Volume 32A. number 3 P H Y S I C S L E T T E R S 29 June 1970

M O B I L I T Y A N D D I F F U S I O N C O N S T A N T O F l l 0 A g I N K B r S I N G L E

L. Bo HARRIS, J . R. HANSCOMB and J , L. S C H L E D E R E R School of Physics . Uttiversity of New SoMh Wales. Ketzsinglot/, N. S, W.. 2033 A/ish'alia

Received 26 May 1970

C R Y S T A L S

Activation enthalpy for mobility of l l0Ag in KBr single crystals between 260 ° and 400°C is 0.67 eV. equal to that for cation vacancy mobility. The corresponding activation enthalpy for diffusion is anomal- ously smal ler . 0.5~ eV.

D i s l o c a t i o n s in a l k a l i ha l ide s ing l e c r y s t a l s can be d e c o r a t e d at low t e m p e r a t u r e s by u s ing an e l e c t r i c f i e l d I1,21; m e t a l ions f r o m e l e c t r o - des m i g r a t e p r e f e r e n t i a l l y a long the d i s l o c a t i o n s u b s t r u c t u r e . Such b e h a v i o u r i n d i c a t e s i n h o m o - geneous c h a r g e t r a n s p o r t , but in g e n e r a l t h e r e i s not m u c h e x p e r i m e n t a l da t a a v a i l a b l e to con- f i r m o r deny any s u g g e s t i o n that s u b s t r u c t u r e can have a m a r k e d in f luence on conduc t ion and d i f fus ion: F o r e x a m p l e , s i l v e r r e a d i l y d e c o r a t e s d i s l o c a t i o n s in K B r at 150°C I1, 2], but conduc t ion and d i f fus ion of s i l v e r in K B r have not a s ye t b e e n c o m p a r e d with that of p o t a s s i u m . As p a r t of a p r o g r a m m e to ob ta in da ta in th i s f i e ld the p r e - s en t w o r k r e p o r t s on s i m u l t a n e o u s m e a s u r e m e n t s of m o b i l i t y and d i f fus ion cons t an t of s i l v e r in the K B r la t t ice° The m e t h o d u s e d was that of Red ing - ton [3] in wh ich a r a d i o a c t i v e i so tope ( l l 0 A g ) is d r i v e n by an e l e c t r i c f i e l d inot a c r y s t a l which is a f t e r w a r d s s e c t i o n e d by m i c r o t o m e and e x - a m i n e d by s c i n t i l l a t i o n counte r° Use of e l e c t r i c f i e l d s up to s e v e r a l t housand vo l t s p e r cm p e r - m i t t e d m e a s u r e m e n t s at r e l a t i v e l y low t e m p e r a - t u r e s ( 2 6 0 ° - 4 5 0 o c ) ; th is i s in the e x t r i n s i c r e - gion, w h e r e few s tud i e s of m o b i l i t y and d i f fus ion cons t an t have b e e n made .

T r a c e r m o b i l i t y t~ (p lo t ted as tl T) and d i f - fus ion cons t an t D a r e g iven in f ig. 1 as a func - t ion of i n v e r s e t e m p e r a t u r e l / T . M e a s u r e m e n t s w e r e m a d e on s p e c i m e n s f r o m one s ing l e c r y s t a l p u l l e d f r o m the m e l t , and w e r e found to be r e - p roduc ib l e ° The a c t i v a t i o n en tha lpy fo r m o b i l i t y of s i l v e r i ons ob ta ined f r o m the l e a s t - s q u a r e s p lo t of f ig. 1 i s 0.67 + 0.01 eV, though s y s t e m - a t i c e r r o r s in the a b s o l u t e m a g n i t u d e of m o b i l i t y m a y be l a r g e r ° The a c t i v a t i o n en tha lpy h v fo r ca t ion v a c a n c y m o b i l i t y in K B r has b e e n found f r o m c o n d u c t i v i t y m e a s u r e m e n t s to be 0.e65 eV [4] o r 0.66 eV [5]; hence the s i l v e r ion m i g r a t e s

