IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. sv-18, NO. 1, JANUARY 1971 1

Polarizing Effect With Piezoelectric Plates and Second-Order Effects

Abstract-Piezoelectric plates vibrating in thickness, change their natural frequencies in proportion to the magnitude of the dc potential applied to their exciting electrodes. This effect, known as the polarizing effect, is interpreted in this paper as a result of changes in dimensions of the plates and in the values of their elastic, piezo- electric, and dielectric constants; these changes being due to the electric field in the plates. The magnitude of the polarizing effect is expressed in terms of the piezoelectric strain tensor and the compo- nents of electroelastic, electrostrictive, and electrooptic tensors. By studying the polarizing effect with plates of various orientations the simultaneous determination of the components of these three tensors becomes thus possible provided their contributions to the polarizing effect do not fall behind the limits of accuracy with which the magnitude of the polarizing effect is measured. An application to alpha-quartz plates made in this paper did not lead to determina- tion of all three tensors. However, a satisfactory interpretation of the polarizing effect was obtained on the assumption that the polar- izing effect was caused, apart from the changes in dimensions of the plates, mainly by the electroelastic effect. The relevant compo- nents of the electroelastic tensor were determined.

INTRODUCTION

HE CHANGE in natural frequency of piezoelectric cuts caused by a dc electric potential maintained across their exciting electrodes is known as the

polarizing effect. The object of this paper is to make an attempt at an interpretation of this effect with thickness vibrations of piezoelectric plates placed between two plane parallel electrodes. The following treatment is based on the general theory of vibrating infinite piezo- electric plates developed by Tiersten [l] slight.ly modi- fied and generalized for a small nonzero dc potent.ia1 across the electrodes. In this form, the analysis of the polarizing effect is not based on any general nonlinear theory of electromechanical media [Z] and, consequently, the terms corresponding to the electric body force and body couples are not considered. This approach is strongly supported hy an analogous though somewhat simpler case of polarizing effect with longitudinally vi- brating piezoelectric rods and contour shear vibrations of alpha-quartz plates where a similar treat,ment [3] lead to a quantitatively successful interpretation of the polarizing effect [4] , [5].

BARIC RELATIONS

A thin rectangular piezoelectric crystal plate of thick- ness t, width W , and length 1 is surrounded by two plane electrodes equal in dimensions and parallel to the main

Manuscript received August 6 , 1969; revised May 14, 1970. The author is with the Department of Physics, University of

Khartoum, Khartoum Sudan.

large faces of the crystal plate as shown in Fig. 1. The separation of the electrodes is d, where t, d << W , 1. A dc electric potential V can be maintained across the electrodes. The thickness, length, and width of the plate are parallel, respectively, to the axes X:, X i , and X; of an auxiliary orthogonal set of reference axes, while another basic orthogonal set of reference axes X,, X,, and X , is defined fixedly in the cryst'alline material of the plate [6]. The orientation of the plate with respect to the basic set of axes is arbitrary and the two sets of reference axes are relat'ed as usual by a matrix of direction cosines

x, x2 x,

Assuming that the dc potential V is sufficiently small the following relations are valid

where

2 = 1, 2 , 3 (10)

b = t o t + €(G! - L ) , (11)

F , = a t a l e s j k , k = l , 2 , 3 (12)

e = aman€,,. (13) All the latin indices used nhore assume values 1 , 2, 3 wherever summation is understood. An exception is made by the superscript, z that is not to be understood as R

2 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, JANUARY 1971

summation index unlcss the summat.ion symbol 8 is explicitly used. The sylnhols cilfLl, and mean com- ponents of the tensor of elastic stiffnesses, piezoelectric stress constants, and dielectric permittivity tensor, re- spectively; C,, is the permittivity of the vacuum, and p the dcnsity of the crystalline material. The symbols Sik

and are the cornponcnts of the Kronecker and Levi- Civita tensors, respectively. All other quantities are formally suficicntly defined by the above (2)-(6) and auxiliary definitions (7 ) - (13 ) .

