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Variable range percolation confirms dispersive hopping transport
Journal of Non-Crystalllne Solids 35 & 36 (1980) 111-115 North-Holland Publishing Company
VARIABLE RANGE PERCOLATION CONFIRMS DISPERSIVE HOPPING TRANSPORT
M. S i l v e r and T. Dat~a Department of P h y s i c s and Astronomy U n i v e r s i t y o f North C a r o l i n a C h a p e l H i l l , North C a r o l i n a U.S.A.
Monte Carlo c a l c u l a t i o n s of hopping t r a n s p o r t i n d i s o r d e r e d systems have been performed. The hopping s i t e s have been placed a t random on a c u b i c l a t t i c e , f o r a number of c o n c e n t r a t i o n s . A l l Jumps up to two l a t t i c e spacings (32 p o s s i b l e s i t e s ) were allowed. The r e s u l t s d i s p l a y d i s p e r s i o n when t h e t r a n s p o r t time i n a g i a n t c l u s t e r i s small compared w i t h the time f o r a p a r t i c l e g e n e r a t e d o u t s i d e t h e g i a n t c l u s t e r to Jump on to i t . The f r a c t i o n of p a r t i c l e s g e n e r a t e d i n s i d e the g i a n t c l u s t e r i s p r o p o r t i o n a l to the p e r c o l a t i o n f r a c t i o n . Thus d i s p e r s i o n i n c r e a s e s with d e c r e a s i n g c o n c e n t r a t i o n of a c t i v e s i t e s . INTRODUCTION
At the l a s t amorphous semiconductor c o n f e r e n c e i n Edinburgh, t h e r e was a l i v e l y d e b a t e on whether d i s p e r s i v e ( n o n - g a u s s i a n ) t r a n s p o r t can r e s u l t from pure h o p p i ~ . Defending t h i s p o s s i b i l i t y were H. Scher and co-workers while the a n t a g o n i s t s were M. S i l v e r , J . M a r s h a l l and M. P o l l a k . At t h a t time t h e q u e s t i o n was n o t r e s o l v e d to e v e r y o n e ' s s a t i s f a c t i o n . The r e s o l u t i o n o f t h i s q u e s t i o n , however, i s imperat i v e for our u n d e r s t a n d i n g of such i m p o r t a n t phenomena as energy and c h a r g e t r a n s p o r t , recombination and p o s s i b l y luminescence i n amorphous systems. Since the l a s t meeting M a r s h a l l 1 p u b l i s h e d some s i m u l a t i o n r e s u l t s which he i n t e r p r e t e d as i n d i c a t i n g the absence of d i s p e r s i o n . However, the r e s u l t s a l s o showed n o n - g a u s s i a n b e h a v i o r because the r a t e of decay of t h e charge a t the a b s o r b i n g boundary was independent of t h e t r a n s i t time. P o l l a k 2 analyzed the time dependence from an a v e r a g e c l u s t e r approach. From t h i s , he claimed a l a c k . o f d i s p e r s i o n f o r p a r t i c l e s s t a r t i n g on a g i a n t c l u s t e r . Schaffman and S i l v e r ~ on t h e o t h er hand, u s i n g a c l u s t e r method showed d i s p e r s i o n i n a random system. However, t h e i r method i s a p p l i c a b l e o n l y f o r d i l u t e c o n c e n t r a t i o n s . L a t e r S i l v e r , Risko and B a s s l e r 4 a n a l y z e d e x c i t o n d i f f u s i o n i n terms of p e r c o l a t i o n and the r e s u l t s were i n accord w i t h e x p e r i m e n t s . Because they r e s t r i c t e d the s i m u l a t i o n to the n e q r e s t neighbor Jumps, d i s p e r s i o n could not be o b s e r v e d . However, i t was c l e a r t h a t the p a r t i c l e s g e n e r a t e d o u t s i d e a g i a n t c l u s t e r would r e q u i r e a t l e a s t one h a r d Jump to r e a c h the a b s o r b i n g boundary and so some d i s p e r s i o n would be expected. The f r a c t i o n o f t h e t o t a l p a r t i c l e s forming t h e g i a n t c l u s t e r would be p r o p o r t i o n a l to the p e r c o l a t i o n f r a c t i o n and t h i s f r a c t i o n tends t o zero a t the p e r c o l a t i o n threshold. Thus one a l s o e x p e c t s d i s p e r s i o n to be c o n c e n t r a t i o n d e p e n d e n t . As a r e s u l t of t h e s e developments, i t was f e l t t h a t t h e i s s u e of d i s p e r s i o n c o u l d hopef u l l y be c l a r i f i e d by performing a d d i t i o n a l c a l c u l a t i o n s i n c l u d i n g n o n - n e a r e s t neighbor t r a n s i t i o n s . Thus, we would be i n v e s t i g a t i n g the time e v o l u t i o n o f v a r i a b l e range p e r c o l a t i o n . Since a l l Jumps a r e a v a i l a b l e (up to t h e n e a r e s t 32 s i t e s ) , t h e r e i s o f c o u r s e no t h r e s h o l d f o r t h e D.C. d i f f u s i o n c o e f f i c i e n t . Our r e s u l t s c l e a r l y show d i s p e r s i o n p r o v i d i n g t h e time to Jump onto a g i a n t c l u s t e r i s long compared .with the time to d i f f u s e a l o n g t h e c l u s t e r . The s i m u l a t i o n t e c h n i q u e s p r e s e n t l y employed have b e e n d e s c r i b e d p r e v i o u s l y ~ and w i l l only be b r i e f l y o u t l i n e d h e r e . As i n d i c a t e d e a r l i e r the p h y s i c a l system t h a t i s being s i m u l a t e d i s t h a t of v a r i a b l e range hopping o f a t e s t p a r t i c l e on a cubic
