RECOMBINATION OF NONEQUILIBRIUM CARRIERS IN THE PRESENCE OF DISLOCATIONS IN SEMICONDUCTORS

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RECOMBINATION OF NONEQUILIBRIUM

CARRIERS IN THE PRESENCE OF DISLOCATIONS IN SEMICONDUCTORS

R. Labusch

To cite this version:

R. Labusch. RECOMBINATION OF NONEQUILIBRIUM CARRIERS IN THE PRESENCE OF DISLOCATIONS IN SEMICONDUCTORS. Journal de Physique Colloques, 1979, 40 (C6), pp.C6- 81-C6-86. �10.1051/jphyscol:1979617�. �jpa-00219033�

JOURNAL DE PHYSIQUE CoZloque C6, s u p p l h e n t au n06, tome 4 0 , j u i ~ 1 9 7 9 , page C6-81

RECOMBINATION OF NONEQUILIBRIUM CARRIERS I N THE PRESENCE OF DISLOCATIONS I N SEMICONDUCTORS

R. Labusch

I n s t i t u t fiir Angewcmdte Pizysik d e r TI1 Clausthal und S F B 126 Gottingen - ClausthaZ, F.R.C.

Resume.- Le problgme general de l a recombinaison e s t r e d u i t c3 une chaine de processus c a r a c t e r i s e s chacun par une simple decroissance e x p o n e n t i e l l e e t une constante de temps unique. Les constantes de temps sont c a l c u l e e s pour l e s spectres de niveaux de d i s l o c a t i o n q u i semblent p l a u s i b l e s . Dans l e cas d ' u n exemple s p e c i f i q u e on compare c e t t e cha9ne de processus avec des v a l e r ~ r s experimenta- l e s obtenues dans l e Ge, e t on i d e n t i f i e l e s mecanismes de recombinaison.

Abstract.- The general recombination problem i s reduced t o a s e t o f processes w i t h a simple exponen- t i a l decay and a s i n g l e time constant. The time constants a r e c a l c u l a t e d f o r those spectra o f d i s - l o c a t i o n l e v e l s t h a t appear t o be p o s s i b l e . I n a s p e c i f i c example t h i s s e t i s compared w i t h e x p e r i - mental values o f Ge and t h e recombination mechanisms a r e i d e n t i f i e d .

1. I n t r o d u c t i o n . - The aim o f t h e f o l l o w i n g i n v e s t i - g a t i o n i s t o d e r i v e r u l e s and equations t h a t a r e as general as p o s s i b l e but, a t the same time, s p e c i f i c enough t o serve as a q u a n t i t a t i v e basis f o r t h e ana- l y s i s o f recombination experiments i n deformed semi

-

conductors. Analyses o f recombination and t r a p p i n g a t d i s l o c a t i o n s have been given before /1,2,3,4/ b u t had the purpose t o e x p l a i n e x i s t i n g experiments and were done w i t h an incomplete knowledge o f the t r u e energy spectrum o f t h e d i s l o c a t i o n s t a t e s and o f the c o n f i g u r a t i o n o f t h e d i s l o c a t i o n core. This knowledge has been improved meanwhile and, although we are f a r from knowing _all p r o p e r t i e s o f even a s i n g l e d i s l o - c a t i o n type i n a s p e c i f i c m a t e r i a l , we have a good i d e a o f the p o s s i b l e f e a t u r e s t h a t d i s l o c a t i o n s i n semiconductors

can

2. R e s t r i c t i o n s and t h e o r e t i c a l basis.- For t h e sake o f s i m p l i c i t y we have t o impose a few r e s t r i c t i o n s on our a n a l y s i s :

