NON-EQUILIBRIUM HOT-CARRIER DIFFUSION PHENOMENON IN SEMICONDUCTORS II. AN EXPERIMENTAL MONTE CARLO APPROACH

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NON-EQUILIBRIUM HOT-CARRIER DIFFUSION PHENOMENON IN SEMICONDUCTORS II. AN EXPERIMENTAL MONTE CARLO APPROACH

P. Lugli, J. Zimmermann, D. Ferry

To cite this version:

P. Lugli, J. Zimmermann, D. Ferry. NON-EQUILIBRIUM HOT-CARRIER DIFFUSION PHE- NOMENON IN SEMICONDUCTORS II. AN EXPERIMENTAL MONTE CARLO APPROACH.

Journal de Physique Colloques, 1981, 42 (C7), pp.C7-103-C7-110. �10.1051/jphyscol:1981710�. �jpa-

00221646�

JOURNAL DE PHYSIQUE

Colloque C7, supplément au n°10, Tome 42, octobre 1981 page c7-i03

NON-EQUILIBRIUM HOT-CARRIER DIFFUSION PHENOMENON IN SEMICONDUCTORS II. AN EXPERIMENTAL MONTE CARLO APPROACH

P. Lugli, J. Zimmermann* and D.K. Ferry

Department of Electrical Engineering, Colorado State University, Fort Collins CO 80525, U.S.A.

Permanent Address : C.H.S., Greco Miaroondes, Universite de Lille, Lille 1, France

Résumé - Nous nous servons d'une méthode-de Monte-Carlo d'ensemble pour étudier les caractéristiques de transport de porteurs chauds en régime transitoire dynamique (TDR) sous l'effet d'un champ électrique stationnaire homogène. Ceci est appliqué au Silicium-N et nous présentons une analyse de l'évolution temp- orelle de quantités essentielles comme les fonctions de corrélation, le coeffi- cient de diffusion et l'énergie moyenne. Nous étudions la façon dont la dis- tribution spatiale des porteurs s'écarte d'une gaussienne par le calcul des cumulants d'ordre élevé. A la lumière des résultats fournis par la méthode de Monte-Carlo, nous discutons les effets de retard qui se manifestent dans les processus de transport sur les échelles de temps et distances très courtes, et vérifions la validité des équations réellement utilisées.

Abstract - An ensemble Monte Carlo technique is used to study the transport characteristics of hot carriers in the Transient Dynamic Response (TDR) to a steady homogeneous electric field. This is applied to n-type Silicon and an analysis of the time evolution of important quantities such as correlation function, diffusion coefficient and average energy is presented. The departure from Gaussian behavior of the spatial distribution function is also studied taking into account high order cumulants of the distribution. Retardation effects involved in the transport processes on the very-short time/space scale and the validity of the equations actually used are discussed in terms of the EMC results.

1. Introduction.- Two main aspects of non-stationary hot carrier transport in semi- conductors are reported in this paper. These are: 1) The evolution of the diffu- sion coefficient in the transient dynamic response (TDR) regime, which is generally associated with velocity overshoot, and 2) The retarded nature of transport pro- cesses on the very short time/space scales, as they are reflected for instance in the Retarded Langevin Equation (RLE) [1], and subsequent retarded balance equations established from non-equilibrium statistical thermodynamics [2]. These two aspects are of great importance to short-channel, submicron devices. In particular, diffu- sion affects the way in which carrier transit through the pinched-off channel occurs.

For a device whose dimensions are of the same order of magnitude as the collision- free flight, it will affect the thermal noise properties of the device. To properly treat retarded transport, which leads to retarded energy and momentum balance equa- tions, drastically improved simulation methods must be used, as current approaches are inadequate as they in general ignore these effects by using empirical relaxa- tion equations [3,4]. Moreover, it is found that the ensemble Monte Carlo (EMC)

x

T h i s work supported by the U.S. Army Research Office.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981710

C7- 1 04 JOURNAL DE PHYSIQUE

method used h e r e n a t u r a l l y i n c l u d e s a l l of t h e s e e f f e c t s , a r e s u l t t h a t was n o t recognized e a r l i e r d e s p i t e t h e c o n s i d e r a b l e amount of l i t e r a t u r e devoted t o non- s t a t i o n a r y t r a n s p o r t on t h e short-time s c a l e .

