,to suppressheat genera~on. Drivingthemotorat rather than at resonance,is also an i n ~ g u onthe p i e ~ ~ eand r the ~ power c s u p p l ~ ~ 2" )h suffers most from strain hysteres thispurpose.Thepulsedrivem quickresponsewithacertainow ap~lication.

X

b

~lassificationof piezoelect~c/elec~os~ctive actuators.

186

displaceme Etectric field

n-t (a)

ent vibration of a bi scale with a unit of half of t Ink ribbon

187 actuators are very impo~antfor improving the 7.30 shows ~ansientvibrations o applied.Therisetime is varied with a unit of To12, where To stands for that the overshoot and ringing o ~splacementis completely suppress^ when the rise time is precisely adj -device(i.e.,forn = 21.43) A flight a c ~ a t o r ent a d a steel ball. A 5 pm an hit a 2 mm steel ball up to 20 m using a flight actuator as se width, the movementof to realize no vibrational ringingor double hit~ng,

~ o n s i ~the e r longitudin m e c h a ~ vibration c~ in a iezw b, width W and length L (bc< d y n ~ i equation: c

n the followingd y n ~ i c

e~ua~on:

seud~stepvoltage, as demons 7.30.

7,8.3),usingtheLaplace z(t) as U(s,x) and

18 S1

p

S2

U(s,x) = ~2U(S,X)/~X2'

is ~ s u m e the s ~ o ~ l o winitial i n ~ conditions: u(t=O,x) = 0, ~u(t=O,x)/~t = 0.

superi~posin~ the d3 1

onse to a pseudo-s

o r n = 1,

0 c t c Liv

For n = 2,

Thus, 0 c t c Liv

u(t,L) = (d31E0v2i4L) [t2 -2 (t -L/v)~] L B, and freely betweenB --> A. If the vibratory piece and the piezo-vibratorare tuned properly, theyform a resonatingstructure,andif the bendingdeformation is sufficiently small compared with the length, the tip locus during the free vibration --> A) is representedby

x = uo sin ( a t + a), y = u1 sin ( a t + p),

(7.47)

which is an ellipticallocus.Therefore,onlytheduration A -->B provides a uni~rectionalforcetotherotorthrough fiction, and,therefore, an intermittent

I

W'

'S.

\

l

l l

ller

l

Al horn

ler ~ l ~ a s o nmotor ic (a) and the

motionof the torsional

202

r

1 \

ca

"~indmill"motor with a dis~-shapedtorsional coupler.

A com~actultrasonic roto^ motor, as tiny as 3 mm in eter er, has been devel at the Pennsylv~niaState University. As shown in Fig. 7.47, the stator cons a p i e z ~ l e c ~ring c and two concav~/convexmetal e n d c ~ swith "windmill" n torslona slots bonded together, so as to n e ~ t ea coupling ofthe u ~ - d o ~and vibration^.^^) Sincethenumber of components is process is muchsimplified,the f a b ~ c ~price ~ nis disposable design becomes feasible. When driven revolution of 600 rpm and a m ~ i m u m t o r ~ofu1e eter er motor. a p i e ~ o e l ~ceramic ~ c cylinderfor 7.48).64) Using interdigital an type electrode with

on the cylinder surface, torsion vibration simple ultrasonic motor.

was

a torsionalvibrator (Fig. a 4 5 O cant angle ch is applicable for a

Ceramic

Jw-Cylinder

(a)

Piezoelectric cylinder torsional vibrator(a) and its electrode pattern (b).

203

BOLT Two-vibration-mode coupled type motor. Uehaproposed a two-vibration-mode coupled type (Fig. 7.49), that is, a torsional Langevin vibrator was combined with three multilayer actuators to generate larger longitudinal and transverse surface displacements of the stator, as well as to control their phasedifferen~e6~) The phase change can change the rotation direction.