Thus we s h a l l o n l y discuss the recombination a f t e r

"weak" e x c i t a t i o n . As a r u l e "weak" e x c i t a t i o n means t h a t t h e response o f t h e system i s l i n e a r i n t h e e x c i t a t i o n r a t e i - e . , t h e l i g h t i n t e n ' s i t y o r t h e r a t e o f i n j e c t i o n and so f o r t h . A somewhat more spe- c i f i c requirement w i l l t u r n up l a t e r on. I n general an e l e c t r o n t h a t recombines w i t h a h o l e goes through

a sequence o f t r a n s i t i o n s , c a l l e d a recombination path. The elementary step i n recombination i s the t r a n s i t i o n between two l e v e l s i n such a path. I t may be assumed w i t h o u t any l o s s o f g e n e r a l i t y t h a t the e i g e n f u n c t i o n s o f such a p a i r a r e l o c a l i z e d a t the same d i s l o c a t i o n and, i f the d i s l o c a t i o n i s disso- c i a t e d , even a t t h e same p a r t i a l because otherwise t h e r e i s no o v e r l a p and t h e t r a n s i t i o n i s so slow t h a t t h e corresponding p a t h would be blocked. As a consequence t h e separation o f t h e two l e v e l s i s n o t changed v i a t h e e l e c t r o s t a t i c i n t e r a c t i o n o f excess e l e c t r o n s o r holes t h a t a r e trapped a t t h e same d i s - l o c a t i o n , although t h i s i n t e r a c t i o n s h i f t s a l l s t a - t e s ( b u t by t h e same amount!) w i t h r e s p e c t t o t h e unperturbed edges o f the valence and t h e conduction band which may serve as r e f e r e n c e energies.

It i s g e n e r a l l y accepted now t h a t t h e d i s l o c a - t i o n s t a t e s form one dimensional bands. The important consequence f o r recombination problems i s t h a t , as a r u l e , t h e exchange o f e l e c t r o n s between s t a t e s i n t h e same d i s l o c a t i o n band i s so r a p i d compared w i t h inter-.

band t r a n s i t i o n s t h a t they a r e alwavs i n thermal e q u i l i b r i u m and t h e d i s t r i b u t i o n w i t h i n each band i s completely described by a quasi-Fermi l e v e l p. T h i s a l l o w s us t o analyse t r a n s i t i o n s between d i s l o c a t i o n bands r a t h e r than between s i n g l e d i s l o c a t i o n l e v e l s , u s i n g t h e general theory o f Landsberg 151.

I n our d i s c u s s i o n we have t o consider t h r e e b a s i c a l l y d i f f e r e n t - t y p e s o f d i s l o c a t i o n bands : i ) P a r t l y f i 1 le d bands t h a t have t h e i r Fermi l e v e l p

i n s i d e t h e band. We denote these by t h e name H ( f o r h a l f f i l l e d , although o c c a s i o n a l l y t h e f i l - l i n g may be a d i f f e r e n t f r a c t i o n o f u n i t y ) .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979617

C6-82 JOURNAL DE PHYSIQUE

i i ) " F u l l " bands, named F, where t h e Fermi l e v e l i s above t h e upper edge EF.

i i i ) "Empty" bands, named E, where t h e Fermi l e v e l i s below t h e l o w e r edge cE.

The d i f e r e n t t y p e s a r e e x e m p l i f i e d i n a sche- m a t i c diagram ( F i g . 1) t h a t has been suggested as a model f o r n e g a t i v e l y charged d i s l o c a t i o n s i n Ge / 2 / . The f u l l d i s l o c a t i o n band i n t h i s diagram i s c l o s e l y r e l a t e d t o t h e f r e e v a l e n c e band s t a t e s and i t s empty s t a t e s may be c o n s i d e r e d as h o l e s , bound b y t h e a t t r a c t i v e d i s l o c a t i o n p o t e n t i a l . A c t u a l l y t h e upper empty band (E) c o n s i s t s o f c o n d u c t i o n band s t a t e s t u n e l i n g i n t o t h e r e p u l s i v e d i s l o c a t i o n p o t e n t i a l . I t s l o w e r edge i s t h e r e f o r e n o t q u i t e sharp b u t t h e t u n e l i n g p r o b a b i l i t y and thus, t h e m a t r i x element o f t r a n s i t i o n s t o o t h e r s t a t e s t h a t a r e l o c a l i s e d a t t h e same d i s l o c a t i o n d r o p s r a p i d l y t o z e r o f o r e n e r - g i e s l o w e r t h a n a few h u n d r e d t h o f an eV below t h e upbent edge o f t h e c o n d u c t i o n band. T h e r e f o r e we may t r e a t t h e t u n e l i n g s t a t e s as a d i s l o c a t i o n band l i k e t h e bound s t a t e s .

