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NONLINEAR STRUCTURES IN SILICON CLUSTERS
N. Flytzanis, A. Mistriotis, S. Farantos
To cite this version:
N. Flytzanis, A. Mistriotis, S. Farantos. NONLINEAR STRUCTURES IN SILICON CLUSTERS.
Journal de Physique Colloques, 1989, 50 (C3), pp.C3-89-C3-93. �10.1051/jphyscol:1989314�. �jpa-
00229455�
Colloque C3, supplément au n°3, Tome 50, mars 1989 C3-89
NONLINEAR S T R U C T U R E S IN S I L I C O N CLUSTERS
N. F L Y T Z A N I S , A . MISTRIOTIS*
x> a n d S . FARANTOS* •
( 1>
Department of Physics, University of Crete, PO Box 470, Heraklio GR-711 10, Crete, Greece
"Department of Chemistry, University of Crete, Po Box 470, Heraklio GR-711 10, Crete, Greece
Résumé - Nous proposons un potentiel interatorv.ique nonlineaire pour le silicium, qui est un développement important du mode! de Sti llinger-l/eber. Notre modèle tiens plus en compte les interactions S quatre-corps, et conduit à une valeur très acceptable pour la température. Les énergies obtenues et les geometries des structures nonlineaires de 1 état fundamental et des états metastables des clusters de silicium sont à bon accord avec des calculs électroniques ab initie
Abstract - We propose a nonlinear interatomic potential for silicon which is a signi- ficant improvement over the Still inner-Weber potential. This model which includes four-body interactions, simulates the melting of the crystal and gives correct geo- metries and energies for the nonlinear ground state and low metastable structures of silicon clusters.
1 - INTRODUCTION
Clusters constitute an intermediate state of matter, between individual molecules and the bulk crystalline or amorphous structures. The changes in the transition between molecules and bulk solids via ordering: in increasing size clusters is an open problem in solid state physics.
From the study of their structure and dynamics we can obtain important information for nucleation and cluster growth as well as the structure of the amorphous material. Due to the large surface to volume atom ratio we can also obtain useful information for surface structure and reconstruction. Investigations on semiconductor clusters like Si, GaAs etc. can be very useful since there is a growing interest for smaller and smaller dimensions (that could reach the size of a cluster in a matrix) in microdevices. Another important application is the
Ionized Cluster Beam (ICB) technique /1/ for the formation of amorphous silicon films with the deposition of large ionised clusters (n>60 atoms) on a substrate. The ion charge and the dyna- mical condition of the clusters is very important (among other parameters) for the production of homogeneous films.
The progress in pulsed laser evaporation and in supersonic jet expansion for the production of clusters, the advances in laser spectroscopy techniques and in vacuum technology have signi- ficantly developed the experimental work. Still, however, little direct information on the structure of n>3 atom clusters has been obtained. The observation of the "magic" numbers (cluster sizes particularly stable) in the photofragmentation of ionised clusters, separated by mass spectroscopy is well established /2,3/. Thus for cluster structure determination, it is very important to complement the experimental work with accurate theoretical predictions for both neutral and ionised clusters. The calculations can be of three types:
a) Electronic first principle calculations, which however are restricted to small clusters due to the difficulty of repeating them for the numerous different ionic confiaurations /4-7/.
b) Phenomenological potentials, the most successful of which is the Still inger-l.'eber potential which gives good results for many properties, but some wrong structures for small clusters / 8 / , with some improvement when introducing a finite temperature /9/.
c) The Car -Parrinello method which combines successfully the local density functional (LDF) method for the electronic part with lattice dynamics /10/.
For the structure determination of large clusters and especially their dynamics calculations
Research Center of Crete, Po Box 1527, Heraklio GR-711 10, Crete, Greece
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989314
C3-90 JOURNAL DE PHYSIQUE
based on e m p i r i c a l i n t e r a c t i o n p o t e n t i a l s a r e v e r y advantageous. They can b e e a s i l y extended t o s t u d y c o l l i s i o n s between c l u s t e r s , t h e i r f r a g m e n t a t i o n , o r t h e d e p o s i t i o n o f c l u s t e r s on a s u b s t r a t e o f t h e same o r o t h e r m a t e r i a l a t d i f f e r e n t s u b s t r a t e t e ~ p e r a t u r e s and i n c i d e n t energy. I n some cases (e.?. r a r e gases) o n l y a two body t e r m i s r e q u i r e d t o d e s c r i b e t h e i n t e r - atomic p o t e n t i a l . T h i s i s n o t t r u e f o r semiconductors due t o t h e d i r e c t i o n a l i t y of t h e s t r o n g c o v a l e n t bonds, and t h e p o s s i b i l i t y o f d i f f e r e n t a t o m i c o r b i t a l h y b r i d i z a t i o n . Thus t h e deve- l o p e d i n t e r a t o m i c p o t e n t i a l s s h o u l d be c o r r e l a t e d and checked w i t h energy s u r f a c e s and s m a l l c l u s t e r s t r u c t u r e d e t e r m i n e d f r o m f i r s t p r i n c i p l e s e l e c t r o n i c c a l c u l a t i o n s .
