ON INTERVALLEY DIFFUSION OF HOT-ELECTRONS

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ON INTERVALLEY DIFFUSION OF HOT-ELECTRONS

R. Brunetti, C. Jacoboni, L. Reggiani

To cite this version:

R. Brunetti, C. Jacoboni, L. Reggiani. ON INTERVALLEY DIFFUSION OF HOT-ELECTRONS.

Journal de Physique Colloques, 1981, 42 (C7), pp.C7-117-C7-122. �10.1051/jphyscol:1981712�. �jpa-

00221648�

JOURNAL DE PHYSIQUE

Colloque C7, supplément au n°10, Tome 42, octobre 1981 page C7-117

ON INTERVALLEY DIFFUSION OF HOT-ELECTRONS

R. Brunetti, C. Jacoboni and L. Reggiani

Gvuppo Nazionale di Struttura delta Materia, Istituto di Fisioa dell'Universita di Modena, Via Campi 213/A, 41100 Modena, Italy

Résumé.- Nous proposons dans cette étude une discussion sur l'applicabilité de l'é- quation de Shockley dans le calcul du coefficient de la diffusion intervallée DÏ pour le cas réel d'un semi-conducteur à plusieurs vallées. En particulier on montre que l'hypothèse simplificatrice posée d'un temps "X. pour les transitions intervallées_des porteurs constants conduit à des valeurs du D

1

très élevées. Pour le calcul de D

1

par la méthode de simulation Monte Carlo, il faut porter attention à la définition et aux valeurs des apports de vallée à la diffusion. Nous présentons des résultats numériques dans le cas de Si pour les températures de T = 77 K et T = 200 K. Les va- leurs sont comparées avec des résultats obtenus par la relation de Shockley.

Abstract. - This paper presents an analysis of the applicability of the Shockley formula to the evaluation of the intervalley diffusion coefficient D

¹

of electrons for a realistic case of ma ny-valley semiconductor. In particular it is shown that the symplifing hypothesis of a const ant time y for carrier intervalley transfer leads to an overestimation of D

¹

. For a determina tion of D

1

from Monte Carlo simulation particular care must be put in the definition and the eva luation of the valley contributions to diffusion. Numerical results are presented for the case of Si at T = 77 and T = 200 K, and compared with the results obtained by means of the Schok_

ley formula.

1. Intervalley diffusion. - If we apply a static uniform electric field E to a many valley serm conductor , electrons in valleys diffently oriented with respect to the field orientation have dif ferent drift velocities. This fact brings about an intervalley diffusion A , 2/ , which must be added in the longitudinal diffusivity if the valley drift velocities have different components pa rallel to E , and/or to the transverse diffusivity if valley drift velocities have different compo nents perpendicular to E. The physical reason of this phenomenon is shown in Fig. 1 for a two-valley semiconductor.

In the first case (Fig. 1-a) if electrons are in x = 0 at t = 0, at increasing time they will spend part of their time in one type of valley and part in the other type, so that they will drift part of their time with velocity v^ and part with v

2

. Thus an initial bunch will spread along the direction of the field even in absence of thermal diffusion.

In the second case, even though the average drift velocity is parallel to E, in each valley electrons have drift velocities not parallel to E, the components of v\ and V2 orthogonal to E being equal in modulus, but with opposite orientations. This fact, together with the coupling of the two valleys, leads to the presence of a transverse intervalley diffusion.

For silicon, which is the material considered in this paper, these two cases occur when E is along a <100) direction or a < l l l ) direction, respectively. If one considers other symme try directions, such as a<110> direction, parallel and transverse intervalley diffusivities are present simultaneously.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981712

JOURNAL DE P H Y S I Q U E

Fig. 1 - Phenomenology of t h e l o n g i t u d i n a l a) and t r a n s v e r s e b) i n t e r v a l l e y d i f f u s i o n i n a s i m p l i f i e d two-valley model. The upper p a r t o f t h e f i g u r e shows t h e o r i e n t a t i o n o f t h e v a l l e y d r i f t v e l o c i t i e s vl and v2

