INFLUENCE OF DISLOCATIONS ON THE HALL EFFECT IN SILICON AND GERMANIUM

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INFLUENCE OF DISLOCATIONS ON THE HALL EFFECT IN SILICON AND GERMANIUM

W. Schröter

To cite this version:

W. Schröter. INFLUENCE OF DISLOCATIONS ON THE HALL EFFECT IN SILI- CON AND GERMANIUM. Journal de Physique Colloques, 1979, 40 (C6), pp.C6-51-C6-57.

�10.1051/jphyscol:1979611�. �jpa-00219027�

INFLUENCE OF D I S L O C A T I O N S ON THE HALL E F F E C T IN SILICON A N D GERMANIUM

W. S c h r o t e r

IV. Physikalisches I n s t i t u t der Univer~sitat Giittingen and Sonderforschungsbereich 1 2 6 CGLtingen Clausthal 3400 Giittingen, F. H. C .

Resume.- Les r e s u l t a t s experimentaux d ' e f f e t H a l l obtenus s u r l e s e c h a n t i l l o n s de s i l i c i u m e t de g e r - manium o n t e t e i n t e r p r e t e s p a r une s t a t i s t i q u e d ' o c c u p a t i o n des e t a t s dans l a q u e l l e on suppose que l e s p e c t r e d ' e n e r g i e a s s o c i e

a

l a d i s l o c a t i o n n e u t r e s u b i t une t r a n s l a t i o n e l e c t r o s t a t i q u e r i g i d e l o r s q u e l e s e t a t s a s s o c i e s a l a d i s l o c a t i o n s o n t occup@s p a r des e l e c t r o n s . Une bande i? une dimension, p a r t j e l - lement r e m p l i e e s t associee aux d i s l o c a t i o n s p a r t i e l l e s n e u t r e s

a

90' dans l e s i l i c i u m e t l e germanium.

Une bande p l e i n e e t une bande v i d e s o n t a s s o c i e e s aux d i s l o c a t i o n s p a r t i e l l e s n e u t r e s

a

30" dans l e germanium.

A b s t r a c t . - E x p e r i m e n t a l H a l l d a t a f o r deformed s i l i c o n and germanium have been a n a l y z e d by means o f o c c u p a t i o n s t a t i s t i c s t h a t assume a r i g i d e l e c t r o s t a t i c s h i f t o f t h e energy spectrum a t a n e u t r a l d i s - l o c a t i o n , when t h e d i s l o c a t i o n s t a t e s a r e o c c u p i e d b y e l e c t r o n s . A p a r t i a l l y f i l l e d one d i m e n s i o n a l band i s a t t r i b u t e d t o n e u t r a l 9 0 ' - p a r t i a l d i s l o c a t i o n s i n S i and Ge and a f u l l and empty band t o neu- t r a l 3 0 ' - p a r t i a l s i n Ge.

I n t r o d u c t i o n . - One b a s i c f e a t u r e which d i s t i n g u i s h e s l o c a l i z e d s t a t e s a t d i s l o c a t i o n s f r o m t h o s e a t s h a l - low donors and a c c e p t o r s i s t h e i r many-electron-cha- r a c t e r . T h e r e f o r e t h e quantum-mechanical d e s c r i p t i a l o f t h e d i s l o c a t i o n c o r e and t h e s t a t i s t i c a l a n a l y s i s o f i t s o c c u p a t i o n w i t h e l e c t r o n s a r e i n general n o t

independent. On1 y under c o n d i t i o n s

,

where t h e one- e l e c t r o n - a p p r o x i m a t i o n p r e s e n t s an a p p r o p r i a t e des- c r i p t i o n o f t h e c o r e s t a t e s , an independent s t a t i s - t i c a l t r e a t m e n t re1 a t e s m i c r o s c o p i c f e a t u r e s o f t h e l o c a l energy spectrum t o measurable q u a n t i t i e s and may t h e n be a p p l i e d t o analyse and i n t e r p r e t e x p e r i -

d i f f e r e n t f r o m t h o s e o b t a i n e d a f t e r d e f o r m a t i o n a t h i g h temoeratures. I s t h i s d i f f e r e n c e a s s o c i a t e d w i t h d i f f e r e n c e s i n t h e c o r e r e g i o n of d i s l o c a t i o n s ? I s h a l l summarize f i r s t arguments.

F i n a l l y i n c h a p t e r 3 I s h a l l d i s c u s s t h e i n f l u - ence o f t h e d i s s o c i a t i o n o f d i s l o c a t i o n s , o f e l e c - t r o n - h o p p i n g a l o n g t h e d i s l o c a t i o n core, o f t h e i n - t r a b o n d Coulomb i n t e r a c t i o n and o f h y b r i d i z a t i o n between l o c a l i z e d s t a t e s and t h e extended band s t a t e s on t h e model o f t h e l o c a l energy spectrum a t d i s l o c - a t i o n s , t h a t we d e r i v e f r o m t h e a n a l y s i s o f e x p e r i - mental H a l l data.

mental d a t a . 1. Simple s t a t i s t i c a l a n a l y s i s . - 1.1.- klupptjgrj

I n t h i s paper I s h a l l c o n f i n e myself t o t h e

s t a t i s t i c s

.-

Simple o c c u p a t i o n s t a t i s t i c s n e g l e c t s d i s c u s s i o n o f experiments, w h i c h have been performed

---

d i s s o c i a t i o n of d i s l o c a t i o n s ( b u t see 3.1 .)

