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Theory of diffusion and ionic conduction in glass
No~h-HoHa~ Amsm~a~
T H E O R Y O F DIFFUSION AND I O N I C C O N D U C T I O N I N GLASS W, SCHIRMACHER
L InU.oduca~ In many non-metallic ~lids ae ~nduetivity data show a frequ~cy dependence ac~rding to * ' ( ~ ) ~ ' ( 0 < s < 1) over ~ n y decades of frequency r ~ # n g from the kHz to the G H z ~glme [ 1-3]. In crystalline fast ionic c o n d u a o ~ such a behaviour ~ n be attributed to intemction effects b e t w ~ n the mobile ions [4]. In ~ o t ~ b o u s materials, on the other hand~ such a b ~ haviour is usually attributed to a b ~ d di~t ribution of microscopic Idn~ie ~eflleient$ [ I ]. For d~uribing ionic jump diffusion or electronic hopping con-
duetionintfisorde~dsolids[5,6]oneeanstartfrom a set of master equations [6,7 ], one ~ n u ~ a ~ n dom walk description [8] or a m a d a m network model [9.10]. All three a p p r ~ c h ~ are equival~t: The microscopic m ~ t e r and ~ n d o m walk equations ~ s p o n d ~o Ki~hboff's equations of the network, The exlsting app~aehes for perFo~ing the configuration average a n d miring for the dynamic diffusivity a n d ~nduetivit y can be divided into three group~ (a) averaging over the kinetic ~efficients, which ~ s p ~ d s to a n equivalent c i v i l o f par. allelimpedanees[ll,12];(b)av~aglngo~theun, reno~a[ized single site p~pagator (single site approximation, SSA) ~ s p o n d i n g to a ~rial equivalent eir~it [8,9,13-15 ] and (c) effective me-
dinm approaches (EMA) [6,7,10,16-20]. It is ~ l l k n o ~ [10,21 ] that both (a) and (b) c ~ I ~ d to grossly ~ n g l~sn[t ~,esper the de ]imit. In three-dim~sional systems w h e ~ tile ki~tic coefficients ( or ~ n d u e t a a ~ s ) tincture stmn#y one cn~ u n t ~ an intrinsic p~colatlou problem [22]. This behaviour is a~ounted for by 1he EIVIA description a l t h ~ g h ~ l y in a m ~ field way [ 6.10], but not by app~ximations (a) a n d (b). F~di~ussingionicjumpdiffusioningla~untll recently 123] only a p p m a c h ~ (a) and (b) have been u ~ d , This led to inconsistencies in diseasing exp~dmental data: while the anomalous frequency d ~ p c n d ~ c e of the ae conductivity saggr a broad dlstrlbutioa of aelivation energi~ oF single jumps one ~ u l d not ~ p l a i n why the diffusivity a~a dc conductivi~in#as~sbo~onlymin~deviafions f~m an Arthenlus law. Within (a) and (b) a n A~heninslike temperature dependen~ can only be proauced using very specific distributions of activation energies, which is unsatisfactory in view of the univer~lity o f the obse~ed Arrhertius behaviour [ 12,24]. F u r t h e ~ o ~ these distributions are not compatible with the ob~e~ed o ' behaviour. h is the purpo~ of the p~sent ~ n t r i b u t i ~ to show t~at within an E M A d~eription anomalous f~qu~cydepend~eeandA~heniusbehavloureanbo r ~ n c i l e d easily: It will be s h ~ that if the sittgle j u m p s a ~ ~ r i ~ e d to a fixed ~nge in which the m t ~
0 167-2738/gg/$ 03.50 9 Elsevier Science Publishe~ B.V. ( North-Holland Physics Publishing D i v i s i ~ )
do not depend on lhe j u m p dlstane~ ("fixed range hopping"[, the EMA predicts an A~he~ius taw fur the dc ~ndu~_~vlt./ independ~t of the activation energy distribution. Explicit msutts a ~ g l e n for fixed-range hopping with constant a~ivation spur. ~ m . This model m n be used to explain de a n d ac ~nduetivity data in s o d i ~ ~ i s i l i ~ gla~ over a large t ~ p e m m ~ ~ g r ~ d ten d ~ d ~ of frequenw'
Let us ~ n s l a e r now ~be motio~ o f I~mlet~ which ~ n p e f f o ~ iastanlaneous jumps I x ~ e e ~ interstitint sit~ in the glass. " I h e ~ sit~ a ~ assumed to be fixed and ~ d o m l y $istributed in space, Such a type of morion is dmcrlbed by the following ~ t o f Markovian master 9 dn;(r) V W n (0 + Z W n (t). = --~ ~ * s ~ ~ (1) n,(t) is the o~upatio~ probability of site L For the hopping motion of small particle~ in g l a s ~ the jump ~tes w,, are assumed to b~ of the folJowingf o ~ {"activated fix~d-~ge h~pplng'): IV,~= w ( e,,, Eo) = O(B--r#) Vo r {2) w h ~ r 0 is the step function, r o is the distance bet w r site i and j, E~ the (f~e) energ~ barrier leading (rum ~ to j, vo tile attempt frequency and T the tempemtu~. The single panicle propagator is formally given as G(k, w) . If ~ dt d~e r} 3
(3)
w h ~ e I is the unit matrix a n d ( ) d ~ o t ~ a ~ n flguratio~laverage. K i s a m a t r i x w i t b diagonale/e, m ~ t s K~t=--~jW o and offMiag~al e / e m i t s K,~=w#. G(r, t) gives the probability for finding a olSgin r = 0, The neutron ~ t t e r i n g law is g l v ~ by
S(k,w)f~-tRe(G(k.~)).
