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STRUCTURAL MODELLING OF THE LIQUID STATE OF TETRAHEDRALLY BONDED
SEMICONDUCTORS
M. Popescu
To cite this version:
M. Popescu. STRUCTURAL MODELLING OF THE LIQUID STATE OF TETRAHEDRALLY BONDED SEMICONDUCTORS. Journal de Physique Colloques, 1981, 42 (C4), pp.C5-1043-C5-1046.
�10.1051/jphyscol:19814228�. �jpa-00220858�
JOURNAL DE PHYSIQUE
CoZZoque C4, suppZ6ment au nOIO, Tome 42, octobre 1981 page C4-1043
STRUCTURAL MODELLING OF THE L I Q U I D STATE OF TETRAHEDRALLY BONDEC SEMICONDUCTORS
M. Popescu
I n s t i t u t e o f Physics and TechnoZogy of Materials, Bucharest, M $ ~ @ Z ~ , P.O.
Box MG-7, Romania.
A b s t r a c t . - The a t o m i c - s c a l e s t r u c t u r e o f l i q u i d germanium and s i l i c o n was modelled. A model w i t h s t r u c t u r e d f i r s t a t o m i c c o o r d i n a t i o n was proposed. The first s h e l l c o n s i s t s o f two s u b s h e l l s : t h e i n n e r one
-
a d i s t o r t e d t e t r a h e d r a l c o n f i g u - r a t i o n , t h e o u t e r one-
a random c o n f i g u r a t i o n o f 1-5 atoms.An e x c e l l e n t f i t t o t h e e x p e r i m e n t a l d i f f r a c t i o n d a t a was o b t a i n e d when weak bond b e n d i n g f o r c e s i n t h e t e t r a h e d r a l s u b s h e l l were c o n s i d e r e d .
I n t r o d u c t i o n . - The s t r u c t u r e o f l i q u i d s e m i c o n d u c t o r s (SC) a n d s e - m i m e t a l s is s t i l l a c h a l l e n g i n g problem. Based on e a r l i e r d i f f r a c - t i o n e x p e r i m e n t s ? K r e b s /l/ proposed f o r l i q u i d germanium ( l i q - G e ) a s t r u c t u r e consisting o f a d i s t o r t e d NaC1-type c o n f i g u r a t i o n w i t h f o u r n e a r e s t n e i g h b o u r s a n d two s l i g h t l y f a r t h e r removed atoms.
Isherwood e t a l . /2/ b u i l t a s p a t i a l model w i t h d i s t o r t e d o c t a h e - d r a , randomly c o n n e c t e d . L a t e r on, O r t o n / 7 , 4 / d e v i s e d a g e n e r a l model f o r m o l t e n SC and SM ( t h e d o u b l e h a r d s p h e r e model) assuming t h e e x i s t e n c e o f two s p e c i e s o f atoms i n t h e l i q u i d , g i v i n g l o n g and s h o r t i n t e r a t o m i c d i s t a n c e s . P o l t a v t s e v /5/ proposed a model f o r l i q u i d germanium based on two t y p e s o f a t o m i c c l u s t e r s : one w i t h s i m p l e c u b i c c o n f i g u r a t i o n and t h e o t h e r w i t h a random n e i g h - bour c o n f i g u r a t i o n . For s i l i c o n m e l t h e proposed a model w i t h d i s - t o r t e d s i m p l e c u b i c c o o r d i n a t i o n s i m i l a r t o t h a t proposed by Krebs.
Waseda and S u z u k i /6/ and G a b a t h u l e r and S t e e b / 7 / i n v e s t i g a t e d t h e s t r u c t u r e o f Ge and S i m e l t s and c o n c l u d e d t h e t t h e y c o n s i s t o f a m i x t u r e of m e t a l - l i k e a t o m i c a r r a n g e m e n t randomly d i s t r i b u t e d and c o v a l e n t - l i k e a r r a n g e m e n t .
All t h e s e models f a i l t o r e p r o d u c e s a t i s f a c t o r i l y b o t h s t r u c - t u r e f a c t o r and r a d i a l d i s t r i b u t i o n f u n c t i o n ( R D F ) o f t h e m e l t s . Consequently we p r o p o s e a new model f o r l i q u i d s i l i c o n and germa- nium e l e m e n t s which gave a n e x c e l l e n t f i t t o t h e e x p e r i m e n t a l d i f - f r a c t i o n d a t a and a l l o w s f o r a s i m p l e e x p l a n a t i o n o f t h e s t r u c t u r a l e v o l u t i o n o f t h e m e l t s w i t h t h e t e m p e r a t u r e .
