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THEORETICAL CALCULATIONS OF ELECTRON STATES ASSOCIATED WITH DISLOCATIONS
R. Jones
To cite this version:
R. Jones. THEORETICAL CALCULATIONS OF ELECTRON STATES ASSOCIATED WITH DISLOCATIONS. Journal de Physique Colloques, 1979, 40 (C6), pp.C6-33-C6-38.
�10.1051/jphyscol:1979607�. �jpa-00219023�
THEORETICAL CALCULATIONS OF ELECTRON STATES ASSOCIATED WITH DISLOCATIONS
R. Jones
Department of Physics, University of Exeter, U.K.
Resume.- L'ensemble des mgthodes t h e o r i q u e s u t i l i s e e s p o u r c a l c u l e r l e s @ t a t s e l e c ~ r o n i q u e s a s s o c i e s aux d i s l o c a t i o n s dans l e s m a t g r i a u x ii l i a i s o n s c o v a l e n t e s , e s t passe en revue. Un r6sume de l a s i t u a - t i o n e x p e r i m e n t a l e e s t f a i t q u i i n c l u t c e q u i a pu O t r e d & d u i t p o u r l e s d i s l o c a t i o n s L3 60" dans l e s i l i c i u m , l e germanium e t l ' a r s e n i c de g a l l i u m , e t l e s d i s l o c a t i o n s p a r t i e l l e s dans l e s i l i c i u m . A b s t r a c t . - A r e v i e w i s g i v e n o f t h e t h e o r e t i c a l methods t h a t have been used t o c a l c u l a t e t h e e l e c t r o - n i c s t a t e s a s s o c i a t e d w i t h d i s l o c a t i o n s i n c o v a l e n t l y bonded m a t e r i a l s . A summary o f t h e e x p e r i m e n t a l s i t u a t i o n i s a l s o i n c l u d e d as w e l l as r e s u l t s t h a t have been d e r i v e d f o r t h e 60" d i s l o c a t i o n s i n s i - l i c o n , germanium, g a l l i u m a r s e n i d e and p a r t i a l d i s l o c a t i o n s i n s i l i c o n .
1. S t r u c t u r e and p r o p e r t i e s o f d i s l o c a t i o n s i n cova- l e n t m a t e r i a l s . - H o r n s t r a /1/ has d i s c u s s e d some o f t h e v e r y l a r g e number o f p o s s i b l e models o f d i s l o c a - t i o n s i n e l e m e n t a l semiconductors such as germanium and s i l i c o n . The s i m p l e r ones a r e t h e screw, 60" and t h e edge. Models o f t h e s e have been b u i l t u s i n g t h e r u l e t h a t t h e number o f broken bonds s h o u l d be a minimum. The screw may n o t possess any d a n g l i n g bonds, whereas t h e 60" has a l i n e o f them. The edge d i s l o c a t i o n has a l i n e o f atoms w i t h two d a n g l i n g bonds, b u t i f t h e s e d i f f u s e o u t , a r e c o n s t r u c t i o n can be made which e l i m i n a t e s a1 1 d a n g l i n g bonds.
O t h e r d i s l o c a t i o n s can be b u i l t o u t o f t h e s e t o g e t h e r w i t h j o g s and k i n k s .
Besides t h e s e p e r f e c t o r complete d i s l o c a t i o n s , t h e r e a r e p a r t i a l d i s l o c a t i o n s which b o r d e r a s t a c k - i n g f a u l t . The 60" d i s l o c a t i o n can " d i s s o c i a t e " i n t o 90" and 30" p a r t i a l s
-
c a l l e d t h e s h u f f l e s e t b y H i r t h and L o t h e /2/ on account o f t h e way t h e l a t t e r g l i d e s . The 30" p a r t i a l a l s o has a l i n e o f d a n g l i n g bonds. H i r t h and L o t h e /2/ have p o i n t e d o u t t h a t a second t y p e o f 60" d i s l o c a t i o n , w i t h t h r e e t i m e s as many d a n g l i n g bonds as t h e one d i s c u s s e d by H o r n s t r a , can a l s o d i s s o c i a t e i n t o 90" and 30" p a r t i a l s ( t h e g l i d e s e t ) . The 30" p a r t i a l , i n t h i s case, i s q u i t e u n l i k e t h e c o r r e s p o n d i n g member o f t h e s h u f f l e s e t , a l t h o u g h t h e o t h e r p a r t i a l s a r e i d e n t i c a l . We s h a l l see i n5
4, t h a t i t i s u n l i k e l y t h a t e i t h e r o f t h e s e p a r t i a l s possess d a n g l i n g bonds. Indeed o u r c o n c l u - s i o n i s t h a t a l t h o u g h H o r n s t r a ' s 60° d i s l o c a t i o n i s p r o b a b l y more s t a b l e t h a n t h e o t h e r t y p e , t h e s h u f f l e s e t o f p a r t i a l s a r e l e s s s t a b l e t h a n t h e g l i d e s e t .Which, i f any, o f t h e s e d i s l o c a t i o n s a c t u a l l y e x i s t i n p l a s t i c a l l y deformed germanium and s i l i c o n i s n o t y e t c o m p l e t e l y known. However e l e c t r o n m i c r o s -
copy /3, 4, 5/ has shown t h a t a l a r g e p r o p o r t i o n o f d i s l o c a t i o n s o f a l l o r i e n t a t i o n s a r e d i s s o c i a t e d , a l t h o u g h some s m a l l segments c o n s i s t o f c o n s t r i c t e d r e g i o n s w h i c h may be u n d i s s o c i a t e d . The s t a c k i n g f a u l t i s o f t e n i n t r i n s i c , a l t h o u g h f o r s m a l l e r o r i e n - t a t i o n s , a more s t a b l e e x t r i n s i c f a u l t i s sometimes observed. Nessel and Alexander /6/ observe a g r e a t d i f f e r e n c e i n t h e m o b i l i t i e s o f d i f f e r e n t p a r t i a1 s and conclude t h a t t h e 30" one may b e l o n g t o t h e shuf- f l e s e t .