The ensemble Monte Carlo used h e r e h a s been developed by Lebwohl and P r i c e [ 5 ] and s u b s e q u e n t l y used and d e s c r i b e d i n g r e a t d e t a i l by one of t h e p r e s e n t a u t h o r s

[6]. The ensemble of 5000 e l e c t r o n s , i n i t i a l l y i n e q u i l i b r i u m , i s s u b j e c t e d t o a s t r o n g e l e c t r i c f i e l d s t e p a t t

0 , and t h e t r a n s i e n t response i s s t u d i e d . T h i s procedure i s a p p l i e d h e r e t o n-Si, b i a s e d a l o n g t h e <Ill> d i r e c t i o n s . Non-parabol- i c i t y i s included w i t h a = 0.5 e v - l ,

The v a l i d i t y of t h e RLE approach, p r e s e n t e d i n a p r e v i o u s paper [ I ] , i s checked u s i n g t h e p r e s e n t r e s u l t s . I n p a r t 2. of t h i s paper, a d i s c u s s i o n of t h e t r a n s i e n t t r a n s p o r t i s p r e s e n t e d w i t h p a r t i c u l a r a t t e n t i o n paid t o t h e s o - c a l l e d b a l l i s t i c regime and t h e r e l i a b i l i t y of t h i s model i s c r i t i c a l l y d i s c u s s e d . P a r t 3 . d e a l s with one of t h e most important q u a n t i t i e s i n TDR, t h e t r a n s i e n t d i f f u s i o n c o e f f i - c i e n t . Here EMC r e s u l t s o b t a i n e d f o r a v a r i e t y of e l e c t r i c f i e l d s a r e p r e s e n t e d and d i s c u s s e d . I n p a r t 4 . , we s t u d y t h e temporal e v o l u t i o n of t h e s p a t i a l d i s t r i b u t i o n f u n c t i o n of t h e ensemble and i t s d e p a r t u r e from Gaussian behavior.

2. T r a n s i e n t Transport.- A n o n - s t a t i o n a r y two time c o r r e l a t i o n f u n c t i o n f o r t h e v e l o c i t y f l u c t u a t i o n s can be d e f i n e d through:

and d i r e c t l y c a l c u l a t e d u s i n g t h e EMC technique. The a n g u l a r b r a c k e t s i n (1) r e p r e - s e n t a n ensemble average. T h i s a l l o w s us t o check an important p r o p e r t y , which i s i n s t r i n s i c t o t h e Retarded Langevin Equation [ I ] , namely t h a t t h e t r a n s i e n t velo- c i t y i s given by

< v ( t ) > = ,, [ @ ' A v ( r . O ) d ~ ⁽²⁾

where 4' (r,O) i s t h e normalized c o r r e l a t i o n f u n c t i o n [O ( ~ , 0 ) / @ ( 0 , 0 ) ] . It i s

Av Av Av

i n t e r e s t i n g t o n o t e t h a t t h i s i s t h e same e x p r e s s i o n o b t a i n e d f o r t h e e q u i l i b r i u m s i t u a t i o n by Kubo [ 7 ] . Both s i d e s of (2) have been c a l c u l a t e d i n t h e s i m u l a t i o n and a r e compared i n F i g . 1. The agreement i s w i t h i n t h e accuracy of t h e EMC method and l e a d s t o t h e conclusion t h a t t h e RLE can c o r r e c t l y d e s c r i b e t h e motion of h o t car- r i e r s i n t h e TDR regime. From (2) we s e e t h a t a l i n e a r i n c r e a s e of t h e v e l o c i t y

