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ATOMISTIC MODELS OF
POINT-DEFECT/DISLOCATION INTERACTIONS IN OXIDES
M. Puls
To cite this version:
M. Puls. ATOMISTIC MODELS OF POINT-DEFECT/DISLOCATION INTERACTIONS IN OX- IDES. Journal de Physique Colloques, 1981, 42 (C3), pp.C3-13-C3-29. �10.1051/jphyscol:1981302�.
�jpa-00220693�
JOURNAL DE PHYSIQUE
Colloque C3, supplément au n°6, Tome 42, juin 1981 page C3-13
ATOMISTIC MODELS OF POINT-DEFECT/DISLOCATION INTERACTIONS IN OXIDES
M.P. P u i s
Materials Science Branch, Atomics Energy of Canada Limited, Whiteshell Nuclear Research Establishment, Pinawa, Manitoba, ROE 1L0, Canada
Résumé.- Nous présentons les méthodes et les modèles utilisés pour calculer les énergies et les configurations des défauts ponctuels situés dans le coeur des dislocations dans les solides ioniques. La base théorique pour évaluer les énergies de formation est une méthode générale de Mott-Little- ton (ML). L'utilisation et la validité de cette méthode pour les défauts ponctuels dans les cristaux ioniques parfaits, incorporée dans le programme d'ordinateur HADES de Norgett, sont maintenant bien établies. Nous présen- tons les modifications de cette approche quand elle est appliquée à des réseaux disloqués. Dans ces réseaux, le calcul de l'interaction Coulombienne représente une complication considérable. Nous avons trouvé une solution à ce problème seulement pour des orientations particulières de la ligne de dislocation. Nous décrivons le programme d'ordinateur PDINT qui a été déve- loppé pour suppléer à la méthode ML. Ce programme n'a été appliqué que dans quelques cas. Nous donnerons un résumé et discuterons de ces résultats en mettant l'accent sur la précision qui peut être atteinte avec les modèles et méthodes actuelles.
A b s t r a c t . - We present the methods and models used to c a l c u l a t e the energies and configurations of point defects s i t u a t e d in the core region of d i s l o c a - tions in ionic s o l i d s . The t h e o r e t i c a l b a s i s for the evaluation of formation energies i s a generalized M o t t - L i t t l e t o n (ML) method. The u t i l i z a t i o n and v a l i d i t y of t h i s method for point defects in perfect ionic c r y s t a l s , as embodied in Norgett's HADES program, i s now well e s t a b l i s h e d . We present the modifications to t h i s approach when applied to the dislocated l a t t i c e . In t h i s l a t t i c e , the computation of the long-ranged Coulomb i n t e r a c t i o n presents a s u b s t a n t i a l complication. We have found a solution to t h i s problem only for special o r i e n t a t i o n s of the d i s l o c a t i o n l i n e . We describe the program PDINT which has been developed to implement the ML method. There have been only a few applications of t h i s program. We summarize and discuss these r e s u l t s paying special a t t e n t i o n to the accuracy t h a t can be obtained with present models and methods.
1 . I n t r o d u c t i o n . - At t e m p e r a t u r e s a b o v e a b o u t h a l f t h e m e l t i n g t e m - p e r a t u r e , p l a s t i c f l o w p r o c e s s e s c a n i n f l u e n c e t h e m e c h a n i c a l p r o p e r - t i e s o f o x i d e c r y s t a l s . The d i s c u s s i o n s o f t h e f i r s t Workshop on p o i n t - d e f e c t / d i s l o c a t i o n i n t e r a c t i o n s a t B e l l e v u e , F r a n c e / l / showed t h a t f o r a t h e o r e t i c a l u n d e r s t a n d i n g o f o b s e r v e d d i s l o c a t i o n b e h a v i o u r , we n e e d t o b e a b l e t o d e v e l o p m o d e l s w h i c h a l l o w u s t o r a t i o n a l i z e 1) t h e o b - s e r v e d s l i p p l a n e s i n some m a t e r i a l s f o r w h i c h s i m p l e a r g u m e n t s a p p e a r t o f a i l , 2) t h e amount o f d i s l o c a t i o n d i s s o c i a t i o n , p a r t i c u l a r l y c l i m b d i s s o c i a t i o n t h a t i s o b s e r v e d i n some m a t e r i a l s a n d , 3) t h e r o l e o f p o i n t d e f e c t s i n i n h i b i t i n g d i s l o c a t i o n m o t i o n . I t i s e v i d e n t t h a t a l l o f t h e s e a r e p r o b l e m s i n v o l v i n g t h e i n t e r a c t i o n o f t h e d i s l o c a t i o n
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981302
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e i t h e r w i t h t h e l a t t i c e ( P e i e r l s - N a b a r r o s t r e s s and d i s l o c a t i o n d i s s o - c i a t i o n ) o r w i t h p o i n t d e f e c t s ( s o l u t i o n h a r d e n i n g , jog f o r m a t i o n and d i s l o c a t i o n c l i r n b ) . I n e i t h e r c a s e , an a c c u r a t e r e p r e s e n t a t i o n o f t h e d i s l o c a t i o n ' s c o r e i s r e q u i r e d . T h i s n e c e s s i t a t e s t h e development o f a t o m i s t i c models, a s t h e s e a r e t h e o n l y models c a p a b l e o f d e a l i n g ade- q u a t e l y w i t h t h e two most i m p o r t a n t f e a t u r e s d e t e r m i n i n g d i s l o c a t i o n c o r e p r o p e r t i e s : t h e c r y s t a l s t r u c t u r e and n o n - l i n e a r i n t e r a t o m i c i n - t e r a c t i o n s .
I n t h i s p a p e r w e g i v e a b r i e f review o f t h e methods and models which have been developed t o accomplish o b j e c t i v e s 1 ) t o 3 ) above and summarize some of t h e r e s u l t s . I t i s c o n v e n i e n t t o c o n c e n t r a t e on t h e point-defect/dislocation i n t e r a c t i o n problem, s i n c e t h e d i s l o c a t i o n / l a t t i c e i n t e r a c t i o n h a s a l r e a d y been d i s c u s s e d e x t e n s i v e l y i n a r e c e n t r e v i e w /2/.
2. Background.- Over t h e l a s t twenty y e a r s , t h e v a s t m a j o r i t y o f a t o - m i s t i c models o f p o i n t - d e f e c t / d i s l o c a t i o n i n t e r a c t i o n s have been deve- l o p e d f o r m e t a l s . T h i s emphasis o n m e t a l s i s a r e s u l t o f t h e t e c h n o l o - g i c a l i m p o r t a n c e o f t h e s e m a t e r i a l s . N o n e t h e l e s s , t h e e a r l i e s t c a l c u l a - t i o n s o f a s s o c i a t i o n e n e r g i e s were c a r r i e d o u t by B a s s a n i and Thomson /3/ f o r c a t i o n v a c a n c i e s and i m p u r i t i e s s i t u a t e d i n t h e c o r e of a 1/2 [110] (110) edge d i s l o c a t i o n i n t h e i o n i c c r y s t a l NaC1. These com- p u t a t i o n s were performed b e f o r e t h e development o f l a r g e , high-speed computers. Consequently, i t was i m p o s s i b l e t o c o n s i d e r e x p l i c i t l y more t h a n a few atoms s u r r o u n d i n g t h e p o i n t d e f e c t i n t h e c o r e o f t h e d i s l o c a t i o n . I n a d d i t i o n t o t h i s l i m i t a t i o n , i t was n e c e s s a r y t o employ a nurnbbr of s i m p l i f y i n g assumptions i n o r d e r t o make t h e e q u a t i o n s g i - v i n g t h e e q u i l i b r i u m p o s i t i o n s of t h e atoms t r a c t a b l e . Attempts t o i m - prove upon t h e s e p i o n e e r i n g c a l c u l a t i o n s were n o t made u n t i l some twen- t y y e a r s l a t e r by Woo e t a l . /4/ and by Granzer e t a l . / 5 , 6 / . The con- t r i b u t i o n s o f Granzer e t a l . were i n t h e form o f c o n f e r e n c e a b s t r a c t s g i v i n g few d e t a i l s a s t o e i t h e r methods o r r e s u l t s . P u l s e t a l . / 7 / , s u b s e q u e n t l y gave a more d e t a i l e d a c c o u n t o f t h e f o r m u l a t i o n o f t h e i r own method, g i v e n e a r l i e r i n r e f e r e n c e /4/. They a l s o p r e s e n t e d r e s u l t s f o r t h e a s s o c i a t i o n e n e r g i e s o f s i m p l e c a t i o n and a n i o n v a c a n c i e s w i t h 1/2 [ l l ~ ] (110) edge d i s l o c a t i o n i n MgO. These c a l c u l a t i o n s were l a t e r e x t e n d e d by P u l s , f i r s t l y t o i n c l u d e a g r e a t e r number of p o s i t i o n s s u r r o u n d i n g t h e d i s l o c a t i o n ' s c o r e /6/ a n d , s e c o n d l y , t o examine t h e a s s o c i a t i o n e n e r g i e s o f t h e (FeMg- V1IMg- ~ e ~t r i m e r g ) ~( ' 1 w i t h t h e same d i s l o c a t i o n /9/.
