DISLOCATION MOBILITY AND MECHANICAL BEHAVIOUR OF Si CRYSTALS

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DISLOCATION MOBILITY AND MECHANICAL BEHAVIOUR OF Si CRYSTALS

K. Sumino

To cite this version:

K. Sumino. DISLOCATION MOBILITY AND MECHANICAL BEHAVIOUR OF Si CRYSTALS.

Journal de Physique Colloques, 1979, 40 (C6), pp.C6-147-C6-150. �10.1051/jphyscol:1979630�. �jpa- 00219046�

JOURNAL DE PHYSIQUE Colloque C6, suppl6ment au n06, tome 40, j u i n 2979, page C6-247

DISLOCATION M O B I L I T Y AND MECHANICAL BEHAVI3UR OF S i CRYSTALS K. Sumino

The Research I n s t i t u t e for Iron, S t e e l and Other Metals, Tohoku U n i v e r s i t y , Sendai 980, J a p m

Resume.- Les c a r a c t 6 r i s t i q u e s du mouvement d'ensemble des d i s l o c a t i o n s oendant l a d e f o r m a t i o n de c r i s - t a u x de s i l ic i u m deformes 2 v i t e s s e de d e f o r m a t i o n c o n s t a n t e s o n t e t u d i e e s a p a r t i r de l a m o b i l i t e des d i s l o c a t i o n s i s o l e e s . Un regime permanent de d P f o r m a t i o n dans l e q u e l l a d e n s i t e e t l a v i t e s s e moyenne des d i s l o c a t i o n s m o b i l e s ne dependent pas de l a d e f o r m a t i o n s ' e t a b l i t a p a r t i r du m i l i e u du stade 0.

Les c a r a c t e r i s t i q u e s de ce regime permanent peuvent se d e d u i r e de l l h y p o t h P s e s u i v a n t e : l e mouvement d'ensemble des d i s l o c a t i o n s s ' a j u s t e de f a ~ o n ce que l a c o n t r i b u t i o n des d i s l o c a t i o n s m o b i l e s a l a c o n t r a i n t e d'ecoulement prenne l a v a l e u r m i n i m a l e c o m p a t i b l e avec l a v i t e s s e de d e f o r m a t i o n imposee.

A b s t r a c t . - The c h a r a c t e r i s t i c s o f t h e c o l l e c t i v e m o t i o n of d i s l o c a t i o n s d u r i n g t h e d e f o r m a t i o n o f s i - l i c o n c r y s t a l s a t c o n s t a n t s t r a i n - r a t e s a r e i n v e s t i g a t e d on t h e knowledge o f t h e m o b i l i t y o f i s o l a t e d d i s l o c a t i o n s . The steady s t a t e o f d e f o r m a t i o n where t h e d e n s i t y and t h e mean v e l o c i t y o f moving d i s l o - c a t i o n s a r e c o n s t a n t w i t h r e s p e c t t o t h e s t r a i n appears i n t h e d e f o r m a t i o n stage a f t e r t h e m i d d l e o f stage 3. The c h a r a c t e r i s t i c s o f t h i s steady s t a t e of d e f o r m a t i o n i s w e l l d e s c r i b e d b y t h e f o l l o w i n g h y p o t h e s i s : t h e c o l l e c t i v e m o t i o n o f d i s l o c a t i o n s i s r e a l i z e d so t h a t t h e component o f t h e f l o w s t r e s s which i s determined by t h e s t a t e o f such c o l l e c t i v e m o t i o n t a k e s t h e minimum v a l u e t o m a i n t a i n t h e e x t e r n a l l y g i v e n s t r a i n - r a t e .

1. I n t r o d u c t i o n . - D i s l o c a t i o n s i n c o v a l e n t c r y s t a l s screw d i s l o c a t i o n s move s l o w e r t h a n 60' d i s l o c a t i o n s w i t h diamond-type s t r u c t u r e show dynamic c h a r a c t e r i s - and i t s v e l o c i t y v can be expressed by a f o l l o w i n g t i c s which a r e v e r y d i f f e r e n t from t h o s e o f d i s l o c a - e q u a t i o n ;

t i o n s i n o t h e r t y p e s o f c r y s t a l s . Such c h a r a c t e r i s - m

v = vo T exp ( - ^Q/ k T ) , t i c s a r e a t t r i b u t e d t o a h i a h P e i e r l s b a r r i e r i n t h i s