111
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l a t t i c e where t h e l a t t i c e i s randomly o c c u p i e d by a c t i v e s i t e s t o t h e d e s i r e d c o n centration. The d i f f u s i n g p a r t i c l e i s a l l o w e d r e s i d e n c e o n l y on t h e s e a c t i v e h o s t sites. S e m i c l a s s i c a l i s o e n e r g e t i c " i n t e r = s i t e " t r a n s i t i o n s between t h e s e l o c a l i z e d s t a t e s a r e p e r m i t t e d . F u r t h e r m o r e , i t i 8 assumed t h a t t h i s i n t e r - s i t e t r a n s p o r t i s decoupled from a n y ~ ' i n t r a - s i t e " d y n a m i c s . The t i m e s c a l e s of i n t e r e s t
are:
(I ) t d i f -/Deff where
2
a ) v i j = v o e ' ~ r i J where r i j i s t h e d i s t a n c e frequency. b) f o r n e a r e s t n e i g h b o r Jumps, Vnn = 1 and D u r i n g th e a c t u a l c a l c u l a t i o n , we c h o s e a n p o t e n t i a l s i t e s , a f r a c t i o n C of which a r e where C i s t h e d e s i r e d c o n c e n t r a t i o n . At t h e i n i t i a l gin.
instant
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r n n = 1 which means v o - e ~. empty c u b i c l a t t i c e c o n t a i n i n g 21x21xZl f i l l e d a t random w i t h a c t i v e s i t e s and
( t = 0) a t e s t p a r t i c l e
(the walker) is placed at the ori-
The c o n d i t i o n a l p r o b a b i l i t y t h a t t h e w a l k e r p r e s e n t l y on t h e t h e J t h s i t e i s g i v e n by P i J ' where
PiJ = (vii)/ (r~il)"
ith
s i t e w i l l go to
(2)
The Jump f r e q u e n c y between two s i t e s k and £ i s d e f i n e d by Vk~ = 0 i f £ i s n o t an a c t i v e s i t e and g i v e n by a) and b) i f £ i s a c t i v e . I t sh o u ld be noted t h a t th e q u a n t i t y 7 c o n t a i n s a l l t h e p r o p e r t i e s of t h e medium. For our purposes 7 w i l l be r e g a r d e d a s a n u m e r i c a l p a r a m e t e r h a v i n g t h e same v a l u e a t a l l o ccu p ied s i t e s . Any c o r r e l a t i o n between d i f f e r e n t w a l k e r s w i l l be i g n o r e d ; t h i s c o r r e s p o n d s to a weak s i g n a l l i m i t . The Jump d e s t i n a t i o n £ i s d e t e r m i n e d by a u n i f o r m d i s t r i b u t i o n of random numbers, X, we i g h ted by t h e Jump p r o b a b i l i t y . The time of s o j o u r n a t k i s t k = (~Vk~)-lln X . or by t h e c o m p u t a t i o n a l l y more a t t r a c t i v e
(3) form
where Y i s o b t a i n e d from a n e x p o n e n t i a l d i s t r i b u t i o n o f random numbers. We f o l l o w t h e w a l k i n g p a r t i c l e t i l l i t g e t s . o u t s i d e a g i v e n r a d i u s R beyond which a l l a c t i v e s i t e s termiuate the motion of the p a r t i c l e . For t h e p r e s e n t c a l c u l a t i o n s R was c h o s e n to be 10. T h i s problem i s t h u s r e l a t e d to t h e c a l c u l a t i o n o f t h e p r o b a b i l i ty t h a t a p a r t i c l e i s s t i l l i n s i d e a s p h e r i c a l volume e n c l o s e d by a p e r f e c t l y a b s o r b i n g b o u n d ary . For a n i s o t r o p i c homogeneous medium, t h e p r o b a b i l i t y t h a t t h e p a r t i c l e i s s t i l l i n s i d e t h e volume Q(t) a t t i m e t i s
Q(t) = 2 Z(-I)m+l exp(-m) 2 Dt/R 2)
(5)
m
where D i s t h e d i f f u s i o n c o e f f i c i e n t , We p l o t our r e s u l t s a s shown i n f i g s 1 and 2 m e r e l y f o r c o m p a r i s o n . I n t h i s way a n y t i m e dependence of D w i l l be a p p a r e n t . The d i s p e r s i o n w i l l b e m a n i f e s t e d by Q (t) b e i n g a non e x p o n e n t i a l f u n c t i o n o f t . The Monte C a r l o algorithm., was t e s t e d w i t h C - 100Z; t h e c a l c u l a t e d D c o n f i r m e d e quat i o n 5 where D = va 2 w i t h t h e v a l u e f o r v s l i g h t l y b i g g e r t h a n t h e n e a r e s t n e i g h b o r
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f r e q u e n c y a s would be e x p e c t e d b e c a u s e we i n c l u d e d non n e a r e s t n e i g h b o r
Jumps.