( < v ( t ) > =eFt/m*), a s expected f o r b a l l i s t i c t r a n s p o r t , can o n l y occur when

i s

AV

c o n s t a n t i n time. I n F i g . 2, we show t h e temporal e v o l u t i o n of 4 ' ( t , t o ) ( t > t o )

AV

f o r t h r e e d i f f e r e n t v a l u e s of to, i n c l u d i n g t

0. It i s e v i d e n t from t h i s f i g u r e t h a t t h e time d u r a t i o n o v e r which @ i s c o n s t a n t , corresponding t o a b a l l i s t i c r i s e

AV

of t h e v e l o c i t y , i s exceedingly s h o r t , perhaps o n l y a few femtoseconds, even though t h e mean f r e e time i s much longer. We a l s o n o t e t h a t t h e MC r e s u l t s show a g e n e r a l t r e n d i n t h a t @ ( t , t ) decays more r a p i d l y f o r l o n g e r to, corresponding t o a h i g h e r

Av o

Fig. 2

Normalized velocity autocorrelation function @Av(t,to) for three initial times to.

Fig. 3

Response of the average energy (in degrees 0 of the ensemble. Thedht'a point are EFIC results, the continu- ous curve represent the forcing term of equation (3) and the dashed curve represent the ballistic rise.

Fig. 1

Comparison of the ensemble drift velocity (A) and integral of the velocity autocorrelation function

(0)

as a function of time.

2

F

h

5

0

- - _- - >=

0 01 0 2 0 3 04

t ( p s )

-

t a . ^{4 4} . ^,

@

N -

.

•

K

J I I I I

C7-106 JOURNAL DE PHYSIQUE

temperature and more r a p i d randomization of t h e ensemble a s time i n c r e a s e s .

Another problem i n t h e u s u a l t r e a t m e n t of b a l l i s t i c t r a n s p o r t i s r e l a t e d t o t h e energy b a l a n c e equation. Normally, t h e c o n t r i b u t i o n of t h e thermal energy, due t o t h e random motion of c a r r i e r , is completely n e g l e c t e d . But, t h i s can be shown t o b e t h e dominant term. A q u a l i t a t i v e l y e x p l a n a t i o n [8] of t h e p r o c e s s e s on v e r y s h o r t - time s c a l e s i s t h e f o l l o w i n g : When a n e l e c t r i c f i e l d i s t u r n e d on a t t

0 , t h e ensemble of e l e c t r o n s b e g i n s t o respond i n s t a n t a n e o u s l y by a s h i f t i n momentum space, causing an i n s t a n t a n e o u s r i s e of t h e v e l o c i t y . However, we d o n ' t have an i n s t a n t a - neous i n c r e a s e of v e l o c i t y f l u c t u a t i o n s , t h a t i s a s p r e a d i n g o f t h e ensemble i n momen- tum space. Only when t h e c o l l i s i o n s begin t o break up t h e c o r r e l a t i o n of t h e ensemble w i t h t h e i n i t i a l s t a t e can t h e energy begin t o evolve. The memory e f f e c t i n t h e evo-

l u t i o n of t h e energy i s n o t taken i n t o account i n t h e e q u a t i o n s normally used f o r b a l l i s t i c t r a n s p o r t [3]. A f i r s t a t t e m p t t o e x p r e s s t h i s i n a formal way l e a d s t o t h e following e q u a t i o n [9]

- 2 ^{- -} q F v d ( t ) [ l - 9 A v ( t , 0 ) l - d t i [ E ( t - t ' ) - E o 1 x e ( t t ) ,

