ON THE MOTION OF DISLOCATION BENDS IN TERMS OF THE KINK MODEL

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ON THE MOTION OF DISLOCATION BENDS IN TERMS OF THE KINK MODEL

H. Gottschalk

To cite this version:

H. Gottschalk. ON THE MOTION OF DISLOCATION BENDS IN TERMS OF THE KINK MODEL.

Journal de Physique Colloques, 1983, 44 (C4), pp.C4-475-C4-477. �10.1051/jphyscol:1983457�. �jpa-

00223077�

JOURNAL DE PHYSIQUE

Colloque C4, suppl6ment a u n09, Tome 44, s e p t e m b r e 1983 page C4-475

ON THE MOTION OF DISLOCATION BENDS IN TERMS OF THE KINK MODEL

H. G o t t s c h a l k

Abt. f. MetalZphysik, II. PhysikaZisches Institut, Universit(it K6Zn, ZUZpicher Str. 77, 0-5000 K6Zn 41, F.R.G.

RCsum6 - ^Le regime permanent de d'eplacement des p a r t i e s courb6es d e s d i s l o c a t i o n s d a n s un c r i s t a l de s i l i c i u m e n c o u r s de d e f o r m a t i o n e s t expliqu'e p a r un modkle de d6crochements dans l e q u e l l e s dCcrochements s ' a n n i h i l e n t au v o i s i r i a g e de l ' a p e x d e l a c o u r b u r e . On d i s c u t e l e paradoxe a p p a r e n t d ' u n e d 6 f o r m a t i o n s o u s f a i b l e c o n t r a i n t e c o n d u i s a n t & d e s d i s l o c a t i o n s c o u r b g e s malgr6 un p o t e n t i e l de P e i e r l s Q l e v 6 .

A b s t r a c t - The s t e a d y s t a t e motion of d i s l o c a t i o n bends i n s i l i c o n t a k i n g p l a c e when t h e c r y s t a l i s b e i n g deformed i s e x p l a i n e d i n t e r m s o f t h e k i n k model assuming a k i n k a n n i h i l a t i o n mechanism i n t h e apex r e g i o n of t h e bend. The a p p a r e n t l y p a r a d o x i c a l b e h a v i o u r t h a t a d e f o r m a t i o n a t low s t r e s s p r o d u c e s ' c u r v e d d i s l o c a t i o n s though t h e P e i e r l s e n e r g y i s h i g h i s d i s c u s s e d .

A f t e r t h e d e f o r m a t i o n of s i l i c o n c r y s t a l s under h i g h s t r e s s a t low t e m p e r a t u r e one t y p i c a l l y f i n d s s t r a i g h t d i s l o c a t i o n segments l y i n g i n t h e < 1 1 0 > - P e i e r l s t r o u g h s . The d i s l o c a t i o n s c o n t a i n b e n t segments when t u r n i n g from one ( 1 1 0 ) - d i r e c t i o n t o a n o t h e r one. The o b s e r v a t i o n s a r e s t i l l t h e same changing t o lower s t r e s s and h i g h e r d e f o r m a t i o n t e m p e r a t u r e , p r o v i d e d t h e l o c a l d i s l o c a t i o n d e n s i t y i s n o t t o o h i g h ( s e e Fig.1 and Fig.3 i n / 1 / ) .

I t i s r e m a r k a b l e t h a t i t i s p o s s i b l e t o c a l c u l a t e t h e a p p l i e d s t r e s s Prom t h e measured r a d i i o f c u r v a t u r e o f t h e bends i n specimens c o o l e d t o room t e m p e r t u r e w i t h a p p l i e d l o a d . I n t h e s e specimens t h e d i s l o c a t i o n s a r e t h e r e f o r e imaged i n t h e s t a t e o f moving. The a s s u m p t i o n t h a t t h e r e s o l v e d s h e a r s t r e s s and t h e b a c k s t r e s s e x e r t e d by t h e l i n e t e n s i o n i n t h e bend a r e e q u a l t h e r e f o r e seems t o be j u s t i f i e d 111.

An

e a s i l y s e e n consequence of a s t e a d y s t a t e motion of a d i s l o c a t i o n bend i s t h a t t h e d i s l o c a t i o n v e l o c i t y of e a c h segment o f t h e bend i s o n l y d e p e n d e n t on t h e v e l o c i t i e s o f t h e two a d j a c e n t s t r a i g h t segments and a g e o m e t r i c a l r e l a t i o n b u t i n d e p e n d e n t on t h e a c t u a l s t r e s s i n t h e bend and t h e c h a r a c t e r o f t h e segment i n q u e s t i o n ,

C o n s i d e r i n g t h e d i s l o c a t i o n motion i n a diamond-like semiconductor a s performed by t h e n u c l e a t i o n and t h e sideways m o t i o n o f k i n k s t h e r e a r i s e some q u e s t i o n s c o n c e r n i n g t h e b e h a v i o u r o f k i n k s i n t h e b e n t d i s l o c a t i o n segments.