Uchinoinvented a m h nearmotor.66)Thislinearmotor is multilayerpiezoelectric r andfork-shapedmetalliclegs as sho Since there is a slight difference in the mechanical resonance fiquency between two legs, the phase Merence between the bending vibrations of both legs can controlledbychangingthedrive fkquency, Thewalkingslidermovesin a way similar to a horse using its fore and hind legs when trotting. A test motor, 20 x 20 x 5 mm3 in dimension, exhibits a m ~ i m u mspeed of 20 c d s and a maximum thrust of 0.2 kgf with a maximum efficiency of 20%, when driven at 98lcHz at 6V (actual power = 0.7 W). Figure 7.51 shows the characteristics of the linear motor. "his motor has been employedin a precision X-Y stage. Tomikawa's rectangular plate motoris also i n t r i g ~ i n g 6 ~As ) shown in Fig. 7.52, a forms an elliptical displacement. The combination of thetwomodesofvibration chosen werethe1st longitudinal mode (L1 mode)and the 8th bending whose resonance frequencies were almost the same. By applying voltages with a phasedifferenceof 90 degreestotheL-modeandB-modedriveelectrodes,

elliptical ~ o t i o nin thesamedirecti

To Oscillator

I

O T 114 T 214 T 314 T

I

?

i

I

1 T

linearultrasonicmotor. (a) cons egree phase ~ f f e of ~ two n ~le

V

f

v

>

10

) .

t-

P

0

0

50 LOAD m (g f )

1OQ

m

otor characteristics of the shape^ motor.

20

........

~ ~ h i and dUeha aet

al.

linear motor as i l l u s ~ ain t~ edat both ends of a steel

s-sec~on. Ass

206

Horn( l: 4 )

Piezo brato or 20 4

Linear motor using a bending vibratio~.

V

= (E I / p A)

d ~ ,

(7.50) (7.5 1)

Usingthebendingvibration,thewavelength h canbeeasilychosenasshort several mm to satisfy a stable surface contact with the slider section areaA or the momentof inertia I of the ~ansmission 7.53, h = 26.8 mm. slider, the c o n ~ cface t of which is CO ssionrodwithan app~op~ate force.Thetran bythevibrationsourcepositionontherod, distance from the free endof the rod to thePO intoaccountthewavephase,thevibration co~espondingto one wavelen th h (i.e., 26.8 mm) slider, madeof a steelc l ~ p e60 r m~ i waves, was driven at a speed of 20 cm/s w problem with this type of motor is foundin its lo t excitedevenwhenonly a the whole rod ~ u s be output. Thus, ringtypemotorswereinventedby be utilized, because the lengthsof the stator arr

as

Whenwedeformthe rod discussedintheprevioussection to make a ring by conn~tingthe two ends topologic~ly,we can make a rotary type motor using a bending vibration. Two types of "ring" motor designs are possible: (a) the bending mode type and (b) the extensional mode t ~ p e . 7 ~ ) A l t h o uthe ~ hprinciple is similar to the linear type, more sop~sticatedstructures are employed with respect to the ceramic poling and the mech~icalsupport mech~ism. a vibration source drives one positian of a closed ring (circular or co~espondin~ to the resonance of this ring, only a standing wave is excited, because the vibration propagates in two directions sy~e~cally from the vibration source and i n ~ ~ e r e n coccurs. e When multiple vibration sources are installedonthering,displacementscanbeobtainedby superi~posingall the waves (two waves from each vibration source). Using the superimposition principle, we can generate a ~ropagatingwave in the closed ring with theprofile of the original s t ~ d i n gwave. Assuming a vibra~onsource of A cos cot at the point 8 = Q of the elastic ring, the nth mode s~ndingwave can be expressed by u(0,t) = A cos ne cos a t ,

(7.52)

and the traveling wave by (7.53)

Sincethetravelingwavecanbeexpressed waves as

as a superimposition of two

cos cot + A cos (ne

- 2) cos ( a t -W2),

(7.54)

nce is m ~ n ~ inn Seace ~

208 ightypercent of theexchangelensesin beenreplacedbythe ul~asoni nic motorsdone in the Uni modi~cationsof Sashida's type.

anon's

I'

i

U

U

Vibration source positions for g~n~rating a propa~atingwave in a rin Slider

Elastic nng

Rotor

/

Rotor

Felt

Stator structure of Sashida's motor.