by the G ~ e n - K u b o identity l i m o G ( k . w [ = [ i w + O ( ~ ) k2I - '
(5)
a n d is ~lated zo thr dynamic ~nduetivity a ( o [ b y the Ein~eia f o ~ u l a , a ( o ) = [nq~/kBT) D ( O [ . (6) n is the n~mber o f vartlcles per volume a n d q the charg~ o f the panicle. The ~lfMiffus~on ~ fllclent
2, l t o # n g model
= ~ <~p(ik'ro ) ( i ~ l - K ) b ~ > ,
The frequ~o/tlt'p~nde~tdiffuslvi W D( eu! is givc~
(4)
D,. wbleh de~/Sbe~ Ihe molion o f a single lagged
particle, is ~ot e x ~ I to O ( o = 0) bat ~ reg.e.ed by eo~elatlon eff~ts. T~e simplest mean-field argu~t [ 6,25 [ gives
D,=(l--nln.JD(o=O), where n~ is the n m b e r of sites per ~ l u m r is p~portional to the S F e c ~ m of velocity fluctuations, If this sb'ectmm is a white noise spe~mm D ( ~ ) = ~ n s t = D , (5) ~duee~ to the ~]ution of the ordinaty diff~io~ equation ("norreal diffusion*'), AS st ated in the beglanlng ~ ( ~ ) in ionic conducting gias~s is stronglyfrequencyd~ [~e~dunl a ~ r a i n g to a ' ( ~ ) ~ ' : Via (5) a ~ d (6} thk implies the I~resen~ of anomalous diffusion: the mean~quarr distan~ walked by a particle i n ~ s e s wilh h m e as t'-~ instead of a l i n e r behavlouy.
D'(w) = R r
3. Effeetice m e d i m al~proxlmali~ A ve~ion of the EMA suitable for a m o r p h ~ s systerns has been develOlmd by M o ~ g h a r el a]. [ 17] by applying the ~ o o ~ a l i z e d perturbation expansion [26] to the resolvent matrix ( 3 ) Omls~ion of p ~ ~ s e s which i~voivr r loops and averaging over the t~o-site sclr~nelgies [17] yields ~he fnllovAng EMA expression for tile averaged k- and r d ~ t ~lf-energy: ~ v ( k , w ) = n , ~ d~efdEp(r) P(E)expfik'r) 1
1
' {7[
lierep(r) isthedislrlbutionofsitedlsta~eLP(E )
zhe barri~ distribution, ~d ap ffiexp( ~ I ) is a ~r~ction for double ~ting [17]. ~(w) ~ o(0, ~) has tile m~ning of a genersii~d frequency-dependent jump f~qu~cy. The p~pagal~ is giv~ as
{8) Insemngcxp~ion(2)forW(r,E)into(7)weob. G(k,~)=[i~-vfk,~)+v(~)]-' .
& The de dh~fusl~ity ~d ~ndu~ivily The ~If~nsistent EMA b~mes in dc limit
eq~ion
(9) for r(m)
the
|
= a Z, j c l E p ( E ) P
o
+
/
1__
1 [o(0) a~olexp(~'kaT)"
lain ihe following EMA e q ~ t i o n
( 13 )
u(~)~Z~dEP(E)j
If we define a quantity
/~(T)= -kaTln[u(O}la~uo] x ( i ~ + ~ (1~ ) l a ~ 4- exp(ElkeT) u~ ) -i '
(9)
03)[=tGZTdEp(E)I+exp{[E/(TO]/kBT~ J ~-P
(14)
whe~ Z~
The F e ~ i funelion in the integrand b ~ m ~ a step ~functl~ for kaT~lt, which means lhat in this limit beeom~ tempemtu~ indel~endenl. The e o ~
=n~4g ~'~dr r:p(r)
spending lem~e~tu~ dependen~ of o(o) is given
is the site coordination number. The propagator is given by
by P(O) =a~P~ e x p ( - M k a T )
G(k,o~)ffili~+f(k)u(~)] -~ ,
with
'
(10)
1=a~Z~"i P(E) dE.
