Classical hopping conduction: a.c.-conductivity in solid electrolytes

Sold State lonics 5 (1981) 1 2 1 - 1 2 4 North-Holland Publishing Company

CLASSICAL HOPPING CONDUCTION:

o,c,-CONDUCTIVITY IN SOLID ELECTROLYTES Tadao Ishii

Applied Physics, School of Engineering, 0kayama University, Okayama 700, Japan

A double role of particles, t r a n s l a t i o n a l and o s c i l l a t o r y m o t i o n , is studied in a model of classical hopping, in terms of the configurational representation of Kimball and Adams. An expression of the conductivity O~ is given by the introduction of a harmonic oscillation around the sites. The hopping rate s controls the motion of the particles to be either diffusion-like ( s ~ ) or oscillator-like (E+0). In this harmonic approximation it can be shown that ~ 2 @ ~ / ~ 2 < 0 . An effect of the anharmonic motion results in ~ 2 o ~ / ~ 2 > 0 as well as <0, and adds some structures where in some cases such as al <~ the conductivity has a broad peak at w-c. s

1.

INTRODUCTION

In most solid electrolytes, temperature dependence of the d.c.-ionic conductivity is best described by the Arrhenius type of diffusion process, where it seems reasonable to interpret the behaviour of mobile ions in terms of a random hopping of them from site to site. It is not clear, however, whether a hopping model can explain the two typical traits of the a.c.-conductivity: ~2Ow/~W2<0 , Z?rude type, and ~20w/3W2>O , non-Drude t~/pc, at w=O. Kimball and Adams [IkA] have discussed the conductivity in their pioneering work, by taking accounts o[ single-ho p and double-hop in the time-averaged current-correlation function and successively transforming the time average of them by intuition to the configurational average. the result they obtained has shown up a nonDrude type of conductivity [i]. On the other hand, Ishii has directly dealt with the oonfigurational average of the current-correlation function to yield an exact conductivity of Drude type, consksting of all orders of the multiple hopping [2]. In this paper we discuss the above two typical traits in detail on the basis of a classical inopping model. The discussion will be made by commencing with the Kubo formula and using the configurational representation of KA. For ~rans~atio~al ho~pin!~ among the sites and harm o n i o os~iTi<~tory m o t i o n around the sites, a series-expansion of the resolvent and recolleclion of all members leads to an expression of :he conductivity. If we take no oscillatory motion into account, we obtain an exact formal conductivity as mentioned before which decreases monotonically with frequency. Generally speaking, the hopping rate g determines the behaviour of particles to be diffusion-like or oscillator -like, for g~) or g÷0, respectively. It turns out, however, that in this harmonic approximation together with the stability

0 167-2738/81/0000-0000/$02.75

condition of oscillators the conductivity manifests itself as the Drude type, while an anharmonic m o t i o n around the sites adds some interesting profiles on the conductivity. Two ways of introducing this effect are tried: One is that the r.m.s, of an extension of the oscillator is given a constant in the numerical calculation of the conductivity obtained in the harmonic approximation, and the other is to consider cubic and quartic terms in the Hamiltonian for the oscillators and to obtain the conductivity to the lowest order of them. The former results in both Drude type and nonDrude type of conductivity where o w has a broad peak at w=g in case of the non-Drude type. The case of the latter results turns out to have some higher harmonic structures on oal.

CLASSICAL HOPPING AND CONDUCTIVITY

2.

It is assumed that a solid electrolyte possesses metastable equilibrium configurations, and that ionic motion consists of oscillations about these local-energy minima with occasional hopping from one configuration to another. Further assumption is that tile average hopping time 7 h is much larger than tile flight time Tf. Define the probability Pa to find the system in the configuration a as p = p o + p 1 p o = p C.p s (i) a a a ~ ~ a a ~ o Pa = g-l"exp[-B(HaC + HAS)]'

z = Xfdr exp[-B(HaC +

HaS)l,

w h e r e H° a n d He a r e t h e e o n f i g u r a t i o n a l and the oscillatory part of the Hamiltonian in an equilibrium state, and the suffixes o and l signify the equilibrium part and the nonequilibrium part, respectively. The m a s t e r equation, presenting the time evolution of P, can be written in the form ~P/~t

© N o r t h - H o l l a n d Publishing C o m p a n y

=

iEtP

+ iSsP +

{H.,P}

(2)

T. lshii / a.c.-conductivity in solid electrolytes

122

[{Ltg] a

Z(F

=

X

- F

P ~Y

Y

iLs P = {Hs"P}"

3.2.1

Harmonic

Ilm=~ (ze)R.E

Res = [-2;L£ ~ + i£

_~ = Z.(<..)a,

= being the c o n f i g u r a t i o n a l coordinate w h i c h is the sum of the c o o d i n a t e R~ of an {th site a particle exists on. Let us define the o p e r a t o r

iZ* = [P°] -1[iz]r °, operates

(3)

a dynamical

2,~ = -yZFya($ a -

variable

where

3.