t y p e o f c r y s t a l s . I t has been e s t a b l i s h e d e x ~ e r i m e n - where v, = 4.3x106 cm/ s - ( l o 7 ~ / m ' ) - ~ , m = 1.1, t a l l y t h a t t h e o f i s o l a t e d d i s l o c a t i o n s in Q = 2. l7 eV, in the temperature range 5202.900"C t h i s t y p e o f c r y s t a l s depends s e n s i t i v e l y on t h e tem- and the range l a 3 0 MN/m2'

perature and rather insensitively on the stress. These 3. I n - s i t u o b s e r v a t i o n s of t h e dynamic b e h a v i o u r of c h a r a c t e r i s t i c s i n t h e d i s l o c a t i o n mobi 1 i t y a r e r e - d i s l o c a t i o n s d u r i n g d e f o r m a t i o n by HVEY.- Sato and fleeted in the mechanical behaviour o f covalent crys- Sumino /5/ have conducted i n - s i t u o b s e r v a t i o n s of tals which is observed to depend sensitively on both t h e dynamic b e h a v i o u r o f d i s l o c a t i o n s d u r i n g d e f o r - t h e t e m p e r a t u r e and t h e s t r a i n - r a t e . The p l a s t i c de- mation in the temperature range 650 ^% 8000C with use f o r m a t i o n of a c r y s t a l proceeds by means o f t h e mo- a high "Itage Operating at tion of numerous djslocations. Thus, the mechanical 1000 kV equipped w i t h a h i g h - t e m p e r a t u r e t e n s i l e s t a - behaviour o f crystals is controlled by the characte- ge. I t i s observed t h a t d i s l o c a t i o n s move s l o w l y and ristics o f the collective motion o f dislocations. The smoothly even on a m i c r o s c o p i c s c a l e d u r i n g deforma- purpose o f t h e p r e s e n t paper i s t o make c l e a r t h e t i o n . No j e r k y m o t i o n i s observed. The v e l o c i t y o f relation between the nature of the collective motion d i s l o c a t i o n s observed i s o f t h e same o r d e r o f magni- o f dislocations during macroscopic deformation and tude as t h a t determined b y macroscooic methods. T h i s that of the motion of isolated dislocations and to shbws t h a t t h e r a t e - c o n t r o l l i n g mechanism o f d i s l o - understand the feature of the mechanical behaviour c a t i o n m o t i o n d u r i n g macroscopic deformation i s t h e o f s i l i c o n c r y s t a l s . same as t h a t f o r i s o l a t e d d i s l o c a t i o n s and i s n o t 2. I n - s i t u measurements o f t h e m o b i l i t y of i s o l a t e d b y t h e i n t e r a c t i o n between d i s l o c a t i o n s . The d i s l o c a t i o n s . - Harada and Sumino /1/ have measured between d i s l o c a t i o n s d u r i n g d e f o r m a t i o n t h e m o b i l i t y o f i s o l a t e d d i s l o c a t i o n s i n s i l i c o n a r e m o s t l y O f e l a s t i c n a t u r e . A dynamic p i l e - u p c r y s t a l s as a f u n c t i o n o f t h e shear s t r e s s and t h e against is ~ ~ ~ e r v e d t o be a t y p i c a l mode temperature T b y means o f i n - s i t u o b s e r v a t i o n s w i t h the c o l l e c t i v e m o t i o n of d i s l o c a t i o n s . Thus, groups X-ray topography and a TV system. Together w i t h t h e d i s l o c a t i o n s move r a t h e r s t e a d i l y on t h e a c t i v e d a t a o f o t h e r a u t h o r s /2-4/, t h e y have shown t h a t 'lip planes.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979630

JOURNAL DE PHYSIQUE C6- 148

4. Measurements o f d e f o r m a t i o n c h a r a c t e r i s t i c s and d e t e r m i n a t i o n o f t h e s t a t e o f t h e c o l l e c t i v e m o t i o n o f d i s l o c a t i o n s . - From o b s e r v a t i o n s mentioned above, i t may be assumed t h a t t h e m o t i o n of screw d i s l o c a - t i o n s i s r a t e - c o n t r o l l i n g i n t h e d e f o r m a t i o n o f s i - l i c o n c r y s t a l s and t h a t t h e mean v e l o c i t y - v o f screw d i s l o c a t i o n s d u r i n g d e f o r m a t i o n i s g i v e n by a f o l l o - w i n g e q u a t i o n ;

