DISPERSIVE TRANSPORT IN R-HOPPING SYSTEMS

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DISPERSIVE TRANSPORT IN R-HOPPING SYSTEMS

H. Imgrund, H. Overhof

To cite this version:

H. Imgrund, H. Overhof. DISPERSIVE TRANSPORT IN R-HOPPING SYSTEMS. Journal de

Physique Colloques, 1981, 42 (C4), pp.C4-83-C4-86. �10.1051/jphyscol:1981413�. �jpa-00220748�

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JOURNAL DE PHYSIQUE

CoZZoque 64, suppZ6ment au nOIO, Tome 4 2 , octobre 1981 page C4-83

DISPERSIVE TRANSPORT IN R-HOPPING SYSTEMS

H. Imgrund and H. Oyerhof

*

I n s t i t u t e for Theoretics Z Physics, Technische Universitat Berlin, F. R. G.

*

Fachbereieh Physik, Universitat-GH-Paderborn, F.R.G.

A b s t r a c t . We have i n v e s t i g a t e d t h e dynamical p r o p e r t i e s o f an i s o e n e r g e t i c random hopping system by a computer s i m u l a t i o n r e s t r i c t i n g ourselves t o 2D systems f o r computational reasons. T h i s allowed us t o study q u a s i - i n f i n i t e systems i n space w i t h o u t boundaries. The r e s u l t s show t h a t t h e d i s p e r s i v e behaviour o f the c u r r e n t i s present b u t l i m i t e d t o a time s c a l e corresponding t o hops t o a few s i t e s o n l y . A f t e r t h i s time t h e system i s governed by a quasi-Gaussian behaviour. No i n d i c a t i o n i s found t h a t t h i s p a t t e r n changes f o r v e r y low d e n s i t i e s n o t a c c e s s i b l e t o our p r e s e n t s i m u l a t i o n approach. Compari- son w i t h a r e c e n t t h e o r y by Movaghar e t a l . f o r t h e frequency dependence o f t h e c o n d u c t i v i t y i s made and near p e r f e c t agreement i s found i f t h e d e n s i t y c o r r e c t i o n f a c t o r i s a l t e r e d f o r t h e 2D case.

I n t r o d u c t i o n . -The d i s p e r s i v e t r a n s p o r t p r o p e r t i e s o f R-Hopping systems have been discussed by several authors. Scher and Lax ( 1 ) u s i n g t h e CTRW formalism o f M o n t r o l l and Weiss ( 2 ) have obtained a long d i s p e r s i v e t a i l extending over a l a r g e t i m e s c a l e i f t h e d e n s i t y o f hopping s i t e s i s s u f f i c i e n t l y small. P o l l a k ( 3 ) on t h e o t h e r hand u s i n g p e r c o l a t i o n arguments concludes t h a t d i s p e r s i o n i s n e g l i g i b l e f o r (isoenerge- t i c ) R-Hopping i n c o n t r a s t t o a paper o f Zvyagin ( 4 ) a l s o based on p e r c o l a t i o n theory. Working o u t a mean-field t h e o r y ( 5 ) Movaghar e t a l . show t h a t t h i s system does e x h i b i t d i s p e r s i o n although on a l i m i t e d time scale o n l y ( 6 ) .

To c l a r i f y t h i s s i t u a t i o n Marshall ( 7 ) has undertaken a computer-based Monte- Carlo c a l c u l a t i o n working on a small 3D sample. I n these c a l c u l a t i o n s t h e average d r i f t c u r r e n t i n t r o d u c e d by an e x t e r n a l f i e l d was obtained as a f u n c t i o n of time.

The r e s u l t was t h a t d i s p e r s i o n disappears a f t e r a s h o r t time scale. The small volume of t h e sample i n t h i s simulation, however, can l e a d t o erroneous r e s u l t s s i n c e t h e d r i f t c u r r e n t i s l i m i t e d by c o n s t r u c t i o n t o a narrow f i l a m e n t . We, t h e r e f o r e , s h a l l present r e s u l t s o f a s i m i l a r s i m u l a t i o n on a sample which i s v i r t u a l l y f r e e o f b o u n - d a r i e s . I n c o n t r a s t t o M a r s h a l l ' s work we s h a l l discuss t h e Brownian motion o f t h e c a r r i e r i n the absence o f an e x t e r n a l f i e l d . I n t h i s approach we r e s t r i c t ourselves t o a 2D system which can be handled more e a s i l y . T h i s a l l o w s us t o extend t h e simu- l a t i o n t o times l i m i t e d o n l y by t h e computational e f f o r t i n v o l v e d .

