FIELD ION ENERGY DEFICITS IN THE ATOM PROBE FIM

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FIELD ION ENERGY DEFICITS IN THE ATOM PROBE FIM

R. Forbes

To cite this version:

R. Forbes. FIELD ION ENERGY DEFICITS IN THE ATOM PROBE FIM. Journal de Physique

Colloques, 1986, 47 (C2), pp.C2-31-C2-36. �10.1051/jphyscol:1986205�. �jpa-00225636�

JOURNAL DE PHYSIQUE

Colloque C2, supplément au n°3, Tome 47, mars 1986 page C2-31

FIELD ION ENERGY DEFICITS IN THE ATOM PROBE FIM

R.G. FORBES

University of Surrey, Department of Electronic and Electrical Engineering, GB-Guildford, GU2 5KH (Surrey), Great Britain

Résume - Nous re-examinons la théorie des énergies d'apparition et de déficits d'énergie s'applicant aux mesures de resolution à haute énergie dans la sonde à atomes puisée par laser. Des commentaires sur quelques récentes études sont présentés.

abstract - The theory of appearance energies and energy deficits, as it applies to pulsed laser atom probe measurements in the limit of high energy resolution, is reviewed, and comments are made on some recent work.

In recent papers [1-3] Tsong has reported elegant time-of-flight measurements, and uses these to discuss energy deficits for field-desorbed gaseous ions and field- evaporated metal ions.

Amongst other things, he has suggested that:

1) The critical energy deficit of field-desorbed or field-evaporated ions can, in principle, be determined to ±0.1 eV.

2) His data are accurate enough to confirm the validity of a formula for the

"critical energy deficit" AEC °f a field-evaporated ion that travels down the atom-probe flight-tube in an r-fold charged state. This formula has the form:

r

AEC = A° + L Ii - Q - r<t>E (1)

i=l

where: A0 is the sublimation energy of the atom, Ii is the i-th free-space ionization energy, Q& is the local emitter work function, and Q is the relevant field evaporation activation energy. (Ky notation differs slightly from that of ref.[l], for reasons that will become clear later.)

However, if this A EC is meant to be interpreted as re8c, where e is the elementary charge and 6C is a critical voltage difference that would be necessary to bring the ion to a halt outside a retarding electrode, and if by "critical" energy deficit is meant the "standard" energy deficit Ds t a n d as defined in the Appendix, then this formula is certainly wrong. It may be shown [4) (also see Appendix) that the correct formula is:

r

AEC (= DStand) = AF + E U - Q - r«R (2)

i=l

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986205

J O U R N A L D E PHYSIQUE

where AP is t h e atomic b i n d i n g e n e r g y i n t h e p r e s e n c e o f t h e a p p l i e d f i e l d , and @R i s t h e work-function o f t h e r e l e v a n t r e t a r d i n g s u r f a c e ; b o t h t h e b i n d i n g energy and t h e work-function terms are d i f f e r e n t h e r e .

The formula f o r d e f i n i t i o n o f a n appearance energy g i v e n i n r e f . [ l ] is a l s o i n c o r r e c t . The s t a n d a r d appearance energy

stand

f o r t h e above r - f o l d i o n is g i v e n i n terms o f t h e s t a n d a r d energy d e f i c Y t by:

stand is given b y t h e f i r s t t h r e e terms i n e q . ( 2 ) , and is determined s o l e l y by t h e emission p r o c e s s .

The t h e o r e t i c a l argUntentS g i v e n b y Tsong [l-31 a r e i n c o r r e c t because t h e y f a i l t o t a k e account o f e f f e c t s due t o c o n t a c t p o t e n t i a l s and p a t c h f i e l d s . I t is s l i g h t l y d i f f i c u l t t o understand how t h e work-function e r r o r a r o s e , s i n c e r e f e r e n c e s [S, 6 1 t h a t a r e c i t e d i n [ l ] do g i v e formulae i n terms o f t h e c o l l e c t o r / r e t a r d e r work-function. However, t h i s e r r o r does o c c u r elsewhere i n t h e l i t e r a t u r e ; s o it h a s been thought u s e f u l t o s u m a r i s e i n t h e Appendix a formal d e r i v a t i o n o f e q . ( 2 ) . I n many circumstances t h e numerical d i f f e r e n c e between