F i g . 1 : Schematic model o f t h e l e v e l s a t a d i s l o c a - t i o n i n Ge, c a r r y i n g a n e g a t i v e l i n e c h a r g e . E, H and F a r e one d i m e n s i o n a l bands. E c o n s i s t s a c t u a l l y o f t u n n e l i n g s t a t e s .

I n most e x p e r i m e n t s t h e bound h o l e s t a t e s and t h e t u n n e l i n g s t a t e s a r e i n t h e r m a l e q u i l i b r i u m w i t h t h e v a l e n c e - and t h e c o n d u c t i o n band r e s p e c t i v e l y , because t h e exchange o f c a r r i e r s w i t h t h e s e bands i s v e r y r a p i d /3/.

A c c o r d i n g t o Landsberg t h e t r a n s i t i o n r a t e o f e l e c t r o n s f r o m one d i s l o c a t i o n band t o a n o t h e r i s /5/

rij = C . .n.p. (1-exp(pj t pi)/kT)

1J 1 J - ^{( 1 )}

ni i s t h e number o f e l e c t r o n s

i n

number' o f h o l e s i n band j and pi and p t h e respec- j

t i v e q u a s i - F e r m i l e v e l s . I f t h e whole system i s i n thermal e q u i l i b r i u m p and pi a r e o f c o u r s e equal t o

j

t h e Fermi l e v e l

c.

1 J

t h e d e n s i t i e s of s t a t e s Ni(€) and N i ( € ' ) i n t h e two d i s l o c a t i o n bands as

Cij =

1

Ni(€)Nj(€' ) f (E-vi) ( l - f ( ~ ' ~ ~ ) ) d c d ~ ' Where f i s t h e Fermi d i s t r i b u t i o n f u n c t i o n and n i , j ( ~ , ~ ' ) t h e quantum mechanical t r a n s i t i o n proba- b i l i t y between a l e v e l a t E i n band i and a 1,evel a t E ' i n band j.

We s h a l l c o n c e n t r a t e o u r d i s c u s s i o n m a i n l y on t h e f a c t o r

R = n . p (1-exp ( p . - u . ) / k T )

1 J J 1 ( 2 )

w h i c h has an e x p o n e n t i a l t e m p e r a t u r e dependence. Cij i s o b v i o u s l y c o n s t a n t i f n . . ( E , E ' ) i s independent

1 J

o f ^E and E ' . However, t h e r e can be n o t i c e a b l e d e v i a - t i o n s f r o m t h i s s i m p l e b e h a v i o u r t h a t y i e l d an expo- n e n t i a l t e m p e r a t u r e dependence o f Ci as we1 1 /3/.

F o r empty and f u l l d i s l o c a t i o n bands we make use o f t h e Bol tzman a p p r o x i m a t i o n s

nE = NNE ' exp ( p E - ~ , - ) / k r ( 3 )

and

PF = NF exp ( E ~ - p F ) / k T ( 4 )

r e s p e c t i v e l y . If t h e band edges a r e a p p r o x i m a t e l y p a r a b o l i c t h e e f f e c t i v e d e n s i t i e s o f s t a t e s NE and NF a r e equal t o

=

zT2 h2

mass o f t h e d i s l o c a t i o n band' and p t h e d i s l o c a t i o n l i n e l e n g t h p e r u n i t volume.

I n a p a r t l y f i l l e d band we d e f i n e c R as t h e p o s i t i o n t h a t t h e q u a s i Fermi l e v e l would have i f t h e number o f e l e c t r o n s i n H were equal t o iH, t h e number i n thermal e q u i l i b r i u m . I f excess c a r r i e r s a r e p r e s e n t i n d i s l o c a t i o n s t a t e s , ^E,, e x p e r i e n c e s t h e same e l e c t r o s t a t i c s h i f t as a l l o t h e r d i s l o c a - t i o n l e v e l s . The p o s i t i o n o f t h e q u a s i Fermi l e v e l i s pH = E~ + AD,,, where ApH i s an a d d i t i o n a l s h i f t due t o t h e presence o f AnH excess e l e c t r o n s ( o r -ApH excess h o l e s ) i n H. F o r s m a l l AnH/" we use t h e expansion

F o r t h e sake o f s i m p l i c i t y we s h a l l o f t e n r e f e r t o d i s l o c a t i o n bands and t o t h e v a l e n c e and t h e c o n d u c t i o n band as " l e v e l s " i n t h e f o l l o w i n g b u t s h o u l d keep i n m i n d t h e i r t r u e meaning and t h e d e f i - n i t i o n s g i v e n h e r e .

