HAL Id: jpa-00223074
https://hal.archives-ouvertes.fr/jpa-00223074
Submitted on 1 Jan 1983
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
BAND STRUCTURE OF VACANCIES AND DISLOCATIONS IN DIAMOND
R. Jones, T. King
To cite this version:
R. Jones, T. King. BAND STRUCTURE OF VACANCIES AND DISLOCATIONS IN DIAMOND.
Journal de Physique Colloques, 1983, 44 (C4), pp.C4-461-C4-464. �10.1051/jphyscol:1983454�. �jpa- 00223074�
BAND STRUCTURE OF VACANCIES AND DISLOCATIONS IN DIAMOND
R . Jones and T.E.G. King
Department of Physics, University of Exeter, Stocker ~ o a d , Ereter EX4 4QL, Devon, U.K.
~ 6 s u m 6 - Nous d e c r i v o n s l e s r e s u l t a t s d ' u n c a l c u l 2 p a r t i r d e s p r i n c i p e s fondaruentaux,des niveaux e t bandes d 1 8 n e r g i e de lacunes i d e a l e s , e t de 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 diamant. Le p o t e n t i e l e s t c o n s t r u i t en superposant l e s p o t e n t i e l s atomiques 2 un p o t e n t i e l d'echange l o c a l . Pour c a l c u l e r l e s niveaux d ' e n e r g i e , nous u t i l i s o n s l a mdthode r e c u r s i v e g e n e r a l i s e e pour l e s i n t e g r a l e s de recouvrement.
A b s t r a c t - The r e s u l t s of a f i r s t p r i n c i p l e s c a l c u l a t i o n o f t h e energy l e v e l s and bands of i d e a l vacancies a s w e l l a s p a r t i a l d i s l o c a t i o n s i n diamond a r e d e s c r i b e d . The p o t e n t i a l i s c o n s t r u c t e d by t h e s u p e r p o s i t i o n of atomic p o t e n t i a l s , t o g e t h e r w i t h a l o c a l exchange p o t e n t i a l . The r e c u r s i o n method, g e n e r a l i s e d t o d e a l w i t h o v e r l a p i n t e g r a l s , i s used t o compute t h e energy l e v e 1s.
We have c a r r i e d o u t c a l c u l a t i o n s from f i r s t p r i n c i p l e s f o r t h e s t a t e s a s s o c i a t e d with vacancies and d i s l o c a t i o n s i n diamond. These c a l c u l a t i o n s s h o u l d b e more r e l i a b l e t h a n corresponding ones f o r s i l i c o n /1,2/ based on S l a t e r - K o s t e r parameters which were o b t a i n e d by f i t t i n g s t o t h e band s t r u c t u r e . T h a t procedure y i e l d s many s e t s of parameters and p a r t i c u l a r c h o i c e s may n o t enjoy any fundamental s i g n i f i c a n c e . Furthermore, t h e p o t e n t i a l n e a r a d e f e c t l i k e a vacancy i s s o d i f f e r e n t from t h a t of t h e b u l k , t h a t t h e r e a l m a t r i x elements between atoms c l o s e t o t h e d e f e c t b e a r l i t t l e r e l a t i o n t o t h e corresponding c r y s t a l l i n e elements. We began by f i n d i n g t h e p o t e n t i a l f o r a c r y s t a l l i n e c l u s t e r , by o v e r l a p p i n g n e u t r a l atom p o t e n t i a l s con- s t r u c t e d from Clementi wavefunctions /3/ using a l o c a l d e n s i t y exchange p o t e n t i a l
( a = 1) and a 2s22p2 c o n f i g u r a t i o n . The choice of c o n f i g u r a t i o n i s n o t c r i t i c a l . This p o t e n t i a l was then f i t t e d t o a l i n e a r combination of simple a n a l y t i c s p h e r i c a l f u n c t i o n s , w ( r ) , l o c a l i s e d around each atom. A b a s i s of s- and p- m O was s e l e c t e d by s o l v i n g t h e Schroedinger e q u a t i o n f o r w ( r ) t o g e t h e r w i t h a q u a d r a t i c r e p u l s i v e p o t e n t i a l t o e n s u r e a l i m i t e d range of t h e s e b a s i s f u n c t i o n s . T h i s p r e s c r i p t i o n a l l o w s a l l m a t r i x elements t o be a n a l y t i c a l l y e v a l u a t e d and a v o i d s t h e problems encountered by Persson and Jones /4/ who e v a l u a t e d t h e m a t r i x elements numerically.