Computational d e t a i l s . - I n t h e f o l l o w i n g we consider t h e random walk o f c a r r i e r s on a twodimensional a r r a y o f hopping s i t e s w i t h average area d e n s i t y equal t o 1. The t r a n s i t i o n r a t e f o r hopping between s i t e s separated by r i n space i s g i v e n by

w = wo exp ( - a r ) (1

where a-' i s t h e i n v e r s e l o c a l i s a t i o n l e n g t h an wois taken t o be 1. The s i m u l a t i o n i s i n i t i a t e d on a f i n i t e a r r a y A o f s i z e 2 0 x 2 0 o r 3 0 x 3 0 by t h e generation o f r a n -

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981413

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JOURNAL DE PHYSIVJE

dom coordinates f o r N hopping s i t e s . The number N i s determined from random numbers d i s t r i b u t e d according t o a Binominal d i s t r i b u t i o n w i t h a mean value equal t o t h e area A. One s i t e i s chosen randomly t o be t h e p o i n t where t h e c a r r i e r i s generated i n i t i a l l y . The coordinates o f t h i s p o i n t serve as o r i g i n f o r t h e reference frame o f the s i m u l a t i o n . From t h i s p o i n t t h e 19 nearest neighbours i n space a r e determined.

The Monte Carlo s i m u l a t i o n i s performed adding t h e t r a n s i t i o n r a t e s t o t h e 1 9 n e i g h - bours t o o b t a i n the t r a n s i t i o n r a t e W . f o r l e a v i n g t h e i n i t i a l s i t e i. The a c t u a l time t a t which t h i s hop occurs i s c a l c u l a t e d from

w i t h t h e a i d o f a random v a r i a b l e x i n t h e 0 < x < l range which provides a stoch- a s t i c element i n t h e c a l c u l a t i o n . The f i n a l s i t e f t o which t h i s hop occurs i s determined by a random procedure weighted by t h e t r a n s i t i o n r a t e o f t h e i n d i v i d u a l hops. For the f i n a l s i t e f t h e process o f determining t h e 19 neighbours i s repea- ted and thus t h e Brownian motion o f t h e c a r r i e r i s monitored.

I f t h e c a r r i e r comes c l o s e t o t h e boundary o f the a r r a y A, i . e . , i f one o f t h e boundaries o f A i s c l o s e r t o the s i t e i a c t u a l l y occupied by t h e c a r r i e r than one o f t h e 19 neighbours o f i t h e a r r a y i s modified: A new a r r a y A ' centered around s i t e i i s constructed. I n t h e p a r t o f A' t h a t i s common w i t h A t h e s i t e coordinates from A are copied. For t h e remaining p a r t o f A ' a new s e t o f N ' s i t e s i s constructed a t random where N ' a g a i n i s a random number from a Binominal d i s t r i b u t i o n w i t h mean value equal t o t h e new area. The s i m u l a t i o n i s then continued on A ' u n t i l t h e c a r r i e r reaches one o f t h e boundaries o f A' and t h e a r r a y i s changed t o A " .

AS l o n g as t h e c a r r i e r proceeds i n one d i r e c t i o n o n l y t h i s s i m u l a t i o n i s s t r i c t - l y e q u i v a l e n t t o a s i m u l a t i o n on an i n f i n i t e a r r a y . Since N ' i s a random v a r i a b l e we f i x t h e s i t e d e n s i t y on t h e average only. Hence the, d i s t r i b u t i o n o f s i t e coordinates i s e q u i v a l e n t t o a random d i s t r i b u t i o n . For t h e r e a l movement o f t h e c a r r i e r , however, i t occurs t h a t t h e c a r r i e r moves from A t o A ' e t c . and l a t e r on moves back t o say A. I n t h i s case o u r present code t r e a t s A as a new area and hence generates new s i t e coordinates. We have no i n d i c a t i o n t h a t t h e n e g l e c t of c o r r e l a t i o n o f s i t e co- o r d i n a t e s gives r i s e t o any a r t e f a c t .