*

with r e s p e c t t o t h e d r i f t i n g f o r c e

4 F. In t h e lower p a r t , t h e time evolutions of t h e s p a t i a l d i t r i b u t i o n s are shown f o r bunches of c a r r i e r s which s t a r t a t t = 0

i n x = 0. The t h r e e cases i n a) r e f e r , from top down, t o t h e absence of any d i f f u s i o n , t o t h e presence of i n t e r v a l l e y d i f f l s i o n alone, and t o t h e presence of both i n t e r v a l l e y and thermal d i f f u s i o n , r e s p e c t i v e l y ; in part b) t h e two cases r e f e r t o the absence of any d i f f u s i o n and t o t h e presence of both i n t e r v a l l e y and thermal d i f f u s i o n .

2 .

Shockley formula.

-

F o r a model with two t y p e s of v a l l e y s the total diffusivity D i s given by the e x p r e s s i o n :

w h e r e n l and n2 a r e the f r a c t i o n o f e l e c t r o n s i n v a l l e y s of type 1 and

2

r e s p e c t i v e l y ,

D l

and D2 a r e the single-valley contributions t o the diffusion, and ~i i s t h e i n t e r v a l l e y diffusion coef ficient. I n t h i s c a s e f o r the evaluation of the i n t e r v a l l e y diffusion a g e n e r a l e x p r e s s i o n h a s been given by Shockley /1/

:

w h e r e 2, i s the i n v e r s e r a t e f o r e l e c t r o n t r a n s f e r from a v a l l e y of type 1 t o any valley of t y p e

2 ,

and s i m i l a r l y f o r Z2 . Eq.(2) h a s been obtained f o r a simplified model i n which the i n t e r - -valley-transfer t i m e s T, and T Z a r e supposed to be constant in time. T h i s h y p o t h e s i s , howev_

e r , i s f a r from r e a l i t y i n a c t u a l s e m i c o n d u c t o r s . I n f a c t , when a c a r r i e r u n d e r g o e s a n inter- valley s c a t t e r i n g i t a r r i v e s in t h e f i n a l valley with a n e n e r g y which i s , in g e n e r a l , l o w e r thm the mean e n e r g y of t h e v a l l e y , b e c a u s e intervalley-phonon a b s o r p t i o n i s l e s s f r e q u e n t than e m i s s i o n , s o that the c a r r i e r e n e r g y g r o w s up d u r i n g the f i r s t time spent in t h e new valley.

T h e r e f o r e , the i n t e r v a l l e y s c a t t e r i n g p r o b a b i l i t y , which i s s t r o n g l y energy-dependent, a l s o v a r i e s with time.

T h i s phenomenon c a n be made p a r t i c u l a r l y a p p a r e n t f o r a semiconductor model in which

the i n t e r v a l l e y s c a t t e r i n g o c c u r s only i f , and a s soon a s , the e l e c t r o n s r e a c h a value of e n e y

gy E , , and i n t r a v a l l e y s c a t t e r i n g i s a b s e n t . I n t h i s c a s e Eq.(2) l e a d s t o a n i n t e r v a l l e y diffu-

sion d i f f e r e n t from z e r o , r, and Zzbeing equal t o the times n e c e s s a r y f o r the c a r r i e r e n e r g y to r e a c h the value E, . The i n t e r v a l l e y diffusion, h o w e v e r , i s z e r o , b e c a u s e t h e e l e c t r o n d y n t mics i s no more stochastic: if we c o n s i d e r , f o r example, a two-valley model with the electro- n s in x

= 0

a t t

= 0

with k

= 0,

a f t e r any time t a l l of them have spent the same time i n e a c h t e pe of v a l l e y , and the indetermination on t h e s e t i m e s , which b r i n g s about i n t e r v a l l e y diffusion, i s l o s t . The sirnplifing hypothesis of a constant

'C

l e a d s tb an indetermination on the time of o c c u r r e n c e of an i n t e r v a l l e y t r a n s i t i o n of the o r d e r of , while for t h e stepwise probability function c o n s i d e r e d above the indetermination i s z e r o . F o r a r e a l physical c a s e the scatter- ing probability again i n c r e a s e s r a p i d l y a t the e n e r g y r a n g e of the i n t e r v a l l e y phonons, and the indetermination on Z i s intermediate between the two c a s e s . S o we c a n conclude that Eq.(2) gives a g e n e r a l o v e r e s t i m a t e of D ~ .

