GROWTH SEQUENCE OF Si-CLUSTERS: FROM A FEW ATOMS TO

JOURNAL DE PHYSIQUE

CoZZoque C4, suppl6ment au nO1O, Tome 42, octobre 1981

GROWTH SEQUENCE OF S i - C L U S T E R S

page C4-245

: FROM A FEW ATOMS TO THE AMORPHOUS

PHASE

B. Mosseri and J.P. Gaspard

X

Laboratoire de Physique des SoZides, CNRS, 2, place A. Briand, 92190 MeudonBe ZZevue, France t I n s t i t u t de Physique, UniversitQ de Li2ge, B-4000 Sart MZman, BeZgim

Abstract

We study t h e growth sequence of c l u s t e r s of atoms

bonded by s and p e l e c t r o n i c i n t e r a c t i o n s . I t i s demonstrated t h a t , l i k e i n t h e c a s e of c e n t r a l f o r c e s , t h e c l u s t e r s bear l i t t l e r e l a t i o n t o t h e s t r u c t u r e s formed a s s u b u n i t s of t r i o r t e t r a v a l e n t c r y s t a l l i n e l a t t i c e s . The s t a b i l i t y of t h e c l u s t e r i n c r e a s e s with t h e number of bonds even i f t h e bond formation i s accompanied by bond angles s u b s t a n t i a l l y d i f f e r e n t from 109

o r 1200. But,in c o n t r a s t with t h e c a s e of c e n t r a l f o r c e s , bond angles Si-Si-Si

s m a l l e r than about 90° produce

an i n c r e a s e i n

bond l e n g t h accompanied by a weakening of t h e c l u s t e r cohesion. We f i n d t h a t i n some c a s e t h e system undergoes a Jahn-Teller d i s t o r t i o n when t h e HOMO (Highest Occupied Molecular O r b i t a l ) i s degenerate.

I. Introduction.-

I t i s important t o study t h e growth sequence of small c l u s t e r s

of covalent atoms not only because i t s i m u l a t e s t h e d e p o s i t i o n of atoms when forming t h e amorphous phase, but because i t g i v e s information on t h e r e l a t i v e importance of t h e r a d i a l and angular p a r t s of t h e i n t e r a t o m i c p o t e n t i a l s , i n d e pendently of t h e e f f e c t of t h e neighbouring atoms occuring i n t h e bulk. I t i s shown i n t h i s paper t h a t t h e most s t a b l e c l u s t e r geometries correspond t o a maximization of t h e number of bonds, a t l e a s t f o r bond angles g r e a t e r than 90°. We f i n d t h a t , l i k e i n t h e c a s e of c e n t r a l forces[l] ( r a r e gases...), t h e growth sequence of Si atoms bear l i t t l e r e l a t i o n t o t h e s t r u c t u r e s formed a s s u b u n i t s on any ( t e t r a v a l e n t ) c r y s t a l l i n e l a t t i c e . I n t h e c a s e of c e n t r a l f o r c e s , t h e r u l e i s j u s t t o maximize t h e number of bonds w h i l s t i n t h e c a s e of covalent systems, a reduction of t h e bond a n g l e below 90° c o s t s energy and may prevent t h e formation of s h o r t r i n g s ( t r i a n g 1 e s ) . with bond angles of 10SO(close t o 109°28')

, are

The f i v e - f o l d ( p 1 a n a r ) r i n g s favoured and g i v e r i s e t o a

l o c a l f i v e f o l d synnnetry.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981452

JOURNAL DE PHYSIQUE

I. Models and methods

I n o r d e r t o c a l c u l a t e t h e formation e n e r-m - of t h e c l u s t e r s . we have t o assume a model o f t h e e l e c t r o n i c ( a t t r a c t i v e ) i n t e r a c t i o n and a p a i r w i s e c e n t r a l r e p u l s i v e p o t e n t i a l V(r) which p r e v e n t s t h e system from c o l l a p s e . We c o n s i d e r o n l y s and p v a l e n c e e l e c t r o n s i n t h e t i g h t binding framework[2] ,[3] The energy l e v e l s Es and E a r e s e p a r a t e d by 7.3 eV. The o v e r l a p of atomic wavefunctions i s n e g l e c t e d 'and t h e t i g h t binding i n t e r a c t i o n parameters a r e , a s g i v e n by Lannoo e t a l . 141 a t a d i s t a n c e of 2.35 fl : s s r = - 1 . 9 3 eV, s p a = 2.23 eV, p p a r 3.22 eV and ppn r-0.83 eV w i t h a uniform e x p o n e n t i a l behaviour w i t h d i s t a n c e : exp(-qr) w i t h q s l . 106 A-

.