,

t h e under c o n d i t i o n s o f thermal e q u i l i b r i u m , and among

i n t e r a c t i o n between a d j a c e n t d i s l o c a t i o n s ( b u t see t h e s e o f measurements o f t h e H a l l e f f e c t . F o r non-

e q u i l i b r i u m s t a t i s t i c s and t h e i r a p p l i c a t i o n t o ex- p e r i m e n t s t h e r e a d e r i s r e f e r r e d t o t h e c o n t r i b u t i o n o f R. Labusch and T. F i g i e l s k i a t t h i s conference.

The v a l i d i t y o f t h e o n e - e l e c t r o n - a p p r o x i m a t i o n i s t h e b a s i s f o r t h e development of s i m p l e occupa- t i o n s t a t i s t i c s , w h i c h w i l l be b r i e f l y r e v i e w e d i n c h a p t e r 1. O n l y a few a n a l y t i c a l s o l u t i o n s have been d e r i v e d t i l l now and o n l y some o f them have been compared w i t h e x p e r i m e n t a l d a t a . The main t a s k of t h e e x p e r i m e n t a t o r i s t h e n t o c a r e f u l l y check t h e l i m i t a t i o n s of t h e s e s o l u t i o n s , and t h i s w i l l be my main p o i n t i n c h a p t e r 1. As we s h a l l see i n c h a p t e r 2, d e f o r m a t i o n a t l o w temperatures (e.g T s 500" C i n Ge) y i e l d s H a l l r e s u l t s , which a r e s i g n i f i c a n t l y

1.2.) and e l e c t r o n c o r r e l a t i o n ( b u t see 3 . 3 . ) , b u t t a k e s i n t o account t h e Coulomb i n t e r a c t i o n o f one added e l e c t r o n w i t h t h e l i n e charge q o f t h e d i s l o - c a t i o n c o r e and t h e s c r e e n i n g charge s u r r o u n d i n g t h e d i s l o c a t i o n .

The use o f t h e o n e - e l e c t r o n - a p p r o x i m a t i o n i s j u s t i f i e d , when t h e c o r e s t a t e s o f t h e d i s l o c a t i o n a r e we1 1 l o c a l i z e d on a s c a l e , t h a t measures t h e ex- t e n s i o n o f t h e s u r r o u n d i n g s c r e e n i n g cloud. Then t h e i n t e r a c t i o n between l o c a l i z e d e l e c t r o n s may be appro- x i m a t e l y d e s c r i b e d b y a r i g i d s h i f t Ee o f t h e s t a t e s a t a n e u t r a l d i s l o c a t i o n on t h e energy s c a l e /I/.

L i n e charge q, w h i c h g i v e s r i s e t o t h e s h i f t Ee, and f r a c t i o n o f o c c u p i e d s i t e s f, w h i c h r e s u l t s f r o m t h e

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979611

C6-52 JOURNAL DE PHYSIQUE

occupation s t a t i s t i c s o f the s h i f t e d d i s l o c a t i o n s t a t e s , have t o be p r o p o r t i o n a l t o each o t h e r t o a r r i v e a t s e l f-consistency.

Table I : E l e c t r o s t a t i c s h i f t Ee o f d i s l o c a t i o n s t a t e s as a f u n c t i o n o f the occupation r a t i o f ( f o r f u r t h e r d e t a i l s see t h e work o f Labusch and Schroter / I / )

' _i

E l e c t r o s t a t i c s h i f t + )

i

Yean s p a t i a l value o f

i

^Validity i ^,

/

conc. screening

i i

t h e f r e e e l e c t r o n j range I

i

d e n s i t y <n>

...

Read /2/

i

n-type 'E

- -

e 2 f

[;

En(f/fc)-0.116]i

(1954)

- -

2 m s

jspace sharge c y l i n - i i<n> = n (I-* f o r

11

i

^-

E<A

j d e r o f i o n i z e d

ifc

⁼

_C~

_{Nd7v3 o i ~ ~ ~ / ~ ~ < l}

jdonors, e l e c t r o n s i , L -

j expel 1 ed j q volume f r a c t i o n

j

e v / k B ~ > l

---; --- ;-tttttttttttttttt-tttttttt-tttit~f -

S P C C _ % - E ~ _ C C ~ C - C _ Y ~ L ~ ~ - - - - ~ Broudy /3/ : i n a d d i t i o n t o ~ e a d /

II i<n>= n

( 1963) j low m o b i l i t y r i n g s

i

jsurrounding space

i i

^{0 =}

^bed,

4 ~ 0 1 . f r a c t . o f

i

^{I 1}

j charge c y l

.

^Fto

...