The symbol f in (10) means the natural frequency of the thickness vibrations of the plate. As a solution of (2) only those values of f are obtained that correspond to t,he odd harmonics. This is in agreement with the pas- sibility to excite only the odd harmonics in the arrange- ment shown in Fig. 1. The c1uantit.y E dcfinetl hy CS) is the time-average value of the electric-field intensit’y in the vibrating plate at) the clc potential V across the elcc- t,rodea. The electric fiel(1 E is parallel to the thicknePs of the plate.

All the above equations c:tn l)? ol)taineci from [ l] using the same procedure as in [ l] and assuming non- zero air gap between the electrodes and nonzero potential across the electrodes. They are valid in the MKSd sys- tem of units. The material constants are referred t,o the l w i c wt of l*eitlrence axes S I , X?: X:$ in the crystal. St,rictly speaking, (2)-113) vcrc tlerivrtl for the ease of an infinite pl:lte vibrating bct\v-ocn two infinite electrodrs. Kevertheleas they a w rrgartled as valid as long as t , d << W , 1.

~ 1 A ( ; S l T t T D E OF T I I E ~ O L . i R I Z I > - ( ; EFFECT

It w ; t ~ fount1 experi~nent:~lly that if an electric po- tential AI/’ is :Ipplied to the I) lntc elcctrotlcs, the come- quent change in t.he natural frequency Af is proportional to the magnitude of the potrntial AV. The linearity of the dependence of A j on AV seems t,o be a general feature of the polarizing effect with any mode of vibrations of piezoelectric cuts varied in dlape, orientation, and crys- t’alline material [3]-[5], [7]-[g]. Therefore, it is con- cluded tha t h f / ~ l r = (df/aV) v=n.

It is assumed that the change in frequency is caused by the presence of a nonzero time-average electric field so tha t

where

F;ventually we obtain

For the sake of clarity it is wortllwhile to point out that the condition V = 0 in (14) refers not only to the crystal plate dimensions whose dependence on V through the converse piezoelectric effect is obvious but also to the values of all material constants as long as they may be regarded as dependent on the electric field E and hence on the potential V.

The quantity [ (l/f) (d f /dE) ] E = O is definied as the magnitude of the polarizing effect. As is now seen clearly from (14) the magnitude of the polarizing effect can be determined directly using the experimentally found value of ilf/hT’ and (11) where we use only material constants and other quantities that are already known.

INTERPRETATION OF THE POLARIZING EFFECT The following treatment is aimed a t obtaining a quanti-

tative phenomenological interpretation of the polarizing effect in terms of crystal material constantb. T o this end the relations ( 2 ) - ( 5 ) and the definitions (7)-(13) are widely used. It must be pointed out that these relations were derived from [ l ] for small potentials V and, con- wquently, small wlues of the electric field intensity E only. However, since these relat>ions anti their differentials are used exclusively in reference to E = 0, this lirnit,a- tion does not, restrict using them in any way. The rela- tion (6) 11as nlrcntly been used in the previous paragraph while the limits to its validity, as may be now ollserved, were not trnnsgresbctl.

I n order to simplify the writing of the following rather long formulas thc suffix notation E = 0 denoting the above mentioned phc~lome~~on may he completely- omitted. Therefore the electric-ficlcl dcpenclcrlt plate dinlensions, nlaterial cond:1nt4q as well :IS the electxic-field intensity tlerivntivea of tloth aw, from now on, automatically un- derstood as taken a t E = 0.

