Coulomb fluctuations and Raman scattering in solid electrolytes

~

Solid State Communications,Vol.52,No.12, pp.945-948, 1984. Printed in Great Britain.

CO~O~B ~A~YON.q

AN~ ~

0038-I098/84 $3.00 + .00

Pergamon Press Ltd.

SCA~.ERI~G 7N SOT Tn RT.RC.T~OT.Y,~,,ES

Y~N~Bamda~ev and A ~ B ~

P~ics

Research Institute, Odessa Univex~it~, Odessa 2TG057,. ~SSR Received 4 October 1984 by V.M.Agranovieh

~ e ~hent~ ~ ~ scattering ( ~ ) b~ au~erlnnic caudu~t-c~a (SG) is ~reaented. l ~ is s h a ~ that the RS cent~a1 ~ e a k i n SC o£ t h e 0( ~ t ~ e i s caused l~y two-tamale relax-af~lun ~acesaea.The ~hanmn. sho,~Id~ is due to the fi~storde~ scattering am the o~tical ~homon made ~n~ is chlefl~v determine@ bx the screening effects in SC.Proposed theox~ gives exRlanati~a of the ~S ex~er~Iments ~hlch had not been i u t e r ~ r e t e d cn~le~ently s.o fan.

A ch~ac~e~ist~4c feature o f Ram='n scattering (RS) by solid eleet~lytes, c~ superlcmie conductors (SC) and elect r o . t i c melts is a depolarized, highintensity centx~l peak 30 to 50 cm-I wide 1,2. As distinct f~cm RS s~ectra in normal cx~stals, thnee in SO do not c o n t ~ n sharp optical phonon peaks,e.g. in the spectra of SO ~ - A g l there is a shoulder centred at a frequency shift o~ about i00 cm -I corresponding to a h i g h - e n e r g y ~ptlcal phanon in t h e orde-

phonon shoulder to the a ~ t u x ~ I dis-order existing in ~-Agl. However, it is obvious that in des crlbing RS in SC d~m~c effects arising c1~e.to the presence of the subsystem of high-m~bilit~ ions should be considered in the first place. The purpose of this paper is tn demonstrate that auly when the d~u~m~CR Of c h ~ g e d fluctuations in electrol~tes is taken into account it is possible to consistentl~ interpret the mentioned features of RS. It will be shown that while the RS central peak in SC of the O( -AgI type is caused by two-mode relaxation processes, the phauon shoulder is due to the first-order scattering on the phonon mode and is chiefl~ dete~mlned by the screening effects in the mobile ion subsystem. To describe "slew" motions in SC, let us introduce the dyD~m~c variable E(~,t) which is the intensity of the longitudinal electric field produced by the local electroneutrality disturban-

red ~-phase oi" Agl. The existing results of theca~tical studies c~ SC RS a~e based amlel~ on model asaumptiaus 1,3 and co~respcnd to the available experimental data but r~ughl~ or, ccntrax~ to these data, predict a comE,lately polarized central peak having a low int-ensit~v proportional to q2 (~ is the exchanged wave vector). 4,5 Besides,the above oae-phonon RS shoulder in a central-symmet~ SO 0(-Ag! has been difficult to interpret u~ to, now. An attempt was made in 6,~ to attribute this oneAll correspondence should be sent to:

V.N.Bondarev, Phys.Res.lnst., Odessa University, 27 ul.Pasteur'a, Odessa, 270057, USSR. 945

946

Vol. 52, No. 12

COULOMB FLUCTUATIONS AND RAMAN SCATTERING IN SOLID ELECTROLYTES

ces a% the space point ~ an~ tha m~ment t. ,R,ri,~ " ~ l e date~mlnes the purely dissipative process - that of electric conductivity in SC. ~ae ion oscillations will he describe~ by the normal coordinate T (~,t) of the optical phonon. Pot the sale of simplicity, this coordinate will be considered longitudinal. The fluctuative polarlzahilit~v tenscur ij causing RS can be expressed as a function of E and . Using only the main E and ~ terms re q ~ foz the object in view, we have