I n t h e c o n c e p t p r e s e n t e d h e r e t h e d i s s o c i a t i o n o f d i s l o c a t i o n s i s n o t c o n s i d e r e d i n p a r t i c u l a r . For low s t r e s s and s m a l l d i s l o c a t i o n s p l i t t i n g t h e motion of k i n k s on b o t h p a r t i a l s i s c o r r e l a t e d , t h e n t h e d i s l o c a t i o n c a n be t r e a t e d l i k e a p e r f e c t one.

For h i g h s t r e s s and wide s p l i t t i n g when t h e k i n k motion i s u n c o r r e l a t e d , t h e c o n c e p t may be a p p l i e d t o e a c h p a r t i a l s e p a r a t e l y .

F i r s t , suppose a s t a t i c d i s l o c a t i o n bend w i t h r a d i u s of c u r v a t u r e R ( F i g . 1 ) . To e s t a b l i s h a 6 0 " - d i s l o c a t i o n bend a c e r t a i n number and a c e r t a i n d i s t r i b u t i o n of k i n k s on t h e d i s l o c a t i o n i s n e c e s s a r y . Assuming t h a t t h e b e n t segment A1A i s formed o f k i n k s b e l o n g i n g t o segment 1 and A2A o f k i n k s b e l o n g i n g t o segment 2 , a s i m p l e g e o m e t r i c a l c o n s i d e r a t i o n y i e l d s t h e t o t a l number Z o f k i n k s c o n t a i n e d i n t h e bend A1A2 :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983457

JOURNAL DE PHYSIQUE

Z

2 (1-cos3O0)R/a

0.27 R/a

( a is the distance of neighbouring Peierls troughs. a(Si)

0.33nm, R

lOOnm yields Z

82).

The local distance y of kinks depends only on the angle between the average dislocation line and the (110)-direction:

y

a/sin/ji, i

1,2 (see Fig.1) and the local kink density:

z

lly

sinp./a these values both being independent of R. ~ s / 3 . m a x

3 0 " ,

ymin

2a, z 112a.

max

- Kink annihilation in a forward moving bend

The kinks present in the segment

A are all kinks with the same sign. Hirth and Lothe 121 have shown that there exists a mutual repulsive force of kinks of the same 1 sign and that the force exerted by the line tension of a bent dislocation segment can be derived from this kink-kink interaction.

Considering the apex of a bend one notices (Fig. 2 ) that it is by no means possible to reduce the distance of neighbouring kinks below the minimal length 2a because of crystallographic reasons. It can be seen, too, that the two sorts of kinks, e.g. BC (or ABCD) on segment 1 and CD (or BCDE) on dislocation 2 cannot be distinguished.

The consequence of the dynamic equilibrium state of a moving bend arrangement is that the total number of kinks lying in the bend and the local kink density must be constant by time. When kinks driven by the applied stress are approaching the bend they must slow down by the effect of the repulsive forces of the kinks still present in the bend. Simultaneously they exert forces o n all kinks of the bend, which will move a small distance towards the apex. A differential change of the kink distance in the apex region, however, is no longer possible. The force o n the apex kinks is increasing by new kinks joining the bend until a kink of double height (Fig.2, C-rC', new line AC'E) may be generated and the whole configuration is rearranged.

This process is equivalent with the annihilation of a kink.

This concept is in good agreement with experimental results obtained for dislocations moving under low stress and for the leading partials in high stress deformed crystals /I/. Interesting, however, is that the radii of curvature for the trailing partials in the high stress deformed crystals in general are found too large, so that the force exerted by the line tension is less than the applied driving force. Perhaps particularly the trailing partial has strong interaction with point defects which make the annihilation of kinks in the apex region more difficult. To overcome these obstacles a n increasing number of kinks is piled up in the bend, the force exerted by the line tension decreases, and part of the driving for&e is now present to stimulate the kink annihilation process.

The relaxation of a frozen in dislocation formation may be considered within the same framework. From experiments it is known that in general the radii of curvature are increasing when the system is relaxing 131. Here two limiting cases must be distinguished: the split dislocation with small separation and the isolated partial.

