DO TUNNELING ELECTRONS PROBE THE IMAGE INTERACTION ?

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DO TUNNELING ELECTRONS PROBE THE IMAGE INTERACTION ?

H. Nguyen, T. Feuchtwang, P. Cutler

To cite this version:

H. Nguyen, T. Feuchtwang, P. Cutler. DO TUNNELING ELECTRONS PROBE THE IM- AGE INTERACTION ?. Journal de Physique Colloques, 1986, 47 (C2), pp.C2-37-C2-44.

�10.1051/jphyscol:1986206�. �jpa-00225637�

JOURNAL D E PHYSIQUE

Colloque C2, supplbment a u n 0 3 , Tome 47, mars 1986 _page c 2 - 3 7

H.Q. NGUYEN, T.E. FEUCHTWANG and P.H. CUTLER

D e p a r t m e n t of P h y s i c s , T h e P e n n s y l v a n i a S t a t e U n i v e r s i t y , Univer- sity P a r k , P e n n s y l v a n i a 1 6 8 0 2 , U . S . A .

Abstract

-

Some years ago Weinberg and Hartstein suggested that the dis- crepancy between their data on internal photoemission and photoassisted field emission was due to the inability of tunneling electrons to respond fully to the classical image potential. Since then much effort has been expended on explaining this as well as the more general problem of the ef- fect of the full dynamical (i.e., velocity dependent quantum) image inter- action on electron transport. However, neither problem has been defini- tively resolved. A major difficulty in the analyses using the classical image barrier has been the inability of over-simplified and inadequate models to explain observations in a manner consistent with the quantum mechanical limitations imposed on a tunneling electron. Specificdlly, the mean barrier approximations and the "image reduced" mean barrier are often imprecise approximations. Recently the logarithmic characteristics lnI(s;V = const) and lnV(s;I = const) have been determined using the STM.

We have calculated lnI(s;V = const) for planar homo-junctions using two models with one-dimensional barriers,

4

= 4 ( z ) : (1) Simmons' mean barrier and hyperbolic approximations of the multiple image interaction, in the low bias limit, V

<<

$0, the work function of the electrodes. (2) An exact integration of the Schrodinger equation to calculate the transmission across the full multiple image barrier. These calculations are discussed and compared with the experimental curves as xell as some reported calcula- tions by Binning et al.

I

-

INTRODUCTION

The purpose of this paper is to discuss the current status of the classical image interaction as the dominant term in the effective one-particle surface potential barrier. In particular we shall consider the evidence for quantum modifications of this potential at short range. One of the most important problems in surface

*This research was supported in part by the Office of Naval Research, Arlington, Virginia, Contract No. NR 619 007.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986206

C2-38 JOURNAL DE PHYSIQUE

p h y s i c s is t h e d e t e r m i n a t i o n of t h e shape of t h e e f f e c t i v e , s i n g l e p a r t i c l e , s u r - f a c e b a r r i e r a c t i n g on an e l e c t r o n moving a c r o s s t h e solid-vacuum i n t e r f a c e . This problem h a s a l r e a d y been considered i n connection with t h e e a r l y i n v e s t i g a t i o n of e l e c t r o n emission phenomena (thermionic- / l / , photo- /2/ and f i e l d - e m i s s i o n / 3 / ) . However, t h e i n c r e a s i n g s o p h i s t i c a t i o n and s e n s i t i v i t y of e l e c t r o n emission and s c a t t e r i n g experiments a s w e l l a s t h e development of more a c c u r a t e t h e o r i e s have r e v e a l e d d i s c r e p a n c i e s of fundamental s i g n i f i c a n c e between c u r r e n t t h e o r e t i c a l models and experimental o b s e r v a t i o n s . C l a s s i c a l l y , an e l e c t r o n a t r e s t o u t s i d e a p o l a r i z a b l e medium i n t e r a c t s with t h e induced f i e l d . The l e a d i n g term of t h i s i n t e r a c t i o n i s t h e image p o t e n t i a l . For a s e m i - i n f i n i t e s l a b with a d i e l e c t r i c c o n s t a n t E, e x t e n d i n g over z < 0 , t h i s a t t r a c t i v e p o t e n t i a l is,