Several a t t e m p t s have been made t o develop an a n a l y t i c a l i n t e r a c t i o n p o t e n t i a l model f o r S i /8,11/ and a p p l y i t f o r t h e s t u d y o f s t r u c t u r a l and dynamical p r o p e r t i e s /9,11,12/.
These models a l t h o u g h t h e y s u c c e s s f u l l y s i m u l a t e p r o p e r t i e s o f t h e b u l k s i l i c o n , f a i l t o g i v e good r e s u l t s f o r t h e small s i l i c o n c l u s t e r s . Ile have developed an e m p i r i c a l p o t e n t i a l which models q u i t e w e l l s m a l l c l u s t e r s f o r n>6 w i t h o u t l o s i n g some i m p o r t a n t p r o p e r t i e s o f t h e c r y s t a l /13/. The n o n l i n e a r i n t e r a c t i o n p o t e n t i a l n i v e s a v a r i e t y o f s t r u c t u r e s (ground and l o w m e t a s t a b l e s t a t e s ) w i t h c o n f i g u r a t i o n s and e n e r o i e s v e r y c l o s e t o t h e ab i n i t i o r e s u l t s . I n t h e n e x t s e c t i o n we w i l l d e s c r i b e b r i e f l y t h e developed e ~ p i r i c a l p o t e n t i a l . I n s e c t i o n 3 we p r e s e n t t h e numerical r e s u l t s , w i t h t h e c o n c l u s i o n and summary i n t h e l a s t s e c t i o n . 2 - INTERACTION PCTEPJTIAL
As a s t a r t i n g p o i n t we used t h e S t i l l i n g e r - N e b e r p o t e n t i a l / 6 / which i s t h e most s u c c e s s f u l f o r b u l k p r o p e r t i e s and m o d i f i e d i t t o f i t c l u s t e r p r o p e r t i e s as w e l l .The S t i l l i n g e r - k . ' e b e r p o t e n t i a l , c o n s i s t s o f a two-body and a three-body term. The two-body t e r m has t h e f o r m
where A,B, a and R a r e parameters p r o p e r l y chosen t o f i t dynamic p r o p e r t i e s o f t h e c r y s t a l . The e x p o n e n t i a l c u t o f f f u n c t i o n g r e a t l y reduces t h e c o m p u t a t i o n t i r e f o r m o l e c u l a r dynamics s i r l u l a t i o n s . The three-body term, i s o f t h e f o r m
V 3( r ,r.,r ) = h ( r . .,rik,Ojik) i j k + h(rji,rjk,Oijk) + h ( r . r .,6ikj)
1 J k l ' k ~
where r . . i s t h e d i s t a n c e between t h e atoms i and j, and 8 . . i s t h e a n g l e between t h e v e c t o r s r . The f u n c t i o n h i s g i v e n by t h e f o r m u l a J 1 k
1 j
where h , and y a r e parameters o f t h e model. T h i s three-body t e r m i s r e p u l s i v e , and expresses t h e tendency o f s i l i c o n atoms t o f o r m f o u r sp3 c o v a l e n t bonds ( v a n i s h e s a t cos8.. =1/3),
~ l k T h i s s i m p l e model g i v e s t h e c o r r e c t ground s t a t e o f t h e c r y s t a l (diamond s t r u c t u r e ) , a p p r o x i - mates t h e m e l t i n g p o i n t o f s i l i c o n / 0 / , and g i v e s good e s t i m a t e s f o r some e l a s t i c p r o p e r t i e s o f t h e b u l k /14/, b u t f a i l s t o r e p r o d u c e t h e e n e m y and t h e c o r r e c t s t r u c t u r e o f t h e ground s t a t e o f s i l i c o n c l u s t e r s /9,12/, which have been f o u n d f r o m ab i n i t i o c a l c u l a t i o n s /4/.
The new p o t e n t i a l which we have developed, r e t a i n s t h e advantages o f t h e S t i l l i n g e r - U e b e r model, b u t a l s o g i v e s good r e s u l t s f o r small c l u s t e r s w i t h t h e f o l l o w i n g two m o d i f i c a t i o n s :
a ) The a n g l e dependence o f t h e t h r e e body t e r m i s m o d i f i e d t o A (I - exp
[
- Q (cos8 + 1/312] )where t h e parameter A d e t e r m i n e s t h e a m p l i t u d e and Q t h e second d e r i v a t i v e n e a r t h e minimum.