3 . On the definition of single-valley diffusion.

- The g e n e r a l c o n s i d e r a t i o n s r e p o r t e d above

s u g g e s t the convenience of an evaluation of Dl independent of Eq.(2), which c a n be achieved by calculating

DI

a s t h e d i f f e r e n c e between the t o t a l diffusivity D and t h e single-valley contri- butions Dv. An a p p r o p r i a t e definition of Dv i s however not t r i v i a l . One c a n think of defining Dv in t h e same way a s D using the e x p r e s s i o n :

wher now x(t) i s the s p a c e c o v e r e d by the c a r r i e r when i t i s i n the valley of i n t e r e s t , and t i s the time spent by the c a r r i e r i n the same valley. T h i s definition however i s not c o r r e c t , be_

c a u s e i f only the time spent by the c a r r i e r i n a valley i s taken into account, i n t e r v a l l e y t r a T s i t i o n s b r e a k t h e c o r r e l a t i o n between "the past" and "the future" of the c a r r i e r : coming back in the v a l l e y , tha c a r r i e r h a s l o s t memory of the values of i t s dynarnical p r o p e r t i e s (namely velocity and e n e r g y fluctuations above t h e i r mean values) b e f o r e it had been s c a t t e r e d away.

I n p a r t i c u l a r some contribution t o the diffusivity, r e l a t e d to velocity fluctuations, i s l o s t . If we remember t h a t i n t e r v a l l e y t r a n s i t i o n s bring about negative c o r r e l a t i o n s of velocity fluctu:

tions /3/ , we c a n expect that Eq.(3) o v e r e s t i m a t e s D.

₊

T h e s e c o n s i d e r a t i o n s have been checked i n t h e c a s e of S i with E p a r a l l e l to a(ll1) d i r e 5 t i o n : i n t h i s c a s e a l l valleys a r e equivalent with r e s p e c t t o t h e field d i r e c t i o n a n d , f o r s y m m c t r y , Dv

=

D. F i g .

2

shows that Dv calculated with Eq.(3) i s , i n s t e a d , g r e a t e r than D. Res- ults a l s o show that when t h e number

N

of valleys c o n s i d e r e d i n the evaluation of Eq.(3) i n c r e a s e s Dv tends t o D a s e x p e c t e d , when a l l the s i x valleys a r e c o n s i d e r e d , the c o r r e l a t i o n bet- ween the v a r i o u s fragments of the p a r t i c l e h i s t o r y i s c o r r e c t l y taken into account and Dv becg

S I -

electrons

T = 7 7 K E / (111)

'\

'. _{-. . - -}

- - -

Fig. 2 - Single v a l l e y longitudinal d i f f u s i v i t y as a function of f i e l d s t r e n g t h a t T = 77 K f o r

'i//

(111)

.

^The upper and lower curves r e f e r t o the cases in which only one valley and a l l s i x valleys are considered, respectively, i n t h e evaluation of Eq.(3).

FIELD S T R E N G T H ( k V

cm-')

C7- 120 JOURNAL DE PHYSIQUE

mes equal t o D.

As f u r t h e r s u p p o r t to this physical i n t e r p r e t a t i o n , the a u t o c o r r e l a t i o n function of the velocity fluctuations:

i ^{where -} i n d i c a t e s time a v e r a g e and ( ) ensemble a v e r a g e , h a s been calculated f o r both the c a s e in which a l l the p a r t i c l e h i s t o r y i s c o n s i d e r e d and the c a s e i n which only the single-valley h i s t o r y i s studied. Reults , r e p o r ted i n F i g . 3 show that the negative p a r t of

C ( % ) ,

which is connected t o t h e e x i s t e n c e of negative c o r r e l a t i o n s due to i n t e r v a l l e y t r a n s i t i o n s , i s r e d u c e d , a s expected from

fI'Jrl the consid$rations When E i s along re p o r t e d a (100) d i r e c t i o n above. we have hot and cold

E '.

