THE THEORY OF CRYSTALLOGRAPHIC DEFECTS IN PERIODIC INTERFACES

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THE THEORY OF CRYSTALLOGRAPHIC DEFECTS IN PERIODIC INTERFACES

R. Pond, A. Bastaweesy

To cite this version:

R. Pond, A. Bastaweesy. THE THEORY OF CRYSTALLOGRAPHIC DEFECTS IN PE- RIODIC INTERFACES. Journal de Physique Colloques, 1985, 46 (C4), pp.C4-225-C4-230.

�10.1051/jphyscol:1985424�. �jpa-00224674�

T H E THEORY OF CRYSTALLOGRAPHIC D E F E C T S IN P E R I O D I C INTERFACES

R . C . Pond and A . Bastaweesy

Dept. o f Metallurgy & Materials Science, The University of Liverpool, P. 0. Box 147, Liverpool L69 3 B X , U.K.

A b s t r a c t

-

A g e n e r a l t h e o r y a b l e t o p r e d i c t t h e g e o m e t r i c a l c h a r a c t e r o f c r y s t a l l o g r a p h i c d e f e c t s i n p e r i o d i c i n t e r f a c e s i s developed. The t h e o r y a p p l i e s t o g r a i n and i n t e r p h a s e boundaries, and t r e a t s t h e t o t a l symmetry of b o t h t h e a d j a c e n t c r y s t a l s . An e x p r e s s i o n i s o b t a i n e d which d e s c r i b e s t h e c h a r a c t e r of a l l p e r m i s s i b l e i n t e r f a c i a l l i n e d e f e c t s . I n a d d i t i o n t o ' d s c ' d i s l o c a t i o n s , it i s shown t h a t o t h e r t y p e s of d i s l o c a t i o n s can a r i s e , f o r example when one c r y s t a l i s nonsymmorphic. I n t e r f a c i a l d i s c l i n a t i o n s , and d e f e c t s which a r i s e when one c r y s t a l i s nonholosymmetric a r e a l s o considered.

1. I n t r o d u c t i o n

-

C r y s t a l l i n e m a t e r i a l s a r e o r d e r e d arrangements of atoms and ex- h i b i t symmetry t o a g r e a t e r o r l e s s e r e x t e n t . D e f e c t s i n such m a t e r i a l s a r e d i s - c o n t i n u i t i e s o f t h i s o r d e r , and t h e i r c h a r a c t e r depends on t h e symmetry p r e s e n t i n t h e unperturbed s t r u c t u r e . B i c r y s t a l s can a l s o b e ordered s t r u c t u r e s , and t h e o b j e c t of t h i s paper i s t o develop t h e t h e o r y o f c r y s t a l l o g r a p h i c d e f e c t s i n p e r i o d i c i n t e r f a c e s . The a d j e c t i v e ' c r y s t a l l o g r a p h i c ' i s used t o d e s c r i b e d e f e c t s whose g e o m e t r i c a l c h a r a c t e r , such a s t h e Burgers v e c t o r of a d i s l o c a t i o n , can b e p r e d i c t e d from c r y s t a l l o g r a p h i c i n f o r m a t i o n o n l y . Thus, e f f e c t s a r i s i n g due t o i n t e r f a c i a l r e l a x a t i o n , such a s d i s s o c i a t i o n of i n t e r f a c i a l d i s l o c a t i o n s i n t o p a r t i a l s , a r e excluded.

Much i s a l r e a d y known about c r y s t a l l o g r a p h i c d e f e c t s i n i n t e r f a c e s ; i n p a r t i c - u l a r ' d s c ' d i s l o c a t i o n s have been s t u d i e d e x t e n s i v e l y . Bollmann (1) showed how such d e f e c t s conserve t h e s t r u c t u r e of p e r i o d i c i n t e r f a c e s . He deduced t h e Burgers v e c t o r s of such d i s l o c a t i o n s by c o n s i d e r i n g t h e r e l a t i v e s h i f t s of p a i r s of co- i n c i d e n c e r e l a t e d l a t t i c e s . However, it can be s e e n t h a t such arguments t r e a t o n l y t h e t r a n s l a t i o n symmetry of t h e a d j a c e n t c r y s t a l s , whereas, a s w i l l be shown, t h e p o i n t symmetry o f t h e c r y s t a l s can a l s o l e a d t o i m p o r t a n t i n t e r f a c i a l d e f e c t s . Thus, it i s i n t e n d e d i n t h e p r e s e n t work t o develop a g e n e r a l t h e o r y which t a k e s i n t o account t h e t o t a l synmetry o f t h e a d j a c e n t c r y s t a l s . I t i s a l s o hoped t h a t t h e approach adopted h e r e w i l l emphasize t h e c l o s e i n t e r - r e l a t i o n s h i p between d e f e c t s which can e x i s t i n t h e b u l k and on t h e s u r f a c e of s i n g l e c r y s t a l s and i n i n t e r f a c e s between c r y s t a l s .