The f i r s t a l l o w s t h e p o s s i b i l i t y o f h a v i n g s m a l l bond a n g l e s ( n e a r
60')
which i s t h e case f o r c l u s t e r s c o n t r a r y t o t h e diamo?d s t r u c t u r e . The parameter Q i s i m p o r t a n t t o f i t t h e m e l t i n c p o i n t o f t h e c r y s t a l .b ) t h e c o n t r i b u t i o n o f t h e r e p u l s i v e three-body t e r m t o t h e energy o f t h e c l u s t e r s i n c r e a s e s s l o w e r t h a n what i s expected f r o m t h e a b i n i t i o r e s u l t s , as t h e average c o o r d i n a t i o n number
i s considered, because t h e number o f f o u r - b o d y terms between N a t o w i s N(N-1) (N-2) (N-3)/24, and i n c r e a s e s w i t h t h e number o f atoms 14 f a s t e r than t h e number o f three-body terms which i s N(N-1) (N-2)/6. F o r t h i s reason we add t o t h e p o t e n t i a l a four-body t e r m which e f f e c t i v e l y i n t r o d u c e s a b e t t e r c o u n t i n g o f t h e three-body terms i n c l u d i n g some c o r r e l a t i o n between therr.
The f o u r - b o d y t e r m i s o f t h e f o r m
+ g(rki'rkj'rkl ,8ikj'8ikl '8jkl + g ( r l i ' r l j ' r l k ' O . l l J .'eilk'Ojlk) where
-
1p ( r . 1J . , r l k , r 11 , B ;lk . ,€I.. ~ 1 1
,ekil 1
= A4 exp[
y [ ( r - R ) - ' 1 J + (rikeR) + (ril-R)-'11
The new p o t e n t i a l p r o v i d e s a pood a p p r o x i m a t i o n f o r s e v e r a l p r o p e r t i e s o f small c l u s t e r s even though i t has o n l y one e x t r a parameter. The parameters A=16.30076, B=11.58113 i n t h e two body i n t e r a t i o n have been s e l e c t e d so t h a t t h e energy p e r atom Eo=-4.63eV and t h e l a t t i c e c o n s t a n t 0=5.43k o f t h e c r y s t a l a t z e r o t e m p e r a t u r e a r e equal t o t h e e x p e r i m e n t a l values, w h i l e t h e t h e o t h e r two parameters a r e k e p t t h e sane as i n t h e S t i l l i n g e r - \ ! e b e r model, n a c ~ e l y a=2.C951 and R=3.77116 A. The parameters A3, Aq, y, and Q a r e chosen such t h a t f i r s t , t h e diamond s t r u c t u r e i s t h e ground s t a t e o f t h e s i l i c o n c r y s t a l , and secondly, t h e m e l t i n g p o i n t o f t h e c r y s t a l i s about 2000 K. Our f i n a l c h o i c e o f these parameters i s A3=4.0, A4=47.0, y=2.4, and Q=5.0.
Csing these v a l u e s o f t h e parameters,the enerpy p e r atom as a f u n c t i o n o f t h e volume o f t h e l a t t i c e i s c a l c u l a t e d f o r s e v e r a l s t r u c t u r e s and t h e diamond s t r u c t u r e corresponds t o t h e l o w e s t energy p e r atom. The f c c and bcc s t r u c t u r e s a l s o have a low energy p e r atom, which i s o n l y O.leV h i g h e r than t h e diamond s t r u c t u r e . Even though t h e enerpy d i f f e r e n c e between t h e diamond l a t t i c e and t h e s t r u c t u r e s w i t h l a r g e c o o r d i n a t i o n number ( f c c , b c c ) i s s m a l l compared t o t h e r e s u l t s found f r o m l o c a l d e n s i t y a p p r o x i m a t i o n c a l c u l a t i o n s , t h i s i s necessary f o r f i n d i n ? a good e s t i m a t e f o r t h e m e l t i n g p o i n t o f s i l i c o n . The m e l t i n g o f a c r y s t a l c e l l w i t h 64 s i l i c o n atoms was s t u d i e d under c o n s t a n t p r e s s u r e o f 1 atm u s i n g t h e Andersen's method /15/.
The dependence o f t h e m e l t i n g p o i n t on two o f t h e parameters, A4 and Q, o f o u r model was examined
.