1l~ 7

g u)

d e

1,4

I03/T(°K) "1 14, 1 -6 1"8 2 " 0 1 0 - 1 0

o i I i i i I , C /.00 300 250

400 300 250 °C I I ~ _ I I 10"12

16 18 2-0 103/T( °K )'1

E

lO -11

Fig, i. Temperature dependence of ~tTand D. Circles. left-hand and bottom scales: crosses, right-hand and

top scales.

in K B r at low t e m p e r a t u r e s by a v a c a n c y m e - c h a n i s m , e x a c t l y as i t does in NaC1 at h igh t e m p e r a t u r e s (> 575°C) [6].

The v a l u e s of d i f fus ion cons t an t D in fig. 1 a r e l e s s a c c u r a t e than those of mob i l i t y , but they c l e a r l y ind ica t e a s m a l l e r a c t i v a t i o n en tha lpy hD; a l e a s t - s q u a r e s plot g i v e s a va lue of 0,54 eVo F u r t h e r , the E i n s t e i n r e l a t i o n p/D = (fkT)/e, w h e r e e is the c h a r g e on the ion and f a c o r r e l a - t ion f a c t o r of o r d e r of m a g n i t u d e unity, i s not obeyed. E x p e r i m e n t a l v a l u e s of ~/D a r e s m a l l e r by f a c t o r s which v a r y f r o m 0 0 7 at 260°C to 0.14 at 400°C. S m a l l v a l u e s of t~/D can be e x p l a i n e d e i t h e r by a s s o c i a t i o n of ca t ion v a c a n c i e s wi th r e s i d u a l d iva l en t i m p u r i t i e s to f o r m n e u t r a l c o m - p l e x e s which c o n t r i b u t e to t h e r m a l d i f fus ion but not to mo t ion u n d e r an e l e c t r i c a l f ie ld , o r by a

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Volume 32A. number 3 P H Y S I C S L E T T E R S 29 June 1970

s m a l l c o r r e l a t i o n f a c t o r f f o r the s i l v e r i m p u r i t y iota S i m i l a r l y , the s m a l l e r v a l u e of h D m u s t r e - s u l t e i t h e r f r o m e n h a n c e d e a s e of m o t i o n of n e u - t r a l c o m p ! e x e s t h r o u g h the l a t t i c e , o r f r o m s o m e l o w - e n e r g y m e c h a n i s m p e c u l i a r to the s i l v e r i m - p u r i t y ion, p e r h a p s a s s o c i a t e d w i th s u b s t r u c t u r e .

V a l u e s of D in fig~ 1 a p p e a r l a r g e when c o m - p a r e d wi th e x t r a p o l a t e d D v a l u e s f o r d i f f u s i o n of K + in K B r a t 450°C and a b o v e 17]. T h i s s u g g e s t s t h a t the s i l v e r ion, due to i t s r e l a t i v e l y l a r g e p o - l a r i s a b i l i t y , o s c i l l a t e s m o r e r a p i d l y on a K B r l a t t i c e s i t e t h a n the h o s t c a t i on ; the s a m e b e h a v - i o u r h a s p r e v i o u s l y b e e n o b s e r v e d f o r s i l v e r in

NaC1 a t h igh t e m p e r a t u r e s 16 I¢ F u r t h e r w o r k on t h i s s u b j e c t i s in p r o g r e s s .

References [1] L .B. Harr is . Appl. Phys. Le t te r s 13 (1968) 154. [21 L. B. Harr i s , J. Appl. Phys . . to be published. [3] R.W. Redington. Phys. Rev. 87 (1952) 1066. [4] J. Rolfe. Canadian J. Phys. 42 (1964) 2195. [5] J .N.Maycock . J. Appl. Phys. 35 (1964) 1512. [6] M. Chelma. Compt. Rend. 238 (1954) 82. [7l J , F . Laurent and J. Bernard . J. Phys. Chem. Solids

'3 (1957) 7.

