AN ANALYSIS OF FIELD EVAPORATION USING INTERATOMIC POTENTIALS

HAL Id: jpa-00225633

https://hal.archives-ouvertes.fr/jpa-00225633

Submitted on 1 Jan 1986

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

AN ANALYSIS OF FIELD EVAPORATION USING INTERATOMIC POTENTIALS

A. Sakai, T. Sakurai

To cite this version:

A. Sakai, T. Sakurai. AN ANALYSIS OF FIELD EVAPORATION USING INTER- ATOMIC POTENTIALS. Journal de Physique Colloques, 1986, 47 (C2), pp.C2-17-C2-21.

�10.1051/jphyscol:1986203�. �jpa-00225633�

AN ANALYSIS OF FIELD EVAPORATION USING INTERATOMIC POTENTIALS

A. SAKAI AND T. SAKURAI

The Institute for Solid State Physics, The University of Tokyo, Minato-ku, Tokyo, Japan

A b s t r a c t

-

The f i e l d s e n s i t i v i t y and t h e t e m p e r a t u r e dependence o f t h e f i e l d e v a p o r a t i o n o f t u n g s t e n i o n s were analyzed w i t h i n t h e framework o f t h e charge exchange model. S u p e r p o s i t i o n o f p a i r p o t e n t i a l s was used t o c a l c u l a t e numer- i c a l l y t h e a t o m i c and i o n i c p o t e n t i a l s and t h e p o s i t i o n o f t h e i r c r o s s i n g p o i n t . C a l c u l a t i o n s were performed f o r t h e k i n k s i t e atom o f W(110) s u r f a c e . I t was found t h a t t h e use o f s i m p l e model p o t e n t i a l s g i v e s good agreement w i t h experiment.

A l t h o u g h t h e t h e o r y o f f i e l d e v a p o r a t i o n has been s t u d i e d b y many a u t h o r s , a sys- t e m a t i c comparison o f t h e t h e o r e t i c a l models w i t h e x p e r i m e n t has on1 y been performed i n r e c e n t years. The b a s i c framework o f t h e t h e o r y was s e t b y two models proposed i n t h e 1 9 5 0 ' s and 19601s, i . e . t h e image f o r c e model /l/ and t h e charge exchange model 121. Recent experiments on t h e f i e l d e v a p o r a t i o n o f m e t a l s have shown t h a t t h e image f o r c e model i s i n a p p r o p r i a t e f o r e x p l a i n i n g t h e observed d a t a /3,4,5,6/.

On t h e o t h e r hand, no such comparison w i t h experiment has been c a r r i e d o u t f o r t h e charge exchange model. A d i f f i c u l t y w i t h t h i s model i s t h a t a knowledge o f t h e form otf t h e a t o m i c and i o n i c b i n d i n g p o t e n t i a l s i s r e q u i r e d t o o b t a i n t h e c r o s s i n g p o i n t

X, o f t h e s e two p o t e n t i a l s , which d e t e r m i n e s t h e a c t i v a t i o n energy o f f i e l d evapora- t i o n . We propose a s i m p l e approach which employs p a i r p o t e n t i a l s t o e v a l u a t e t h e b i n d i n g p o t e n t i a l s . T h i s makes i t p o s s i b l e t o c a l c u l a t e t h e e v a p o r a t i o n r a t e and t h e e v a p o r a t i o n f i e l d w i t h o u t any a p p r o x i m a t i o n s on xc, and t o compare t h e c a l c u l a t e d r e s u l t s w i t h experiment.

I 1

-

METHOD OF CALCULATION

I n t h e charge exchange model, t h e f i e l d - e v a p o r a t i n g atom makes a t r a n s i t i o n from t h e a t o m i c p o t e n t i a l U ( x , ~ ) t o t h e i o n i c p o t e n t i a l U. ( x , ~ ) th r o u g h t h e c r o s s i n g p o i n t

( F i g . ) . F o r atoms i n i t i a l l y s i t t i n g a t t h e mjnimum o f Ua ( a t x=xo), t h e a c t i v a - i f o n b a r r i e r becomes

A dependence o f Q on Ui comes i m p l i c i t l y t h r o u g h xcwhich i s determined from t h e e q u a t i o n

Ua(xc,F) = Ui (xc,F).

We used f o l l o w i n g e x p r e s s i o n s f o r Ui and Ua

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986203

JOURNAL DE PHYSIQUE

where In, $, a

,

and X denote t h e n - t h i o n i z a t i o n energy, t h e work f u n c t i o n , t h e p o l a r i z a b i l i t y ! and t h e p e n e t r a t i o n depth, r e s p e c t i v e l y , and ne i s t h e charge o f t h e e v a p o r a t i n g i o n . For t h e f i e l d - f r e e p o t e n t i a l U. and t h e s u r f a c e - i o n r e p u l s i v e p o t e n t i a l U , we employed a s u p e r p o s i t i o n o f Morse p o t e n t i a l s and Born-Mayer poten- t i a l S, r e s p 6 $ ? i v e l y,

where ( X . , y., z . ) a r e t h e c o o r d i n a t e s o f t h e s u r f a c e atoms, and t h e summation ex- t e n d s uplto t h e tecond neighbour atoms. The e s s e n t i a l p o i n t o f t h i s approach i s t h e use o f s i m p l e Morse and Born-Mayer p o t e n t i a l s i n t h e above e q u a t i o n s . T h i s g i v e s a n a l y t i c forms f o r U, and Ui and enables us t o n u m e r i c a l l y o b t a i n x c b y s o l v i n g Eq.