Perhaps t h e c l e a r e s t e v i d e n c e f o r d a n g l i n g bonds comes from e . s . r . experiments on deformed s i - l i c o n (annealed a t temperatures below 700°C) /7, 8, 9/. The s i g n a l i s a p p a r e n t l y due t o groups o f i n - t e r a c t i n g l o c a l i s e d s p i n s . H a l l e f f e c t s t u d i e s show t h a t t h e d i s l o c a t i o n g i v e s r i s e t o amphoteric s t a t e s . I n germanium
/ l o ,
11/ a p a r t i a l l y f i l l e d band i s f o u n d n e a r t h e v a l e n c e edge Ev w i t h t h e f e r m i l e v e l , EF, a t EV+
0.09eV. C a v a l l i n i e t a l . /12/ f i n d t h a t t h i s band drops as t h e d e f o r m a t i o n temperature i n - creases. d e l Penino and Mantovani /13/ show t h a t t h e w i d t h o f t h e l o w e r band i s about 0.18eV i n r o u g h agreement w i t h schaumburg and Willmann /14/. Some o t h e r experiments show t h a t t h e r e a r e two bands a s s o c i a t e d w i t h t h e d i s l o c a t i o n s /15, 16, 17, 18/ : one c l o s e t o t h e v a l e n c e edge and t h e o t h e r i n t h e upper h a l f o f t h e gap. I n s i l i c o n , G r a z h u l i s e t a l . /19, 20/ show t h e r e a r e two l e v e l s a s s o c i a t e d w i t h t h e d i s l o c a t i o n , one a t EV+
042eV, and t h e o t h e r a t Ec-
0.43eV. E a r l i e r H a l l e f f e c t s t u d i e s analysed by S c h r o t e r and Labusch/ l o /
f o u n d one band whose f e r m i l e v e l o c c u r s a t EF = EV + 0.3eV. O t h e r workers /21, 22, 23/ f i n d d i f f e r e n t l e v e l s which may r e f l e c t d i f f e r e n t d e f o r m a t i o n c o n d i t i o n s o r d i f f e r e n t e f f e c - t i v e l e v e l s .Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979607
C6-34 JOURNAL DE PHYSIQUE
We s h a l l see i n
5
2 t h a t H o r n s t r a ' s 60°, and p r o b a b l y t h e 30" s h u f f l e d i s l o c a t i o n , a r e expected t o l e a d t o two bands i n t h e gap; one p a r t i a l l y f i l l e d c l o s e t o EV and a s s o c i a t e d w i t h d a n g l i n g bonds and t h e o t h e r , c l o s e r t o Ec, due t o s t r e t c h e d bonds i n t h e d i s l o c a t i o n c o r e . The l a t t e r band i s s e n s i t i v e t o t h e d i s o r d e r i n t h e core. Thus many o f t h e above experiments can be q u a l i t a t i v e l y unders- tood. However, as has a l r e a d y been s t r e s s e d , these d i s l o c a t i o n s a r e l e s s s t a b l e t h a n t h o s e o f t h e g l i d e s e t , which have o n l y empty bands c l o s e t o Ec. I t may be t h a t t h e observed d a n g l i n g bonds a r e a s s o c i a t e d w i t h k i n k s and j o g s . T h i s would e x p l a i n t h e beha- v i o u r o f s i l i c o n upon a n n e a l i n g a t a h i g h temperatu- r e /8/ and t h e o b s e r v a t i o n o f C a v a l l i n i e t a l . /12/as w e l l as t h e changes observed i n T.E.M. s t u d i e s /24/.
Less i s known a b o u t d i s l o c a t i o n s i n c r y s t a l s o f t h e Z i n c - b l e n d e t y p e . There a r e many more p o s s i - b i l i t i e s f o r t h e c o r e s t r u c t u r e /25/ and so f a r o n l y t h e s i m p l e s t t y p e o f 60" models have been analysed.
I t w i l l be shown i n
5
5, t h a t i n g a l l i u m a r s e n i d e , we may e x p e c t d a n g l i n g bonds on a r s e n i c atoms t o l e a d t o a p a r t i a l l y f i l l e d band f o u n d c l o s e t o EV.Furthermore, t h e r e i s a n e t t r a n s f e r o f e l e c t r o n s t o t h e s e a r s e n i c atoms f r o m t h e o u t e r r e g i o n s o f t h e c o r e . On t h e o t h e r hand, g a l l i u m atoms, which have d a n g l i n g bonds, have empty s t a t e s i n t h e m i d d l e o f t h e gap and t h e s e atoms have l e s s e l e c t r o n i c charge t h a n g a l l i u m atoms i n r e g i o n s o f good c r y s t a l . The c e n t r e o f t h e c o r e must t h e n be p o s i t i v e l y charged.