where, f o r a m e t a l , = m. S i n c e V i C d i v e r g e s a s z + 0, i.e., a t t h e image plane t h i s e x p r e s s i o n cannot b e v a l i d f o r small v a l u e s of z, of t h e o r d e r of i n t e r a t o m i c d i s t a n c e s . One might expect t h e " c o r r e c t " i n t e r a c t i o n t o b e approximated by a 'truncated' p o t e n t i a l of t h e form

where V. i s t h e i n n e r p o t e n t i a l of t h e medium.

I t was f i r s t noted by Bardeen / 4 / t h a t t h e image p o t e n t i a l i s t h e asymptotic l i m i t , a s z + m, of t h e quantum mechanical exchange and c o r r e l a t i o n energy, of an e l e c t r o n , on t h e vacuum s i d e of s u r f a c e /5/. The r e s u l t h a s been c o r r o b o r a t e d by Lang /6/, and o t h e r s , who demonstrated t h a t t h e lowest o r d e r quantum c o r r e c t i o n of Vie, f o r a j e l l i u m model of a m e t a l , i s g i v e n by a displacement of t h e image plane (from t h e j e l l i u m edge) i n t o t h e vacuum by a d i s t a n c e z i

<

-5 f t . The e x a c t c o r r e - l a t i o n energy f o r r e a l i s t i c models, (e.g., t h e inhomogeneous e l e c t r o n g a s ) is g e n e r a l l y not c a l c u l a b l e a s a f u n c t i o n of p o s i t i o n /7/. Thus any g i v e n expres- s i o n f o r a quantum mechanical "image p o t e n t i a l " r e f l e c t s both a p h y s i c a l model and t h e mathematical approximation which have t o b e made i n i t s s o l u t i o n . It i s , i n g e n e r a l , v e r y d i f f i c u l t t o produce r e l i a b l e e s t i m a t e s of t h e e r r o r s i n any such c a l c u l a t e d p o t e n t i a l . The problem is moot u n l e s s experimental probes of t h e e l e c - t r o n - s u r f a c e i n t e r a c t i o n a t i n t e r a t o m i c s e p a r a t i o n s r e v e a l s i g n i f i c a n t discrepan- c i e s with e i t h e r

Vie

o r Lang's quantum m o d i f i c a t i o n of t h i s q u a n t i t y . Over t h e l a s t decade t h r e e groups, u s i n g d i f f e r e n t probes, suggested j u s t t h a t /E-10/. Most r e c e n t l y , t h e development of t h e scanning t u n n e l microscope /11/ (STM) h a s provided a s e n s i t i v e probe of t h e image p o t e n t i a l , o r more p r e c i s e l y t h e quantum " m u l t i p l e image" p o t e n t i a l . + A t t h i s time, t h e STM o b s e r v a t i o n s do n o t seem t o r e q u i r e any quantum m o d i f i c a t i o n s of t h e image p o t e n t i a l . However, t h i s may b e due more t o t h e c u r r e n t s t a t e of t h e t h e o r y of t h e STM t h a n t o t h e c o r r e c t n e s s of t h e c l a s s i c a l m u l t i p l e image i n t e r a c t i o n a t s h o r t range.

