PECULIAR HOT ELECTRON EFFECTS IN CROSSED ELECTRIC AND MAGNETIC FIELDS

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PECULIAR HOT ELECTRON EFFECTS IN CROSSED ELECTRIC AND MAGNETIC FIELDS

T. Kurosawa

To cite this version:

T. Kurosawa. PECULIAR HOT ELECTRON EFFECTS IN CROSSED ELECTRIC AND MAGNETIC FIELDS. Journal de Physique Colloques, 1981, 42 (C7), pp.C7-377-C7-386.

�10.1051/jphyscol:1981746�. �jpa-00221683�

JOURNAL DE PHYSIQUE

Colloque C7, supplément au n°10, Tome 42, ootobre 1981 page C7-377

PECULIAR HOT ELECTRON EFFECTS IN CROSSED ELECTRIC AND MAGNETIC FIELDS T. Kurosawa

Department of Physics, Chuo University, Tokyo, Japan

Résumé. - Les effets de porteurs chauds qui sont attendus (ou quelquefois observés) dans les champs croisés électrique (aussi statiques que de micro-onde) et magnétique (pas quantique) ont été étudiés. Ce qui nous intéresse le plus est le cas où l'in- teraction entre le porteur et le phonon optique est plus forte que les autres processus de diffusion. Dans les champs électri- ques statiques l'effet principal est l'accumulation de porteurs dans la région spéciale de l'espace de la quantité de mouvement qui produit, par exemple, l'inversion de population, la conduc- tivité différentielle négative, la supraradiance et le déplace- ment caractéristique de résonance du cyclotron du germanium de type-p. Dans le champ intense de micro-onde, remarquables sont les caractères systématiques qui se montrent dans le mouvement de porteur et donnent quelques effets particuliers comme le

"bunching" dans l'espace de quantité de mouvement.

Abstract. - Hot carrier effects expected (in some cases observed) in crossed electric (both static and microwave) and magnetic

(not quantum) fields are surveyed. The case that the carrier- optical phonon interaction is strong compared with the other scattering processes is of main concern. In static electric fields, the principal effect is the carrier accumulation in a special region of the momentum space which leads to, e.g., population inversion, negative differential conductivity, superradiance and specific shift of cyclotron resonance in p-type germanium. In intense microwave fields, remarkable is systematic characters exhibited by the carrier motion which bring about some peculiar effects such as the bunching in the momentum space.

$1. Introduction

One of the interests in the study of physics on hot carriers in semiconductors is the possibility of finding peculiar phenomena, which reflect the characteristic features of the scattering processes, the motion, and the distribution function of the carrier. A good hunting field for such phenomena can be found with hot carriers in the situation that the carrier-optical phonon interaction is strong compared with the other scattering processes and the thermal energy kT is much smaller than the optical phonon energy "Hid : The collision time of the carrier is very long when the energy e is smaller than

•Kw , and the carrier is accelerated rather freely until it arrives

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981746

C7-378 JOURNAL DE PHYSIQUE

a t t h e o p t i c a l phonon e n e r g y , t h e n r a p i d l y e m i t s t h e o p t i c a l phonon.

The d i s t r i b u t i o n f u n c t i o n i s q u i t e d i f f e r e n t from t h a t o f a s i m p l e h o t c a r r i e r c o n c e p t w i t h a d e f i n i t e e l e c t r o n t e m p e r a t u r e and l e a d s t o some s t r a n g e e f f e c t s .

The a p p l i c a t i o n o f magnetic f i e l d makes t h e phenomena more

f r u i t f u l . S e v e r a l p e c u l i a r h o t c a r r i e r e f f e c t s e x p e c t e d o r o b s e r v e d i n t h i s c a s e a r e s u r v e y e d ; i n $2 t h e e f f e c t s i n s t a t i c e l e c t r i c f i e l d s , and i n $3 t h o s e i n i n t e n s e microwave f i e l d s .

$2.