&

and @, which is n e a r l y always less than 1 eV, is q u i t e unimportant. However, i t is n o t t r i v i a l i n t h e c o n t e x t o f e n e r g y - d e f i c i t measurement v i a p u l s e d - l a s e r atom-probe (PLAP) t e c h n i q u e s when a p o t e n t i a l r e s o l u t i o n o f ^i0.1 e V i s b e i n g d i s c u s s e d . To see t h i s , l e t u s re-examine how appearance e n e r g y ( A E ) t h e o r y should be a p p l i e d t o the atom-probe s i t u a t i o n .

Consider a "standard" i o n as d e f i n e d i n t h e Appendix t h a t approaches t h e f l i g h t - t u b e e n t r a n c e i n a n r-fold-charged state, and suppose that a v o l t a g e s s t a n d would be n e c e s s a r y it t o b r i n g t o a h a l t j u s t o u t s i d e t h e e n t r a n c e t o t h e f l i g h t t u b e . I f t h e a c t u a l a p p l i e d v o l t a g e between t h e e m i t t e r and t h e f l i g h t - t u b e w a l l s i s ^V, t h e n t h e i o n e n t e r s the f l i g h t t u b e w i t h a k i n e t i c energy stand g i v e n by:

And, s i n c e stand = r e s s t a n d by d e f i n i t i o n , we o b t a i n by u s e o f e q . ( 3 ) :

where @ is a l o c a l work-function a s s o c i a t e d w i t h t h e f l i g h t - t u b e e n t r a n c e .

I n r e a l i t y , it is clear from the work o f t h e B e r l i n group [7,8] t h a t some i o n s are c r e a t e d b y p r o c e s s e s i n which t h e t r a n s f e r r e d e l e c t r o n s e n t e r t h e e m i t t e r at l e v e l s below t h e f e r m i l e v e l , and the e s c a p i n g i o n s may a l s o have some t h e r m a l energy. I n consequence, some i o n s have i n d i v i d u a l v o l t a g e d e f i c i t s s l i g h t l y less t h a n s s t a n d , and hence i n d i v i d u a l AEs s l i g h t l y less t h a n stand. The i o n energy d i s t r i b u t i o n t h u s h a s a n o n s e t , on t h e low energy d e f i c i t s i d e , corresponding t o some AT3 s l i g h t l y lower t h a n stand, and we w r i t e t h i s lower AE i n t h e form:

stand -

AA).

An i o n w i t h t h i s lower A E e n t e r s t h e f l i g h t t u b e w i t h s l i g h t l y h i g h e r k i n e t i c energy (K*) t h a n t h e s t a n d a r d i o n , where K* i s g i v e n by:

K* =

rev

- stand

+

m

+

+@R _{( 6 )}

It is necessary t o understand t h a t t h i s AA i s n o t well-defined u n t i l we have s p e c i f i e d e x a c t l y how o n s e t i s t o be d e f i n e d , - and that i n g e n e r a l t h i s q u e s t i o n cannot b e e p i s t e m l o g i c a l l y w e l l s e p a r a t e d from o t h e r q u e s t i o n s , s u c h as how t h e d i s t r i b u t i o n is t o be measured. Also, s c a t t e r i n g e f f e c t s d u r i n g beam t r a n s p o r t ( f o r example e f f e c t s due t o coulombic r e p u l s i o n ) , i f any such e f f e c t s e x i s t i n a p a r t i c u l a r s i t u a t i o n , would c o n t r i b u t e t o observed v a l u e s o f o n s e t appearance energy, and hence t o

a.

However, it is known from t h e work o f t h e B e r l i n group t h a t t h e r m a l e f f e c t s a l o n e might c o n t r i b u t e AA v a l u e s o f some t e n t h s o f a n ev, even

i n " o r d i n a j " ( i . e . n o t PLAP) f i e l d e v a p o r a t i o n c o n d i t i o n s , s o l i k e l y v a l u e s o f &A are n o t n e g l i g i b l e i n t h e c o n t e x t o f h y p o t h e s i s e d p o t e n t i a l r e s o l u t i o n o f k0.1 eV.