The o u t l i n e of t h e paper i s a s follows: I n S e c t i o n I1 we review t h e t h r e e experiments which s u g g e s t quantum m o d i f i c a t i o n s of t h e c l a s s i c a l image p o t e n t i a l . I n S e c t i o n I11 we d i s c u s s t h e STM d a t a /11/ which e r r o n e o u s l y was taken t o confirm t h e f i n d i n g s of H a r t s t e i n and c o l l a b o r a t o r s /g/. I n a d d i t i o n we c o n s i d e r t h e r e c e n t o b s e r v a t i o n of bound s t a t e s i n t h e image p o t e n t i a l , and t h e use of t h e s e s t a t e s a s probes of t h e image i n t e r a c t i o n . I n S e c t i o n IV we p r e s e n t our conclu- s i o n s .

+1n t h e STM c o n f i g u r a t i o n t h e c l o s e n e s s of t h e two m e t a l l i c e l e c t r o d e s r e q u i r e s t h e use of t h e f u l l m u l t i p l e image i n t e r a c t i o n .

I1

-

EXPERIMENTAL DISCREPANCIES WITH THE CLASSICAL IMAGE POTENTIAL 11.1

-

E l e c t r o n i c S u r f a c e Resonances

McRae and c o l l a b o r a t o r s /8/ have observed s h a r p resonances i n t h e e l a s t i c s c a t t e r i n g of e l e c t r o n s i n c i d e n t from t h e vacuum on a m e t a l l i c s u r f a c e . These resonances a r e t y p i c a l l y observed a t low e n e r g i e s j u s t below Eg,(f,,), t h e t h r e s h o l d s f o r t h e d i f f r a c t e d beams indexed with t h e r e c i p r o c a l - n e t v e c t o r 3, ( s e e Fig. l a ) . Here

g,,

i s t h e reduced wave v e c t o r of t h e i n c i d e n t e l e c t r o n beam p a r a l l e l t o t h e s u r f a c e . More p r e c i s e l y , t h e resonances occur a t Eg

- l

^enl

,

where

(1

enl a r e t h e binding e n e r g i e s of s t a t e s bound i n t h e image p o t e n t i a l , i n t h e absence of e x t e r n a l f i e l d s . I n f i t t i n g t h e observed energy dependence of t h e s p e c u l a r l y s c a t t e r e d c u r r e n t , McRae adopted a modified image p o t e n t i a l which is l i n e a r f o r 0 ( z ( 2 z i and j o i n s smoothly with t h e image p o t e n t i a l , - e 2 / 4 ( z - z i ) , f o r l a r g e r v a l u e s of z ( s e e Fig. l b ) . He emphasizes t h a t he c o u l d n o t f i t t h e d a t a with a t r u n c a t e d image p o t e n t i a l given by Eq. ( 2 ) .

Fig. 1. McRae's Modified Image P o t e n t i a l /8/.

ENERGY (ev)

Fig. l a

-

S p e c u l a r l y r e f l e c t e d c u r r e n t

-

0 - from Cu(001) s u r f a c e , p l o t t e d a s a >

f u n c t i o n of i n c i d e n t energy. P o i n t s :

normalized experimental d a t a . Lines: ⁹ -

c a l c u l a t e d c u r v e s u s i n g z i = 3.8 au, V ( o ) = -7 eV, ( s o l i d l i n e ) z i = 3 au, V(o) = -7 eV, (broken l i n e ) .

( I -

11.2

-

P h o t o a s s i s t e d I n t e r n a l F i e l d Emission

I n a s e r i e s of experiments H a r t s t e i n and c o l l a b o r a t o r s / 9 / have compared i n t e r n a l photoemission and p h o t o a s s i s t e d i n t e r n a l f i e l d emis- s i o n i n MOS and M I M s t r u c t u r e s , u s i n g photons of energy c l o s e t o t h e metal-oxide b a r r i e r height.

