ON THE USE OF CURVE INTERSECTION FORMALISMS IN FIELD EVAPORATION THEORY

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ON THE USE OF CURVE INTERSECTION

FORMALISMS IN FIELD EVAPORATION THEORY

K. Chibane, R. Forbes

To cite this version:

K. Chibane, R. Forbes. ON THE USE OF CURVE INTERSECTION FORMALISMS IN FIELD EVAPORATION THEORY. Journal de Physique Colloques, 1984, 45 (C9), pp.C9-99-C9-104.

�10.1051/jphyscol:1984918�. �jpa-00224396�

JOURNAL DE PHYSIQUE

Colloque C9, supplCment au n012, Tome 45, dkcembre 1984 page C9-99

ON THE USE OF CURVE INTERSECTION FORMALISMS I N FIELD EVAPORATION THEORY

K. Chibane and R.G. Forbes*

Department of Mathematics and Physics, f i e U n i v e r s i t y o f Aston i n Birmingham, Gosta Green, Birmingham B4 7ET, U . K .

+Department of Electronic and EZectricaZ Engineering, University of Surrey, Guildford, Surrey GU2 5XH, U.K.

RQsumd

-

Abstract

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1 - INTRODUCTION

For materials that field evaporate via a Gomer-type escape mechanism, an activation energy formula has been derived by Forbes / I / , assuming a parabolic shape for the atomic bonding well. In the lowest approximation, it is given by:

where F is the evaporation field, Fe its value at zero activation energy and C2 a field-dependent parameter with the dimensions of energy, given approximately by:

where K is the vibration force-constant; a the distance of the well bottom from the emitter's electrical surface; c, and cn the atomic and ionic F2 energy-term coefficients respectively; e the elementary (proton) charge; nn the 'purely chemical7 component of the ion-surface interaction (approximated by image-potential, repulsion and F~ energy terms) and $ its partial derivative with respect to distance. The first bracket is a correction due to field-induced effects (polarization and partial ionization in the bonding state) and the second bracket is a correction due to correlation and repulsion interactions of the ion with the surf ace.

For field-evaporation the emission equation is:

where J is the evaporation flux; nhr the amount of material at high risk of evaporation; A the rate-constant pre-exponential and k the Boltzmann constant. A formula for the temperature dependence of evaporation field is obtained by eliminating Q and combining ( 1 ) and (3) to give:

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984918

C9-100 JOURNAL DE PHYSIQUE

where 8 = Silk En (nhrA/.J0) (5)

8 is a parameter with the dimensions of temperature. Hence, equation ( 4 ) suggests a linear relationship between T2 and I/F, as long as the variation of D with evaporation field is negligible.

This result was supported by the Wada et al. /2/ experimental measurements of temperature-dep ndence of evaporation field, in the case of W and Mo. The corresponding 'Sf vs L/F plots 131 were linear but with deviations occurring at very low temperatures (around 50 K for W and 35 K for Mo) attributed to ion-tunnelling effects.

However, experimental measurements of temperature dependence of evaporation field, carried out by Kellogg / 4 / for W, Mo and Rh show T 2 vs 1/F plots that deviate at high temperatures. Among other possibilities, this could be due to the failure of the parabolic approximation at high temperatures. We have therefore investigated the validity of this approximation. The use of a Morse-potential form for the atomic curve has also been investigated.

2

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The position of the crossing-point xP of the atomic and ionic curves is evaluated numerically to give actiyation energy, as a function of evaporation field.

Consequently, theoretical QT vs 1/F plots are produced.

The standard potential energy un(x,F) of an ion at a distance x outside the emitter's electrical surface is given by /5/:

where: % is the sum of the first n free-space ionization energies, $E the emitter's work-function, and Sn(x,F) the 'variable part' of the ion potential energy.

Consider now an atom vibrating around its bonding point, in an atomic bonding state a. Let V(x) describe the shape of the atomic potential curve, measured relative to the bottom of the well. The total potential energy of the atom, at a distance x, is:

where: A0 is the zero-field binding energy.

In the context of a curve-intersection formalism, the position of the crossing point xP is given by setting Un = Ua

.

where: Kn denotes the 'configurational' energy term (H,

-

.

equation can be solved in a variety of approximations, to give a value for xP. The activation energy Q is then v(xP)

.