2 . Notation

-

We c o n s i d e r two c r y s t a l s , one d e s i g n a t e d w h i t e and t h e o t h e r b l a c k . L e t t h e spacegroup of t h e w h i t e c r y s t a l ' s l a t t i c e b e d e s i g n a t e d @ * ( A ) , and we note t h a t t h e o p e r a t i o n s o f t h i s group can b e expressed ( 2 ) a s

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C4-226 JOURNAL DE PHYSIQUE

where D (1) r e p r e s e n t s t h e ith orthogonal operation ( r o t a t i o n , mirror, inversion, roto-inversion)

,

^and ( A )

.

r e p r e s e n t s t h e jth t r a n s l a t i o n vector. A s i m i l a r ex- pression can be w r i t t e n f a r t h e spacegroup of t h e black c r y s t a l ' s l a t t i c e ,

@* ( u ) .

^We

note t h a t t h e spacegroups of l a t t i c e s a r e always symorphic, i . e . do n o t contain screw-rotation axes o r mirror-glide planes. C r y s t a l s may o r may n o t e x h i b i t the same symmetry a s t h e l a t t i c e s on which they are based, and we designate t h e i r space- groups @(A) and 0 ( u ) f o r t h e white and black r e s p e c t i v e l y . I n t h e case of symmor- p h i c holosymmetric c r y s t a l s , the p o i n t symmetry of t h e c r y s t a l i s t h e same a s t h a t of i t s l a t t i c e , hence

@*

( A ) = @ (1) and

@*

(p) = @ (p)

.

I n nonholosymmetric symmorphic c r y s t a l s some of t h e p o i n t symnetry operations, [D (1)

. 1 01,

a r e suppressed. Non- symmorphic c r y s t a l s can be t r e a t e d r e a d i l y using t h e &ame matrix formulation; t h e spacegroup @ (1) i s then w r i t t e n a s

where ~ ( 1 ) i s t h e supplementary t r a n s l a t i o n a s s o c i a t e d with t h e ith operation. ( I t i s most i m h r t a n t t o note t h a t t h e orthogonal p a r t of a n o n s y m r p h i c o p e r a t i o n i s taken a s a c t i n g through a chosen o r i g i n r a t h e r than through its a c t u a l p o s i t i o n i n space, the vector ~ ( 1 ) w i l l t h e r e f o r e depend on t h e choice of o r i g i n . )

The p a t t e r n created by the i n t e r p e n e t r a t i o n of t h e black and white l a t t i c e s i s c a l l e d a dichromatic p a t t e r n ( 3 ) . Its spacegroup includes a l l the operations which are common t o t h e black and white l a t t i c e s , i . e . t h e operations i n t h e group

@*

(p) =

@*

( A ) A

@*

(p)

.

This i n t e r s e c t i o n may be extended by antisymmetry o p e r a t i o n s f o r s p e c i a l forms of t h e transformation, R, which r e l a t e t h e white coordinate system to t h a t of t h e black. However, we s h a l l n o t be concerned with antisymmetry operations i n t h e p r e s e n t work, and note t h a t t h e operations i n t h e above i n t e r s e c t i o n can be e s t a b l i s h e d using matrix operator n o t a t i o n by finding t h e s o l u t i o n s t o t h e following equation (using t h e white coordinate system)

In the case of coincidence s i k l a t t i c e s , c s l ' s , t h e r e a r e s o l u t i o n s t o equation ( 3 ) of t h e form

where E i s t h e i d e n t i t y matrix. This equation s i m p l i f i e s t o ~ ( c s l l = T ( A )

.