S t a t i c E l e c t r i c F i e l d s

F i g u r e 1 shows a t y p i c a l example o f t h e p e c u l i a r h o t c a r r i e r d i s t r i - b u t i o n f u n c t i o n on t h e pZ=O p l a n e i n t h e momentum s p a c e o b t a i n e d from a Monte C a r l o c a l c u l a t i o n under t h e c r o s s e d

e l e c t r i c and magnetic f i e l d s o f a p p r o p r i a t e s t r e n g t h s : The f r e e a c c e l e r a t i o n o f a c a r r i e r i n t h e p-space i s r e p r e s e n t e d by a c i r c u l a r motion w i t h t h e c y c l o t r o n f r e q u e n - cy

(=eHZ/m*c) around an a x i s d e n o t e d by C

C

i n t h e f i g u r e where px=O and p =pc-mXcEx/HZ.

The c a r r i e r a c c u m u l a t e s i n a c i r c u l a r region Y

around C where t h e e n e r g y i s always s m a l l e r F i g . 1.

t h a n t h e o p t i c a l phonon e n e r g y , w h i l e t h e

circular motion i s i n t e r r u p t e d v e r y r a p i d l y by t h e o p t i c a l phonon e m i s s i o n when t h e c a r r i e r i s o u t s i d e o f t h e r e g i o n [ l ] .

A

d e t e c t a b l e e f f e c t o f t h e a c c u m u l a t i o n i s a l a r g e r H a l l a n g l e t h a n t h a t e x p e c t e d from t h e e s t i m a t i o n n e g l e c t i n g it. T h i s h a s been

o b s e r v e d i n AgCl and AgBr by Komiyama, Masumi and K a j i t a [ 2 ] and i n p-type germanium by Komiyama and S p i e s [ 3 ] . Another e v i d e n c e f o r t h e a c c u m u l a t i o n which seems more c o n c l u s i v e h a s been o b t a i n e d a l s o by Komiyama e t a 1 [ 2 ] : F i g u r e 2 shows t h e c u r r e n t r e s p o n s e J Z t o a weak e l e c t r i c f i e l d a d d i t i o n a l l y a p p l i e d a l o n g t h e magnetic f i e l d , p l o t t e d against H Z f o r some f i x e d v a l u e s o f

Ex.

P h y s i c a l l y , t h e o r d i n a t e i s p r o p o r t i o n a l t o t h e e f f e c t i v e

c o l l i s i o n t i m e (namely, t h e a v e r a g e -

d u r a t i o n o f t h e f r e e motion) o f t h e

510

_.---

c a r r i e r , and t h e d o t t e d l i n e s

980

----

r e p r e s e n t t h o s e e x p e c t e d from t h e

,

assumption t h a t t h e c a r r i e r r e p e a t s

I I

t h e s t r e a m i n g motion a l o n g t h e

t r a j e c t o r y

a s shown i n t h e i n s e t . Fig.2. An e x p e r i m e n t a l e v i d e n c e

f o r t h e c a r r i e r a c c u m u l a t i o n

The v e r t i c a l a r r o w s i n d i c a t e t h e ( s e e t e x t ) .

magnetic f i e l d where t h e e n e r g y o f t h e p o i n t C (=mX (cE /H ) 2 / 2 ) i s

X

z e q u a l t o f i w

OP'

A s

i n c r e a s e s above t h i s v a l u e ( i . e . , t h e p o i n t C e n t e r s t h e c i r c l e o f

, t h e c u r r e n t

becomes much l a r g e r t h a n t h e v a l u e o f t h e d o t t e d l i n e s . OP T h i s means e v i d e n t l y t h a t t h e c a r r i e r spends t h e c o n s i d e r a b l e f r a c t i o n o f t h e t i m e i n t h e a c c u m u l a t i o n r e g i o n .