Analyzing t h e i r f i e l d emission d a t a i n terms of t h e Fowler-Nordheim t h e o r y /12/ t h e y concluded t h a t it c o u l d only b e f i t t e d by t h e expres- s i o n f o r t h e t o t a l c u r r e n t which

Fig. l b

-

The range of v a l u e s of t h e sur- f a c e b a r r i e r parameters z i and V(o) t h a t g i v e a good f i t t o t h e d a t a i n Fig. l a

( s o l i d l i n e ) and t h e v a l u e s used i n c a l - c u l a t i n g t h e c u r v e s i n Fig. l a ( c i r c l e s ) . The s o l i d and open c i r c l e s r e f e r respec- t i v e l y t o t h e f u l l and broken c u r v e s i n Fig. l a . The i n s e r t shows t h e p o t e n t i a l chosen ( s o l i d l i n e ) and t h e t r u n c a t e d image p o t e n t i a l (broken l i n e ) .

n e g l e c t s t h e image p o t e n t i a l . They noted i n p a r t i c u l a r t h a t t h e magnitude of t h e s l o p e of t h e y i e l d / ( f i e l d I 2 v e r s u s ( f i e l d ) - l , a t high f i e l d s , p r e d i c t e d by t h e t h e o r y i n c l u d i n g t h e image p o t e n t i a l , t e n d s t o d e c r e a s e . See F i g 2b. This seemed t o c l e a r l y c o n f l i c t with t h e i r d a t a . T h i s conclusion i s d i s t u r b i n g , because i n a l a t e r p u b l i c a t i o n /9c/ H a r t s t e i n e t a l . p r e s e n t , without comment, d a t a t h a t shows a s l o p e of d e c r e a s i n g magnitude ( s e e Fig. 2 c ) ; t h a t i s , t h e new d a t a i s i n

JOURNAL DE PHYSIQUE

Fig. 2

-

P h o t o a s s i s t e d I n t e r n a l F i e l d Emission /g/.

Fig. 2a

-

^A schematic i l l u s t r a t i o n of t h e photon a s s i s t e d t u n n e l i n g ex- periment.

+

^is t h e aluminum Fermi l e v e l , hv i s t h e photon energy,

h

i s t h e A1-Si02 b a r r i e r h e i g h t , and

@ i s t h e reduced t u n n e l i n g b a r r i e r . V ( X ) r e p r e s e n t s t h e conduction band edge i n t h e SiO2 and F i s t h e a p p l i e d e l e c t r i c f i e l d i n t h e Si02.

Fig. 2 c

-

Photo-assisted-tunneling c h a r a c t e r i s t i c s of t h e A1-SiO2 i n t e r - f a c e . Each c u r v e i s f o r a d i f f e r e n t l a s e r wavelength. The s o l i d c u r v e s a r e t h e t h e o r y and t h e p o i n t s a r e t h e experimental d a t a .

Fig. 2b

-

Comparison of t h e model includ- i n g t h e image f o r c e p o t e n t i a l (dashed c u r v e s ) and t h e t r i a n g u l a r b a r r i e r model ( s o l i d c u r v e s ) f o r t h e two h i g h e s t photon e n e r g i e s . The corresponding d a t a i s a l s o shown. The c l a s s i c a l image f o r c e b a r r i e r lowering f o r s e v e r a l f i e l d s i s i n d i c a t e d a t t h e t o p of t h e f i g u r e .

agreement with t h e s t a n d a r d t h e o r y , in- c l u d i n g t h e image p o t e n t i a l . I n f a c t , i n t h i s most r e c e n t p u b l i c a t i o n t h e y p r e s e n t a r a t h e r i n d i r e c t argument t o j u s t i f y t h e i r o r i g i n a l conclusion. Namely, t h a t t u n n e l i n g e l e c t r o n s a s opposed t o photo- e m i t t e d ones, e x p e r i e n c e a r a t h e r s h a r p l y reduced image p o t e n t i a l a l r e a d y a t ener- g i e s j u s t s l i g h t l y below t h e t o p of t h e b a r r i e r . Quantum e f f e c t s may very w e l l c a u s e t u n n e l i n g e l e c t r o n s t o e x p e r i e n c e a reduced image p o t e n t i a l /13/. However, H a r t s t e i n e t a l . d a t a and e x p l a n a t i o n a r e n e i t h e r c o n s i s t e n t with J o n s o n ' s quantum t h e o r y /13/ nor c o n c l u s i v e by themselves.

