SOUND ATTENUATION IN FERROELECTRIC SOLIDS

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SOUND ATTENUATION IN FERROELECTRIC SOLIDS

U. Naithani, B. Semwal

To cite this version:

U. Naithani, B. Semwal. SOUND ATTENUATION IN FERROELECTRIC SOLIDS. Journal de

Physique Colloques, 1981, 42 (C5), pp.C5-659-C5-664. �10.1051/jphyscol:19815101�. �jpa-00220969�

CoZZoque C5, suppZdrnent au nO1O, Tome 42, o c t o b r e 1981

SOUND ATTENUATION I N FERROELECTRIC SOLIDS

U.C. Naithani and B.S. Semwal

Department o f Physics, D r . B.G.R. C o n s t i t u e n t CoZZege, P a w i (GarhwaZI- 246001, I n d i a .

Abstract:- An expression for the sound-attenuation constant in doped displacive ferroelectrics, in the presence of an external electric field, is obtained by using the double-time thermal- Green's -functions technique. The mass and force constant changes between the impurity and the host lattice atoms are taken into account in the Silverman Hamiltonian augmented with higher -order anharmonic and electric-moment terms. The defect-dependent, elec- tric-field-dependent, and anharmonic contributions to the atte- nuation constant are discussed seperately. An anomalous increase in the attenuation constant is predicted in the vicinity of the Curie temperature. The soft mode is held responsible for this effect. The effect of defect

,

anharmonicity, and electric-field parameters on the stabilization of the soft mode frequency is also studied.

1. Introduction:- Ultrasonic measurements provide a s e n s i t i v e t o o l f o r t h e study of phase t r a n s i t i o n s l a solids. 1 ~t i s n o w revealed both ex~erimentrally and t h e o r e t i c a l l y t h a t t h e soft, o r f e r r o e l e c t r i c , mode pleys e s s e n t i d r o l e i n displacive f e r r o e l e c t r i c s . as t h e temperature approaches t h e Curie tenperature Tc, t h e s o f t mode Srequ- ency fl becomes small

(R-T-T,)

2 r e s u l t i n g is an increase i n

i t s

ampli-

tu&, which should influence t h e acoustic mode via t h e phonola-phonor Iwteractioh md i s expected t o give an aaomalous2 behaviour of m d near T,. Here we have obtained a genercd. expression f o r t h e sound atteriuation constant, by augmenting t h e Silverman Hamiltoni tm with anhiumoni c

t

e m s upto f ourth-order and higher-order e l e c t r i c - ~ o m e n t terms a i s i n g due t o t h e deformation of e l e c t r o n s h e l l s i n

tm

exter- nal e l e c t r i c field. The effect of mass change and force-constant

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19815101

C5-660 JOURNAL DE PHYSIQUE

change between t h e impurity and host l a t t i c e atoms due t o the i n t r o - duction of defects 3 i s a l s o tarken i n t o account. The defects, e l e c t r i c

f i e l d , temperature and frequency dependence of t h e at benuation const- ant i s d l scussed. An anomalous behaviour of sound i n t h e v i c i n i t y of T c is predicted,

2, Hs\mfltonian and Green's Functions:- We consider t h e following Greenr s function f o r t h e acoustic phonons, 4

G , ,

, ct-tl, = <a",ct) ; a:cct',)).

The modified SilvePm~m Hsmiltonian used In the present study i s exact- l y

similar

t o equation (8) of a preVlous paper? Writing t h e equation of motion f o r the Green1 s function ( 1) with the help of thts modified Hamiltonian, Fourier traznsforming and writing i t i n t h e Dysongs equation form, one o b t a i n s 6

G

( o C ~ E ) =

E ( ~ - E 2 + i r ( w ) l ,

⁽²⁾

wheTe r~w) i s t h e damping constant;

Fa)

i s also defined as t h e half width of t h e response of a c o u s t i c phonons mode k. The expression f o r

becomes 6

rw) ^{= Gw + Lea)}

^4-

p'),

⁽³⁾

where

Kco) = . ( - c [ k : o ) ~ ( - ~ ~ C ) + D ( ~ ~ C ) D [ - ~ ~ ~ ) ) [ S ( O - Q )

- g c o + n j j + 4 n i ~ ~ ( a ; , - k ~ ) ~ ( k ~

1

' @)+cc

( * . ; , - b Q ) ~ ( k : , b 4 j

% 4

(at /a:) [S

^C-

- s t ) - 6

^(a

+ Ga )] +lr { Z

9

\*9

^6:

^{) D C- kL, k : )}

1 1 4% k,

Q i s t h e e f f e c t i v e frequency of t h e s o f t mode. I n t h e high-tempera- t ~ r e l i m i t , t h e temperature dependence of _R i s gf ven by3n2,

X

(T-Tc);

K being a temperature independent coefficient. This frequency fi i s s t a b i l i z e d i n t h e presence of defect, e l e c t r i c field and @ha~?~onfc

3. Attenuation constant$- The a t t e n u a t i o n constant i s given by 2

d (0) =

G13>/c ,

⁽¹¹⁾

where t h e damping c o n s t a n t f ~ a ) i s given by eq.(3) and c i s t h e sound velocity. We can r e w r i t e t h e expression f o r a t t e n ~ a t l o n constant, using sqs.(3) and (11) as

-

4 p ( W ) + ~ A < ~ ) + - - ( E ( ~ ) >

O((w)

-

C5-662 JOURNAL DE PHYSIQUE

where

4

p(o)

,

o ( A ( ~ ) and < E [ ~ ) depend d i r e c t l y on def e c t e , anharmonid- t y and e l e c t r i c - f i e l d psrametars, respectively.