=

m

-

R L ( ~ ) where ~ ( c s l ) ~ i s t h e m* t r a n s l a t i o n vector of t h e c s l . ^J

For t h e treatment o f n o n s p o r p h i c and nonholosymmetric b i c r y s t a l s , it has been found advantageous t o define t h e dichromatic complex (4) which can be regarded a s the i n t e r p e n e t r a t i o n o f t h e black and white c r y s t a l s t r u c t u r e s . The spacegroup of t h e complex w i l l include a l l of t h e operations i n the group O* ( c ) = 8

(1)n

@ ( p )

-

^We

note t h a t

@*

^{(p) and 6*} ( c ) have t h e same t r a n s l a t i o n symmetry, b u t t h e p o i n t symmetry of

@*

( c ) may be lower than t h a t of

@*

(p) i f e i t h e r o r both of t h e c r y s t a l s i s non- holosymmetric. Furthermore, @ * ( c ) may be nonsymmorphic i f t h e component c r y s t a l s a r e n o n s y m r p h i c .

3. I n t e r f a c i a l Defects - I n t h e p r e s e n t work we a r e concerned with d e f e c t s i n p e r i o d i c i n t e r f a c e s ; i n o t h e r words we a r e concerned with c r y s t a l s having c s l orien- t a t i o n r e l a t i o n s h i p s . (Crystals with r e l a t i v e o r i e n t a t i o n s c l o s e l y approaching c s l

deviation by appropriate a r r a y s of d e f e c t s which, according t o theory, can e x i s t i n t h e i n t e r f a c e s of i n t e r e s t . ) The s t a r t i n g p o i n t i s t o consider a p e r i o d i c i n t e r f a c e with normal

;

taken t o p o i n t i n t o t h e black c r y s t a l . This configuration can be re- garded a s the unperturbed reference s t a t e , and we can c o n s t r u c t , f o r example, a closed Burgers o r Nabarro c i r c u i t w h i c h c r o s s e s c k i n t e r f a c e a t two coincidence points.

I t i s convenient f o r purposes of i l l u s t r a t i o n now t o s e p a r a t e t h e two c r y s t a l s a t t h e i n t e r f a c e , thereby c r e a t i n g 'matching' s u r f a c e s on t h e white and black c r y s t a l s . We designate t h e surface normal ;(A) i n t h e white coordinate system and ~ ( p ) i n t h e black.

Now consider a l l t h e o p e r a t i o n s t h a t can be c a r r i e d o u t on t h e white c r y s t a l which r e l a t e the o r i g i n a l s u r f a c e t o an equivalent one; t h e s e operations a r e those belonging to the group -3 (.A)

.

Pure t r a n s l a t i o n s ,

[ E I ~ ( x )

^{w i l l} leave t h e o r i e n t a - t i o n of %(A) i n v a r i a n t b u t w i l l s h i f t i t s o r i g i n by an amount

n(X)

. 1 ( X ) .

.

I f these two s u r f a c e s a r e contiguous i n t h e same white c r y s t a l , they w i l l be joiked by a s t e p t h e h e i g h t of which i s designated hS and i s equal t o ~ ( h )

.x(X).

( t h e s u b s c r i p t , c , i n d i c a t e s t h a t t h e s u r f a c e s on e i t h z r s i d e of t h e s t e p a r e i d e k t i c a l except f o r l o c a t i o n i n space, and t h e s t e p i s r e f e r r e d t o a s being 'complete'.) Such a s t e p is i n d i c a t e d schematically i n f i g . 1 ; (a) shows a s l a b of p e r f e c t c r y s t a l , and ( c ) shows the s i t u a t i o n a f t e r t h e i n t r o d u c t i o n of a s t e p . We note t h a t such a s t e p can be re- garded a s having been introduced by the passage of a p e r f e c t d i s l o c a t i o n , with Burgers vector equal t o

r ( A ) . ,

through t h e s l a b . Figure 1 a l s o i l l u s t r a t e s equiva- l e n t s u r f a c e s r e l a t e d t o E(x? by pure r o t a t i o n , ( b ) , and by a combination of r o t a - t i o n and t r a n s l a t i o n , (d)

.

