Theory of relaxation of hot electrons and oscillatory photoconductivity in polar semiconductors

Physma 117B & 118B (1983) 247-250 North-HollandPubhslungCompany

247

THEORY OF RELAXATION OF OSCILLATORY PHOTOCONDUCTIVITY R.

Institut

Evrard,

de

Ph.

Lambin

HOT ELECTRONS AND IN P O L A R S E M I C O N D U C T O R S a n d J.

Schmit

P h y s i q u e B5, U n i v e r s i t ~ B - 4 0 0 0 L i e g e I, E e l g l u m .

de

Liege,

T h e E o l t z m a n n e q u a t i o n is s o l v e d f o r t h e r e l a x a t i o n of o u t - o f - e q u i l l b r i u m e l e c t r o n s d u e to o p t i c a l a n d a c o u s t i c a l p h o n o n s . T h e f i r s t s t a g e o f t h e r e l a x a t i o n , g o v e r n e d b y t h e o p t i c a l p h o n o n s , is q u i t e f a s t a n d l e a d s to a n o n M a x w e l l i a n d i s t r i b u t i o n . T h e a c o u s t i c a l p h o n o n s are r e s p o n s i b l e f o r t h e s e c o n d s t a g e o f the r e l a x a t i o n , m u c h s l o w e r a n d l e a d i n g to a Maxwellian distribution. C o m p a r i s o n is m a d e w i t h the e x p e r i m e n t a l res u l t s f o r G a A s . T h e s t a t i o n a r y s i t u a t i o n e n c o u n t e r e d in t h e s t u d y o f t h e oscillatory photoconductlvlty and photoluminescence is a l s o s t u d i e d . It is s h o w n t h a t t h e e l e c t r o n p o p u l a t i o n h a s an o s c i l l a t o r y b e h a v i o u r v e r s u s the e n e r g y at w h i c h t h e y are p h o t o - i n j e c t e d .

1.

INTRODUCTION

W h e n e l e c t r o n s are i n j e c t e d i n t o t h e c o n duction band of a semiconductor (or h o l e s into a valence band), their initial e n e r g y d i s t r i b u t i o n is u s u a l l y f a r f r o m the equilibrium distribution. Among the q u e s t i o n s t h a t c o m e n a t u r a l l y to m i n d are the f o l l o w i n g : H o w d o e s the e l e c t r o n p o p u l a t i o n r e l a x to t h e e q u i l i b r i u m distribution ? W h a t is t h e r o l e o f t h e i n t e r a c t i o n o f the e l e c t r o n s w i t h the different phonons ? These questions have an o b v i o u s i n t e r e s t in s t a t i s t i c a l m e c h a n i c s of i r r e v e r s i b l e p r o c e s s e s . T h e y a r e a l s o i m p o r t a n t f r o m the p r a c t i c a l v i e w p o i n t , d u e to the d e v e l o p m e n t o f m o r e and more numerous hot-electron devices in s e m i c o n d u c t o r technology. T h e w o r k r e p o r t e d h e r e is d e v o t e d to t h e s t u d y o f t h e s e q u e s t i o n s in t h e frame-' w o r k o f the f o l l o w i n g s i m p l e m o d e l : T h e c o n d u c t i o n b a n d h a s a s i n g l e m i n i m u m at t h e c e n t e r of t h e B r x l l o u i n z o n e . T h i s m i n i m u m is i s o t r o p i c a n d p a r a b o l i c . M o r e o v e r , t h e e l e c t r o n d e n s i t y is l o w e n o u g h f o r the e l e c t r o n - e l e c t r o n interactions to be n e g l i g i b l e . T h e e f f e c t s of t w o t y p e s o f e l e c t r o n phonon interactions are studied. The f i r s t t y p e is the p o l a r i n t e r a c t i o n w i t h l o n g i t u d i n a l o p t i c a l (LO) p h o n o n s as g i v e n b y F r S h l i c h ' s p o l a r o n t h e o r y (I) , a n d t h e s e c o n d o n e is a d e f o r m a t i o n potentlal interaction with the longitudinal acoustical (LA) p h o n o n s . It is w e l l k n o w n t h a t , for i n t r a b a n d t r a n s i t i o n s as An o u r m o d e l , the p h o n o n s e m i t t e d o r a b s o r b e d b y the c h a r g e c a r r i e r s n e c e s s a r i l y h a v e a l a r g e w a v e l e n g t h . T h i s is r e q u i r e d b y the l a w s o f m o m e n t u m a n d e n e r g y c o n s e r v a t i o n . T h i s l e a d s to the u s u a l c o n -