209 into 16 pos~~vely and negati ions so as to generate a 9th pe was c o m ~ ofs a~brass ring 2.5 mmin thickness, c e r ~ i ring c of 0.5 mm in thic~nesswith di shows S~hida'smotor characteristics. ng" motor for a camera automa~c is ins~llingtheringmotorcompactlyinthelensfiame.It stator elastic ring has many ich can ma~nifythe S lens position can be S displacement and improve the with a screwmechanism.The advan~gesof this motoroverthe e l ~ ~ o m a ~ motor n e ~ care:

conv~ntional

1. Silent drive dueto the ul~asonic ~equency drive and no gear mech *

,more s u i ~ b l eforvideo n motor design and tion no

as withmicrophones).

mec~anismsuch as

rv0 sin ot v0 cos ot

Ground

"

E "

f

-

Torque (gf cm >

otor characteristics of Sashida's motor.

=44 kHz

A general problem encountered for these traveling wave type motors is the support of the stator. In the case of a s t a n ~ nwave ~ motor,thepoints or lines %11p: generallysupported;thiscausesminimumeffectsontheresonancevibration. A travelingwave,however,doesnothavesuchsteadynodalpoints or lines. Tlt.us, special considerations are necessary. In Fig. 7.55, the stator is basically s u p p o ~ verygentlyalongtheaxial dimtion on felt so as not to suppressthebending vibration. It is important to note that the stop pins which latch onto the stator teeth only provide high rigidity against the rotation. atsushitaBlectricproposed a nodal line support method using a higher vibrationmode [see Fig.7.57(b)].73)Figure7.57(a)shows the stator structure, where a wide ring is supported at the nodal circular line and "teeth" the maximum~ p l i t u d ecircle to get larger revolution, Seiko I n s ~ m e miniaturized n~ the ultrasonic motor to dimensionsas tiny as 10 mm in diameter using basically the same ~ ~ n c i p l e . ~Figure ~) 7.58 shows the c o n s ~ c t i o nof one of these small motors with a 10 mm diameter and a 4.5 mm thickness. A driving voltage of 3 V and a current of 60 mA produces 6~ rev/min (no-load) with a torque of 0.1 mN*m.AlliedSignaldevelultrasonicmotors ~ for laun~hin similar to Shinsei's, which are utilized as m ~ h a n i cswitches missiles.75)

VI

(

OUTPUT POSl T ION

,~ O ~ A L T

0 I S P L A ~ ~ ~D~I STR N TI BUT I ON IN RADIAL DIR~CTION

(a) T o o t h - s h a ~stator and (b) a higher nodal line for~ x i n g .

POS I ION

zoelectric Devices

OTOR,

' ~ ~ P P O PR T FOR !SIXTO

ruction of Seiko's motor.

21 1

21

r7

SlON

inn in^" type motor by To

Analo~y

V

V

l

214

most suita~lemethod for achievingo~timum

verthewhole

be ~

0

vi~ration

~for driving l the i u l ~ ~~ o nmotor. i ~c

000

40

1500

30

1000

20

500

10

kc. ._.

0 0.01

0 0.02 0.1 0.05

~ i ~ r athe t veloci iofo ~

0.

0.5

l

q u ~ factor i ~ ) reson~cesof a PZT

Servo acement e

215

1.

Piezoelectricfigures of merit: (a) p~ezoelectricstrain c o n s ~dt x = d E (b) piezoelectric voltage constantg E = g X (c) e l ~ t r o m e c hcoupling ~ c ~ factork k2 = (stored m ~ h ~energy i c I~input electrical energy) = d2 / E O € - S (d) mechanical quality factor Qm = ~ 2 A ~

--

--

--

2.Piezoelectric equations: ( i j = 1,2,..., 6 ;m,k = 1,224

k.

and an~esonancemodes are both ttancemaximumandminimumcorrespond ~ ~ e s o n ~respectively. c e , re son^^

m~h~cal ~son~ces. to resonance

4. C l ~ s i f i c a ~ o nceramic of actuators:

~is~lacement Technique Actuator Category aterials ve gid ~spl~ment

~ ~ sEdl ue c ~ e ro s ~ c t o r motor drive Pulse ~ ~ ~ omotor n i c

demerits of the u l ~ ~ o nmotors: ic

oft p i e z ~ l e c ~ c

216

ow to generate a traveling wave on an elastic ring: n-th mode standing wave: u(8,t)= A cos n8 cos cot n-th mode ~ a v e l i nw~ A propagatingwave whose phases differb