(,_~)
We therefo~ arrive at the following imporla~t stal~ m ~ l : Within the EMA tr~tment a n d the fixed range hopping model (2) we expect an A ~ h ~ i ~ s behaviour of the de dlffusivit y and conductivity, indepen-
'
Ifp(r) ~~(r-d),
dent of the details of the barrierdistribution P(E).
d~k)= 1 - s i ~ ( k d ) l k d , which is the Cnudley-Elliot t fun~ion [27]. The difo fusion ~efl~eient is given by
D(m)=~(r~>v(m),
(11)
w~th ~
(16)
o
d
f(k)=4~,~vdrr~p(r) ~[
s (r=>=y
(15)
-z
drr+p(r)(fdrr~p(r))
.
(12)
These expre~ions ~ n be u~ful in an alyzing nuclear ~pin-lat lice r e h ~ t i o n and n e u t ~ n scattering data [23,28].
This offe~ an explanafi~ for the fact 112-14,241 thal diffusivity a n d ~nduclivlt y data in g l a ~ exhibit u n i v e ~ l l y A~henius behaviour. We can perhaps undc~tand the physics described by eqs, (15) a n d (t6) by the following pc~nlatinn ~ a s t ~ c t i o n [6,221: Let us ~ n n e e t all SitES which have a distance 1 ~ than R a n d a barri~ betw~n o c h other smatl~ than E*. If the level E* is c h ~ to be ..................................... netted sil~. The s i ~ of these clust e ~ will i~crease if E* is inerea~d until at E* =/t there exists a per~lallon path lhmugh the system. The resulting conductivity will be p~porlional to the " b o n d " exp(--plkeT ) which closes the pere~
latlon patl~ ~ itself involves only p a l a o f sltes ~4th b a m ~ E < # a n d the corcesp~ding statistical d i s t n~u fion, h is i m p o ~ n I lo note thet *hese per~3lati~ aspects ~ n o t be obtainvd b y the a~proximati~ schemes labelled (a) and {b) in ~ lion 1. S i n ~ in t h ~ sehem~ the nr is ~ placed by a paraLlel or s e d ~ equivalent ci~uit, ~ p e a ~ v d y . the petcoIar p r o p e ~ ~ e ellmi. nate~L The ~sulting exp~$1ons for the d~ ~ n d u r tivi~y depend s~sitively ~n the details of P(E), in particular on the distribution o f high barnr whereas ~he tree sys~em~ lake the lowest possiblc C~mblnation of b a r n e n which leads ~o a ~nduCt~on path t h o u g h the glass,
P(E)=P,. E < E m ~ . =D
(17)
E > Era,,,
w h ~ e Ib e n h ~ ion condition ~ q u i ~ s Po E ~ , . From (I 6) we obtain F = I/aaZ~Po~l / ~ a s dc activation energy~ For l ~ p e m t u r ~ w h ~
e~pI-E~,,(Z~-a;')/Z.ksT]~l h~lds the up~e~ integration bound~.Iy in (9) ~ n be pushed lowards Jofintty a n d we obtain v ( ~ ) =/~/~e T ~ ( w ) ] n [ l + ~ J v ~ ( ~ ) ] , (18) with ~1(~) = p ( ~ ) + i w . l f ~ define a n o ~ a l l z e d complex impedance as z(~) =v(o)/~,(~),
S, The ~ diffes~viq, tnd cen4uc~ltity For cal~lating t he frequency dependence or'#( ~ ) from eq. (9) we must c b ~ a specific model for P(E). The simpl~t a~ump~ion is
we find 1hat Z ls a universal r u n . i o n of the normalized frequency d~=~/~(O). From the ~ m p l e x / m p e d a n ~ plot s h ~ in the insert o f fig 1 it is ~ that Z(o~) behaves almost like a Co]~Cole function, i.e.