=

Qy).

° di' B(z~) ~z/

(4)

[ r ° ~]

-iL*=-iLt*+iL s

[_;/._;,

'~.){[;][-iL

(8) arc

- [tj]-I ]}n.;,

;=[_[L

, zi~,~]-7,

(9)

can be obtained in a compact form provided that the two L i o u v i l l i a n s may be replaced by a set of eigea values [i], such that

,~]

(s)

which

implies

Henee

we

have

Res[GSC;'] = -[:: + L~IResS] -7 { C;) ],';+i,~,;;,:, (;):':) i ; ~, ( ; ) = [ I + ~

and

and

:~ to obtain

E q u a t i o n (4) g u a r a n t e e s =>J~aP,.,°=O. Linearic~ zation of eq. (2) and u s e o f eq.(3) formally give rise to an e x p r e s s i o n of the conductivity, the c l a s s i c a l Kubo formula: f~

~co

motion

if the h a r m o n i c terms in eqs.(7) of our interest, the resolvent

where

which

oscillatory

yaps)

#=iL*R.

2 ;::]-/,

Thus we get an a d d i t i o n a l conductivity:

TRANSLATION AND OSCILLATION

o0 E q u a t i o n (5) includes a double role of particles, simple hopping m o t i o n (called as ~ r ~ n s ~ i t { o n a 7 ~io~fon) and o s c i l l a t o r y m o t i o n about the sites.

,

,C(;)=[7+,j,:!;:l-J

contribution

= ~<:=~>~L<;>l,JrS'lJ>-],, ,) o F, '7- ~i,~ u

to the

+ o (T- g ,J)~'+:4 -

'/ - i Lb

3.1 Simple

hopping

When the o s c i l l a t o r y part of the H a m i l t o n i a n is out of consideration, we find the c o n d u c t i v i t y in the exact form for R=¢: O0j

=

w h i c h c a n be d e r i v e d

straightforwardly

3. '2.2 A n h a r m o n i c

from

cq. (5) w i t h the help of -iLt*=%-IT%, following the r e p r e s e n t a t i o n of KA. This formal equation clearly shows that it d e c r e a s e s m o n o t o n i c a l l y w i t h frequency, inconsistent w i t h the results of KA and P i e t r o n e r o - S t r a s s l e r [3], and that the usual d . c . - c o n d u c t i v i t y follows in the limit co=O. 3.2 H o p p i n g

with oscillatory

motkon

In introducing v i b r a t i o n a l coordinates, it is simply assumed that an o s c i l l a t i o n about a c o n f i g u r a t i o n is independent of all the others, and the E i n s t e i n o s c i l l a t o r is taken for it. T h e r e f o r e we have the v i b r a t i o n a l part of the H a m i l t o n i a n in a c o n f i g u r a t i o n as

H s = :-~7" "" 2" ~

S

=

R

-

+

~co zS2 •

-

i 03 '

-

iris

(7)

:~,

S and f being the c o e f f i c i e n t s of the a n h a r m o n i c parts w h i c h must be quite important in solid e l e c t r o l y t e s [4.5]. Hence the L i o u v i l l i a n is

-iD

w h e r e the first and the second terms origiilate from the fluctuations while the third [erm [s due to the o p t i c a l l y active vibrations [6]. it is interesting to note that the real part of the c o n d u c t i v i t y v a n i s h e s for T=0 in eqs. (6) und (U)

= ~'~

~s"~

b~2 8 +~p ~ +

,,f.3 8

oscillatory

motion

A s e r i e s - e x p a n s i o n is done it] the same way a s in eq. (9) and f- and d - l i n e a r terms aru recollected on the a s s u m p t i o n that the anharmonic parts in the H a m i l t o n i a n (7) are somu per[urbative values. F u r t h e r m o r e we are confined t o recollect terms o f the o r d e r 0{,gS~,'7£,/7,[;,,72,{;}, and truncatL, the series to the two successive leading terms for each order. ['hen we find th~ c o n d u c t i v i t y c o n t r i b u t e d to by the a n h a r m o n icities as

o

7~coZ

i

';=,%(:~ ) [ .'.'.'.'.'.'.'.'.V~.' v ,tl-

~,,:~ur~

'

,'.