These e l e c t r i c a l f i e l d s a r e bound t o p l a y an impor- t a n t r o l e i n t h e i n t e r p r e t a t i o n o f experiments.
Stoneham /26/ has suggested t h a t t h e s e f i e l d s would quench t h e luminescent c e n t r e s /27/
-
an e f f e c t which a p p a r e n t l y has a1 r e a d y been seen /28/.2 . C a l c u l a t i o n s o f t h e E l e c t r o n i c S t a t e s . - Shockley /29/ f i r s t p o i n t e d o u t t h a t d a n g l i n g bonds i n s i l i - con and germanium were expected t o l e a d t o l e v e l s l y i n g i n t h e gap. There i s a s i m p l e p r o o f o f t h i s /3/, based on t h e min-max theorem /31/ and a p p l i e d t o t h e Weaire h a m i l t o n i a n d e s c r i b i n g a c o v a l e n t semi- c o n d u c t o r /32/. I n t h i s h a m i l t o n i a n , t h e v a l e n c e bands a r i s e f r o m bonding o r b i t a l s broadened by i n - t e r a c t i o n between bonds on t h e same atom. S i m i l a r l y t h e c o n d u c t i o n band i s made up o f broadened a n t i - bonding o r b i t a l s . A d a n g l i n g bond must l e a d t o p a r - t i a l l y occupied s t a t e s , l y i n g i n t h e m i d d l e o f t h e gap, i f t h e i n t e r a c t i o n w i t h t h e r e m a i n i n g bonds on t h e same atom i s weak enough. I n a d d i t i o n , weakening a bond forms a s t a t e l y i n g near t h e c o n d u c t i o n band.
Brown /33/ has a l s o argued t h a t edge d i s l o c a t i o n s l e a d t o two bands i n t h e gap as w e l l as s c a t t e r i n g resonances.
The f i r s t d e t a i l e d arguments used a v a r i a t i o n a l method a p p l i e d t o an e f f e c t i v e mass h a m i l t o n i a n w i t h a s t r a i n p o t e n t i a l o f t h e f o r m A s i n ( @ ) / r . These c a l c u l a t i o n s /34, 35/ show t h a t t h i s long-ranged s t r a i n p o t e n t i a l produces s t a t e s l y i n g w i t h i n 0.05eV o f t h e c o n d u c t i o n band. S i m i l a r c o n c l u s i o n s have been reached b y workers employing more s o p h i s t i c a t e d h a m i l t o n i a n s /36, 37/. Deeper l e v e l s a r i s e f r o m t h e more r a p i d l y v a r y i n g p o t e n t i a l a s s o c i a t e d w i t h t h e c o r e . Here t h e e f f e c t i v e mass a p p r o x i m a t i o n i s sim- p l y n o t v a l i d .
There have been t h r e e approaches t o t h i s p r o - blem. A l l have used s e m i - e m p i r i c a l t i g h t - b i n d i n g t h e o r y t o d e s c r i b e t h e c r y s t a l l i n e s t a t e . T h i s by i t s e l f i s an assumption s i n c e t h e l a t t e r i s b e s t t r e a t e d b y p s e u d o p o t e n t i a l t h e o r y . N e v e r t h e l e s s t h e r e i s a growing consensus i n s p i r e d by develop- ments i n s u r f a c e s t a t e c a l c u l a t i o n s /38, 39/, t h a t t i g h t - b i n d i n g c a l c u l a t i o n s , u s i n g i n t e r a c t i o n s which e x t e n d o u t t o t h e second s h e l l , a r e n o t i n a p p r o p r i a t e f o r t h e c o v a l e n t l y bonded m a t e r i a l s . A more f o r m a l p r o o f i s found i n Dean and Jones /40/ and r e s t s on t h e r e s u l t t h a t t h e i n t e r a c t i o n s between two d i s t a n t atoms have e i g e n f u n c t i ons which a r e o r t h o g o n a l , o r n e a r l y so, t o t h e s t a t e s i n t h e v a l e n c e band and low- l y i n g c o n d u c t i o n bands.
Labusch and S c h r i j t e r /41/ s e t up a symmetry adapted b a s i s s e t o f a t o m i c l i k e f u n c t i o n s . By con- c e n t r a t i n g on t h e t o p o l o g y of t h e screw and 60" d i s l o - c a t i o n , t h e y showed t h a t t h e f o r m e r possesses two bands i n t h e g a p - o n e f u l l and t h e o t h e r empty
-
w h i l s t t h e l a t t e r l e a d s t o a p a r t i a l l y f i l l e d band.
