DIFFUSION OF GALLIUM IN GERMANIUM ALONG DISLOCATIONS

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DIFFUSION OF GALLIUM IN GERMANIUM ALONG DISLOCATIONS

K. Ahlborn

To cite this version:

K. Ahlborn. DIFFUSION OF GALLIUM IN GERMANIUM ALONG DISLOCATIONS. Journal de

Physique Colloques, 1979, 40 (C6), pp.C6-185-C6-188. �10.1051/jphyscol:1979638�. �jpa-00219054�

JOURNAL DE PHYSIQUE CoZZoque C6, s u p p l h e n t au n06, tome 4 0 , juin 1 5 7 9 , page C6-185

K. Ahlborn

I V PhysikaZisches I n s t i t u t der Universiti3t Gb'ttingen, BunsenstraBe 11-15, 3400 Ciittingen, F.R.C.

Resume.- Une etude de l a d i f f u s i o n du g a l l i u m dans l e germanium a P t @ e f f e c t u e e ,

a

680°C e t 710°C, sur des e c h a n t i l l o n s contenant une d e n s i t e de d i s l o c a t i o n s de 1,5x106

a

3x107 cm-'. Les p r o f i l s de c o n c e n t r a t i o n o n t

e t e

determines

a

l ' a i d e d ' u n analyseur i o n i q u e . Pour l a d i f f u s i o n en volume nous trouvons une energie d ' a c t i v a t i o n Q = 3,5 eV 2 0,3 eV e t un p r e - f a c t e u r Do Z 1,5x103 cmZ/s. Pour des densites de d i s l o c a t i o n s superieures a 3x106 ~ m - ~ , on observe, dans un repere I n c ( x ) , des queues de p r o f i l s l i n e a i r e s dues

a

l a d i f f u s i o n l e l o n g des d i s l o c a t i o n s .

Les r e s u l t a t s experimentaux o n t @ t & in t e r p r e t e s 3 p a r t i r d'un modele phi.nor&nologique des d i s l o c a - t i o n s , e t de l a s o l u t i o n a n a l y t i q u e donnee p a r Mimkes / 7 / . Yous avons determine de facon independan- t e l e rayon a de l a d i s l o c a t i o n e t l e r a p p o r t A e n t r e l e c o e f f i c i e n t de d i f f u s i o n dans l e s d i s l o c a - t i o n s D' e t l e c o e f f i c i e n t de d i f f u s i o n en volume D.

Abstract.- D i f f u s i o n o f g a l l i u m i n germanium was i n v e s t i g a t e d i n samples c o n t a i n i n g a d i s l o c a t i o n d e n s i t y o f 1.5x106 t o 3x10' ~ m - ~ , a t temperatures o f 680°C and 710°C. The c o n c e n t r a t i o n p r o f i l e s were obtained by means o f

SIMS.

For t h e volume d i f f u s i o n an a c t i v a t i o n energy o f Q = 3.5 eV i 0.3 eV and a p r e - f a c t o r o f Do : 1.5x103 cm2/s i s found. F o r e t c h p i t d e n s i t i e s l a r g e r than 3x106 c K 2 we observe t h e c h a r a c t e r i s t i c t a i l s due t o p i p e d i f f u s i o n wich a r e l i n e a r on an I n c-x

-

^scale.

The r e s u l t s have been analyzed i n terms ofaphenomenological model o f t h e d i s l o c a t i o n and the analy- t i c a l s o l u t i o n given by Mimkes / 7 / . The p i p e r a d i u s a and the r a t i o A o f the d i f f u s i o n c o e f f i c i e n t i n the p i p e D' t o t h a t i n t h e volume D a r e determined independently.

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

.-

D i f f u s i o n along d i s l o c a t i o n s may be more r a p i d than d i f f u s i o n i n the volume due t o an a t t r a c t i v e i n t e r a c t i o n between p o i n t d e f e c t s and d i s l o c a t i o n s and a reduced m i g r a t i o n enthalpy o f p o i n t d e f e c t s along the core. Thus p i p e d i f f u s i o n experiments a r e o f i n t e r e s t f o r understanding o f i n - t e r a c t i o n between p o i n t d e f e c t s and t h e d i s l o c a t i o n core.

The c o n c e n t r a t i o n p r o f i l e s obtained i n crys- t a l s c o n t a i n i n g d i s l o c a t i o n s show a l i n e a r p a r t i n t h e I n c ( x ) curves a t l a r g e distances from t h e sur- face. For a phenomenological treatment o f p i p e d i f f u - s i o n a model o f the d i s l o c a t i o n i s commonly used, which describes t h e d i s l o c a t i o n by a tube o f r a d i u s a and a d i f f u s i o n c o e f f i c i e n t

D'

= A.D, where D i s t h e d i f f u s i o n c o e f f i c i e n t i n t h e volume. The two Fick-equations i n c y l i n d r i c a l coordinates d e s c r i b i n g t h e i n n e r and t h e o u t e r r e g i o n have been solved i n d i f f e r e n t approximations.