3. D e f i n i t i o n o f t h e r a t e d e t e r m i n i n g t r a n s i t i o n . - F o r any system o f more t h a n two l e v e l s t h e r e a r e s e v e r a l p o s s i b l e p a t h s a l o n g which t h e r e c o m b i n a t i o n t a k e s p l a c e . F o r i n s t a n c e , an e l e c t r o n can proceed f r o m E t o F i n f i g u r e 1 t h r o u g h t h e sequence E-H-F

R. L a t u s c h C6-83

b u t a l s o t h r o u g h E-F. However, s i n c e t h e t r a n s i t i o n r a t e s u s u a l l y have an e x p o n e n t i a l t e m p e r a t u r e depen- dence t h e r e e x i s t s ( w i t h v e r y few e x c e p t i o n s ) a p a t h t h a t i s , by f a r , t h e f a s t e s t one w h i l e r e c o m b i n a t i o n a l o n g a l l o t h e r s i s n e g l i g i b l e . The o n l y s y s t e m a t i c exemptions f r o m t h i s r u l e a r e a few r a t h e r s h a r p l y d e f i n e d c r o s s o v e r t e m p e r a t u r e s where t h e system chan- ges i t s r e c o m b i n a t i o n p a t h .

L e t us c o n s i d e r now a steady s t a t e e x p e r i m e n t i n which e l e c t r o n s and h o l e s a r e generated a t t h e same r a t e G i n t h e l e v e l s E, and r e s p e c t i v e l y and recombine v i a t h e s i n g l e o r d e r e d sequence ^{E ~ ,} i = 1,

...,

Again t h e f a c t o r s Ci,i+lnipi+l a r e u s u a l l y exponen- t i a l f u n c t i o n s o f T so t h a t , a p a r t f r o m a few c r o s s - o v e r s , one o f t h e v a l u e s (Ci ,i+lnipi+l)-' i s

=

g r e a t e r t h a n a l l t h e o t h e r s . L e t t h i s b i g g e s t v a l u e be assumed f o r i = k. We r e q u i r e now t h a t t h e genera- t i o n G i s "weak" enough so t h a t

I L I ~ + ~ - ~ ~

l

and Ipi+l-ui

I<<\

A l l l e v e l s w i t h i ( k have t h e same q u a s i Fermi e n e r - gy p k and t h e c o r r e s p o n d i n g v a l u e f o r t h e l e v e l s w i t h i = k + 1,

...,

...,

i = n t 1,

...,

F i g . 2 : Schematic example o f a r e c o m b i n a t i o n p a t h , i n v o l v i n g f o u r l e v e l s and a t r a p ( l e v e l 5 ) a t t a c h e d t o l e v e l 1. The r a t e d e t e r m i n i n g t r a n s i t i o n i s 2-3.

The t o t a l number o f excess e l e c t r o n s i s

Antot = !Ani where t h e sum i s t a k e n o v e r t h e l e v e l s 1 t o k and t h e t r a p s a t t a c h e d t o them. The t o t a l number of excess h o l e s i s Ap = An = :Api where now t h e sum i s t a k e n o v e r t h e l e v e l s k = 1 t o n and t h e i r t r a p s . I f t h e g e n e r a t i o n i s s w i t c h e d o f f , t h e l e v e l s w i t h a comnon Fermi l e v e l s t a y i n e q u i l i b r i u m and t h e recom- b i n a t i o n p r o c e s s i s d e s c r i b e d by t h e s i m p l e r a t e e q u a t i o n

which means t h a t t h e t r a n s i t i o n k ⁺ k ^t 1 i s r a t e d e t e r m i n i n g . T h i s h o l d s a l s o i f a t r a p i s p r e s e n t , u n l e s s t h e v a l u e (Ci,t ni

-

The s o l u t i o n o f e q u a t i o n ( 7 ) under t h e c o n d i - t i o n s o f l i n e a r response i s an e x p o n e n t i a l decay w i t h a s i n g l e t i m e c o n s t a n t . Such a decay can be observed i n most e x p e r i m e n t s i f o n l y t h e g e n e r a t i o n i s made weak enough and whenever i t i s found, chances a r e v e r y good t h a t o u r approach i s v a l i d .