go ~.2
10C]O
~,-
-10 <2
~o9 [~ (Uz)] ~i$. l. Full li,es. ~ . l~n~es and sq~r~: ~c c~nd~iv~ty data of Na.O 2S{O2~ s s ~S@mpiled by Wo~ a al. [ [ ] ; dasM liaes:
z ( ~ ) ~ [ 1 + (i~)~1 - i , with s= 0.8 7. This m ~ n s that the conductivity b ~ haves m
[4] ~ Fon~, $ofidSlalr I o ~ 1~19 (1986) 183. [51 N.F.M~Uand E.A.Da,~s,E l ~ i e p r ~ i * ~on<~ m]li~m~tcfials(on-ne on ~ s ~ Oaord, 197I)!61 I~. ~ l g c r a~d Y.V. B r r ~ ~ o ~ ; ~ d u ~ i o n in s~lld, Rev Mod.Ph~. 530981 ) Z?S
i n ago'cement wlth the expclSmenlal o b v i a t i o n s [ 15,13,141. The m o m a l o u s frequencv d e p c n d c n ~ predicted b y eq. (18) is ~ m p ~ d w i t h ar ~ n d u ~ ~ v d f l ~ o~ ~ a l O , 3 S i O2 r 1~ 5g, [ w ~ P - ' = 0 . 7 e v has b ~ n taken. The a g ~ m e n t over such a large frequ~cy and l ~ p e m t u r e ~ale is quite ~ t l s f a ~ o ~ . A s i a the ~ d i ~ r [ 1-3,13. 15] ~ ~ ~eha~out ari~s as a ~ n s e q ~ e e o~ a very b ~ a d das~tibution of ncti~lion energies. Un-
likethcothettmatmemstheEMAisahletode~n'b~ the t m n s i l i o n between the d~spe~ive ~ d non-dlspersive region, w h ~ the percolation effects a ~ d o m i ~ n t , c o ~ e l l y , and therefore l o r e . n o l l e lhe ~* behaviour wlth the Ardlenius-likc t ~ p e r a t u r e dependen~ of a'(0).
Aekn~ledgera~t It is a p leasu ~ to acknowtedge he]plat discussions with Professor Dr. B.U. Fel d~hof, Pl~fessor Dr. W. G 6 ~ e and Dr. A. S e h i ~ .
[g~l~.~cherandM.~e~Rev.~tO(~973)4491,4502 [9]A'MillcrandLAbm~m~P~s ~,120(19fi0) 745. [ 101S ~rkpaldck' g e v ' M ~ " ~YS" 4~ 11973) 574" [ I I ] ~ E , ~ e I ~ and J,C. K e ~ , J , AppI Phi,45 ([974) ]45g, [12]H,jai~,J, Non~,Sol~d~d6(1984)517, [13]D.~i.gandJ.L~uqu~,l Chim.~,5(1974)693 Il~] P B M ~ d g CT. M ~ ihaa and R. BO~, PhYS ~ e ~ G ~ s ~ Z3 (t972)17t ~15]~.C ~ m , P h ~ LCt1~ A10g(l~gs] 45~ ph~g ~0 (1979)425~. [ i 71 B MovagMr,K Poh~mann~ d O Saul. p h ~ StaresSo. Iidi (b) 97 (I 980) 553: gM~ SS9; W. ~ h i ~ a ~ , SolidS1~r ~omman. 39 fl 9Sl ) g93 [ isi T. ~ a ~ k i ~ d M. ~ ph~ Rev. n24 (i 9gl ) S2~ [ 191~ $ammc~dd, ~ l l d S~alrCommon.39 (19gl } 401. 1:01 t Wc~a~, P ~ . Roy ~ t r 47 (I 9Sl ) 1496. ~2t]lA Mcl~n~ P.~. Balch,c a~d J D. OarZ, Phil M,g B41 S4 (1971)2fi12. [ ~ ] w. $r in: Amorpho~ ~ d liquid marcels, ~ s ~ L~her' G"Ffimh and G" Jacu~i (Ni2h~ ~ Ha~c' t9S7). 1241B. Csntor. in: ~pidl~ ~ n c h e d m~ l s . cds. S. S~cb an~ a W~llmoat ( ~se~cr, A~St~dam, 1985)p. 595. [23] K, lr RA. Tahlr.Khdl and R I ElZioU,J. ~ y s C]~ O982) 209, [27) G.~ Cb~d[e~and R3. ~l]i~t, P I ~ Ph?s SO~,77 096J )
(Dekker,New VorL 1976). [21 A.~ Jonz~cr, Natu~ 267 {I979) 673. [3] K.L,N~i, ~ m m . SolidSlate pays, 9 (~979) t:7,
128] w. $cbi~acher end A. $ c h i ~ . SalidState Iooim,28-J0 (I988) 134.