I n most o f t h e former works /I/, /2/, /3/ t h e authors made r a t h e r r e s t r i c t i v e assumptions so t h a t t h e i r s o l u t i o n s a l l o w o n l y f o r an a n a l y s i s o f t h e slope o f the t a i l s which y i e l d s t h e combined para- meter d.a2.

An exact s o l u t i o n f o r d i f f u s i o n along d i s l o c a - t i o n s i s given by S t a r k /4/ and Brebec / 5 / . However both s o l u t i o n s seem t o be t o o complicated f o r a p p l i - Cation.

*Supported by SF6 126 ( P r o j e c t ~ 9 ) .

Benoist and M a r t i n /6/ proposed an atomic mo- d e l f o r grain-boundary d i f f u s i o n . This model permits t o a d j u s t t h e atomic g r a i n boundary s t r u c t u r e and atomic jump frequencies t o p r o f i l e s o f d i f f u s i o n i n g r a i n boundaries. This model has n o t y e t been ap- p l i e d t o p i p e d i f f u s i o n .

A s o l u t i o n o f t h e two Fick-equations which a l l o w s f o r the f i r s t t i n e t o determine A and a inde- pendently, has been given by Mimkes / 7 / . However, when f i t t i n g t h e model o f tlimkes t o c o n c e n t r a t i o n p r o f i l e s measured f o r metals /8/, / 9 / , 10/ one f i n d s p i p e r a d i i v a r y i n g from 100 A t o 1 bm. A t h e o r e t i c a l e s t i m a t i o n /11/ o f a on t h e b a s i s o f the e l a s t i c i n t e r a c t i o n between vacancy and d i s l o c a t i o n y i e l d s a : 20-50 A. The t o o l a r g e experimental values o f a may r e s u l t from a r e l a x a t i o n o f the d i s l o c a t i o n a r - rangement d u r i n g d i f f u s i o n annealing and a r e l a x a t i o n o f t h e boundary c o n d i t i o n i n experiments, which work w i t h a f i n i t e source. We have t r i e d t o a v o i d these d e f i c i e n c i e s .

D i f f u s i o n experiments were c a r r i e d o u t i n ger- manium w i t h a constant s u r f a c e c o n c e n t r a t i o n o f g a l - lium, because s t a b l e , w e l l d e f i n e d d i s l o c a t i o n a r - r a y s can be produced. G a l l i u m was chosen as an impu- r i t y because o f i t s h i g h s o l u b i l i t y i n germanium ( 5 ~ 1 0 ~ ~ c m - ~ a t 700°C) and i t s small s i z e e f f e c t i n i n t h e germanium m a t r i x . Thus we have c o n d i t i o n s s i m i l a r t o a s e l f d i f f u s i o n experiment.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979638

C6-186 JOllRNAL DE PHYSIQUE

2. Experimental method

.-

Specimen ( 2 0 x 4 ~ 4 mm3 and t o co = 5 ~ 1 0 ' ~ c m - ~ (Fig. 2) 3 0 x 1 0 ~ 4 mm3, lO''~m-~n-doped) w i t h probe a x i s i n

<123> and <112> d i r e c t i o n r e s p e c t i v e l y and w i t h (1111 _cGa,c m-3 sidefaces were c u t from m o n o c r y s t a l l i n e germanium and

mechanically and chemically polished. They were de- formed by s t a t i c compression (T = 0.5

-

1 kg mm-2;

E = 5 %

-

10% a t 730°C) o r by dynamic f o u r p o i n t ben- d i n g (0.01 cm/min ; ra d i u s o f c u r v a t u r e 20

-

50 mm) (see f i g u r e s l a and l b ) .

F i g . 1 : a) S i n g l e s l i p geometry f o r deformation by F i g . 2 : Gallium c o n c e n t r a t i o n vs. p e n e t r a t i o n depth s t a t i c compression. - b ) Specimen o r i e n t a t i o n f o r f o r a d i s l o c a t i o n - f r e e germanium sample.

deformation by dynamic f o u r p o i n t bending. D i f f u s i o n temperature : 710°C D i f f u s i o n time : 4.92 s o as obtained by SIMS

To a v o i d a rearrangement of d i s l o c a t i o n s d u r i n g

-

complementary errorfunction, giving = . 4 9 x 1 0 ~ ⁵ diffusion-procedure t h e specimens were annealed f o r cm2/s and co = 4.4x1019Ga

24 h a t 750°C which i s higher than t h e temperature chosen f o r t h e d i f f u s i o n studies. The e t c h p i t den- s i t i e s Np determined by B i l l i g - e t c h a n t on t h e (1111 sidefaces were between Np = 106cm-2 and Np = 2x107 cmm2.