0

where x ( t ) i s a decay f u n c t i o n r e l a t e d t o t h e energy c o r r e l a t i o n f u n c t i o n . I n Fig. 3 we p l o t t h e i n i t i a l i n c r e a s e of t h e energy a s c a l c u l a t e d by t h e EMC. For comparison, we a l s o show a c u r v e corresponding t o t h e f o r c i n g term i n (3) (contin- uous curve) and t o a b a l l i s t i c regime (dashed curve). It is e v i d e n t t h a t a t v e r y s h o r t t i m e s , t h e r o l e of t h e memory f u n c t i o n i s s i g n i f i c a n t , and t h a t c o l l i s i o n s e f f e c t s a r e a l r e a d y e v i d e n t i n t h e EMC d a t a f o r t 5 .04 psec. Equation (3) was d e r i v e d under t h e assumption t h a t t h e energy a s s o c i a t e d w i t h t h e d r i f t of t h e ensem- b l e i s much s m a l l e r t h a n t h e thermal energy [ 9 ] . U n f o r t u n a t e l y , t h i s i s n o t t h e c a s e f o r S i f o r times s h o r t e r t h a n .1 p s , a s shown i n F i g . 4 where t h e two e n e r g i e s a r e c a l c u l a t e d from t h e EMC f o r a f i e l d of 50 kV/cm (E - ^m*vd/2). 2 Thus, i n o r d e r t o

d

f o r m a l l y s o l v e t h e problem of TDR, we w i l l have t o go back t o t h e o r i g i n a l equa- t i o n s [ 9 ] and t r y t o o b t a i n decoupled e q u a t i o n s f o r energy and momentum.

3. T r a n s i e n t Diffusion.- It h a s been shown t h a t a time dependent d i f f u s i o n c o e f f i - c i e n t can be d e f i n e d i n t h e n o n - s t a t i o n a r y regime a s [ l o ]

1 d

~ ( t )

- - <[Ax(t)]

2 d t ( 4 )

Using t h e EMC method, 9,(t1,t) and <(Ax(t)12> can b e c a l c u l a t e d independently and t h e v a l i d i t y of (4) can be checked. I n t h e p r e s e n t work, t h e i n t e g r a l i s c a l c u l a t e d by computing 25 c o r r e l a t i o n f u n c t i o n s a t e q u a l l y spaced times i n t h e range 0 - 0.4 psec. This r e s u l t i s i l l u s t r a t e d i n F i g . 5 , where t h e two s i d e s of (4) have been c a l c u l a t e d f o r a n e l e c t r i c f i e l d of 50 kV/cm. The agreement i s good, w i t h i n t h e e r r o r due t o t h e numerical i n t e g r a t i o n and d i f f e r e n t i a t i o n performed.

S i n c e t h e s t e a d y - s t a t e d i f f u s i o n c o e f f i c i e n t can be o b t a i n e d a s l i m i t i n g c a s e

of ( 4 ) , t h i s e x p r e s s i o n can b e assumed t o b e a g e n e r a l d e f i n i t i o n of t h e d i f f u s i o n

c o e f f i c i e n t f o r both s t a t i o n a r y and n o n - s t a t i o n a r y regimes. The time e v o l u t i o n of

Fig. 4

Evolution of the drift ( T ~ ) and thermal (T )energy of the ensemble (leftescale) and the ratio of the two (right scale) as a function of the time.

Fig. 6

Diffusion coefficient as a function of time for two electric fields as obtained from EMC results (dashed curves) and RLE results (solid curves).

FPg. 5

Comparison of the diffusion coeffi- cient as a function of tine, obtain- ed from the correlation function

(x) and from the time derivative of the average square displacement

15

-

ul N

E

- 10- .-

.-

- -

0

c

.-

I0 .A-

= _n

0 0.1 0.2 0.3 0.4

Time (

) N-Si

- ^E: 50 k~cm"//<lll>

300 K

, ; L "

- ⁶

6 8

0

6 ; 8 &

-.

I I I I .