7

where a,b,c,d are some_of the elec~roopt+cal constants. ~ r c simplloi~y, $J~e iso,T~Topic cent~all~ s~matrlcal SC model is used. ~ote ~hat e l e c ~ i c field space derivatives inva~ian~s, e.g. ~ijdivE, should be incorporated in the expression 6 E i j" Howe~e~,the contribution of such terms into the RS intensity is as small as q2 (.see 2), while the terms (I) do not include this low-value factor. Substituting (I) into the general expression 8, the following formula will be obtained for the RS tensor: ¢,o

Iijkl (~,~)= " I dt e - i C ° t l d { ei~F • < ~ E i j (r,t)

kl(r=O, t=O )

,

where the angle brackets denote the thermal average. To derive the necessary equations of motion and calculate the thermal averages in (2) let us write down the free energy of fluctuations E = - ~ and ~" in the form

•

ODo~=

(3)

The fluctuating variables ~ (Coulomb potential) and ~ (local charge density)

are related by the Poiss~n equation

whale ~ i s ~ e SO background d i e l e c ~ l c c o n s t a n t ( n o t de~ez~tned~ d ,i~.e¢¢1~ b~ the ion t~anspo~¢), ~3° is the f~equenc~ of the longi~udinal optical phonon un~e~ consldera~ion, ~ > 0,g >0 a~e phenemenological parameters. Pot • simp llcit~, the _~p_~~ i n t e r a c t i o n terms are omitted, i n ( 3 ) . Note t h a t the use o f the f i r s t , two t e ~ s ~ (3) is sufficient t~ o b t a i n the Debye-Huckel equatium with the

so.

golf) 2

g leith

is taken into accmun~ in (3) an approximation more general th~n tha Dehye~ c k e l one can be derived 9. Let us introduce the dlssipati~e £uncCion

('~

+

~x i

-

where J is the local densit~ of the fluctnating eleat~ic cu~z~n%, whlc~ is included in the contlnult~ equation

~flat+~Y=o

;

(6)

6-> 0 is the SO comductivlty, ~ > 0 , ~ > 0 are the efZectlve viscoslt~ factors characterizing the inhomogenai~ of the current of mobile ions that form a llq. uid.like system and actuall$ determining the SO conductivity spatial dispe~siren. Pox slmpllcit~, in the eq.(5) %he optical phonon dampimg was neglected. T~ obtain the Eule~-La~a~Ee equations o~ marion for the ~ourier components of the Coulomb potential ~ (k~,t) and for the phonon coordinate ~ (k,t), let us conform to (3)-(6) the variational principle. It gives @ W(k'ot)=_ 1 / ~ ÷ ( ~ + 4 ~ / 3 ) k 2 a ~ 4~/£ +/k2+gk 4

a 2 ~ (k',t) a t2

+~2~ ~ ° %( - - ' t )

= o ;

(?)

~(E,t)=~ ~ (E,t)/k since opt±cal phonon under consideration is longitudinal.

COULOMB FLUCTUATIONS AND RAMAN SCATTERING IN SOLID ELECTROLYTES

Vol. 52, No. 12 ~o ~ .

t~e cc~rrelato=s < E i ( ~ , t )

~¢uz~ed~

i n tJaa ~S ~enso~, tha oon-

vamtlamal p~o~edume 8 i s u s e d . Sol~Ing

t h e ec,~.(~7) f o r the co~espo~_a~-g ~ t i~l c C ~ t i ~ and s u b s e q u e n t l y calcul s t ; t u ~ the tJaezmal ave~a~ea by use o f (3), the following expEession is obt a i n e d i n t h e ~ - - , 0 linLtt:

•l(S3m) (~9). gij I klI(SCal) ( ~u ) ,

(8)

where the s~mmetr~cal HS Intensit$ I(S3U")C~ }

4 T~

(~)5/'

{a2i,(cO)÷

z5 w~ 3/2 CI' ' (w)÷I' ' (- ~))3}

(9)

,

soala~ P=B i n t e n s i t ~ I(SCal)(~).