As t h e c o n c e n t r a t i o n was l o w e r e d we found t h a t not a i l p a r t i c l e s c o u l d emerse J u s t by t h e n e a r e s t n e i g h b o r Jumps. T h i s f r a c t i o n v s c o n c e n t r a t i o n i s shown i n f i g u r e 3. We s e e t h e r e a n e x t r a p o l a t e d t h r e s h o l d around 26.5Z. The i n f i n i t e volume v a l u e from p e r c o l a t i o n t h e o r y i s a b o u t 30Z; hence our (21) 3 volume a p p e a r s Co g i v e reasonable results. One s h o u ld r e c o g n i z e , however, t h a t our c u r v e i n f i g u r e 3 r e p r e s e n t s t h e u p p e r l i m i t or t h e maximum p r o b a b i l i t y t h a t t h e r e i s a n e a r e s t n e i s h b o r p e r c o l a t i o n p a t h from t h e o r i g i n a c r o s s t h e e n t i r e volume. For i n s t a n c e a t 35Z c o n c e n t r a t i o n a t l e a s t 32Z of t h e p a r t i c l e s a r e g e n e r a t e d o u t s i d e t h e g i a n t c l u s t e r ; f o r t h e s e to emerge, a t l e a s t one h a r d Jump would ~ e r e q u i r e d . If the t i m e f o r t h i s h a r d Jump i s long compared to t~e t i m e to d i f f u s e t h r o u g h t h e g i a n t c l u s t e r , t h e n a s i n d i c a t e d e a r l i e r , one would e x p e c t d i s p e r s i o n . The h a r d Jump f r e q u e n c y i s d e t e r m i n e d by 7. Comparison of the r e s p o n s e Q(t) f o r t h r e e d i f f e r e n t v a l u e s of Y a r e s e e n i n f i g u r e 1. As s h o u l d be e v i d e n t , t h e d e v i a t i o n from e xpon e n t i a l i t y i s more pronounced f o r t h e l a r g e r y=s. A l s o , n o t i c e t h a t l i t t l e d i s p e r s i o n i s s e e n f o r Y " 1 1 . 1 2 . At y - 11.12 the d i f f u s i v e t i m e i s c o m p a r a b l e to
I00
90 80 70
60 5O 4-0 :50 20 I0
.:50
.40 Concentration Figure 3. Pf vs Concentration
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||5
t h e time to wake a h a r d l e a p a l o n g a f a c e d i a g o n a l d i r e c t i o n . However, f o r 7 - 20 t h e f a c e d i a g o n a l Jump t i m e i s much l o n g e r tha n f o r 7 " 11.12 (~40) a n d d i s p e r s i o n i s o b v i o u s . From t h e s e r e s u l t s one c a n u n d e r s t a n d why M a r s h a l l d i d n o t o b t a i n d i s p e r s i o n , a s he used a much too s m a l l 7 ( 6 - 1 4 ) . We e x p e c t to r e p e a t h i s f o r c e f i e l d c a s e w i t h l a r g e r 7 and p a r a l l e l p l a n e ge ome try. The t i me to d i f f u s e th ro u g h a g i a n t c l u s t e r i s a l s o s t r o n g l y c o n c e n t r a t i o n d e p e n d e n t . As p r e d i c t e d from p e r c o l a t i o n t h e o r y ( s e e S i l v e r , Risko and B a s s l e r 4) . T h i s e f f e c t i s s e e n i n t h e p r e s e n t c a s e where a l l Jumps a r e a l l o w e d . F i g u r e 2 shows our r e s u l t s f o r two c o n c e n t r a t i o n s - - 4 0 X and 29Z (7 " 1 1 . 1 2 ) . The e f f e c t i v e d i f f u s i o n c o n s t a n t a s d e t e r m i n e d by t h e s l o p e o f t h e s e c u r v e s i s p r o p o r t i o n a l to (C - E O ) 1 . 8 a s p r e v i o u s l y o b t a i n e d f o r o n l y n e a r e s t n e i g h b o r Jumps.4 DISCUSSION