There a r e f u r t h e r s p e c i a l d i f f i c u l t i e s w i t h t h e atom-probe s i t u a t i o n , a r i s i n g from t h e p o s s i b i l i t y o f work-function v a r i a t i o n s a l o n g t h e i n s i d e s u r f a c e o f t h e f l i g h t t u b e . Any such v a r i a t i o n s w i l l g i v e rise t o a p a t c h - f i e l d d i s t r i b u t i o n i n s i d e t h e f l i g h t tube, and t h e k i n e t i c energy a t any p o i n t o f t h e i o n p a t h w i l l be determined by some k i n d o f " e f f e c t i v e work-function" r e l a t i n g t o t h e s i t u a t i o n o f t h e w a l l s i n t h e n e a r surroundings o f t h e i o n . The v a l u e o f t h e " e f f e c t i v e work-function" may v a r y from p o i n t t o p o i n t a l o n g t h e i o n p a t h . I n a time-of-flight atom probe t h e v a l u e o f k i n e t i c energy d e r i v e d from t h e measurements i s n e c e s s a r i l y some S o r t o f average o v e r t h e f l i g h t ; s o , even i f f l i g h t t i m e c o u l d be p r e c i s e l y measured w i t h o u t d i f f i c u l t y , and a corresponding "mean k i n e t i c energy a t onset" determined, it would b e n e c e s s a r y t o know t h e "mean e f f e c t i v e work-function" b e f o r e t h i s k i n e t i c energy could be converted i n t o a corresponding appearance e n e r g y (and t h e n c e i n t o i n f o r m a t i o n a b o u t t h e e m i s s i o n p r o c e s s ) .

I n p r a c t i c e , t h e v a l u e o f t h e "mean e f f e c t i v e work-function" would p r o b a b l y be determined by some form o f c a l i b r a t i o n experiment u s i n g a s p e c i e s o f presumed known o n s e t appearance energy, p o s s i b l y helium. But it would o f c o u r s e be necessary t o e n s u r e t h a t t h e r e was no change i n t h e s t a t e o f t h e w a l l s ( d u e f o r example t o t h e a d s o r p t i o n o r d e s o r p t i o n o f contaminant g a s e s d u r i n g vacuum c y c l i n g ) between t h e c a l i b r a t i o n run and t h e main e x p e r i m e n t a l runs.

I n r e a l i t y , as i n d i c a t e d by Tsong and Kinkus [ l ] , a main f a c t o r c u r r e n t l y l i m i t i n g t h e energy r e s o l u t i o n o f t h e PLAP is t h e a b i l i t y t o measure f l i g h t times p r e c i s e l y , and f u r t h e r t e c h n o l o g i c a l development i s t o hoped for. I f the c u r r e n t p r e c i s i o n o f k i n e t i c energy measurement is n o t b e t t e r t h a n about 0.5 eV, as Tsong ( p r i v a t e communication) s u g g e s t s , t h e n the d i f f i c u l t i e s I have d e s c r i b e d above a r e o f l i m i t e d p r a c t i c a l s i g n i f i c a n c e . However, t h e c e n t r a l aim o f t h i s n o t e is t o p o i n t o u t t h a t , i f t e c h n o l o g i c a l development i n t h e p r e c i s i o n o f t i m e measurement t a k e s p l a c e , t h e r e a r e f a c t o r s t h a t may p r e v e n t a p r e c i s i o n o f i 0 . 1 e V i n d e r i v e d k i n e t i c energy from b e i n g converted t o a p r e c i s i o n o f t 0 . 1 eV i n t h e measurement o f o n s e t appearance energy, and hence i n t o h a r d i n f o r m a t i o n a b o u t t h e emission p r o c e s s . Some f u r t h e r p o i n t s d e s e r v e mention. The f i r s t r e l a t e s t o t h e measurements o f the energy d i s t r i b u t i o n half-width f o r N i + i o n s . F i g . 1 1 i n R e f . [ l ] shows a PLAP measured FWHM ( f u l l width at h a l f maximum) o f approximately 5 eV. By c o n t r a s t , c o n v e n t i o n a l r e t a r d i n g - p o t e n t i a l measurements b y E r n s t e t al. [ g ] show a N i + FWHM o f about 2 eV. T h i s seems t o s u g g e s t e i t h e r t h a t t h e c u r r e n t r e s o l u t i o n o f t h e PLAP is n o t a s good as is b e l i e v e d , o r that t h e emission p r o c e s s i n t h e PLAP d i f f e r s s i g n i f i c a n t l y from t h a t o f c o n v e n t i o n a l f i e l d e v a p o r a t i o n . ( I t may b e r e l e v a n t t h a t the observed half-width o f emission from a liquid-metal f i e l d - i o n s o u r c e o p e r a t i n g i n t h e low-current regime is about 4.5 t o 5 eV (101.) E i t h e r way, it seems that s t u d i e s aimed at comparing PLAP emission FWHMs and "conventional"