In order to obtain a suitable expression for the magni- t,ude of the polarizing effect (2) is now formally differ- entiated with respcct to the parameter E , the dependence on E being assumed wherever it cannot be ruled out for logical reasons. After differentiation and rearrangement of terms it is obtained

H R U G K A : POLARIZISG EFFECT WITH PIEZOELECTRIC PLATES 3

where

= eabiederQodQbc, i, 2 = 1 , 2 , 4 . (17)

As can be seen from (15) the magnitude of the polar- izing effect is given by eight terms, each of them origi- nating from a different part of the transcendental equa- tion (2 ) . The first three terms appearing on the right-hand side of (15) represent the contribution of t'he converse piezoelectric effect,, the third being obviously nonzero only if d - t # 0, i.e., if an air gap exists be- tween the plate and the electrodes. The fourth term in (15) corresponds to t.he electric-field dependence of the effective elastic moduli of the three modes coupled in the vibration. The requirement that the surfaces of the plate be traction free gives rise to the remaining four terms, t,he last of them being due to the change in amplitude direc- tion of the waves associated wit,h the vibration. It is only at this stage that the origin of different contributions to the polarizing effect can he easily traced.

The above expression (15) will b s now reduced to simpler terms of the piezoelect>ric-strain tensor and the electric-field derivatives of elastic stiffnesses, piezoelec- tric stress constants, and dielectric permittivities.

Realising t)hat d is the electrode separation and t the thickness of the plate, we may write

- 1 a(d - 1) 1 at t aE = ( 6 ' d . l ) - 1) - - t ax

= l , rl = t

= 0, rl # t .

Using the known relation hetween the components of the sOrain tensor and the electric-field intensitmy correspond- ing to the converse piezoelect,ric effect it may be written

Putting further p = m/B where 7n is thc nmss of the plate and B its volume, we may write

= - a % ( f f ) f f k + P i P k + Y i Y k ) d z i k (20)

where d:ik and dl,,,,, (i, j, k , l , m, n = 1, 2, 3) are the piezo- electric strain constants referred to the auxiliary and the basic set' of coordinate axes, respectively. Using (18)-(20) the first three terms in (15) result in

ffi(AajffL + + $ Y ~ Y k ) ~ l i l l c (21) where

In the case of alpha-quartz (21) is of t'he order of magnitude of 10-12 V-'.m while the magnitude of the polarizing effect (l/f) (Af /AE) is known to be ten times greater. Similar relations are valid for both DKT anti ADP piezoelectric crystals [g], [S]. Therefore, the re- maining terms must be regarded as essential for the interpretation of the polarizing effect in (15) . Their existence, as will be shown, is closely linked with the electroelastic, electrostrictive, and electrooptic tensors. Our further aim will be to an:dyse the remaining five terms in (15) and resolve then1 into these fundamental components.

Differentiating (12) :m1 (13') we have in1rntrdiately

aF,/aE = a,a,(ae,,,/aE) (23)

ae/aE = a,mffn(aEmn/aE) (24)

that will be substituted in the fifth and sixth terms of (15). I n order to find a suitable expression for the com- ponents appearing in the last term of (15), an auxiliary vector notation is introduced and vectors X'"' and R[' ) , i, z = 1, 2, 3, are defined by their components

x;) = aA:"'/aE, x: = 1, 2, 3 (25)

RI;' = I ' i k - & { k r " ) , k = 1, 2 , 3. (26)

Differentiating (4j and (5) and using the new notation we map write

(X"), R["') = - ( a R ~ ~ ) / d h ' ) A ~ ) , i, z = 1, 2, 3 (27)

X"') = 0, z = 1, 2 , 3. (28) where

4

With respect to the secular equation (3) that can be rewritten in the form of a determinant lZ2!;)1 = 0, z = 1, 2, 3 , only two of the three vectors R:') may be linearly independent for each value of z. Assuming now that the value of z is fixed let it be vectors R:) and R:' where A and B are two definite fixed values of the index i. The two vectors R:' and R:' form a plane also con- taining the vector X'" as follows from (4) and ( B ) . Thus, it may be written

X ( " ) = j,V)RV>, y = A , B (29)

wllcre k y ) and IC;' are two real numbers. Using the former equations (27) and (29j and forming tn.0 scalar products (R:), X'") and (R:), X'") i t is obtained

(R:) , kt") R:)) = - ( ~ R Y ; / ~ E ) A Y , = A , B

(R:"', kF'R:') = -(dR:"2/dEjAkz), y = A , B .