Modelling.- S t a r t i n g from t h e i d e a t h a t t h e c o v a l e n t f o r c e s a r e n o t c o m p l e t e l y l o s t i n l i q u i d SC and SM b u t o n l y weakened, we f i r s t l y b u i l t a model f o r l i q - G e w i t h 7 1 atoms a s a network o f f l a t t e n e d t e t r a h e d r a i n t e r c o n n e c t e d i n s i x - f o l d r i n g s o f atoms. Two l o n g e r bonding d i s t a n c e s f o r e v e r y atom were a l l o w e d on t h e open s i d e s o f i t s t e t r a h e d r o n . Using t h e bond s t r e t c h i n g f o r c e c o n s t a n t , kr, o f c r y s t a l l i n e germanium (c-Ge) and a t r i a l s e t o f e q u i l i b r i u m d i s t a n - c e s a s g i v e n i n / 6 / , t h e s t r u c t u r e was r e l a x e d by a Monte-Carlo p r o c e d u r e /8/ s o a s t o minimize t h e bond d i s t o r t i o n e n r The a t o m i c i n t e r a c t i o n was i n t r o d u c e d i n t h e form: V =k
(rq-Fj2
a sr e s u l t e d from t h e v a l e n c e f o r c e f i e l d t h e o r y o f $eaFing/9/ (r, is
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19814228
C4-1044 JOURNAL DE PHYSIQUE
t h e e q u i l i b r i u m i n t e r a t o m i c d i s t a n c e ) . A f t e r w a r d s , t h e l o n g bonds c o n f i g u r a t i o n was m o d i f i e d a c c o r d i n g t o t h e t e n d e n c y o f t h e atoms t o w a r d s a more d e n s e p a c k i n g and t h e n t h e model was r e l a x e d a g a i n f o r v a r i o u s s e t s o f bonding d i s t a n c e s and v a r i o u s v a l u e s o f t h e bond b e n d i n g f o r c e c o n s t a n t , k
,
f o r t h e n o n - c e n t r a l f o r c e s i n t h e t e t r a h e d r a l s u b s h e l l . An i n t e r % $ t i o n p o t e n t i a l o f t h e t y p e sugges- t e d by K e a t i n g / g / , V2=ke(Oi-do),
do b e i n g t h e e q u i l i b r i u m a n g l e , w a s c o n s i d e r e d . The b e s t f i t t o t h e s t r u c t u r e f a c t o r s . I ( s ) where s = 2 , s i n Q / % was o b t a i n e d w i t h t h e f o l l o w i n g p a r a m e t e r s : t h e e q u i l i - brium d i s t a n c e s rl=2.702
and r a = 2 . 9 08
and t h e bond b e n d i n g f o r c e c o n s t a n t ke=O.lxk ( G e ) , where k 1 (Ge) is t h e bond b e n d i n g f o r c e cons- t a n t f o r c r y s t a l l ? n e germanium,Bcentred on t h e t e t r a h e d r a l a n g l e( f i g u r e 1 and 2 ) .
I
e : Model 171/
-. -.
-
KO =O 1 K@IGe) d r = 0 . 3 4 I I .. ... A r --0.26 II
----
A r = O A( l
I I II
l
experimenf Ge (980°ClF o r A r = r l - r i z O . 2
8
t h e i n - t e g r a l w i d t h f t h e first peak i n RDF ( ~ 0 . 4W)
c o r r e c t l y r e - p r o d u c e s t e e x p e r i m e n t a l va- l u e (0.43k
/2/). Moreover, a s i n t h e e x ~ e r i m e n t a l RDF. t h e f i r s t peak i n t h e RDF o f t h e model d o e s n o t s p l i t .The e x p e r i m e n t a l d a t a on l i q u i d s i l i c o n were r e p r o - duced w i t h h i g h e r s t r e n g t h o f t h e bending f o r c e : k =0.25k (Gel These r e s u l t s a r e Sif tqualifa- t i v e agreement w i t h t h e o b s e r - ved h i g h e r v a l u e o f t h e bond bending f o r c e c o n s t a n t i n c-Si t h a n i n c-Ge /g/.
The r o u g h e s t i m a t i o n o f t h e mean c o o r d i n a t i o n number o f Ge from t h e model i s m=6.6, i n f a i r l y good agreement w i t h t h e e x p e r i m e n t a l d a t a : 6.8 / 6 / , 6.5 /7/, 7.1 / 2 / . The o u t e r s u b s h e l l o f t h e Ge-surrounding c o n t a i n s 1-5 atoms ( t h e mean number i s 2.6 a t o m s ) .
The s u c c e s o f t h e above model a l l o w e d us t o c o n c l u d e t h a t a g e n e r a l random model f o r m o l t e n SC and S M must t a k e i n t o a c c o u n t t h e s p l i t t i n g o f t h e f i r s t c o o r d i n a t i o n s h e l l and weak n o n - c e n t r a l f o r c e s .
1 ..'