7.1

Calculatethe vibrator for (a) (b)

electromechanic~couplin~factor thefollow in^ vibration mode: Length extension mode Shear modeon the p1

(a) A multilayer a c ~ a t o ris

217

verify thatthe follow in^ approxim couplin~piezoelectric material:

-

k3l2 /(1 k312) = (7t2/4) (6f lfR). (6f = fA

-

(b) Using a pulse drive technique, the ~ansientdisplacement was as a function of time, and the displacement curve was obtai how to determine thek3 1d3 1and Qpvl values from the data.

ve method is analte~ativemethod c~aracte~stics. By apply in^ a step electric field piezoelectricsample,the ~ansient vibra~on correspondi mode (extentional, ~ n d i n getc.) is measured. The s t a b i l i ~displacement and damping constant are obtain fromwhichtheelasticcompliance9piezoelectric cons~t,

Applied Voltage CV)

7.

a rectan~ul~ p i e ~ ~ l e c tplate r and i the c ~ctuator(pinball machine).

~ ~ s v e r ~

n e ~ a ~ vpulse e (-Eo) is appli xed rigidly at oneend, verify

to a

Chapter 7

218 other end is given by2ld3111Egv (v: sound velocity of independent of the length.

the ceramic), and is

(b) Supposethatthisvelocity is ~ o u g ha smallsteelball (mass: M) without loss. Calculate the m ~ i height m ~of the steel ball, when the ball is hit exactly vertically. 7.5

Fromthestraindistribution xl(x) for a low e l ~ ~ m ~ h coupling ~ c a l material pictured in Fig. 7.13, draw the ~splacement dis~bution u(x) for both the resonance and an~esonancestates, and discuss the nce between the two states.

7.

For the equivalent circuit of a piezoelectric ~ s d u at~ther a n ~ s o n a n c e intrinsic physical state [Fig. 7.15(b)], derive the relations of L and C to parameters such as p, d, sE and the dimensions of the transducer.

7.7

general principle for u n i m o ~ hs ~ c ~ e . modeu(8,t) = A cos (28 c o n ~ g ~ a t i o ntos beapplied t mode of vibration. (There wil

-

Jaffe: ~iez~electric C ~ r ~ ~ London: i c s , A ~ a d e Press ~c

1114 (1982).

~ y ~ ~ o s697 i u(1990). ~ ,

I990 ~ l t r a s ~ n i c s

219

V?. A. Smith: Proc. 1989 IEEE ~ l t r ~ o nSymposiu~, ic 755 (1989). Newham:Jpn. J. Appl. tee on Barium titan at^,

XXX~-171-1067 (1983). .A. Auld: A c o ~ t i c~ i eand l Waves ~ in Solids,' 2nd ed., elb bourne: *

Imaging and AnalogSignal Processing, (1987). no: IEEE ~rans.Sonics ~ltrason.,SU-25,

C. Campbell: S ~ Acoustic ~ Wave ~ Devices e and ~ ~Signal i Processing r Applications, San Diego, Calif, Academic Press (1989). atthews: Su~aceWave ~ilters, New York: Wiley Interscience (1977).

r ~r~cisi~ o no s i ~ ~ o no n t ~Edit. ~ l iltl, Chief

23(3),187 (1980).

.Uchino: CeramicData

ook '88 (Chap.:CeramicActuators),Inst. ~ufacturingTech., Tokyo (1988).

( 1 990).

,U. To~ikawaand T. Takano:

Indust~al

45)

Ota, T.~ ~ h i k a wand a T. ~izutani:Jpn.J. Appl. Phys., (1 985).

7'.