Solving these tn.0 equations with respect to k y ) and IC;:) and substitut,ing back in (29) the vector X"' is fOUTLd

IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, JANUARY 1971

Equation (4) can be rewritten

(rcZ)) ' - p l ( r(z))z + p z - p a = 0 (34)

where

p1 = 8 , J r k

p , = +cabctd., aodrherc, (35)

p , = B c a h c € d e , r o d r * e r c ,

and differentist'ed with respect t o E . Eventually it is obtained

( 3 2)

Therefore

6 = 1 , 2 , 3 , z fixed. (33)

A similar expression can be obtained for :uly vdue assumed by the index z .

In very rare cases it happens that t,he t l~ree vectors RI", i = 1, 2, 3, z fixed, defined in (26) are colinear so that t.he former procedure to det,ermine the vector X'" cannot be used. On the ot,ller hand i t would be possible to shorn that in these cases the vector X"' has no influence on the magnitude of the polarizing effect. However, the vibration modes where this occurs cannot be excited electrically so that the polarizing effect cannot be studied with them either. Therefore, these cases lie beyond the limits of our interest and wilt not be discussed further jn this paper.

The expression for ar(")/aE, z = l , 3, 3, will be found now in terms of d r i k / d E using (4). The calculation is done for a fixed value of z; the result will be valid, as before, for arbitrary z .

( r ( d ) 2 aPl I"" *? aP3

aE + - ____- - aE 3 ( r ( Z J ) Z - 2 p , r ( z l + P2

(36)

Since a r'"'/dE in (36) is to be referred to E = 0 as well as all derivatives in this paper, i t is always the solution of (3) with respect to I"" that is to be substituted for

in (:X). If the three vectors R!') , i = 1, 2, 3, defined by (26) are not colinear, as is the case under consideration, the three roots of (3) are different. Therefore the de- nominator of (36), equal to the partial derivative of (34) with respect t o l?(*) is always different from zero. Thus (36) always makes sense.

Substituting now from ( 3 5 ) into ( X ) we obtain finally

-.

dr")/a6 = ~ t ; ) ( a r , , / a ~ ) (37)

where

= ( 6 , , 9 ~ " ' D::' - 6 . , i . 4 ~ z ) ) N ~ z ~ , y = A , R. (44)

The final fornlul:\ (40) gives the interpretation of the magnitutle of the polarizing effect in terms of four basic contributions. These are, as arrangcd on the right-hand side of (40) ; converse piezoelcctric effect), clectroelastic, electroelastic effect, electrostriction and electrooptic ef- fect. These four effects arc represented by the material constants diik, acijh,/aE,, ae,,,,,/8Ep, and ae,,,,,/aE,, respec- tively.

All the other quantities appear in (40) in a form of coefficients in front of the ahove four types of constants. These Coefficients depend on material constants of the first order, i.e., elastic st,iffnesFes, piezoelectric stress con- stants, and dielectric permittivities, dimensions of the cryst,al plate, and separation of the eleetrodeP. The orientation of the plate thickness with respect to the axes of t.he hasic coordinate system is of decisive importance here as well as the nlotle of vibration depending on the choice of frequency f connected with the threc possihle solutions of (2) . The magnitude of the coefficients also depends, though to a relatively lesser extent, on the order of harmonics of the mode chosen. All these coefficients can he calculated from the data that are generally k n o m heforehand using the fornlulas given in this paper.