0 378-4363/83/0000-0000/$03.00 © 1983 North-Holland

clusion that the acoustical phonons are h i g h l y i n e f f i c i e n t f o r r e l a x i n g the e n e r gy of out-of-equillbrium electrons. T h e r e f o r e t h e m a i n c o n t r i b u t i o n to t h e e n e r g y r e l a x a t i o n c o m e s f r o m the i n t e r a c tion with the optical phonons, since these latter have a relatively high energy at z e r o m o m e n t u m . 2.

EFFECTS OF THE POLAR WITH LO PHONONS

INTERACTION

In the f i r s t p a r t o f t h i s w o r k , we c o n s i der the case of a polar interaction with LO phonons, assuming that their energy d i s p e r s i o n is n e g l i g i b l e . T h i s is a s e n sible approximation for the long-wave p h o n o n s we c o n s i d e r h e r e . We a l s o r e s t r i c t o u r s e l v e s to an J s o t r o p i c i n i t i a l electron momentum distribution. For inst a n c e , t h i s is t h e c a s e f o r e l e c t r o n s photoexcited in i s o t r o p i c b a n d s . T h e c a s e of a n i s o t r o p i c d i s t r i b u t i o n s c a n be t r e a t e d in a s i m i l a r w a y . F o r c r y s t a l s w h i c h a r e n o t t o o i o n i c , as the I I I - V o r t h e I I - V I s e m i c o n d u c t o r s , the polar scattering of electrons by LO p h o n o n s c a n be t r e a t e d b y a B o r n a p p r o x i m a t i o n . T h u s , t w o m e c h a n i s m s ere p o s s i ble : absorption or emission of a single p h o n o n . In t h e s e p r o c e s s e s , the e n e r g y of the electron changes always by the same amount : the energy of a phonon ~ w h e r e ~ is t h e f r e q u e n c y o f t h e L O p h o nons. This means that the different terms in t h e B o l t z m a n n e q u a t i o n a p p r o p r i a t e t o t h e p r e s e n t s i t u a t i o n c o u p l e the d i s t r i bution f(c,t) measured for the electron e n e r g y c a n d at t i m e t w i t h the s a m e d i s t r i b u t i o n m e a s u r e d at t h e s a m e t i m e at E + ~ ( a b s o r p t i o n of a p h o n o n ) a n d at ~-~ (emission of a phonon). Therefore, t h e B o l t z m a n n e q u a t i o n is n o l o n g e r an

248

R Evrard et al / Theory of relaxatzon o f hot electrons

integral equation, but becomes a simple al~ebra~c equation. This was noticed by F l e t c h e r a n d B u t c h e r (2) in t h e i r s t u d y o f e l e c t r o n m o b i l i t y in G a A s a n d b y Brosens, Devreese, Kartheuser, Van welzenis and one of the authors (Evrard) in t h e i r w o r k o n p o l a r o n d r i f t m o b i l i t y , H a l l m o b i l i t y a n d i m p a c t i o n i z a t i o n in semiconductors (3). W r i t i n g the e l e c t r o n e n e r g y as £ = (n+x) ~ where

0 ~ x < 1 a n d n is an i n t e g e r ( n = 0 , 1 , 2 , to i n t r o d u c e a v e c t o r G ( x , t ) , n e n t s of w h i c h a r e

) allows the compo-

T h e e q u i l i b r i u m d i s t r i b u t i o n is r e a c h e d a f t e r a t i m e o f t h e o r d e r o f 1 / a ~ (a is t h e F r S h l l c h c o u p l i n g c o n s t a n t ) , the e x a c t v a l u e d e p e n d i n g on the l a t t i c e t e m p e r a t u r e T. F o r G a A s , t h e v a l u e t = l / a ~ c o r r e s p o n d s to 0 , 2 7 ps, so t h a t t h e r e l a x a t i o n due to t h e p o l a r i n t e r a c t i o n w i t h LO p h o n o n s zs q u i t e f a s t . The curves giving the equillbrlum distribution versus the electron energy have discontinuities in the s l o p e f o r c = n ~ ( n = 1 , 2 , .). An e x a m p l e o f s u c h a d i s t r i b u t i o n can be f o u n d in F i g . 1 . It c o r r e s p o n d s to t h e c u r v e w i t h a v a l u e o f a t e e q u a l to 2 . 1 3 .