1

^11.3

^-

P h o t o a s s i s t e d F i e l d Emission I n a series of papers, Reifenberger, Lee and c o l l a b o r a t o r s /11/ have r e p o r t e d an o s c i l l a t i o n i n t h e e l e c t r i c f i e l d de- pendence of t h e t o t a l p h o t o - a s s i s t e d f i e l d emission c u r r e n t . S p e c i f i c a l l y t h e y found t h a t ,

where F i s t h e e l e c t r i c f i e l d and

F

i s t h e f i e l d dependent frequency o f o s c i l - l a t i o n . They t r i e d t o i n t e r p r e t t h i s r e s u l t i n terms of t r a n s m i s s i o n resonances a c r o s s t h e s u r f a c e b a r r i e r /lob/. They considered f i r s t t h e s t a t i c t r u n c a t e d image p o t e n t i a l Vie, i.e.,

A numerical i n t e g r a t i o n of t h e one-dimensional Schrodinger e q u a t i o n revealed t r a n s - mission resonances which a r e p e r i o d i c i n p - l l 4 . This l e d Reifenberger t o c o n s i d e r a

time

dependent ( i . e . , dynamic) image b a r r i e r . He p o s t u l a t e d t h a t t h e t r a n s m i s s i o n p r o b a b i l i t y has a term which i s p e r i o d i c i n an a p p r o p r i a t e l y d e f i n e d b a r r i e r t r a n s i t time. T h i s v a r i a b l e was found t o b e approximately p r o p o r t i o n a l t o F-lI2. He f u r t h e r assumes t h a t t h e o s c i l l a t i o n i n t h e t r a n s m i s s i o n p r o b a b l i t y i s a t t h e s u r f a c e

plasmon frequency wsp /lOb,c/. T h i s phenomenological model w a s d e v i s e d t o f i t t h e d a t a and t h e r e f o r e y l e l d s s a t i s f a c t o r y agreement with experiment. However, t h i s r a i s e s t h e q u e s t i o n , what i s t h e underlying, fundamental, t h e o r y of t h i s phenom- enon? To answer t h i s q u e s t i o n , we i n t e r p r e t t h e time F o u r i e r transform of a dynamic image p o t e n t i a l a s an energy dependent exchange and c o r r e l a t i o n i n t e r a c t i o n . I n f a c t , t h e r e i s no reason t o assume t h a t t h i s e f f e c t i v e i n t e r a c t i o n should be inde- pendent of t h e energy ( s t a t e ) of t h e e l e c t r o n . Jonson /13/ d e r i v e d j u s t such an energy dependent exchange and c o r r e l a t i o n i n t e r a c t i o n . Jonson i m p l i e s t h a t h i s c a l c u l a t i o n does not j u s t i f y R e i f e n b e r g e r ' s model. The same conclusion i s reached by Young / 1 4 / . However, i t appears t h a t R e i f e n b e r g e r ' s l a t e s t d a t a e x h i b i t a much s m a l l e r amplitude of t h e o s c i l l a t i o n i n /IS/, which may be i n b e t t e r agree- ment with t h e above t h e o r i e s . F i n a l l y , Lee r e c e n t l y i d e n t i f i e d t h e o s c i l l a t i n g term observed by him i n t h e f i e l d e m i t t e d c u r r e n t a s a s p u r i o u s e f f e c t due t o t h e e a r t h ' s magnetic f i e l d /16/. By c o n t r a s t , R e i n f e n b e r g e r ' s a p p a r a t u s was not s e n s i - t i v e t o t h i s f i e l d . We conclude t h a t t h e c o r r e c t form of t h e dynamical image p o t e n t i a l and its r e l e v a n c e t o t h e s e t u n n e l i n g experiments a r e s t i l l open ques- t i o n s .