I t i s c l e a r t h a t f o r t h e s i t u a t i o n s b e i n g s t u d i e d a d i s p e r s i v e t r a n s p o r t c a n oc c ur even i n a p u r e l y h o p p i n g s y s t e m . The u n d e r s t a n d i n g t h a t i s e m e r g i n g o u t of t h e s e ongoing c a l c u l a t i o n s i n d i c a t e s t h a t t h e d e t e r m i n i n g f a c t o r s r e l a t e to t h e t i m e s c a l e of t h e w o r s t Jump on a g i v e ~ p~th compared w i t h t h e time to d i f f u s e a l o n g the giant cluster. The lower t h e c o n c e n t r a t i o n , t h e l a r g e r i s t h e f r a c t i o n o f t h e random s i t e s t h a t would r e q u i r e a t l e a s t o n e i n t e r c o n n e c t i n g " b r i d g e " l o n g e r t h a n the lattice constant. The w a l k e r on t h e s e s i t e s c a n n o t t h e r e f o r e a v o i d a h a r d leap. Initially, f o r a n i n t e r v a l l e s s t h a n t h e d i f f u s i o n time a c r o s s a r e p r e s e n t a t i v e sub c l u s t e r a l l hops c o n t r i b u t e s i n c e t h e w a l k e r c a nnot a t t h i s s t a g e d i s t i n g u i s h between b e i n g on or o f f t h e g i a n t c l u s t e r . Consequently the apparent d i f f u s i o n c o e f f i c i e n t i s l a r g e . At l o n g e r t i m e s t h e p a r t i c l e s on t h e s u b c l u s t e r s no l o n g e r c o n t r i b u t e to and t h e d i f f u s , i o n c o n s t a n t d e c r e a s e s . At s t i l l l o n g e r t i m e s some h a r d Jumps onto t h e g i a n t c l u s t e r may be made. This w i l l be b a l a n c e d by t h e o f f - c l u s t e r Jumps and a d . c . d i f f u s i o n c o e f f i c i e n t (much l e s s t h a n t h e i d e a l l a t t i c e v a l u e ) i s r e a c h e d . M i t e s c u e e t a l . 5 have c a l c u l a t e d i n a c u b i c l a t t i c e for v a r i o u s c o n c e n t r a t i o n s , a l l 0 w i n g only n e a r e s t neighbor t r a n s i t i o n s . N e v e r t h e l e s s t h e y too o b s e r v e a s i m i l a r e f f e c t . T h e i r _ r e s u l t s a r e shown s c h e m a t i c a l l y i n f i g u r e 4. T h e i r c o m p u t a t i o n s c l e a r l y show i n c r e a s i n g more r a p i d l y a t s h o r t t i m e s t h a n a t l o n g e r t i m e s . U n d e r s t a n d a b l y a consequence o f t h i s m u l t i p l i c i t y i n e v e n t t i m e s i s t h a t t h e t r a n s p o r t c o e f f i c i e n t i s time d e p e n d e n t and t h e r e f o r e r e s u l t s i n th e n o n - g a u s s i a n " s t r a g g l i n g " of t h e c a r r i e r s . For a p u r e l y random medium t h e d e f i n i t i o n of a c l u s t e r i s not o b v i o u s , b u t w i l l l i k e l y be a f u n c t i o n of t i m e . At some s h o r t time, t s a c l u s t e r m i g h t be r e g a r d e d a s t h a t c o l l e c ~ i o n o f s i t e s f o r w hic h t h e n e a r e s t n e i g h b o r Jump f r e q u e n c y i s greater than 1/ts. The c o n c e n t r a t i o n Cs of such s i t e s w i l l d e t e r m i n e t h e a v e r a g e c l u s t e r s i z e , and t h i s c o n c e i v a b l y w i l l be a power law f u n c t i o n o f Cs . As time goes on, t h e c o n c e n t r a t i o n o f s uc h " l i n k e d " s i t e s w i l l grow u n t i l a l l s i t e s a r e included i n a super c l u s t e r . S i n c e t h e l i m i t i n g Jump f r e q u e n c y d e c r e a s e s w i t h t i m e , so w i l l t h e e f f e c t i v e d i f f u s i o n c o e f f i c i e n t . T h i s i n t e r p r e t a t i o n i s c ons i s t e n t w i t h t h a t proposed by Schaffman and S i l v e r . 3 We a r e c o n t i n u i n g w i t h t h e c a l c u l a t i o n s to g a t h e r enough d a t a to t e s t t h e v a l i d i t y _ of t h e m o d i f i e d s i n g l e s i t e d i s t r i b u t i o n f u n c t i o n p u t forward by P f i s t e r and Sc he r ,b and a l s o to r e v e a l any a n a l y t i c e x p r e s s i o n s t h a t would r e p r e s e n t our s i m u l a t i o n results. REFERENCES
1) J. H. Marshall, P h i l . Mag. B, 1978, ,Vol. 38, No. 4, page 335. 2) M. Pollak, P h i l Puig. 1977, V o l . 36, No. 5, page 1157. 3) M. J . Sch~ffman and M. S i l v e r , P h y s . Rev. B, V o l . 19, No. 8, 1979, page 4116. 4) M. S i l v e r , K. Ris k o and H. B a s s l e r , To be p u b l i s h e d P h i l . Mag.