T H E O R Y O F DIFFUSION AND I O N I C C O N D U C T I O N I N GLASS W, SCHIRMACHER
L InU.oduca~ In many non-metallic ~lids ae ~nduetivity data show a frequ~cy dependence ac~rding to * ' ( ~ ) ~ ' ( 0 < s < 1) over ~ n y decades of frequency r ~ # n g from the kHz to the G H z ~glme [ 1-3]. In crystalline fast ionic c o n d u a o ~ such a behaviour ~ n be attributed to intemction effects b e t w ~ n the mobile ions [4]. In ~ o t ~ b o u s materials, on the other hand~ such a b ~ haviour is usually attributed to a b ~ d di~t ribution of microscopic Idn~ie ~eflleient$ [ I ]. For d~uribing ionic jump diffusion or electronic hopping con-
duetionintfisorde~dsolids[5,6]oneeanstartfrom a set of master equations [6,7 ], one ~ n u ~ a ~ n dom walk description [8] or a m a d a m network model [9.10]. All three a p p r ~ c h ~ are equival~t: The microscopic m ~ t e r and ~ n d o m walk equations ~ s p o n d ~o Ki~hboff's equations of the network, The exlsting app~aehes for perFo~ing the configuration average a n d miring for the dynamic diffusivity a n d ~nduetivit y can be divided into three group~ (a) averaging over the kinetic ~efficients, which ~ s p ~ d s to a n equivalent c i v i l o f par. allelimpedanees[ll,12];(b)av~aglngo~theun, reno~a[ized single site p~pagator (single site approximation, SSA) ~ s p o n d i n g to a ~rial equivalent eir~it [8,9,13-15 ] and (c) effective me-
dinm approaches (EMA) [6,7,10,16-20]. It is ~ l l k n o ~ [10,21 ] that both (a) and (b) c ~ I ~ d to grossly ~ n g l~sn[t ~,esper the de ]imit. In three-dim~sional systems w h e ~ tile ki~tic coefficients ( or ~ n d u e t a a ~ s ) tincture stmn#y one cn~ u n t ~ an intrinsic p~colatlou problem [22]. This behaviour is a~ounted for by 1he EIVIA description a l t h ~ g h ~ l y in a m ~ field way [ 6.10], but not by app~ximations (a) a n d (b). F~di~ussingionicjumpdiffusioningla~untll recently 123] only a p p m a c h ~ (a) and (b) have been u ~ d , This led to inconsistencies in diseasing exp~dmental data: while the anomalous frequency d ~ p c n d ~ c e of the ae conductivity saggr a broad dlstrlbutioa of aelivation energi~ oF single jumps one ~ u l d not ~ p l a i n why the diffusivity a~a dc conductivi~in#as~sbo~onlymin~deviafions f~m an Arthenlus law. Within (a) and (b) a n A~heninslike temperature dependen~ can only be proauced using very specific distributions of activation energies, which is unsatisfactory in view of the univer~lity o f the obse~ed Arrhertius behaviour [ 12,24]. F u r t h e ~ o ~ these distributions are not compatible with the ob~e~ed o ' behaviour. h is the purpo~ of the p~sent ~ n t r i b u t i ~ to show t~at within an E M A d~eription anomalous f~qu~cydepend~eeandA~heniusbehavloureanbo r ~ n c i l e d easily: It will be s h ~ that if the sittgle j u m p s a ~ ~ r i ~ e d to a fixed ~nge in which the m t ~
0 167-2738/gg/$ 03.50 9 Elsevier Science Publishe~ B.V. ( North-Holland Physics Publishing D i v i s i ~ )
do not depend on lhe j u m p dlstane~ ("fixed range hopping"[, the EMA predicts an A~he~ius taw fur the dc ~ndu~_~vlt./ independ~t of the activation energy distribution. Explicit msutts a ~ g l e n for fixed-range hopping with constant a~ivation spur. ~ m . This model m n be used to explain de a n d ac ~nduetivity data in s o d i ~ ~ i s i l i ~ gla~ over a large t ~ p e m m ~ ~ g r ~ d ten d ~ d ~ of frequenw'
Let us ~ n s l a e r now ~be motio~ o f I~mlet~ which ~ n p e f f o ~ iastanlaneous jumps I x ~ e e ~ interstitint sit~ in the glass. " I h e ~ sit~ a ~ assumed to be fixed and ~ d o m l y $istributed in space, Such a type of morion is dmcrlbed by the following ~ t o f Markovian master 9 dn;(r) V W n (0 + Z W n (t). = --~ ~ * s ~ ~ (1) n,(t) is the o~upatio~ probability of site L For the hopping motion of small particle~ in g l a s ~ the jump ~tes w,, are assumed to b~ of the folJowingf o ~ {"activated fix~d-~ge h~pplng'): IV,~= w ( e,,, Eo) = O(B--r#) Vo r {2) w h ~ r 0 is the step function, r o is the distance bet w r site i and j, E~ the (f~e) energ~ barrier leading (rum ~ to j, vo tile attempt frequency and T the tempemtu~. The single panicle propagator is formally given as G(k, w) . If ~ dt d~e r} 3
(3)
w h ~ e I is the unit matrix a n d ( ) d ~ o t ~ a ~ n flguratio~laverage. K i s a m a t r i x w i t b diagonale/e, m ~ t s K~t=--~jW o and offMiag~al e / e m i t s K,~=w#. G(r, t) gives the probability for finding a olSgin r = 0, The neutron ~ t t e r i n g law is g l v ~ by
S(k,w)f~-tRe(G(k.~)).
