“Universal” frequency response of disordered conductors and related problems: a novel approach

26 December 1994 PHYSICS LETTERS A

ELSEVIER

Physics Letters A 196 (1994) 247-252

"Universal" frequency response of disordered conductors and related problems: a novel approach V.N. Bondarev, P.V. Pikhitsa Physics Institute of the I.l. Mechnikov Odessa University, Ul. Pasteur'a, 27, 270100 Odessa, Ukraine

Received 1 August 1994;accepted for publication 5 October 1994 Communicated by V.M. Agranovich

Abstract

The Coulomb fluctuations of the field of mobile defects must influence decisively the transport characteristics of a disordered ionic conductor. In the framework of such an approach a closed-form expression for the averaged complex impedance of the system is obtained. First derived from the most general ideas, this expression has rich analytical properties and allows one to give a complete quantitative description of a large number of experimental data on the relaxation dynamics of disordered conductors.

Long ago, in the classical experiments of R. Kohlrausch (their results are reproduced, for example, in Ref. [ 1 ] ) it has been discovered that the dynamics o f the relaxation of a residual charge in the glass body o f a Leyden jar can be approximately described by a stretched exponential function exp [ - (t/rK)a], where rK and 0 < fl< 1 are the characteristic time and exponent, respectively (the so-called Kohlrausch-Williams-Watts law). Presently it is established that the conductivity a(to) of disordered materials contains a nonanalytic contribution tr(to) - t r ( 0 ) ~tos ( 0 < s < l ) in a wide range of the frequency o9, and there is a'large n u m b e r of works, in which a dependence o f such a type was found experimentally (see, for example, Refs. [ 2-5 ] ). Following Jonscher [ 5 ], this behaviour of a(to) can be considered as a manifestation of the "universal frequency response" of disordered conductors. The problem o f the derivation of the dependences like the above ones has been considered in Ref. [ 6 ] with respect to disordered semiconductors. However, in attempts to extend the ideas of Ref. [ 6 ] to arbitrary (including ionic) disordered conductors suppositions were used, which were hardly grounded seriously (like the introduction of curt-off factors for the integral convergency [ 7 ] ). On the other hand, models [ 8,9 ] allowing one to analyze the dynamics of the ionic transport in the disordered materials only numerically and fractal models [ l 0,11 ] having a strictly mathematical nature, do not give a possibility to find the dependences of the exponents fl, s on the physical characteristics o f the medium (for example, on temperature). In the present paper we propose an analytical approach, which allows us to get closed expressions for the "universal" response functions of disordered ionic conductors (the results obtained below, in principle, are applicable to other systems with hopping conductivity as well). It is based on the idea that Coulomb fluctuations in the subsystem of mobile defects distributed over interstitials of the host (ordered) lattice play a decisive role 0375-9601/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0375-9601 (94) 00855-8

I'.N. Bondarev, P. I'. Pikhlts'a

248

Phvstcs [.elter~ .I 19fi ( I794~ 2 4 7

!"

in the transport characteristics of disordered conductors. The tYuitfulness of such an idea was successfull b proved by Bondarev and Zhukov 12] who used it to study the peculiarities of de conduclivit\ of disordered s},stcms ( like superionic glasses), .vet to the dynamic case this analytical approach has never been applied. Its utilization for the description o f relaxation p h e n o m e n a in such systems l'or the firsl lime allo~ss us to give a commoi/ interpretation o f the total of experimental data on the problems under considcralion The experimental data on the transport characteristics o f conducting systems arc usuall> extracted from measurements o f its complex impedance (see, for example, Refs. [ 5, 13 ] ). Therefore ¢)ur aim is to obtain an expression for { :(~o) ), where the angle brackets denote averaging of the "'specific" impedance .:(c.9) over the spatial fluctuations o f the local conductivity c~of a disordered system, <:(ro)>=(4~i/co~(r,)),

,

~(~,))=~, +4;zwi/~,J

~I :

Here e<~ is the "high-frequency" dielectric constant in the complex dielectric function e(¢,a i. ~l ~o= () the rcsuh of averaging reduces to that found in Rcf. [ 12 ] for the dc average resisti\.ity, :-(f! ) = i ."a Let us turn to the temporal report o f the impedance by introducing the relaxation t'unction 1{1)- 4 ~ 2 ~ J d~oexp(-i~ot),:iz(~,))

exp[-4~//e,z(())]"

J t~

,,=o

n!