.

The r e p u l s i v e p a i r p o t e n t i a l i s g i v e n by: V(r)= Vo exp(-pr) w i t h p = 2.094 A-' Vo i s determined i n o r d e r t o g e t a n i n t e r a t o m i c d i s t a n c e f o r t h e S i 2 molecule equal t o 2

.

.

Let u s s t r e s s t h a t we do n o t assume a p r i o r i a p a r t i c u l a r t y p e of s-p hybrid i n t h e m o l e c u l a r bond: t h e d i r e c t i o n n a l i t y o f t h e bond i s t h e consequence of t h e e l a c t r o n i c i n t e r a c t i o n . The t o t a l energy of t h e system i s g i v e n by

I ETe, +;zE:+ i; =

V,,l(rid) - L N ( E . c E p )

1

where E . a r e t h e energy l e v e l s of t h e h a m i l t o n i a n and N t h e number of atoms. The l a s t t e h n i s t h e s u b s t r a c t i o n of t h e atomic sZp2 energy. I n t h e c a s e of small c l u s t e r s , one o r two geometric parameters a r e v a r i e d c o n t i n u o u s l y i n o r d e r t o g e t t h e e q u i l i b r i u m geometry. No c h a r g e t r a n s f e r i s taken i n t o account a t t h i s s t a g e of t h e c a l c u l a t i o n : i t could modify t h e c o n c l u s i o n s i n c e t h e atoms a r e no l o n g e r n e u t r a l . 111. R e s u l t s on small S i molecules

1)

g3

Let u s f i r s t observe t h a t a simple p a i r i n t e r a c t i o n approximation would i n any c a s e s t a b i l i z e t h e equil a t e r a l t r i a n g l e : t h e l a r g e r t h e numb e r of bonds, t h e more s t a b l e t h e c o n f i g u r a t i o n . The c a l c u l a t i o n t h a t we d i d using t h e method d e s c r i b e d i n I1 t a k e s i n t o account t h e quantum mec h a n i c a l c h a r a c t e r of t h e c o v a l e n t bond. Fig 1. shows t h e v a r i a t i o n of t h e t o t a l energy a s a f u n c t i o n of t h e bond a n g l e i n t h e range from 600(equil a t e r a l t r i a n g l e ) t o 180°( l i n e a r c h a i n ) . The most s t a b l e s i t u a t i o n corresponds t o an angleof about 1000 w i t h a- r a t h e r f l a t minimum( e s p e c i a l l y when l a r g e r v a l u e s of p p r a r e assumed). This means t h a t a n g u l a r d i s t o r t i o n i s n o t so much expensive i n energy provided t h e a n g l e being n o t l e s s t h a n about 80°. I n c r e a s i n g t h e ppx v a l u e ( i . e i n c r e a s i n g t h e s t r e n g t h of the7'tbond) c o n t r i b u t e s t o t h e s t a b i l i z a t i o n o f t h e l i n e a r chain.

-12

-14

-16

m:shows Si3

The energy vers u s bond a n g l e c l e f t s c a l e ) . The lower c u r v e shows t h e e q u i l i b r i u m distance f o r each a % l e ( r i g h t s c a l e ) . the

-16

2) S i , -q.

The most s t a b l e s i t u a t i o n i s n o t t h e t e t r a h e d r o n a s i t would be i n

Fig.2: S i 4 molecule. T o t a l energy versus bond angle.