I n some r e s p e c t s , t h e s e r e s u l t s a r e s i m i l a r t o t h e c o n c l u s i o n s o b t a i n e d f r o m t h e Weaire h a m i l t o n i a n mentioned e a r l i e r . However t h e p o s i t i o n o f these bands can o n l y be determined u s i n g r e a l i s t i c h a m i l - t o n i a n s . We now r e v i e w t h e s e i n v e s t i g a t i o n s . 3. The 60" d i s l o c a t i o n i n s i l i c o n and germanium.- A1 s t r u p and Mark1 und /42/ have a p p l i e d e m p i r i c a l extended-Huckel t h e o r y , u s i n g parameters determined b y A l s t r u p /43/, t o a c l u s t e r o f 36 s i l i c o n atoms
( p e r i o d i c a l l y c o n t i n u e d on a p a i r o f f a c e s ) c o n t a i - n i n g a 60" d i s l o c a t i o n . Many l e v e l s were f o u n d i n t h e gap; many due t o t h e s u r f a c e d a n g l i n g bonds. The h i g h e s t one, however, had an e i g e n s t a t e which was s u b s t a n t i a l l y peaked around t h e c e n t r a l d a n g l i n g bond atom, and has c o n s i d e r a b l e s-p o r b i t a l charac- t e r ; t h e p - o r b i t a l p o i n t i n g i n t h e d i r e c t i o n o f t h i s
m i t y o f t h e c l u s t e r ' s s u r f a c e t o t h e d i s l o c a t i o n c o r e ( 3 a t o m i c s h e l l s ) . The o v e r l a p o f t h e s u r f a c e s t a t e s w i t h t h e core, f o r c e s t h e d i s l o c a t i o n s t a t e s upwards i n t o t h e gap.
Am improved c a l c u l a t i o n was made by ~ d r k l u n d /44/. Here two o p p o s i t e 60" d i s l o c a t i o n s were em- bedded i n 56 ( a l s o 88) atom u n i t c e l l s . The sepa- r a t i o n between t h e d i s l o c a t i o n c o r e s i s about 8 (12) atomic s h e l l s . Because o f t h e imposed p e r i o d i c boundary c o n d i t i o n s , t h e r e a r e no s u r f a c e d a n g l i n g bonds t o m i x w i t h t h e s t a t e s o f i n t e r e s t . The h a m i l - t o n i a n was c o n s t r u c t e d by u s i n g t h e S l a t e r - K o s t e r /45/ parameters taken f r o m Pandey and P h i l i p s /38/.
Changes i n bond l e n g t h s , 6R, c o u l d be t a k e n i n t o account by s c a l i n g t h e s e parameters by exp ( 6 6 R ) where 6 has s e v e r a l v a l u e s . The a t o m i c p o s i t i o n s were deduced f r o m l i n e a r e l a s t i c i t y t h e o r y . F o r each k, value, t h e quantum number d e s c r i b i n g t h e p e r i o d i - c i t y i n t h e d i r e c t i o n p a r a l l e l t o t h e d i s l o c a t i o n a x i s , t h e r e a r e p a i r s o f c l o s e l y spaced l e v e l s . These a r i s e f r o m t h e d i s l o c a t i o n i n t e r a c t i o n and were s u f f i c i e n t l y s m a l l t o be s u r e t h a t t h e l o w e s t 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 has w i d t h 0.2eV and t h e f e r m i l e v e l o c c u r r e d a t EV
+
0.28eV. These q u a n t i t i e s a r e i n good agreement w i t h t h e experimen- t a l values quoted i n5
1. The s t a t e s o f t h i s band a r e made up o f s-p o r b i t a l s as expected. A second band was found n e a r Ec. I t s p o s i t i o n depended on t h e v a l u e o f 6 and i t s s t a t e v e c t o r was s t r o n g l y l o c a - l i s e d about a d i l a t e d bond, and, as we have seen e a r l i e r , i s a consequence o f bond weakening.Another approach has been developed by t h e p r e s e n t a u t h o r /46/ and uses t h e c o n t i n u e d f r a c t i o n r e p r e - s e n t a t i o n o f t h e Green's f u n c t i o n /47/. L e t @l ,a (T) denote a complete orthonormal s e t o f atomic o r b i t a l s , 1 = s, px, py, p,, c e n t r e d on atom a . The method c a l c u l a t e s t h e l o c a l charge d e n s i t y pl ,a (E), where
P1 , a ( ~ ) =
4
6 ( E - E ~ )1 1 ~ 7
(r)ml , a ( r ) d r IEi and Q i ( r ) denote t h e e x a c t energy v a l u e s and wave f u n c t i o n s r e s p e c t i v e l y . The l o c a l and t o t a l d e n s i t y o f s t a t e s a r e
C C
1'1 ,a(E)
'
1 ,a pl ,a(E) r e s p e c t i v e l y . The t o t a l one- . Ce l e c t r o n energy i s
d i f f e r e n c e between t h e t o t a l o n e - e l e c t r o n energy o f an atom w i t h a d e f e c t and a normal atom can e a s i l y be found f r o m pl
.
We s h a l l use t h i s i n5
4 t o,a
draw c o n c l u s i o n s about t h e r e l a t i v e s t a b i l i t y o f d i s l o c a t i o n s . The l o c a l charge d e n s i t y , pl 3 0. (E) i s
1/E-al - b : / ~ - a ~ - b t / ~ -
. . . .
Where t h e c o e f f i c i e n t s ai, bi a r e e a s i l y found f r o m a r e c u r s i o n f o r m u l a /47/. T y p i c a l l y t w e n t y c o e f f i - c i e n t s a r e e v a l u a t e d and t h e l a s t two r e p e a t e d i n d e - f i n i t e l y . The energy E i s made complex t o smooth t h e p r o f i l e s o f pl ,a(E).