(') I n t h i s and t h e following, we use t h e KrEger-Vink / l o / n o t a t i o n i n d e s c r i b i n g p o i n t d e f e c t s and t h e i r complexes.
The programs developed by P u l s e t a l . /7/ employ an i o n i c i n t e r - a c t i o n model f o r t h e s o l i d . They a r e s u i t a b l e f o r s t u d y i n g d e f e c t i n t e r - a c t i o n s i n t h e a l k a l i h a l i d e s . They can a l s o b e used t o d e t e r m i n e a s s o - c i a t i o n e n e r g i e s i n a v a r i e t y of o x i d e s a s l o n g a s i t i s e s t a b l i s h e d t h a t t h e s e o x i d e s have a predominantly i o n i c i n t e r a c t i o n . I n c o n s i d e r i n g t h e a p p l i c a b i l i t y o f hhese programs t o a wide v a r i e t y of o x i d e s , i t i s , however, i m p o r t a n t t o u n d e r s t a n d t h e l i m i t a t i o n s and s t r e n g t h s of t h e
i o n i c model and t h e t e c h n i q u e used t o d e t e r m i n e d e f e c t e n e r g i e s . With t h i s i n mind, we w i l l o u t l i n e below t h e models and methods used t o de- t e r m i n e a s s o c i a t i o n e n e r g i e s and we w i l l examine p o s s i b l e s o u r c e s o f e r r o r s and u n c e r t a i n t i e s i n h e r e n t i n t h e p r e s e n t approach.
3 . P o t e n t i a l model.- The b a s i c assumption of i o n i c i n t e r a c t i o n s i m - p l i e s t h a t t h e i o n s have i n t e g r a l ( o r c l o s e t o i n t e g r a l ) c h a r g e w i t h a s p h e r i c a l l y symmetric non-overlapping e l e c t r o n d i s t r i b u t i o n . T h i s means t h a t t h e t o t a l e n e r g y o f t h e c r y s t a l c a n be approximated a s t h e sum o f c e n t r a l - f o r c e p a i r p o t e n t i a l s , @ ( r ) , f o r which we assume a s u i t a b l e a n a l y t i c form. The most commonly used p a i r p o t e n t i a l s have what a r e g e n e r a l l y r e f e r r e d t o a s t h e Buckingham form c o n s i s t i n g o f t h e long- ranged e l e c t r o s t a t i c (Coulomb) i n t e r a c t i o n s o f t h e u s u a l r-' form and s h o r t e r ranged r e p u l s i v e and a t t r a c t i v e t e r m s ( 2 ,
.
These l a t O e r t e r m s a r e u s u a l l y made up o f a r e p u l s i v e ( e x p o n e n t i a l ) Born-Mayer term r e p r e - s e n t i n g o v e r l a p and exchange c o n t r i b u t i o n s and some a t t r a c t i v e r-6-
8and r van d e r Waals d i s p e r s i o n t e r m s . An e x t e n s i v e c o m p i l a t i o n o f some o f t h e more r e c e n t l y used p o t e n t i a l s ( i - e . , used i n d e f e c t c a l c u - l a t i o n s ) i s g i v e n i n a handbook by Stoneham /11/. The unknown c o n s t a n t s i n t h e s h o r t - r a n g e d p a r t o f @ ( r ) a r e d e r i v e d by f i t t i n g t o a number o f e x p e r i m e n t a l l y measured c o n s t a n t s such a s t h e l a t t i c e p a r a m e t e r , t h e e l a s t i c c o n s t a n t s and t h e c o h e s i v e e n e r g y . For some m a t e r i a l s , p a r t i - c u l a r l y o x i d e s and f l u o r i d e s , s u p p l e m e n t a r y quantum mechanical
c a l c u l a t i o n s a r e used t o deduce t h e second n e i g h b o u r anion-anion r e - p u l s i v e i n t e r a c t i o n . Two f u r t h e r r e f i n e m e n t s t o t h e s i m p l e p o i n t i o n model (PIM) a r e o f t e n i n t r o d u c e d . These a r e t h e s h e l l and b r e a t h i n g s h e l l models (SM and BSM, r e s ? e c t i v e l y ) which a t t e m p t t o i n c l u d e i o n i c p o l a r i z a t i o n (SM) and many-body e f f e c t s (BSM)
.
The St1 i s now s t a n d a r d i n p o i n t d e f e c t c a l c u l a t i o n s , whereas t h e BSM h a s , s o f a r , been used o n l y r a r e l y . I n t h e SM e a c h i o n i s d i v i d e d i n t o a s e p a r a t e l y c h a r g e d c o r e and s h e l l t o s i m u l a t e t h e i o n i c p o l a r i z a t i o n . There i s no Coulomb i n t e r a c t i o n between c o r e and s h e l l o f t h e same i o n , o n l y an harmonic s p r i n g c o u p l i n g . Because t h e i n t e r a t o m i c f o r c e s a c t t h r o u g h t h e s h e l l s ,(') I n t h e following, we w i l l r e f e r t o t h e s e a s t h e short-ranged i n t e r a c t i o n s .
c3- 16 JOURNAL DE PHYSIQUE
t h e s h o r t - r a n g e d f o r c e s a r e m o d i f i e d by i o n i c p o l a r i z a t i o n . T h i s s i m p l e s h e l l . model i s c a p a b l e o f d e s c r i b i n g b o t h t h e e l a s t i c and d i e l e c t r i c p r o p e r t i e s o f an i o n i c s o l i d . The SM p a r a m e t e r s a r e o b t a i n e d by f i t t i n g t o t h e h i g h and low f r e q u e n c y d i e l e c t r i c c o n s t a n t s p l u s t h e t r a n s v e r s e o p t i c f r e q u e n c y . O f t e n , t o o b t a i n a b e t t e r f i t t o t h e known d i s p e r s i o n c u r v e s , t h e t o t a l i o n i c c h a r g e i s a l s o v a r i e d . However, t h i s c a n r e s u l t i n t o o low a c o h e s i v e e n e r g y /12/. S i n c e t h e S M i s a c e n t r a l - f o r c e , p a i r i n t e r a c t i o n model, i t s a p p l i c a b i l i t y i s s t r i c t l y l i m i t e d t o s o l i d s f o r which t h e Cauchy c o n d i t i o n C12 = C q 4 on t h e second-order e l a s t i c c o n s t a n t s i s obeyed. D e v i a t i o n from t h i s c o n d i t i o n c a n b e a c c o u n t e d f o r by t h e BSI.1 /13, 1 4 / i n which t h e e f f e c t i v e i o n s e p a r a t i o n i s modi- f i e d by t h e symmetric b r e a t h i n g d e f o r m a t i o n o f t h e s h e l l s ; t h e s h e l l r a d i u s responds h a r m o n i c a l l y t o t h e t o t a l e f f e c t i v e o v e r l a p p r e s s u r e o f a l l s u r r o u n d i n g i o n s . I n t h i s way, many-body f o r c e s may be s i m u l a t e d and t h e model need n o t s a t i s f y t h e Cauchy c o n d i t i o n .