SchrBter j s c r e e n i n g by f r e e iEe =

and jcharge c a r r i e r s j o f p o i n t ;

i

p-type

Labusch /4/

i

charges i<n> =

-

r

(1969) j % > A

e RH e 2 f En

X

:E =

-

f o r l i n e :

i

pNa/pp < 1

t e 2 a ~ s r, charge j

i

eV/kBT < 1

+) E d i e l e c t r i c constant

-

B magnetic f i e l d v e c t o r

s d i s t a n c e between adjacent dangling bonds - i e l e c t r i c c u r r e n t d e n s i t y v e c t o r

Nd(Na) d e n s i t y o f shallow donors (acceptors) ro ra d i a l w i d t h o f core s t a t e s a t d i s l o c a t i o n D d i r e c t i o n o f t h e d i s l o c a t i o n l i n e element

"

e l e c t r o n d e n s i t y b e f o r e deformation

- AD = ( E K ~ T / ~ ~ < ~ > ) % Debye screening l e n g t h

Some a n a l y t i c s o l u t i o n s , which are o b t a i n e d w i t h i n t h i s approximation and have been compared w i t h experimental data, a r e summarized i n t a b l e I. T h e i r range o f v a l i d i t y may be c h a r a c t e r i z e d by t h r e e para- meters :

(1) The r a t i o of t h e mean f r e e p a t h o f f r e e charge c a r r i e r s R i n t h e undeformed m a t e r i a l and o f the r a d i u s A of t h e screening c l o u d surrounding t h e d i s l o c a t i o n , i n d i c a t e s whether t h e d i s l o c a t i o n i s described more a p p r o p r i a t e l y as a s c a t t e r i n g c e n t e r ( %>A ) o r as a macroscopic inhomogeneity ( a<x ).

(2) The r a t i o pNa/pp (e.g. f o r p-type m a t e r i a l : pNa/pp = (N,/Nv) exp ( E a / k ~ T ) ; Na d e n s i t y of shallow acceptors w i t h a l e v e l a t Ea above t h e valence band edge ; Nv e f f e c t i v e d e n s i t y - o f - s t a - t e s o f t h e valence band) t e l l s us, whether scree- n i n g o f t h e l i n e charge of t h e d i s l o c a t i o n i s m a i n l y achieved by a r e d i s t r i b u t i o n o f f r e e charge c a r r i e r s ( pNa/" << 1) o r by a space charge o f i o n i z e d doping atoms ( pNa/pp>>l ).

(3) The value e V ( r ) / k B a T, where V ( r ) i s t h e screened e l e c t r o s t a t i c p o t e n t i a l o f t h e 1 in e

charge q, may a l l o w f o r a l i n e a r i z a t i o n o f t h e r'oisson equation ( eV/kBT< 1 ) and thereby f o r a l i n e a r response, o r n o t ( eV/kBT> 1 ) . T h i s i s t h e parameter, which i s most d i f f i c u l t t o e s t i - mate because o f i t s space dependence and, because t h e exact s o l u t i o n has t o be known f o r i t s eva- l u a t i o n .

Read /2/ and l a t e r Broudy /3/, i n t h e i r s t a t i s - t i c a l a n a l y s i s o f deformed n-type m a t e r i a l , t r e a t e d t h e case corresponding t o !, ?

/

? ,

<

I

,

pNa/pp>l and

eV/kBT > I i n a space r e g i o n around t h e d i s l o c a t i o n (space charge c y l i n d e r ) , from which a l l f r e e e l e c - t r o n s have been expelled, and eV/kBT = 0 o u t s i d e t h i s region. T h i s space charge c y l i n d e r o f r a d i u s 1, which i s d e r i v e d from t h e n e u t r a l i t y c o n d i t i o n q = rA2eNd (Nd d e n s i t y o f shallow donors) i s assumed t o have sharp boundaries i n Read's treatment, and i s assumed t o be surrounded by a r i n g o f reduced mobi- l i t y i n Broudy's treatment. The e v a l u a t i o n o f t h e d e n s i t y of f r e e e l e c t r o n s n from t h e measured H a l l constant RH has t o take i n t o account t h a t (a) t h e d e n s i t y o f f r e e e l e c t r o n s vanishes i n s i d e t h e space charge c y l i n d e r s , and (b) t h a t t h e value o f RH

t h e d i r e c t i o n 3 f t h e e l e c t r i c c u r r e n t and the ma- s i v e comparison o f equation (3) w i t h experimental da- g n e t i c f i e l d

B

i n a m a t e r i a l w i t h c y l i n d r i c a l voids. t a o f t h e H a l l e f f e c t has been performed f o r e d g e - W e

Read's and Broudy's d e s c r i p t i o n s a r e a p p r o p r i a t e d i s l o c a t i o n s i n Ge. I f t h e h o l e d e n s i t y p as a func- a t low temperatures, below the exhaustion range o r t i o n o f t h e i n v e r s e temperature 1/T i s measured be- f o r s u f f i c i e n t l y l a r g e values o f t h e l i n e charge q. f o r e and a f t e r i n t r o d u c t i o n o f d i s l o c a t i o n s , a cross-

The case L/X>1

,

pNa/pp< 1 and eV/kBT<l, so t h a t over o f t h e two curves i s found, which marks a c o n t i - t h e Poisson equation may be l i n e a r i z e d , has been nuous t r a n s i t i o n from a donor-type t o an acceptor- t r e a t e d by Schrater and Labusch /4/. The screening type behaviour /8/. I t has been i n t e r p r e t e d i n terms i s determined by t h e Debye screening l e n g t h AD, and o f a ~ a r t i a l l y f i l l e d d i s l o c a t i o n band. The occupation t h e d e n s i t y o f f r e e holes o r e l e c t r o n s i s g i v e n by l i m i t o f t h e o a r t i a l l y f i l l e d band c o i n c i d e s w i t h p = r/eRH, where r i s t h e Hal 1 f a c t o r and 1 ie s , de- the Fermi l e v e l p = kBT l n ( N v / p ) . From t h e cross- pendent on s c a t t e r i n g mechanism and magnetic f i e l d , over temperature To the occupation l i m i t Ee o f the between 1 and 2 / I / . To determine r t h e Hal 1 constant n e u t r a l d i s l o c a t i o n has been obtained : Eo=u(To ,Na).