The m e l t i n g p o i n t which corresponds t o o u r f i n a l c h o i c e o f parameters, i s about 2050K. T h i s v a l u e i s s i m i l a r t o t h e one f o u n d by S t i l l i n g e r and lleber. P:oreover, o u r e s t i m a t e f o r t h e l a t e n t h e a t f o r m e l t i n g i s 0.3eV/atorn, w h i l e S t i l l i n g e r - l ! e b e r ' s v a l u e i s 0.3leV/atom, and t h e e x p e r i m e n t a l v a l u e i s 0.52eV/atom. These r e s u l t s show t h e s i m i l a r i t i e s o f o u r model t o t h e one o f S t i l l i n g e r and \.!eber. The d i f f e r e n c e s w i t h t h e e x p e r i m e n t a l v a l u e s ( t h e m e l t i n g p o i n t o f s i l i c o n i s 1683 K ) may be a t t r i b u t e d t o b o t h t h i s s t r a t e g y f o r c o n s t r u c t i n g a model f o r s i l i c o n , and t h e u n r e a l i s t i c c o n d i t i o n s f o r t h e n u m e r i c a l s i f i u l a t i o n o f m e l t i n g . Indeed, t h e p e r i o d i c boundary c o n d i t i o n s which a r e v a l i d even a f t e r t h e m e l t i n g , f o r a s m a l l s i z e c r y s t a l can cause a c o n s i d e r a b l e o v e r e s t i m a t i o n f o r t h e m e l t i n g p o i n t . I t i s a l s o i n t e r e s t i n g , t h a t o u r model reproduces some p r o p e r t i e s o f l i q u i d s i l i c o n . More s p e c i f i c a l l y , t h e s p e c i f i c heat o f t h e l i q u i d i s found t o be 6.8kcal/mole, w h i l e t h e e x p e r i m e n t a l v a l u e i s 7.4kcal/mole.Another i n t e r e s t i n g p r o p e r t y o f t h e l i q u i d i s t h a t i t s atoms have an average number o f f i r s t n e i g h b o r s g r e a t e r t h a n s i x , w h i l e t h e f i r s t n e i g h b o r s f o r an atom i n t h e s o l i d i s f o u r . 3
-
GROUND STATE STRUCTURES OF SEIALL SILICON CLUSTERSThe g r e a t advantage o f o u r p o t e n t i a l over t h e S t i l l i n g e r - l l e b e r model and t h e o t h e r model p o t e n t i a l s , i s t h a t i t produces ground s t a t e s f o r t h e s n a l l c l u s t e r s (n>6) w i t h e n e r p i e s and s t r u c t u r e s s i m i l a r t o those found f r o m Hartree-Fock / 4 / c a l c u l a t i o n s . !loreover, i t g i v e s e n e r g i e s f o r o t h e r s t r u c t u r e s o f m e t a s t a b l e s t a t e s which a r e a l s o i n apreement w i t h t h e ab i n i t i o r e s u l t s .
Using t h e method o f s i m u l a t e d a n n e a l i n g we determine t h e energy and t h e s t r u c t u r e o f t h e ground s t a t e f o r 7 t o 15 atoms. F i g u r e 1 shows t h e s t r u c t u r e o f t h e a b s o l u t e minima f o r t h e c l u s t e r s w i t h 7, 8, 9, 10 atoms. These a r e v e r y s i m i l a r t o t h e s t r u c t u r e s p u b l i s h e d i n
/ l o / ,
where t h e ground s t a t e s were found b y t h e C a r - P a r i n e l l o method. The ground s t a t e o f S i 8 can be d e s c r i b e d as a bicapped t e t r a g o n a l b i p y r a m i d , b u t i n o u r r e s u l t t h e two e x t r a atoms which cap two o f t h e s i d e s o f t h e b i p y r a m i d a r e c l o s e t o each o t h e r so t h e r e i s an i n t e r a c t i o n between them. The ground s t a t e o f S i g i s d e s c r i b e d i n / I @ / as a s t r o n c l y d i s t o r t e d t r i c a p p e d
C3-92 JOURNAL
DE
PHYSIQUEoctahedron. The sane f i g u r e can a l s o be described a s a d i s t o r t e d tetracapped t r i a g o n a l b i - pyramid, which i s a s t r u c t u r e s i m i l a r t o what we found a s t h e ground s t a t e of Sig.
F i n a l l y , t h e a b s o l u t e minimum of S i l o which i s given in / l o / i s a tetracapped t r i a g o n a l prism.