221

222

hapter

Electric field

on-linear polarizability of fenoelec various electrooptic and optical par^ , problems still r e m ~ nin prep crystals and, hence,manufact~ingc epolycrystallinemicros ctrooptic effectif it is si fenoelec~csare of specialinte extraordin~lylarge app~en material is in its so-called p electrooptic properties fenoelec~cs.

seful fe~oelectricelectrooptic material tra~tionallycome from the Ti)O3 system; they generally have sparency in a wavelength ge extending from the visible to infraredy and exhibit optical anisotropy with an applied electric voltage. Figure 8.2 shows the phase d i a of~the~ ~ b l - x L a x ) ( ~ l y T i y ) l - ~ 4 system, ~3 onwhich is indicatedtheelectrooptic effects m a ~ f e s for t~ variousphaseregions.Noticethatthevalence of lanthanumion (3-t) inthea-site (2-t) generates the vacancyof the b-site.

~ ) e PLZT solid solution exhibits boththe Pockels ( p ~ and ceffects, depending on the cition.Some e x ~ p l e sof typical An es are shown in Fig. 8.3. electrooptic coefficien uch larger than the values ntional crystals SUC (SBN) (see Table 8.1)¶which means that the voltage electrooptic shutteris much less for the PLZT.

223

PbZrO,

[mol *h] PbZrO,

PbTi FE,Tat

10

20

30

application.

elation ~ e t w ~PLZT n compostionand

s ~ c t u r eand e l e c ~ o o p ~ c

-20 -10 0 10 2( -20 -10 0 10 Electric field Electric field E [kVlcm] E [kVlcm]

2c

Pol~izationP and b i ~ ~ n ~ e An n c as e a ~ n c t i o nof electric field E for

materials.

Pockels (1st) and aterial

~maryelectrooptic coefficient

0.52 PLZT 8/65/35 ( ~ ~ = 3 ~ m 6.12 )

Secondary electrooptic coef~cient

KTa0.65~b0.35~3 5.30 9/65/35 ( ~ ~ = 2 ~ m 9.1) 2 PL2X 10/65/35 ( ~ ~ = 2 ~ 1.07 m)

>

\

1

P

."-A-A-

El

1. 7km

0 0 -

R D

COEFFICIENT COEFFICIENT

-

BY C. H. H e a r t I i n o \

1 1 . 0 2.0 3.0 4.0 5.0 Grain s i z e

[pm l

rain size de~enden~e of the electroo~tic coe~~cients,

3 x 1.1 x 10-16x 1 x 10-3)

226

5

~ ~ n g e n celectric vs. e field response of

possible phenomeno1ogic.al analysis of this peculiar phenomenon is based on the modelthatthe crystal is composedof coexistingferroelectric andparaelectric phase^.^) Suppose that the volume fractionof the paraelectric phasex(") is given by an cumulated ~aussiandistribution with 'respect to temperature, the b ~ ~ n g e n c e An is estimated by the summationof the linear and quadraticelectrooptic effects4)

An = [1 - x(T)] n3(r33

-r13) E/2+ x(T) n3

(8.3)

where n is the refractive index, andr and R represent theel coefficien~, respectively. Even the if ex~~men~l phenomenologically, the actual situationmay not be so si as x(T) is also a function of the applied electric fieldE. noth her more realisticdesc~ptionis found interns of a m mechanism. very 3 has spindle-l small ~ b i g u o u sb p e ~exn the ~ ctou l ~ field greater than 0.5 kWmm1s applied, the domain walls within a certain region of the sample moves together, such that micro-domains respond to the appli perative manner (See Fig. 8.6).5) Itis n o t e w o ~ ythat the stripe p and bright domains (correspon~ngto up and down p o l ~ ~ t i o n will s ) not '

y domain reversal, and that each domain area changes zero net pol~zationat zero field. The relaxor cry poled easily when an electric field is a p p l i ~aroundthe ~ a n s i t i ~ten eratur depoled c o ~ l e t e l ywithout any remanentp o l ~ ~ a t i o n . “apparent” secondarynodinear effects such as electros phenomena, which occur without any hysteresis.

Domain reversal mechanism in Pb(Zn1/3Nb2/3)03.