has been measured as a f u n c t i o n o f t h e magnetic f i e l d The d i s l o c a t i o n d e n s i t y has been measured independen- i n some cases /5,6/. The s o l u t i o n , given by S c h r o t e r t l y , so t h a t t h e l i n e charge q = e f / s of t h e d i s - and Labusch /4/, may be a p p l i e d a t n o t too low tem- l o c a t i o n i s f i x e d by the n e u t r a l i t y c o n d i t i o n q = peratures and f o r small l i n e charges q. I t describes e(p-Nca)/N.

most a p p r o p r i a t e l y smzll d e v i a t i o n from t h e n e u t r a l p ( q )

-

Eo was found i n q u a n t i t a t i v e agreement w i t h s t a t e of d i s l o c a t i o n s . t h e f u n c t i o n Ee(q)

,

as c a l c u l a t e d by S c h r o t e r and

L e t f be the number o f e l e c t r o n s p e r d i s l o c a t i o n Labusch, i f t h e e l e c t r i c a l l y a c t i v e d i s l o c a t i o n den- s t a t e added t o the core ; i f t h e d i s l o c a t i o n i s neu- S i t y was taken t o be about equal t o t h e value d e r i v e d t r a l , we have f = 0. The f-dependent p a r t o f t h e f r e e from e t c h p i t c o u n t i n g /1,4/.

energy p a r dangling bond may be w r i t t e n : I n v e s t i g a t i o n s o f t h e H a l l e f f e c t i n n-type Ge F ( f ) = f(Eo-u ) ^t Eel(f)

-

T S ( f )

+

A E ( f ) ( 1 ) w i t h edge-type d i s l o c a t i o n s have shown, t h a t p ( f ) where e. g. f o r a p a r t i a l l y f i l l e d d i s l o c a t i o n band Eo v a r i e s c o n t i n u o u s l y w i t h o u t any step f o r t h e whole i s the occupation l i m i t a t the n e u t r a l d i s l o c a t i o n , f-range a c c e s s i b l e t o these experiments /9/. The oc- 11 i s t h e Fermi l e v e l , Eel i s t h e e l e c t r o s t a t i c energy cuoation o f t h e d i s l o c a t i o n s t a t e s w i t h Eo=0.09 eV o f the l i n e charge and o f t h e screening charges, AE v a r i e s between -0.03 and +0.30 w i t h o u t any i n d i c a t i o n measures t h e s h i f t o f t h e occupation l i m i t w i t h i n the o f s a t u r a t i o n , when t h e Fermi l e v e l p runs across t h e

d i s l o c a t i o n band, and S i s the entropy of an e l e c t r o n band gap. Even, i f we admit t h a t e t c h p i t counting i n t h e d i s l o c a t i o n core. A l l energies are counted underestimates t h e d i s l o c a t i o n d e n s i t y up t o f a c t o r from t h e undisturbed valence band edge. The i n t e r p r e - 3, a t l e a s t 10 % o f a l l core atoms are e l e c t r i c a l l y t a t i o n of equation ( 1 ) f o r an empty o r a f u l l band a c t i v e . From t h i s r e s u l t s we may exclude j o g s a t d i s - i s s t r a i g h t f o r w a r d . l o c a t i o n s as p o s s i b l e centers t o account f o r t h e H a l l

It can be shown t h a t AE i s n e g l i g i b l e compared data, b u t n o t ( g e o m e t r i c a l ) kinks.

t o t h e o t h e r terms /8/. I n thermal e q u i l i b r i u m we F o r screw d i s l o c a t i o n s i n Ge, Uagner /10,11/

have

a ~ / a

f = 0 d e r i v e d from H a l l data, by an a n a l y s i s s i m i l a r t o

p = E~

+ a~~~ / a t -

T (2) t h a t described above, a f u l l band and an empty band, separated by a band gap o f dbout 0.55 eV. The energy The entropy S contains an important c o n f i g u r a t i o n a l

positions of these bands are given in figure p a r t Sconf, which r e s u l t s from d e v i a t i o n s o f t h e

F o r edge-type d i s l o c a t i o n s i n S i Schrb'ter and charge d i s t r i b u t i o n along the d i s l o c a t i o n core from / 4 / , analysing Hall data obtained by Weber a p e r i o d i c array. N e g l e c t i n g c o r r e c t i o n s , which are /12/ for p-type Si, derived a partially filled band due t o changes i n e l e c t r o s t a t i c and k i n e t i c energy wiht Eo 0 . 3 eV. This result was supported by nea-

with configuration j 7 / , may be surements of t h e H a l l effect performed by3Scheibe

evaluated and, i n s e r t e d i n t o equation

( Z ) ,

leads t o :