4. E v a l u a t i o n o f r e c o m b i n a t i o n times.- F o r an e v a l u a - t i o n o f t h e t r a n s i t i o n r a t e f a c t o r R i n e q u a t i o n ( 2 ) we use t h e i d e n t i t y p . - p . = ( p . - E . ) - ( u . - E . ) -I

J 1 0 J J 1 1

( E . - f i ) - ( Z . - E ) ^{.I 1} + (E.-g.) J J - ( E ~ - E ~ ) .

The l a s t t w o - t e r m s a r e t h e e l e c t r o s t a t i c s h i f t s o f E~ and ^{E .} which a r e a p p r o x i m a t e l y equal f o r a l l

J

d i s l o c a t i o n l e v e l s and t h e r e f o r e cancel each o t h e r . I n s e r t i n g f o r uj - ui we o b t a i n :

exp(ci-gi ) / k ~ . exp(u .-E . ) / k ~

e x p ( ( p . - u . ) / k T ) = - J -

J 1 exp(ui-")/kT exp(fi-F.)/kT J (8)

We can e x p r e s s now t h e r a t e f a c t o r R i n terms o f e l e c t r o n - and h o l e d e n s i t i e s f o r any p a i r o f d i s - l o c a t i o n bands, k and k t l , and e v a l u a t e t h e recombi- n a t i o n t i m e under t h e assumption t h a t t h i s p a r t i c u - l a r p a i r i s r a t e d e t e r m i n i n g . There a r e f i v e p o s s i b l e c o m b i n a t i o n s o f w h i c h we have t o a n a l y s e t h e f o l l o w - i n g t h r e e :

i ) The upper band i s empty and t h e l o w e r one p a r t l y f i l l e d .

i i ) The upper band i s empty and t h e l o w e r one f u l l . i i i ) B o t h d i s l o c a t i o n bands a r e empty.

The r e m a i n i n g combinations o f two f u l l bands and o f a p a r t l y f i l l e d and a f u l l band a r e o b t a i n e d f r o m i i ) and i i i ) i f t h e energy s c a l e i s i n v e r t e d and e l e c t r o n s a r e exchanged f o r h o l e s and v i c e v e r s a .

C6-84 JCURNAL DE PHYSIQUE

Ne s h a l l now discuss t h e cases i ) t o i i i ) i n t u r n :

i ) Remembering t h a t E~ = u we o b t a i n from equations ( 2 ) , ( 3 ) and ( 5 ) REH = nEPH - ( i E / n E ) . exp(- hpH/NHk~)]. Linear response i m p l i e s t h a t ApH must be small enough f o r a l i n e a r expansion o f the exponential.

Furthermore

~

~ ~ / i ~

where a = ;H/~HL(:H) LT End o n l y l i n e a r terms i n AnE and ApH have been taken i n t o account. NH i s t h e d e n s i t y o f s t a t e s i n t h e H-band.

An estimate o f a, assuming a reasonable bandwith o f ( 0 . 1 t o 1 eV) y i e l d s values o f t h e order o f 10 t o 100, depending on T.

I n n-type m a t e r i a l we have aApH/EH >> AnE/iE because a >> 1. Therefore RE,H = a i E ApH. The quasi Fermi l e v e l o f t h e valence band i s equal t o uH. Therefore the f r e e h o l e d e n s i t y i s pv = Nv exp(6,-p)/kT w h i l e

6,

A p v . = ApH a Pv and, n e g l e c t i n g holes i n o t h e r P u

d i s l o c a t i o n stayes below H ( i f t h e r e a r e any), A ~ t o t = ApH(l+ apV/oH).

Under most c o n d i t i o n s , since pv i s t h e m i n o r i t y c a r r i e r d e n s i t y , apv/pH c c 1 i . e . most excess holes a r e i n H and o n l y a small f r a c t i o n i n t h e valence band. Then the recombination time i s T = ( c ~ , ~ ~ ~ 1 - l exp(gE - E H ) / k ~ ) (lea) where we have used equation ( 3 ) f o r iE.