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5) C. D. Mitescu, H. Ottavl and J. Roussenq, in:
3. C. Garland and D. B. Tanner
(eds), E l e c t r i c a l T r a n s p o r t and O p t i c a l P r o p e r t i e s of Inhomogeneous Media, AIP Conf. Proc. No. 40, page 377 (1978). 6) G. P f i s t e r and H. Scher, Adv. i n Phys. (1978), Vol. 27, No. 5, page 747.
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VARIABLE RANGE PERCOLATION CONFIRMS DISPERSIVE HOPPING TRANSPORT
M. S i l v e r and T. Dat~a Department of P h y s i c s and Astronomy U n i v e r s i t y o f North C a r o l i n a C h a p e l H i l l , North C a r o l i n a U.S.A.
Monte Carlo c a l c u l a t i o n s of hopping t r a n s p o r t i n d i s o r d e r e d systems have been performed. The hopping s i t e s have been placed a t random on a c u b i c l a t t i c e , f o r a number of c o n c e n t r a t i o n s . A l l Jumps up to two l a t t i c e spacings (32 p o s s i b l e s i t e s ) were allowed. The r e s u l t s d i s p l a y d i s p e r s i o n when t h e t r a n s p o r t time i n a g i a n t c l u s t e r i s small compared w i t h the time f o r a p a r t i c l e g e n e r a t e d o u t s i d e t h e g i a n t c l u s t e r to Jump on to i t . The f r a c t i o n of p a r t i c l e s g e n e r a t e d i n s i d e the g i a n t c l u s t e r i s p r o p o r t i o n a l to the p e r c o l a t i o n f r a c t i o n . Thus d i s p e r s i o n i n c r e a s e s with d e c r e a s i n g c o n c e n t r a t i o n of a c t i v e s i t e s . INTRODUCTION
At the l a s t amorphous semiconductor c o n f e r e n c e i n Edinburgh, t h e r e was a l i v e l y d e b a t e on whether d i s p e r s i v e ( n o n - g a u s s i a n ) t r a n s p o r t can r e s u l t from pure h o p p i ~ . Defending t h i s p o s s i b i l i t y were H. Scher and co-workers while the a n t a g o n i s t s were M. S i l v e r , J . M a r s h a l l and M. P o l l a k . At t h a t time t h e q u e s t i o n was n o t r e s o l v e d to e v e r y o n e ' s s a t i s f a c t i o n . The r e s o l u t i o n o f t h i s q u e s t i o n , however, i s imperat i v e for our u n d e r s t a n d i n g of such i m p o r t a n t phenomena as energy and c h a r g e t r a n s p o r t , recombination and p o s s i b l y luminescence i n amorphous systems. Since the l a s t meeting M a r s h a l l 1 p u b l i s h e d some s i m u l a t i o n r e s u l t s which he i n t e r p r e t e d as i n d i c a t i n g the absence of d i s p e r s i o n . However, the r e s u l t s a l s o showed n o n - g a u s s i a n b e h a v i o r because the r a t e of decay of t h e charge a t the a b s o r b i n g boundary was independent of t h e t r a n s i t time. P o l l a k 2 analyzed the time dependence from an a v e r a g e c l u s t e r approach. From t h i s , he claimed a l a c k . o f d i s p e r s i o n f o r p a r t i c l e s s t a r t i n g on a g i a n t c l u s t e r . Schaffman and S i l v e r ~ on t h e o t h er hand, u s i n g a c l u s t e r method showed d i s p e r s i o n i n a random system. However, t h e i r method i s a p p l i c a b l e o n l y f o r d i l u t e c o n c e n t r a t i o n s . L a t e r S