2

~ o ~ p o s i t i o xn

e n

=

2.49 of

0.08 0.12 0.10

0. I4

Ti Fraction x

0.16

tive in~exas a f ~ n c ~ Q ofcQm~Qsition n x for (1

- x)

Composition

1/3

(x1

230

The data indicate that the 0.88Pb(Mg1/3Nb2/3)03-0.12PbTiO3has the potential to be a better electrooptic ceramic than PLZT with high m~hanicaltoughness. ~ g h e r optimi~ingthefabrication optical transmit~cemust be achieved,however,by process.

Oneoftheearliestapplications is Ferpic(FerroelectricPicture er no^ Device). Figure 8.10 shows the principle of the Ferpic?) Initally, a PLZT 7/65/35 ceramic plate is uniformly DC-poled laterally [see Fig. 8.10(a)]. Then, storage is achieved by switching domains at points corresponding to the image's high-intensity regions. To switch domains, a high-contr~ttransparency is placedin front ofthe Ferpic a d illuminated [Fig. 8.10 (b)], creating low-impedance regions in the photoconduc~ve film. The writing voltage supply will then cause switching in these regions only. Viewin~readingthe memorized image is accomplishedby passing polarized light ~ o ~ the g Ferpic h andan analyzer as shown in Fig. 8.10(c). a n a l y ~ rareparallel,theregionswithremnantpolarizationnormaltotheplate produce a bright image, and the other regions produce a dark image.

SandiaNationalLaboratoriesdesignedPLZTgogglesforthe U.S. AirForce to provide thermal and ashb blindness protection for aircraft personneL8) The goggle is basically a transverse-mode shutter using an i n t e r ~ g i ~surface l elec con~gurationsimilar to that shown in Fig.8.1 1

PLZT eye glasses for stereoTV (see Fig. 8.11) have been fkbricated using the light shutter principleg) Thelensesconsist of a pair of opticallyisotropicPLZT (9/65/35) discssandwichedbetweentwocrossedpolarizers. en zero voltage is present between the electrodes, light will not be t r a n s ~ ~ e dThe . transmitted light intensity increases with increasing applied voltage, and reaches a m ~ i m u mwhen a phase difference on) of 180° is i n d u d in the PLZT disc (at the half-wave voltage). Stereo TV images of an object are taken by two video cameras co~espondingto the two eyes and the signal from each camera is mixed alternately to makea frame. for the right and left eyes. When viewing, the right and left PLZT shutters are triggered synchronously to each image frame, resultingain stereo image.

231

Electroo~ticDevices

VER

POLA~IZER FERPIC ANALYZER

(c)

~rincipleof Ferpic: (a)initial DC poling, (b) writingprocessusing film,(c) reading process using a pair of parallel polarhers. ~hotocon~uctive

a

2

left

A stereo TV system using a pair of P

current require men^ for high defin been proposed. One of the promisi one-dimensi*nallO) or two- dimension^ utilizing two-dimensional PL pment of a simple mass

n

cal display: (a) a m a ~ se x

e l ~ c o~n ~~~ uer a ~ofo n a (lox 10) device in the figure rep re sen^ one inn

e l ~ ~ Figure ~ ~ 8,14(b) s . shows a picture of anactualdisplay. ayer thickness is about 0.35 mm.

Chapter 8

234

I

i

Fabrication process for the two-dimensional PLZT optical display.

The driving circuit for the display is shown schematicallyin Fig. 8,16(a). terminals of the device areaddressed as shown in Fig. 8.16(b), the image appearingin Fig. 8.16(c) (letter " F ) is generated on the screenel

235 Vertical Electrode External

Picture

l

, Electrode

T

d

1

Vi

Side

View

front

m

ew

chema~ce l ~ con~guration ~ ~ e of a (lox 10) matrix PL e. (b) Top view photograph of a PLZT light valve array with external electrodes.

rightness on a screen vs. applied voltage for red, green or blue light. Note that the half-wave voltage differs for these three lights.

23

1ms

H-l H-2 H-3 H-4 H-§ H-6 H-7 W-8 W-9

W-l 0

v-3 V. 4.