D e g r e e

of AccurnuZation

To e s t i m a t e t h e d e g r e e o f t h e a c c u m u l a t i o n , we a d o p t a s i m p l e r a t e e q u a t i o n approach. We d i v i d e t h e momentum s p a c e i n t o t h r e e r e g i o n s :

(1) The r e g i o n where t h e accumulation t a k e s p l a c e , ( 2 ) t h e r e g i o n o u t s i d e of t h e accumulation r e g i o n and o f ~ < f i w and ( 3 ) t h e r e g i o n

OP' o f €>Flu

OP'

I f we d e n o t e t h e number o f c a r r i e r s i n t h e s e r e g i o n s by nl,n2 and n3, we may w r i t e t h e f o l l o w i n g approximate e q u a t i o n s :

where

i s t h e o p t i c a l phonon e m i t t i n g r a t e ,

t h e a v e r a g e c o l l i s i o n t i m e i n t h e r e g i o n ( l ) , OP t h e a v e r a g e r o t a t i n g a n g l e i n t h e r e g i o n

( 2 1 , and

i s t h e p r o b a b i l i t y t h a t t h e c a r r i e r i s s c a t t e r e d i n t o t h e r e g i o n (1) a f t e r t h e o p t i c a l phonon e m i s s i o n .

I n t h e s t e a d y s t a t e , we have

n3/n2

uc/W0p4'2

l (4)

and nl/nZ

.

⁽⁵

Thus, t h e l a t t e r e x p r e s s i o n g i v e s t h e d e g r e e of t h e accumulation.

Here, $2 i s u s u a l l y 2 ~ 3 , w h i l e uc.r s t r o n g l y depends on t h e e x p e r i m t a l c o n d i t i o n s b u t may b e e s t i m a t e d r a t h e r e a s i l y i n e a c h c a s e . A s f o r

t h e r e i s no s i m p l e way t o e s t i m a t e i t , b u t t h e r e s u l t s o f numerical c a l c u l a t i o n s can b e r e p r e s e n t e d w e l l by u s i n g o n l y two p a r a m e t e r s when w c ~ > > l .

The v a l u e s of

a r e p l o t t e d i n Fig.3 f o r p o l a r o p t i c a l s c a t t e r i n g and i n Fig.4 f o r n o n p o l a r s c a t t e r i n g , a g a i n s t two p a r a m e t e r s Bop/wc and pc/pop. The p a r a m e t e r pc/p d e n o t e s t h e p l a c e o f t h e p o i n t C r e l a t i v e t o t h e momentum a t t h e o p t i c a l phonon e n e r g y p OP

OP

On t h e o t h e r hand,

i s t h e c o e f f i c i e n t f o r t h e s c a t t e r i n g r a t e by t h e o p t i c a l phonon, which OP i s w r i t t e n a s

f o r n o n p o l a r s c a t t e r i n g , and

JOURNAL DE PHYSIQUE

f o r p o l a r s c a t t e r i n g , where t h e o c c u p a t i o n number o f t h e o p t i c a l phonon i s p r a c t i c a l l y 0 i n t h e p r e s e n t c a s e and o n l y t h e spontaneous e m i s s i o n i s c o n s i d e r e d . A s &-tfiw ( 7 ) i s reduced t o ( 6 ) . When w c ~ i s l a r g e ,

o p t

Pk i s n e a r l y i n d e p e n d e n t o f t h e d e t a i l s o f s c a t t e r i n g p r o c e s s e s f o r E<RW

OP'

The f o l l o w i n g f e a t u r e s a r e remarkable: (1) A s B /wc i n c r e a s e s , t h e r e s i d u a l e n e r g y o f c a r r i e r s a f t e r t h e phonon e m ~ s s i o n ?P d e c r e a s e s and Pk becomes s m a l l e s p e c i a l l y f o r l a r g e pc/p

OP'

( 2 ) Because o f t h e forward s c a t t e r i n g n a t u r e o f t h e p o l a r s c a t t e r i n g , Pk i s g e n e r a l l y 1.5%3 t i m e s l a r g e r t h a n t h a t f o r t h e n o n p o l a r c a s e .