"

T-[,w

o (T_~ko)~,+j,~,j

(T-::,:,:,) ~ (T- ;:~) ':+:~,~-/

I~'>~

~J

,,

i %' "

Io>2

(8)

Fig 1

Configurations

of 4-sites

to X - p a r t i c l e s

T. lshii / a.c.-conductivity in solid electrolytes

2.0,f l s ~ 1.0~ 0

which has a peak value at y = ~ in the repulsive regime. Hereafter an attention is payed to the a.c.-conductivity.

1.5

4.1 Harmonic

Fig.2

~2

~w=ow/Od.c=[7~2 + 2 (1-~s2)+(l+as2)2 a [~2-~s2+l]2+4~s 2 4 ~2 + - (16) M~a2~2 [~2_~s2+112+4~s2]'

1.0 fl

10

where physical quantities are normalized by such that ~=w/g, and M and ~ are the ionic mass and the inverse of the temperature times Boltzmann's constant, respectively. The r.m.s., /~2-7, can be evaluated to give ~ = in the harmonic approximation.

~s=Ws/g,

w-dependence of ~w, eq.(16); harmonic

approximation.

oscillatory motion

The eigen values and eigen vectors of eq.(14) are substituted in eqs.(6) and (Ii) to lead us to

0.5

0.1

123

~ in cm -I.

[MBWs2]-i/2

iwr SsI¢> °w,~n~=B(ze)2[M~¢'S~-iw (T_iw)2+4Ws

_ ~<¢i ~

1 iw 2S2~I~>],(13) (T-iw)2 (T-iw)2+8ws

where cross terms of f and g have been omitted.

4.

CONDUCTIVITY FOR SOME ONE-DIMENSIONAL RINGS

Kimball and Adams have applied their general results to a hopping on one dimensional rings. Here we consider a ring of M-sites occupied by E-identical ions [M-sites to N-particles]. The sites are separated by a distance a. The ions hop, one at a time, only to nearest neighbor unoccupied sites, and the hopping rates are chosen as in the following: The hop, (i) which does not change the number of nearest-neighbor mobile ion pairs proceeds at a rate F,(ii) which decreases the number of nearest-neighbor pairs by one occurs at a rate yF, and (iii) which increases the number of nearest-neighbor pairs by one occurs at a rate y-iF.

NUMERICALPROFILESOF THE FREQUENCY-

5.

DEPENDENT CONDUCTIVITY 5.1 Harmonic

oscillatory motion

The stability condition of the local oscillators may limit an extention of them by

[<$2>o]1/2 < a/2,

(17)

for solid electrolytes. Accordingly the following condition must be satisfied for

32ow/3w2>O : 3~s2-I 2. 2_ ! MBa2 ~s211+~s212>~ >MBa2~s2 .

(18)

Equation (18) gives no solution so that d w is a Drude type of the conductivity as shown in Figs.2 and 3, but with a resonant peak at w = Ws, where the physical quantities are chosen as 6-I=500K, a=l.8A and iV= 1.8x10-22g in Figs.2

2.0 E=IO

The problems we have considered as examples are 4-sites to 2-particles, 5-sites to 2- and 3particles and 6-sites to 4-particles. KA have taken the 6-sites to 3-particles as one of the examples, but it seems to be difficult to obtain exact eigen values and eigen vectors. We only write out here the case of 4-sites to 2particles: eigen value

~1.3

I~

1.5 1.0

I

~ 2.

c=2(y+2y-J)F : I~>=C[yl1>-ffl2>], where C=[2+X2]-I/2. Use of eq.(14) in

(14)

eq.(6)

to

dd.c.=~(zea)2"4yF/[2+y2] ,

I I

eigenvector

[¢>=c[~I1>+ylS>]

0

reduces

1.5 ~

0.1

(15) Fig.3 w-dependence approximation.

1.0 fl of ~w' eq.(16);

10 harmonic

71 lshii

124

2"°

g=o, =o.25

in solid electrolytes

(ns, e)

( g ,as)

1.0 -

-(0.5,6.0)

I

5[ ; [

/a.c.-conductivity

r~

"% \ 'v ir(o. ,o.s)

\

"(1.0,6.0)

\ \,,((0.3,4.0)

~::D ~ J " / / ~ - - ( 1 . 5 , 6 . 0 )

0.5-

I--

0.1

~

5 =0.25

1.0

10

0.1

1.0

I0

fl Fig.4

~-dependence

of d~,

eq.(16);

~ =

<~.

through 5. It is clearly u n d e r s t o o d that a damping of the o s c i l l a t i o n about a c o n f i g u r a tion, w h i c h may be r e i n t e r p r e t e d as a d a m p i n g of the phonons in a c o n f i g u r a t i o n , increases w i t h an increment of the hopping rate. This explicitly shows up a d o u b l e role of particles, t r a n s l a t i o n a l and o s c i l l a t o r y motion, w h i c h is one of the most r e m a r k a b l e characters in solid electrolytes. 5.2 A n h a r m o n i c

oscillatory

Fig.5 ~-dependence 5 × 10-12 erg/,~ 3

o f ij<,), e q . ( 1 6 )

[)/us eq.(13).