V-§ V-6 v-7 v-l .2.8.9.10

1

CTO

r

n

~

~ r o s s t a test l ~ system.lightthrough ~ ~ .

a slitfocused onthe s~reenis

1

~ f f e ~ innt ~ combinations: ~ t (a) v

Chapt

238

~ r o s s ~was l k monitored on the 2-D display using the setup shown in Fig.8.1'7 with made by keepingonevertical te~inal monoc~omaticlight?) Thetestwas (=P el=@&) on (i.e., Ground) and applying high v01 h o ~ z o n ~ t e ~ i(continuous nals plate-ugh el~trodes)simultaneously.There three different cr~sstuZ~ patterns: vertical, h o ~ z o nand ~ oblique types; that is, light leakage observed at vertically,ho~zontallyand obliquely adjacent pixels. The results are shown in Fig. 8.18(a)-(c) for three ~ f f e ~input nt c and bottom of figures ina pixel indicate the li ht inten the ON and OFF state, respectively verticaland horizon~l crosstal~i intensity, respectively, which does the other hand, oblique type cros & ~ n ~ on n gthe applied voltage and the n u m ~ of r continuous e l ~ t r ~ e s ' address) (called combination type c onfig~ationsis necessary to eliminate

In Fig. 8.15, thefirstmaximuminthelightensity voltages for red, green and blue light; 160 V for 150 blue.

is obtainedat

V for greenand 1

(1) Explain the reason physically.

osing that the ~ ~ c t i index v e n (= 2.49) and the e l ~ ~ o o p t ci c R12)(3.6 x [m2N2]) does not changesignificantlyfor e calculate the wavelength of these thee lights.

-

e half wave voltage is calculated from 2 r , = ( m " o 3 E3(R11 -

=n.

(P8.2.1)

(1) Sincethehalfwavevoltage is providedbyEq. (P8.2.1), accor~ngto the illumination light wavelength, the required voltage differs: for shorter wavelengths, a smaller electric field is required.

(2) Taking into account the electrode gap of 0.4 mm, E3 = 3.55, 3.33 and 2.89 x ,respectively, and a pat~engthL given by (1.O 0.1) mm (note that the surface depth 0.1 mm is an inactive layer):

-

239

h = 2.4g3 x ( 3 . 5 5 ~ 1 0 (3.6 ~ ) ~x 10m16) (0.9 x low3) = 630 [nm] (for red), h = 555 [nm] (for green), h = 418 [nm] (for blue).

(P8.2.3)

Lightwaveguidescanbe f a b r i c a ~by deposit in^ a ~ g h - r e ~ t i index v e layeron a substrate.Theprinciple of the wave~ideis shownschematicallyin Fig. 8.19. 2, Like an optical fiber,the light tends to bend toward high ~fractive-indexside, so that the light should be confined in the narrow high refractive-index layer fabricated on the LiNbO3 single crystals are commonly used. Figures8.~O(a)and 8.20(b) are typical planar and ridge type electrooptic waveguides. 3, The fabrication ofapplied electric field of a planar type is easy, but the nonunifo~ dis~butionthe isproblem. a On the other hand, as you can imagine, the ridge type sophistic~tedmanufacturin~technology, but the device functionis close to t Thetransmittedlightintensity is easily m ~ u l by aapplying ~ a relativelylow voltage. Phase modulation by 1 hieved by applying a voltage of 0.3. V with power consumption of S *

*

240

rod

C

Electrooptic waveguides:(a) plan~-typeand (b) ridge-type.12)

l.

Relaxorferroelectrics are widelyapplicablefor el~troopticlight valve/display applications. "he superior c h ~ a c ~ r i s t i cof sthese materials are athribu p ~ m ~ i to l ythe easy poling of thef e ~ ~ l e micro"dom~ns. c ~ c

2. A electrooptic new cerarnic 0.88Pb( g 1 2 P ~ T i ~with high 3 mech~icaltoughness is one of the ~romisingnew mate~alsfor fori lifetime display applications. 3. A newtypeofPLZTtwo-dimensionallightvalve,fabric

by a tapecasting technique, is one excellent example of a design well-suit~dto mass-produc~onat a low ~an~facturing cost.