f i e l d - e v a p o r a t i o n FtVHMs should be undertaken.

Second, it must now be c l e a r t h a t it is i n c o r r e c t t o suppose that observed changes i n " c r i t i c a l energy d e f i c i t " can be a t t r i b u t e d t o changes i n e m i t t e r work f u n c t i o n . The d i s c u s s i o n i n r e f . [l] concerning t h e appearance energy I " ( H ~ + ) o f t h e H ~ + i o n i s t h e r e f o r e p a r t l y i n e r r o r .

T h i r d , n o t e t h a t t h e t h e o r y s e t o u t above and i n t h e f o l l o w i n g Appendix a p p l i e s s t r i c t l y o n l y t o m e t a l f i e l d evaporation. S l i g h t l y d i f f e r e n t t h e o r y a p p l i e s t o t h e f i e l d - i o n i z a t i o n o f i n e r t g a s e s (which a r e n o t bound t o t h e s u r f a c e p r i o r t o ionization), and t o t h e f i e l d e v a p o r a t i o n o f semi-conductors (where t h e r e may be no a c c e s s i b l e e m i t t e r s t a t e s at t h e f e r m i l e v e l ) ; b u t t h e same need t o g e t t h e work-function and

a

terms c o r r e c t e x i s t s .

Fourth, n o t e t h a t a d i s t i n c t i o n has t o b e made between t h e "standard" q u a n t i t i e s , ( e . g . energy d e f i c i t s ) as d i s c u s s e d i n t h e Appendix, t h a t a r e t h e o r e t i c a l l y w e l l d e f i n e d and f o r which good formulae c a n be given, and t h e "onset" o r "peak"

JOURNAL

DE

PHYSIQUE

q u a n t i t i e s t h a t a r e t h e ones u s u a l l y measured. The q u a n t i t i e s t e n d t o d i f f e r from each o t h e r b y amounts o f t e n o f t h e o r d e r o f t e n t h s o f an eV. The terminology

" c r i t i c a l energy d e f i c i t " is o f t e n a p p l i e d , somewhat i n d i s c r i m i n a t e l y , t o a l l t h e s e t y p e s o f energy d e f i c i t . When p o t e n t i a l energy r e s o l u t i o n o f t0.1 eV is under d i s c u s s i o n it is n e c e s s a r y t o b e c a r e f u l i n making a p p r o p r i a t e d i s t i n c t i o n s .

I n c o n c l u s i o n , t h i s n o t e should n o t b e t r e a t e d as a c r i t i c i s m o f t h e t e c h n i q u e o f measuring appearance e n e r g i e s by P U P : r a t h e r , it is a n a p p e a l f o r c a u t i o n and t h e o r e t i c a l c a r e i n t h e i n t e r p r e t a t i o n o f t h e r e s u l t s o f t h e t e c h n i q u e , s i n c e it promises t o p r o v i d e i n t e r e s t i n g i n f o r m a t i o n about t h e PLAP emission p r o c e s s .