^We note t h a t t h e shape change depicted i n c r y s t a l blocks

(b) and (d) could have been produced by t h e passage of a d i s c l i n a t i o n (5) and a dis- p i r a t i o n (6) r e s p e c t i v e l y through t h e o r i g i n a l c r y s t a l s l a b . A l t e r n a t i v e l y , t h e shaped s l a b s ( b ) , ( c ) and (d) can be regarded a s having been obtained from a block of s i n g l e c r y s t a l by c u t t i n g along s u r f a c e s ;(A) and

ID

^{( A )}

^. 1

^a ( X I i

+

1 ( A ) .];(A)

.

Similarly, s u r f a c e s ~ ( p ) and [ ~ ( ! J ) ~ l & ( p ) ~

x(u)Jn(u)

can b& prepared fro& a block of black c r y s t a l (where ~ ( p ) i s t h e inward p o i n t i n g normal expressed i n t h e black coordinate system)

.

A l l t h e white c r y s t a l s u r f a c e s produced i n t h e way described above a r e equiva- l e n t t o t h a t with normal n(X)

,

and a l l t h e black a r e equivalent t o n ( p ) . I t follows t h a t we can r e j o i n t h e s u r f a c e s of shaped s l a b s , by introducing t h e necessary d i s - t o r t i o n , i n order t o form i n t e r f a c e s with s t r u c t u r e s equivalent to t h a t o f t h e o r i g i n a l i n t e r f a c e . This i s depicted schematically i n f i g . 2 which shows t h e corr- esponding s u r f a c e s before joining i n t h e cases where both t h e new black and white s u r f a c e s have been produced by a) pure t r a n s l a t i o n , and b) pure r o t a t i o n . We can e s t a b l i s h t h e geometrical c h a r a c t e r of the r e s u l t i n g i n t e r f a c i a l d i s c o n t i n u i t y by constructing t h e o r i g i n a l Burgers o r Nabarro c i r c u i t around t h e d e f e c t . Any disloca- t i o n c h a r a c t e r w i l l correspond t o a closure f a i l u r e i n t h e Burgers c i r c u i t , and any d i s c l i n a t i o n c h a r a c t e r t o an o r i e n t a t i o n f a i l u r e i n t h e Nabarro c i r c u i t . (We note t h a t any adjustment of t h e d i r e c t i o n of t h e l i n e of t h e i n t e r f a c i a l d i s c o n t i n u i t y necessary i n t h e joining process w i l l n o t modify t h e d i s l o c a t i o n o r d i s c l i n a t i o n content of t h e d e f e c t . )

It i s now necessary t o examine the i n t e r f a c i a l d i s c o n t i n u i t i e s described above i n order t o e s t a b l i s h those which a r e permissible l i n e d e f e c t s . This can be c a r r i e d o u t by expressing, i n t h e coordinate system of one c r y s t a l , t h e nature of t h e opera- t i o n which r e l a t e s the o r i g i n a l i n t e r f a c i a l s t r u c t u r e t o t h e new one. Thus, i f

[ D ( x ) ~ ~ ~ ( x ) ~ + r ( X ) A

i s t h e operation c a r r i e d o u t on t h e white c r y s t a l , and

[ ~ ( p )

IU(?.I)~ ⁺ ~ ( u ) " 1

^{i s} the operation c a r r i e d o u t on t h e black, t h e combined opera- t i o n Axpressed i n tEe coordinate system of t h e white c r y s t a l i s given by

Expanding t h i s expression we o b t a i n

C4-228 JOURNAL DE PHYSIQUE

P e r m i s s i b l e l i n e d i s c o n t i n u i t i e s correspond t o o p e r a t i o n s which have t h e form

[ E I ~

where i s a displacement, i . e . p u r e d i s l o c a t i o n s , [D[o] where D i s a p r o s e r r o t a t i o n , i . e . p u r e d i s c l i n a t i o n s , and combinations o f t h e s e .