For the bonding potential V(x), we have used the two forms:

vl(x) =

+

V2(x) = D (1

-

D is the Morse well depth and p a constant associated with the well width. For small vibration amplitudes, p is related to K via:

We have a l s o used d i f f e r e n t a p p r o x i m a t i o n s f o r t h e i o n i c term S n ( x , F ) a s f o l l o w s :

P r i m i t i v e c a s e Sn = -neFx ( 1 1 4

Simple c a s e Sn = -neFx

-

+

Normal c a s e S = -neFx - n2e2/16neox ( 1 1 ~ )

F o r comparisons i n v o l v i n g rhodium, a n F~ e n e r g y term was a l s o i n c l u d e d .

F i n a l l y we summarize i n t a b l e ( 1 ) t h e d i f f e r e n t p a r a m e t e r v a l u e s u s e d i n t h e c a l c u l a t i o n s .

n A" Y1 ^{a K P} G I ~ O - ~ t

( e v ) ( e v ) (nm) (evInm2) (nm-') ( e v nm9)

T a b l e 1

The A', $ and a v a l u e s come from T s o n g ' s / 6 / t a b u l a t i o n s ; K(W) from F o r b e s / 7 / u s i n g t h e same Tsong's t a b u l a t i o n s .

K ( R ~ ) i s a n e s t i m a t e from F o r b e s e t a l . 1 8 1 , w h i l s t K(Mo) i s c a l c u l a t e d by u s i n g t h e

1

s l o p e of t h e Mo Ti v s 1 / F p l o t of K e l l o g 141.

The r e p u l s i o n t e r m G i s c a l c u l a t e d a s i n Biswas and F o r b e s 191.

3

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( 3 a )

-

F i r s t c o n s i d e r t h e p r i m i t i v e p a r a b o l a c a s e obtafned from e q s ( 8 ) , ( 9 a ) and ( l l a ) ( o m i t t i n g t h e F~ term from eq. ( 8 ) ) . A p l o t of Q Z vs 1/F , f o r t h e W d a t a , i s shown a s F i g . ( 1 ) . The i m p o r t a n t f e a t u r e s of t h i s c u r v e a r e : ( 1 ) It i n t e r s e c t s t h e 1 / F a x i s a t 1 /

.

.

( 3 b ) - Comparison of d i f f e r e n t w e l l s h a p e s

F i g . (1) a l s o shows, f o r t h e p r i m i t i v e model c a s e , t h e e f f e c t s of u s i n g t h e Morse p o t e n t i a l , eq ( 9 b ) . Two d i f f e r e n t D v a l u e s , namely 8.66 eV and 1.0 eV, a r e used.

The f i r s t of t h e s e h a s D e q u a l t o t h e t u n g s t e n b i n d i n g e n e r g y A0 , and i s t h e ' c o n v e n t i o n a l ' Morse w e l l . The lower v a l u e of 1 eV i s used t o s i m u l a t e a t u n g s t e n atom is a l o c a l bonding w e l l : s u c h an atom would d i f f u s e a c r o s s t h e s u r f a c e , r a t h e r t h a n e v a p o r a t e , a s t e m p e r a t u r e i s i n c r e a s e d .

JOURNAL DE PHYSIQUE

Figure 1.

I

Theoretical Q' vs 1/F plots chosen to illustrate the effect of varying bonding- well shape. All plots use tungsten parameters and the primitive form, eq. (lla) ,

for Sn(x,F)

.

The curves shown correspond to bonding wells as follows:

(a) parabola; C

Cb) Morse well of depth equal to binding energy;

Cc) "shallow" Morse well.

Figure 2.

Theoretical Q' vs 1/F plots chosen to illustrate the effects of using different approximations for S,(x,F).

All plots use the convent- ional Morse well for V(x).

The curves represent:

(a) The primitive (rt-hand) and normal (left-hand) cases for tungsten;

(b) The simple case for tungsten;

(c) For rhodium, the normal case (rt-hand), and the normal case + repulsive +

terms

Figure 3. I R - I

(+) T-F, Wada et al, 1980 4.

(A) T-F, Kellogg, 1981 3.

(X) T-F, Kellogg, 1984 2.

(m) Q-F, Kellogg, 1984 I.

12

Comparison of experiment

I I.

with theory, for tungsten.

Theoretical plots are the

'"-

normal-case ion potential, ^s.

for parabolic Cupper) and 8.

Morse (lower) bonding 7.

potentials. Experimental 5.

results are: 5.