With t h i s r e s t r i c t i o n i n mind we can m o n i t o r t h e motion o f a c a r r i e r over an u n l i m i t e d t i m e s c a l e i n p r i n c i p l e . Unfortunately, however, we r e q u i r e s t a t i s t i c a l averages over many successive and independent s i m u l a t i o n s i n order t o c a l c u l a t e pro- per averages. Thus t h e p r a c t i c a l computations where terminated a f t e r some 105 hop- p i n g events.

For l a r g e r values o f a t h e s t r o n g dependence o f w on t h e i n t e r s i t e d i s t a n c e causes an i n c r e a s i n g number o f back and f o r t h hops i n small groups o f nearby s i t e s . This e f f e c t i s a l r e a d y described by Marshall ( 7 ) who reduced i t by a s p e c i a l t r e a t - ment of p a i r s o f nearby s i t e s . I n o u r s i m u l a t i o n the hopping r a t e s a r e symmetric

(no e x t e r n a l f i e l d ) and hence the e l i m i n a t i o n o f p a i r s d i d n o t l e a d t o a s i g n i f i c a n t r e d u c t i o n o f CPU times. An attempt t o e l i m i n a t e l a r g e r c l u s t e r s o f nearby s i t e s f a i l e d because i t i n f l u e n c e d t h e dynamical behaviour o f t h e system i n t h e intermed- i a t e time regime. We, t h e r e f o r e , s h a l l r e s t r i c t our d i s c u s s i o n t o s m a l l e r values o f a where a s p e c i a l treatment o f such c l u s t e r s can be avoided on t h e expense o f CPU time.

Results.

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The system under s i m u l a t i o n can be regarded as a system o f i s o e n e r g e t i c s i t e s a t a f i n i t e temperature. Since the s i t e coordinates a r e determined by a random procedure t h e r e w i l l be s i t e s t h a t a r e embedded i n t o a c l u s t e r o f c l o s e s i t e s as w e l l as s i t e s t h a t a r e r a t h e r i s o l a t e d . Elementary s t a t i s t i c s t e l l s us t h a t b o t h types o f s i t e s must i n e q u i l i b r i u m have t h e same occupation p r o b a b i l i t y because t h i s p r o b a b i l i t y i s i n f a c t independent o f t h e environment of t h e s i t e . I n our case where the c a r r i e r i s placed i n i t i a l l y on a s i t e chosen a t random we do a l r e a d y s t a r t w i t h an e q u i l i b r i u m d i s t r i b u t i o n . This i s most e a s i l y seen i f we i n t e r p r e t t h e motion o f the c a r r i e r s i n successive s i m u l a t i o n runs as a simultaneous motion o f many c a r r i e r s i n d i f f e r e n t p a r t s o f an i n f i n i t e medium. Consequently t h e occupation p r o b a b i l i t y i n our system f o r a l l times resembles t h a t o f an e q u i l i b r i u m d i s t r i b u t i o n . As a conse-

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quence q u a n t i t i e s l i k e average hopping time o r average hopping d i s t a n c e are independent of time. We have checked t h a t e v a l u a t i n g average hopping frequencies and d i s - tances d i r e c t l y a f t e r the c r e a t i o n o f t h e c a r r i e r i n each s i m u l a t i o n r u n and a f t e r the c a r r i e r has moved f o r some time. The independence o f these q u a n t i t i e s from the

"age" of t h e system was taken as an i n d i c a t i o n t h a t t h e s i m u l a t i o n t r e a t s t h e prob- lem adequetely.

A t i m e e v o l u t i o n i n our model system can occur o n l y f o r such q u a n t i t i e s t h a t r e f e r t o b o t h time and space p o i n t o f the c a r r i e r generation. I n t h e f o l l o w i n g we s h a l l discuss the time e v o l u t i o n o f t h e spread o f t h e c a r r i e r packet

as a f u n c t i o n o f the reduced time t = t - 2 a / a L

.

I n t h i s reduced time t h e average hop- p i n g time i s equal t o 1. Results o f s i m u l a t i o n s f o r fi2 a r e shown i n f i g . 1 f o r several values o f a. The abscissa r e f e r s t o a = 3 and subsequent curves a r e s h i f t e d by one t o the r i g h t f o r c l a r i t y . The curves can be parameterized by

where the parameter E i s given by E = 1

-

^0.04a f o r the range o f a values con- sidered.

I .