A s

a n u m e r i c a l example, i f we c o n s i d e r t h e c a s e t h a t Bop/wc=5 and pc/pop=0.65, we have Pk=0.12 and 0.074 f o r p o l a r and n o n p o l a r c a s e s r e s p e c t i v e l y . Furthermore i f we assume w ~ = 1 5 and @ 2 = 2 , nl/n2

C

becomes 0.90 and 0.56, which a r e a p p r e c i a b l e a c c u m u l a t i o n s i n d e e d when compared w i t h t h e volume r a t i o o f t h e two r e g i o n s i n t h e momentum s p a c e , 0.086. I n t h e c a s e o f F i g . 1 , pc/pop=0.55, B /wc=20 (nonpolar)

OP Pkz0.20, w c ~ - 3 0 , 1 $ ~ = 2 . 1 and nl/n2=2.9.

N e g a t i v e DifferentiaZ Conductivity

The accumulation o f t h e c a r r i e r and i t s d i s a p p e a r a n c e w i t h t h e i n c r e a s e o f

l e a d t o n e g a t i v e d i f f e r e n t i a l c o n d u c t i v i t y (NDC) [ 1 , 4 ] . Simply t h i s i s e x p l a i n e d i n t e r m s o f t h e d e c r e a s i n g d r i f t v e l o c i t y due t o t h e c a r r i e r t r a n s f e r from t h e accumulation r e g i o n w i t h a v e r a g e v e l o c i t y pc/mX t o t h e s t r e a m i n g s t a t e w i t h a b o u t p /2m* (<pc/m*)

OP w i t h i n c r e a s i n g

I n t h e p r e s e n t c a s e , however, t h e c u r r e n t makes a l a r g e a n g l e t o t h e e l e c t r i c f i e l d and t h e NDC must be d e f i n e d d i f f e r e n t l y from t h e u s u a l c a s e [ 5 ]

The r a p i d d e c r e a s e i n t h e H a l l a n g l e w i t h Ex a l s o c a u s e s o r enhances t h e NDC. However, we summarize o n l y t h e r e s u l t s h e r e , s i n c e t h e s i t u a t i o n i s r a t h e r c o m p l i c a t e d .

-

polar s c a t t e r i n g

-

100 10

Bap/ ¹

s c a t t e r i n g

-

100 10

Bop''JJc 1

F i g . 3. The p r o b a b i l i t y Pk ( s e e F i g . 4 . The p r o b a b i l i t y Pk

e q . (1) f o r poLar c a s e . f o r n o n p o l a r c a s e .

There i s a lower l i m i t f o r w

t o c a u s e t h e NDC. The lower l i m i t

C

i s 15a20 and r a t h e r s l o w l y dependent on B /mc i n t h e p o l a r c a s e , while it i s g e n e r a l l y s m a l l e r i n t h e n o n p o l a r c a s e ; OP it i s a b o u t 10 a t

B /wc=5a10 and approaches t h e p o l a r v a l u e s f o r l a r g e r Bop/wc.

OP

A s s e e n from f i g . 1 , t h e c a r r i e r d i s t r i b u t i o n i s t h i n i n t h e v i c i n i t y o f t h e p o i n t

T h i s tendency becomes more pronounced w i t h t h e i n c r e a s e o f B o p / ~ c . Such a d i s t r i b u t i o n c o r r e s p o n d s t o t h e population i n v e r s i o n o v e r Landau l e v e l s , and a p o s s i b i l i t y of t h e NDC a t wc h a s been p o i n t e d o u t

.