by the G ~ e n - K u b o identity l i m o G ( k . w [ = [ i w + O ( ~ ) k2I - '
(5)
a n d is ~lated zo thr dynamic ~nduetivity a ( o [ b y the Ein~eia f o ~ u l a , a ( o ) = [nq~/kBT) D ( O [ . (6) n is the n~mber o f vartlcles per volume a n d q the charg~ o f the panicle. The ~lfMiffus~on ~ fllclent
2, l t o # n g model
= ~ <~p(ik'ro ) ( i ~ l - K ) b ~ > ,
The frequ~o/tlt'p~nde~tdiffuslvi W D( eu! is givc~
(4)
D,. wbleh de~/Sbe~ Ihe molion o f a single lagged
particle, is ~ot e x ~ I to O ( o = 0) bat ~ reg.e.ed by eo~elatlon eff~ts. T~e simplest mean-field argu~t [ 6,25 [ gives
D,=(l--nln.JD(o=O), where n~ is the n m b e r of sites per ~ l u m r is p~portional to the S F e c ~ m of velocity fluctuations, If this sb'ectmm is a white noise spe~mm D ( ~ ) = ~ n s t = D , (5) ~duee~ to the ~]ution of the ordinaty diff~io~ equation ("norreal diffusion*'), AS st ated in the beglanlng ~ ( ~ ) in ionic conducting gias~s is stronglyfrequencyd~ [~e~dunl a ~ r a i n g to a ' ( ~ ) ~ ' : Via (5) a ~ d (6} thk implies the I~resen~ of anomalous diffusion: the mean~quarr distan~ walked by a particle i n ~ s e s wilh h m e as t'-~ instead of a l i n e r behavlouy.
D'(w) = R r
3. Effeetice m e d i m al~proxlmali~ A ve~ion of the EMA suitable for a m o r p h ~ s systerns has been develOlmd by M o ~ g h a r el a]. [ 17] by applying the ~ o o ~ a l i z e d perturbation expansion [26] to the resolvent matrix ( 3 ) Omls~ion of p ~ ~ s e s which i~voivr r loops and averaging over the t~o-site sclr~nelgies [17] yields ~he fnllovAng EMA expression for tile averaged k- and r d ~ t ~lf-energy: ~ v ( k , w ) = n , ~ d~efdEp(r) P(E)expfik'r) 1
1
' {7[
lierep(r) isthedislrlbutionofsitedlsta~eLP(E )
zhe barri~ distribution, ~d ap ffiexp( ~ I ) is a ~r~ction for double ~ting [17]. ~(w) ~ o(0, ~) has tile m~ning of a genersii~d frequency-dependent jump f~qu~cy. The p~pagal~ is giv~ as
{8) Insemngcxp~ion(2)forW(r,E)into(7)weob. G(k,~)=[i~-vfk,~)+v(~)]-' .
& The de dh~fusl~ity ~d ~ndu~ivily The ~If~nsistent EMA b~mes in dc limit
eq~ion
(9) for r(m)
the
|
= a Z, j c l E p ( E ) P
o
+
/
1__
1 [o(0) a~olexp(~'kaT)"
lain ihe following EMA e q ~ t i o n
( 13 )
u(~)~Z~dEP(E)j
If we define a quantity
/~(T)= -kaTln[u(O}la~uo] x ( i ~ + ~ (1~ ) l a ~ 4- exp(ElkeT) u~ ) -i '
(9)
03)[=tGZTdEp(E)I+exp{[E/(TO]/kBT~ J ~-P
(14)
whe~ Z~
The F e ~ i funelion in the integrand b ~ m ~ a step ~functl~ for kaT~lt, which means lhat in this limit beeom~ tempemtu~ indel~endenl. The e o ~
=n~4g ~'~dr r:p(r)
spending lem~e~tu~ dependen~ of o(o) is given
is the site coordination number. The propagator is given by
by P(O) =a~P~ e x p ( - M k a T )
G(k,o~)ffili~+f(k)u(~)] -~ ,
with
'
(10)
1=a~Z~"i P(E) dE.