gn(X,t)=f(Cn,t) with c =(n+x)~w. n Then, the Boltzmann equation can w r i t t e n in t h e f o l l o w i n g f o r m

be

atw 572

~G(x,t) ~t

= LG(x,t)

,

285

1/.7

w h e r e L is a n o n s y m m e t r i c t r i d i a g o n a l matrix. The elgenvalues -Ai(x) and eigenv e c t o r s Hi(x) of t h i s m a t r i x are e a s i l y obtained. All the eigenvalues are negat i v e , e x c e p t o n e (let s a y A 0) w h i c h is z e r o . T h e i n i t i a l d i s t r i b u t i o n G ( x , 0 ) is expanded into a linear combination of the e i g e n v e c t o r s G(x,0) and by

the

=

time

G(x,t)=

359 10.9 /..39

2.13 1.33

Z ci(x)Hi(x) i=0 evolution

Z i=0

""--'~...-~. ~

is o b v i o u s l y

ci (x) exp[-~i

93&

given

( x ) t ] H i ( x ) • (11

T h e r e f o r e , t h e r e l a x a t i o n is d e s c r i b e d b y an i n f i n i t e n u m b e r o f r e l a x a t i o n t i m e s Ti(x)=Ai-1(x) d e p e n d i n g on t h e v a l u e o f x. A f t e r a t i m e l o n g e n o u g h , t h e o n l y c o m p o n e n t r e m a i n i n g in E q . 1 is t h a t c o r r e s p o n d i n g to i = 0 ( A 0 = 0 ) . C o n s e q u e n t l y the equilibrium distribution given by

G(X,=)

72.t.

o

1

2

3

4

s

6

E~OJ

F x g u r e 1 : T i m e e v o l u t i o n of t h e e l e c t r o n energy distribution ( a r b i t r a r y u n i t s ) in G a A s at 4 2 0 K due to s c a t t e r i n g b y the LO a n d LA p h o n o n s .

= C0(x)H0(x)

It c a n be s h o w n t h a t , at e q u i l l b r l u m , the r a t i o o f t h e d i s t r i b u t i o n m e a s u r e d at E + ~ a n d C is g i v e n b y the B o l t z m a n n factor exp(-~/kT). H o w e v e r , t h i s is n o longer true for the distribution computed f o r t w o v a l u e s o f the e n e r g y w h i c h d i f f e r by less than ~. T h i s is d u e to the f a c t t h a t t h e p r o b l e m is n o t r e a l l y e r g o d i c , s i n c e an e l e c t r o n h a v i n g an i n l t i a l "energy £ can o n l y r e a c h the s t a t e s w i t h e n e r g i e s e q u a l to £ + n ~ and not the other s t a t e s . (4)

3.

EFFECTS OF THE INTERACTION ACOUSTICAL PHONONS

WITH

We s t i l l c o n s i d e r the c a s e o f an I s o t r o p i c d i s t r i b u t i o n . T h e n the a c o u s t i c a l p h o n o n s do n o t c o n t r i b u t e to the e l e c t r o n r e l a x a t i o n if t h e c o l l i s i o n s are c o n s i d e r e d as s t r i c t l y e l a s t i c . T h e r e f o r e we m u s t h e r e i n c l u d e the e f f e c t s o f i n e l a s t i c i t y . T h i s p r e c l u d e s the use o f a relaxatlon-time approximation and leads us to r e l y on a n u m e r i c a l m e t h o d to s o l v e the Boltzmann equation. This latter

R Evrard et al / Theory ofrelaxatzon of hot electrons

d e s c r i b e s a l o w - d e n s i t y e l e c t r o n g a s interacting with LO phonons having no energy dispersion and LA phonons obeying the dispersion law

~1~)

- qv s

w h e r e ~ is t h e L A p h o n o n w a v e v e c t o r a n d v s the speed of sound for longitudlnal waves. The LA phonon-electron interaction is d e s c r i b e d b y t h e u s u a l d e f o r m a t i o n p o t e n t i a l t h e o r y . (5). The Boltzmann w r i t t e n as ~f(ctt)

equation

can

formally

Lf(c,t)