4 . I n t e r f a c i a l D i s l o c a t i o n s

-

P e r f e c t i n t e r f a c i a l d i s l o c a t i o n s a r i s e when t h e oper- a t i o n s c a r r i e d o u t on b o t h t h e b l a c k and w h i t e c r y s t a l s a r e , f o r example, p u r e t r a n s - l a t i o n s , i . e . when t h e o p e r a t i o n s a r e [E IL(1)

.]

and [E l E l l ( p ) A r e s p e c t i v e l y . Sub- s t i t u t i n g t h e s e i n t o e x p r e s s i o n 6 t h e combine& o p e r a t i o n t h e n becomes

This e x p r e s s i o n w i l l b e recognised a s t h e s e t o f t r a n s l a t i o n s which c o n s t i t u t e t h e ' d s c ' l a t t i c e , and i t i s s t r a i g h t f o r w a r d t o show t h a t t h i s s e t i s i t s e l f a group. We n o t e t h a t ' d s c ' d i s l o c a t i o n s can b e regarded a s a r i s i n g from t h e c o a l e s c e n c e i n t h e i n t e r f a c e o f white and b l a c k c r y s t a l d i s l o c a t i o n s . I n a d d i t i o n , we n o t e t h a t t h o s e o p e r a t i o n s given by e x p r e s s i o n 7 which a r e a l s o e q u a l t o an o p e r a t i o n

[EI

~ ( c s l ) )

,

i . e . a s o l u t i o n o f e q u a t i o n 4, correspond t o t h e Burgers v e c t o r s of d i s l o c a t i o n s which can, i n p r i n c i p l e , e x i s t n o t o n l y i n t h e i n t e r f a c e b u t a l s o i n b o t h c r y s t a l s . F i n a l l y , we note t h a t t h e h e i g h t of t h e i n t e r f a c i a l s t e p a s s o c i a t e d w i t h a ' d s c ' d i s l o c a t i o n i s given by )I (h: (1)

+

^{hz (p) = )I} (_n ( A ) . ~ ( 1 )

+

2 (p) .'(p) ,)

,

where t h e Burgers v e c t o r o f t h e d i s l o c a t i o n i s e q u a l t o ( A )

-

^:R ( ~ 1 ~ . The climb component o f t h e d i s l o c a t i o n i s given by ( h z ( A ) -hz

(u)

) ( 7 )

.

i

5. I n t e r f a c i a l D i s c l i n a t i o n s

-

I n t e r f a c i a l d i s c l i n a t i o n s a r i s e when

D ( X ) R D - l ( p ) , R-1 corresponds t o a p r o p e r r o t a t i o n , and t h e a s s o c i a t e d t r a n s l a t i o n i s zkro. The a e f e c t s can b e r e g a r d e d a s b e i n g formed, f o r example, by t h e c o a l e s - ence i n t h e i n t e r f a c e o f a b l a c k and white c r y s t a l d i s c l i n a t i o n . Moreover, i n t h e c a s e s where D (1) = R D ( ~ ) ~ R - ~ , i . e . where t h e r o t a t i o n o p e r a t i o n i s common t o t h e two c r y s t a l s , th'k corresponding d i s c l i n a t i o n can e x i s t i n t h e i n t e r f a c e and a l s o i n both c r y s t a l s . D i s c l i n a t i o n s a r e n o t g e n e r a l l y observed i n three-dimensional c r y s t a l s because t h e s t r a i n s a s s o c i a t e d w i t h allowed r o t a t i o n s a r e l a r g e ( 5 ) . How- e v e r , i t can be s e e n from e x p r e s s i o n (6) t h a t i n t e r f a c i a l d i s c l i n a t i o n s can b e asso- c i a t e d w i t h s m a l l e r r o t a t i o n s . The s i t u a t i o n i s analogous t o ' d s c ' v e c t o r s , t h e magnitudes o f which diminish a s t h e p r i m i t i v e t r a n s l a t i o n v e c t o r s o f t h e c s l i n c r e a s e i n magnitude. We n o t e t h a t i n t e r f a c i a l s t e p s a r e n o t a s s o c i a t e d w i t h i n t e r f a c i a l d i s c l i n a t i o n s , although t h e p l a n e o f t h e i n t e r f a c e w i l l b e buckled t o an e x t e n t de- pending on t h e e l a s t i c accommodation i n t h e a d j a c e n t c r y s t a l s .