f /13/ and by Duster &14/, who found Eo = 0.33 eV, b u t

KBT (In(l-f) (a)empty band Or states was questioned by G r a z h u l i s e t a l . /15/, who, u s i n g l+f (b)half-fil led band (3) a d i f f e r e n t analysis, a r r i v e d a t a donor-acceptor- KBT ( c ) f u l l band

l e v e l w i t h El = Ev

+

0.42 eV and an a c c e p t o r - l e v e l a t E2 = EC

-

0.43 eV, b o t h associated w i t h edge-type

C6-54 JOURNAL DE PHYSIQUE

't

conduction

edge-type

1 0

s c r e w

b

F i g . 1 : ( a ) Fermi l e v e l p as a f u n c t i o n o f t h e occu- p a t i o n r a t i o f, as d e r i v e d f r o m H a l l d a t a and e t c h p i t counts, f o r edge-type ( 0 ) /8/ and f o r screw d i s - l o c a t i o n s (+) i n Ge /11/.

( b ) The l o c a l energy s p e c t r a a t edge-type and screw d i s l o c a t i o n s , as d e r i v e d from e x p e r i m e i l t a l p ( f )

-

curves.

d i s l o c a t i o n .

Common t o a l l s o l u t i o n s , w h i c h a r e summarized i n t a b l e I i s t h e assumption, t h a t d i f f e r e n t d i s l o c a - t i o n s may be t r e a t e d i n d e p e n d e n t l y o f each o t h e r . A l r e a d y t h e f i r s t measurements on S i r e v e a l e d l o n g t a i l s i n t h e p ( f ) - c u r v e s f o r f < 0 /12/, which, a c c o r d i n g t o a proposal o f S c h r 6 t e r and Labusch / 4 / , r e s u l t from an o v e r l a p o f t h e s c r e e n i n g c l o u d s o f a d j a c e n t d i s l o c a t i o n s . By an e x t e n s i o n o f t h e s t a t i s - t i c a l t r e a t m e n t , t h e a u t h o r s were a b l e t o d e s c r i b e q u a n t i t a t i v e l y t h e i n s e t o f o v e r l a p between a d j a c e n t s c r e e n i n g clouds.

In t h e double s c r e e n i n g l e n g t h 2 h exceeds t h e mean d i s t a n c e betweed a d j a c e n t d i s l o c a t i o n s dN = (IT N ) - % , an a n a l y s i s even by t h e extended p ( f ) - r e - l a t i o n , d e r i v e d b y S c h r d t e r and Labusch, i s meanin- g l e s s ;

Graz h u l i s e t al. /15/ i n v e s t i g a t e d t h e Hal 1 e f f e c t o f l i g h t l y doped n - t y p e (2x101" ~ m - ~ ) and p - t y p e

(1013 S i c o n t a i n i n g a h i g h d e n s i t y o f d i s l o c a - t i o n s ( % l o 9 cm-'). I n a s e m i l o g a r i t h m i c p l o t o f t h e H a l l c o n s t a n t versus t h e i n v e r s e temperature t h e y found a straight-line-s art, f r o m which t h e y deduced energy l e v e l s Eo o f t h e d i s l o c a t i o n . I n S i , which remained n- t y p e a f t e r n o t t o o severe deformation, t h e d e t e r m i n a t i o n o f Eo needs an e s t i m a t i o n o f Ee, which would be s p e c u l a t i v e because o f 2X>dN. I n p- t y p e S i one m i g h t r e a l i z e E << Eo n e a r fzO, so t h a t

e

one o b t a i n s p = Eo a c c o r d i n g t o e q u a t i o n ( 3 b ) and f r o m t h i s p = N,,-exp(-Eo/kB-T). G r a z h u l i s e t a1 have demonstrated t h a t t h e l a s t e q u a t i o n y i e l d s an aporo- p r i a t e d e s c r i p t i o n o f t h e i r r e s u l t s and have d e r i v e d a donor-acceptor l e v e l f o r d i s l o c a t i o n a t Eo= 0.42 eV.

I t s h o u l d be n o t e d t h a t f r o m a s t r a i g h t - l i n e - p a r t o f I n p ( l / T ) one d e r i v e s Eo(T = OK) and f r o m t h e c r o s s - o v e r temperature To,Eo (T = To). By compa- r i n g t h e two v a l u e s one a r r i v e s a t a temperature s h i f t o f t h e d i s l o c a t i o n s t a t e s a E 0 / a ~ " 3 x l ~ - * e ~ k - ' w i t h r e s p e c t t o t h e valence band edge, which i s o f t h e same o r d e r as t h a t f o r many deep p o i n t c e n t e r s . 2. I n t r o d u c t i o n o f d i s l o c a t i o n s a t low temperatures.- C a v a l l i n i e t a1 /16/ and, independently, A l b e r s e t a l . /17/ found t h a t d e f o r m a t i o n a t temperatures below 400" C r e s u l t s i n I n p ( l / T ) - c u r v e s , which d i f f e r s i - g n i f i c a n t l y f r o m t h o s e o b t a i n e d a f t e r d e f o r m a t i o n a t 580" C (see F i g . 2 ) . The I n p ( l / T ) - c u r v e s a f t e r d e f o r - m a t i o n a t T 6 400" C a r e s h i f t e d t o much l a r g e r h o l e d e n s i t i e s and a r e r a t h e r f l a t compared t o I n p ( l / T ) - curves o b t a i n e d a f t e r i n t r o d u c t i o n of d i s l o c a t i o n s a t 580" C.