~/2(6~--) 5/2 {h2I'C~u)+.

dZe

(z'E)3~)=. o

• z 2 . ~ x 4 ) 2 , ( / ~ / 2 ) 2 ( 1 +Sx2) 2 ] -~ dx,

z-(,b)= ] xz( I +s=2) [ ( I .xz+a.4) z + o

• (,.Q-~o)2( I -,-sx2) 2] -~ az;

(ll)

( %. The expression (8) describes the depolarized RS spectra of the second an~ first order on the conductiwlt~ and optical phonon modes,respectively, the RS intensity belng finite w i t h ~ - * O . As can be seen from the above formulae, the central peak effective width D D c ~ S T C ~ / E . Substituting the characteristic values ~ 5, Y = 1(Ohm'cm) -I 1 we obtA1,~ cO c = 2 5 am -I which, like the qualitative results of (8)-(ll) a~e in full agreement with the experimental data 1,2,10.

947

~Ig,l shows the experimental data f a ~ B S in ~ - A g ~ i0 and the results of calcu!a~iona using the formula

l(~~z'

(~u)-~.o4

[I"(~)÷z"C-ml]12)

for S=I/2,G=I/4,CDo=120 cm -I, ~ =6, =l(Ohm.am) -1 (see (9)-(ll)).In th~is formula the scaling factor is omitted. ~he numerical coefficient in (12) is a combination of elect~ooptical constants. It was chosen to agree the theoretical results with the experlmental data. It is worthwhile noting that at largo Co values I ( ~ ) decreases as ( ~ --000)-3/2 , i.e. aver a wider frequence range than the Lorentzian (see (ll), (12)).The low rate of I(~O) decrease is in qualitative agreement with the unusuallE broad RS spectra in SC that were observed experimentallE 1. Besides, that while the central RS component reflects the described two-mode relaxation process the shoulder at CO~LO a represents one-phonon scattering which (as can be seen from our results) is caused by the optical mode -relaxation mode relationship (see (1)) characterizing the dynnm!c disorder in SC. In 1,7 it was suggested that

T=7qgK D >-

f-.

3 0

SO

"tOO

15o

co (cm'O

Fig.l. Raman spectra o f O(-AgI. Dote are the experimental data i0 and solid line represents our calcula~b~ns based on t h e foz~ula ( 1 2 ) .

948

COULOMB FLUCTUATIONS AND RAMAN SCATTERING IN SOLID ELECTROLYTES

such one-phonon processes in RS are caused by phonons whose wave vectors ca~ assume any values within the Brillouin zone. However, our results indicate that the optical phonons with wave vectors line,ted by the inverse screening length are efficient in RS. ~ h ~ is in fact confi=med by the strong convergence of integrals in (ll). Th;~, ~Lr' apRroach enables the princlpal features of RS in SO to be in-

Vol. 52, No.

tea~reted and ma$ s e r ~ e a s a basis fo~ develo,p~ng a d e ~ ! l e d t,heoz"y of RS im solid and liquid electrolytes.

Acknowledgements - The: authors wish to express their gratitude @o Prof, V.M. Agrame~ich £ o r useful discussions. The$ also wish t~ thsn1~ Prof. A.]P.Andresev fo~ his im~e~est in this research.

REFERENCES

i.

2.

3. 4. 5. 6.

Physics of Supmriomic Conductors, ed. b y M.B.Sa!smo~, Springer, Berlin (1979). R.J.Nemauich, R.M.Msmtin, J.C. Mikkelsen, Solid State Commun. 32, 79 (1979). T.Geisel: ibid. 24, 155 (1977). B.A.Paylthorpe, D.A. McQuarrie, J. Chem. Phys. 67, 1838 (1977). R.Zeyher, Z.Physik ~ 127 (1978). G.Burms, F.H.Dacol, R.Alben, Soli& State Commnn. /2j 71 (1979).

7.

V~azzacurati, G.Smocco, O.Signoralli, E.Cazzauelli, A . P o a 1 ~ a , G. Mariotto, Phse .Rev .B26,2216 (1982). 8. V~N.Bondarev, Fiz.Tverd.Tela 23, 2413 (IS81); Zhurn.Eksp.Teo~-.Fiz. 2042 ( 1 9 8 2 ) . 9. L.D.Lamdau, E.M.iifshitz, ~lektrodln~m~ka s p l o a ~ k h szed, Na~ka, Mos c~w (1982). lO. S.Ushieda, M.J.Delame~, Solid State Commum. ~ 67 (1979).

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