The above approach t o p o t e n t i a l i n t e r a c t i o n s i s h i g h l y e m p i r i c a l . Although t h e form o f t h e p o t e n t i a l roughly d e r i v e s i t s j u s t i f i c a t i o n
from quantum mechanical models, t h e p a r a m e t e r s g o v e r n i n g t h e s t r e n g t h of t h e i n t e r a c t i o n a r e d e t e r m i n e d from e x p e r i m e n t a l d a t a . P o t e n t i a l p a r a m e t e r s d e r i v e d by means o f such a p r o c e d u r e c a n n o t be unique and i n o r d e r t o check t h e v a l i d i t y o f a p o t e n t i a l s o d e r i v e d , i t i s ne- c e s s a r y t o e s t a b l i s h d i f f e r e n t v e r s i o n s by v a r y i n g t h e f i t t i n g proce- d u r e . The s e n s i t i v i t y o f c a l c u l a t e d d e f e c t p r o p e r t i e s must t h e n b e t e s t e d a g a i n s t t h e s e v a r i a t i o n s i n t h e p o t e n t i a l . X f u r t h e r c r i t e r i o n f o r t h e r e l i a b i l i t y of t h e p o t e n t i a l i s assumed t o b e i t s a b i l i t y t o r e p r o d u c e a d e f e c t p r o p e r t y , such a s a vacancy m i g r a t i o n e n e r g y , f o r which r e l i a b l e e x p e r i m e n t a l r e s u l t s a r e deemed t o e x i s t /15/. Because
o f t h e h i g h l y e m p i r i c a l n a t u r e o f t h e s e models, i t i s d i f f i c u l t t o make r i g o r o u s e s t i m a t e s o f t h e magnitudes o f t h e p r o b a b l e e r r o r s due
t o t h e a p p r o x i m a t i o n s made and hence t o gauge t h e r e l i a b i l i t y of t h e models. Fundamental q u e s t i o n s such a s t h e d e g r e e and n a t u r e o f i o n i c i t y ,
f o r i n s t a n c e , and i t s e f f e c t on t h e n a t u r e o f t h e bonding can o n l y b e t a c k l e d i n an a d hoc manner w i t h i n t h e above framework. T h i s l a s t ques- t i o n s e e m s t o b e o f l e s s e r i m p o r t a n c e i n t h e a l k a l i h a l i d e s b u t c o u l d p l a y a n i m p o r t a n t f a c t o r i n many o f t h e o x i d e s and remains t o be e x p l o - r e d i n more d e t a i l .
4 . Method o f c o m p u t a t i o n .
-
4 . 1 gpegqy f o r m u l a t i o n .-
The approach t o c a l c u l a t i n g t h e f o r m a t i o n e n e r g y o f p o i n t d e f e c t s i n e i t h e r a d i s l o c a t e d o r a p e r f e c t l a t t i c e h a s been p r e v i o u s l y d e s c r i b e d i n some d e t a i l by P u l s e t a l . /7/. I t s u f f i c e s h e r e t o expand upon t h i s e a r l i e r d e r i v a - t i o n . I n a d d i t i o n , we g i v e a b r i e f o u t l i n e o f t h e computer program PDINTwhich h a s been d e s i g n e d t o implement t h i s method.
The o b j e c t i v e o f t h e PDINT program i s t o c a l c u l a t e t h e f o r m a t i o n e n e r g y , EFD o r EF P
,
o f a p o i n t d e f e c t i n a d i s l o c a t e d o r a p e r f e c t l a t t i c e , r e s p e c t i v e l y . The i n t e r a c t i o n ( o r a s s o c i a t i o n ) e n e r g y Ei n t
'
t h e n f o l l o w s a s t h e d i f f e r e n c e i n f o r m a t i o n energy o f t h e p o i n t d e f e c t i n t h e d i s l o c a t e d and i n t h e p e r f e c t l a t t i c e ; t h a t i s
T h i s i s n e g a t i v e f o r a t t r a c t i v e i n t e r a c t i o n s f o l l o w i n g t h e same s i g n c o n v e n t i o n a s i s employed i n t h e u s u a l e l a s t i c i n t e r a c t i o n model /16/
b u t o p p o s i t e t o t h a t u s e d by B a s s a n i and Thomson /3/. The c a l c u l a t i o n
D P
o f EF o r EF f o l l o w s a g e n e r a l i z e d M o t t - L i t t l e t o n scheme /17/ whereby t h e ( e f f e c t i v e l y i n f i n i t e ) c r y s t a l i s d i v i d e d i n t o a r e g i o n 1, which i n c l u d e s t h e d e f e c t and i n which r e g i o n i n t e r a c t i o n s between t h e i o n s a r e c o n s i d e r e d e x p l i c i t l y . T h i s r e g i o n i s s u r r o u n d e d by an h a r m o n i c a l l y r e l a x e d r e g i o n 2 o f which o n l y a s m a l l f i n i t e boundary l a y e r needs t o be e x p l i c i t l y c o n s i d e r e d .
Formally, t h e t o t a l f o r m a t i o n e n e r g y i s made up of t h e sum o f t h r e e terms: t h e e n e r g y o f r e g i o n 1, E l
(6) ,
t h e i n t e r a c t i o n e n e r g y between r e g i o n s 1 and 2 , E 2 ( & , & ) , and t h e harmonic r e l a x a t i o n energy o f r e g i o n 2 , E3 ( & ) ; t h a t i s ,where
&
a r e t h e i o n c o o r d i n a t e s i n r e g i o n 1 and$,
t h e (harmonic) d i s - placements i n r e g i o n 2 . For t h e PIM:i e r e g . 1 i e r e g . 1
JOURNAL DE PHYSIQUE
s u b l a t t i c e s R 2 ~
where
t h e Madelung ( e l e c t r o s t a t i c ) 9 .
By(r;i)
= q i- -
h o t e n t i a l o f a n i o n a t a n y p o i n t6i
i n t h e R - l a t t i c e jl ~ ~ - @ , l
a l l i o n s
i # i
M 9 . t h e Madelung p o t e n t i a l o f a n
Bi(Ei)
= 4 = i o n a t a n y p o i n tEi
i n t h eI E ~ - E ~ 1
R - l a t t i c e a l l i o n sj # i
&i
= d e f e c t l a t t i c e s i t e s ,Ei
= r e f e r e n c e l a t t i c e s i t e s ,i j - ~ j
='.
