CALCULATIONS OF DISLOCATION PIPE DIFFUSION

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CALCULATIONS OF DISLOCATION PIPE DIFFUSION

J. Mimkes

To cite this version:

J. Mimkes. CALCULATIONS OF DISLOCATION PIPE DIFFUSION. Journal de Physique Collo-

ques, 1979, 40 (C6), pp.C6-181-C6-183. �10.1051/jphyscol:1979637�. �jpa-00219053�

JOURNAL DE PHYSIQW C o ~ ~ o q u e C6,suppldment au n06, tome 40, juin 1979, page C6-181

CALCULATIONS OF DISLOCATION P I P E DIFFUSION

J. Mimkes

Faehbereich Physik, CescznthochschuZe Paderborn, F.R.C.

Resume.- Les mesures de p r o f i l s de d i f f u s i o n o n t e t e i n t e r p r e t e e s 2 p a r t i r de l a s o l u t i o n complete de l ' e q u a t i o n de l a d i f f u s i o n dans l e canal des d i s l o c a t i o n s . Deux parametres o n t e t e a j u s t e s : l a d i f f u - s i v i t e dans l e canal des d i s l o c a t i o n s D', e t l e rayon du canal des d i s l o c a t i o n s a. Les v a l e u r s obte- nues pour D' se t r o u v e n t en bon accord avec c e l l e s d e j a donnees p a r d ' a u t r e s auteurs. Par c o n t r e l e s valeurs obtenues pour a sont souvent p l u s elevees, a cause des e r r e u r s experimentales dans l a d e t e r - m i n a t i o n de l a d e n s i t e de d i s l o c a t i o n s .

Cependant, l a zone de c o n c e n t r a t i o n @levee n ' e s t pas l i m i t e e au coeur des d i s l o c a t i o n s ; e l l e depend de l a d i f f u s i v i t e en volume D, e t peut Gtre c a r a c t e r i s e e p a r l e rayon e f f e c t i f de l a d i f f u s i o n l e

long des d i s l o c a t i o n s , R = 2m.

Abstract.- Experimental d i f f u s i o n p r o f i l e s have been analyzed by a complete s o l u t i o n o f the p i p e d i f - f u s i o n equations, u s i n g two f i t t i n g parameters, pipe d i f f u s i v i t y Dl and d i s l o c a t i o n p i p e r a d i u s a.

The p i p e d i f f u s i v i t y D' agrees w e l l w i t h values obtained by o t h e r authors. The p i p e r a d i u s o f t e n t u r n s o u t very l a r g e due t o the inaccuracy o f t h e ,experimental d i s l o c a t i o n d e n s i t y .

However, t h e area of h i g h c o n c e n t r a t i o n along t h e d i s l o c a t i o n i s n o t c o n f i n e d t o t h e d i s l o c a t i o n core, b u t depends on t h e d i f f u s i v i t y D o f the b u l k and may be c h a r a c t e r i z e d by an e f f e c t i v e d i f f u s i o n p i p e r a d i u s R = 2 d E .

1. I n t r o d u c t i o n . - I n d i f f u s i o n experiments d i s l o c a - t i o n s have o f t e n been observed /1/ t o form a s h o r t c u t f o r the d i f f u s i o n process. As t h e d i f f u s i v i t y i n t h e d i s l o c a t i o n p i p e may be up t o 1000 times h i g h e r than i n t h e bulk, p i p e d i f f u s i o n i s o f considerable i n t e r e s t t o m a t e r i a l s research.

2. Pipe d i f f u s i o n model.- I n order t o c a l c u l a t e p i p e d i f f u s i o n data t h e d i f f e r e n t i a l equations o f p i p e d i f f u s i o n have t o be solved.

I n a simple model i s o l a t e d d i s l o c a t i o n s may be approximated by i n f i n i t e pipes o f r a d i u s a and d i f - f u s i v i t y Dl p e n e t r a t i n g t h e c r y s t a l perpendicular t o t h e surface. The c o n c e n t r a t i o n c ' o f d i f f u s i n g mate- r i a l i s assumed t o be o f r a d i a l s y m e t r y i n t h e p i - pe :

y i s t h e i n i t i a l c o n c e n t r a t i o n o f an instantaneous source a t t h e s u r f a c e z = o.

I n t h e b u l k d i f f u s i o n i s determined by t h e d i f f u s i v i t y D :

At t h e boundary o f t h e d i s l o c a t i o n p i p e we have

These equations may be solved by F o u r i e r - Lap1 ace t r a n s f o r m a t i o n s s i m i l a r t o t h e approach t o grainboundary d i f f u s i o n by Whipple /2/ and Suzuoka /3/

and w i l l y i e l d a s i m i l a r r e s u l t f o r c ( r , z , t ) /4/.

For t h e e v a l u a t i o n of s e c t i o n i n g experiments t h e so- l u t i o n has t o be i n t e g r a t e d w i t h r e s p e c t t o t h e r a - d i u s r :

Q ( z , t ) = l

c(r,Z,t,) 2rrrdr = _m (Qo(2,t)+Ndna2 .

QI ( z , t ) ) (4)

Q o i s t h e t o t a l ( ' ) amount o f m a t e r i a l t h a t has d i f f u - sed the d i s t a n c e z from t h e s u r f a c e F :

i s t h e amount o f m a t e r i a l t h a t has d i f f u s e d i n t o t h e b u l k through t h e d i s l o c a t i o n . A

D'/D i s t h e r a t i o o f d i f f u s i v i t i e s i n p i p e and bulk, r e s p e c t i v e l y , a i s the r a d i u s o f t h e pipe. S e c t i o n i n g experiments may be evaluated by f i t t i n g the two parameters D' and a t o t h e experimental data, which must c o n t a i n t h e d i f f u s e d amount Q, t h e s e c t i o n i n g depth z, d i f f u s i o n time t, and t h e d i s l o c a t i o n d e n s i t y Nd.