S u p e r r a d i a n c e and L i n e a r Higher Harmonic G e n e r a t i o n

Consider a d i s t r i b u t i o n a s F i g . 1 and assume t h a t Ex i s suddenly removed a t t = O . A f t e r t = O , t h e whole p a t t e r n o f t h e d i s t r i b u t i o n f u n c t i o n b e g i n s t o r o t a t e w i t h wc around t h e p,-axis. T h i s c o r r e s - ponds t o a c o h e r e n t c i r c u l a r motion of t h e c a r r i e r s and b r i n g s a b o u t s u p e r r a d i a n c e . S i n c e t h e a v e r a g e induced d i p o l e moment p e r c a r r i e r i s epc/m*wc, t h e r a d i a t i o n e m i t t e d p e r u n i t t i m e i s , -

i f

c a r r i e r s r o t a t e c o h e r e n t l y . The r a d i a t i o n l o s s c a u s e s t h e damping o f t h e c o h e r e n t motion; t h e r e l a x a t i o n t i m e Tr i s g i v e n by

4Ne2wt

-

Tr 3m*c3 , ⁽⁹⁾

s i n c e t h e e n e r g y a s s o c i a t e d w i t h t h e c o h e r e n t motion i s ~ ~ : / 2 m * . For i n s t a n c e , i f we p u t N=1o1',

=2x10

C

llsll and m*=O . ^{l m ,} we g e t a v e r y s t r o n g damping:

T r = 2x10-l'

,

o r wCTr 4 .

Namely, Tr may be even s h o r t e r t h a n t h e c o l l i s i o n t i m e

More g e n e r a l l y , an a p p r o p r i a t e r a p i d change i n t h e e l e c t r i c f i e l d

can a l s o c a u s e t h e s i m i l a r e f f e c t . However, it i s n o t a technically e a s y t a s k t o r e a l i z e t h e r a p i d change o f

w i t h i n a f r a c t i o n of

A

more f e a s i b l e way i s t o a p p l y an a l t e r n a t i n g f i e l d w i t h a f r e q u e n c y w=wc/n ( n i s an i n t e g e r ) a l o n g t h e x - a x i s i n a d d i t i o n t o t h e s t a t i c f i e l d Ex, a l t h o u g h t h e r e s u l t i n g e f f e c t i s a l i t t l e d i f f e r e n t from t h e above one: The t o t a l e l e c t r i c f i e l d i s

E ' ( t ) _X = E _X

+ A c o s w t , ( 1 0 ) and t h e f r e e motion o f a c a r r i e r i s g i v e n by

where a and

a r e a r b i t r a r y c o n s t a n t s . I n t h i s c a s e , t h e c a r r i e r s

C7-382 JOURNAL DE PHYSIQUE

accumulate i n a r e g i o n a s h a t c h e d i n F i g . 5 . The r e g i o n r o t a t e s w i t h w around t h e moving p o i n t

t h a t makes t h e e l l i p t i c motion ( c o r r e s -

C

ponding t o t h a t o f a=O i n eq. (11)) around C a s shown i n t h e f i g u r e . Because t h e p o i n t C ' does n o t c o i n c i d e w i t h t h e c e n t e r o f mass of t h e accumulation r e g i o n , a d i p o l e moment r o t a t i n g w i t h w i s induced. I f

C

w e assume t h e d i s t r i b u t i o n t o be uniform o v e r t h e r e g i o n , t h e a v e r a g e momentum on t h e p,=O p l a n e i s g i v e n by

2

2 s i n - s i n w c t }

p x ( t ) ^c eA - s i n o t +-

-

n 2 - 1 n w(n -1)

2 n 2

I ⁽¹²⁾

and Py ^{( t )}

{n c o s o t -r s i n -cos n o c t +pcr

w (n2-1) n -1

when

i s n o t t o o l a r g e . The second t e r m r e p r e s e n t s a p e c u l i a r h i g h e r harmonic g e n e r a t i o n which i s

v e r y s t r o n g and p r o p o r t i o n a l t o

i n c o n t r a s t w i t h t h e o r d i n a r y c a s e

p r o p o r t i o n a l t o

The l i n e a r X

I swop

dependence comes from t h e a p p r o x i -

mation t h a t t h e edge o f t h e d i s t r i -

1 H @ E-- I

b u t i o n i s s h a r p . I n r e a l i t y , t h e

edge h a s some s m a l l w i d t h Ap. I f F i g . 5 . C a r r i e r accumulation i n s t a t i c p l u s weak a l t e r n a t i n g ( o f eA/wc becomes s m a l l e r t h a n Apt t h e _fields.