Appendix: The S t a n d a r d Energy D e f i c i t o f f i e l d - e v a p o r a t e d metal i o n s

F o r s i m p l i c i t y , c o n s i d e r a n o t i o n a l i o n i z a t i o n p r o c e s s i n which an r - f o l d i o n is c r e a t e d from a bound atom, i n the h i g h e l e c t r i c f i e l d n e a r a n e m i t t e r s u r f a c e , b y a charge-hopping t y p e mechanism. F u r t h e r , suppose t h a t t h e bound atom i s i n i t i a l l y s t a t i o n a r y , and that t h e act o f i o n i z a t i o n t a k e s p l a c e i n t h e r e l e v a n t c r i t i c a l s u r f a c e , s o t h e removed e l e c t r o n s t r a n s f e r d i r e c t l y t o t h e e m i t t e r f e r m i l e v e l . The i n i t i a l l y s t a t i o n a r y i o n i s a c c e l e r a t e d away by t h e f i e l d . Suppose t h a t it is t h e n r e t a r d e d , and s t o p s i n a r e l a t i v e l y f i e l d - f r e e r e g i o n s l i g h t l y o u t s i d e t h e s u r f a c e o f an e l e c t r o d e ( t h e " r e t a r d e r " ) o f work-function

G.

T h i s is achieved by having t h e r e t a r d e r n e g a t i v e r e l a t i v e t o t h e e m i t t e r , b y a v o l t a g e o f magnitude

stand.

I f e d e n o t e s t h e elementary ( p o s i t i v e ) c h a r g e , t h e p o s i t i v e q u a n t i t y stand d e f i n e d by:

i s termed t h e s t a n d a r d ( e l e c t r o c h e m i c a l ) e n e r g y d e f i c i t f o r t h i s f i e l d - i o n emission p r o c e s s . The d e s c r i p t i o n "standard" h e r e i m p l i e s t h a t t h e i o n was c r e a t e d i n t h e r e l e v a n t c r i t i c a l s u r f a c e , w i t h z e r o k i n e t i c energy, w i t h a l l t h e e l e c t r o n s i n t h i s p r o c e s s and any subsequent p o s t - i o n i z a t i o n p r o c e s s g o i n g d i r e c t l y t o t h e f e r m i l e v e l , and t h a t t h e i o n s u f f e r s no s c a t t e r i n g i n t e r a c t i o n s d u r i n g t r a n s p o r t from t h e e m i t t e r t o the r e t a r d e r . The d e s c r i p t i o n " e l e c t r o c h e m i c a l " and t h e symbol D i n d i c a t e t h a t t h e e n e r g y d e f i c i t is d e f i n e d b y r e f e r e n c e t o v o l t a g e d i f f e r e n c e s , i . e . d i f f e r e n c e s i n e l e c t r o c h e m i c a l p o t e n t i a l f o r t h e e l e c t r o n s i n t h e retarder and t h e e m i t t e r . (Another c a t e g o r y o f e n e r g y d e f i c i t , d e f i n e d b y r e f e r e n c e t o d i f f e r e n c e s i n t h e energy l e v e l E o f i o n s i n s p a c e , and f o r which t h e symbol m is a l s o used, is o c c a s i o n a l l y employed i n the l i t e r a t u r e , b u t t h i s l a t t e r c a t e g o r y o f d e f i c i t is l e s s u s e f u l t h e o r e t i c a l l y . )

I n p r a c t i c e , some i o n s formed by normal f i e l d - i o n emission reach t h e r e t a r d e r f o r v o l t a g e s d i f f e r e n c e s s l i g h t l y lower t h a n 6 s t a n d . T h e i r energy d e f i c i t s a r e d e f i n e d by formulae analogous t o ( 2 ) . The lowest d e f i c i t s y s t e m a t i c a l l y measured i s t e m p t h e o n s e t energy d e f i c i t and i s denoted by

ons set.

The v a l u e d e r i v e d e x p e r i m e n t a l l y f o r

onset

c a n depend on the s i g n a l a n a l y s i s method. T h e o r e t i c a l l y i t is best t o first c o n s i d e r stand, and t h e n c o n s i d e r c o r r e c t i o n s .

Consider t h e f o l l o w i n g electrothennodynamic c y c l e , which starts w i t h t h e n e u t r a l atom s t a t i o n a r y i n t h e c r i t i c a l s u r f a c e :

(1) I o n i z e t h e atom by t a k i n g r e l e c t r o n s t o t h e e m i t t e r fermi l e v e l . ( 2 ) Move r e s u l t i n g i o n t o a p o s i t i o n s l i g h t l y o u t s i d e the r e t a r d e r .