F i g . 2 : H o l e d e n s i t y p as a f u n c t i o n o f t h e i n v e r s e temperature 1/T f o r a Ge-specimen ( i n i t i a l s t a t e A ) t h a t was deformed a t 400" C and quenched a f t e r d e f o r - m a t i o n ( 0 ) /17/. F o r comparison t h e I n p ( l / T ) - c u r v e o b t a i n e d a f t e r d e f o r m a t i o n a t 580' C and s l o w c o o l i n g i s a l s o shown (El) /8/.

F u r t h e r d e t a i l s o f t h i s e f f e c t have been g i v e n b y Schoen /18/. The I n p ( l / T ) - c u r v e , o b t a i n e d a f t e r

corresponds t o d e f o r m a t i o n a t 580" C, by a n n e a l i n g o f e l e c t r o n - h o p ~ i n g a l o n g t h e c o r e and

-

g o i n g beyond t h e specimen a t 432" C f o r 3 h

.

As f a r as can be de- t h e o n e - e l e c t r o n - a p p r o x i m a t i o n

-

t h e i n f l u e n c e o f t h e duced f r o m e t c h - p i t counts, t h e d r a m a t i c change i n i n t r a b o n d Coulomb i n t e r a c t i o n and o f h y d r i d i z a t i o n I n p ( l / T ) cannot r e s u l t f r o m a decrease o f t h e d i s l o - between l o c a l i z e d and extendend s t a t e s .

c a t i o n d e n s i t y d u r i n g a n n e a l i n g . By quenching t h e 3.1.

- ~ j s ~ o c i a g j o n - g f - ~ j ~ 1 g c ~ g i g ~ s . -

I t i s now w e l l specimen a f t e r deformation,a c l e a r e f f e c t f o r d e f o r - e s t a b l i s h e d , by means o f t h e weak-beam-technique i n m a t i o n temperatures up t o 480' C was o b t a i n e d . The e l e c t r o n microscopy, t h a t d i s l o c a t i o n s a r e d i s s o c i a - u ( p ) - c u r v e s a f t e r v a r i o u s t r e a t m e n t s a r e compared i n t e d i n t o p a r t i a l s on t h e main p a r t o f t h e i r l e n g t h

f i g u r e 3. (see F i g . 4) /I/.

V' .fl

,

.

..

^/

^.

0 del o m

0 1 A ' G e 9 . v

Ge 18

.

Ge 16 A

Ge 19 a W 3 x

r

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0.5 1 0

,

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x10 crn

P

F i g . 3 : Fermi l e v e l p as a f u n c t i o n o f t h e h o l e den- s i t y p f o r f o u r specimens deformed a t 400' C (Ge 9V), 450' C (Ge 16 A ) , 462" C (Ge 19 A ) and 480" C (Ge 18 r ) r e s p e c t i v e l y , and quenched a f t e r d e f o r m a t i o n /18/. The u ( p ) - c u r v e f o r a specimen, deformed a t 400" C (Ge 9;and t h e n annealed a t 432" C f o r 3 h ( V ) , i s a l m o s t i d e n t i c a l w i t h t h a t o f a specimen, w h i c h was deformed a t 580" C and s l o w l y c o o l e d (x, N, = 2 . 4 ~ 1 0 ' ~ c m - ~ , ~ = 1. 1x108 cme2)

Using a q u a n t i t a t i v e s t a t i s t i c a l a n a l y s i s , Schoen has demonstrated, t h a t n e i t h e r a model, which com- b i n e s edge-type d i s l o c a t i o n s w i t h a homogeneous d i s - t r i b u t i o n o f a p o i n t d e f e c t , n o r t h a t o f two 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 s w i t h d i f f e r e n t d e n s i t i e s and Eo-values g i v e s a c o n s i s t e n t d e s c r i p t i o n o f h i s e x p e r i m e n t a l r e s u l t s .

A s i g n i f i c a n t v a r i a t i o n w i t h d e f o r m a t i o n tempe- r a t u r e has a l s o been found i n t h e EPR-signal /19/

and i n t h e DLTS-spectrum /20/ f o r S i . The t r a n s i t i o n t e m p e r a t u r e was found between 700 and 800" C.

Though we do n o t have a u n i q u e e x p l a n a t i o n f o r t h i s s t r o n g dependence o f I n p ( l / T ) on d e f o r m a t i o n temperature, i t m i g h t i n d i c a t e , t h a t a t l o w tempera- t u r e s p o i n t d e f e c t s s u r r o u n d i n g t h e d i s l o c a t i o n c o r e l e a d t o a change i n t h e p h y s i c a l p r o p e r t i e s o f d i s - l o c a t i o n s .

3. Local energy spectrum a t edge-type d i s l o c a t i o n s . - The b a s i s o f o u r s t a t i s t i c a l t r e a t m e n t , g i v e n i n c h a p t e r 1, was a s h u f f l e - s e t d i s l o c a t i o n w i t h t h e most s i m p l e c o r e c o n f i g u r a t i o n , and a o n e - e l e c t r o n - a p p r o x i m a t i o n t o d e s c r i b e t h e l o c a l energy spectrum a t d i s l o c a t i o n s . I n t h i s second p a r t we s h a l l d i s c u s s

F i g . 4 : G e o m e t r i c a l c o r e c o n f i g u r a t i o n f o r a 6O3- d i s l o c a t i o n o f t h e s h u f f l e s e t ( a ) and f o r t h e 3O0- p a r t i a l and 9 0 " - p a r t i a l ( b ) .