D i f f u s i o n o f gal 1 ium i n t o t h e deformed specimen and a d i s l o c a t i o n f r e e c o n t r o l specimen was c a r r i e d o u t i n a q u a r t z ampoule, which was evacuated t o

t o r r and then sealed. A c o n s t a n t g a l l i u m vapour pressure was maintained by keeping a p i e c e o f g a l - l i u m w i t h i n t h e tube a t a temperature lower than d i f f u s i o n temperature g i v i n g r i s e t o a constant g a l - 7 ium c o n c e n t r a t i o n a t t h e germanium s u r f a c e d u r i n g d i f f u s i o n time. T h i s boundary c o n d i t i o n of an i n f i - n i t e source leads t o c o n c e n t r a t i o n p r o f i l e s o f t h e form c = co e r f c ( x / 2 m ) .

When t h e s u r f a c e c o n c e n t r a t i o n co i s l a r g e r than t h e number o f i n t r i n s i c c a r r i e r s a t d i f f u s i o n temperature (ni = 4 . 9 ~ 1 0 l ~ c m - ~ a t 700°C) one expects d e v i a t i o n s from t h i s simple behaviour due t o an

i n n e r e l e c t r i c f i e l d i n t h e d i f f u s i o n f r o n t . However our p r o f i l e s f i t an complementary e r r o r f u n c t i o n up

The c o n c e n t r a t i o n p r o f i l e s were obtained by means o f SIMS ( I ) . The (111)-fates o f t h e specimen were bombarded w i t h a p r i m a r y i o n beam (O+-ions, a c c e l e r a t e d t o 10 keV)

.

The 1 owest d e t e c t a b l e gal

-

l i u m c o n c e n t r a t i o n i s o f t h e order o f 1016Ga The p r i m a r y i o n beam i s scanned over t h e spe- cimen. The bombarded aerea ( 1 x 1 mm2) i s f a i r l y grea- t e r than the analyzed surface i n the c e n t e r o f t h e c r a t e r (300 um diameter) ( F i g . 3a). This avoids e r r o r s i n t h e cGa(x)-curves due t o g a l l i u m i o n s s p u t t e r e d from t h e border o f t h e c r a t e r .

3. Results.- The temperature dependence o f t h e d i f f u - s i o n c o e f f i c i e n t o f g a l l i u m i n germanium D i s i n accordance w i t h t h e values given by Dunlap /12/ f o r h i g h e r temperatures. However, Dunalp's value f o r T = 65OoC appears t o be t o o l a r g e . From o u r data we o b t a i n as a c t i v a t i o n energy Q

*

3.5 eV k 0.3 eV and as p r e f a c t o r Do = 1.5x103 cm2/s ( F i g . 4 ) .

( ' ) ~ n a l y s e u r i o n i q u e CAMECA;

i n

L a b o r a t o i r e de Phy- sique du Solide, Nancy, France.

K. Ahlborn _C6-187

F i g . 3 : a ) Schematic bombardment and a n a l y z i n g con- d i t i o n s o f SIMS. The specimen i s bombarded w i t h 10 keV 0' - i o n s w i t h p r i m a r y i o n d e n s i t y Ip. The focussed beam i s r a s t e r e d o v e r t h e s u r f a c e t o o b t a i n a c r a t e r o f 1 x 1 mm2. The s p u t t e r e d secondary i o n s , g i v i n g r i s e t o a secondary i o n d e n s i t y Is a r e ana- l y z e d w i t h r e s p e c t t o mass/charge.

b ) Only a zone o f 300 vm d i a m e t e r i n t h e c e n t e r o f t h e c r a t e r i s analyzed. W i t h i n t h i s zone e t c h p i t s a r e counted a f t e r t h e p r o f i l e measurement t o o b t a i n 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 Nd=cg.f.Np.

F i g . 4 : Temperature dependence o f t h e d i f f u s i o n c o e f f i c i e n t i n t h e volume D ( i n cm2/s) f o r t h e d i f - f u s i o n o f g a l l i u m i n germanium.

0 v a l u e s g i v e n b y Dunlap /12/

+

o u r v a l u e s

The c o n c e n t r a t i o n p r o f i l e s o f t h e deformed spe- cimens c o n s i s t o f an e r f c - p a r t near t h e s u r f a c e f r o m w h i c h

D

i s o b t a i n e d , f o l l o w e d by a t a i l , which i s l i n e a r on t h e l n c - x p l o t (see f i g u r e 5 ) . For d i f f e -

r e n t r e g i o n s analyzed on t h e ( 1 1 1 ) - s i d e f a c e o f t h e same specimen, t h e t a i l s may show d i f f e r e n t s l o p e s . As demonstrated i n f i g u r e 5, t h i s b e h a v i o u r o f t h e t a i l s i s q u a l i t a t i v e l y r e l a t e d t o a c o r r e s p o n d i n g v a r i a t i o n o f t h e l o c a l e t c h - p i t d e n s i t i e s Np. W i t h i n c r e a s i n g v a l u e s o f Np t h e t a i l s a r e s h i f t e d t o h i g h e r c - v a l u e s .