= harmonic d i s p l a c e m e n t s i n r e g i o n 2, +IRZB = o u t e r r a d i u s o f r e g i o n 2 A , q . = i o n i c c h a r g e s ,
Q = n e t c h a r g e o f t h e d e f e c t ,
Kk = M o t t - L i t t l e t o n d i s p l a c e m e n t f a c t o r , Vc = u n i t - c e l l volume,
Note t h a t t h e c o n t r i b u t i o n s due t o t h e e l e c t r o s t a t i c i n t e r a c t i o n s o f t h e i o n s i n E l
( E )
and E 2 (&,&) have been d i v i d e d i n t o two components:a Madelung t e r m , r e p r e s e n t i n g t h e e l e c t r o s t a t i c energy o f i n t e r a c t i o n o f an i o n w i t h a l l t h e o t h e r i o n s i n t h e c r y s t a l , p l u s f o u r o t h e r terms which, combined, a r e e q u i v a l e n t t o a d i p o l e - d i p o l e i n t e r a c t i o n . T h i s
l a t t e r c o n t r i b u t i o n a l s o r a n g e s o v e r a l l i o n s i n r e g i o n 2 ( i . e . , t h e r e s t o f t h e c r y s t a l ) b u t a t a r e a s o n a b l e d i s t a n c e away from t h e c e n t r e o f r e g i o n 1, t h e d i s p l a c e m e n t p a t t e r n r e s e m b l e s a c o n t i n u o u s s p h e r i c a l d i p o l e d i s t r i b u t i o n which does n o t i n t e r a c t w i t h d i s p l a c e m e n t d i p o l e s i n r e g i o n 1 . The o n l y r e m a i n i n g long-range i n t e r a c t i o n w i t h r e g i o n 2 i s due t o t h e n e t d e f e c t c h a r g e @. T h i s i n t e r a c t i o n , l i s t e d under E3(,$,), r e a l l y c o n s i s t s o f t h e sum of two t e r m s , one o f which b e l o n g s t o E3(k), t h e o t h e r coming from E2(&,&,). Note t h a t t h e summations i n t h e above e x p r e s s i o n s a r e o v e r two l a t t i c e s , t h e r e f e r e n c e ($) and t h e d e f e c t
( 5 )
l a t t i c e . When c a l c u l a t i n g E~ t h e r e f e r e n c e l a t t i c e i sF ' P
t h e d i s l o c a t e d l a t t i c e , w h i l s t when c a l c u l a t i n g E F , t h e r e f e r e n c e l a t - t i c e i s t h e p e r f e c t l a t t i c e . The d i s l o c a t e d r e f e r e n c e l a t t i c e c o o r d i - n a t e s a r e t h e e q u i l i b r i u m ( r e l a x e d ) c o o r d i n a t e s from an a t o m i s t i c mo- d e 1 ( 3 ) o f t h e s i n g l e d i s l o c a t i o n c o n t a i n i n g no p o i n t d e f e c t s . A s p e c i a l
M M .
d i f f i c u l t y i s p r e s e n t e d by t h e Madelung sums, $i
(zi)
and $i(Ei) ,
s i n c e u n l i k e t h e o t h e r sums i n EF, t h e s e e x t e n d o v e r a l l i o n s of t h e ( i n f i - n i t e ) c r y s t a l . I t i s t h u s n e c e s s a r y t o f i n d r a p i d l y c o n v e r g e n t e x p r e s - s i o n s which c a n r e d u c e t h e summations t o a f i n i t e and manageable num- b e r o f t e r m s . T h i s i s accomplished i n o u r c a s e by e x p l o i t i n g t h e p e r i o - d i c i t y a l o n g t h e d i s l o c a t i o n l i n e d i r e c t i o n u s i n g t h e i n f i n i t e i o n s t r i n g method /18,19/. T h i s method works f o r b o t h t h e p e r f e c t and t h e( ' ) s u c h a s , s a y , g e n e r a t e d by o u r DIPOS program 118-211.
c3- 20 JOURNAL DE PHYSIQUE
d i s l o c a t e d l a t t i c e . However, t h e method o n l y c o n v e r g e s w e l l f o r a l t e r - n a t e l y c h a r g e d s t r i n g s , p r e s e n t l y r e s t r i c t i n g t h e a p p l i c a t i o n o f t h e program t o d i s l o c a t i o n l i n e s o r i e n t e d a l o n g t h e s e f a v o u r a b l e d i r e c - t i o n s ( f o r i n s t a n c e t h e 1/2 [ 1 1 0 ] ( l i 0 ) d i s l o c a t i o n i n t h e NaCl s t r u c t u r e )
.
The p r e s e n t method a l l o w s f r e e v a r i a t i o n o n l y o f t h e c o o r d i n a t e s o f i o n s i n r e g i o n l w h i l s t t h e r e g i o n 2 c o o r d i n a t e s a r e h e l d r i g i d l y i n p l a c e . The i o n d i s p l a c e m e n t s i n r e g i o n 2 a r e e s t i m a t e d , f o l l o w i n g t h e method o f Wott and L i t t l e t o n / 2 2 / , by assuming t h e m a t e r i a l r e s p o n d s a s a d i e l e c t r i c continuum. S p e c i f i c a t i o n o f t h e d i s p l a c e m e n t s i s
s t r a i g h t f o r w a r d b e c a u s e we can d e t e r m i n e t h e s t r e n g t h o f t h e d i s p l a c e - ment f i e l d s i m p l y from t h e n e t c h a r g e , Q , o f t h e d e f e c t ( o r d e f e c t s ) i n r e g i o n 1. A n p r o p r i a t e e x p r e s s i o n s a r e g i v e n by N o r g e t t /17/. S t r i c t - l y s p e a k i n g , t h e r e s h o u l d a l s o b e d i s p l a c e m e n t s due t o t h e e l a s t i c r e s p o n s e of t h e medium t o t h e d e f e c t . I n t r o d u c i n g t h i s l a t t e r f i e l d however, i s l e s s s t r a i g h t f o r w a r d , a s t h e s t r e n g t h o f t h e i n t e r a c t i o n i s n o t known b e f o r e h a n d /23/. A d d i t i o n a l l y , even i n c u b i c c r y s t a l s , t h e d i s p l a c e m e n t s can b e s t r o n g l y a n i s o t r o p i c . So f a r , we have assumed t h a t f o r a c h a r g e d d e f e c t , t h e most s i g n i f i c a n t c o n t r i b u t i o n t o t h e d i s p l a c e m e n t s f i e l d comes from t h e induced p o l a r i z a t i o n and t h i s h a s a l s o been t h e p r a c t i c e i n t h e m a j o r i t y o f d e f e c t c a l c u l a t i o n s done f o r o t h e r w i s e p e r f e c t i o n i c c r y s t a l s /23/. A d d i t i o n a l l y , t h i s a p p r o x i m a t i o n h a s been used f o r p o i n t d e f e c t complexes which c a r r y no n e t c h a r g e / 2 4 , 25/ b u t , i n t h i s c a s e , t h e e r r o r s i n n e g l e c t i n g t h e e l a s t i c f i e l d may be s i g n i f i c a n t . A f l e x i b l e e l a s t i c t r e a t m e n t which can t a k e f u l l a c c o u n t o f t h e e l a s t i c a n i s o t r o p y o f t h e boundary r e g i o n w i l l b e n e c e s s a r y t o f u l l y e x p l o r e t h e consequences of t h i s a p p r o x i m a t i o n .
The e x p r e s s i o n s f o r Ep g i v e n by e q u a t i o n s ( 2 ) t o ( 5 ) a r e f o r t h e P I N . To e x t e n d t h e s e t o t h e SM, t h e Coulomb i n t e r a c t i o n s a r e simply sub- d i v i d e d i n t o s e p a r a t e i n t e r a c t i o n s due t o t h e c o r e and s h e l l c h a r g e s , w h i l s t s h o r t - r a n g e i n t e r a c t i o n s a r e r e s t r i c t e d t o i n t e r a c t i o n s between s h e l l c e n t r e s o n l y . The o n l y term f o r m a l l y n o t i n c l u d e d i n e q u a t i o n s
( 2 ) t o ( 5 ) i n t h e SM v e r s i o n , t h e n , i s t h e s p r i n g - c o n s t a n t e n e r g y due t o t h e s e p a r a t i o n o f t h e s h e l l and c o r e of t h e same i o n . T h i s added e n e r g y i s t h e SM c o n t r i b u t e s t o El (&) and E j
( 6 )
a s f o l l o w s :i e r e g . 1
where ki = s h e l l - c o r e s p r i n g - c o n s t a n t and c and s r e f e r t o c o r e and s h e l l c o o r d i n a t e s , r e s p e c t i v e l y . The sum i n e q u a t i o n ( 7 ) s h o u l d , s t r i c t - l y s p e a k i n g , r a n g e o v e r a l l i o n s i n r e g i o n 2 b u t , a t a r e a s o n a b l e d i s - t a n c e from t h e d e f e c t c e n t r e , t h e i o n i c p o l a r i z a t i o n s have g e n e r a l l y become n e g l i g i b l e and can t h u s be s a f e l y n e g l e c t e d . I n t h e S M , i n a d d i - t i o n t o t h e above e n e r g y , t h e r e i s a l s o a s m a l l a m b i g u i t y i n a s s i g n i n g s e p a r a t e harmonic d i s p l a c e m e n t s t o t h e s h e l l and t h e c o r e i n r e g i o n 2.