The r a t i o o f t h e l e n g t h s o f d i s s o c i a t e d ld t o u n d i - s s o c i a t e d segments lu i s ld/lu >> 1. By a s i m p l e e s t i m a t e , Labusch and S c h r B t e r /1/ have e x c l u d e d t h e p o s s i b i 1 i t y t h a t o n l y d i s s o c i a t e d segments a r e e l e c - t r i c a l l y a c t i v e . On t h e o t h e r hand, t h e y expect, t h a t t h e o c c u p a t i o n s t a t i s t i c s f o r two p a r t i a l s , which a r e s e p a r a t e d b y s t a c k i n g f a u l t of w i d t h d, a r e a l m o s t t h e same as f o r a complete d i s l o c a t i o n , if d i s s m a l l compared w i t h t h e e x t e n s i o n X o f t h e s c r e e n i n g charge : d<<h. F o r r o / d 10-I t h e r a t i o V ( r o ) / V ( d ) i s of t h e o r d e r o f two, so t h a t a modi- f i c a t i o n of o c c u o a t i o n s t a t i s t i c s i s expected, when t h e Fermi l e v e l r u n s t h r o u g h s t a t e s of b o t h p a r t i a l s .

C 6 - 5 6 JOURNAL DE PHYSIQUE

3.2.

-

Electrgp=h_gpping_plgn_g-lh_e_-~gre.- The response t i m e o f f r e e charge c a r r i e r s t o a l o c a l charge f l u c -

EE 0

t u a t i o n i s t h e d i e l e c t r i c r e l a x a t i o n t i m e ~~~l =-- 0 0

(ao e l e c t r i c a l c o n d u c t i v i t y ) . I f t h e hopping frequen- c y o f e l e c t r o n s a l o n g t h e d i s l o c a t i o n c o r e i s s m a l l compared t o t h e i n v e r s e d i e l e c t r i c r e l a x a t i o n t i m e , t h e e l e c t r o s t a t i c p o t e n t i a l o f t h e d i s l o c a t i o n may be d e s c r i b e d by a s u p e r p o s i t i o n o f t h e p o t e n t i a l s o f screened p o i n t charges, w h i c h l e a d s t o :

A0 If1

e 2 f i n ( l - e x ~ ( - ~ - ) ) p - E o = E e = - 2nc s

I f t h e h o p p i n g f r e q u e n c y i s l a r g e compared t o t h e i n v e r s e d i e l e c t r i c r e 1 a x a t i o n time, t h e e l e c t r o s t a t i c p o t e n t i a l o f t h e d i s l o c a t i o n i s t h a t o f a l i n e charge

where ro i s t h e r a d i a l w i d t h o f t h e charge d i s t r i b u - t i o n a t t h e d i s l o c a t i o n core.

Scheibe /13/ and i n d e p e n d e n t l y Ouster /14/ have c a r e f u l l y compared b o t h f o r m u l a s w i t h e x p e r i m e n t a l r e s u l t s o f p ( f ) f o r S i w i t h edge-type d i s l o c a t i o n s n e a r f = 0 and have f o u n d t h a t t h e s e d i s l o c a t i o n s a r e a s s o c i a t e d w i t h a l i n e charge i n t h e sense spe- c i f i e d above. As a l o w e r l i m i t f o r t h e w i d t h o f t h e d i s l o c a t i o n band one o b t a i n s

R e c e n t l y o t h e r e x p e r i m e n t a l methods

-

1 i ke s p e c t r a l p h o t o c o n d u c t i v i t y /21/, c a p a c i t y a t S c h o t t k y diodes /22/ and microwave c o n d u c t i v i t y /23/

-

have e s t a b l i - shed t h e e x i s t e n c e o f a dangling-bond-band a t edge- t y p e d i s l o c a t i o n s , whose w i d t h i s o f t h e o r d e r o f 0.2 eV i n Ge and 0.4 eV i n S i /24/.

3.3. -~ntya_bgpd-Cgulgmb-~n_~e_ra_gt_Ign_-a_n_~-h_yb_ri~igg-

t j g n . - F o r t h e f r e e atom t h e i n t r a - a t o m i c Coulomb i n t e r a c t i o n U l e a d s t o an energy s e p a r a t i o n of two d i f f e r e n t charge s t a t e s o f t h e o r d e r 1

-

10 eV. I n S i and Ge some deep p o i n t c e n t e r s r e a l i z e d two d i f f e - r e n t charge s t a t e s w i t h i n an energy range o f a b o u t 0 . 1 eV, w h i c h p o s s i b l y r e s u l t s f r o m a s t r o n g h y b r i - d i z a t i o n o f t h e l o c a l i z e d l e v e l s w i t h t h e s t a t e s o f t h e v a l e n c e and c o n d u c t i o n band, whereby U i s redu- ced t o a< v a l ue Ueff < U.