-> 8

6

I t i s e a s i l y s e e n t h a t a l l t h r e e c u r v e s have t h e same g e n e r a l s h a p e , b u t t h e turn- over l e v e l i s much lower f o r t h e c o n v e n t i o n a l Morse p o t e n t i a l (0.5 eV) t h a n f o r t h e p a r a b o l a (1.8 eV), and i s lower s t i l l f o r t h e ' l o c a l ' Morse p o t e n t i a l (0.35 eV).

( 3 c )

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Fig. ( 2 ) u s e s t h e c o n v e n t i o n a l Morse p o t e n t i a l f o r W , and i l l u s t r a t e s t h e e f f e c t s of u s i n g t h e d i f f e r e n t approximations i n eq. (11) f o r Sn

.

The e f f e c t of i n c l u d i n g a v a r i a b l e r e p u l s i v e t e r m h a s a l s o been i n v e s t i g a t e d . The r e s u l t s a r e e s s e n t i a l l y i d e n t i c a l w i t h t h o s e f o r t h e normal c a s e , s o we s h a l l make comparisons u s i n g t h e normal case.

( 3 d )

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I n g e n e r a l terms we s e e t h a t : ( 1 ) Choosing d i f f e r e n t forms f o r V(x), ( b u t keeping ^K c o n s t a n t ) mainly i n f l u e n c e s t h e turn-over l e v e l . ( 2 ) Choosing d i f f e r e n t forms f o r Sn mainly i n f l u e n c e s t h e s l o p e and i n t e r c e p t i n t h e l i n e a r r e g i o n .

I n c l u d i n g F~ energy t e r m s , a n d / o r a l t e r i n g t h e v a l u e of K , a g a i n a f f e c t s t h e s l o p e and i n t e r c e p t .

We a l s o show i n f i g . ( 2 ) t h e e f f e c t of i n c l u d i n g an F energy term and a r e p u l s i v e term a s v a r i a b l e , i n t h e c a s e of Rh. Compared t o t h e normal c a s e , we s e e t h a t : ( 1 ) t h e r e i s a s h i f t i n Fe ( i n c r e a s e by about 30%); ( 2 ) a t h i g h e v a p o r a t i o n f i e l d s , t h e s h i f t s i n Q-values a r e i m p o r t a n t , b u t d i s a p p e a r c o m p l e t e l y a t h i g h t e m p e r a t u r e s . The most obvious e f f e c t of changing m a t e r i a l s i s t o reduce t h e ' t u r n - o v e r ' l e v e l .

4

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F i g . ( 3 ) shows v a r i o u s r e s u l t s f o r t u n g s t e n . We i n c l u d e : ( 1 ) The normal c a s e , u s i n g t h e v a r i a n t s of V(x). ( 2 ) K e l l o g g ' s (1984) v a l u e s of Q and F / l o / . ( 3 ) The Wada (1980) v a l u e s of T and F. ( 4 ) The Kellogg (1981) v a l u e s of T and F. ( 5 ) The Kellogg (1984) v a l u e s of T, and F. I n c a s e s ( 3 )

-

Note t h a t no a t t e m p t t o ' s t a n d a r d i s e ' t h e s e r e s u l t s h a s been made. D i f f e r e n c e s i n a p p a r e n t s l o p e and i n t e r c e p t have v e r y l i t t l e p h y s i c a l meaning. Our i n t e r e s t l i e s i n t h e ' t u r n - o v e r ' behaviour of t h e v a r i o u s p l o t s . The f o l l o w i n g f e a t u r e s of t h e r e s u l t s d e s e r v e comment: ( 1 ) A l l t h e e x p e r i m e n t a l p l o t s have a l i n e a r r e g i o n , between about 0.35 eV and 0.8 eV. I n t h i s r e g i o n eq. ( 1 ) should be a good approximation. ( 2 ) The Wada (1980) r e s u l t s d i v e r g e a t low t e m p e r a t u r e s , due t o i o n t u n n e l l i n g . One of t h e Kellogg (1984) T-F p o i n t s may a l s o l i e i n t h i s r e g i o n . (3) The Kellogg (1984) Q-F r e s u l t s g i v e r i s e t o a s t r a i g h t l i n e p l o t t h a t seems t o be l i n e a r up t o a l e v e l s l i g h t l y h i g h e r t h a n might be expected on t h e b a s i s of our numerical c a l c u l a t i o n s . ( 4 ) The Wada e t a 1 (1980) r e s u l t s f a l l w i t h i n t h e l i n e a r regime of o u r numerical c a l c u l a t i o n . ( 5 ) The Kellogg (1984) T-F r e s u l t s s u p p o r t t h e Kellogg (1981) T-F r e s u l t s , i n t h a t t h e y ' t u r n - o v e r ' a t about Q = 0.9 eV. Although 'turn-over' i s common t o b o t h t h e s e r e s u l t s and our numerical t h e o r y , t h e e x p e r i m e n t a l and t h e o r e t i c a l r e s u l t s a r e n o t r e a l l y i n agreement w i t h each o t h e r . The most i m p o r t a n t t h i n g about t h e s e r e s u l t s i s t h a t t h e T-F r e s u l t s 'turn-over' b u t t h e Q-F r e s u l t s do n o t . I f d i v e r g e n c e of t h e bonding w e l l from p a r a b o l i c were a prime f a c t o r i n breakdown of t h e t h e o r y , t h e n we should e x p e c t b o t h t h e T-F and Q-F r e s u l t s t o g i v e p l o t s t h a t t u r n over. Hence t h e d i v e r g e n c e between t h e s e r e s u l t s