F i g . 5 : G a l l i u m c o n c e n t r a t i o n v s . p e n e t r a t i o n d e p t h i n germanium o b t a i n e d on one bended specimen.

( T = 713"C, d i f f u s i o n t i m e t = 5 . 7 8 - I U ' s. The d i f f e - r e n t t a i l s a r e due t o d i f f e r e n t l o c a l e t c h p i t den- s i t i e s Np.

ONp = 1.8x106 c r 2

+

^N

-

3.7x106

~ N E ¹

^6.7x106

b e s t f i t a c c o r d i n g t o t h e model o f Mimkes / I ? / . We have f i t t e d t h e f o r m u l a , which Mimkes d e r i - ved w i t h i n h i s model o f a p i p e o f r a d i u s a and d i f - f u s i o n c o n s t a n t D1 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 , t o o u r e x p e r i m e n t a l d a t a and d e r i v e d t h e r a t i o A=D1/D and t h e p i p e r a d i u s a. We found t h a t t h e combined f a c t o r h a 2 i s m a i n l y d e t e r m i n e d by t h e s l o p e o f t h e t a i l s i n agreement w i t h r e s u l t s o f Pavlov e t a1 ./2/

b u t t h a t t h e v a l u e s f o r a and A a r e r a t h e r s e n s i t i v e t o t h e v a l u e chosen f o r t h e d i s l o c a t i o n d e n s i t y (see f i g u r e 6 ) .

The d i s l o c a t i o n l e n g t h p e r u n i t volume Nd i s r e l a t e d t o t h e number o f e t c h p i t s p e r u n i t a r e a N b y Nd = cg.f

.N ,

where l/f i s t h e f r a c t i o n o f

P P

d i s l o c a t i o n s t h a t a r e marked b y an e t c h - p i t and c g i s a f a c t o r which accounts f o r t h e a n g u l a r d i s t r i - b u t i o n o f l i n e elements r e l a t i v e t o t h e e t c h e d s u r -

C6-I 88 JOURNAL DE PHYSIQlTE

f a c e . c has been estimated t o about 1.5 f o r t h e 9

s l i p geometry used i n our experiments. For t h e e t c h a n t , which we used, S p r i n g e r /13/ has compared N p and N d , a s measured by transmission e l e c t r o n mi- croscopy and found a value of f between 1.5 and 2 . We note t h a t Springer measured s i g n i f i c a n t l y l a r - ger values of f f o r etch p i t d e n s i t i e s of t h e order l o 7 cm-', but this r e s u l t s i n our opinion may be an a r t e f a c t of t h e method.

Fig. 6 : Pipe r a d i u s a , a s obtained by f i t t i n g t h e model of Mimkes /12/ t o d i f f e r e n t concentration pro- f i l e s vs. d i s l o c a t i o n d e n s i t y Nd =.cg.f.Np.

1.5. The arrows mark d i s l o c a t i o n d e n s i t i e s :%r;esponding t o f = 3 .

I f we t a k e cg.f-3 ( p o i n t marked by an arrow i n f i g u r e 6 ) we o b t a i n values f o r a between 10 and 180 A and f o r A between l o 3 and 10'. The e r r o r of e t c h p i t counting i s about 20% and t h e r e f o r e cannot account f o r t h e v a r i a t i o n i n a and A . A l a r g e r va- l u e f o r f , a s been proposed by Springer, would lead t o a decrease of a and an i n c r e a s e of A , but would not reduce t h e v a r i a t i o n f o r t h e s e parameters (see f i g u r e 6 ) . However, from t h e experimental p o i n t of view t h e d i s t r i b u t i o n of d i s l o c a t i o n s cannot be ap- p r o p r i a t e l y incorporated i n t o t h e a n a l y s i s a t pre- s e n t , so t h a t i s must remain open whether o r not t h e model underlying t h e s o l u t i o n of Mimkes gives a f a i r d e s c r i p t i o n of pipe d i f f u s i o n .

Acknowledgment.- The author would l i k e t o thank P r o f . W . S c h r o t e r and Dr. G . Edelin f o r helpful d i s - cussions. He would a l s o l i k e t o thank Prof. G . Cham- p i e r f o r t h e p o s s i b i l i t y of carrying o u t the SIMS- measurements i n h i s l a b o r a t o r y and f o r helpful sup- p o r t of Dr. S. Scherrer and S. Weber. This work was p a r t l y sponsored by t h e CNRS (ATP i n t e r n a t i o n a l e 1975).

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