A c l u s t e r o f -700 atoms (-1500 atoms o f germa- nium) a r e b u i l t c e n t r e d on t h e d i s l o c a t i o n c o r e . Sur- f a c e s t a t e s now have n e g l i g i b l e i n f l u e n c e a t t h e c l u s t e r ' s c e n t r e and l e a v e t h e l o c a l d e n s i t y o f s t a - t e s t h e r e r e l a t i v e l y u n a f f e c t e d ( b u t see below). I n t h e case o f c r y s t a l l i n e s i l i c o n /46/ a r e a s o n a b l e d e n s i t y o f s t a t e s i s o b t a i n e d , except t h a t t h e band edges Evl, (Ecl) o c c u r 0.2eV below (above) t h e theo- r e t i c a l v a l u e s EV(Ec). Thus t h e band gap i s 0.4eV t o o l a r g e . Why a r e t h e s e v a l u e s o f Evl, Ecl so d i f f e r e n t f r o m t h e t r u e v a l u e s ? We t h i n k t h e s u r - f a c e d a n g l i n g bonds i n t e r a c t w i t h t h e band s t a t e s and r e p e l them, e n l a r g i n g t h e gap. F o r example, u s i n g a c l u s t e r o f 700 atoms o f germanium, we found an e x t r e m l y wide gap i n w h i c h a s m a l l peak o c c u r r e d
-
d o u b t l e s s due t o a s u r f a c e s t a t e . On i n c r e a s i n g t h e c l u s t e r s i z e t o -1500 atoms, t h i s peak disappeared and t h e gap narrowed c o n s i d e r a b l y :
Evl = Ev
-
0.3eV and Ecl = Ec+
0.2eV /30/.F o r t h e 60' d i s l o c a t i o n , t h e l o c a l d e n s i t y o f s t a t e s on a s i l i c o n atom w i t h a d a n g l i n g bond gave a l a r g e band c e n t r e d a t E,,
+
0.05eV and of w i d t h 0.4eV. T h i s band was h a l f - f u l l . Now, we be1 i e v e i f Evl was c o i n c i d e n t w i t h Ev, t h e n t h e d a n g l i n g bond s t a t e s would b e r e p u l s e d upwards and so i t i s b e t t e r t o measure i t s e n e r g i e s f r o m E T h i s means t h a t t h ev1'
v a l u e s quoted i n Jones /30,46/ need some a d j u s t m e n t The f e r m i l e v e l i s now f o u n d a t E
+
0.25eV and t h ev 1
band extends upwards i n t o t h e gap u n t i l Evl
+
0.45 eV.
These v a l u e s a r e i n c l o s e r agreement w i t h Marklund /44/. We observe t h e r e p u l s i o n between t h e band l e v e l s and t h e d i s l o c a t i o n l e v e l s s i n c e t h e c o n d u c t i o n band edge i s pushed upwards by 0.25eV i n t h e r e g i o n o f t h e d i s l o c a t i o n core. T h i s e f f e c t would m a n i f e s t i t s e l f b y t h e appearance o f a n t i - resonances i n t h e b o t t o m o f t h e c o n d u c t i o n band and s u p p o r t s t h e p r e d i c t i o n o f Brown /33/. The s t a t e s o f t h e d i s l o c a L i o n band a r e made up o f s-p o r b i t a l s w i t h t h e l a t t e r p o i n t i n g i n t h e d i r e c t i o n o f t h e d a n g l i n g bond. The e f f e c t o f r e l a x a t i o n , o r a l t e r i n g t h e s t r e n g t h o f t h e S l a t e r - K o s t e r parameters f o r atoms i n t h e core, i s n o t i m p o r t a n t f o r t h e d a n g l i n g bond s t a t e s . I t i s i m p o r t a n t t o n o t e t h a t t h e p o s i -
C6-36 JOURNAL DE PHYSIQUE
i o n s o f s t a t e s deep i n t h e v a l e n c e band a l s o undergo changes; t h i s has i m p l i c a t i o n s f o r t h e t o t a l energy ( s e e
5
4 . ) .F o r germanium, t h e p a r t i a l l y f i l l e d d a n g l i n g bond band has a f e r m i - l e v e l a t Evl
+
0.2eV and ex- tends t o Evl+
0.4eV. I t o v e r l a p s t h e v a l e n c e band.Changing the7atom cl t o one which has a d i l a t e d bond, and d e c r e a s i n g t h e s t r e n g t h o f t h e bond b y 20%, draws a band o u t o f t h e c o n d u c t i o n s t a t e s . I t s po- s i t i o n i s v e r y s e n s i t i v e t o t h e bond weakening f a c - t o r . T h i s s t a t e i s a g a i n an s-p h y b r i d i s e d o r b i t a l w i t h c o n s i d e r a b l e p - o r b i t a l p o i n t i n g i n t h e Burgers v e c t o r d i r e c t i o n I t o f f e r s an e x p l a n a t i o n o f t h e o p t i c a l p o l a r i s e d a b s o r p t i o n experiments o f B a r t h e t a l . /16, 17/, ( s e e a l s o Kamienieki /48/) as w e l l as t h e second band seen b y G r a z h u l i s e t a1
.