The values of the piezoelectric, strain constants d i j s arc well known, thus making it possible to calculate directly the magnitude of the first term in (40). The contribution of the converse piezoelectric effect to the magnitude of the polarizing effect thus may be easily evaluated. The other t,llree basic contributions are quant,itatively un- known since the values of the respective quantities acijl;;JaE,, ae,,,ii/aEp, d ~ , , , , , / d E ~ have not yet been all sys- matically mcasured. These quantities form tensors of the fifth, fourth and third ranks, respectively, having, in the case of the lowest crystal symmetry, 63, 36, and 6 inde- pendent components. However, these numbers are re- duced substantially with increasing crystal symmetry. With typical piezoelectric crystals such as alpha-quartz > ~ n d ADP they all total 17 and 16 only.

411 the quantities appearing in (40) and throughout t,his paper are, as previously mentioned, referred to zero average internal electric-field intensity E = 0, i.e., E , =

5

0, p = I , 2, 3. With respect to t,he arrangement shown in Fig. 1 as well as the thertnodynamic condit,ions under vibrations, they should be further specified as constants a t zcro st'ress and conshnt, ent'ropy.

-~PPLIC.XTIOS TO ALPHA-QUARTZ Several nleasurenlents of t'he polarization effect were

carried out on alpha-quartz plates including the measure- ment of frequency of all three fundamental thickness modes and the cllange in thcsc frequencies under the effect of a dc potential :lpplicd to the exciting electrodes of the plates. The platcs were excited by means of a Heegner oscillator. In order to protect the Heegner oscil- lator from t,he dc potcntial applied to the plate electrodes, two con(1cnscrs were used, c' = 1 p F each, scparating the plate on both sides from the Hecgncr oscillator. The fre- quencies w r e determined with a iLlarconi counter-frc- quenc.y meter, ~r~oclel TF 1417X. 411 measurements were done a t roon1 temperature of 20°C and normal air pressure.

Eleven alpha-quartz plate specimens supplied by Tesla Works, Czechoslovakia, were investigated. The plates were square, 25 mm in size and thicknesses of 0.600 * 0.001 n m . Bevelling was used t'o avoid coupling wit'lr other modes. The orientation of the platcs was (xyw)O with the angle 6 ranging from 6 to 30". The accuracy of the orientat,ion angle was always better than 20 minutes. The specimens were platcd ancl the side length of the square electrodes was 18 mm. Thc plates were mounted by means of four picccs of thread forming a bipyramitlal suspension.

The change in frequency due to the applied dc electric field was measurcd a t 600 V in both directions with re- spect to the tllickncss of the plates. For opposite direc- tions of the electric field, the changes in frequency were found opposite in sign antl equal in absolute value. The linearity of this effect was establislled reliably through measuring the change in frequency corresponding to the applied dc voltage of 450, 300, antl l50 V in a number of

Ex1)t:rimcntal values of the magnitude of the polarizing effect were finally obtained as defined before. These val- ues are, for the plates and modes investigated, shown in Table I. The accuracy of the experimental values of af/if.AE) was calculated based on the dispersion of the rcadings taken of frequency, its change, applied voltage, and the plate thickness. The mean-probable error did not exceed 5 percent. However, the accuracy also might be impaired by the presence of some parasite resonances whose influence, in the case of polarizing effect, is very likely magnified.

When (40) was applied to plates of the above orienta- tion, the polarizing effect was obtained a,s a function of

ae,,/aE,, and a ~ , , / d E , , all regarded as unknown in this paper. Before this was done (40) was adjusted with due respect to the symmetry of alpha-quartz, while at the same time, all quantities appearing in it were rewritten

c i1scs.

ac, , /aEl, ac22/aEll ac,,/aE,, a(cI4 - cz4)/aEI, ae,,/dE1,

m 9

z T:

d

IEEE TRANSACTIONS O N SONICS AND ULTRASONICS, JANUARY 1971

in terms of the so-called matrix-index notation, reducing the number of indices used. All other material constrants in (40) were well known and their values were taken over from [lo].