-5 2

I

I

I

3

4

5

atoms

6

-

w :Molecular

O r b i t a l Energy Levels f o r t h e p l a n a r Si5 c l u s t e r Fig.3: Comparison between isomer s t a b i l i t y versus bond angles.~ h ~ d o t t e d f o r N = 3 t o 6. l i n e d e s c r i b e s t h e Lower Unoccup i e d Molecular Level. t h e c a s e o f a c e n t r a l p a i r p o t e n t i a l ( c.f Hoare) because t h e a n g u l a r d e f o r m a t i o n i s p r o h i b i t i v e . Fig.2. shows t h e energy v a r i a t i o n f o r t h e p l a n a r a r r a y o f f o u r atoms. The ( p e r f e c t ) s q u a r e i s not t h e most s t a b l e c o n f i g u r a t i o n s i n c e t h e HOMO i s d e g e n e r a t e : t h e system undergoes a J a h n - T e l l e r d i s t o r t i o n , t h r e e edges a r e s h o r t e n e d due t o t h e i n c r e a s e of t h e bond a n g l e s whereas t h e f o u r t h edge i s s u b s t a n c i a l l y i n c r e a s e d . Let u s remark t h a t t h e e n e r g y c u r v e d e c r e a s e s l i n e a r l y a t e = 900 a s i t i s expected i n t h e J a h n - T e l l e r i n s t a b i l i t y . S t a b i l i z i n g t h e l i n e a r S i 4 m o l e c u l e r e q u i r e s a more d r a m a t i c i n c r e a s e of p p n t h a n i n t h e c a s e of S i 3

.

3) g 5 The c o h e s i v e energy o f a l a r g e r number o f p l a u s i b l e isomers have t o b e compared. Fig.3 shows t h e isomers c o n s i d e r e d i n t h i s c a l c u l a t i o n s e t a c c o r d i n g t o t h e i r c o h e s i v e energy. The most s t a b l e isomer( among t h o s e c o n s i d e r e d ) i s t h e pentagon( s l i g h t l y ) d i s t o r t e d a g a i n by a Jahn-Tellei- e f f e c t . The a n g u l a r d i s t o r t i o n i s 4 O (f)= 112O). The s q u a r e pyramid i s l e s s s t a b l e ; l e t u s n o t i c e t h a t t h e i n t r o d u c t i o n of a n atom above t h e s q u a r e d i l a t e s t h e s q u a r e , j u s t t h e o p p o s i t e way a c e n t r a l p o t e n t i a l would do. The m o l e c u l a r o r b i t a l energy l e v e l s a r e p l o t t e d a g a i n s t t h e bond a n g l e on Fig.4

.

4) Sf6 We have compared t h e c o h e s i v e energy o f t h e l i n e a r c h a i n , t h e s q u a r e bipyramid and t h e s i x - f o l d r i n g (Fig.3). The s i x - f o l d r i n g h a s been c o n t i n u o u s l y d i s t o r t e d o r "corrugated" s t a r t i n g from t h e p l a n a r c o n f i g u r a t i o n toward " c h a i r " c o n f i g u r a t i o n s o f v a r i a b l e bond a n g l e . The minimum o c c u r s f o r a n a n g l e o f * O l e . 5) s 3 t C l u s t e r s of 8 S i atoms a r e i n s t r u c t i v e . We stability have compared h e r e f o u r c l u s t e r s o f d i f f e r e n t d i m e n s i o n a l i t y : -ID, t h e l i n e a r c h a i n ( bond a n g l e e q u a l s t o 180°) -2D, a p a i r of f i v e - f o l d r i n g s w i t h a common edge t a k e n from a dodecahedron(2D curved s u r f a c e ) w i t h a l l a n g l e s equal t o 108O. -3D, a cube and a s o - c a l l e d "small b a r r e l a n " . We o b s e r v e t h a t t h e most s t a b l e c l u s t e r s a r e not p r e s e n t i n t h e c r y s t a l l i n e diamond s t r u c t u r e . An e i g h t Fig.5: Comparison between atoms p i e c e o f diamond i s n o t e n e r g e t i c a l l y compeisomers s t a b i l i t y f o r 8 t i t i v e , b e c a u s e o f t h e lower a v e r a g e c o o r d i n a t i o n atoms cluster.

-

JOURNAL DE PHYSIQUE

number. However t h e small b a r r e l a n appears i n t h e w u r t z i t e s t r u c t u r e . I n f a c t , t h e average c o o r d i n a t i o n number i s n o t t h e o n l y parameter governing t h e s t a b i l i t y a s i t i s shown on Fig.5 The l i n e a r c h a i n (7 bonds) i s l e s s s t a b l e t h a n t h e c u b e ( l 2 bonds), but t h e most s t a b l e c l u s t e r s a r e t h e small b a r r e l a n ( 9 bonds) and t h e p a i r of 5-fold r i n g s w i t h a common edge( 9 bonds).