/19, 20/.4. P a r t i a l D i s l o c a t i o n s i n S i l i c o n . - R e c e n t l y , b o t h methods have been a p p l i e d t o a s t u d y o f t h e g l i d e p a r t i a l s i n s i l i c o n , /49, 50/ and came t o s i m i l a r c o n c l u s i o n s . F i r s t we comment on t h e t o t a l one- e l e c t r o n e n e r g i e s d i s c u s s e d i n § 2. An atom w i t h a d a n g l i n g bond i n t h e 60" d i s l o c a t i o n , has 2.5eV more energy t h a n an atom i n a r e g i o n o f good c r y s t a l . T h i s demonstrates t h a t t h e d a n g l i n g bond c e r t a i n l y a f f e c t s s t a t e s deeper i n t h e v a l e n c e band i .e. t h e o t h e r e l e c t r o n s on t h e same atom. T h i s f i g u r e i s comparable t o t h e bond energy f o u n d by Manca /51/
and t h e a c t i v a t i o n energy f o r d i s l o c a t i o n m o t i o n /24/. I t a l s o suggests t h a t t h e o t h e r s o r t o f 60"
d i s l o c a t i o n , w h i c h has t h r e e t i m e s as many d a n g l i n g bonds, i s much l e s s s t a b l e .
The r e l a t i v e t o t a l o n e - e l e c t r o n energy f o r an atom on an i n t r i n s i c s t a c k i n g f a u l t i s 0.018eV, w h i c h l e a d s t o a s t a c k i n g f a u l t energy o f 45mJ/m2 /5/. S a t i s f i e d t h a t we g e t reasonable v a l u e s f o r t h e s e cases, we t u r n now t o t h e g l i d e s e t o f p a r - t i a l s .
Models o f t h e 90" p a r t i a l /50/ show t h a t a d o u b l e c h a i n o f o f f s e t , n e a r l y p a r a l l e l , d a n g l i n g bonds l i e a l o n g t h e d i s l o c a t i o n a x i s . These l e a d /49/ t o two o v e r l a p p i n g bands c e n t r e d a t Evl- O.leV and Ev,
+
0.6eV w i t h EF = Evl+ 0.2eV. The energy d i f f e r e n c e between an atom w i t h a d a n g l i n g bond and a normal atom i s 2.13eV. A s l i g h t r e l a t i v e d i s p l a - cement o f t h e c h a i n s b r i n g s t h e d a n g l i n g bonds i n t o s t r o n g e r mutual i n t e r a c t i o n . T h i s s p l i t s t h e two bands f u r t h e r , l o w e r i n g EF and pushing t h e empty band f u r t h e r up i n t o t h e gap. The r e l a t i v e t o t a l o n e - e l e c t r o n energy i s now o n l y 0.13eV-
a r e d u c t i o n o f 2eV. To t h i s energy must be added t h e coulombic i n t e r a c t i o n between t h e e l e c t r o n s ( w h i c h i s a b o u tt h e same as i n a normal bond) and t h e e x t r a i n t e r a c - t i o n between t h e a t o m i c c o r e s caused by t h e d i s t o r - t i o n . A r o u g h e s t i m a t e based on t h e bond bending and s t r e t c h i n g e n e r g i e s f o r bonds,between normal atoms /52/, gave 0.3eV p e r atom. Thus t h e r e appears t o be a c o n s i d e r a b l e energy t o be g a i n e d t h r o u g h t h i s d i s - t o r t i o n .
The 30" p a r t i a l /50/ has a l i n e o f d a n g l i n g bonds n e a r l y para1 l e l t o t h e a x i s . These g i v e r i s e t o a band c e n t r e d a t Evl
+
0.6eV and each d a n g l i n g bond atom has a r e l a t i v e t o t a l o n e - e l e c t r o n energy o f 1.96eV. C a r k l u n d /50/ has suggested t h a t a favou- r a b l e r e c o n s t r u c t i o n , i n t h i s case, i n v o l v e s t h e r o t a t i o n o f e v e r y odd atom w i t h a d a n g l i n g bond. T h i s i n v o l v e s 1 i t t l e permanent s t r a i n , b u t couples t h e d a n g l i n g bonds t o g e t h e r i n p a i r s . An a n a l y s i s /49/shows t h a t t h i s s p l i t s t h e d i s l o c a t i o n band i n t o a f i l l e d one l y i n g c l o s e t o Ev and an empty one i n t h e upper h a l f o f t h e gap. Once a g a i n t h e r e seems t o be c o n s i d e r a b l e energy s a v i n g t h r o u g h t h i s r e c o n s t r u c - t i o n .
Thus i t appears t h a t t h e g l i d e s e t o f p a r t i a l s have no d a n g l i n g bonds, o n l y empty bands l y i n g i n t h e upper p a r t o f t h e gap.
5. D i s l o c a t i o n s i n G a l l i u m Arsenide.- P o s s i b l e d i s - l o c a t i o n s i n c r y s t a l s o f t h e z i n c b l e n d e s t r u c t u r e s a r e more numerous owing t o t h e two t y p e s o f atom p r e s e n t . Only t h e s i m p l e s t models have been analysed so f a r : namely t h e 60" d i s l o c a t i o n s analogous t o H o r n s t r a ' s . There a r e two s o r t s /25/ owing t o t h e p o s s i b i l i t y t h a t a l i n e o f d a n g l i n g bonds may be occupied by c a t i o n s o r anions. Jones /53/ and i j b e r g /54/ ( u s i n g t h e method o f Marklund /44/) have o b t a i n e d s i m i l a r r e s u l t s c o n c e r n i n g t h e e l e c t r o n i c s t r u c t u r e . B o t h used S l a t e r - K o s t e r parameters d e r i - ved b y Lowther /55/ which e x t e n d o n l y t o n e a r e s t neighbours. These parameters g i v e a band s t r u c t u r e w h i c h i s n o t c o m p l e t e l y s a t i s f a c t o r y because t h e c o n d u c t i o n band i s t o o f l a t . F o r t h e s e parameters, t h e s u r f a c e d a n g l i n g bonds have l i t t l e e f f e c t and Evl i s v e r y c l o s e t o Ev.