C7- 1 08 JOURNAL DE PHYSIQUE

D ( t ) , a s o b t a i n e d from EMC methods and from t h e RLE approach, i s shown i n Fig. 6 , f o r two d i f f e r e n t e l e c t r i c f i e l d s . The most important f e a t u r e i s t h a t t h e d i f f u s i o n c o e f f i c i e n t e x h i b i t s an overshoot. This i s r e l a t e d t o t h e o s c i l l a t o r y shape of t h e c o r r e l a t i o n f u n c t i o n O ( t l , t ) , which a r i s e s from a combination of momentum and

AV

energy r e l a x a t i o n . This a l s o l e a d s t o a v e l o c i t y overshoot e f f e c t . T h i s i s p r e s e n t both i n t h e t h e o r e t i c a l and MC r e s u l t s , which a r e found t o be i n f a i r l y good q u a l i - t a t i v e agreement. The presence of a n overshoot i s a l s o e v i d e n t from Fig. 7 where we p l o t t h e d i f f u s i o n c o e f f i c i e n t a s a f u n c t i o n of f i e l d a t t h r e e d i f f e r e n t times.

Here, t = 0 . 4 ps corresponds t o s t e a d y s t a t e . We n o t i c e t h a t a t v e r y s h o r t times ( t

.02 p s ) , t h e response of t h e ensemble i s independent of t h e v a l u e of t h e e l e c t r i c f i e l d . T h i s means t h a t t h e ensemble s t i l l h a s a memory of i t s i n i t i a l s t a t e and c o l l i s i o n s have n o t y e t broken up t h i s c o r r e l a t i o n .

4 . S p a t l a l D i s t r i b u t i o n Function.- It i s w e l l known t h a t F i c k ' s law, c o n s i s t e n t with a s t a t i o n a r y Markovian random p r o c e s s , i m p l i e s t h a t t h e d i s t r i b u t i o n f u n c t i o n i s Gaussian f o r a l l times [11,12]. Due t o t h e non-Elarkovian and n o n l i n e a r n a t u r e of t h e p r o c e s s e s involved i n t h e TDR regime, we expect d e v i a t i o n s from t h e Gaussian behavior f o r t h e s p a t i a l d i s t r i b u t i o n f u n c t i o n , a s has been p r e v i o u s l y r e p o r t e d [13].

The EMC technique a l l o w s us t o c a l c u l a t e t h e time e v o l u t i o n o f a l l moments of t h e d i s t r i b u t i o n f u n c t i o n , and t h u s t o t a l c u l a t e t h e s e d e v i a t i o n s a s t h e ensemble evolves. The i n i t i a l d i s t r i b u t i o n f o r t h e ensemble i s taken t o be a &-function i n space. A Gaussian d i s t r i b u t i o n i s c h a r a c t e r i z e d by t h e f a c t t h a t a l l cumulants of o r d e r g r e a t e r t h a n 2 a r e zero. The t h i r d cumulant, which i s i d e n t i c a l t o t h e t h i r d c e n t r a l moment v

<(x - <x>) 3 >, and t h e f o u r t h cumulant, which can be expressed

3

a s a f u n c t i o n d f t h e second and f o u r t h c e n t r a l moments by k

Y4 - 3p25 have been c a l c u l a t e d i n t h e s i m u l a t i o n . The r e s u l t s Are shorn i n Fig. 8 , where t h e normalized q u a n t i t i e s

< ( x - < x > ) ~ > / < x > ~ and

= ( V 4 - lu>/!J:are p l a t t e d a s a f u n c t i o n

3

of time. The t h i r d cumulant s t a r t s with an a p p r e c i a b l e n e g a t i v e v a l u e and, a s t h e ensemble approaches s t a t i o n a r i t y , r e a c h e s a v a l u e very c l o s e t o zero. This seems t o i n d i c a t e t h a t t h e s p a t i a l d i s t r i b u t i o n f u n c t i o n m a i n t a i n s a pronounced symmetry around cx,. However,the f o u r t h cumulant d e v i a t e s c o n s i d e r a b l y from zero. T h i s shows a non-Gaussian b e h a v i o r , even Uhen s t a t i o n a r i t y i s reached. Yet, t h e t r a n - s i e n t e v o l u t i o n of t h e d i s t r i b u t i o n f u n c t i o n i s c h a r a c t e r i z e d by n e g a t i v e v a l u e s of t h e t h i r d and f o u r t h cumulant. What l e a d s t o t h i s d e v i a t i o n from Gaussian behavior and which e q u a t i o n should be s u b s t i t u t e d f o r F i c k ' s law h a s y e t t o be understood.