, C o n s e q u e n t l y , we b u i l t a model
3 L 5 6 w i t h 1 6 5 atoms l i n k i n g t o g e t h e r 93 u n i t s c o n s i s t i n g o f a cen- F i g . 2 : The RDPs of liq-Ge t r a l atom t e t r a h e d r a l l y bonded
model ( 7 1 a t o m s ) r e - t o o t h e r f o u r atoms. A f t e r a l a x e d f o r v a r i o u s s e t s p r e l i m i n a r y r e l a x a t i o n w i t h o f bonding d i s t a n c e s . o n l y t h e bond s t r e t c h i n g f o r c e
c e n t r e d on t h e e x p e r i m e n t a l mean bonding d i s t a n c e of l i q - G e (2.82
2
/ 6 / ) , t h e bonds were r e a r r a n - ged a c c o r d i n g t o t h e t e n d e n c y shown by t h e atoms t o w a r d s a b e t t e r s u r r o u n d i n g . The f i n a l r e l a x a t i o n was performed f o r t h e s e t o f e q u i - l i b r i u m d i s t a n c e s a l r e a d y o b t a i n e d i n t h e model w i t h 7 1 atoms and f o r v a r i o u s bond bending f o r c e s i n t h e t e t r a h e d r a l s u b s h e l l ( f i g u r e 3 ) . When weak bond b e n d i n g f o r c e was used i n t h e c o m p u t a t i o n s , ke=O.l Xk e ( G e ) , t h e s t r u c t u r e f a c t o r of liq-Ge was s a t i s f a c t o r i l y r e p r o d u c e d . The e x i s t e n c e o f two bonding d i s t a n c e s d e t e r m i n e s a r a p i d damping o f t h e s t r u c t u r e f a c t o r a t h i g h S , a s i s e x p e r i m e n t a l l y observed. A l o - wer c o o r d i n a t i o n number i n t h i s model, Rz5.9, a c c o u n t s f o r t h e f i n e s h i f t o f t h e l e f t component o f t h e main d i f f r a c t i o n peak from i t s r i g h t p o s i t i o n .
When t h e t e m p e r a t u r e o f t h e Ge-melt i s r a i s e d from 1 0 0 0 ~ ~ t o 1 3 0 0 ~ ~ t h e l i q u i d germanium l o s s e s t h e r e s i d u a l bond b e n d i n g f o r c e s i n t h e t e t r a h e d r a l s u b s h e l l and t h e s t r u c t u r e a p p r o a c h e s t h a t of a m e t a l - l i k e m e l t , a s can be s e e n by t h e d i s a p p e a r a n c e of t h e s h o u l d e r on t h e f i r s t d i f f r a c t i o n peak /10/.
C4-1046 JOUXNAL DE PHYSIQUE
I I
F i g . 3 : The s t r u c t u r e f a c t o r o f t h e random model w i t h s t r u c - t u r e d s h e l l ( 1 6 5 a t o m s ) o f Ge-melt, c a l c u l a t e d f o r va- r i o u s r e l a x a t i o n p a r a m e t e r s .
C o n c l u s i o n s . - On t h e b a s i s o f a computer m o d e l l i n g o f t h e s t r u c t u r e o f l i q u i d germanium and s i l i c o n i t was c o n c l u d e d t h a t t h e m e l t o f a t e t r a h e d r a l l y bonded s e m i c o n d u c t o r c o n s i s t s o f a random n a c k i n g o f atoms e x h i b i t i n g a s t r u c t u r e d c o o r d i n a t i o n : a n i n n e r s u b s h e l l w i t h a d i s t o r t e d t e t r a h e d r a l c o n f i g u r a t i o n and a n o u t e r s u b s h e l l w i t h 1-5 atoms randomly d i s t r i b u t e d . I n t h e t e t r a h e d r a l s u b s h e l l weak bond b e n d i n g c o v a l e n t f o r c e i s a c t i n g , i t s s t r e n g t h d e p e n d i n g on
t h e t y p e o f e l e m e n t and o n t h e t e m p e r a t u r e o f t h e l i q u i d . R e f e r e n c e s .
/l/ K r e b s H . , Fundamentals o f i n o r g a n i c c r y s t a l c h e m i s t r y , McGraw- - H i l l , 1968.
/2/ Isherwood S.P., O r t o n B.R., ? J 5 n 3 i l 3 R.
,
J. Non-Cryst. S o l i d s , 8-10( 1972 ) 691./3/ O r t o n B.R., 2. N a t u r f o r s c h . , 7Oa(1975)1500; 9 1 a ( 1 9 7 6 ) 9 9 7 ; 3 2 a ( 1 9 7 7 ) 7 7 2 ; 3 4 a ( 1 9 7 9 ) 1 5 4 7 .
/4/ O r t o n B.R., J . P h y s i q u e , 41(1980)C8-280.
/5/ P o l t a v t s e v Yu.G.? K r i s t a l l o g r a f i a ( S o v . J ) , 1 6 ( 1 9 7 1 ) 4 5 6 . /6/ Waseda Y., S u z u k l K . , 2. P h y s i k , B20(1975)379.
/7/ G a b a t h u l e r J . P . , S t e e b S., Z . N a t u r f o r s c h . , 9 4 a ( 1 9 7 9 ) 1 3 1 4 . /8/ Popescu M., Ph.D. T h e s i s , B u c h a r e s t 1975.
/9/ K e a t i n g P.N., Phys.Rev., 1 4 5 ( 1 9 6 6 ) 6 9 7 .
/10/ G l a z o v V.M., C h i z h e v s k a i a S.N., G l a g o l e v a N . N . , L i q u i d Semi- c o n d u c t o r s , Plenum P r e s s , 1 9 6 9 , p.78.