(,_~)
We therefo~ arrive at the following imporla~t stal~ m ~ l : Within the EMA tr~tment a n d the fixed range hopping model (2) we expect an A ~ h ~ i ~ s behaviour of the de dlffusivit y and conductivity, indepen-
'
Ifp(r) ~~(r-d),
dent of the details of the barrierdistribution P(E).
d~k)= 1 - s i ~ ( k d ) l k d , which is the Cnudley-Elliot t fun~ion [27]. The difo fusion ~efl~eient is given by
D(m)=~(r~>v(m),
(11)
w~th ~
(16)
o
d
f(k)=4~,~vdrr~p(r) ~[
s (r=>=y
(15)
-z
drr+p(r)(fdrr~p(r))
.
(12)
These expre~ions ~ n be u~ful in an alyzing nuclear ~pin-lat lice r e h ~ t i o n and n e u t ~ n scattering data [23,28].
This offe~ an explanafi~ for the fact 112-14,241 thal diffusivity a n d ~nduclivlt y data in g l a ~ exhibit u n i v e ~ l l y A~henius behaviour. We can perhaps undc~tand the physics described by eqs, (15) a n d (t6) by the following pc~nlatinn ~ a s t ~ c t i o n [6,221: Let us ~ n n e e t all SitES which have a distance 1 ~ than R a n d a barri~ betw~n o c h other smatl~ than E*. If the level E* is c h ~ to be ..................................... netted sil~. The s i ~ of these clust e ~ will i~crease if E* is inerea~d until at E* =/t there exists a per~lallon path lhmugh the system. The resulting conductivity will be p~porlional to the " b o n d " exp(--plkeT ) which closes the pere~
latlon patl~ ~ itself involves only p a l a o f sltes ~4th b a m ~ E < # a n d the corcesp~ding statistical d i s t n~u fion, h is i m p o ~ n I lo note thet *hese per~3lati~ aspects ~ n o t be obtainvd b y the a~proximati~ schemes labelled (a) and {b) in ~ lion 1. S i n ~ in t h ~ sehem~ the nr is ~ placed by a paraLlel or s e d ~ equivalent ci~uit, ~ p e a ~ v d y . the petcoIar p r o p e ~ ~ e ellmi. nate~L The ~sulting exp~$1ons for the d~ ~ n d u r tivi~y depend s~sitively ~n the details of P(E), in particular on the distribution o f high barnr whereas ~he tree sys~em~ lake the lowest possiblc C~mblnation of b a r n e n which leads ~o a ~nduCt~on path t h o u g h the glass,
P(E)=P,. E < E m ~ . =D
(17)
E > Era,,,
w h ~ e Ib e n h ~ ion condition ~ q u i ~ s Po E ~ , . From (I 6) we obtain F = I/aaZ~Po~l / ~ a s dc activation energy~ For l ~ p e m t u r ~ w h ~
e~pI-E~,,(Z~-a;')/Z.ksT]~l h~lds the up~e~ integration bound~.Iy in (9) ~ n be pushed lowards Jofintty a n d we obtain v ( ~ ) =/~/~e T ~ ( w ) ] n [ l + ~ J v ~ ( ~ ) ] , (18) with ~1(~) = p ( ~ ) + i w . l f ~ define a n o ~ a l l z e d complex impedance as z(~) =v(o)/~,(~),
S, The ~ diffes~viq, tnd cen4uc~ltity For cal~lating t he frequency dependence or'#( ~ ) from eq. (9) we must c b ~ a specific model for P(E). The simpl~t a~ump~ion is
we find 1hat Z ls a universal r u n . i o n of the normalized frequency d~=~/~(O). From the ~ m p l e x / m p e d a n ~ plot s h ~ in the insert o f fig 1 it is ~ that Z(o~) behaves almost like a Co]~Cole function, i.e.