The a r s e n i c d a n g l i n g bonds g i v e a p a r t i a l l y f i l l e d band c e n t r e d around Ev and t r a i l i n g i n t o t h e gap t o Ev
+
0.25eV. R e l a x a t i o n o f t h e m a t r i x e l e - ments produces a band below Ec due t o t h e d i l a t e d bonds. The g a l l i u m atoms w i t h d a n g l i n g bonds p r o - duce an empty band l y i n g c l o s e t o t h e c e n t r e o f t h e gap, a t about Ev+
0.7eV. The s t a t e s o f t h i s band a r e m a i n l y composed o f s - o r b i t a l s on t h e g a l l i u m atom w i t h a s m a l l amount o f p - h y b r i d i s a t i o n .The c o n t i n u e d f r a c t i o n method a l s o i n f o r m s us
o f t h e charge accumulated on each atom. Thus i n t h e c r y s t a l l i n e case, each a r s e n i c atom r e c e i v e s about O.le f r o m t h e g a l l i u m atoms. The a r s e n i c d a n g l i n g bond atom, however, r e c e i v e s 0.4e ( t a k i n g EF=Ev)
,
whereas t h e g a l l i u m atoms w i t h d a n g l i n g bonds h o l d -0.5e l e s s t h a n g a l l i u m atoms i n a r e g i o n o f good c r y s t a l . I t i s n o t y e t known f r o m which atoms these charge t r a n s f e r s m a i n l y t a k e p l a c e . Hence t h e d i s l o - c a t i o n s i n g a l l i u m a r s e n i d e have h i g h l y charged c o r e s even i n t h e undoped case, and t h i s i s bound t o i n f l u e n c e t h e e l e c t r o n i c p r o p e r t i e s o f t h e s e d i s l o - c a t i o n s t o a c o n s i d e r a b l e e x t e n t .
6. Conclusions.- We have seen t h a t t h e d i s l o c a t i o n s t r u c t u r e a f f e c t s and i s a f f e c t e d by t h e e l e c t r o n i c s t r u c t u r e . Perhaps a f u l l y s e l f - c o n s i s t e n t c a l c u l a - t i o n i s necessary t o understand f u l l y t h e s e two as- p e c t s . However t h i s appears t o b e beyond o u r r e s o u r - ces a t t h e p r e s e n t t i m e . It i s w o r t h s t r e s s i n g t h a t many o f t h e p r o p e r t i e s o f d a n g l i n g bonds seem v e r y l o c a l i s e d and do n o t depend upon h a v i n g a l i n e o f them. F o r example, Joannopoulos and Cohen /56/ ob- t a i n e d v e r y s i m i l a r r e s u l t s f o r s u r f a c e s t a t e s i n g a l l i u m a r s e n i d e , even though t h e s t r u c t u r e s a r e v e r y d i f f e r e n t . T h i s i s one o f t h e reasons we e x p e c t d a n g l i n g bonds i n j o g s and k i n k s t o have s i m i l a r p r o p e r t i e s t o t h o s e i n t h e 60' d i s l o c a t i o n . On t h e o t h e r hand t h e p o s s i b i l i t y o f p a i r i n g o f bonds seems t o i n v i t e a r e l a x a t i o n o f t h e s t r u c t u r e w i t h conse- q u e n t e l i m i n a t i o n o f t h e u n p a i r e d e l e c t r o n s . We have a l r e a d y argued t h a t t h i s occurs f o r t h e g l i d e s e t o f p a r t i a l s . T h i s r a i s e s t h e q u e s t i o n o f t h e l o c a t i o n o f t h e d a n g l i n g bonds, seen i n experiments, and i s one w h i c h we cannot y e t answer w i t h c o n f i d e n c e .
Acknowledgement.- The a u t h o r g r a t e f u l l y acknowledges u s e f u l c o n v e r s a t i o n s w i t h A. Claesson, S. Marklund, S. Winter, S.
berg,
A.M. Stoneham and P.B. H i r s c h .References
/1/ H o r n s t r a , J., J. Phys. Chem. S o l i d s - 5 (1958) 129.
/2/ H i r t h , J.P., L o t h e , J., Theory o f D i s l o c a t i o n s , (McGraw H i l l , New York) 1968.
/3/ Ray, I.L.F., Cockayne, D.J.H., Proc. R. Soc.
A= (1971) 534.
/4/ HaUsserman, F., Schaumburg, H., P h i l o s . Mag.
z,
(1973) 745.
/5/ Gbmez, A., Cockayne, D.J.H., H i r s c h , P.B., V i t e k , V., P h i l . Mag.
31
(1975) 105./6/ Wessel, K., Alexander, H., P h i l o s . Mag.
2
(1977) 1523./8/ G r a z h u l i s , V.A., Yu.A Osip'yan, Sov. Phys. JETP 33 (1971) 623.
-
/9/ Alexander, H., Labusch, R., Sander, Y., S o l i d S t a t e Comun.