I ? \

Once X i s determined, t h e a c t i v a t i o n energy can be o b t a i n e d from Eq. ( l ) . Then t h e f i e l d $ e n s i t i v i t y o f t h e ( r e l a t i v e ) e v a p o r a t i o n r a t e k / k and t h e t e m p e r a t u r e de- pendence o f t h e e v a p o r a t i o n f i e l d F(T) a r e g i v e n b y t h e

Pal

l o w i n g e q u a t i o n s

where R i s a gas c o n s t a n t and F. i s a r e f e r e n c e e v a p o r a t i o n f i e l d a t t e m p e r a t u r e To and e v a p o r a t i o n r a t e kg.

I 1 1

-

^{RESULTS AND} DISCUSSION

C a l c u l a t i o n s were c a r r i e d o u t f o r a k i n k s i t e atom on t h e (100) l e d g e o f W(110).

T h i s atom has f o u r n e a r e s t neighbours and t h r e e o f them f o r m a small (111) p l a n e as shown i n F i g . 2. From t h e symmetry o f t h e a t o m i c arrangement, we assumed t h a t t h e k i n k s i t e atom e v a p o r a t e s a l o n g t h e normal o f t h i s plane. We t h e r e f o r e chose t h e x - a x i s a l o n g t h e normal d i r e c t i o n and p l a c e d t h e o r i g i n on t h e ( 1 1 1 ) plane. The i n i t i a l p o s i t i o n o f t h e k i n k s i t e atom t h e n becomes x0=a0/3, where a. i s a n e a r e s t neighbour d i s t a n c e .

Values o f t h e parameters, e x c e p t h , i n Eqs. ( 3 ) - ( 6 ) were t a k e n o r c a l c u l a t e d from t h e l i t e r a t u r e /7,8,9/. The f o l l o w i n g i s a summary o f t h e parameters used i n t h e c a l c u l a t i o n :

We assumed a d o u b l y charged i o n and p u t n=2 i n Eq.(3). For t h e r e f e r e n c e evapora- t i o n , we used an e x p e r i m e n t a l v a l u e /10/,

o b t a i n e d w i t h X=1.14 A. A l t h o u g h we used o n l y one a d j u s t a b l e parameter, t h e c a l c u - l a t e d e v a p o r a t i o n r a t e shows e x c e l l e n t agreement w i t h t h e d a t a . I n t h e p r e v i o u s work/3,11,12/ two o r t h r e e parameters were needed t o o b t a i n s i m i l a r agreement. Thus t h e good f i t i n Fig.3 i n d i c a t e s t h a t o u r c a l c u l a t i o n scheme works w e l l f o r f i e l d e v a p o r a t i o n .

Another i m p o r t a n t q u a n t i t y i s t h e t e m p e r a t u r e dependence o f t h e e v a p o r a t i o n f i e l d g i v e n by Eq. ( 8 ) . I n t h i s case t h e c a l c u l a t i o n c o n t a i n s no a d j u s t a b l e parameters so t h a t t h e comparison w i t h experiment p r o v i d e s a good t e s t f o r t h e charge exchange model. I n Fig.4, we p l o t t h e c a l c u l a t e d t e m p e r a t u r e dependence o f t h e e v a p o r a t i o n f i e l d w i t h r e c e n t measurements /5,13/. A d i s c r e p a n c y among t h e d a t a p o i n t s a t h i g h temperatures i s due t o t h e d i f f e r e n c e i n t h e e v a p o r a t i o n r a t e /5/, t h e e f f e c t o f w h i c h i n c r e a s e s w i t h t e m p e r a t u r e . A l t h o u g h such an e f f e c t obscures a q u a n t i t a t i v e comparison, t h e agreement between t h e c a l c u l a t e d r e s u l t and t h e experiments i s q u i t e s a t i s f a c t o r y . T h i s r e s u l t and t h e good f i t i n Fig.3 demonstrate t h a t t h e charge exchange model consistentZy e x p l a i n s both t h e f i e l d s e n s i t i v i t y o f t h e e v a p o r a t i o n r a t e and t h e t e m p e r a t u r e dependence o f t h e e v a p o r a t i o n f i e l d , w h i c h were s e p a r a t e l y analyzed i n t h e p r e v i o u s work.