T h i s ambiguity can b e circumvented a s d i s c u s s e d i n more d e t a i l by Nor- g e t t /17/.
4.2 MjqjIjzai.jo~.- Given t h e r e l a t i o n s ( 2 ) t o ( 7 ) , t h e problem i s t o f i n d t h e c o o r d i n a t e s ,
zi,
which minimize t h e t o t a l f o r m a t i o n e n e r g y , EF. T h a t i s , we need t o f i n d t h e minimum o f a f u n c t i o n o f many v a r i a b l e s . Two p a r t i c u l a r l y e f f i c i e n t methods f o r a c c o m p l i s h i n g t h i s a r e t h e v a r i a - b l e m e t r i c and t h e c o n j u g a t e g r a d i e n t methods /26,27/. The v a r i a b l e m e t r i c method i s by f a r t h e more e f f i c i e n t o f t h e two, b u t r e q u i r e s s t o r a g e o f t h e m a t r i x o f second d e r i v a t i v e s o f t h e e n e r g y . T h i s i s manageable f o r p o i n t d e f e c t s i n p e r f e c t c r y s t a l s , s i n c e t h e h i g h d e g r e e of l a t t i c e symmetry can b e e x p l o i t e d t o r e d u c e t h e t o t a l number o f m a t r i x e l e m e n t s t h a t a c t u a l l y need t o be s t o r e d . I t h a s been implemen- t e d i n N o r g e t t ' s HADES program /17/, c o n t r i b u t i n g s i g n i f i c a n t l y t o t h e e f f i c i e n c y of t h i s program. F o r d i s l o c a t e d l a t t i c e s , on t h e o t h e r hand, t h e c o n j u g a t e g r a d i e n t method i s more a p p r o p r i a t e , s i n c e o n l y f i r s t d e r i v a t i v e s o f t h e e n e r g y need t o b e s t o r e d . We have used a v e r s i o n of t h e c o n j u g a t e g r a d i e n t method developed by S i n c l a i r ( a s c i t e d by P u l s /19/)which r e q u i r e s no c a l c u l a t i o n o f t h e e n e r g y d u r i n g t h e minimiza- t i o n p r o c e d u r e . We have mainly used t h i s v e r s i o n i n t h e PDINT c a l c u l a - t i o n s b e c a u s e t h e u s u a l v e r s i o n , u t i l i z i n g b o t h f o r c e s and t o t a l e n e r - gy g e n e r a l l y t e r m i n a t e s t o o e a r l y due t o rounding e r r o r d i f f i c u l t i e s . I n t h o s e c a s e s ( 4 ) where b o t h v e r s i o n s were a b l ~ t o r e d u c e t h e f o r c e s t o t h e s p e c i f i e d l e v e l ( u s u a l l y a b o u t 0 . 0 0 3 eV/A), t h e EF-values ob- t a i n e d w i t h e i t h e r method were w i t h i n 0.01 eV.4 . 3 S t r u c t u r e o f theepgggg-rL- A f e a t u r e o f t h e p r e s e n t f o r m u l a t i o n which c o n t r i b u t e s t o t h e c o m p l e x i t y o f PDINT, i s t h e need t o p r o v i d e a c o r r e s p o n d e n c e between t h r e e l a t t i c e s . I t i s e v i d e n t t h a t a t l e a s t
( 4 ) ~ l l perfect l a t t i c e minimizations and a few dislocated l a t t i c e cases.
c3-22 JOURNAL DE PHYSIQUE
two l a t t i c e s a r e n e e d e d b e c a u s e of t h e p r e s e n c e , i n e q u a t i o n s ( 2 ) t o ( 7 ) , o f terms depending on b o t h
g ,
t h e c u r r e n t l a t t i c e andE,
t h e r e - f e r e n c e l a t t i c e . T h i s l a t t e r l a t t i c e , however, must a l s o , a s r e q u i r e d , t a k e on a n o n - c y c l i c o r a c y c l i c form. The c y c l i c form i s n e c e s s a r y when e v a l u a t i n g t h e Madelung e n e r g i e s and f o r c e s . I n o u r method o f eva- l u a t i n g t h e s e t e r m s , each i o n i n o r5
i n t e r a c t s w i t h an i n f i n i t e l y l o n g row o f i o n s i nE.
( I t i s i m p l i c i t i n t h i s approach t h a t t h e i o n s i n t h e row have m a i n t a i n e d t h e i r l a t t i c e p e r i o d i c i t y ) . Thus f o r t h e d i s l o c a t e d l a t t i c e , t h e row d i r e c t i o n must c o i n c i d e w i t h t h a t o f t h e d i s l o c a t i o n l i n e . I f t h e i o n c h a r g e s do n o t a l t e r n a t e i n s i g n a l o n g t h a t row, t h e r e w i l l be convergence d i f f i c u l t i e s f o r f o r c e v a r i a t i o n s p e r p e n d i c u l a r t o t h e row and t h i s p a r t i c u l a r method o f e s t i m a t i n g Ma- d e l u n g sums b r e a k s down.The program i s o r g a n i z e d a s f o l l o w s . The f i r s t s t e p i s t o s e t up c y c l i c and n o n - c y c l i c l a t t i c e s i n c l u d i n g a m a ~ p i n g scheme ~ r h i c h makes i t p o s s i b l e t o go from a l o c a t i o n i n t h e n o n - c y c l i c l a t t i c e t o a c o r r e s - ponding l o c a t i o n i n t h e c y c l i c l a t t i c e . The r e f e r e n c e l a t t i c e i s
s t o r e d i n t h e s e two c o n f i g u r a t i o n s . The n o n - c y c l i c l a t t i c e i s subse- q u e n t l y r e a r r a n g e d i n t o a s p h e r i c a l c o n f i g u r a t i o n c o n s i s t i n g o f a s p h e r e of r e g i o n 1- atoms s u r r o u n d e d by a s p h e r e o f boundary atoms ( r e g i o n 2A). T h i s i s t h e d e s i r e d c o n f i g u r a t i o n f o r which t h e c a l c u l a t i o n o f t h e d e f e c t f o r m a t i o n e n e r g i e s w i l l be c a r r i e d o u t a s o u t l i n e d i n s e c - t i o n 4 . 1 . The i n i t i a l c o o r d i n a t e p o s i t i o n s i n t h i s f i n a l arrangement a r e g i v e n by t h e r e f e r e n c e l a t t i c e l o c a t i o n s p o l a r i z e d ( d i s p l a c e d ) a c c o r d i n g t o t h e e l e c t r i c d i s p l a c e m e n t f i e l d due t o t h e n e t c h a r g e o f t h e d e f e c t l o c a t e d a t t h e c e n t r e o f s p h e r i c a l r e g i o n 1. I n c o n j u n c t i o n w i t h s e t t i n g up t h e s p h e r i c a l r e g i o n s , a n o t h e r mapping scheme i s g i v e n which makes i t p o s s i b l e t o f i n d , g i v e n an atom i n t h e s p h e r i c a l l a t t i c e , i t s c o r r e s p o n d i n g r e f e r e n c e l a t t i c e l o c a t i o n a s s t o r e d i n t h e c y c l i c and n o n - c y c l i c l a t t i c e s . Note t h a t when t h e r e f e r e n c e l a t t i c e i s d i s - l o c a t e d , t h e atom p o s i t i o n s f o r t h e s i n g l e d i s l o c a t i o n c o n t a i n i n g no p o i n t d e f e c t s must a l s o be s u p p l i e d . I n o u r c a s e t h e s e c o o r d i n a t e s a r e g e n e r a t e d u s i n g t h e f l e x i b l e boundary package DIPOS /IS