3
(1965) 357./ l o /
S c h r o t e r , W., Labusch, R., Phys. S t a t u s S o l i d i b36 (1969) 539. -/11/ Labusch, R., S c h e t t l e r ,
U.,
Phys. S t a t u s S o l i d i a9 (1972) 455.-
/12/ C a v a l l i n i , A., Gondi, P., C a s t a l d i n a , A., Phys.
S t a t u s S o l i d i
-
a43 (1977) K205./13/ d e l Penino, U., Mantovani, S., Phys. S t a t u s S o l i d i
-
a38 (1976) 109./14/ Schaumburg, H., Willman, F., Phys. S t a t u s S o l i d i a34 (1976) K173.
-
/15/ !4eber, H.R., Phys. S t a t u s S o l i d i - a25 (1974) 445.
/16/ B a r t h Y., GUth, W., Phys. S t a t u s S o l i d i - 38 (1970) K141.
/17/ B a r t h , W., E l s l s s e r , K., Phys. S t a t u s S o l i d i b48 (1971) K147.
-
/18/ J a s t r z e k s h a , M., F i g i e l s k i , T., Phys. S t a t u s S o l i d i
32
(1969) 791./19/ G r a z h u l i s , V.A., Kveder, V.V., Yu'Mukina, V., Phys. S t a t u s S o l i d i
a44
(1977) 107./20/ G r a z h u l i s , V.A., Kveder, V.V., Yu'Mukina, V., Phys. S t a t u s S o l i d i
a43
(1977) 407./21/ Mantovani, S., d e l Penino, U., Phys. S t a t u s S o l i d i a30 (1975) 747. -
/22/ Neubert, D., Kos, H.J., J . Appl. Phys. (Japan) Suppl. 2 (1974) 401.
/23/ Drozdov, N.A., P a t r i n , A.A., Tkachev, V.D., JETP. L e t t .
3
(1976) 597./24/ Alexander, H., Haasen, P., S o l i d S t a t e Phys. 22 E d i t e d by H. Ehrenreich,, F. S e i t z , D. T u r n b u n (Academic Press, New Y o r k ) , 1968 p65 - 67.
/25/ H o l t , D.B., J. Phys. Chem. S o l i d s
23
(1962) 1353./26/ Stoneham, A.M., ( p r i v a t e communication).
/27/ Tasker, P.W., Stoneham, A.M., J . Phys. C10
(1977) 5131. -
/28/ Boehm, K., 1978 Phys. S t a t u s S o l i d i , t o be p u b l i s h e d .
/29/ Shockley, W., Phys. Rev.
91
(1953) 229./30/ Jones, R., P h i l o s . Mag.
2
(1977) 677./31/ W i l kinson, J .H., "The A l g e b r a i c Eigenval ue Problem" (Clarendon Press, Oxford) 1965.
/32/ Weaire, D., Phys. Rev. L e t t .
26
(1971) 1841 /33/ Brown, R.A., Phys. Rev.156
(1967) 889./34/ Landauer, R., Phys. Rev. - 94 (1954) 1386.
/35/ C e l l i , V., Gold A., Thomson, R., Phys. Rev.
L e t t .
8
(1962) 96./36/ Claesson, A., Phys. S t a t u s S o l i d i - b61 (1974) 599.
/37/ Winter, S., Phys. S t a t u s S o l i d i - b79 (1977) 637.
/38/ Pandey, K.C., P h i l i p s , J.C., Phys. Rev. B13
(1976) 750. -
/39/ Chadi, D.J., Cohen, M.L., Phys. S t a t u s S o l i d i b68 (1975) 405.
-
/40/ Dean, C., Jones, R., J . Physique C o l l o q . (1978) 3415.
/7/ G r a z h u l i s , V.A., Yu A Osip'yan, Sov. Phys. JETP 31 (1970) 677.
-
JOURNAL DE PHYSIQUE
/41/ Labusch, R . , S c h r o t e r , W . , " D i s l o c a t i o n Theory"
e d . by F. R . N . Nabarro (North Holland) 1976.
/42/ A l t r u p , I . , Marklund, S . , Phys. S t a t u s S o l i d i b80 (1977) 301.
-
/43/ A l s t r u p , I . , S u r f . S c i .
g
(1970) 335./44/ Marklund, S . , Phys. S t a t u s S o l i d i - b85 (1978) 673.
/45/ S l a t e r , J . C . , K o s t e r , G.F., Phys. Rev.
2
(1954) 1498.
/46/ J o n e s , R . , P h i l o s . Mag.
35
(1977) 5 7 ./47/ Haydock, R . , Heine, V . , K e l l y , M.J., J . Phys. C5 (1975) 2845.
/48/ Kamienieki, E . , J . Phys. - C9 (1976) 1211.
/49/ J o n e s , R . , ( u n p u b l i s h e d ) .
/50/ Marklund, S . , Phys. S t a t u s S o l i d i , t o be pu- b l i s h e d .
/51/ Nanca, P . J . , Phys.Chem. S o l i d s
20
(1969) 268./52/ K e a t i n g , P.N., Phys. Rev.
145
(1966) 637./53/ J o n e s , R . , P h i l o s . Flag. ( t o be p u b l i s h e d ) . /54/ Oberg, S . , Phys. S t a t u s S o l i d i ( t o be p u b l i s h e d ) . /55/ Lowther, J . E . , Phys. Rev. $J (1977) 3928.
/56/ Joannopoulos, J . D . , Cohen, M.L., Phys. Rev.
(1974) 5075.