For low d i s l o c a t i o n d e n s i t i e s and a t temperatures close t o t h e i n t r i n s i c temperature a; /fl ^{>> 1 i s}

a l s o possible, so t h a t T =

:

(Bv/fE,HyE~H)

0 0

exp(cE-aH)/kT. With pv = (NcNv/nc)exp(-~G/kT), where EG i s t h e gap energy, t h i s can be t r a n s f o r - med t o

NcNv exp ( ( E ~ - t H )

-

= CEaHNEF

6 _ i s - t h e - e l e c t r o n d e n s i t y i n t h e conduction band.

1; p-type m a t e r i a l we have REH = 6,,AnE, Antot = An .(It N exp ( z E - g c ) / k ~ and t h e r e f o r e

E

r = (CE,HbH)- l(1tNc/NE ~ X ~ ( ; ~ - ; ~ ) / X T .

With cE-cC = ( c E -cH) ^t ( E ~ - E ~ ) - ^E~ t h e second term i n t h e brackets can be transformed t o (NcNV/

NE$) exp((EE-gH)

-

The recombination time i s t h e r e f o r e

NcNv 1 ⁰ 0

T =

---

-

'E ,HPH~E Pv

Equations ( l l a ) and ( l l b ) h o l d i f t h e excess mino- r i t y c a r r i e r s ( e l e c t r o n s ) a r e trapped i n t h e E- band o r r e s i d e i n the conduction band r e s p e c t i v e l y . i i ) To d i s t i n g u i s h the energy l e v e l s i n v o l v e d i n t h e

t r a n s i t i o n E-F from those o f t h e process E-H we use now t h e symbols qE and nF. The a n a l y s i s pro- ceeds as under i :

0 0 A'lE A p ~

REF = ~ E P F

(H ' F)

and, using t h e same algebra as before, we o b t a i n t h e f o l l o w i n g recombination times. I n n-type ma- t e r i a l :

T = (CE,F ~ ~ ) - ' e x p ( ; ~ - : ) / k T (12a)

i n p-type m a t e r i a l :

T = (CEFNF)- exp

(E

Again equations (12a) and (13a) are v a l i d i f the excess m i n o r i t y c a r r i e r s are trapped i n E and F r e s p e c t i v e l y , equations (12b) and (13b) h o l d i f these c a r r i e r s a r e i n f r e e band s t a t e s .

i i i ) This r e q u i r e s the presence o f two empty d i s l o c a - t i o n bands a t t h e same p a r t i a l d i s l o c a t i o n . I t i s a r a t h e r u n l i k e l y case because t h e t r a n s i t i o n pro- b a b i l i t y i s expected t o be extremely h i g h f o r two l e v e l s o f s i m i l a r c h a r a c t e r t h a t are c l o s e t o each o t h e r on t h e energy scale. The r e s u l t s are iden- t i c a l w i t h those under i ) i f pH i s replaced by NE2 and ^EE by cE1/ Therefore t h i s process i s h a r d l y d i s t i n g u i s h a b l e from "E-H".

5 . Trapping.- We know t h a t most d i s l o c a t i o n s i n semi- conductors are d i s s o c i a t e d and we have t o expect t h a t each p a r t i a l has i t s own spectrum o f energy l e - v e l s . Therefore we have t o consider t h e p o s s i b i 1 i t y t h a t recombination takes p l a c e a t one p a r t i a l w h i l e t h e o t h e r one a c t s as a t r a p . The case o f i n t e r e s t i s the combination o f a H-band, t o g e t h e r w i t h F and E, a t one p a r t i a l andan E- and an F-band w i t h a gap between oE and qF a t t h e o t h e r one. Excess m i n o r i t y c a r r i e r s t h a t want t o recombine v i a l e v e l cr a t one d i s l o c a t i o n can be trapped i n a l e v e l E~ a t t h e o t h e r p a r t i a l . E can t r a p e l e c t r o n s , F can t r a p holes and H can t r a p e l e c t r o n s and holes I f c t and E, are i n thermal e q u i l i b r i u m , as they must be under steady s t a t e c o n d i t i o n s , t h e f r a c t i o n o f c a r r i e r s i n er i s o n l y ( I +

$

~

-

K . Labusch C6-85

holes. Nt and Nr are e f f e c t i v e d e n s i t i e s o f s t a t e s . The expressions i n t h e brackets e n t e r t h e correspon- d i n g recombination times as f a c t o r s .