I11 - THE SCANNING TUNNEL MICROSCOPE (STM)

The STM i s i d e a l l y s u i t e d t o determine t h e l o c a l I-S (and V-S) c h a r a c t e r i s t i c s , i.e., l n I ( s ; V = c o n s t l (and lnV(s:I = c o n s t ) ) . Here s is t h e vacuum gap width, i . e . , t h e d i s t a n c e between t h e image planes. A s S ranges from l a r g e v a l u e s (50 A ) t o very s m a l l ones ( - 2 A ) t h e d e v i c e goes from a f i e l d emission t o a t u n n e l i n g regime. I n both regimes t r a n s m i s s i o n resonances / l l a , 1 7 / may provide a s e n s i t i v e probe of t h e a c t u a l shape of t h e image b a r r i e r . Simmons t u n n e l i n g t h e o r y /18/ p r e d i c t s , i n t h e z e r o b i a s l i m i t , a c u r r e n t d e n s i t y

V 'I2 exp {-1.025A s

CC

G

' e f i eff

where, A s = (sR-sL) i s t h e S-dependent width of t h e t u n n e l i n g b a r r i e r a t t h e Fermi energy of t h e l e f t e l e c t r o d e , i n d i c a t e d i n Fig. 3 . $eff i s an e f f e c t i v e b a r r i e r h e i g h t , r e f e r r e d t o t h e Fermi energy of t h e l e f t e l e c t r o d e ,

Fig. 3

-

Energy l e v e l diagram f o r c l a s s i c a l image p o t e n t i a l model of STM. sR

-

^{sL =} A s i s t h e e f f e c t i v e b a r r i e r width ( a t t h e Fermi energy EFIL. V ( z )

i s t h e t o t a l p o t e n t i a l b a r r i e r , ^{E F L} i n c l u d i n g t h e m u l t i p l e image

i n t e r a c t i o n .

0

which i n t h e aknence of t h e imave p o t e n t i a l , reduces t o t h e average work f u n c t i o n ($L + ⁼ 4. The d a t a of Rinniny e t a l . / l l a / was f i t t e d by

~ 2 - 4 2 JOURNAL DE PHYSIQUE

J V exp C-1.025 A s @ 112 ] e f f

-

where, f o r small b i a s e s , $'eff

- @.

This f i n d i n g was e v i d e n t l y m i s i n t e r p r e t e d a s confirming H a r t s t e i n ' s o b s e r v a t i o n t h a t t u n n e l i n g e l e c t r o n s do not experience t h e image p o t e n t i a l . I n f a c t , i n a l a t e r p u b l i c a t i o n / l l c / Binning e t a 1 suggest t h a t t h i s i s not n e c e s s a r i l y so. T h e i r argument seems t o depend on s e v e r a l approxima- t i o n s such a s t h e Sirrmons mean b a r r i e r approximation, and t h e use of an unusual approximation of t h e c l a s s i c a l m u l t i p l e image i n t e r a c t i o n . Simmons' t h e o r y is based on t h e WKB approximation and i s t h e r e f o r e u n r e l i a b l e f o r such narrow b a r r i e r s ,

( < 5 i?) /19/. Furthermore one should use Sixnmons' i n t e r m e d i a t e b i a s formula, which e x p r e s s e s t h e c u r r e n t a s a sum of..exponentials. To g a i n b e t t e r i n s i g h t i n t o t h i s d i f f i c u l t y we i n t e g r a t e d t h e Schrodinger e q u a t i o n numerically, and c a l c u l a t e t h e e x a c t c u r r e n t d e n s i t y i n a p l a n a r tWK j u n c t i o n , i n c l u d i n g t h e complete m u l t i p l e irnage i n t e r a c t i o n . T h e s e . r e s u l t s a r e presented i n Fig. 4a. We found t h a t , even f o r a moderate b i a s (-0.1 V ) t h e I-s c h a r a c t e r i s t i c was remarkably l i n e a r . F u r t h e r - more, Simmons' t h e o r y approximates t h e e x a c t I-S c h a r a c t e r i s t i c s u p r i s i n g l y well.