O R B I T A L M & G N E T I S M I N T H E T I G H T - B I N D I N G L I M I T

M . SHIMIZU a n d Y TAKAHASHI Deparlmenl qf Applied Physics. Nagoya UMt,ersily. Nagoya. Japan

Received 25 April 1970

Orbital magnet ic suscept ibi l i ty is calculated in the t ight-binding limit for sc. bec, and fcc s t ruc tu re s for a s ingle-band model. It is shown that this suscept ibi l i ty somet imes becomes paramagnet ic .

T h e g e n e r a l and e x a c t r e s u l t of the m a g n e t i c s u s c e p t i b i l i t y due to o r b i t a l m o t i o n of e l e c t r o n s in a m e t a l w i t h i n the l i m i t s of the B l o c h m o d e l w a s o b t a i n e d by H e b b o r n a n d S o n d h e i m e r I1 I. T h e i r r e s u l t w a s c o n v e n i e n t l y w r i t t e n a s the s u m of fou r t e r m s . X = XI+X2+X3+X4 , ( e q s . ( 5 3 ) - (56) in [11)~ H e r e , Xl i s the s u m of the L a n d a u - P e i e r l s t e r m XLp a n d a new t e r m . a s g i v e n by

e 2 .ie2Em~2Ern ~2Em 2

3{ PEme3Em + ~ Erne3 Em){ ~ f (Em)dk ' (1)

w h e r e E m i s the e i g e n v a l u e of the m t h s u b - b a n d , fog m) t he F e r m i d i s t r i b u t i o n func t i on , and the i n t e g r a t i o n w i t h r e s p e c t to the w a v e v e c t o r k e x - t e n d s o v e r the f u n d a m e n t a l B r i l l o u i n zone . X2, X3, and )(4 a r e e x p r e s s e d in t e r m s of not on ly

164

the e n e r g y d e r i v a t i v e s bu t a l s o d e r i v a t i v e s of the p e r i o d i c p a r t of the B l o c h f u n c t i o n w i t h r e - s p e c t to k and c o - o r d i n a t e s .

In the t i g h t - b i n d i n g a p p r o x i m a t i o n of a s i n g l e band . ×1 of (1) i s n u m e r i c a l l y c a l c u l a t e d a s a f u n c t i o n of n, t he n u m b e r of e l e c t r o n s p e r a t o m , f o r s i m p l e cub ic , fcc , and b c c s t r u c t u r e s ° T h e E m in t h i s a p p r o x i m a t i o n fo r sc , fcc , and bcc s t r u c t u r e s a r e g iven , r e s p e c t i v e l y , by

1 2 + 2 + 2 _1 + -~w(cx Cv Cz), -~w(cxcv CvCz +C~Cx), -½ w c.vc vc z , w h e r e C i = cos½ dk i with i = x , y , a n d z, a i s the l a t t i c e c o n s t a n t , a n d W i s the b a n d width . F o r the c o n v e n i e n c e of n u m e r i c a l c a l c u l a t i o n s the c a l c u l a t i o n s of ×1 h a v e b e e n c u r r i e d out a t f i n i t e t e m p e r a t u r e s , and i t i s found t h a t the r e s u l t s a r e a l m o s t c o n s t a n t a b o v e the t e m p e r a t u r e T = 3 × 10 -3 W/k a n d t h e r e a r e s m a l l t e m p e r a t u r e v a r i a t i o n s a t l o w e r t e m p e r a - t u r e s * . T h e n u m e r i c a l r e s u l t s of X L p a n d Xl a t T = 8.2 × 10 -3 w/k f o r sc , fee , a n d b e e s t r u c - t u r e s a r e s h o w n in fig° l a , w h e r e b r o k e n c u r v e s

* Details are shown in ref. [2] and will be published e lsewhere .