The i n f l u e n c e o f t h i s i n t e r a c t i o n on t h e o n e - d i - mensional a r r a y o f d a n g l i n g bonds may be b e s t ex- p l a i n e d f o r tw6 l i m i t i n g cases /21/. I f t h e d i r e c t and i n d i r e c t ( v i a conducf;ion and valence band) cou- p l i n g between' d a n g l i n g bonds i s s m a l l , a f i n i t e ener- gy s e p a r a t i o n Ueff i s expected.between t h e s t a t e s o f one e l e c t r o n and o f two e l e c t r o n s o c c u ~ y i n g t h e dangl i n g bond, r e s p e c t i v e l y . Ueff i s t h e i n t r a b o n d

Coulomb i n t e r a c t i o n U, reduced b y c o n t r i b u t i o n s r e - s u l t i n g f r o m h y b r i d i z a t i o n , exchange and r e c o n s t r u c - t i o n . I n t h e p ( f ) - c u r v e s one expects a s t e p o f h e i g h t Ueff n e a r f = 0, as has i n d e e d been f o u n d f o r screw d i s l o c a t i o n , w h i c h d i s s o c i a t e s i n t o two 3 0 ' - ~ a r t i a l s .

I f t h e o v e r l a p between d a n g l i n g bonds i s s u f f i - c i e n t l y s t r o n g , t h e i n f l u e n c e o f Ueff may be d e s c r i - bed b y an a d d i t i o n a l t e r m i n t h e e l e c t r o s t a t i c s h i f t Ep(q) o f t h e dangling-bond-band. T h i s l i m i t has been

L

c a l c u l a t e d b y Usadel and S c h r o t e r /25/, who used an Anderson-type H a m i l t o n i a n and e v a l u a t e d t h e i n t r a b o n d Coulomb i n t e r a c t i o n by i t s v a l u e UHF i n t h e H a r t r e e - Fock-approximation and h y b r i d i z a t i o n e f f e c t s parame- t r i c a l l y b y t h e t r a n s i t i o n r a t e A t o and f r o m t h e va- l e n c e and c o n d u c t i o n band s t a t e s . They compared t h e i r r e s u l t w i t h H a l l d a t a f o r edge-tyoe d i s l o c a t i o n s i n Ge and f o u n d a r a t h e r s m a l l v a l u e f o r A 5 0.3 eV.

T h i s v a l u e i m p l i e s t h a t a s i m p l e d a n g l i n g bond con- c e p t g i v e s a f a i r d e s c r i p t i o n o f t h e l o c a l i z e d s t a t e s a t d i s l o c a t i o n s . The second r e s u l t , v i z . UHF < 4 eV remains somewhat vague due t o t h e u n c e r t a i n t y i n t h e d i s l o c a t i o n d e n s i t y N. I n t o t a l , t h i s d e s c r i p t i o n f i t s r a t h e r w e l l t h e e x p e r i m e n t a l u ( f ) - c u r v e s f o r edge-type d i s l o c a t i o n s , w h i c h d i s s o c i a t e i n t o a 90"- p a r t i a l and a 3 0 " - o a r t i a l o r two 6 0 ' - p a r t i a l s .

4. Conclusions.- The one-electron-a~proximation y i e l d s a c o n s i s t e n t model o f t h e c o r e s t a t e s a t d i s l o c a t i o n s and t h e i r o c c u p a t i o n s t a t i s t i c s . The a n a l y s i s o f H a l l d a t a i n t e r m o f t h e s e s t a t i s t i c s determined b a s i c parameters o f t h e energy s o e c t r a a t edge-type d i s l o - c a t i o n s i n S i and Ge and a t screw d i s l o c a t i o n s i n Ge.

N e u t r a l screw d i s l o c a t i o n s a r e a s s o c i a t e d w i t h one f u l l and one emoty band, s e p a r a t e d b y an energy gao.

N e u t r a l edge-type d i s l o c a t i o n s a r e a s s o c i a t e d w i t h p a r t i a l l y f i l l e d s t a t e s , whose number i s o f t h e o r d e r o f t h e number o f c o r e atoms, and between w h i c h e l e c - t r o n - h o p p i n g occurs. I n agreement w i t h independent i n v e s t i g a t i o n s , w h i c h a p p l y more r e f i n e d methods, we a t t r i b u t e a p a r t i a l l y f i l l e d dangl ing-bond-band t o edge-type d i s l o c a t i o n s .

One may conclude then, t h a t 3 0 ' - p a r t i a l s have t o t o be d e s c r i b e d i n t h e a t o m i c l i m i t , 9 0 ' - p a r t i a l s i n t h e band l i m i t i n t h e sense o u t l i n e d i n c h a p t e r 3.3.

T h i s view i s s u p p o r t e d by an i n s p e c t i o n o f t h e geome- t r i c a l arrangement o f t h e c o r e atoms and t h e i r bon- d i n g t o b u l k atoms ( s e e F i g . 4 b ) , f r o m w h i c h one d e r i - ves a s i g n i f i c a n t l y l a r g e r d i r e c t and i n d i r e c t cou- p l i n g between d a n g l i n g bonds a t 9 0 ' - p a r t i a l s t h a n a t 3 0 " - o a r t i a l s . B u t t h e s e r e a s o n i n g s need s u p p o r t f r o m a t h e o r e t i c a l t r e a t m e n t o f d i r e c t and i n d i r e c t cou- p l i n g between d a n g l i n g bonds.

p o r t o f N i s s S c h e l l h a a s and H. Bock i n p r e p a r i n g t h e manuscript.

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