¹ ^I

0 2 4 l o g ( f l

F i g . 1 s i m u l a t i o n r e s u l t s f o r 6 ( t ) f o r d i f f e r e n t values of 2 a.

From f i g . 1 i t i s e v i d e n t t h a t a f t e r a v e r y s h o r t i n i t i a l p e r i o d equal t o t h e mean hopping time we observe a d i s p e r s i v e spread o f t h e c a r r i e r packet x h i c h i s more pro- nounced f o r t h e l a r g e r a values. A f t e r some lo3 hops f o r a = 3 ( 1 0 f o r a = 9 ) , however, t h i s d i s p e r s i v e spread g r a d u a l l y turns over i n t o a Gaussian behaviour w i t h 62- t, S i m u l a t i o n over even longer times ( w i t h corresponding lower s t a t i s t i c a l weight) c o n f i r m t h a t 62 - t holds i n t h e l i m i t o f v e r y l o n g times.

We can c a l c u l a t e the frequency dependence o f t h e c o n d u c t i v i t y from 6 2 ( t ) u s i n g the f l u c t u a t i o n - d i s s i p a t i o n theorem

2 2 m 2

e

*

o(w) = 6 ( t ) e x p ( i w t ) d t

0

I n f i g u r e 2 we compare o(w) thus obtained ( f u l l l i n e s ) w i t h the r e s u l t s o f a theory o f Movaghar e t a l . (6) a p p l i e d t o a

2 D

hopping system (8) (broken l i n e s ) . I n t h i s

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C4-86 JOURNAL DE PHYSIQUE

f i g u r e t h e d e n s i t y c o r r e c t i o n f a c t o r , a

,

i n t h e mean f i e l d theory was decreased by 30%. T h i s i s necessary because t h e p e r c g l a t i o n number, B c , i n c r e a s e s by about t h e same amount i f we s w i t c h from 3D t o 2D problems (9). The d ~ f f e r e n c e s between our

I . . .

-5 -L -3 -2 -1 0 1 2

log(=)

F i g . 2 ~ ( w ) o f t h e present c a l c u l a t i o n ( f u l l F i g . 3 Average Number o f s i t e s l i n e s ) compared w i t h r e f . 6 (broken l i n e s ) Nr v i s i t e d by a c a r r i e r r e s u l t s and those o f t h e e f f e c t i v e medium t h e o r y f o r small a show t h a t t h e t r e a t - ment o f s e l f a v o i d i n g pathes by a constant d e n s i t y c o r r e c t i o n f a c t o r i s n o t s t r i c t l y c o r r e c t . The e x c e l l e n t agreement f o r the l a r g e r values b o t h i n a b s o l u t e value and i n frequency dependence show t h a t t h e e f f e c t i v e medium theory gives a near exact d e s c r i p t i o n o f t h e dynamics of a hopping system.

We have a l s o monitored t h e t o t a l number o f d i f f e r e n t s i t e s v i s i t e d by a c a r - r i e r , Nr, as a f u n c t i o n o f time. Results a r e summarized i n f i g . 3 . S u r p r i s i n g l y N (T) decreases as a i s increased. The increase o f

f

i s due t o t h e r a p i d growth o f bacK and f o r t h hops b u t Gaussian d i s p e r s i o n i s observed a f t e r t h e c a r r i e r has v i s i t e d some 40 s i t e s f o r a = 9 (156 f o r u = 3 ) .

References

1 H. Scher, M. Lax,Phys.Rev.B7 (1973) 4491, Phys.Rev. B7 (1973) 4502 2 E.M.Montrol1, G.H.Weiss, JGrn.Math.Phys. - 6 (1965)

lm

3 M.Pollack, Phil.Mag.

3

(1977) 1157

4 I.P.Zvyagin, Phys.Stat.Sol.(b) 95 (1979) 227 5 B.Movaghar, Journ.Phys. C13 ( 1 9 m ) 4915

6 B.Movaghar, B.Pohlmann, n a u e r , phys.stat.so1. ( b ) - 97 (1981' 533 7 J.M.Marshal1, Phil.Mag. 38 (1978) 335

8 D.Wurtz p r i v a t e c o m m u n i c ~ i o n . We are indepted t o D.Wurtz f o r performing t h e numerical e v a l u a t i o n o f t h e formulae i n r e f . ( 6 ) i n t h e 2D case 9 G.E.Pike, C.H.Seager, Phys.Rev. B10 (1974) 1421 -