Obviously hole t r a p s a r e i n e f f i c i e n t i f ct i s below cr because otherwise (Nt/Nr)exp(ct-cr)/kT<<l.

On the o t h e r band, qF must be below cH. Therefore t h e r e i s no e f f e c t o f t r a p p i n g by t h e F-band a t nF i f , i n n-type m a t e r i a l , t h e process E-H i s r a t e de- termining. The same holds f o r p - m a t e r i a l and t h e pro- cess H-F.

I n p - m a t e r i a l , w i t h ct = nE and cr = qE, t r a p p i n g i s p o s s i b l e and equation ( l l a ) i s replaced by

T = ( ~ ~ ~ p ~ ) - ' ~ ~ ~ ( 2 ~ - nE)/kT ( I 1 c ) provided t h a t cE i s above nE and E-H t h e r a t e d e t e r - mining t r a n s i t i o n . As an example we have i l l u s t r a t e d t h i s case i n f i g u r e 3.

F i g . 3 : Model o f recombination and t r a p p i n g a t a d i s s o c i a t e d d i s l o c a t i o n . I n t h i s example o f p-type m a t e r i a l t h e r a t e determining t r a n s i t i o n c o u l d be from cE t o uH w h i l e most excess e l e c t r o n s are s t o r e d a t uE.

I f recombination takes p l a c e v i a the l e v e l s a t nE and qF, t h e H-band a t t h e companion d i s l o c a t i o n i s always an e f f i c i e n t t r a p because E~ i s above qF and below nE. The corresponding recombination times i n n- and p-material are obtained by m u l t i p l i c a t i o n o f e u a t i o n s (12a) and (12b) w i t h

#

I,

and exp(nE - E ~ ) / ~ T . T h i s y i e l d s i n both cases.

T = --- exp (:E H - ; F ) / k ~

EF E F ( 1 2 ~ )

6. Discussion and a p p l i c a t i o n t o Ge.- For comparison w i t h experiments we a r e mainly i n t e r e s t e d i n t h e a c t i v a t i o n energy Q o f T which c o n s i s t s o f t h e a c t i - v a t i o n energy Qi o f t h e f a c t o r C i l t h e extrapo-

, j Y

l a t e d value o f the energy d i f f e r e n c e i n t h e exponen- t i a l a t T = 0 and, i n some cases, the f r e e c a r r i e r c o n t r i b u t i o n Qc = 61n(Nc/nc)/6(l/kT) o r Q v = 61n ( N , , / P ~ ) / G ( ~ / ~ ~ ) . A l i n e a r e x t r a p o l a t i o n o f t h e

energy d i f f e r e n c e t o T = 0 i s necessary because any p a r t t h a t i s l i n e a r i n T does n o t c o n t r i b u t e t o Q. I n our r e s u l t s , w i t h t h e exception o f equation ( l l c ) , t h e r e appear o n l y energy d i f f e r e n c e s a t the same par- t i a l d i s l o c a t i o n so t h a t t h e i r dependence on T i s weak, b u t nevertheless the a c t i v a t i o n energies i n the f o l l o w i n g t a b l e should be considered as the extrapo- l a t e d r a t h e r than t h e t r u e values. The temperature dependence o f some o f t h e energy d i f f e r e n c e s has been c a l c u l a t e d by W. Schroter /2/. I n the same paper a l s o p o s s i b l e mechanisms t h a t y i e l d an a c t i v a t i o n energy f o r C . have been discussed. Qi . t u r n s o u t

I ,j 1 J

t o be small b u t n o t n e g l i g i b l e ( o f the order o f 0.1 eV). The columns o f t h i s t a b l e r e f e r t o t h e r a t e determining t r a n s i t i o n s "E-F", "E-H" and "H-F1' . The expressions under "H-F" can be obtained w i t h o u t c a l - c u l a t i o n from "E-H" by the conversion o f e l e c t r o n s and holes. The l i n e s a, b and c r e f e r t o t h e c o r r e s - ponding l e t t e r s a t equations (10a) t o (13b). The excess m i n o r i t y c a r r i e r s are s t o r e d i n d i f f e r e n t l e v e l s f o r a, b and c :

I n t h e recombination l e v e l i t s e l f f o r a, i n f r e e band s t a t e s f o r b and i n a t r a p p i n g l e v e l a t t h e companion p a r t i a l f o r c.