^-0

v a l l e y s , and consequently , a n i n t e r v a l l e y term is added

3

Q

to the longitudinal diffusivity. F o r t h i s r e a s o n c a r e must

0 0.5 1 1 5 2

be put in o r d e r t o obtain an estimate of the D,, ' s , without

T I M E ( p s e c )

having i n t e r f e r e n c e with i n t e r v a l l e y diffusion. Eq .(3) can

be used f o r a single cold o r hot v a l l e y , thus obtaining D I P

A L

F i g . 3 - Rutocorrelation f u n c t i o n of

ve and D l h , r e s p e c t i v e l y ; f u r t h e r m o r e Eq.(3) c a n b e used

locity fluctuations for a single valley

with the collection of a l l equivalent cold o r hot v a l l e y s ,

(broken curve) and f o r a l l s i x v a l l e y s

thus obtaining D2c and Dqh , r e s p e c t i v e l y . If we assume

(continuous curve).

that e a c h f o r g s c a t t e r i n g event plays the same r o l e i n

the determination of Dx,, then D,,,which c o l l e c t s inform%

tion only from the c o l d ~ v a l l e y s , c o r r e c t s Dl, f o r a l l g s c a t t e r i n g and no f s c a t t e r i n g , while Dqh,which collects information only from hot valleys, c o r r e c t s Dlh f o r a l l g and half f s c a t t e r i n g s .

T h u s an estimate of Dvc and Dvh c a n be obtained a s follows:

where Nf and Ng a r e the number of f and g s c a t t e r i n g e v e n t s , a s obtained from t h ~ M o n t e C a r l o simulation, r e s p e c t i v e l y . The above p r o c e d u r e h a s been checked i n the c a s e of E along a (111) direction: Dv , evaluated from Eq.(5) , w h e r e D2 and D4 have been obtained collecting r e s u l t s on two valleys and four valleys r e s p e c t i v e l y , i s equal to the total diffusion coefficient D, a s expected.

4. Numerical r e s u l t s .

- Once the valley diffusion c o n s t a n t s have been obtained by means of

Eq (5) with t h e p r o c e d u r e d e s c r i b e d i n the p r e v i o u s s e c t i o n , the i n t e r v a l l e y diffusivity D l c a n be obtained by d i f f e r e n c e , from Eq.(l).

Numerical r e s u l t s have been obtained from a Monte C a r l o s i l i c o n model whose c h a r a c t e r i s - t i c s a r e d e s c r i b e d i n d e t a i l in /4/ . ^{F i g . 4} shows t h e r e s u l t s f o r D1 compared with the pre- dictions of Eq.(2) f o r which Z h a s been obtained from t h e simulation /4/ . F o r comparison, the t o t a l longitudinal diffusivity D h a s been r e p o r t e d .

At 200 K the difference between Dl, a s evaluated from Eq . ( l ) , and from Eq.(2) i s small,

comparable t o the s t a t i s t i c uncertainty of the r e s u l t s . At 77 K , on t h e c o n t r a r y , the differen-

c e i s quite l a r g e , and d e c r e a s e s a s the field s t r e n g t h i n c r e a s e s . T h i s behaviour c a n be u n d e r

k - % + - Y @

FIELD STRENGTH ( V c m - ' )

Fig. 4 - Longitudinal d i f f u s i o n c o e f f i c i e n t (dot- -dashed l i n e s ) , i n t e r v a l l e y d i f f u s i o n c o e f f i c i e n t as evaluated from Eq.(2) (dashed l i n e s ) ,and from Eqs.(l) and ( 5 ) (continuous l i n e s ) a s f u n c t i o n of f i e l d s t r e n g t h .

F i g . 6 - Autocorrelation f u n c t i o n of v e l o c i t y f l u c t u a t i o n s f o r a s i m p l i f i e d s i l i c o n model (see t e x t ) .