.

IV.

Larger S i c l u s t e r s

We c a l c u l a t e t h e c o h e s i v e energy of t h r e e l a r g e r c l u s t e r s u s i n g t h e same method t h a n f o r t h e s m a l l e r ones. We c o n s i d e r a 29 atoms s u b u n i t of t h e diamond l a t t i c e c o n t a i n i n g a l l t h e 6-fold r i n g s p a s s i n g throw a c e n t r a l v e r t e x , a dodecahedron and a 21 atoms c l u s t e r which c o n s i s t s i n t h e packing of 4 small b a r r e l a n s around a c e n t r a l v e r t e x , t h e l a t t e r two being drawn on Fig.6 The t o t a l energy of t h e c l u s t e r s h a s been c a l c u l a t e d a t each s t e p of t h e i r c o n s t r u c t i o n , s t a r t i n g from a few atoms t o t h e f u l l c l u s t e r . The r e s u l t s f o r t h e dodecahedron and t h e b a r r e l a n c l u s t e r s a r e shown on Fig.7 where t h e e l e c t r o n i c ( a t t r a c t i v e ) p a r t of t h e energy has been p l o t t e d a g a i n s t t h e number of atoms. I t has t o b e noted t h a t t h e energy curve i s v e r y s i m i l a r t o t h e p l o t of a rough count of t h e average number of dangling bonds p e r atom which shows t h e cruc i a l r o l e of bond formation when b u i l d i n g t h e c l u s t e r . Moreover, t h i s i s support e d by t h e g a i n i n energy c l e a r l y v i s i b l e a t each r i n g c l o s u r e . The 21 atoms b a r r e l a n c l u s t e r has i t s own i n t e r e s t i n t h e l i g h t of r e c e n t r e s u l t s on m i c r o c r y s t a l l i n e S i . Indeed i t can be shown t h a t i t i s an int e r m e d i a t e s t a g e of r e c o n s t r u c t i o n when going from a Connell-Ternkin l i k e model t o a wurtzite lattice. The dodecahedron i s t h e most s t a b l e c o n f i g u r a t i o n ( among t h e t h r e e considered h e r e ) . However s t r a i n i s supposed t o i n c r e a s e when adding new dodecad r a s h a r i n g f a c e s w i t h t h e f i r s t one. lw,, I b Q t . l

.

a

Fig.6: Larger c l u s t e r s . a ) The dodecahedron(20 atoms) b) The "4 small b a r r e l a n s " c l u s t e r ( 2 1 atoms)

-

Fig. 7 : Energy v e r s u s number of atoms. The lower c u r v e s show t h e a t t r a c t i v e p a r t of t h e t o t a l energy ( l e f t s c a l e ) . The upper c u r v e s show t h e number o f dangling bonds p e r atom ( r i g h t s c a l e ) . a ) dodecahedron b ) b a r e l a n cfuster. V. Conclusions. The main c o n c l u s i o n s can be s m e r i z e d a s follow: I n s and p bonded S i c l u s t e r s : i ) c r y s t a l l i n e arrangements a r e not favoured( l i k e i n r a r e g a s e s c l u s t e r s ) . i i ) a n approximate r u l e i s t o maximize t h e number of bonds provided t h a t t h e bond a n g l e s a r e g r e a t e r t h a n about 90°. i i i ) t h e r e i s an open c o m p e t i t i o n between isomers. I t has t o be noted t h a t hydrogenation of t h e c l u s t e r s could change t h e c o n c l u s i o n s s i n c e Si-H bond i s s t r o n g e r t h a n Si-Si bond.

-

111 [2]

[3]

[4]

Hoare,M.R. and PAL,P., Adv. Phys. 20 (1971) 161 Harrison,W.A. " E l e c t r o n i c s t r u c t u r e and t h e p r o p e r t i e s of S o l i d s " ( Freeman,San Francisco, 1980). Friede1,J. J. de Physique 2 (1978) 651 Lannoo ,M.and Allan,G. p r i v a t e communication. s e e a l s o Lannoo,M. J.de Physique %(1979)461