The f o l l o w i n g experimental f a c t s are known about d i s l o c a t i o n s (probably m a i n l y o f the 60'-type) i n Ge from p h o t o c o n d u c t i v i t y measurements : I n a p l o t o f T versus t h e photon energy hv t h e r e i s , a t low temperatures, a s t e p a t hv z 0.45 t o 0.5 eV w i t h t h e h i g h e r value o f T above 0.5 eV /6/. Obviously t h i s must correspond t o a change o f the recombination p a t h e i t h e r from "H-F" t o "E-H" o r from "H-F" t o

"E-F", because E s t a y s empty and cannot be on the recombination p a t h f o r small phtoon energies. This holds f o r n- and p - m a t e r i a l . For t h e h i g h e r photon energies the a c t i v a t i o n energy i s Q = 0.4 t o 55 eV i n n-Ge and Q around 0.25 eV i n p-Ge.

From o t h e r experiments we know t h a t a H-band e x i s t s /7,8/ so t h a t f o r t h e p a t h "E-F", H-trapping must be expected. The d i f f e r e n c e o f Q i n n-Ge and p-Ge then r u l e s o u t t h e p a t h "E-F" because i t would y i e l d t h e same a c t i v a t i o n energy f o r n-Ge and p-Ge.

Consequently t h e recombination must be "E-H". The a c t i v a t i o n energy o f 0.25 eV i n p-Ge appears t o be t o o h i g h t o be accounted f o r by QE,H alone. There- f o r e l i n e "c" i n t h e lower h a l f o f column "E-H" i s more l i k e l y than l i n e "a". A t h i g h e r temperatures t h e a c t i v a t i o n energy i s o f t h e order o f - 0.22.eV i n p-Ge and n-Ge. Obviously t h e l i n e s "b" apply t o t h i s case. With t h e o u t s i d e i n f o r m a t i o n t h a t (cH - ^{E ~ )} z 0.1 eV / 7 / we can r u l e o u t column "H-F".

JOLIRNAL DE PHYSIQUE

Table I : A c t i v a t i o n energies o f recombination times

The remaining two cases "E-H" and "E-F" cannot be References

n-type

d i s t i n g u i s h e d a t present. I n t h e t a b l e the cases /1/ F i g i e l s k i , T., Phys. S t a t u s S o l i d i 2 (1965) 555.

t h a t t u r n o u t t o be v a l i d i n Ge from t h i s a n a l y s i s / 2 / Schroter, W., Phys. S t a t u s S o l i d i (a) - 19 (1973)

are i n d i c a t e d by double l i n e s . 159.

c not possible

Q ~+ ,"E ~- ^'IF

QE,F + qE

-

QE F + nE - vF

-

P-type

/3/ Jastrzebska, M. and F i g i e l s k i , T., Phys. Status S o l i d i

32

/4/ Labusch, R. and Schroter, W . , 1nst.PHys. Conf.

Ser. No 23 (1975) 56.

QH,F

QH,F ⁺ EH

-

-

a b

QE,F +

-

Q~ + nE - vF - E~ +

Qv

Q ~+ ,"E ~

-

/5/ Landsberg, T., Festkoperprobleme I V , ( V e r l a g F.

Vieweg, Braunschweig) 1967.

/6/ Mergel, D. and Labusch R., Phys. Status S o l i d i ( a )

41

/7/ Schroter, W., Phys. Status S o l i d i - 21 (1967) 211.

Q + E - E

QE,H + e E - c H - eG + QC

a b C

/8/ Labusch, R. and S c h e t t l e r , R., Phys. Status So- l i d i ( a ) 9 (1972) 455. -

QE,H

Q ~+ ,'E ~

-

-

Qv

Q ~+ ,'E ~

-

QH,F + EH - EF - cG + QV not p o s s i b l e

I