-exp[(t-t

thermally excited and we disregard their e f f e c t s in o u r c a l c u l a t l o n s . T h e r e l a x a t i o n due t o L A p h o n o n s is t h e n e v a l u a t e d taking the distribution that Ulbrich obt a i n e d at a t i m e J u s t a f t e r t h e e x c i t a t i o n as i n i t i a l d i s t r i b u t i o n . T h e r e s u l t s f o r d i f f e r e n t t i m e s a r e s h o w n in F i g . 2 . T h e y a r e to be c o m p a r e d w i t h t h o s e o f Fig.2 of Ulbrich's paper. The general t r e n d o f t h e r e l a x a t i o n we o b t a i n is s i m i l a r to the e x p e r i m e n t a l results. However the theoretical and experimental characteristic t i m e s are q u i t e d i f f e r e n t .

GoAs

(2)

w h e r e L is n o w an i n t e g r a l o p e r a t o r . f o r m a l s o l u t i o n of E q . 2 is o b v i o u s l y

f(c,t)

be

o)L] f(E,t

o) •

249

/..2 K

The

time Ins)

(3)

&80 190

S m a l l i n c r e m e n t s t - t 0 are c o n s i d e r e d so t h a t o n l y t h e f i r s t f e w t e r m s in t h e T a y l o r e x p a n s i o n o f the o p e r a t o r e x p [ ( t t 0 ) L~_ are k e p t in t h e c a l c u l a t l o n o f t h e r i g h t h a n d s i d e of E q . 3 . T h e r e s u l t s are p r e s e n t e d in Fig. l, w h e r e t h e e l e c t r o n e n e r g y d i s t r i b u t i o n is d r a w n v e r s u s the d i m e n s i o n l e s s variable E/~ a n d f o r d i f f e r e n t t i m e s . T h e v a l u e s of t h e p a r a m e t e r s c o r r e s p o n d t o the c a s e o f G a A s at 4 2 0 K (i.e. at k T ~ 1 ~ ) . It is c l e a r that one can distinguish two characterist i c t i m e s . T h e f i r s t o n e c o r r e s p o n d s to values of the dimensionless parameter ~ t ~ b e t w e e n 0 a n d a b o u t 2 (0.6 ps f o r G a A s ) . D u r i n g t h i s t i m e t h e r e l a x a t i o n is d u e a l m o s t e n t i r e l y t o t h e L O p h o n o n s , an a n d at the e n d , t h e e l e c t r o n d i s t r i b u t i o n is in e q u i l i b r i u m w i t h t h e s e LO p h o n o n s . The second part of the relaxation, which is f a r s l o w e r , is d u e to the w e a k i n e l a s t i c i t y of t h e c o l l i s i o n s w i t h the L A p h o n o n e . T h e e q u i l i b r i u m is n o t r e a c h e d b e f o r e a t i m e o f t h e o r d e r o f T = I 0 0 0 (0,3 ns f o r GaAs) , at l e a s t at the t e m p e r a t u r e considered here. The distribution evolves asymptotically towards a Maxwell-Boltsmann dzstrlbution. 4.

COMPARISON

WITH

EXPERIMENT

As s h o w n b y U l b r i c h (6} ~ t h e t i m e e v o l u t i o n o f t h e l u m i n e s c e n c e o f G a A s d u e to recombination of electrons photo-lnjected in the c o n d u c t i o n b a n d allows the determ i n a t i o n of the t i m e e v o l u t i o n o f the e n e r g y d i s t r i b u t i o n of t h e s e e l e c t r o n s . T h e r e l a x a t i o n d u e to LO p h o n o n s is so f a s t t h a t it is n o t o b s e r v e d b y U l b r i c h . T h e r e f o r e , t h e d i s t r i b u t i o n o b s e r v e d at t h e i n i t i a l t i m e is in e q u i l i b r i u m w i t h the L O p h o n o n s . As t h e e x p e r i m e n t is p e r f o r m e d at 4 . 2 K , t h e LO p h o n o n s a r e n o t

120 97 8/.