< z,, '2,

(19)

where

<,'];]>=<$'k'>o[].+0i°o2:72<~/.'o'~] <.92>:<.g2>o[l+.J@]'o? ], Further conditions that does not diverge impose

~

.;;

for

for

(2()) (2L)

j.

tile s e r i e s - e x p a n s i o n the restriction of

Ifo/>:c1<~. for a I,,<~:>.."7.-'; <:.

~or ,,.

(2~,)

motion

Let us consider two ways of i n t r o d u c i n g its effect, gate m o d e l and p e r t u r b a t i o n theory. One of the most s i g n i f i c a n t m a t t e r s of ionic m o t i o n s may be c o n c e r n e d w i t h g e o m e t r i c largeness of ions. Taking a Ag + ion in ~-Agi as an example, we simply conjecture that it is not easy to escape from its m e t a s t a b l e site to another a v a i l a b l e site on account of the interference of I- ions, owing to a g e o m e t r i c limitation. If a geometric c o n d i t i o n happens to be satisfied enough to make a Ag + ion escape from one tetrahedral site, which i s the case that the d i s t a n c e of the two I- ions b e c o m e larger titan the radius of Ag + ion, the Ag + ion has a p r o b a b i l i t y to escape. Let us consider a hard core p o t e n t i a l at S - ± ~ / 2 and a h a r m o n i c p o t e n t i a l for I S I < ~ / 2 in one d i m e n s i o n a l space. In this case the ion localizes such that ~ ~ ~ / i # ~ for any values of d=[,gB~s2~2/8] I/2. For larger values of d>>l, <$2> a ~ p r o a c h e s [,"f$~s2]-I. This example shows us that is almost independent of <{. Some constant values of <$2>!~22/12 yield profiles of the c o n d u c t i v i t y as in Fig.4, d i s p l a y i n g b o t h Drude type and n o n - D r u d e type. A c h a r a c t e r i s t i c point of this n o n - D r u d e type of b e h a v i o u r is a broad peak at ~ - g for ~%<8. This is interpreted hl a sense as a resonance to a hopping frequency. The other way is to c a l c u l a t e A g a i n the s t a b i l i t y c o n d i t i o n

eq. (12) or eq. (13). is

The results give rise to no notable structural change in o~ but some q u a n t i t a t i v e values. A r e p l a c e m e n t of a constant value for ~7L72>o % again in eq.(12) or (I]), and ~:q.(16) turns out to have some higher harmonic structures, w h i c h is shown in Fig.3 for ;,.

REFERENCES [1]

Kimball,

.1. C. a n d A d a m s ,

1.. W., . J r . ,

P h y s . R e v . B18 ( 1 9 7 8 ) 5 8 5 1 - 5 8 3 8 . ishii, T., TJt~_r',~. . j r v '. ,. .~.,. ~ ~ " "~ ' <£~Z _fJ~ g ~%<~'...rJi.& .; L/ Z z. ,';.,'g ~*zg. : J . ; ~£>Lz .,',z ;7>,{,~/ /i~.c .~,S_,' ,{:f6;a~ ( N o r t h Holland, Amsterdam, 1981). [3] Pietronero, L. a n d S t r ~ s s L e r , S., ':*a/g,.e~.;'tov' U < j . : ~ ,'n ~ ' ~ ; e :U~bc.,,>zqZ <~/r
[2]

:j'..~¢t:;bn:? ;"a2 cz-/1j. 7, S o l id S L a t e Commun. 22 (1977) 763-763. [5] Hatsubara, T., fJo$c ,J~' r~£',',/z,g.;~ Sr, g n ~ / > / ,n .[n ,':{T, d. Phys. Soc. 5span 38 (1975) [0761079. [6] Ish~i, T., unpublished: Equation (il) i~ not exact. An e x a c t e x p r e s s i o n call safely be o b t a i n e d [n a f o r m a l r e p r e s e n t a t i o n a s ill e q . ( l l ) w h e r e / ( 1 ( 7 ) a n d ~2(.7) a r c r e p l a c e d by ]'~l(/d') and .~2(.i,') w i t h .77 the d[ago,,al elements of 7.