The r e s u l t i n g band s t r u c t u r e /5/ i s i n good agreement wit!! o t h e r c a l c u l a t i o n s . *.e i n d i r e c t band gap i s 5.3 eV. To d e a l w i t h d e f e c t s we c o n s t r u c t e d a c l u s t e r of about 300 atoms c e n t r e d on t h e d e f e c t . Then we e v a l u a t e d t h e d i f f e r e n c e between t h a t p o t e n t i a l c o n s t r u c t e d from f i r s t p r i n c i p l e s , as above, and t h a t due t o a super- p o s i t i o n o f t h e a n a l y t i c a l p o t e n t i a l s w ( r - R). I n g e n e r a l t h i s d i f f e r e n c e was So s m a l l t h a t i t s e f f e c t s can b e t r e a t e d by p e r t u r b a t i o n theory. The l o c a l d e n s i t y of s t a t e s on a c e n t r a l atom f o r t h i s c l u s t e r can t h e n be o b t a i n e d by t h e r e c u r s i o n method, modified t o d e a l w i t h a non-orthogonal s e t of b a s i s f u n c t i o n s /5,6/.
T y p i c a l l y 40 c o e f f i c i e n t s were e v a l u a t e d and t h e l o c a l d e n s i t y of s t a t e s e v a l u a t e d . The charge d e n s i t y around t h e d e f e c t can a l s o be found /6,7/.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983454
JOURNAL DE PHYSIQUE
2. TKE DIAMOND VACANCY
The p o t e n t i a l of t h e i d e a l vacancy d i f f e r s by a t most about 10% from t h a t found from a s u p e r p o s i t i o n of w ( r - R) e v a l u a t e d from t h e c r y s t a l , (excluding t h e vacancy s i t e R) and t h i s d i f f e r e n c e c a u s e s only a s m a l l s h i f t (0.2 eV) i n t h e T2 l e v e l . We found t h e T2 l e v e l t o b e a t Ev + 2.56 eV and an A1 resonance a t Ev - 0.7 eV. These a r e t o be compared w i t h more s o p h i s t i c a t e d c a l c u l a t i o n s by B a c h e l e t , Baraff and S c h l b t e r /8/, who found t h e Tp l e v e l a t Ev + 1.53 eV and t h e Al resonance a t Ev - 2 eV. An atomic c o n f i g u r a t i o n o f 2s 2p3 c a u s e s an almost r i g i d s h i f t downwards of t h e energy bands a s w e l l a s t h e vacancy l e v e l s . We b e l i e v e t h e d i f f e r e n c e between o u r r e s u l t s and B a t c h e l e t e t a 1 /8/ i s due t o t h e l i m i t e d b a s i s s e t we employ. I t i s i n t e r e s t i n g t o observe t h a t i f we u s e t h e c r y s t a l l i n e m a t r i x elements f o r t h e atoms i n t h e c l u s t e r w i t h a vacancy, we f i n d a T2 l e v e l about Ev + 1.53 eV. This h a s a l s o been r e p o r t e d i n /9/. This i s p r i n c i p a l l y due t o t h e e f f e c t of t h e t h r e e - c e n t r e c o n t r i - b u t i o n o f t h e removed atom t o t h e d i a g o n a l m a t r i x elements o f t h e atoms c l o s e s t t o t h e vacancy. T h i s agreement i s p u r e l y f o r t u i t o u s ,
3. DISLOCATIONS
The s t a c k i n g f a u l t energy has been measured by P i r o u z , Sumida, Lang and Hirsch /lo/
t o be about 300 mJ/m2. This r e s u l t s i n a s e p a r a t i o n between p a r t i a l s of about 30 8.
T h i s v a l u e i s i n agreement with a r e c e n t c a l c u l a t i o n by Persson /11/ who e v a l u a t e d t h e t o t a l e l e c t r o n i c energy of a s u p e r c e l l c o n t a i n i n g an i n t r i n s i c s t a c k i n g f a u l t . The p o t e n t i a l and b a s i s f u n c t i o n s were c a l c u l a t e d from f i r s t p r i n c i p l e s b u t a l l m a t r i x elements were e v a l u a t e d numerically.
Fig. 1 The ( i l l ) p l a n e s o f atoms c o n t a i n i n g t h e 90° and 30°
g l i d e p a r t i a l s . The move- ments n e c e s s a r y f o r recon- s t r u c t i o n a r e i n d i c a t e d .