arrow shows a s i t u a t i o n w i l l approach t h e ordinary r E t a t i n g d i p o l e moment which

i n d u c e s s u p e r r a d i a n c e .

c a s e . * T y p i c a l l y , t h e corresponding v a l u e o f

i s e s t i m a t e d t o be 1%10 V/cm.

Even i n t h e l a t t e r c a s e , t h e r a d i a t i o n damping i s s t i l l g i v e n by t h e same r e l a x a t i o n time a s ( 9 ) . Thus, i f f a v o u r a b l e c o n d i t i o n s a r e p r e p a r e d , one can e x p e c t t o o b s e r v e a s u s t a i n e d s u p e r r a d i a n c e pumped by t h e f i e l d E& w i t h a p p r e c i a b l e damping t h a t i s o c c a s i o n a l l y

comparable w i t h t h e o r d i n a r y c o l l i s i o n r e l a x a t i o n i n c r y s t a l s .

S e m i c o n d u c t o r s

I n t h e c a s e o f t h e warped band, t h e t r a j e c t o r y o f a c a r r i e r i n t h e momentum s p a c e u n d e r c r o s s e d e l e c t r i c and magnetic f i e l d s d i f f e r s c o n s i d e r a b l y from t h a t i n t h e s p h e r i c a l band. Furthermore i n t h e c a s e o f p-type s i l i c o n , t h e band n o n p a r A b o l i c i t y h a s pronounced effects on t h e motion o f h o l e s . R e c e n t l y , a n e g a t i v e H a l l e f f e c t which presumably r e l a t e s t o t h a t h a s been o b s e r v e d by K a j i t a

*There i s a doubt a b o u t t h i s , because o f t h e s i n g u l a r i t y a t t h e edge o f t h e d i s t r i b u t i o n f u n c t i o n ( f o r A=O) c a u s e d by t h e s i n g u l a r n a t u r e o f t h e o p t i c a l phonon e m i t t i n g r a t e which i s p r o p o r t i o n a l t o /G%--

OP'

p-type germanium

I

F i g . 6 . T r a j e c t o r i e s o f a heavy Fig.7. S p e c t r a l d i s t r i b u t i o n of h o l e i n Ge. The numbers d e n o t e t h e c y c l o t r o n mass o f t h e accu- t h e c y c l o t r o n mass i n t h e u n i t mulated h o l e s i n Ge.

o f f r e e e l e c t r o n mass.

A n e x p e c t e d e f f e c t i n p-type germanium a s s o c i a t e d w i t h t h e band warping and t h e c a r r i e r a c c u m u l a t i o n i s a s p e c i f i c s h i f t of c y c l o t r o n

r e s o n a n c e . F i g u r e 6 shows t h e momentum s p a c e t r a j e c t o r i e s o f a h o l e on t h e p,=O p l a n e under e l e c t r i c (170) and magnetic (001) f i e l d s i n germanium. Values o f c y c l o t r o n mass a l o n g t h e t r a j e c t o r i e s a r e a l s o shown i n t h e u n i t o f f r e e e l e c t r o n mass. The band p a r a m e t e r s a r e t h o s e g i v e n by Hensel and Suzuki

The u s u a l c y c l o t r o n mass on t h e pZ=O p l a n e i s 0.28m. Thus, i f t h e c y c l o t r o n r e s o n a n c e e x p e r i m e n t i s performed on t h e accumulated c a r r i e r s ,

a p p r e c i a b l e s h i f t w i l l be found. F i g . 7 f

shows t h e e x p e c t e d s p e c t r a l d i s t r i b u t i o n o f t h e c y c l o t r o n mass c a l c u l a t e d by summing o v e r a l l o r b i t s ( w i t h appropriate w e i g h t ) i n t h e accumulation r e g i o n , f o r t h r e e v a l u e s o f p /p

c op (pop i s t h e momentum a t i n (110) d i r e c t i o n ) .