go ~.2
10C]O
~,-
-10 <2
~o9 [~ (Uz)] ~i$. l. Full li,es. ~ . l~n~es and sq~r~: ~c c~nd~iv~ty data of Na.O 2S{O2~ s s ~S@mpiled by Wo~ a al. [ [ ] ; dasM liaes:
z ( ~ ) ~ [ 1 + (i~)~1 - i , with s= 0.8 7. This m ~ n s that the conductivity b ~ haves m
[4] ~ Fon~, $ofidSlalr I o ~ 1~19 (1986) 183. [51 N.F.M~Uand E.A.Da,~s,E l ~ i e p r ~ i * ~on<~ m]li~m~tcfials(on-ne on ~ s ~ Oaord, 197I)!61 I~. ~ l g c r a~d Y.V. B r r ~ ~ o ~ ; ~ d u ~ i o n in s~lld, Rev Mod.Ph~. 530981 ) Z?S
i n ago'cement wlth the expclSmenlal o b v i a t i o n s [ 15,13,141. The m o m a l o u s frequencv d e p c n d c n ~ predicted b y eq. (18) is ~ m p ~ d w i t h ar ~ n d u ~ ~ v d f l ~ o~ ~ a l O , 3 S i O2 r 1~ 5g, [ w ~ P - ' = 0 . 7 e v has b ~ n taken. The a g ~ m e n t over such a large frequ~cy and l ~ p e m t u r e ~ale is quite ~ t l s f a ~ o ~ . A s i a the ~ d i ~ r [ 1-3,13. 15] ~ ~ ~eha~out ari~s as a ~ n s e q ~ e e o~ a very b ~ a d das~tibution of ncti~lion energies. Un-
likethcothettmatmemstheEMAisahletode~n'b~ the t m n s i l i o n between the d~spe~ive ~ d non-dlspersive region, w h ~ the percolation effects a ~ d o m i ~ n t , c o ~ e l l y , and therefore l o r e . n o l l e lhe ~* behaviour wlth the Ardlenius-likc t ~ p e r a t u r e dependen~ of a'(0).
Aekn~ledgera~t It is a p leasu ~ to acknowtedge he]plat discussions with Professor Dr. B.U. Fel d~hof, Pl~fessor Dr. W. G 6 ~ e and Dr. A. S e h i ~ .
[g~l~.~cherandM.~e~Rev.~tO(~973)4491,4502 [9]A'MillcrandLAbm~m~P~s ~,120(19fi0) 745. [ 101S ~rkpaldck' g e v ' M ~ " ~YS" 4~ 11973) 574" [ I I ] ~ E , ~ e I ~ and J,C. K e ~ , J , AppI Phi,45 ([974) ]45g, [12]H,jai~,J, Non~,Sol~d~d6(1984)517, [13]D.~i.gandJ.L~uqu~,l Chim.~,5(1974)693 Il~] P B M ~ d g CT. M ~ ihaa and R. BO~, PhYS ~ e ~ G ~ s ~ Z3 (t972)17t ~15]~.C ~ m , P h ~ LCt1~ A10g(l~gs] 45~ ph~g ~0 (1979)425~. [ i 71 B MovagMr,K Poh~mann~ d O Saul. p h ~ StaresSo. Iidi (b) 97 (I 980) 553: gM~ SS9; W. ~ h i ~ a ~ , SolidS1~r ~omman. 39 fl 9Sl ) g93 [ isi T. ~ a ~ k i ~ d M. ~ ph~ Rev. n24 (i 9gl ) S2~ [ 191~ $ammc~dd, ~ l l d S~alrCommon.39 (19gl } 401. 1:01 t Wc~a~, P ~ . Roy ~ t r 47 (I 9Sl ) 1496. ~2t]lA Mcl~n~ P.~. Balch,c a~d J D. OarZ, Phil M,g B41 S4 (1971)2fi12. [ ~ ] w. $r in: Amorpho~ ~ d liquid marcels, ~ s ~ L~her' G"Ffimh and G" Jacu~i (Ni2h~ ~ Ha~c' t9S7). 1241B. Csntor. in: ~pidl~ ~ n c h e d m~ l s . cds. S. S~cb an~ a W~llmoat ( ~se~cr, A~St~dam, 1985)p. 595. [23] K, lr RA. Tahlr.Khdl and R I ElZioU,J. ~ y s C]~ O982) 209, [27) G.~ Cb~d[e~and R3. ~l]i~t, P I ~ Ph?s SO~,77 096J )
(Dekker,New VorL 1976). [21 A.~ Jonz~cr, Natu~ 267 {I979) 673. [3] K.L,N~i, ~ m m . SolidSlate pays, 9 (~979) t:7,
128] w. $cbi~acher end A. $ c h i ~ . SalidState Iooim,28-J0 (I988) 134.