A n a l t e r n a t i v e e q u a t i o n t o F i c k ' s law suggested f o r s t a t i o n a r y c o n d i t i o n s , was pro-

posed by Jacoboni e t a l . [14]. This l e a d s t o an e x p r e s s i o n f o r t h e time d e r i v a t i v e

of a l l high-order c e n t r a l moments a s a f u n c t i o n of t h r e e c o n s t a n t c o e f f i c i e n t s ( t h a t

can b e r e l a t e d t o d r i f t v e l o c i t y and d i f f u s i o n ) and of t h e moments of lower o r d e r .

This e q u a t i o n can b e g e n e r a l i z e d t o n o n - s t a t i o n a r y c o n d i t i o n s , f o r example, by

assuming time-dependent c o e f f i c i e n t s . EMC r e s u l t s can be used t o check t h e v a l i d i t y

of t h i s approach. We found t h a t f o r times s h o r t e r t h a n 0 . 3 psec, t h e d a t a

s u r p r i s i n g l y f i t t h e g e n e r a l i z e d e q u a t i o n f a i r l y w e l l . However, f o r times l o n g e r t h a n 0.3 p s e c , d e v i a t i o n from t h e e x p e c t e d v a l u e s can b e n o t i c e d . The i n f l u e n c e of t h e i n i t i a l c o n d i t i o n s on t h e time e v o l u t i o n of t h e d i s t r i b u t i o n f u n c t i o n i s n o t known i n d e t a i l , and t h e t i m e r e q u i r e d t o r e a c h s t a t i o n a r i t y i s n o t known y e t . These f a c t o r s probably s t r o n g l y a f f e c t t h e r e s u l t s .

5. Conclusions.- The EMC i s one of t h e most r e l i a b l e c a l c u f a t i o n a l t e c h n i q u e s t h a t can b e used t o s t u d y t r a n s p o r t i n t h e TDR regime. It a l l o w s u s t o wroid s i m p l i f y i n g assumptions p r e s e n t i n most c u r r e n t t h e o r e t i c a l approaches t o t h i s problem ( b u t s e e r e f . 9 ) . It can b e a l s o shown t h a t t h e EMC i s a s o l u t i o n t o g e n e r a l i z e d t r a n s p o r t e q u a t i o n s t h a t a r e t h o u g h t t o a c c u r a t e l y d e s c r i b e t h e f a s t ; n o n - s t a t i o n a r y TDR r15].

From t h e r e s u l t s p r e s e n t e d h e r e , i t i s e v i d e n t t h a t b a l l i s t i c c o n d i t i o n s i n S i a t 300 K e x i s t o n l y over a t i m e of t h e o r d e r of a few femtoseconds. Overshoot e f f e c t s p r e s e n t i n b o t h d r i f t v e l o c i t y and t h e d i f f u s i o n c o e f f i c i e n t a r i s e from a combina- t i o n of momentum and energy r e l a x a t i o n e f f e c t s , and n o t due t o a b a l l i s t i c r i s e of t h e s e q u a n t i t i e s . A d e p a r t u r e from Gaussian b e h a v i o r i s found i n t h e t i m e e v o l u t i o n of t h e s p a t i a l d i s t r i b u t i o n f u n c t i o n , b u t a n i n t e r p r e t a t i o n of t h e c a u s e s of t h i s has s t i l l t o b e found. However, t h i s i s n o t unexpected. From t h e c o n v e r s e o f Doob's theorem [ 1 6 ] , t h e g e n e r a l l y non-exponential b e h a v i o r of $ ( t , t l ) f o r t h e n o n l i n e a r ,

AV

n o n - s t a t i o n a r y , non-equilibkium t r a n s p o r t i n t h e TDR regime s h o u l d s i g n a l a depar- t u r e from Gauss-Markov b e h a v i o r f o r t h e s p a t i a l d i f f u s i o n .