Fig. 4

-

C a l c u l a t e d I-S c h a r a c t e r i s t i c s f o r a p a r a l l e l plane model of t h e STM. @ = 4.5 eV, E F , ~ = E F , ~ 7.88 eV, V = 0.1 Volt. S o l i d curve, c a l c u l a t e d from e x a c t s o l u t i o n of ~ c h r o d i n ~ e r e q u a t i o n with f u l l m u l t i p l e image i n t e r a c t i o n . Dashed curve, c a l c u l a t e d from Simmons' approximation (VKB a ~ p r o x i r n a t i o n and h y p e r b o l i c m u l t i p l e image approximation). Dotted curve, e x a c t r e s u l t without image i n t e r a c t i o n . A t s = 3 i?, Simmons' approximation f a i l s . A t 2.2 It t h e t o p of t h e m u l t i p l e image b a r r i e r c o i n c i d e s with E F , ~ . A t 0.8 t h e t o p of t h e m u l t i p l e image b a r r i e r c o i n c i d e s with t h e bottom of t h e conduction band of t h e l e f t e l e c t r o d e .

lnJ(S;V = . l V ) Fig. 4b

- -

_ds

J = current density in ~ / m ~ .

However the I-S characteristic obtained by neglecting the image potential differs significantly from the others. These results are further understood by a plot, in Fig. 4b, of the derivative of the I-S characteristics. At large distances all three curves converge to -1.025 However, the slope of the I-S character- istics, including the image potential, is monotonically decreasing, while the char- acteristic without the image potential is monotonically increasing. The differ- ence becomes dramatic as the top of the image reduced barrier approaches the Fermi energy of the emitter. The Simmons' approximation fails completely because there are no longer any classical turning points in the approximate WKB theory (i.e, at the Fermi energy). That is, the effective barrier width for the approximate muitiple image interaction vanishes at S , 3 A. On the other hand the slope of the exact I-s characteristic rises rapidly to zero as an increasing number of electrons have an energy larger than the potential barrier separating the electrodes. Evidently I is no longer a tunneling current with an exponential dependence on S. Rather it tends to saturate at a high value, and the junction exhibits a conductivity approaching that of a metallic contact.

The near linearity of the measured I-S characteristic is to be expected. The approximately constant slope --l .025 proves that the image potential affects the tunneling electron, rather than the converse.

IV

-

CONCLUSIONS

There is plausible evidence that the full dynamical as well as the quantum mechanical image potential can be probed in a variety of electron emission experi- ments. However, it is essential to carefully verify that the evidence is not spuri- ous due either to uncertainities in the experiment or the misinterpretation and misapplication of classical models such as discussed above in reference to the S'l'M.

A promising device to probe the corrections to the classical image potential is the STM, which could be used to study the I-S characteristic over a wide range of gap

JOURNAL DE PHYSIQUE

widths and, in particular, for small vacuum gap widths. The observation of a drastic change in the slope of the I-S characteristic, such as indicated in Fig.

4b, would be interesting because it might provide a reproducible reference for the measurement of vacuum gap widths. The STM can also be used as a spectroscope to study the S-dependence of the Rydberg spectra of the states bound in the two image potential wells /20,21/. Finally, the study of the S-dependence of transmission resonances in field emission is also of interest in the search for deviations from the classical image barrier.

REFERENCES

/l/ For a comprehensive review see C. Herring and M. H. Nichols, Rev. Mod. Phys.

21, 155 (1949).

-

/2/ (a) W. Schottky, Z. Physik

3,

63 (193.3); (h) L. W. Nordheim, Proc. Roy. Soc.