6. D e f e c t s A s s o c i a t e d w i t h Nonholosymmetric and Nonsymmorphic C r y s t a l s - I n t h i s s e c t i o n we i n d i c a t e t h a t t h e above a n a l y s i s r e a d i l y p r e d i c t s t h e c h a r a c t e r o f d e f e c t s which have been observed i n i n t e r p h a s e boundaries where one o f t h e c r y s t a l s i s non- holosymmetric o r nonsymmorphic. Consider f i r s t t h e c a s e of N i s i :Si i n t e r f a c e s . These c r y s t a l s can be grown e p i t a x i a l l y and have very s i m i l a r l a g t i c e parameters and p a r a l l e l o r i e n t a t i o n ( 8 ) . I n o t h e r words we may t a k e R=E, and @*(A) =

@*

( p ) = ~ m 7 r n . Now N i s i 2 , having t h e f l u o r i t e s t r u c t u r e , i s holosymmetric and symmorphic ( @ ( I ) = Fm3m), whereas S i , having t h e diamond s t r u c t u r e , i s holosymmetric b u t n o n s ~ r p h i c

(@AN) = Fd3m). The corresponding o p e r a t i o n s i n t h e two c r y s t a l spacegroups Fm3m and Fd3m have been l i s t e d i n t h e Table. I t - c a n be s e e n t h a t 24 o f t h e p o i n t symm- e t r y o p e r a t i o n s , corresponding to t h e group 43m, a r e common t o t h e two c r y s t a l s , whereas t h e o t h e r 24 o p e r a t i o n s have a supplementary displacement +[111] i n t h e diamond c a s e . These l a t t e r o p e r a t i o n s a r e e i t h e r nonsymmorphic (diamond m i r r o r - g l i d e p l a n e s e.g.1 o r symmorphic b u t do n o t a c t through t h e chosen o r i g i n (2 p a r a l l e l t o

<110> e - g . ) .

According t o e x p r e s s i o n 6 p e r m i s s i b l e d i s l o c a t i o n s a r i s e f o r c o h i n e d o p e r a t i o n s of t h e form [E

14 ,

^where

t

i s a displacement. The ' d s c ' v e c t o r s , i .e. t h e s o l u t i o n s t o e x p r e s s i o n 7, correspond t o t h e v e c t o r s x ( X ) . , s i n c e t h e s e a r e common t o t h e two c r y s t a l s i n t h e p r e s e n t c a s e . However, additioAa1 d i s l o c a t i o n s can e x i s t , w i t h Burgers v e c t o r s e q u a l t o

I;/.

T h i s can be shown by, f o r example, s u b s t i t u t i n g c o r r -

t o b e l t h e diamond glide-plane p a r a l l e l t o (100) i n t h e ( b l a c k ) 5'1. The c o d i ~ e d o p e r a t i o n becomes

I n o t h e r words, d i s l o c a t i o n s w i t h Burgers v e c t o r magnitudes e q u a l t o

lcrl

can e x i s t i n such i n t e r f a c e s . The o r i g i n of such d i s l o c a t i o n s can be viewed i n t h e f o l l o w i n g way. The N i s i 2 c r y s t a l can n u c l e a t e from e i t h e r o f t h e two atoms which form t h e b a s i s of S i and which occupy symmetry r e l a t e d p o s i t i o n s ( i . e . 0 , 0 , 0 and

I,L,L).

Where i n t e r f a c i a l domains, having n u c l e a t e d from t h e two o r i g i n s , impinge a d i s -

l o c a t i o n i s n e c e s s a r y t o accommodate t h e change o f o r i g i n . However, such d i s l o c a - t i o n s cannot e x i s t i n a l l i n t e r f a c e s ; t h e d i s l o c a t i o n s must s e p a r a t e d e g e n e r a t e i n t e r f a c i a l s t r u c t u r e s , and hence can o n l y e x i s t i n i n t e r f a c e s n w h o s e o r i e n t a t i o n i s l e f t i n v a r i a n t by t h e i n d i v i d u a l c r y s t a l o p e r a t i o n s . I n o t h e r words

CD

( A )

.