\~/

,According to Ref. [ 12 ], the local value of'z( 0 t can be presented as l~)llo~ ~. /~;:,--{D°+ q [ ( o ( r + a ) - ~ ( r ) ] ) , z(0) =z exp\__ T ,

t ~i

where the product o f the ionic charge q and the random potential difference of a mobile ~on at points r and r + a ( 21a[ is the length o f the an elemenlar}, hop ) should be considered as the additix e to the initial (in the absence of d i s o r d e r ) activation energy Do: T is temperature. - is a preexponential f a c l o r The averaging in Eq. ( 2 ) is performed over r a n d o m ~pwith the distribution function cxp [ - FI ~0*,/ TI, which is expressed through the free energy of the C o u l o m b fluctuations,

where p = - ( e¢,~/4Jr)V:~0 is the fluctuation charge density, R is the Debyc screening length in the "'plasma" of mobile defects. The average m Eq. ( 2 ). <,~-- " ( 0 ) > =

f

Drpz - " ( 0 ) e x p ( - / , ~ p ; i l ' )

(j

'; 1 i l)~pcxp( - / ,"~¢~,

)

"~)

is represented by the normalized functional integral over the local fluctuations olq), 1

q~(r)= u , ~

(okexp(ik'r),

~p ~=¢~'~ .

(~)

where V is the volume o f the system. Then, as in Ref. [ 12 ], there is no need for an explicit functional integration in the Gaussian integrals ( 5 ) since the variables u~= Re[(0~ e x p ( i k . r ) ] , change from - ~

z,~=lm [q~ e x p ( i k - r t ]

to ~ . Indeed, introducing new ~ariables

KN. Bondarev,P. V. Pikhitsa / PhysicsLettersA 196 (I 994) 247-252 4nqn c o s ( k . a ) - 1 U'k=Uk+ ¢~x/~k2(l+RZkZ ) ,

249

4nqn sin(k.a) V'k=Vk-- E~x/~k2(l+RZk2 ) ,

in the numerator of Eq. (5) we can write

-nq[~°(r+a)-~°(r) ] - ½ ~ d V ( p~°+ 4Z -

4nqZn z 1 - cos(k.a) ,o~ ~k2(l+R2k2)

E~

~

k2(l+R2k2)(u'*2+v~2)"

Then after reduction of the fraction in Eq. (5) of common multiplier we obtain ( z - n ( O ) ) =aS exp(n26D/T) ,

(7)

where the substantially positive quantity 4nq2 V 1 - c o s ( k . a )

(8)

6D= eo~V "V k2(l+R2k2) is the resulting additive [ 12 ] to the activation energy of conductivity due to random fields, and ao = z - l e x p ( - D o ~ T )

is the initial (in the absence of disorder) conductivity of the system. Substituting the summation for the integration according to the general rules in Eq. (8) we get 6D = (q2/~oo a) [a / R - 1 + exp ( - a/R) ] [ 12 ]. It is remarkable that series (2) with account of Eq. ( 5 ) can be exactly summed up and as a result we arrive at the elegant formula for the relaxation function f(t)= ~

1 i

du e x p [ - u 2 - (t/r) exp(x/~u) l ,

t>0,

u=46D/T,

r=e~/4nao

(9)

--oo

(the summation procedure will be presented in a detailed paper; here we only point out that the correspondence between the series and the integral can be proved by expansion of exp[ - ( t / r ) exp (x/~u) ] in powers of t and by termwise integration). Returning to the frequency report we arrive at the main result (z(~,) > =

4x/~ ~o

i exp(-u2) r - ~ du _ i w r + exp(x/~u)

(10)

whence the expression for the frequency-dependent conductivity of the disordered system follows,

1

B(w,v)

a(og) = R e
1 w= -ln(ogZ)v "

(11)

The function introduced here (apparently it is not in the literature), GO

1

B(w, v) = 2x/r~ -o~

d y e x p [ - y 2 / 4 v + ( ½-w)Y] = B ( - w , v)

(12)

1+ e x p ( y )

at the values w=n+½, n=0, + 1, +2, ... can be calculated exactly (so that B(½, v)=½) and the functional equation

B ( w - ½, v) + B(w+ ½, v) =exp(vw 2)

(13)

250

l ~N. Bondarev, P. l'. Pikhitsa / Physics Letters .-1 190 (1994) 24 ~- 25 ?

holds. With the help of Eq. ( 13 ) one may write B( u. u) as a c o n t i n u o u s fraction {a formula which is analogous to ( 13 ) a n d r e p r e s e n t a t i o n s via c o n t i n u o u s fractions exist also t'oi .: z( ~,J 1 , ). ()n other h a n d there are the tyro t'ollowing representations.

2,fv

B(w+{,v)=~

k~,,

f

I,-!

(4~} v du -a cosl v w ;

" ~- ~ "

(-l)"~exp[v(n-n')"][l-O(

x u {u-w)

4 !

n =: I i

+exp[v(n+w+l)2][l-(/)(x.,V(n+w+l

Ili

.