I n t h e absence of a n h a m o n i d t y and e x t e r n a l e l e c t r i c f i e l d , considering t h e allowed ( energy-conserving) processes, t h e d e f e c t dependent contribution t o t h e sound a t t e n u a t i o n constant i s given by

dI,

=TDlc ,

where i s given by eq.(4). I n

ey&I

replacing t h e sUmmation by i n t e g r a t i o n , a s usual, i n t h e Debye approximation we g e t

4,cmO;, ⁵ (024 1 2)

S

^sin ~

de

~

d4 LC

~ ^(-

ka,

~

k4;

)

c

(

CO;

iz; )

where t h e I n t e g r a t i o n has been carried o u t over constant energy surfaces given by a+= a*,

.

^If f o r c e constant change i s ignored f o r l n s t *rice, t h e n

I n case of random d i s t f i b u t i o n of i m p u r i t i e s , averaging t h e

terms

i n b r a c k e t s i n eq.(14) over a l l p o s s i b l e atomic configurations and

t&-

ng t h e d i r e c t i o n a l average we g e t

which shows t h a t the atf e m a t i o n constant due t o s c a t t e r i n g of phono- ns by mass defect i s proportional t o t h e f o u r t h power o f phonon e e q - uency and t h e square of t h e mass change (M'-M). The f i r s t i n eq.( W ) which I s proportional t o

(ad4

g i v e s t h e u s u a l Hayleigh Law, 798 but t h e o t h e r term i n eq.( B ) , arising due t o modification of t h e h m o n l

c

f o r c e constant, g i v e s a frequency independent contribution t o

.(D

The pmur./b&onic c o n t r i b u t i o n t o t h e attexiuation c o n a t a t i. given by o < ~ ( w ) C =

rA

( U ) / C

i n

eq.( 121, where rA(w)

i s

given by eq.(5). The c a l m l a t i o f i s show t h a t t h e frequency dependence i s given by W'QZ+ oa,

,

^{where t h e} 02dependence ariaes &ue t o q u a r t i c anharmonic i n t e r a c t i o n s while t h e o dependence arises due t o

t h e previous by considering t h i r d - o r d e r anharmonic i n t e r - actions only. I n t h e high t a p e r a t m e 1iaf.t t h e tenperature depend.

ence of 4,Cw) from eqsb(5), (9) and (10) can be expressed a s

1t i s c l e a r from eq.(16) t h a t i n t h e v i c i n i t y of t h e Curie tempaa-

ture

Tc(T--ST,), t h o a t t e r n a t i o n constant i n c r e a s e s anomalously i n agreement with previous

results?

Let us now consider t h e e f f e c t of e l e c t r i c f i e l d on the cltt-ua- t i o n const ant. The e l e c t r i c-field-dependent contribution t o

4

Cw)

i s g i v e by o ( , C ~ )

( = T E ~ ~ ) / c ) i n

eq.( 12), where rECw) i s given by eq.(

6). ~t

any temperature

w e l l

above

T , ,

t h e temperature depend- ence of o ( ~ C W ) i s given by

4

(w) C4

+ el^

-t C ~ L T /

IT-% {I2 1

^E~

. ⁽¹⁷

The above

result

shows t h a t a(E C a ) V a r i e s a s t h e s p a r e of t h e applied e l e c t r i c field,and t h e attenuation constant i n c r e a s e s with t h e f i e l d , i n agreement with t h e experimental results;' The treatment adopted here, leads one t o see the comparative v a r i a t i o n of t h e a t t e n u a t i o n constant with t h e v a r i a t i o n of defect and e l e c t r i c - f i e l d p a a m e t ~ s ,

i n

presence of anharmordcity. Recently, we have found

an

i n t e r a c t i o n term

(<,,

^{l w )}

)

of defect with e l e c t r i c f i e l d i n t h e presence of anhannonicity i n t h e damping constant end s h a l l be reported s e p ~ a t e l y i n a forthcoxung paper.

Ref @ a c e s r

-

1

C.K. Jones and 3.K. Holm Phys.Lett. A26, 182 ( 1968).

2

KbTani and IT.Tsuda, J.Phys.Soc. Jpn.

26,

113 ( 1969).

C5-664 JOURNAL DE PHYSZQUE

3

Bita Bahadur and

P.K.

S h m a , Phys. Rev. B

g,

448 !1975).

4

D.N. Zubarev, Sov. PWs. Usp. 3_, 320 (1960).

5

U,c, N d t h a n i R.P. Gairola and B.S. Sawa.1, J. PWs, S O C ~ Jgn.

43 204 (19773.

-9

6

B.P.

G d r o l a and B.S. Semwal, J.' PWs. Soc. Jpn,

9,

975 (Ud??>j

2,

954 (1977).

7

P,G. Klemens, Solid State PhysicsL edited by F.Seitz and B.Turnbull (~cademic,N.Y. 1958)

,z

pp.l-98.

8 P. Ceruthers, Rev, Mod. PhYs.

33,

92 (1961).

9

C.Mavroyanrds and K.N.Pathak, Phys.Rev. @2, 872 (1969).

10 N.P.Heuter e31d D.P. NeUhaUs, J . ~ c o u s t ~ S o c . ^Am.

27,

292 (1955).