OD

Another e x p e c t e d e f f e c t i n p-type / \

m a t e r i a l s a s s o c i a t e d w i t h t h e accumu-

l a t i o n comes from t h e band degeneracy; Fig.8. Accumulation in light t h e e x i s t e n c e of l i g h t and heavy h o l e h o l e band and i n d u c e d photon bands. I f t h e r a t i o E*/H= i s suitable, e m i s s i o n .

- -

t h e accumulation t a k e s p l a c e o n l y i n t h e l i g h t h o l e band and i n c r e a s e s

t h e p o r t i o n o f t h e l i g h t h o l e . I n t h i s c a s e , p o p u l a t i o n i n v e r s i o n

between two bands and e m i s s i o n o f i n f r a r e d r a d i a t i o n ( F i g . 8 ) a r e

e x p e c t e d

.

C7-384 JOURNAL

PHYSIQUE

$ 3 . Microwave F i e l d s

I f s u f f i c i e n t l y s t r o n g microwave f i e l d s a r e a p p l i e d , t h e c a r r i e r i s r e p e a t e d l y a c c e l e r a t e d up t o t h e o p t i c a l phonon e n e r g y w i t h i n a h a l f c y c l e of t h e f i e l d s . We c a l l t h i s s i t u a t i o n " s t r e a m i n g cyclotron motion", which i s c o m p l e t e l y d i f f e r e n t from t h e u s u a l c y c l o t r o n

resonance c o n d i t i o n . Some e x p e r i m e n t a l e v i d e n c e s o f such a motion have been found i n AgBr [ 1 0 ] .

r a p i d i n c r e a s e o f t h e l i n e w i d t h and a pronounced peak s h i f t w i t h t h e f i e l d i n t e n s i t y have been o b s e r v e d

(Figs.9 and 1 0 ) .

Monte C a r l o c a l c u l a t i o n h a s reproduced t h e e x p e r i - mental r e s u l t s v e r y w e l l and a l s o s u p p o r t e d t h e c o n c e p t o f t h e

s t r e a m i n g c y c l o t r o n motion.

I '

microwave a m p l i t u d e (V/cm) microwave amplitude (V/cm)

(V

810

N

1

Fig.9. Half w i d t h o f resonance Fig.10. V a r i a t i o n i n t h e l i n e v s . microwave i n t e n s i t y . peak p o s i t i o n .

l l 1

AgBr 3 5 G H z

0 O

- P 0

-

8'.

A

8'0

However, t h e d e t a i l s r e v e a l e d by t h e c a l c u l a t i o n a r e even more i n t e r e s t i n g [ l l ] . The motion o f c a r r i e r s i s q u i t e d i f f e r e n t from t h a t e x p e c t e d from b r u t a l a c c e l e r a t i o n by such i n t e n s e f i e l d s b u t much more o r d e r e d . The d i s t r i b u t i o n f u n c t i o n and t h e shape o f t h e r e s p o n s e c u r r e n t e x h i b i t p e c u l i a r f e a t u r e s . Some o f them a r e

p r e s e n t e d i n Figs.11-16, which show t h e d i s t r i b u t i o n f u n c t i o n s on t h e p =O p l a n e . I n t h e i n s e t s , t h e r e s p o n s e c u r r e n t

( t ) (sometimes a l s o

t o t h e f i e l d

( t )

c o s w t a r e p l o t t e d t o g e t h e r w i t h t h e indication of t h e c o r r e s p o n d i n g t i m e a t which t h e d i s t r i b u t i o n v i s r e c o r d e d ( t h e

v e r t i c a l a r r o w s ) .