(01and [ ~ ( p ) la (p) must l e a v e t h e o r i e n t a t i o n o f

q

i n v a r i a n t . I n s p e c t i o n of thi l a s t 24 opera&i;ns I n t h e Table below shows t h a t such d i s l o c a t i o n s can e x i s t o n l y i n {hko) i n t e r f a c e s ( 9 , l O ) . These d i s l o c a t i o n s w i l l have a climb component, a s s o c i a t e d with a s u r f a c e s t e p on t h e S i r w i t h magnitude 11.". We n o t e t h a t such d i s l o c a t i o n s can accommodate t h e s m a l l m i s f i t which i s a c t u a l l y p r e s e n t , and a r e g l i s s i l e only on 1110) i n t e r f a c e s where 11.5can be zero. These p r e d i c t i o n s a r e i n d e t a i l e d agreement w i t h experimental o b s e r v a t i o n s ( 9 ) .

We now c o n s i d e r t h e i n t e r f a c e between Ge ( b l a c k ) and GaAs (white)

.

^{A s} i n t h e p r e v i o u s example, t h e s e c r y s t a l s can b e grown e p i t a x i a l l y w i t h p a r a l l e l o r i e n t a t i o n and have-very s i m i l a r l a t g i c e parameters. Thus, we have O* ( A ) = O* (p) = Fm3m, and O ( X ) = F43m and 4

(u)

= Fd3m. I t i s convenient to r e g a r d t h e s t r u c t u r e o f GaAs a s b e i n g a d e r i v a t i v e of t h e diamond s t r u c t u r e ; i n o t h e r words it i s a nonholosymmetric s t r u c t u r e i n which some o f t h e o p e r a t i o n s which l e a v e t h e diamond s t r u c t u r e i n - v a r i a n t a r e suppressed. These l a t t e r o p e r a t i o n s a r e t h e 24 which r e l a t e atomic s i t e s a t 0,0,0 t o atomic s i t e s a t

LrL,L

and vice-versa, and, s i n c e t h e s e s i t e s a r e

4 4 4

occupied by d i f f e r e n t atomic s p e c i e s i n GaAs, t h e o p e r a t i o n s a r e suppressed i n t h i s m a t e r i a l . We r e f e r t o t h e s e o p e r a t i o n s i n G a A s a s exchange o p e r a t i o n s and d e s i g n a t e them I[ (A)

. 1

^{a ( A )}

.] ,

and n o t e t h a t t h e y r e l a t e one G a A s c r y s t a l t o an a n t i - s i t e c r y s t a l , i l e T on& where t h e Ga and A s s p e c i e s a r e i n t e r c h a n g e d ( 1 0 , l l ) .

S u b s t i t u t i n g i n t o e x p r e s s i o n 6 , we s e e t h a t i n t e r f a c i a l d e f e c t s w i t h o u t d i s - l o c a t i o n o r d i s c l i n a t i o n c h a r a c t e r can a r i s e , f o r example

I n o t h e r words, an a n t i - s i t e domain boundary i n t h e G a A s o v e r l a y e r can t e r m i n a t e a t c e r t a i n boundaries and s e p a r a t e s e n e r g e t i c a l l y d e g e n e r a t e domains o f i n t e r f a c i a l s t r u c t u r e . A s i n t h e N i s i : S i c a s e , this can only o c c u r on i n t e r f a c i a l p l a n e s where n i s l e f t i n v a r i a n t by b o d t h e white and black o p e r a t i o n s , i . e . on

I1101

i n t e r - -

f a c e s i n t h e p r e s e n t c a s e . T h i s is i n d e t a i l e d agreement w i t h experimental obser- v a t i o n s ( 1 0 ) . We a l s o n o t e t h a t t h e i n t e r f a c i a l d e f e c t may have an a s s o c i a t e d s t e p , w i t h h e i g h t g i v e n by

&.a,

i . e . t h e r e w i l l be complementary s t e p s on t h e Ge and GaAs c r y s t a l s u r f a c e s . F i n a l l y , we n o t e t h a t no c l o s u r e f a i l u r e of a Burgers c i r c u i t O r

o r i e n t a . t i o n i n c o m p a t i b i l i t y i n a Nabarro c i r c u i t a r i s e s f o r such a d e f e c t . This i s because such c i r c u i t s a r e c o n s t r u c t e d u s i n g t h e c r y s t a l l a t t i c e s , and t h e o r t h o g o n a l p a r t s of t h e exchange o p e r a t i o n s , IJJ ( A ) i , l e a v e a c r y s t a l ' s l a t t i c e i n v a r i a n t .