72 60 /,8 36

0 Figure energy

1

2

3

¢ (meV)

2 : S a m e as in f i g u r e I f o r e l e c t r o n s in G a A s at 4 . 2 K .

low

Indeed, our predictions for the time r e q u i r e d f o r t h e r e l a x a t i o n a r e an o r d e r of magnitude larger than those observed by Ulbrlch. Either the value of the def o r m a t i o n p o t e n t i a l u s e d in t h e p r e s e n t w o r k is t o o s m a l l , or o t h e r i n e l a s t i c p r o c e s s e s , as e l e c t r o n - e l e c t r o n interact i o n s a n d i n e l a s t i c c o l l i s i o n s on i m p u r i t i e s , are n o t n e g l i g i b l e . Other experimental results which could be u s e d as t e s t s f o r o u r c a l c u l a t i o n s c o n c e r n t h e o s c i l l a t o r y b e h a v i o u r of photolumlnescence (7) a n d o f p h o t o c o n d u c t i v i t y (8) w i t h r e s p e c t to t h e f r e q u e n c y of the incldent radiation. These experim e n t s a r e p e r f o r m e d in a s t a t i o n a r y s t a t e T h e e l e c t r o n s are b r o u g h t in t h e c o n d u c t i o n b a n d w i t h a k i n e t i c e n e r g y E. B y e m i t t i n g L O p h o n o n s , t h e y r e l a x to an e n e r g y c l o s e to the b o t t o m o f t h e c o n d u c tion band where they ar~ finally trapped

R Evrard et at / Theory o f relaxatton o f hot electrons

250

a n d r e c o m b i n e w i t h a hole. We h a v e s o l v e d the B o l t z m a n n e q u a t i o n for t h i s s t a t i o n a ry s i t u a t i o n , t h e r e f o r e a d d i n g a s o u r c e t e r m for the p h o t o - i n j e c t i o n of the e l e c t r o n s a n d a sink t e r m to d e s c r i b e the t r a p p i n g at low k i n e t i c e n e r g y . The total e l e c t r o n p o p u l a t i o n is t h e n d e d u c e d as a f u n c t i o n of the i n i t i a l e l e c t r o n e n e r g y . T h i s p o p u l a t i o n h a s an o s c i l l a tory behavlour, as e x p e c t e d . A t y p i c a l r e s u l t is s h o w n in F i g . 3 . w o r k is in p r o g r e s s to c a l c u l a t e e i t h e r the i n t e n s i t y of the l u m i n e s c e n t l i g h t , or the c o n d u c t i v i t y . Indeed, these quantities are t h o s e m e a s u r e d in the experiments cited above and determining them theoretically w o u l d p r o v i d e us w i t h a possibility of d i r e c t c o m p a r i s o n w i t h experiment. ACKNOWLEDGMENT T h i s w o r k is p a r t of a r e s e a r c h red by the N A T O g r a n t nr 1089.

sponso-

REFE RENCES

(I)

D e v r e e s e , J.T. (ed.), P o l a r o n s in I o n i c C r y s t a l s and P o l a r S e m i c o n d u c tors (North-Holland, Amsterdam 1972) (2) F l e t c h e r , K. a n d B u t c h e r , P . N . , J. P h y s . C 5 (1972) 2 1 2 - 2 2 4 . (3) D e v r e e s e , J . T . , Van W e l z e n l s , R.G. and E v r a r d R., p a p e r p r e s e n t e d at this conference. (4) D e v r e e s e , J . T . a n d E v r a r d , R., L i n e a r and N o n l i n e a r E l e c t r o n T r a n s p o r t in Solids (Plenum, N.Y.,1976) (5) C o n w e l l , E.M., H i g h F i e l d T r a n s p o r t in S e m i c o n d u c t o r s , in S e l t z F., T u r n b u l l , D. a n d E h r e n r e i c h , H. (eds) S o l i d St. Phys. S u p o l . 9 ( A c a d . p r e s s , New York 1976) I05-111. (6) U l b r l c h , R., Phys. Rev. B8 (1973) 5719-5727 (7) U l b r i c h , R., Phys. Rev. L e t t e r s 27 (1971) 1 5 1 2 - 1 5 1 4 . (8) N i c h o l a s , R.J., C a r t e r , A . C . , Fung, S., S t r a d l l n g , R.A., P o r t a l , J.C. a n d H o u l b e r t , C., J. P h y s . C13 (1980) 5215-5231.

Figure 3 : Total number sus the energy at which

of electrons verthey are photo-

i n j e c t e d in the c o n d u c t i o n band. T r a p p i n g and r e c o m b i n a t i o n o c c u r at the b o t t o m of the band.