4. Light waveguides can be f a b r i c ~ ~ byddepositing a high re a subs~atesuch as LiNb03.

8.1

Let us considera PLZT thinfilm platewiththe follow in^ for lateral electric fieldan

(l

ht

iscuss the merits config~ations.

md d e m e ~of~ the above two elec

~onsiderthe ~ire~ingence and the S electric el^.

e l e c ~ o o p coef~cient ~c matrix for this sy

242

Chapter

"he electrooptic coefficient matrix is given as

E K. Tokiwa and K. Uchino: Ferroelectrics 94,87 (1989). K. Uchino and T. Takasu: Inspec. 10, 29 (1986). F. Kojima, J. Kuwataand S. Nomura: Proc, 1st.Mtg.on Ferroelectric Mater. h Appl. (Kyoto)p.155 (1977). wata, K. Uchino and S. Nomura: Ferroelectrics 2 iie and K. Uchino: Proc. IEEE Ultrasonic Sympl K. Uchino:CeramicsInternational 21, 309 (1995). L. M. Levinson edit.: Electronic Ceramics, Marcel Dekker (New York), Chap.7, p.371 (1988). J. T. Cutchen: Roc, 49thAnnual Sci. Mtg.AerospaceMedical Assoc., New Orleans, May (1978). A. Kumada, K. Kitta, K. Kat0 and T.Komata: Proc. Ferroelectric Mater. h Appl., 2, p.205 (1977). K, Murano: Ceramic Transactions 1 ~ a t e r i a l sp.283 , (1990). K. Uchino, K, Tokiwa, J. Giniewicz, Y, Murai and K. ~ h m ~ Ceramic a: Transactions 14 EZectro-Qptics and onl linear Optic ~ a t e r ~ a p.297 l s ; (1990). M. E. Lines andA.M.Glass:PrinciplesandApplications of Ferroelectrics and Related Materials, p. 604, Clarendon Press, Oxford (1977). I. P. Kaminov: Trans, IEEE; M. T. 2T. 3, 57 (1975).

aTi03) is doped withlanthanumatlevels less than 0.3 S semi conduct in^ with a resistivity i resistivity is drasticallyincreased, t e r n ~ ~ around t ~ e the Curie point. was discovered in 1954,and is referred to as thePTC or P ~ C ~ c o e ~ c ~of e~ ~ et s i s t i v i ~ ~ e ~Since e c t . then it has been investigated intensively by 9.1 showstheimpact of variousdopantsonthe eramic resisti as a function of tempera~re.

100 Ml -

Mn 0.127

10 M

1MI -

F

CI.

~

t100k

e v

1-

cz

*#

10 k

U l k

l00

10

, .

100

200

300

Temperature (“C)

esistivity as a function of temperaturc for several dopedBaTi03 PTC c e ~ c s ~. o p a nconcen~ations t are indicated near each curve. 243

244

esistivi~vs, t isovalent su~stitution d o ~ ~ t s t y have ~ i c a~highe ly ions such as La, §m, Ce or Cd) or the T host s ~ c t u r e . Sincethe t e m ~ r a ~ e closelyrelated with theCurie ~oint,

Pb (resistivity curve ec

is ex

to

e

to

.3.

246

-e

Conduction band

Fermi level

Ns

Grain 't/oundary

Energy-level diagram near

a grain

bound^ of

Inorder to explain the PTC or PTC phenomenon, the most Fig.illustrated in the prop initially 9.4, was which particles (m-type) ceramic barrier) (Schottky er ven is ge by the following equation:

.Q, = e ~ s 2 / 2 & 0 &,~ d where Nd is the concen~ationofdonor atoms at the p e ~ i t t i E~obeys i ~ th &=C/(T-To), above Tc, and that the low resistance at T c is thus accoun the ~ o t e n tbarrier i~ due to the increase in p e ~ t t i v as i~ elowTCthe pe~ittivity falls, but t ~ spontaneo e controlstheelectron concen~ationto

lectronic properties in ceramics are strongly grain bound^. Supposethat a grainbou grains possesses acceptor impurities, and that er is generated as shown dis~butionmodel represented in Fig.9.5: (x) = e =O

(0