i l v e r , Risko and B a s s l e r 4 a n a l y z e d e x c i t o n d i f f u s i o n i n terms of p e r c o l a t i o n and the r e s u l t s were i n accord w i t h e x p e r i m e n t s . Because they r e s t r i c t e d the s i m u l a t i o n to the n e q r e s t neighbor Jumps, d i s p e r s i o n could not be o b s e r v e d . However, i t was c l e a r t h a t the p a r t i c l e s g e n e r a t e d o u t s i d e a g i a n t c l u s t e r would r e q u i r e a t l e a s t one h a r d Jump to r e a c h the a b s o r b i n g boundary and so some d i s p e r s i o n would be expected. The f r a c t i o n o f t h e t o t a l p a r t i c l e s forming t h e g i a n t c l u s t e r would be p r o p o r t i o n a l to the p e r c o l a t i o n f r a c t i o n and t h i s f r a c t i o n tends t o zero a t the p e r c o l a t i o n threshold. Thus one a l s o e x p e c t s d i s p e r s i o n to be c o n c e n t r a t i o n d e p e n d e n t . As a r e s u l t of t h e s e developments, i t was f e l t t h a t t h e i s s u e of d i s p e r s i o n c o u l d hopef u l l y be c l a r i f i e d by performing a d d i t i o n a l c a l c u l a t i o n s i n c l u d i n g n o n - n e a r e s t neighbor t r a n s i t i o n s . Thus, we would be i n v e s t i g a t i n g the time e v o l u t i o n o f v a r i a b l e range p e r c o l a t i o n . Since a l l Jumps a r e a v a i l a b l e (up to t h e n e a r e s t 32 s i t e s ) , t h e r e i s o f c o u r s e no t h r e s h o l d f o r t h e D.C. d i f f u s i o n c o e f f i c i e n t . Our r e s u l t s c l e a r l y show d i s p e r s i o n p r o v i d i n g t h e time to Jump onto a g i a n t c l u s t e r i s long compared .with the time to d i f f u s e a l o n g t h e c l u s t e r . The s i m u l a t i o n t e c h n i q u e s p r e s e n t l y employed have b e e n d e s c r i b e d p r e v i o u s l y ~ and w i l l only be b r i e f l y o u t l i n e d h e r e . As i n d i c a t e d e a r l i e r the p h y s i c a l system t h a t i s being s i m u l a t e d i s t h a t of v a r i a b l e range hopping o f a t e s t p a r t i c l e on a cubic
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l a t t i c e where t h e l a t t i c e i s randomly o c c u p i e d by a c t i v e s i t e s t o t h e d e s i r e d c o n centration. The d i f f u s i n g p a r t i c l e i s a l l o w e d r e s i d e n c e o n l y on t h e s e a c t i v e h o s t sites. S e m i c l a s s i c a l i s o e n e r g e t i c " i n t e r = s i t e " t r a n s i t i o n s between t h e s e l o c a l i z e d s t a t e s a r e p e r m i t t e d . F u r t h e r m o r e , i t i 8 assumed t h a t t h i s i n t e r - s i t e t r a n s p o r t i s decoupled from a n y ~ ' i n t r a - s i t e " d y n a m i c s . The t i m e s c a l e s of i n t e r e s t
are:
(I ) t d i f -
2
a ) v i j = v o e ' ~ r i J where r i j i s t h e d i s t a n c e frequency. b) f o r n e a r e s t n e i g h b o r Jumps, Vnn = 1 and D u r i n g th e a c t u a l c a l c u l a t i o n , we c h o s e a n p o t e n t i a l s i t e s , a f r a c t i o n C of which a r e where C i s t h e d e s i r e d c o n c e n t r a t i o n . At t h e i n i t i a l gin.