7. Conclusions

-

^We have developed a g e n e r a l t h e o r y f o r p r e d i c t i n g t h e geometrical c h a r a c t e r o f c r y s t a l l o g r a p h i c d e f e c t s i n p e r i o d i c i n t e r f a c e s . T h i s t h e o r y t a k e s i n t o account t h e t o t a l symmetry of t h e a d j a c e n t c r y s t a l s , and can b e a p p l i e d t o g r a i n and i n t e r p h a s e boundaries. The t h e o r y i n d i c a t e s t h a t i n t e r f a c i a l d i s c l i n a t i o n s a s w e l l a s d i s l o c a t i o n s can e x i s t , and t h a t both may be i m p o r t a n t i n i n t e r f a c i a l p r o c e s s e s . I t h a s been shown t h a t t h e Burgers v e c t o r s of ' d s c ' d i s l o c a t i o n s can be o b t a i n e d from t h e t h e o r y , and t h a t d i s l o c a t i o n s w i t h non ' d s c ' Burgers v e c t o r s can a r i s e i n i n t e r f a c e s where one c r y s t a l i s nonsymmorphic. I n a d d i t i o n , t h e t h e o r y can be a p p l i e d r e a d i l y t o t h e a n a l y s i s o f i n t e r f a c i a l s t e p s , and t o d e f e c t s which a r i s e when one o r both of t h e a d j a c e n t c r y s t a l s i s nonholosymmetric.

JOURNAL

DE

PHYSIQUE

F i g . 1. Schematic i l l u s t r a t i o n o f t h e shape change Fig.2. Schematic i l l u s t r a t i o n induced i n a block o f p e r f e c t c r y s t a l , ( a ) , by t h e o f c r y s t a l s having e q u i v a l e n t passage of (b) a d i s c l i n a t i o n , ( c ) a d i s l o c a t i o n , s u r f a c e s p r i o r t o b e i n g

and (d) a d i s p i r a t i o n . welded i n t o b i c r y s t a l s . The

i n t e r f a c i a l d i s c o n t i n u i t y a r i s e s due t o t r a n s l a t i o n o p e r a t i o n s a p p l i e d t o b o t h c r y s t a l s i n ( a ) and r o t a t i o n o p e r a t i o n s i n (b)

.

References.

Bollmann, .W., C r y s t a l D e f e c t s and C r y s t a l l i n e I n t e r f a c e s , Springer-Verlag, B e r l i n (1970)

.

Shubnikov, A.V. and Koptsik, A.V., Symmetry i n S c i e n c e and &t, Plenum P r e s s , N.Y. (1974).

Pond, R.C. and Bollmann, W., P h i l . Trans. R. Soc. Lond.

292

(1979) 449.

Pond, R.C. and Vlachavas, D.S., Proc. R. Soc. Lond.

A386

(1983) 95.

Kleman, M. D i s l o c a t i o n s i n S o l i d s 5, North-Holland, Amsterdam (1980) 243.

H a r r i s , W.F., P h i l . Mag.

22

(1970) 949.

King, A.H. and Smith, D,A,, Acta C r y s t .

A36

(1980) 335.

Cherns, D., A n s t i s , G.R., Hutchison, J.L. and Spence, J.C.H., P h i l . Mag.

A46

(1982) 849.

Cherns, D. and Pond, R.C., Mat. Res. Soc. Syrnp. Proc.

2

(1984) 423.

Pond, R.C., P o l y c r y s t a l l i n e Semiconductors Springer-Verlag, B e r l i n , i n p r e s s . Pond, R.C., Gowers, J.P., Holt, D.B., Joyce, B.A., Neave, J.H., and

Larsen, P.K., Mat. Res. Soc. S p p . P ~ o c .

25

(1984) 273.

Table Symmetry o p e r a t i o n s i n t h e group Fd?m; t h e corresponding o p e r a t i o n s i n ~ m ? m a r e t h e same e x c e p t t h a t t h e a s s o c i a t e d t r a n s l a t i o n i s z e r o i n a l l c a s e s . P o s i t i v e and n e g a t i v e s u p e r s c r i p t s s i g n i f y clockwise and a n t i c l o c k w i s e r o t a t i o n s .