7 ⁼ v o (-ra - T ~ ) m exp ( -Q/kT)

= m exp (-Q/kT) ( 2 )

where -ra i s t h e a p p l i e d s t r e s s , -ri t h e back s t r e s s due t o t h e i n t e r a c t i o n between d i s l o c a t i o n s , and

T~~~ t h e s o - c a l l e d e f f e c t i v e s t r e s s . T h i s t y p e o f v e l o c i t y e q u a t i o n has been used b y Alexander and Haasen /8/ i n t h e a n a l y s i s o f a macroscopic b e h a v i o u r o f c o v a l e n t c r y s t a l s . One can d e t e r m i n e t h e magnitu- des of ^T~~~ and ^-ii a t v a r i o u s d e f o r m a t i o n stages by s t a i n - r a t e change t e s t s w i t h use o f e q u a t i o n /2/.

And from them, t h e v e l o c i t y i and t h e d e n s i t y N, o f moving d i s l o c a t i o n s a r e known. The p o s s i b l e e r r o r

i n t h e d e t e r m i n a t i o n o f ^T~~~ due t o t h e change o f t h e i n t e r n a l parameters d u r i n g t h e s t r a i n - r a t e change i s supposed t o be l e s s t h a n 10% i n t h e p r e s e n t expe- r i m e n t s .

F i g u r e 1 shows t h e t e m p e r a t u r e dependence o f s t r e s s - s t r a i n b e h a v i o u r o f speciflens w i t h an i n i t i a l d e v s i t y o f d i s l o c a t i o n s o f 2 x l o 4 cm-'subjected t o [123] t e n s i l e d e f o r m a t i o n a t = 1.2 x lo-+ s - l ,

l b e i n g t h e a p p a r e n t shear s t r a i n - r a t e a s s o c i a t e d w i t h t h e cross-head speed o f t h e t e n s i l e machine.

Shear strain

.

F i g . 1 : Temperature dependence of t h e s t r e s s - s t r a i n c h a r a c t e r i s t i c s and t h e b e h a v i o u r o f t h e e f f e c t i v e

i n t h e d e f o r m a t i o n f o r E = 1.2 x lO-"s-'.

%%ii?~~

(Yonenaga and Sumino ( u n p u b l i s h e d ) ) .

Specimens a r e c u t f r o m s i n g l e c r y s t a l s grown by t h e f l o a t i n g - z o n e method. W i t h i n c r e a s e i n t e m p e r a t u r e t h e s t r e s s - s t r a i n c u r v e s h r i n k s i n s i z e m a i n t a i n i n g

a s i m i l a r shape. The b e h a v i o u r o f -reff a g a i n s t t h e s t r a i n i s a l s o shown i n t h e f i g u r e . -reff decreases w i t h t h e s t r a i n f r o m t h e upper y i e l d p o i n t t h r o u g h t h e l o w e r y i e l d p o i n t as i s expected f r o m a Johnston- Gilman t y p e t h e o r y o f y i e l d i n g /8/. The magnitude of -reff becomes c o n s t a n t w i t h r e s p e c t t o t h e s t r a i n a f t e r t h e m i d d l e o f s t a g e 0. T h i s i s i n t e r p r e t e d as t h a t themagni t u d e s o f 7 and Nm a r e c o n s t a n t w i t h r e s p e c t t o t h e s t r a i n i n such a d e f o r m a t i o n stage. Such a s t a t e i s c a l l e d " t h e s t e a d y s t a t e o f d e f o r m a t i o n " . The e f f e c t i v e s t r e s s i n t h e s t e a d y s t a t e o f

* t

d e f o r m a t i o n i s denoted b y -reff. -reff i s observed t o be a f u n c t i o n o f t h e d e f o r m a t i o n c o n d i t i o n , namely, o f and T. I t i s p l o t t e d a g a i n s t f o r v a r i o u s tem- p e r a t u r e s i n f i g u r e 2 and i s w e l l expressed by

-rEff = A el/n exp (U/LT) ( 3 )

w i t h n = 3.2 + 0.2 : m ^t 2 and U =0.72 + 0.04 eV z Q / ( m + 2 ) .

F i g . 2 :-rZff p l o t t e d a g a i n s t f o r v a r i o u s tempera- t u r e s . (Yonenaga and Sumino ( u n p u b l i s h e d ) ) .