instant
between t h e s i t e s
and v i i i s t h e Jump
r n n = 1 which means v o - e ~. empty c u b i c l a t t i c e c o n t a i n i n g 21x21xZl f i l l e d a t random w i t h a c t i v e s i t e s and
( t = 0) a t e s t p a r t i c l e
(the walker) is placed at the ori-
The c o n d i t i o n a l p r o b a b i l i t y t h a t t h e w a l k e r p r e s e n t l y on t h e t h e J t h s i t e i s g i v e n by P i J ' where
PiJ = (vii)/ (r~il)"
ith
s i t e w i l l go to
(2)
The Jump f r e q u e n c y between two s i t e s k and £ i s d e f i n e d by Vk~ = 0 i f £ i s n o t an a c t i v e s i t e and g i v e n by a) and b) i f £ i s a c t i v e . I t sh o u ld be noted t h a t th e q u a n t i t y 7 c o n t a i n s a l l t h e p r o p e r t i e s of t h e medium. For our purposes 7 w i l l be r e g a r d e d a s a n u m e r i c a l p a r a m e t e r h a v i n g t h e same v a l u e a t a l l o ccu p ied s i t e s . Any c o r r e l a t i o n between d i f f e r e n t w a l k e r s w i l l be i g n o r e d ; t h i s c o r r e s p o n d s to a weak s i g n a l l i m i t . The Jump d e s t i n a t i o n £ i s d e t e r m i n e d by a u n i f o r m d i s t r i b u t i o n of random numbers, X, we i g h ted by t h e Jump p r o b a b i l i t y . The time of s o j o u r n a t k i s t k = (~Vk~)-lln X . or by t h e c o m p u t a t i o n a l l y more a t t r a c t i v e
(3) form
where Y i s o b t a i n e d from a n e x p o n e n t i a l d i s t r i b u t i o n o f random numbers. We f o l l o w t h e w a l k i n g p a r t i c l e t i l l i t g e t s . o u t s i d e a g i v e n r a d i u s R beyond which a l l a c t i v e s i t e s termiuate the motion of the p a r t i c l e . For t h e p r e s e n t c a l c u l a t i o n s R was c h o s e n to be 10. T h i s problem i s t h u s r e l a t e d to t h e c a l c u l a t i o n o f t h e p r o b a b i l i ty t h a t a p a r t i c l e i s s t i l l i n s i d e a s p h e r i c a l volume e n c l o s e d by a p e r f e c t l y a b s o r b i n g b o u n d ary . For a n i s o t r o p i c homogeneous medium, t h e p r o b a b i l i t y t h a t t h e p a r t i c l e i s s t i l l i n s i d e t h e volume Q(t) a t t i m e t i s
Q(t) = 2 Z(-I)m+l exp(-m) 2 Dt/R 2)
(5)
m
where D i s t h e d i f f u s i o n c o e f f i c i e n t , We p l o t our r e s u l t s a s shown i n f i g s 1 and 2 m e r e l y f o r c o m p a r i s o n . I n t h i s way a n y t i m e dependence of D w i l l be a p p a r e n t . The d i s p e r s i o n w i l l b e m a n i f e s t e d by Q (t) b e i n g a non e x p o n e n t i a l f u n c t i o n o f t . The Monte C a r l o algorithm., was t e s t e d w i t h C - 100Z; t h e c a l c u l a t e d D c o n f i r m e d e quat i o n 5 where D = va 2 w i t h t h e v a l u e f o r v s l i g h t l y b i g g e r t h a n t h e n e a r e s t n e i g h b o r
113
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f r e q u e n c y a s would be e x p e c t e d b e c a u s e we i n c l u d e d non n e a r e s t n e i g h b o r
Jumps.
As t h e c o n c e n t r a t i o n was l o w e r e d we found t h a t not a i l p a r t i c l e s c o u l d emerse J u s t by t h e n e a r e s t n e i g h b o r Jumps. T h i s f r a c t i o n v s c o n c e n t r a t i o n i s shown i n f i g u r e 3. We s e e t h e r e a n e x t r a p o l a t e d t h r e s h o l d around 26.5Z. The i n f i n i t e volume v a l u e from p e r c o l a t i o n t h e o r y i s a b o u t 30Z; hence our (21) 3 volume a p p e a r s Co g i v e reasonable results. One s h o u ld r e c o g n i z e , however, t h a t our c u r v e i n f i g u r e 3 r e p r e s e n t s t h e u p p e r l i m i t or t h e maximum p r o b a b i l i t y t h a t t h e r e i s a n e a r e s t n e i s h b o r p e r c o l a t i o n p a t h from t h e o r i g i n a c r o s s t h e e n t i r e volume. For i n s t a n c e a t 35Z c o n c e n t r a t i o n a t l e a s t 32Z of t h e p a r t i c l e s a r e g e n e r a t e d o u t s i d e t h e g i a n t c l u s t e r ; f o r t h e s e to emerge, a t l e a s t one h a r d Jump would ~ e r e q u i r e d . If the t i m e f o r t h i s h a r d Jump i s long compared to t~e t i m e to d i f f u s e t h r o u g h t h e g i a n t c l u s t e r , t h e n a s i n d i c a t e d e a r l i e r , one would e x p e c t d i s p e r s i o n . The h a r d Jump f r e q u e n c y i s d e t e r m i n e d by 7. Comparison of the r e s p o n s e Q(t) f o r t h r e e d i f f e r e n t v a l u e s of Y a r e s e e n i n f i g u r e 1. As s h o u l d be e v i d e n t , t h e d e v i a t i o n from e xpon e n t i a l i t y i s more pronounced f o r t h e l a r g e r y=s. A l s o , n o t i c e t h a t l i t t l e d i s p e r s i o n i s s e e n f o r Y " 1 1 . 1 2 . At y - 11.12 the d i f f u s i v e t i m e i s c o m p a r a b l e to