In the first case the stress acting o n the straight segments nearly vanishes and

only the force exerted by the line tension is acting o n the apex of the bend. In the

Fig.3 - Kink g e n e r a t i o n when a d i s -

- l o c a t i o n bend r e l a x e s

l a t t e r case t h e f o r c e a c t i n g on t h e s t r a i g h t segments i s t h e s t a c k i n g f a u l t e n e r g y w h i l e i n t h e bend t h e l i n e t e n s i o n i s a d d i t i o n a l l y e f f e c t i v e . In b o t h c a s e s f i r s t a sideways m o t i o n o f k i n k s from t h e apex may t a k e place ( F i g . 3 , f u l l l i n e changing t o pointed l i n e ) and t h e n a new k i n k may be c r e a t e d b y r e v e r s i n g t h e a n n i h i l a t i o n process d e s c r i b e d b e f o r e ( A 4 A ' ) . In t h e case o f an i s o l a t e d p a r t i a l which f o r m e r l y was a l e a d i n g one an a d d i t i o n a l problem a r i s e s a s k i n k s o f t h e o p p o s i t e s i g n move t o t h e bends on t h e s t r a i g h t segments and a n n i h i l a t e k i n k s i n t h e bend. T h i s a n n i h i l a t i o n o f k i n k s i s i n concurrence w i t h t h e g e n e r a t i o n o f k i n k s i n t h e apex. As t h e s t r e s s a t t h e a p e x , which d r i v e s t h e d i s l o c a t i o n i n t h e r e v e r s e d i r e c t i o n , i s h i g h e r t h a n a t any o t h e r part o f t h e d i s l o c a t i o n t h e p r o b a b i l i t y o f k i n k f o r m a t i o n i s enhanced t h e r e t o overcome t h e l o s s o f k i n k s by a n n i h i l a t i o n a t t h e o u t e r part o f t h e bend. The former t r a i l i n g p a r t i a l a t t h e beginning o f t h e r e l a x a t i o n process again moves i n t h e former forward d i r e c t i o n . T h e r e f o r e t h e mechanism d e s c r i b e d f o r t h e forward m o t i o n can b e a p p l i e d .

F i n a l l y t h e problem o f curved d i s l o c a t i o n s i n m a t e r i a l s w i t h h i g h P e i e r l s e n e r g y s h a l l be d i s c u s s e d . As a d i s l o c a t i o n l y i n g i n a P e i e r l s t r o u g h i s e n e r g e t i c a l l y favoured one should assume t h a t a low s t r e s s would n o t be a b l e t o form a curved d i s l o c a t i o n .

i s , however, w e l l known t h a t i f t h e d e f o r m i n g s t r e s s i s low ( i . e . t h e temperature i s h i g h ) most o f t h e d i s l o c a t i o n s a r e curved becoming t h e more s t r a i g h t t h e h i g h e r t h e a p p l i e d s t r e s s i s .

T h i s paradoxical behaviour can b e e x p l a i n e d by t h e c o n c e p t o u t l i n e d b e f o r e . The k i n k s generated on t h e s t r a i g h t d i s l o c a t i o n segments o f a d i s l o c a t i o n loop which i s e x t e n d i n g on t h e g l i d e plane a r e t r a v e l l i n g t o t h e bends and a r e c o l l e c t e d t h e r e . If t h e s t r e s s i s low, t h e number o f k i n k s i n t h e e q u i l i b r i u m s t a t e i s h i g h ; i f t h e s t r e s s i s h i g h , t h e k i n k s a r e more and more forced towards t h e apex and t h e i r number i s low because o f t h e a n n i h i l a t i o n p r o c e s s e s . The r e s u l t t h e r e f o r e i s t h a t t h e p o r t i o n o f s t r a i g h t d i s l o c a t i o n segments f o r a loop o f g i v e n diameter i n c r e a s e s w i t h i n c r e a s i n g s t r e s s . The i n t r i n s i c r e a s o n f o r t h e f o r m a t i o n o f curved d i s l o c a t i o n s i s based on t h e p r o p e r t y o f t h e d i s l o c a t i o n t o be m o b i l e , and t h a t i s e q u i v a l e n t t o t h e p r o p e r t y o f n u c l e a t i n g double k i n k s . S i n c e t h e s e k i n k s which a l r e a d y had overcome t h e P e i e r l s h i l l s are always p r e s e n t and t h e i r f o r m a t i o n e n e r g y must n o t be brought up b y t h e b e n t d i s l o c a t i o n segments, no i n f o r m a t i o n on t h e P e i e r l s energy can be obtained from t h e c u r v a t u r e o f t h e b e n d s .

Acknoledgements - 'Ihe author w i s h e s t o thank P r o f . D r .

Alexander and Dip1.- Phys.

Tonnessen f o r s t i m u l a t i n g d i s c u s s i o n s on t h e s e problems.

R e f e r e n c e s

111 G o t t s c h a l k , H . , t h i s c o n f e r e n c e

/ 2 / H i r t h , J . P . and L o t h e , J . , Theory o f D i s l o c a t i o n s (McGraw H i l l ) , i 9 6 8 , Chapt.8-8

I31 G o t t s c h a l k , H . , Proc. 1 0 t h I n t . Congr. E l e c t r . Micr. H a m b u r g , ( 1 9 8 2 ) , 5 2 7