5

where q)(x) = ( 2 / , / ~ ) .fi~ exp ( - y - ) d v is lhe error integral. It is seen from Eq. ( 1 0 ) that as ~o ,() the c o n d u c t i v i t y of the disordered system tends to its low-frequency limit a ( 0 ) = ao exp ( - ~ v ), a n d as (o ,,x= it tends to the "'high-frequency "" one a( z ) -: ao exp ( ~ v ) ( it is a s s u m e d that the value a ( o c ) is reached at frequencies lower t h a n the frequencies of the optical p h o n o n s of the ionic c o n d u c t o r ) . But in the i n t e r m e d i a t e range o f (,~ an analytical e x p a n s i o n of the i m p e d a n c e is absent, reflecting the e x p e r i m e n t a l l y observed n o n a n a l y t i c i t y of the - u n i v e r s a l " : frequency response of disordered conductors. T h u s d e r i v a t i o n of the e x p o n e n t of nonanalyticit~ d In a ( ( o ) d In (,J is of essential interest. A c c o r d i n g to o u r results, .~ ,{) as ~,~ ~ 0 a n d as ¢,~ • and at some ¢,)rthe e x p o n e n t reaches its m a x i m u m value. W h e n u , 0 ( " w e a k " d i s o r d e r ) o~f= r *, .s'((of) = ~ v = , $ D / T . In the limil o f " s t r o n g " disorder ( v >> l ), s u b s t i t u t i n g the m a i n terms of series ( 14 ), ( 15 ) in Eq. ( 1 I ), we find from Eq. ( 16 ) s(o)t=l-

-

P(~-]n((ur)}.

v~v \
/

2 exp(-{ z) t'(¢)=~+,/~r l+O(O =V/I÷~+4(~-'~'t

+

~1-,

where ~t-=0.389 d e t e r m i n e s the valuc ¢,,~= r ' e x p ( V v{t ). W h e n c e il ibllows, lhut in the frequency range r ~exp( - 1 . 2 7 v / v ) << (o << r t e x p ( 2 . 0 5 , / i i ) e x p o n e n t i a l l y wide with respect to x/'u. the e x p o n e n t s is practl% F cally i n d e p e n d e n t of the frequency a n d is d o s e to s(¢,**.) = I - =. 146/x. t,. Therefore, the f u n c t i o n a(¢o) i n d e e d has the form ,: ~o'. m the frequency range which is rapidly e x t e n d e d as , j v grows. Such a b e h a v i o u r o f a ( o , ) is j u s t observed in e x p e r i m e n t s on d y n a m i c c o n d u c t i v i t y of disordered materials [3-8, t 1 ]. II is useful to give an a p p r o x i m a t e expression for the e x p o n e n l at the point of contrar.~ flexure {or at a r b i t r a o ' v, 2(2+v)

(4+v)v/]+ v

,

which is obtained by the steepest descent method in integral ( 12 ) (compare with thc above limit values of ~ i Let us stress that the calculated exponent depends only on v, and the conductivity being divided by a~ is doscribed by a universal function in terms of the variable u)~ (or w) at given ~,. Let us use the explicit form o f 61) at R >> a. a s s u m i n g that the Debye length is d e t e r m i n e d , m a i n l y , by lhc c o n c e n t r a t i o n x ( x < < l ) o f extrinsic defects (like the o v e r s t o i c h i o m e t r i c calions Na ~ in Na f l - a l u m i n a ) . R=x/~.ooT/8gnoxq 2 [ 12], where n o - 102~ cm -~ is the density of interstitials over which " e x t r i n s i c " ions are d i s t r i b u t e d in the host lattice. S u b s t i t u t i n g the o b t a i n e d expression for r~l) into the t b r m u l a for s, we find s ~ 1 -- -1"/7~,

~ 19 )