London

E ,

626 (1925).

/3/ (a) Surface effects in photoemission are reviewed in Photoemission and Electronic Properties of Surfaces, B. Feuerbacher, R. Fitton and R. F. Willis, eds. (Wiley, New York 1978); (b) W. L. Schaich in Photoemission in Solids I, M. Cardona and L. Ley, eds. (Springer-Verlag, Berlin 1978).

/4/ J. Bardeen, Phys. Rev.

49,

653 (1936).

/5/ V. Sahni and K. R. Bohnen, Phys. Rev.

e,

1045 (1985), have recently argued that the image interaction is the limit of only the correlation interaction, and that the exchange does not contribute to the image charge.

/6/ N. D. Lang and W. Kohn (a) Phys. Rev.

z,

4555 (1970); (b) Phys. Rev.

g,

1215 (1971); (c) Phys. Rev.

z,

3541 (1973).

/7/ (a) J. W. Gadzuk, Surf. Sci.

11,

469 (1965); (b) J. C. Inkson, J. Phys.

E ,

2143 (1973).

/8/ (a) E. G. McRae, Rev. of Mod. Phys.

21,

541 (1979); (b) R. E. Dietz, E. G. McRae and R. L. Campbell, Phys. Rev. Lett. %, 1280 (1980).

/9/ (a) 2 . A. Weinberg and A. Hartstein, Solid State Comm.

2,

179 (1976); (b) A.

Hartstein and Z. A. Weinberg, Phys. Rev.

X,

1335 (1979); (c) A. Hartstein, Z. A.

Weinh~rg and S. J. DiMaria, Phys. Rev.

g,

7174 (1982).

/10/ (a) M. J. C. Lee and R. Reifenberger, Surf. Sci.

10,

114 (1978); (b) R.

Reifenberger, D. L. Haavig and C. M. Egert, Surf. Sci.

109,

276 (1981); (c) D. L.

Haavig and R. Reifenhqrgcr, Phys. Rev.

g,

6408 (1982); (d) R. Reifenberger, C. M.

Egert and D. L. ~aavig, J. Vac. Sci. and Tech.

z,

927 (19841.

/11/ (a) G. Binning and H. Rohrer Helvetica Physica Acta

5,

726 (1983); (b) Surf.

Sci.

126,

236 (1983); (c) G. Binning, N. Garcia, H. Rohrer, J. M. Soler and F. Flores Phys. Rev.

E ,

4816 (1984).

/12/ E. L. Murphy and R. H. Good, Phys. Rev.

102,

1464 (1956).

/13/ M. Jonson, Solid State Comm.

33

743 (1980).

/14/ R. A. Young, Solid State Comm.

45,

263 (1983).

/IS/ R. Reifenberger, private communication.

/16/ M. J. C. Lee, to be published in Surf. Sci. (1985).

/17/ D. Straub an? F. J. Himpsel, Phys. Rev. Lett.

52,

1922 (1984).

/18/ J. 0 . Simmons, J. Appl. Phys. (a) 34, 1793 (1963); (b)

34,

2581 (1963);

(C)

35,

2472 (1964). Note that image interaction in (a) and (bl is in error and should be multiplied by 1/2 as corrected in (c).

/19/ H. Nguyen, P. H. Cutler, T. E. Feuchtwang, M. N. Miskovsky and A. A. LuCa.5, to be published in Surf. Sci. (1985).

/20/ J. B. Pendry and P. E. Echenique, J. Phys.

E,

2065 (1979).

These authors consider the states bound in the image potential at the surface of a semi-infinite solid. In the STl4 there is one such set associated with each

electrode. The two sets have a nonnegligible overlap, this complicates the observed spectra and their analysis.

/21/ K. Giesen, F. Hage, F. J. Himpsel, H. J. Riess and W. Steinman, Phys. Rev. Lett.

55, 300 (1985).