There i n d e e d s t i l l r e m a i n some disagreements as seen i n Fig.5 where we p l o t 1 / F as a f u n c t i o n o f R . A c c o r d i n g t o t h e t h e o r y o f Chibane and Forbes 1141, t h i s p l o t would y i e l d a s t r a i g h t l i n e i f t h e a t o m i c p o t e n t i a l i s harmonic. T h i s i s n o t t h e case f o r t h e observed d a t a which show l a r g e c u r v a t u r e . Our c a l c u l a t e d e v a p o r a t i o n f i e l d has a s l i g h t upward c u r v a t u r e b u t t o o s m a l l t o account f o r t h e experiment. E v i d e n t l y t h e k i n k s i t e atom i s w e a k l y bound t o t h e s u r f a c e and i t s b i n d i n g p o t e n t i a l has l a r g e r u n h a r m o n i c i t y t h a n t h e s i m p l e Morse f u n c t i o n 141.

F i n a l l y we make one comment on t h e v a l u e o f X w h i c h appears t o be l a r g e compared t o t h e s c r e e n i n g l e n g t h o f m e t a l s . We n o t e t h a t o u r A i s measured f r o m t h e x=O p l a n e w h i c h does n o t c o i n c i d e w i t h t h e image p l a n e from w h i c h t h e c o r r e c t X s h o u l d be mea- sured. i f we assume t h e c o r r e c t X t o be 0.5 A, t h e image p l a n e s h o u l d l o c a t e a t X=-0.64 A. T h i s s i t u a t i o n i s n o t p h y s i c a l l y unreasonable because t h e s t r o n g p o s i t i v e f i e l d t e n d s t o push t h e image p l a n e i n s i d e t h e c r y s t a l .

I V

-

SUMMARY

We have shown t h a t t h e charge exchange model combined w i t h s i m p l e model p o t e n t i a l s g i v e s a good d e s c r i p t i o n o f t h e e x p e r i m e n t a l d a t a on t h e f i e l d e v a p o r a t i o n o f W(110).

We b e l i e v e t h a t a d e t a i l e d q u a n t i t a t i v e a n a l y s i s o f t h e f i e l d e v a p o r a t i o n w i l l be p o s s i b l e b y t h e use o f more r e a l i s t i c i n t e r a t o m i c p o t e n t i a l s .

REFERENCES

/l/ M u e l l e r , E. W., Phys. Rev. 102 (1956) 618.

1 2 1 Gomer, R. and Swanson, L. M, J. Chem. Phys.

2

(1963) 1913.

1 3 1 Tsong, T. T., J. Chem. Phys. 54 (1971) 4205.

1 4 1 Ernst, N., S u r f . S c i . 87 (197- 469.

1 5 1 Kellogg, G. L., J. A p p r Phys. 52 (1981) 5320.

1 6 1 Kellogg, G. L., Phys. Rev. B29 v 9 8 4 ) 4304.

/7/ G i r i f a l c o , L. A. and W e i z e r 3 . G., Phys. Rev.

114

(1959) 687.

/8/ Abrahamson, A. A., Phys. Kev. 178 (1969) 76.

/9/ Konishi, K., Wada, M., and N i s m a w a , O., S u r f . S c i . 107 (1981) 63.

/ l 0 1 S a k u r a i , T. and M u e l l e r , E. W., J. Appl. Phys.

48

( 1 9 m 2618.

/l11 Tsong, T. T., J. Phys. F8 (1978) 1349.

1121 Vesely, m. and E h r l ich,%.

,

S u r f . S c i . 34 (1973) 547.

1 1 3 1 Wada, M., K o n i s h i , M., and Nishikawa,

O z

S u r f . S c i . 100 (1980) 439.

1141 Chibane, K. and Forbes, R. G., S u r f . S c i .

122

(1982)

m.

JOURNAL DE PHYSIQUE

( ENERGY

I

^{X 0 xc}

^\

^{Fig. 2}

^-

^A kink s i t e atom(hatched) and

i t s four n e a r e s t neighbours. Three Fig. 1

-

Schematic p o t e n t i a l energy neighbours form t h e (111) plane from diagram f o r t h e charge exchange model. which t h e d i s t a n c e X i s measured.

Fig. 3

-

Comparison of t h e c a l c u l a t e d and experimental evaporation r a t e of W(110) a t 78 K.

Fig. 4

-

C a l c u l a t e d t e m p e r a t u r e dependence o f t h e e v a p o r a t i o n f i e l d of W(110) Experimental r e s u l t s a r e a1 s o p l o t t e d f o r comparison.

5.0

o a

\ >

IA

4,0

3.0

o Kellogg ( k

-

I 1oyer.s-' ) Kellogg ( k

-

^{10' i0ns.s-' )}

A Wada et al. ( k

-

0.1 1ayer.s-'1 -

W ( 1 1 0 )

0

- ⁰ Kellogg ( k I 1ayer.s-' ) o Kellogg ( k

--

10' i0ns.s-' )

0 A Wada et al. ( k 0.1 1ayer.s-' )

-

calculated

0

-

calculated

0

0 100 2 0 0 300 400 500

T ( K )

F i g . 5

-

l / F ( T ) v e r s u s fl p l o t o f t h e d a t a o f F i g . 5 .