C u r r e s u l t s show a s l i g h t l y d i f f e r e n t ground s t a t e f o r S i l o . iiore s p e c i f i c a l l y , t h e s t r u c t u r e we found i s a d i s t o r t e d tricanned t r i a n a u l a r nrism with one more atom i n t e r a c t i n g with t h e two atoms of an edqe of t h e prism and t h e two nearby e x t r a atoms which a r e not elements of t h e prism. The s t r u c t u r e of t h e ground s t a t e of S i l o i s b a s i c a l l y t h e same a s t h a t found by t h e Car-Parinello method, but t h e t e n t h atom i s not placed over one of t h e t r i a n g u l a r b a s i s of t h e prism, p r e f e r i n g t o i n t e r a c t with f o u r atoms i n s t e a d of t h r e e . Note t h a t i n f i g u r e 1 l i n e s a r e r'rawn between atoms, which a r e l e s s than ~ = 3 . 7 7 1 1 8 1 a p a r t .
Fig. 1
-
The ground s t a t e of s i l i c o n , c l u s t e r s with 7,8,9 and 10 atoms. Lines a r e drawn between a t o m which a r e l e s s than R=3.77118 A a p a r t .The e n e r g i e s per atom f o r t h e c l u s t e r s of s i z e 5 t o 15 a r e shown in f i g u r e 2. Our r e s u l t s a r e noted by a +, while
*
marks t h e corresponding values f o r t h e Stillinger-l!eber p o t e n t i a l a s i t i s given i n /9/. The mark x shows t h e values found by t h enew
Biswas-Hamann model. The mark#notes t h e ab i n i t i o values found by Raghavashari, and o notes t h e same r e s u l t s a f t e r they have been scaled t o f i t t h e experimental values f o r Si2 and Si3. O u r r e s u l t s not only approach very much t h e scaled ab i n i t i o v a l u e s , but a l s o they show a s i m i l a r t o t h e ab i n i t i o r e s u l t s i n c r e a s e f o r t h e energy per atom a s t h e s i z e of t h e c l u s t e r s grows.
4
-
CCI'ICLUSICNSThe new p o t e n t i a l which i s presented in t h i s paper, i s a n o d i f i c a t i o n of t h e S t i l l i n g e r - l l e b e r model. In o t h e r words a four-body term i s added t o t h e i r nodel, and t h e angle dependence of t h e three-body and four-body terms has been changed. These modifications r e s u l t i n a q u a l i t a - t i v e improvement over t h e previous models f o r s i l i c o n . Fore s p e c i f i c a l l y , good values f o r t h e energy of t h e ?round s t a t e s of t h e c l u s t e r s a r e obtained f o r a l a r g e ranse of t h e parameters of t h e three-body terms. Therefore t h e s e parameters can be s e l e c t e d t o f i t p r o p e r t i e s of t h e c r y s t a l .
I t i s very important t h a t by introducing only two e x t r a parameters we have succeeded t o f i t a l a r g e number of new p r o p e r t i e s , i . e . t h e s t r u c t u r e and e n e r g i e s of the p o u n d s t a t e and low metastable s t a t e s of small c l u s t e r s .
Re developed a p o t e n t i a l which d e s c r i b e s t h e s t r u c t u r e and energy of s i l i c o n c l u s t e r s more s u c c e s s f u l l y than any of t h e o t h e r a n a l y t i c a l models proposed so f a r . b!e plan t o use t h i s p o t e n t i a l f o r doinq molecular dynamic simulations f o r c l u s t e r s t h e r e s u l t s of which nay be
f e r e n t c l a s s i c a l models and t h e ab i n i t i o r e s u l t s . The marks o and correspond t o t h e ab i n i t i o r e s u l t s a f t e r s c a l i n g and b e f o r e s c a l i n g r e s p e c t i v e l y . The marks
*,
x and + correspond t o t h e S t i l l i n g e r - W e b e r p o t e n t i a l , Biswas-Hamann p o t e n t i a l and t h e new p o t e n t i a l proposed i n t h i s paper r e s p e c t i v e l y .compared w i t h f u t u r e experimental data. It i s a l s o o f i n t e r e s t t o a p p l y o u r model p o t e n t i a l f o r t h e s t u d y o f r e c o n s t r c t i n on Si surfaces, w i t h o r w i t h o u t an o v e r l a y e r o f o t h e r atoms. F o r t h e f i r s t case t h e GeY1117 s u r f a c e i s s t a b i l i s e d t o a J3 x J3 two dimensional s t r u c t u r e w i t h an antimony o r l e a d o v e r l a y e r /16/. One can a l s o compare w i t h t h e r e s u l t s o f o t h e r p o t e n t i a l s t h a t have been s p e c i f i c a l l y developed f o r s u r f a c e s t u d i e s /11,17/.
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