R e f e r e n c e s

1. J. Zimmermann, P. L u g l i , D. K. F e r r y , t h e s e proceedings.

2. D. K. F e r r y , P. L u g l i , J . Zimmermann, "The Ensemble Monte C a r l o Method and G e n e r a l i z e d T r a n s p o r t E q u a t i o n s f o r T r a n s i e n t Dynamic Response i n Semiconductors;' s u b m i t t e d f o r p u b l i c a t i o n .

3 . M. S. Shur, L. F. Eastman, IEEE T r a n s . E l e c t r o n Dev., ED-26 (1979) 1677.

4 . J. P. Nougier, J. C. V a i s s i e r e , D. Gasquet, J. Zimmermann, E. C o n s t a n t , J. Appl.

Phys. 52 (1981) 825.

5. P. A. Lebwohl, P. J. P r i c e , Appl. Phys. L e t t . 1 9 (1971) 530.

6. D. K. F e r r y , J. R. B a r k e r , Phys. S t a t . Sol. (b) 100 (1980) 683.

7. R. Kubo, J. Phys. Soc. J a p a n , 1 2 (1957) 570.

8. D. K. F e r r y , J. Zimmermann, P. L u g l i , H. Grubin, " L i m i t a t i o n t o B a l l i s t i c Trans- p o r t i n Semiconductors," s u b m i t t e d f o r p u b l i c a t i o n .

9. D. K. F e r r y , i n v i t e d p a p e r , t h e s e p r o c e e d i n g s .

10. J . Zimmermann, P. L u g l i , D. K. F e r r y , "Hot C a r r i e r D i f f u s i o n i n t h e T r a n s i e n t Regime," s u b m i t t e d f o r p u b l i c a t i o n .

11. W. E. A l l e y , B. J. A d l e r , Phys. Rev. L e t t . 43 (1979) 653.

12. R. Kubo, K. Matsuo, K. K i t a h a r a , J. S t a t . Phys. 9 11973) 51.

13. D. K. F e r r y , J. R. Barker, J. Appl. Phys. 52 (1981) 818.

14. C. J a c o b o n i , G. G a g l i a n i , L. R e g g i a n i , 0. T u r c i , S o l i d S t . E l e c t r o n i c s 2 1 (1978)

653. T h i s e q u a t i o n i s s i m i l a r t o one a l s o d i s c u s s e d by I. M. 9 e Schepper,

C 7 - 1 1 0 JOURNAL DE PHYSIQUE

H. van Beyeren, M. H. E r n s t , Physica 75 (1974) 1. These l a t t e r a u t h o r s p o i n t

4 6

o u t t h a t V n , V n , . . . terms cannot b e used f o r d i f f u s i o n even i n t h e l i n e a r c a s e ; due t o a tl" t a i l on D ( t ) , a behavior seen even i n semiconductors, a s shown by D. K. F e r r y , Phys. Rev. L e t t e r s 45 (1980) 758.

15. R. W. Zwanzig, i n L e c t u r e s i n T h e o r e t i c a l P h y s i c s , Ed. by W. E. B r i t t i n , B. W. Downs, and J. Down ( I n t e r s c i e n c e , New York, 1961) p. 106.

16. J. L. Doob, Ann. Math. S t a t . 43 (1942) 351.

Fig. 7

D i f f u s i o n c o e f f i c i e n t a s a f u n c t i o n of t h e e l e c t r i c f i e l d , f o r t h r e e d i f f e r e n t times.

Fig. 8

Normalized t h i r d cumulant a = 3

<(x - < x > ) ~ > / < x > ~ (x) and f o u r t h 2 2

cumulant a 4

₍₁₁₄ - 3 ~ ~ ) / ~ ~ ( 0 ) as a f u n c t i o n of time.

0 2

0 -02- -04- -06-

0 0 1 0 2 0 3 0 4

t ( P S )

-

- - - a -g - & - + - - x -

0

;

0 O X, N - S i

o

E=BOkV/cm//~lll~

x

300

I I I