Experimental r e s u l t s have been analyzed /5/

according t o equations ( 4 ) - ( 6 ) f o r v a r i o u s mate- r i a l s .

-

( ' ) 1 n o r d e r t o e v a l u a t e non homogeneous d i s l o c a t i o n d i s t r i b u t i o n s equation ( 6 ) d i f f e r s s l i g h t l y from r e f . /4/.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979637

C6-182 JOURNAL DE PHYSIQUE

The d i f f u s i v i t i e s agree well with the expected values and lead t o s a t i s f y i n g r e s u l t s f o r t h e f i r s t f i t t i n g parameter A = D 1 / D .

The second f i t t i n g parameter, the radius a of the dislocation pipe has given l e s s s a t i s f y i n g re- s u l t s , the pipe radius f o r self-diffusion in gold / 6 / i s calculated a t the order of 50 f i .

This i s , however, due t o the f a c t , t h a t in contrast t o the high accuracy of the diffusion data the accuracy of the dislocation density often has been neglected. As the dislocation density Nd enters d i r e c t l y i n t o t h e calculations of the pipe radius, N~~ a

const

the pipe radius a will be t h e smaller t h e higher the dislocation density i s assumed /8/.

Only great care in determining the true d i s l o - cation density will y i e l d a correct pipe radius a .

However, i t may be noted, t h a t the diffusion p r o f i l e around a s i n g l e dislocation i s almost inde- pendent of the pipe radius a , and will extend over large distances i n t o t h e bulk of the c r y s t a l . This will lead t o a very large e f f e c t i v e radius f o r pipe diffusion data of Nb i n t o Ta by Pawel and Lundy /7/

in figure 1.

Fig. 1 : P r o f i l e of Nb diffusion i n t o Ta. Open c i r - c l e s give experimental data of a c t i v i t y as function of the square of penetration depth z 2 according t o r e f . /7/. The sol id line' corresponds t o the f i t Nd = ID7 u r 2 , a = 170 1, A = 1700, D = 1,5x10-l5 cm2 s-' in equation ( 4 ) .

The data may be f i t t e d by Nd = l o 7 cm2 ( a fac- t o r of 100 l a r g e r than given by the authors), A=1700 and a = 170 i. This pipe radius s t i l l i s too large

and a higher dislocation density may have t o be assu- med.

However, the diffusion p r o f i l e around a d i s l o - cation turned out t o be almost independent of t h e pipe radius ( o r the dislocation density assumed).

Figure 2 shows the concentration c ( r , z , t ) as a func- tion of r t o extend over more than s i x times the assumed pipe radius of 170 8.

Fig. 2 : P r o f i l e of Nb diffusion i n t o Ta. The con- centration c ( r , z , t ) has been calculated as a func- tion of distance r from the dislocation a t a pene- t r a t i o n depth z = 7000 8 f o r the same f i t t i n g para- meters as in figure 1.

As we have t o integrate c ( r ) . 2 ~ r over r in order t o obtain the t o t a l concentration per section the function f ( r ) = c ( r ) - 2 a r has been plotted in figure 3.

0 200 LOO 600 800 1000 1200 1400

Fig. 3 : P r o f i l e of Nb diffusion i n t o Ta. f ( r , z , t ) =

2 ~ ~ r * c ( r , z , t ) has been calculated f o r t h e penetra-

tion depth z = 7000 and the same f i t t i n g parame-

t e r s a s in f i g u r e 1.

T h i s f i g u r e r e v e a l s two r e s u l t s :

1. t h e m a t e r i a l i n s i d e t h e d i s l o c a t i o n p i p e does n o t c o n t r i b u t e much t o t h e t o t a l amount o f a section, 2. t h e m a t e r i a l d i f f u s e d through t h e pipes extends

over a l a r g e d i s t a n c e i n t o t h e b u l k o f the crys- t a l . T h i s d i s t a n c e w i l l be c o n t r o l l e d by t h e d i f - f u s i v i t y o f t h e bulk, and w i l l be o f t h e o r d e r R = 2 m , o r i n f i g u r e 3, R = 650 1.

Independently o f t h e r a d i u s a o f t h e d i s l o c a - t i o n core t h e d i f f u s i o n e f f e c t i v e p i p e r a d i u s R =

2m w i l l govern pipe d i f f u s i o n and wi 11 cause a l a r - ge amount o f m a t e r i a l t o t r a v e l along d i s l o c a t i o n s as d i f f u s i o n s h o r t c u t s .

An a p p l i c a t i o n o f the s o l u t i o n s o f pipe d i f f u - s i o n s t o t e t r a h e d r a l l y coordinated semiconductors w i l l be given i n t h e f o l l o w i n g paper.

References

/1/ Hofmann, R.E., and T u r n b u l l , D., J. Appl. Phys., 22 (1951) 634.

-

/2/ Whipple, R.T.P., P h i l o s . Mag., 45 (1954) 1225.

/3/ Suzuoka, T., J. Phys. Soc. Japan, 19 (1964) 839.

/4/ Mimkes, J . and Wuttig, M., Phys. Rev., B 2 (1970) 1619.

/5/ Mimkes, J., Thin S o l i d Films, - 25 (1975) 221.

/ 6 / Gupta, D., Phys. Rev., B 1 (1973) 586.

/7/ Pawel, R.E. and Lundy, T.S., Acta M e t a l l . , 13

(1965) 345.

/8/ Ahlborn, K. and Schroter, W., t h i s conference.