t h e c a l c u l a t i o n s a r e c a r r i e d o u t f o r AgBr, i n which t h e p o l a r o p t i c a l , a c o u s t i c and n e u t r a l i m p u r i t y s c a t t e r i n g p r o c e s s e s a r e c o n s i d e r e d . The c o l l i s i o n f r e q u e n c y by t h e i m p u r i t y i s f i t t e d t o g i v e t h e e x p e r i m e n t a l l y o b s e r v e d low f i e l d m o b i l i t y a t 4.2K, 2 ~ 1 0 ~ c m ~ / ~ - s , and w / 2 ~ i s t a k e n t o be 3 5 G H z . I n t h e c a s e o f Fig.11, HZ=O and t h e c a r r i e r r e p e a t s t h e s i m p l e s t r e a m i n g motion between

m ---

-

4

W

C 7 - 3 8 5

F i g . l l . A=1200V/cm, u = 0 . F i g . 1 2 . A=1200V/cm, <o /io=0 . 8 .

F i g . 1 3 . A=1200V/cm, ui /ID=0 . 8 .

F i g . 1 4 .

=780V/cm, ou /ou=3.

F i g . 1 5 . A=1200V/cm, ou / u = 3 . 5 . F i g . 1 6 . A=1200V/cm, ou /oo=4.

C7-386 JOURNAL DE PHYSIQUE

p,=-pop and pop. F i g u r e s 12 and 1 3 a r e o f two d i f f e r e n t t i m e s i n t h e same s e r i e s and show t h e bunching e f f e c t o f t h e c a r r i e r s i n t h e

p-space, t h e motion o f t h e bunched c a r r i e r s l e a d s t o sawtooth c u r r e n t

1111 . The c a s e o f Fig.14 h a s some resemblance t o t h a t o f F i g . 5 ; t h e r o t a t i n g d i p o l e moment a s i n d i c a t e d by t h e arrow i n d u c e s t h e t h i r d harmonic g e n e r a t i o n . I n t h e c a s e s o f F i g s . 1 5 and 1 6 , t h e motion i s r a t h e r d i s o r d e r e d and t h e d i s t r i b u t i o n e x h i b i t s c o m p l i c a t e d p a t t e r n s , which a r e n o t t h e f l u c t u a t i o n due t o t h e Monte C a r l o c a l c u l a t i o n b u t r e p r o d u c i b l e .

R e f e r e n c e s

1) H.Maeda and T.Kurosawa: Proc. Z l t h I n t . C o n f . Phys. Semiconductors, Warsaw 1972, p.602.

2 ) S.Komiyama, T.Masumi and K - K a j i t a : Phys. Rev. B20 (1979) 5192.

3) S-Komiyama and R.Spies: t o be p u b l i s h e d i n t h i s c o n f e r e n c e . 4 ) A.A.Andronov, V.A.Valov, V.A.Kozlov and L.S.Mazov: S o l i d S t a t e

Commun. 36 (1980) 603.

5 ) T.Kurosawa, H.Maeda and H.Sugimoto:

Phys. Soc. Jpn 36 (1974) 491.

6 ) Y a . I . A l l b e r , A.A.Andronov, V.A.Valov, V.A.Kozlov and 1-R-Ryazantseva:

S o l i d S t a t e Commun. 19 (1976) 955.

7) T.Kurosawa: S o l i d S t a t e Commun. 24 (1977) 357.

8 ) K . K a j i t a : S o l i d S t a t e Commun. 31 (1979) 573.

9 ) J.C.Hense1 and K-Suzuki: Phys.Rev. B9 (1974) 4219.

10) S.Komiyama, T-Masumi and T-Kurosawa: Proe. 1 4 t h I n t . Conf. Phys.

S e m i c o n d u c t o r s , Edinburgh 1978, p.335.

11) T-Kurosawa: Proc. 2 5 t h I n t . Conf. Phys. S e m i c o n d u c t o r s , Kyoto

1980, p.345.