Fig. 5 - Longitudinal d i f f u s i v i t y (dot-dashed l i n e ) , i n t e r v a l l e y d i f f u s i v i t y a s evaluated from Eq.(2) (dashed l i n e ) and from Eqs.(l) and (5) (continuous l i n e ) as a function of f i e l d s t r e n g t h f o r a s i m p l i f i e d s i l i c o n model ( s e e t e x t ) .

T I M E ( p s e c )

s t o o d by c o n s i d e r i n g that a t h i g h e r e l e c t r o n e n e r g i e s t h e time dependence of

Z

, c a u s e d by the e n e r g y - l o s s due t o i n t e r v a l l e y e m i s s i o n , i s r e d u c e d .

I n o r d e r t o c h e c k the physical i n t e r p r e t a t i o n of t h e above r e s u l t s , a simple model h a s be- e n a n a l y s e d in which a c o u s t i c i n t r a v a l l e y s c a t t e r i n g i s s t r o n g l y d e p r e s s e d and a v e r y s t r o n g i n t e r v a l l e y s c a t t e r i n g i s p r e s e n t , with equivalent t e m p e r a t u r e To= 5 0 0 K . The l a t t i c e temp:

r a t u r e h a s been kept low, in o r d e r t o eliminate phonon a b s o r p t i o n . I n t h i s way we a p p r o a c h t h e limiting situation d i s c u s s e d i n s e c t i o n 2 .

F i g . 5 s h o w s the a u t o c o r r e l a t i o n function of velocity fluctuations obtained f o r a field of

500 V/cm along a (111) d i r e c t i o n , whose long o s c i l l a t o r y c h a r a c t e r i n d i c a t e s a s t r o n g z t r e a -

ming b e h a v i o u r , which, in t u r n , c o r r e s p o n d s to a s t r o n g time dependence

of 'C

. ^{F o r E} p a r a 2

l e l t o a (100)direction the Shockley formula y i e l d s , with t h i s model, a n i n t e r v a l l e y d i f f u s i v i

ty which i s even g r e a t e r than the total D ( F i g . 6 ) . T h e c o r r e c t evaluation of D l from E q . ( l )

C7-122 JOURNAL DE PHYSIQUE

on t h e c o n t r a r y , g i v e s v a l u e s well below D , a s expected.

We h a v e s e e n that i n Eq.(2)

'C

p l a g t h e r o l e of t h e uncertainty i n the lime of o c c u r r e n c e of i n t e r v a l l e y t r a n s i t i o n s ; by c o n s i d e r a t i o n of the connection between n o i s e and diffusion and of t h e Wiener-Khintchine t h e o r e m , i t s e e m s r e a s o n a b l e that a b e t t e r approximation would be obtained by usingZ i n Eq.(2) a s t h e a u t o c o r r e l a t i o n time of t h e fluctuations of the number of c a r r i e r s i n hot and cold v a l l e y s , defined a s the i n t e g r a l of t h e i r a u t o c o r r e l a t i o n function C L Q ) d i v i d e d by i t s value a t 9

=

0 .

Acknowledgments.

- We wish t o thank f o r s u p p o r t t h e C e n t r o di Calcolo E l e t t r o n i c o of the

Modena U n i v e r s i t y , w h e r e the computations have been performed.

R e f e r e n c e s .

/1/

W .

Shockley,

J

.A. Copeland and R . P . J a m e s , in Quantum T h e o r y of Atoms, Molecules and T h e Solid S t a t e , p. 537, Academic P r e s s , New York (1966).

/2/ T . Ohmi and S. Hasuo, Elect. and Comm. J a p a n 52 , 137 (1968).

/3/

W .

F a w c e t t and

H .

D . R e e s , P h y s . L e t t e r s

A=

, 578 (1969);

R . Fauquembergue , J . Zimmermann, A . K a s z y n s k y , E . C o n s t a n t , and G . Microondes ,

J.App1. P h y s . 2 , 1065 (1980).

/ 4 / R . B r u n e t t i , C . J a c o b o n i , F , Nava, L . Reggiani, G. Bosman and R . J . J . Z i j l s t r a , J .

Appl. P h y s . , t o be published.