JOURNAL DE PHYSIQUE

F i g u r e 4.

Comparison of experiment w i t h t h e o r y , f o r rhodium.

T h e o r e t i c a l p l o t s a r e t h e normal-case i o n p o t e n t i a l , f o r p a r a b o l i c (upper) and Morse (lower) bonding p o t e n t i a l s . E x p e r i m e n t a l r e s u l t s a r e :

(+) Q-F, E r n s t , 1979 (A) T-F, K e l l o g g , 1981

must have some o t h e r c a u s e . The most ~ l a u s i b l e h y p o t h e s i s i s t h a t f i e l d dependence i n t h e f l u x p r e - e x p o n e n t i a l , a s d i s c o v e r e d e x p e r i m e n t a l l y by Kellogg / l o / , i s r e s p o n s i b l e f o r t h e d i v e r g e n c e .

5

-

F i g . 4 shows t h e c o r r e s p o n d i n g r e s u l t s f o r Rhodium, i n c l u d i n g : ( 1 ) T h e o r e t i c a l c u r v e s a s c a l c u l a t e d by u s ; ( 2 ) The E r n s t (1979) Q-F r e s u l t s /11/; (3) The K e l l o g g

(1981) T-F r e s u l t s . Broadly t h e same b e h a v i o u r i s d i s p l a y e d a s f o r Tungsten. T h i s s u g g e s t s t h a t we s h o u l d e x p l o r e t h e p o s s i b i l i t y of flux-dependence i n t h e pre-expon- e n t i a l h e r e t o o .

6 - GENERAL COMMENTS

T-F r e s u l t s f o r a number of o t h e r m a t e r i a l s a r e a v a i l a b l e i n t h e l i t e r a t u r e /12/.

However, f u r t h e r a n a l y s i s of t h e s e s h o u l d p e r h a p s proceed w i t h c a u t i o n u n t i l we have d i r e c t measurements of t h e F dependence of Q and n rA f o r more m a t e r i a l s , and more t h e o r e t i c a l u n d e r s t a n d i n g of t h e o r i g i n of F-depenience i n t h e p r e - e x p o n e n t i a l . ACKNOWLEDGEMENTS

One of us (KC) w i s h e s t o thank t h e M i n i s t r y of Higher E d u c a t i o n and S c i e n t i f i c Research of t h e R e p u b l i c of A l g e r i a f o r p e r s o n a l f i n a n c i a l s u p p o r t .

REFERENCES

1. FORBES R.G., S u r f a c e S c i . 116 (1982) L195.

2. WADA M., KONISHI M. and NIKKAWA ^0. , S u r f a c e S c i .

100

3. CHIBANE K. and FORBES R.G., S u r f a c e S c i .

122

4. KELLOGG G.L., J . Appl. Phys.

2

5. FORBES R.G., J. Phys. D: Appl. Phys. 15 (1982) 1301.

6. TSONG T.T., S u r f a c e S c i . - 70 (1978) 311.

7. FORBES R.G., S u r f a c e S c i . 70 (1978) 239.

8. FORBES R.G., CHIBANE K. ERNST ST N . , S u r f a c e S c i .

141

9. BISWAS R.K. and FORBES R . G . , J . Phys. D: Appl. Phys.

15

10. KELLOGG G.L., Phys. Rev. (1984) 4304.

11. ERNST N . , S u r f a c e S c i . 87 (1979) 469.

12. WADA M., UEMORI R. and SSHIKAWA O., S u r f a c e S c i . 13$ (1983) 17.