V.N. Bondarev, P. V. Pikhitsa /Physics Letters A 196 (1994) 247-252

251

at temperatures T<< To, where To= 2 q 2 ~ / ~ is the characteristic temperature. Note the formal analogy between Eq. (19) and those obtained in phenomenological models of ionic transport in systems with static disorder [ 14,7 ]. It is essential that at "high" temperatures the exponent s, according to Eq. (18), decreases by the law s ~ T 2 / T 2, unlike the unphysical cross-over of s to negative values, that formally follows from models [14,7] at T>To. As an example, in Fig. 1 we display the experimental dependence or(co) for superionic Na fl-alumina at 113 K [4,15] together with the derived one according to our theory at or(0) =2.51 X l0 -7 f~-i cm-~ and v = 15.65 ( To = 223.5 K). Notice the good agreement between the calculated and experimental values at least within four orders on to. Some discrepancy in the high-frequency region is evidently due to the degrees of freedom o f optical phonons (their contribution will be taken into account later). Therefore, with the help of the function B (w, v), which together with compactness possesses rich analytical properties, one can construct formulae which correctly reproduce all details of the "universal" frequency response. As to the relaxation function, its asymptotic behaviour at t / z >> 1 according to Eq. (9) can be shown to have the form f ( t ) ~ exp [ - v-~ ln2( t~ z) ], i.e. f ( t ) is represented by one of a variety of exponential-logarithmic forms [ 11 ]. As t-, 0 one gets f ( t ) ~ exp ( - t/z) from Eq. (9) (the necessity of a cross-over to a pure exponential form o f f ( t ) at small times was remarked on the empirical level in Ref. [ 4 ], see also Ref. [ 8 ] ). In Fig. 2 the solid line shows the profile of the relaxation function in dependence of log ( t / z ) for u = 4 according to our theory; besides, for comparison, the graph of the Kohlrausch function exp [ - (t/ZK) # ] for zK= 0.75 z and fl= 0.58 is given by the dashed line. Since experimental data on non-Debye relaxation are usually fitted by the Kohlrausch function with proper parameters, the close agreement between the dependences shown in Fig. 2 confirms the applicability o f our theory to a quantitative analysis of the relaxation phenomena in disordered conductors. Note that such an analysis could be hardly realized in the framework o f the approach of Ref. [ 8 ] which demands cumbersome computer calculations. Summarizing, we claim that in the present paper there has been developed an approach which allows one to obtain closed-form expressions for the response functions of disordered conductors without demanding farreaching model notions about the concrete peculiarities of particle hops. So our results seem to be quite general

t(t) ,

E

.o

"\ 0.5

1o 2"

\~

~,

° ° °

g

,

1~

0.2

1~' ......... ~ ......... '......... ' ......... ' ......... ' lo lO 1 1.o lo to ~/Z~

{Hz)

~'~

-1

o

¢' ,~. . x" __ 1

2

5

)g(t/.~)

Fig. 1. Frequency dependences of conductivity of disordered systems. Points: experimental tr(co) for superionic Na//-alumina at 113 K [4,15 ]. Solid line: results of calculation of a(co ) according to our theory for the followingvalues of parameters: a(0 ) = 2.51 X l 0 - 7 f~cm-' and v= 15.65 (To=223.5 K). Fig. 2. An example of the relaxation functions of disordered systems. Solid line: calculation off(t) according to our theory for u = 4. Dashed line: Kohlrausch function for rK=0.75r andfl=0.58.

252

|'.N. Bondarev. P V. Pikhitsa / Ph.vsics Letters ..I /96 (1994) 24 r 2";2

a n d c a n m a k e b e a b a s i s o f a c o n s i s t e n t d e s c r i p t i o n o f t h e t r a n s p o r t d y n a m i c s in s y s t e m s o f t h e t y p e o f d i s o r d e r e d ionic conductors. O n e o f t h e a u t h o r s ( V . B . ) is g r a t e f u l to t h e I n t e r n a t i o n a l

Science Foundation

lot financial support.

References [ 1 ] J. J~ickle, Philos. Mag. B 56 ( 1987 ) 113. [ 2 ] J.C. Wang and J.B. Bates, Solid State lonics 50 ( 1992 i 7:, [3] K. Pollak and T.H. Geballe, Phys. Rev. 122 ( 1061 ) 1742 [4] K.I. Ngai and U. Strom, Phys. Re,,. B 38 (1988) 10350. [5] A.K. Jonscher, Phys. Stat. Sol. (a) 32 (1975) 665. [6] H. Scherand M. Lax, Phys. Rev. B 7 (1973) 4491,4502 [7] J.C. Dyre. J. Phys. C 19 (1986) 5655. [ 8 ] K. Funke and R. Hoppe, Solid State lonics 40/41 ( 1990 ) 200. [9] P. Maass. J. Petersen, A. Bunde, W. Dieterich and H.E. Roman, Phys. Rev. [_ell. 06 ( I~t~l i ~ [ 10] S. Havlin and D. Ben-Avraham, Adv. Phys. 36 (1987) 695. [ 1 I ] L. Pietronero and E. Tosatti, eds., Fractals in physics {North-Holland. Amsterdam, 1~)~6 ! { 12 ] V.N. Bondarev and V.M. Zhukov, Fiz. Tvcrd. Tela 33 ( I 9 t~I ) 846. [ 13 ]L.D. Landau and E.M. Lifshitz, Electrodynamics o|'contintlous media ( Nauka. Moscow. I ~,~82) [ m R usslan 1 114] J. Bernasconi, H.U. Beyeler and S. Str~issler, Phys. Rev. Len. 42 (197g) 819. [ 15 ] D.P. ,Almond, A.B. West and R.J. Grant, Solid State (kmnnun. 44 ( 1982 ) 1277