Electromechanical resonance in selenium determination of the piezoelectric coefficient d11

Volume 37A, number 1

ELECTROMECHANICAL OF THE

PHYSICS

RESONANCE PIEZOELECTRIC

LETTERS

25 October 1971

IN SELENIUM COEFFICIENT

DETERMINATION dll

J. B O U A T and J. M . T H U I L L I E R

Groupe de Physique des Solides de l'Ecole Normale Sup~rieure, Paris, France Received 16 July 1971

The variation of impedance for a thin rod of selenium is calculated near a mechanical resonance frequency of the sample. F r o m the observed variation, we may deduce the conductivity, the dielectric constant ~11 and the piezoelectric constant d l l . S e l e n i u m b e l o n g s to the point g r o u p 32, and so exhibits piezoelectric properties. The piezoe l e c t r i c t e n s o r has only two c o m p o n e n t s , d l l and d14. d l l is d e d u c e d f r o m m e a s u r e m e n t s of the v a r i a t i o n of i m p e d a n c e f o r r o d - l i k e s a m p l e s . T h e c a l c u l a t i o n s m a d e by Q u e n t i n and T h u i l l i e r r] ] f o r thin r o d s of t e l l e r i u m , and by A r l t [2] f o r p l a t e s , a r e t r a n s p o s e d to m o n o c r y s t a l l i n e s e l e nium. T h e s a m p l e a r e p a r a l l e l to a b i n a r y a x i s . We a s s u m e that the e n d s a r e m e c h a n i c a l l y f r e e . T h e only c o m p o n e n t of the s t r e s s , T l l = T, is p a r a l l e l to the l e n g t h 2l of the r o d : T = T O exp (jcot) (cos k x - c o s kl) T h e s t r a i n is g i v e n by:

O2T/Ox2 = pco2 S We c a l c u l a t e D and E f r o m the e q u a t i o n s of s t a t e :

Z=R

1 - 8/(1 - j WolW) ( t g k l / k l )

1 +j (co/~o) (1 - 8)

R is the r e s i s t a n c e of the s a m p l e w i t h o u t p i e z o e l e c t r i c e f f e c t . N e a r the f u n d a m e n t a l m e c h a n i c a l r e s o n a n c e f r e q u e n c y cor = H / 2 I ( p s ) l / 2 , Z c a n be written: coo+'corL - ( a c o 0

co-oJ r + ~ ) + . l a +Wo[co-cor]\

2cor

Jr2

cor

~

) r

a is e m p i r i c a l l y i n t r o d u c e d a s a c o r r e c t i v e t e r m to the e l a s t i c c o n s t a n t s w h i c h b e c o m e s s(1 - j a ) to t a k e into a c c o u n t any k i n d of m e c h a n i c a l l o s s e s e x c e p t the p i e z o e l e c t r i c o n e s . We can w r i t e Z = = Z o ( E + j F ) w h e r e Z o = R Wo/(Wo+ jWr) and o n l y E and F g i v e i n d i c a t i o n s about p i e z o e l e c t r i c e f f e c t .

S = s T + dE D = d T + eE T h e c o n d i t i o n d i v J = 0, w h e r e the c u r r e n t d e n s i t y c o n t a i n s the d i s p l a c e m e n t c u r r e n t :

R

J = ~ E - U ( k B T / q ) (a2D/~x2) + jcoD R to

l e a d s to the r e l a t i o n :

k2 1 - j w/co D pco2----s- 1 : - 0 1 + j (CO/COD+ COO/W)

R"

C~

with

coo = e / E

COD = c o 2 q / ~ k B T k 2

Fig. 1. Equivalent circuit

~ = d2/Es

In o u r e x p e r i m e n t s , co/coD is n e g l i g i b l e w i t h r e s p e c t to 1, and we c a l c u l a t e t h e e l e c t r i c a l i m p e d a n c e Z f r o m the e x p r e s s i o n s of J and E:

C=-- 1 RCOo L -

C' =

H2Rwo 80~2

~0 __V~RCO I o

H2Ra ~o R' =

C" -

8C II2 - 8

R" = ! R ([12-8) 8

r

71

Volume 37A, n u m b e r 1

PHYSICS

LETTERS

25 October 1971

'E

1C

l/

-

i~ ~

~

Real

---

Imagi.~ry part

part

//I/]ii

5

11b

1~o

I I I

-x

.5

pomP"

J

l

I

V/13o

120

130

F(~Hz~

1~,o

1

¢ x p e . r i me..n i'~.l Lurv¢ rh¢orcL'i¢..~l

t,40 * F(ks,}

'F

Fig. 2. Real and i m a g i n a r y p a r t s of the admittance. T h e e q u i v a l e n t c i r c u i t i s s h o w n in fig. 1. C r y s t a l s a r e g r o w n f r o m a m e l t of e i t h e r p u r e o r t h a l l i u m - d o p e d s e l e n i u m . A c o n s t a n t a.c. v o l t a g e i s a p p l i e d to t h e s a m p l e . T h e i n t e n s i t y through the rod is measured by a synchronous d e t e c t o r w h i l e t h e f r e q u e n c y i s v a r i e d . It g i v e s t h e v a r i a t i o n s of r e a l a n d i m a g i n a r y p a r t s of t h e a d m i t t a n c e a s a f u n c t i o n of f r e q u e n c y (fig. 2). F r o m t h e c a l c u l a t e d c u r v e s of E a n d F (fig. 3), we may deduce the piezoelectric constant. T h e m e c h a n i c a l r e s o n a n c e f r e q u e n c y i s in t h e r a n g e 100 to 150 kHz. T h e r e l a t i v e w i d t h of t h e l i n e i s a b o u t 2%, a n d t h e r e l a t i v e v a r i a t i o n of t h e r e a l p a r t of t h e a d m i t t a n c e i s a b o u t 10. In o r d e r to h a v e a good a g r e e m e n t b e t w e e n t h e e x p e r i m e n t a l a n d t h e t h e o r e t i c a l c u r v e s (fig. 3), w e h a d to a s s u m e t h a t t h e d i e l e c t r i c c o n s t a n t a n d t h e c o n d u c t i v i t y d e p e n d on f r e q u e n c y , a s s h o w n b y t h e m e a s u r e m e n t s of S a l o e t al. [3] a n d B a r b o t e t al. [4]. F r o m t h e a s y m p t o t i c v a r i a t i o n of t h e admittance, we deduce that the conductivity is a b o u t 1 0 - 6 f~- l c m - 1, a n d t h e r e l a t i v e d i e l e c t r i c c o n s t a n t i s 29. (~ d e p e n d s on e; i t s m e a n v a l u e i s 0.06. T h e p i e z o e l e c t r i c c o n s t a n t , w h i c h i s i n d e p e n d a n t of e, i s d l ] = 4 . 1 × 1 0 -11 C.N - 1 , a n d

72

.5

X x

0 X

Fig. 3. E and E curves.

i s c a l c u l a t e d w i t h a n i n c e r t i t u d e of 8%, due to t h e v a r i a b l e c r y s t a l l i n e q u a l i t y f r o m one s a m p l e to a n o t h e r .

References [1] G. Quentin and J. M. Thuiller, Solid State Comm. 2 (1964) 115. [2] G. Arlt, J. Acoust. Soc. Am. 37 no 1 (1965} 151. [3] T. Salo, T. Stubb and E. Suosara, The physics of Se and Te, (Pergamon P r e s s , 1969}. [4] J. Bardot, J. Bouat, A. Launay and J. C. Thui[[ier. to be published.