-
20/. Given t h i s f a i r l y complex scheme f o r l o c a t i n g atoms i n t h e i r v a r i o u s l a t t i c e s , t h e r e m a i n i n g p a r t o f t h e program c o n s i s t s o f s t r a i g h t f o r w a r d e n e r g y and f o r c e c a l c u l a t i o n s and t h e i r r e s p e c t i v e m i n i m i z a t i o n . Note, though, t h a t i n t h e p r e s e n t g e n e r a l i z e d M o t t - L i t t l e t o n scheme, o n l y c o o r d i n a t e s i n r e g i o n 1 o f t h e s p h e r i c a l r e g i o n s need t o b e v a r i e d .5 . R e s u l t s . - A c t u a l a p p l i c a t i o n s o f t h e PDINT program h a v e , s o f a r , been f a t r l y l i m i t e d . Using m o s t l y t h e s h e l l model v e r s i o n , t h e program h a s been a p p l i e d t o d e t e r m i n e t h e f o r m a t i o n energ-- o f a n i o n and c a t i o n
v a c a n c i e s s u r r o u n d i n g t h e c o r e of 1/2 [110] ( l i 0 ) edge d i s l o c a t i o n i n MgO /7,8/. Eint v a l u e s f o r vacancy p o s i t i o n s up t o 4b from t h e c o r e
(where b i s t h e B u r g e r s v e c t o r ) have been c a l c u l a t e d and t h e r e s u l t s compared w i t h t h e p r e d i c t i o n s o f t h e u s u a l i s o t r o p i c e l a s t i c model assuming t h a t t h e p o i n t d e f e c t a c t s a s a m i s f i t t i n g inhomogeneity
( t h e l a t t e r r e f e r r i n g t o t h e d i f f e r e n c e i n e l a s t i c p r o p e r t i e s o f t h e d e f e c t and t h e m a t r i x )
.
A s e x p e c t e d , n e a r t h e d i s l o c a t i o n ' s e d g e , t h e r e was n e i t h e r q u a l i - t a t i v e nor q u a n t i t a t i v e agreement between t h e p r e d i c t i o n s of t h e two models. The l a c k o f agreement a t d i s t a n c e s a s g r e a t a s 4b from t h e d i s l o c a t i o n ' s c e n t r e was, however, s u r p r i s i n g . T h i s d i s a g r e e m e n t may r e f l e c t a s much t h e u n c e r t a i n t i e s e v i d e n t i n some o f t h e p a r a m e t e r s o f t h e e l a s t i c m o d e l ( i . e . , t h e r e l a x a t i o n volume and t h e e f f e c t i v e e l a s t i c c o n s t a n t s of t h e d e f e c t ) a s i n t h e e l a s t i c model i t s e l f . I n t h e absence o f r e l i a b l e e x p e r i m e n t a l d e t e r m i n a t i o n s , t h e s e p a r a m e t e r s a r e n o t a c c e s s i b l e w i t h i n t h e e l a s t i c model b u t must b e d e r i v e d u s i n g a t o m i s t i c methods. Recent d i s c u s s i o n s i l l u s t r a t e some o f t h e d i f f i c u l - t i e s and u n c e r t a i n t i e s i n h e r e n t i n e s t i m a t e s o f r e l a x a t i o n volumes b a s e d on a t o m i s t i c c a l c u l a t i o n s / 2 3
-
30/.Following a s u g g e s t i o n t h a t n e u t r a l p o i n t d e f e c t complexes may b e t h e more i m p o r t a n t d i s l o c a t i o n p i n n e r s /31,32/, we a l s o looked a t t h e i n i s r a c t i o n o f t h e ( F e '
-
V"-
Fe' ) X t r i m e r ( ' ) w i t h t h e aboveM9 Mg M9
edge d i s l o c a t i o n and r e p o r t e d some preliminary r e s u l t s / 9 / . The work i n hand i n c l u d e s e x t e n s i o n o f t h e s e t y p e s of s t u d i e s t o NaCl /34/ and a more d e t a i l e d a n a l y s i s o f t h e I7e3+ - t r i m e r s t u d y i n MgO / 3 5 / .
The r e s u l t s t o d a t e on MgO f o r b o t h t h e s i m p l e v a c a n c i e s and t h e t r i m e r g i v e a maximum b i n d i n g e n e r g y which i s i n t h e r a n g e -1 t o -2 eV
( s e e below f o r a d i s c u s s i o n of some c o r r e c t i o n s t o t h e p u b l i s h e d r e - s u l t s ) , b u t t h e r e s u l t s f o r t h e t r i m e r a r e q u i t e s e n s i t i v e t o t h e i m - p u r i t y p o t e n t i a l s used. We have n o t found any p a r t i c u l a r l y s t e e p v a r i a - t i o n s o f Eint v e r s u s d i s t a n c e between d e f e c t and d i s l o c a t i o n c e n t r e s a s a n t i c i p a t e d by w h i t w o r t h /36/ o r i n f e r r e d by R i t c h i e and Sprungmann /37/. I n f a c t , f o r t h e s i m p l e v a c a n c i e s , t h e x - v a r i a t i o n ( 6 ) i s s i m i l a r t o what one might e x p e c t from t h e e l a s t i c model. We have a l s o s o u g h t , b u t have n o t found, any i n d i c a t i o n , a s s u g g e s t e d by Whitworth /38/, t h a t a vacancy l o c a t e d a t t h e edge o f t h e e x t r a p l a n e of atoms ( i . e . , a t t h e
(5) A t t h e p r e s e n t workshop i t was p o i n t e d out (Brian Henderson, p r i v a t e communica- t i o n ) t h a t t h i s t r i m e r had a c t u a l l y n o t been e x p e r i m e n t a l l y observed. However, r e c e n t r e s u l t s suggest t h a t t h i s d e f e c t may now have been i d e n t i f i e d 1 3 3 1 .
(6) x i s t h e d i s t a n c e between d e f e c t c e n t r e s .
C3-24 JOURNAL DE PHYSIQUE
maximum b i n d i n g p o s i t i o n , t h e 'D' p o s i t i o n shown i n f i g u r e 1 of r e f e - r e n c e /8/ may b e a b l e t o move a l o n g w i t h t h e c o r e i n a n a t h e r m a l manner a s t h e d i s l o c a t i o n g l i d e s a l o n g t h e s l i p p l a n e . From t h e Eint v e r s u s r a d i a l d i s t a n c e p r o f i l e s , i t i s p o s s i b l e t o o b t a i n a rough e s t i m a t e of t h e i n t e r a c t i o n f o r c e between d e f e c t and d i s l o c a t i o n . T h i s i s o f t h e o r d e r o f 1 0 - l o N f o r b o t h t h e v a c a n c i e s and t h e t r i m e r and i s a b o u t an o r d e r o f magnitude s m a l l e r t h a n i n f e r r e d from s o l u t i o n h a r d e n i n g ex- p e r i m e n t s /32/. The r e s u l t s t o d a t e s u g g e s t t h a t , i n terms o f p i n n i n g s t r e n g t h , it i s n o t p o s s i b l e t o d i s t i n g u i s h between t h e two t y p e s o f d e f e c t s .