( 3 ) Take r e l e c t r o n s from e m i t t e r t o r e t a r d e r f e r m i l e v e l , v i a t h e " b a t t e r y " . ( 4 ) Take r e l e c t r o n s from r e t a r d e r f e r m i - l e v e l o n t o t h e i o n , hence n e u t r a l i s i n g it.

( 5 ) Return t h e n e u t r a l atom t o its s t a r t i n g p o i n t i n t h e c r i t i c a l s u r f a c e .

Consider t h e works W done b y a h y p o t h e t i c a l e x t e r n a l a g e n t i n e a c h s t e p . w(1) is z e r o , because t h e e l e c t r o n s t r a n s f e r d i r e c t t o t h e f e r m i l e v e l . w(2) is also zero, because t h e i o n is s t a t i o n a r y a t beginning and end. F o r t h e remaining s t e p s w e have :

Because t h e r e t a r d e r is n e g a t i v e r e l a t i v e t o t h e emitter, t h e e x t e r n a l a g e n t does work i n s t e p ( 3 ) , s o W( 3 ) is p o s i t i v e . I n s t e p ( 4 ) . Er d e n o t e s the sum of t h e f i r s t r free-space i o n i z a t i o n e n e r g i e s f o r t h e atom i n q u e s t i o n . I n s t e p ( 5 ), U E e t d e n o t e s t h e ~ L e n t i a l energy o f t h e n e u t r a l atom when s l i g h t l y o u t s i d e t h e r e t a r d e r s u r f a c e , and U0 its p o t e n t i a l energy when i n remote f i e l d - f r e e space; o u r

s p e c i f i c a t i o n of t h e c y c l e c o n d i t i o n s means t h a t

uget

⁼

G.

@ is t h e p o t e n t i a l e n e r g y o f t h e atom a t its s t a r t i n g and ending p o i n t , which i n t h e c y c l e j u s t c o n s i d e r e d is i n t h e c r i t i c a l s u r f a c e .

The above c y c l e is c l o s e d , s o t h e t o t a l work done b y t h e e x t e r n a l a g e n t must be z e r o . Hence we o b t a i n :

The b r a c k e t e d q u a n t i t y is t h e d i f f e r e n c e between t h e v p o t e n t i a l e n e r g i e s o f a e v a p o r a t i n g e n t i t y when i n a n e u t r a l s t a t e i n remote f i e l d - f r e e space and when a t t h e t o p of t h e a c t i v a t i o n energy hump; as i n d i c a t e d i n F i g . 1 , t h i s b r a c k e t c a n be i d e n t i f i e d w i t h AF-Q, where d is t h e t o t a l binding energy o f t h e atom t o t h e s u r f a c e ( i n c l u d i n g a l l field-dependent e n e r g y terms), and Q 1s t h e a c t i v a t i o n e n e r g y needed f o r a bound atom t o escape. We t h u s o b t a i n t h e s t a n d a r d formula:

The above proof is s t r i c t l y v a l i d f o r metal evaporation by t h e charge-hopping mechanism. S l i g h t m o d i f i c a t i o n o f t h e p r o o f is n e c e s s a r y f o r t h e c h a r g e - d r a i n i n g mechanism, i n which t h e e v a p o r a t i n g i o n may e s c a p e o v e r t h e a c t i v a t i o n e n e r g y hump i n a n i l l - d e f i n e d c h a r g e state. There are more g e n e r a l ( b u t less obvious) p r o o f s i n r e f . [ 4 ]

.

These show r e s u l t ( 1 0 ) still t o b e v a l i d i f U P is i n t e r p r e t e d as t h e p a r t i c l e p o t e n t i a l energy a t t h e "pass" a t the t o p o f t h e a c t i v a t i o n - e n e r g y hump, whatever t h e d e t a i l e d mechanism o f escape.