Both t h e upper and t h e l o w e r y i e l d s t r e s s e s and t h e f l o w s t r e s s i n t h e f o l l o w i n g d e f o r m a t i o n stage a r e a1 1 observed t o depend on t h e i n i t i a l den- s i t y o f d i s l o c a t i o n s No. However, -rZff does n o t de- pend on No and i s determined o n l y b y E and T. The magnitudes o f Nm and 7 i n t h e s t e a d y s t a t e o f d e f o r - m a t i o n a r e c a l c u l a t e d by u s i n g e x p e r i m e n t a l l y d e t e r - mined -rZff and a r e shown i n f i g u r e 3 as a f u n c t i o n o f T f o r v a r i o u s s t r a i n - r a t e s . I t i s seen t h a t Nm decreases and i n c r e a s e s as T i n c r e a s e s , w h i l e b o t h Nm and i i n c r e a s e w i t h t h e i n c r e a s e i n E. T h i s beha- v i o u r o f Nm w i t h r e s p e c t t o and T accords w e l l w i t h t h a t of t h e d e n s i t y o f d i s l o c a t i o n e t c h p i t s revea- l e d i n s t a g e 0 and s t a g e I on t h e specimen s u r f a c e . 5. On t h e N a t u r e o f t h e c o l l e c t i v e m o t i o n o f d i s l o - c a t i o n s i n t h e s t e a d y s t a t e o f deformation.- The

-

c h a r a c t e r i s t i c s o f t h e c o l l e c t i v e m o t i o n o f d i s l o c a - t i o n s r e a l i z e d i n t h e d e f o r m a t i o n o f t h e c o n s t a n t s t r a i n - r a t e shown above can be d e s c r i b e d q u a n t i t a t i -

K. Sumino C 6 - 149

v e l y on t h e b a s i s o f a f o l l o w i n g hypothesis : t h e c o l - l e c t i v e motion of d i s l o c a t i o n s i n t h e steady s t a t e o f deformation i s r e a l i z e d so t h a t the component o f the f l o w s t r e s s which i s determined by t h e s t a t e o f such a c o l l e c t i v e motion takes a minimum value neces- sary t o m a i n t a i n the e x t e r n a l l y given s t r a i n - r a t e .

T , 'C

900 800 710

- 1

1 o - ~

Fig. 3 : The d e n s i t y N and t h e v e l o c i t y i o f moving d i s l o c a t i o n s i n t h e stgady s t a t e o f deformation p l o t - t e d a g a i n s t t h e i n v e r s e o f t h e temperature 1/T f o r various s t r a i n - r a t e s .

T may be d i v i d e d i n t o two components ; one i s i

due t o the i n t e r a c t i o n between moving d i s l o c a t i o n s themselves and t h e o t h e r due t o t h e i n t e r a c t i o n w i t h - .

immobile d i s l o c a t i o n s . They a r e denoted by T; and

;A,

where G i s t h e shear modulus, B a parameter characte- r i z i n g t h e e l a s t i c i n t e r a c t i o n between moving d i s l o- c a t i o n s and O i s t h e f u n c t i o n o f the s t r a i n . The f l o w s t r e s s o f t h e c r y s t a l s i s w r i t t e n as r a = (;p,vob) 'Im N;"~ exp (g/mk T) + G b Nm 112 / B + 0 ^{( E )} ,

( 5 ) where i s t h e p l a s t i c shear s t r a i n - r a t e and t h e

f i r s t term on t h e r i g h t hand s i d e P T ~ The hypothe- ~ ~ .

s i s o f t h e steady s t a t e o f deformation means the steady value of Nm t o be given by

-.

T~~~ and T; i n t h e steady s t a t e of deformation are obtained as

(7)

T~~~ t given by equation (7) shows a good agreement w i t h experimental r e s u l t s o f equation (3) w i t h respect t o both t h e s t r a i n - r a t e and t h e temperature dependences.

The o n l y a d j u s t a b l e parameter i n t h e present model i s 6 and can be determined e x p e r i m e n t a l l y t o be 3.3 + 0.4.