Relating (40) to all modes and plate orientation? in- vwtigated, an attempt was made to determine t'he previ- ously mentioned seven unknown material constants so that they may give the best fit over the whole set of experimental results available. The mean-probable errors of these constants were cnlculatcd sinlult'aneously with their values. However, the mean-probable errors were very large, rendering the results unacceptable.

An assumption was then made that out of the three effects to which the material constants dcxp/dE,, de,,/dE,, and d~, , , , , /dE~ correspond, the electroelastic effect is the one that is responsible nnainly for the polarizing effect. I n agreement with this, the terms containing del l /dEl , de14/dEl, and dcll/dEI were dropped from (40). A new attempt was made, in the same manner as before, to de- termine the component,s of the electroelastir tensor that remained in the simplified version of (40). The results obtained are presented in Table 11. The relative mean- probable errors in their values did not, exceed 30 percent except in d(c14 - C - ~ ) / ~ E ~ where i t was about. 70 percent.

The found values of dcA,/dE, were then substituted hack in (40j and the theoretical values of the magnitude of the polarizing effect determined. They are also p c - sentecl in Table I showing a good agreement with experi- ments for all three vibration modes.

CONCLUSION The interprdation of the polarizing effect clescri1)ctl in

this paper resulted in finding the relation between the magnitude of the polarizing effcct and the electroelastir, electrostrictive, and electrooptic tensors. -4 random analysis showed that the majoritay of existing independent, components of these tensors actually prtici1)ate in (40). In the case of alpha-quartz, 16 out of the total of 17 independent components, for ADP their full number. Consequently, experimental investigation of the polariz- ing effect presents a possibility, on principle a t least, of determining simultaneously (which means under the same conditions) almost conlplete electroelastic, electro- strictive, and electrooptic tensors. However, practically, only those tensors can be determined whose contribution to the polarizing effect is suffiriently great to exceed in magnitude the experimental error with which the polar- izing effect can be determined.

The accuracy of the polarizing effect measurement,s as carried out here to illustrate the theoretical part of this paper, was limited by the experimental equipment avail- able. Apart from that,, the used modes could not be checked for their purity. This permitted determination of the components of the electroelastic tensor only. These are, together with the converse piezoelectric effect, obvi- ously the main contributors to the polarizing effect.

For the time being it is inlposrihle to comment on the differences between the value of dcii,/dE,, published in

HRUkKA: POLARIZING EFFECT WITH PIEZOELECTRIC PLATES

TABLE I1

ALPHA-QCARTZ AT 20°C. v.LLUES FOR SOME COMPOSENTS OF ELECTROELASTIC TENSOR OF

~ _ _

acdaE , , , N . (1’ . 7)%)-’

as l , a& 2 .55

ac22/aEI a c d a E l - 1.33

J c l d l d E l - ac,d/aE, -0.35 1.07

- ~

[ I l l and those determined here. The former were ob- tained indirectly, being calculated by means of ds~,JdE,, determined from the polarizing cffect with rods vibrating in length.

In the majority of cases ;1 fairly good agreement was reached between the theoretical and experimental results. This implies that the polarizing effect with alpha-quartz plates vibrating in thickness can be interpreted on the basis of the approach adopted in this paper wit11 a rea- sonable degree of success.

A4CKNOWLEDC;ME?;T

The author is greatly intlehtecl t o J. Recs, Computer Manager, for his extraordinary understanding and per- sonal interest and to the staff of the Computer Center, University of Khartoum, for its very cooperative atti- tude. In particular, the author wishes to thank Dr. 0. Blahnik, formerly a Mcn1l)er of the Staff of the Unirer-

7

sity of Khartoum, for valuable assistance in program- ming. Dr. J . Zelenka ~ ~ 7 a s very kind to supervise manu- facturing of the crystal resonators supplied by Tesla Works, Czechoslovakia.

L11

L21

131

L41

r51

161

C71

C81

C91

[l01

L111

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