Ref. [43 a l s o shows that e q . ( l O ) remains v a l i d i n form i f f i e l d e v a p o r a t i o n i n i t i a l l y t a k e s p l a c e i n t o an n+ i o n i c state, and t h i s i o n is subsequently p o s t - i o n i z e d i n t o a n r+ s t a t e . I n t h i s case Q must b e i n t e r p r e t e d as t h e a c t i v a t i o n energy a s s o c i a t e d w i t h t h e i n i t i a l e s c a p e p r o c e s s .

R e s u l t ( 1 0 ) i s t h u s g e n e r a l l y v a l i d , provided that AF is i n t e r p r e t e d as t h e t o t a l b i n d i n g energy ( r e l a t i v e t o t h e e n e r g y o f a n e u t r a l atom i n remote f i e l d - f r e e Space) of t h e atom i n its as-bound e l e c t r o n i c s t a t e .

Eqns ( 9 ) and ( 1 0 ) c o n t a i n t h e work-function o f t h e r e t a r d e r . Experimental papers sometimes f a i l t o g i v e i n f o r m a t i o n about t h i s . R e s u l t s are better expressed s o l e l y i n terms o f t h e emission process, by using t h e s t a n d a r d f i e l d - i o n appearance energy as d i s c u s s e d e a r l i e r . I n t h e o r e t i c a l d i s c u s s i o n , however, it i s sometimes convenient t o l a b e l t h e v a r i o u s q u a n t i t i e s involved w i t h subscripts showing t h e c h a r g e states w i t h which t h e y are a s s o c i a t e d . With a t o d e n o t e t h e bound s t a t e , n the s t a t e immediately a f t e r e s c a p e , and r t h e s t a t e on a r r i v a l a t t h e r e t a r d e r , we o b t a i n :

The i d e a of an appearance energy, i n t h e above s e n s e , was o r i g i n a l l y i n t r o d u c e d b y

JOURNAL

DE

PHYSIQUE

G o l d e n f e l d and c o - w o r k e r s [ll]. E q u a t i o n ( 1 0 ) i n i t s g e n e r a l f o r m w a s first given by t h e present a u t h o r [ 4 ] i n 1976.

Some energy r e l a t i o n s h i p s i m p l i c i t i n the above f o r m u l a e are i l l u s t r a t e d i n Fig.1.

Figure 1. A schematic d i a g r a m t o i l l u s t r a t e s o m e of the e n e r g y r e l a t i o n s h i p s i n h e r e n t i n the f o r m u l a e derived i n the A p p e n d i x .

( T h e d i a g r a m is not t o s c a l e )

TSONG, T . T . , KINKUS, T . J . , P h y S . R e v . B 29 ( 1 9 8 4 ) 5 2 9 . TSONG, T . T . , P h y s . R e v . B 30 ( 1 9 0 4 ) 4 9 4 6 .

TSONG, T. T., MC-, S. B., LIOU, Y . , J. de P h y s i q u e 45 ( 1984 ), C o l l o q u c C9, p.71.

FORBES, R.G., Surface Sci. 61 ( 1 9 7 6 ) 2 2 1 .

INGHRAM, M.G., GOMER, R . , J. Chem. P h y s . 22 ( 1 9 5 4 ) 1 2 7 9 .

TSONG, T.T., SCHMIDT, W.A., FRANK, O., Surface Sci. 65 ( 1 9 7 7 ) 109.

DOMKE, M., HUMMEL, E . , BLOCK, J., Surface Sci. 78 ( 1 9 7 8 ) 3 0 7 . HUMMEL, E . , DOMKE, M,, BLOCK, J . , 2. N a t u r L o r s c h . 34a ( 1 9 7 8 ) 46.

ERNST, E . , BOZDECH, G., BLOCK, J., Surface Sci. 80 ( 1 9 7 4 ) 645.

MAIR, G.L.R., MULVEY, T . , FORBES, R.G., J. de P h y s i q u e , 45 ( 1 9 8 4 ) , C o l l o q u e C 9 , p . 1 7 9 .

GOWENFELD,

I.v.,

KOROSTYSHEVSKY,

L.z.,

MISCHANSHUK, B . G . , I n t . J. =ss.

S p e c t r o m . I o n P h y s . 13 ( 1 9 7 4 ) 2 9 7 .