I t may be concluded t h a t t h e hypothesis o f t h e steady s t a t e o f deformation gives a c o r r e c t descrip- t i o n o f t h e experimental r e s u l t s on t h e c o l l e c t i v e motion o f d i s l o c a t i o n s i n the deformation stage a f t e r

the middle o f stage 0 i n s i l i c o n c r y s t a l s . The hypo- t h e s i s has a l r e a d y been shown t o be v a l i d a l s o f o r t h e deformation o f germanium c r y s t a l s 161. Thus, i t seems t o be a general r u l e governing t h e c o l l e c t i v e motion o f d i s l o c a t i o n s d u r i n g deformation of c r y s t a l s i n which d i s l o c a t i o n s move i n viscous manner. With denoting t h e s t r e s s component determined by t h e s t a t e o f t h e c o l l e c t i v e motion o f d i s l o c a t i o n s by

- i

T ( = T ? ~ ~ + T ~ ) , a f u n c t i o n a l r e l a t i o n s h i p holds b e t - ween E N and T a t a given temperature. I t can ea-

P ' m

s i l y be shown t h a t the c o n d i t i o n o f t h e steady s t a t e o f deformation (a? / aNm)l = 0 i s e q u i v a l e n t t o

(a; / aNm) ⁼ 0 which means t h a t Nm i s c o n t r o l l e d so as t o g i v e the maximum s t r a i n - r a t e under a given stress. The r e l a t i o n & , T and Nm a t a given tempe-

P

r a t u r e i s described by t h e p o i n t s on t h e s u r f a c e shown i n f i g u r e 4.

- .

Fig. 4 : The r e l a t i o n between tp,= + r: and N i n t h e deformation a t a given temperature. The bvoken l i n e shows the locus o f the steady o f defor- mation. (Sumino / 7 / )

The hypothesis means t h a t the steady s t a t e o f defor-

C6- 1 5 0 JOURNAL DE PHYSIQUE

m a t i o n i s g i v e n by t h e p o i n t s a l o n g t h e r i d g e o f t h e s u r f a c e shown by a broken l i n e where ED has t h e maxi- mum f o r c o n s t a n t T and r has t h e m i n i k m f o r c o n s t a n t

E P

6. Y i e l d b e h a v i o u r as a t r a n s i e n t phenomenon.- I f t h e steady s t a t e o f d e f o r m a t i o n i s regarded as a dynami- c a l e q u i l i b r i u m s t a t e of a c r y s t a l deformed a t a c o n s t a n t s t r a i n - r a t e , t h e y i e l d r e g i o n o f t h e s t r e s s - s t r a i n curves may b e t h o u g h t t o be a t r a n s i e n t s t a g e

f r o m an undeformed s t a t e t o t h e steady s t a t e o f d e f o r - m a t i o n . Haasen and h i s co-workers /8/ developed a t h e o r y o f y i e l d i n g on t h e b a s i s o f t h e dynamical cha- r a c t e r o f d i s l o c a t i o n s i n c o v a l e n t c r y s t a l s . Suezawa, Sumino and Yonenaga ( u n p u b l i s h e d ) have analysed t h e y i e l d b e h a v i o u r o f s i l i c o n c r y s t a l s on t h e model o f Haasen e t a l . I t has been shown t h a t t h e b e h a v i o u r o f t h e upper y i e l d s t r e s s i s w e l l d e s c r i b e d b y t h e model. However, some disagreements a r e found on t h e l o w e r y i e l d s t r e s s . The model assumes t h a t Nm i n c r e a - ses s t e a d i l y w i t h t h e i n c r e a s e o f E. Consequently, a minimum i n r e f f +

-rii

rii

hi

References

--

/1/ Sumino, K., Oyo B u t s u r i , 5 (1977) 1107 ; Harada, H., V a s t e r ' s Thesis (1976), Tohoku Univ.

/2/ George, A., Escaravage, C., Champier, G. and S c h r o t e r , W . , Phys. S t a t u s Sol i d i ( b ) , 3 (1972) 483.

/3/ Erofeev, A. and N i k i t e n k o , V . I . , Sov. Phys. JETP, 33 (1971) 963.

-

/ 4 / I m a i , M., ?laster's T h e s i s (1977), Tohoku Univ.

/5/ Sato, M. Sumino, K., Proc. 5 t h I n t e r n a t i o n a l Conf. on HVEW, (Kyoto, Imura, T. and Hashimoto A. Ed.) (1977) 459.

/ 5 / Sumino, K., Mater. S c i . Eng.,

13

/7/ Sumino, K., Nihon B u t s u r i Gakkai-shi,

30

/8/ Alexander, Y. and Haasen, P., S o l i d S t a t e Phys.

22, S e i t z , F. and T u r n b u l l , D. Ed. (Academic Press, New York) (1968) 28.