The s t a c k i n g f a u l t energy o f diamond and s i l i c o n s c a l e s roughly with t h e band gaps and s u g g e s t s t h a t t h e m a t r i x elements f a l l o f f i n t h e same way. To a n a l y s e t h e band s t r u c t u r e s of t h e d i s l o c a t i o n s , t h e p o t e n t i a l s w ( r - R) were e v a l u a t e d f o r t h e atoms along t h e c o r e of t h e d i s l o c a t i o n b u t t h e s e were h a r d l y d i f f e r e n t from t h e bulk p o t e n t i a l s . Consequently a l l t h e c a l c u l a t i o n s d i s c u s s e d below used t h e same a n a l y t i c p o t e n t i a l s w ( r ) found f o r t h e c r y s t a l . The c o o r d i n a t e s o f t h e atoms were d e r i v e d by Marklund f o r s i l i c o n and i n t h e p r e s e n t c a l c u l a t i o n s simply s c a l e d by t h e r a t i o of l a t t i c e parameters.
3.1 The 30° P a r t i a l
The 30° p a r t i a l gives r i s e t o a broad p a r t i a l l y - f i l l e d band which completely o v e r l a p s t h e gap (Fig. 3 ) . D i s p l a c i n g t h e atoms i n Fig. 1 by u = 0.3 R, where R i s t h e c r y s t a l l i n e ban? l e n g t h , caused a s u b s t a n t i a l gap t o appear, Fig. 4. D i f f e r e n t r e c o n s t r u c t i o n s caused s m a l l e r gaps t h a n Fig. 4. We have n o t found i t p o s s i b l e t o e v a l u a t e t h e t o t a l energy change of t h i s r e c o n s t r u c t i o n , b u t g e n e r a l arguments favour some r e c o n s t x u c t i o n /12/. The s o l i t o n , Fig. 5, g i v e s r i s e t o a l e v e l a t EV + 2.43 eV, r a t h e r c l o s e t o t h e T2 vacancy l e v e l .
6 : 90° p a r t i a l , 7 : Reconstructed 90° p a r t i a l . The c r o s s e s denote the c r y s t a l l i n e band edges.
JOURNAL DE PHYSIQUE
3.2 The 90° P a r t i a l
Fig. 6 shows t h e l o c a l d e n s i t y of s t a t e s of t h e u n r e c o n s t r u c t e d 90' p a r t i a l . Again t h e r e is a l a r g e band o v e r l a p p i n g t h e gap a s t h e s e p a r a t i o n between d a n g l i n g bonds i s only 1.33 R. Fig. 7 shows t h e e f f e c t of a r e c o n s t r u c t i o n w i t h u = 0.3 R which h a s p a r t i a l l y c l e a r e d t h e gap, l e a v i n g some a c c e p t o r l e v e l s n e a r Ec - 1.8 eV. The
r e c u r s i o n method broadens energy bands and t h e s e a c c e p t o r l e v e l s may r e a l l y r e s i d e c l o s e r t o Ec. Again o t h e r v a l u e s of u we have used gave s m a l l e r gaps.
The p i c t u r e t h a t h a s emerged from t h e s e c a l c u l a t i o n s i s t h a t t h e p r o p e r t i e s of d i s l o c a t i o n s i n diamond and s i l i c o n a r e s i m i l a r e x c e p t f o r s c a l e . This means, f o r example, t h a t t h e kink m i g r a t i o n energy i s about 7 eV and t h e a c t i v a t i o n energy f o r t h e d i s l o c a t i o n v e l o c i t y i s a b o u t 1 0 eV.
5. ACKNOWLEDGMENTS
We wish t o thank M . I . Heggie f o r u s e f u l d i s c u s s i o n s .
REFERENCES
JONES R, J d e Physique, 40 (1979) C6-33.
HEGGIE M I , JONES R, T h i s Conference.
CLEPIEWTI E , T a b l e s of Atomic Functions, I B M Corp. r e p r i n t , (1965).
PERSSON A , JONES R, Phys. S t a t . Sol., ( b ) 112 (1982) 641.
JONES R , K I N G T, P h i l . Mag., t o be p u b l i s h e d (1983).
JONES R, KING T, Proc. Amsterdam Conf. on L a t t i c e D e f e c t s , (1982).
JONES R, Diamond Conference, Oxford, (1982).
BACHELET G B, BARAFF G A , SCHLUTER M, Phys.
TALWAR D N , TING C S, J Phys. C , S o l i d S t a t e Phys. 15 (1982) 6573.
PIROUZ P, SUMIDA N , LANG A R , HIRSCH P B, Diamond Conference, Oxford,(1982).
PERSSON A, P h i l . Mag. t o be p u b l i s h e d (1983).
JONES R, I n s t . o f Phys. Conf. S e r . , 6 0 (1981) 45.