R e c e n t l y we have a l s o found a n e r r o r i n t h e s h e l l model v e r s i o n o f t h e program i n v o l v i n g t h e s p r i n g - c o n s t a n t e n e r g y . We found t h a t t h e r e f e r e n c e energy t e r m i n e q u a t i o n ( 6 ) had been i n a d v e r t e n t l y o m i t t e d . T h i s o m i s s i o n was n o t n o t i c e d when comparing o u r r e s u l t s o f vacancy f o r m a t i o n e n e r g i e s i n t h e p e r f e c t l a t t i c e w i t h t h o s e o f C a t l o w e t a 1 . / 2 3 / , b e c a u s e i n t h a t c a s e t h i s t e r m i s z e r o . However, when t h e d i s l o c a t e d l a t t i c e i s t h e r e f e r e n c e l a t t i c e , t h i s s i t u a t i o n no l o n g e r a p p l i e s . Moreover, t h e s p r i n g - c o n s t a n t e n e r g y v a r i e s w i t h p o s i t i o n ; t h a t i s , it, depends on where t h e c e n t r e o f r e g i o n 1 i s l o c a t e d w i t h r e s p e c t t o t h e d i s l o c a t i o n ' s c e n t r e . T h i s r e f e r e n c e - l a t t i c e s p r i n g - c o n s t a n t e n e r g y c o n t r i b u t e s an a d d i t i o n a l energy o f a b o u t -0.35 eV t o -0.11 eV t o Eint.
S i n c e ~ . t i s a c o n s t a n t t e r m i n EF, o m i s s i o n of i t , f o r t u n a t e l y , had no e f f e c t on t h e p r e v i o u s l y d e t e r m i n e d r e l a x e d c o o r d i n a t e s . D e t a i l s o f t h e c o r r e c t e d r e s u l t s w i l l b e r e p o r t e d e l s e w h e r e /39/ i n c l u d i n g a r e a n a l y s i s of t h e comparison between t h e r e s u l t s o f t h e a t o m i s t i c and e l a s t i c models. T h i s r e a n a l y s i s was f u r t h e r prompted by t h e r e c e n t improved t h e o r e t i c a l e s t i m a t e s o f t h e volume o f f o r m a t i o n i n i o n i c c r y s t a l s /28- 30,'.
I n r e f e r e n c e s / 7 / and / 0 / , a t t e m p t s were made t o e s t i m a t e t h e a c c u r a c y o f t h e c a l c u l a t i o n by l o o k i n g a t t h e e f f e c t s o f v a r y i n g a ) t h e s i z e s o f t h e r e g i o n s , b ) t h e d e s i r e d r e d u c t i o n i n t h e f o r c e s , and
C ) t h e p o t e n t i a l . I t w i l l be i n s t r u c t i v e t o a m p l i f y on t h i s h e r e by l o o k i n g a t t h e v a r i o u s components which combine t o g i v e E A s can b e
F
-
s e e n from e q u a t i o n s ( 3 ) t o ( 5 ) f o r t h e PIM, t h e t o t a l e n e r g y c o n s i s t s o f two t y p e s o f i n t e r a c t i o n s , e l e c t r o s t a t i c and s h o r t - r a n g e . F o r t h e SM t h e r e i s a l s o t h e s p r i n g - c o n s t a n t e n e r g y g i v e n by e q u a t i o n s ( 6 ) and
( 7 ) , which may be viewed a s simply a n o t h e r c o n t r i b u t i o n t o t h e s h o r t - r a n g e i n t e r a c t i o n e n e r g y . I n t h e examples shown i n t a b l e I, we have t h u s s e p a r a t e d t h e e n e r g y i n t o Coulomb t e r m s from El and E2 ( t h e s e a r e f u r t h e r s u b d i v i d e d i n t o Madelung and d i p o l e - d i p o l e c o n t r i b u t i o n s ) , s h o r t - r a n g e terms from El and E2 p l u s t h e two main t e r m s making up E3
Table I.-
E n e r g i e s o f formation*of a c a t i o n vacancy i n MgO
* A l l c a l c u l a t i o n s employ a r e g i o n 1 r a d i u s of 3.1re r e g i o n 2 r a d i u s o f 5 . l r o where r i s t h e i n t e r i o n i c d i s t a n c e = 0.2106 nm.
P o s i t i o n
P.L.
t
A 7%
t
P e r f e c t l a t t i c e+ c e n t r a l atom above j o g edge f o r a t y p e I symmetry 112 [I 1 0 1 ( l i 0 ) edge d i s l o c a t i o n ( i l l u s t r a t e d i n r e f e r e n c e s / 7 / and 181).
E n e r g i e s (eV)
Table 11.-
I n t e r a c t i o n energys, Eint, f o r a c a t i o n vacancy i n MgO T o t a l : Ep
23.92 24.49
*
The same c o n d i t i o n s were u s e d a s t h o s e t o g e n e r a t e the d a t a o f t a b l e I.Madelung
47.82 49.91
P o s i t i o n t
A A2 A3 A4
A-row f o r a type-I s y m e t r y 112 [110] (170) edge d i s l o c a t i o n ( i l l u s t r a t e d i n r e f e r e n c e 1 8 1 ) .
Dipole- d i p o l e
-
23.12-
23.601
E n e r g i e s (eV) Net Coulomb
1.61 2. 16 2.32 0.82
S h o r t - range
-
0.03-
1.14Short-range
-
1 . 1 1 - 1.75-
2.60-
0.89Spring- c o n s t a n t
0.26 0.33
E23A E23B
1.38 - 2.39 1.38
-
2.39S p r i n g - c o n s t a n t
0.07 0.07 0.07 0.07
T o t a l : E i n t 0.65 0.48
-
0.21 0.00c3- 26 JOURNAL DE PHYSIQUE
( t h e o n e i n v o l v i n g t h e b r a c e s , t h e o t h e r t h e i n t e g r a l ; we have c a l l e d t h e s e E23A and E23B, r e s p e c t i v e l y ) . I n a d d i t i o n , f o r t h e S M , we keep t r a c k o f t h e s p r i n g c o n s t a n t e n e r g i e s i n r e g i o n 1 ( t h e c o r r e s p o n d i n g e n e r g y i n r e g i o n 2 i s n e g l i g i b l e ) . I n t a b l e I1 we show t h e same e n e r g i e s which add up t o g i v e t h e i n t e r a c t i o n e n e r g y o f a c a t i o n vacancy a t va- r i o u s p o s i t i o n s p a r a l l e l t o t h e s l i p p l a n e a l o n g t h e A-row ( i l l u s t r a t e d i n / 8 / ) o f a 1 / 2 [110] (1i0) edge d i s l o c a t i o n o f t y p e - I symmetry i n MgO.
Table I shows t h a t , a s e x p e c t e d , t h e l a r g e s t c o n t r i b u t i o n t o t h e f o r - mation e n e r g y comes from t h e Coulomb i n t e r a c t i o n , t h e s h o r t - r a n g e and s p r i n g - c o n s t a n t e n e r g i e s c o n s t i t u t i n g o n l y a b o u t 1 t o 3% o f t h e t o t a l e n e r g y ( )
.