I00
90 80 70
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.:50
.40 Concentration Figure 3. Pf vs Concentration
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M. Silver, T. Datta / Dispersive Hopping Transport
||5
t h e time to wake a h a r d l e a p a l o n g a f a c e d i a g o n a l d i r e c t i o n . However, f o r 7 - 20 t h e f a c e d i a g o n a l Jump t i m e i s much l o n g e r tha n f o r 7 " 11.12 (~40) a n d d i s p e r s i o n i s o b v i o u s . From t h e s e r e s u l t s one c a n u n d e r s t a n d why M a r s h a l l d i d n o t o b t a i n d i s p e r s i o n , a s he used a much too s m a l l 7 ( 6 - 1 4 ) . We e x p e c t to r e p e a t h i s f o r c e f i e l d c a s e w i t h l a r g e r 7 and p a r a l l e l p l a n e ge ome try. The t i me to d i f f u s e th ro u g h a g i a n t c l u s t e r i s a l s o s t r o n g l y c o n c e n t r a t i o n d e p e n d e n t . As p r e d i c t e d from p e r c o l a t i o n t h e o r y ( s e e S i l v e r , Risko and B a s s l e r 4) . T h i s e f f e c t i s s e e n i n t h e p r e s e n t c a s e where a l l Jumps a r e a l l o w e d . F i g u r e 2 shows our r e s u l t s f o r two c o n c e n t r a t i o n s - - 4 0 X and 29Z (7 " 1 1 . 1 2 ) . The e f f e c t i v e d i f f u s i o n c o n s t a n t a s d e t e r m i n e d by t h e s l o p e o f t h e s e c u r v e s i s p r o p o r t i o n a l to (C - E O ) 1 . 8 a s p r e v i o u s l y o b t a i n e d f o r o n l y n e a r e s t n e i g h b o r Jumps.4 DISCUSSION
I t i s c l e a r t h a t f o r t h e s i t u a t i o n s b e i n g s t u d i e d a d i s p e r s i v e t r a n s p o r t c a n oc c ur even i n a p u r e l y h o p p i n g s y s t e m . The u n d e r s t a n d i n g t h a t i s e m e r g i n g o u t of t h e s e ongoing c a l c u l a t i o n s i n d i c a t e s t h a t t h e d e t e r m i n i n g f a c t o r s r e l a t e to t h e t i m e s c a l e of t h e w o r s t Jump on a g i v e ~ p~th compared w i t h t h e time to d i f f u s e a l o n g the giant cluster. The lower t h e c o n c e n t r a t i o n , t h e l a r g e r i s t h e f r a c t i o n o f t h e random s i t e s t h a t would r e q u i r e a t l e a s t o n e i n t e r c o n n e c t i n g " b r i d g e " l o n g e r t h a n the lattice constant. The w a l k e r on t h e s e s i t e s c a n n o t t h e r e f o r e a v o i d a h a r d leap. Initially, f o r a n i n t e r v a l l e s s t h a n t h e d i f f u s i o n time a c r o s s a r e p r e s e n t a t i v e sub c l u s t e r a l l hops c o n t r i b u t e s i n c e t h e w a l k e r c a nnot a t t h i s s t a g e d i s t i n g u i s h between b e i n g on or o f f t h e g i a n t c l u s t e r . Consequently the apparent d i f f u s i o n c o e f f i c i e n t i s l a r g e . At l o n g e r t i m e s t h e p a r t i c l e s on t h e s u b c l u s t e r s no l o n g e r c o n t r i b u t e to
1) J. H. Marshall, P h i l . Mag. B, 1978, ,Vol. 38, No. 4, page 335. 2) M. Pollak, P h i l Puig. 1977, V o l . 36, No. 5, page 1157. 3) M. J . Sch~ffman and M. S i l v e r , P h y s . Rev. B, V o l . 19, No. 8, 1979, page 4116. 4) M. S i l v e r , K. Ris k o and H. B a s s l e r , To be p u b l i s h e d P h i l . Mag.
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5) C. D. Mitescu, H. Ottavl and J. Roussenq, in:
3. C. Garland and D. B. Tanner
(eds), E l e c t r i c a l T r a n s p o r t and O p t i c a l P r o p e r t i e s of Inhomogeneous Media, AIP Conf. Proc. No. 40, page 377 (1978). 6) G. P f i s t e r and H. Scher, Adv. i n Phys. (1978), Vol. 27, No. 5, page 747.
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Supported by ARO Grant DAAG29-776003!
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