T h i s p i c t u r e i s , however, s i g n i f i c a n t l y changed when l o o k i n g a t t h e i n t e r a c t i o n e n e r g y , g i v e n i n t a b l e 11. Now b o t h t h e Coulomb and t h e s h o r t - r a n g e c o n t r i b u t i o n s a r e o f comparable magnitude. S i n c e t h e y a r e o f o p p o s i t e s i g n , t h e i n t e r a c t i o n e n e r g y i s much s m a l l e r t h a n e i t h e r o f t h e s e q u a n t i t i e s . T h i s l a t t e r r e s u l t , w i t h some e x c e p t i o n s , h a s been found t o b e t h e r u l e f o r most of t h e o t h e r p o s i t i o n s s t u d i e d . The l a r g e c o n t r i b u t i o n t h a t t h e s h o r t - r a n g e p o t e n t i a l makes t o t h e i n t e r a c t i o n e n e r g y means t h a t we c a n n o t b e anywhere n e a r l y a s c o n f i d e n t of t h e accu- r a c y o f E a s we can of t h e f o r m a t i o n energy. T h i s was a l s o e v i d e n ti n t
i n o u r e a r l i e r s t u d y /8/ where we compared E - v a l u e s c a l c u l a t e d u s i n g i n t
two d i f f e r e n t p o t e n t i a l s and found d i f f e r e n c e s o f a s much a s 0.4 eV between t h e two models, t h e l a r g e s t d i f f e r e n c e s b e i n g f o r some atom po- s i t i o n s l o c a t e d c l o s e t o t h e d i s l o c a t i o n ' s edge. I t i s a t t h e s e p o s i - t i o n s t h a t some o f t h e i n t e r i o n i c d i s t a n c e s a r e most s e v e r e l y a l t e r e d from t h e i r p e r f e c t l a t t i c e s e p a r a t i o n s and t h e v a l i d i t y o f t h e model p o t e n t i a l , b a s e d on p r o p e r t i e s p e r t i n e n t t o p e r f e c t l a t t i c e s e p a r a t i o n s , i s most s u s p e c t .
I t i s d i f f i c u l t , w i t h t h e p r e s e n t models and methods, t o a r r i v e a t a v e r y p r e c i s e e s t i m a t e o f t h e a c c u r a c y o f t h e E - v a l u e s . The
i n t
o n l y p r a t i c a l method seems t o b e t o r e p e a t t h e c a l c u l a t i o n s u s i n g d i f f e - r e n t p o t e n t i a l s and t o compare t h e r e s u l t s . I n t h e p r e s e n t c a s e , t h i s i s an e x p e n s i v e and time-consuming p r o c e s s which, a s a r e s u l t , i s un- l i k e l y t o be e x h a u s t i v e . Based on t h i s approach, t h e p r e v i o u s s t u d y by P u l s /8/ i n d i c a t e s t h a t t h e maximum i n t e r a c t i o n e n e r g y f o r s i m p l e va- c a n c i e s a t t h e edge o f t h e e x t r a p l a n e can be e s t i m a t e d t o no b e t t e r t h a n a 4 0 % . Although t h i s i s much l e s s t h a n t h e a c c u r a c y we can a c h i e v e i n t h e f o r m a t i o n e n e r g y , i t i s s t i l l a g r e a t improvement compared t o t h e o n l y o t h e r p r a c t i c a l a l t e r n a t i v e , t h e e l a s t i c continuum i n t e r a c t i o n
( 7 ) T h i s i s l e s s t h a n t h e u s u a l 10% o b t a i n e d i n t h e a l k a l i h a l i d e s , because t h e i o n i c charge i n t h e a l k a l i n e e a r t h o x i d e s i s twice t h h t of the h a l i d e s .
model. Moreover, a l t h o u g h a b s o l u t e v a l u e s o f t h e i n t e r a c t i o n energy have a l a r g e u n c e r t a i n t y a s s o c i a t e d w i t h them, v a r i a t i o n s of Eint w i t h p o i n t - d e f e c t / d i s l o c a t i o n s e p a r a t i o n seem t o b e much l e s s s e n s i t i v e t o u n c e r t a i n t i e s i n t h e p o t e n t i a l .
6. Conclusions.- We have d e s c r i b e d t h e methods and models developed t o e v a l u a t e p o i n t - d e f e c t f o r m a t i o n e n e r g i e s i n p e r f e c t and d i s l o c a t e d i o n i c s o l i d s . The methods a r e implemented by means o f t h e computer pro- gram P D I N T . The t h e o r e t i c a l b a s i s f o r t h e e v a l u a t i o n o f f o r m a t i o n e n e r - g i e s i s a g e n e r a l i z e d M o t t - L i t t l e t o n method. T h i s method h a s found wide a p p l i c a t i o n i n t h e s t u d y o f p o i n t d e f e c t s i n p e r f e c t i o n i c c r y s t a l s t h r o u g h N o r g e t t ' s program HADES. The computer program PDINT r e p r e s e n t s a n a t u r a l e x t e n s i o n o f t h e b a s i c formalism o f HADES t o a more g e n e r a l t y p e o f l a t t i c e such a s a d i s l o c a t e d l a t t i c e . I t i s e v i d e n c e d from t h e l i m i t e d s t u d i e s p u b l i s h e d t o d a t e t h a t , compared t o p o i n t d e f e c t c a l c u - l a t i o n s i n p e r f e c t c r y s t a l s , c o r r e s p o n d i n g e f f o r t s i n d i s l o c a t e d c r y s - t a l s a r e s t i l l i n t h e i r i n f a n c y . A s i d e from t h o s e r e s u l t s g e n e r a t e d w i t h PDINT / 7 - 9 / and t h e e a r l i e s t e v a l u a t i o n s o f B a s s a n i and Thomson /3/, t h e r e a p p e a r t o be no o t h e r p u b l i s h e d a c c o u n t s o f a t o m i s t i c c a l - c u l a t i o n s o f point-defect/dislocation i n t e r a c t i o n s i n i o n i c s o l i d s . The c a l c u l a t i o n s r e p o r t e d a r e shown t o b e complex, r e q u i r i n g two d i f f e r e n t computer programs. They a r e a l s o t i m e consuming, b o t h i n t e r m s o f p r e -
( 8 ) p a r a t i o n and computer t i m e
.
There i s t h u s an onus t o choose t h e t o p i c s t o b e s t u d i e d j u d i - c i o u s l y . I n p a r t i c u l a r , f o r t h e more complex d e f e c t s , it seems unpro- d u c t i v e t o c o v e r a l l t h e p o s s i b l e o r i e n t a t i o n s and d e f e c t - d i s l o c a t i o n s e p a r a t i o n s . The o b j e c t , i n t h e s e c a s e s , would a p p e a r t o b e t o u s e t h e r e s u l t s o f t h e a t o m i s t i c model p r i m a r i l y a s a g u i d e t o improve upon t h e more approximate models.
ato or
t h e p u b l i s h e d r e s u l t s , t y p i c a l CPU time p e r E - v a l u e i s f r o m 0.5 h - 1 hon a CDC CYBER 170/175 computer. i n t
JOURNAL DE PHYSIQUE
R e f e r e n c e s
/1/ M i t c h e l l , T.E., Hobbs, L.W., H e u e r , A . H . , C a s t a i n g , J . , Cadoz,
J.
and P h i l i b e r t ,J . ,
A c t a M e t a l 111
(1979) 1677 /2/ P u l s , M.P., P r o c e e d i n g s o f a n I n t e r n a t i o n a l C o n f e r e n c e onD i s l o c a t i o n M o d e l l i n g o f P h y s i c a l S y s t e m s , G a i n e s v i l l e , F l o r i d a , J u n e 1 9 8 0 , A c t a / S c r . M e t a l l . , i n p r e s s
/3/ B a s s a n i , F. a n d Thomson, R . , Phys. Rev.
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(1956) 1264 /4/ koo, C.H., P u l s , M.P. a n d N o r g e t t , M . J . , J . P h y s i q u e C o l l o q .37 (1976) C7-557
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/5/ G r a n z e r , F . , B e l z n e r , V. a n d P o t s t a d a , H . , J . P h y s i q u e C o l l o q .
37
(1976) C7-557/6/ G r a n z e r , F., B e l z n e r , V . , Biicher, M., P e t r a s c h , P . a n d P o t s t a d a , H . , I n t . Conf. D e f e c t s i n I n s u l a t i n g C r y s t a l s , G a t l i n b u r g 1 9 7 7 , A b s t r a c t s , p. 155
/7/ P u l s , M.P., Woo, C.H. a n d N o r g e t t , M . J . , P h i l o s . Mag.
36 ( 1 9 7 7 ) 1457
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