COMMENTS ON THE THEORY OF THE RESOLUTION IN THE SCANNING TUNNELING MICROSCOPE (STM) AND THE STRUCTURE OF THE TUNNELING BARRIER

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COMMENTS ON THE THEORY OF THE

RESOLUTION IN THE SCANNING TUNNELING MICROSCOPE (STM) AND THE STRUCTURE OF

THE TUNNELING BARRIER

T. Feuchtwang, P. Cutler, E. Kazes

To cite this version:

T. Feuchtwang, P. Cutler, E. Kazes. COMMENTS ON THE THEORY OF THE RESOLU- TION IN THE SCANNING TUNNELING MICROSCOPE (STM) AND THE STRUCTURE OF THE TUNNELING BARRIER. Journal de Physique Colloques, 1984, 45 (C9), pp.C9-111-C9-118.

�10.1051/jphyscol:1984920�. �jpa-00224399�

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Colloque C9, suppl6ment au

_n012,

Tome 45, ^d6cembre 1984 page C9-11 I

COMMENTS ON T H E THEORY OF THE RESOLUTION I N THE SCANNING T U N N E L I N G

MICROSCOPE

(STM) AND THE

STRUCTURE OF

THE TUNNELING BARRIER

T.E. Feuchtwang, P.H. Cutler and E. Kazes

Department

of

Physics, The PennsyZvania S t a t e U n i v e r s i t y , University Park, PennsyZvania 16802, U . S.A.

Resume - Une revue du STM et de son fonctionnement est presentee. LrBvolution regulisre et parfois deconcertante des modsles et de la machine et l'interpre- tation des resultats qui en rdsulte sont revus /1,3/. I1 est demontre qu'une theorie realiste du STM, qui serait consistente avec les theories multidimen- sionnelles habituelles de l'effet tunnel, doit inclure plusieurs points essentiels

:

11 Une barri6re de potentiel realiste,

d

trois dimensions non separables, qui doit tenir compte des interactions-images multiples et de la geomdtrie non plane de la jonction tunnel /4,5/.

2) Une definition operationnelle du plan de surface de reference

?

partir

i

duquel les deplacements de la pointe sont mesures.

3) L'identification des quantites physiques reelles qui S0nt sond6es par le STM

/6/.

4) La definition de la resolution et l'analyse de ses limites pratiques.

Dans ce papier, nous nous focalisons sur les points 4) et 1). Une revue des definitions conventionnelles de la resolution des microscopes /7/ r6vGle leur inapplicabiliteau STM

d

cause de l'absence d'effets d'aberration et de grandis-

sement dans les jonctions tunnel. Les trois definitions recemment proposees de la resolution du STM /1,3/ sont discutees de manisre critique. I1 est suggb- re que contraste et resolution sont interdependants et que les deux sont forte- ment affect& par la structure du potentiel tunnel. Bien plus, le diamGtre effectif du "faisceau tunnel" et/ou de la structure emissive ne represente pas necessairement la limite inferieure de resolution. L'impact de ces conside- rations sur la theorie et la conception du STM est considerde.

Abstract - A review of the STM and its operation is presented. The steady and occasionally baffling evolution of the models of the device and consequent interpretation of the data are reviewed

/1,3/.

It is argued that a realistic theory of the STM, which should be consistent with current multidimensional tunneling theory, must include several essential points

:

1) A realistic and non-separable three-dimensional tunneling potential barrier, which has to account for the complete multiple image interaction and the non-planar geometry of the tunneling junction /4,5/.

2)

An operational definition of the reference or surface plane, i.e., the surface from which the vertical displacement of the tip is measured.

3) ~dentification of the actual physical quantities being probed by the STM

/ 6 / .

4) Definition of resolution and analysis of its practical limits.

In this paper we concentrate on points 4) and 1). A review of the conventio-

nal definition of microscopes /7/ underscores their inapplicability to the STM

because of the absence of lense aberrations and magnification effects in tunne-

ling junctions. The three recently proposed definitions of the resolution in

the STM /1,3/ are critically discussed. Alternatively, it is suggested that

resolution and contrast are interrelated and that both are strongly affected

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984920

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by the structure of the tunneling potential. Furthermore, the effective diame- ter of the "tunneling beam" and/or the emitting structure (e.g., whisker) does not necessarily represent a lower bound for the resolution. The relevance of these considerations to the theory and design of the STM is considered.

I. INTRODUCTION

The purpose of this paper is to discuss the vacuum tunneling barrier in the scanning tunneling microscope and its effect on the resolution of this device. A scanning tunneling microscope (STM) was recently developed by Binnig et al. /la,b/. The microscope is basically a scanning device which monitors the current tunneling across a vacuum potential barrier between a field emission tip and a planer surface. The microscope is operated with a relatively small bias of about lo-*

V

and at

a

cons- tant current of about low9

A.

The current is controlled by a piezo-electric feed- back device which displaces the tip in direction normal to the scanned surface.

Although the first observations of electron tunneling in a metal-vacuum-metal junction were made over a decade ago by Young et al. /8a,8b/ and more recently by Teague /8c/, Binnig et al. were the first to construct a device which is capable of controlled lateral scans at a nominal distance of nanometers above the sample surface. It is not surprising that this development has been received with great enthusiasm. However, it soon became clear that before the potentialities of the STM are reaslized and tunneling microscopy becomes a routine and reliable labora- tory technique several questions of both experimental and theoretical nature have to be resolved. Of particular concern is the strong dependence of the interpreta- tion of the experimental data on the model of the device. We shall illustrate this point in terms of the four models published in the literature /1,3,6/.

An important detail of the theoretical models is the description of the vacuum tunneling barrier. This is particularly obvious in the analysis presented by Feuchtwang et al. /6/ in which this barrier directly affects the "instrument func- tion", which has to be computed in order to extract the surface characteristics from the raw output of the STM. In order to implement such a computation one has in effect to calculate, within a reasonable approximation, the Green function (or Feynman propagator which determines the wave function) /9/ for the vacuum barrier.

To lowest order, this potential can be represented by the barrier between the pla- nar surface of a semi-infinite free electron metal and free electron model of the tip. Our analyses of tunneling across such barriers

/4/

has demonstrated the inade- quacy of one-dimensional modes1 derived from separable potentials in a planer geome- try. Furthermore, we have documented the dominant contribution of the full (classi- cal) multiple image interaction to the effective vacuum barrier in point (micro) contact systems

/4/.

These conclusions were confirmed by the calculation, by Luca et al.

/5/,

of the field distribution in a planar metal-vacuum-metal barrier with a hemispherical protrusion representing an emitting microstructure (e.g., atomic cluster) on the much larger tip (represented by a plane). The potential discussed by Lucas is, like all realistic models of barriers for the STM, non-separable. The practical (as opposed to formal) analysis of multidimensional tunneling in non-

separable potentials is significantly more difficult than the familiar one-dimensio- nal problem. However, recent progress in the development of multidimensional exten- sions of --type (semi-classical) approximations / l o / may provide a practical and intuitively appealing method for analysing the STM and improving its resolution.

The outline of the paper is as follows. In Section

I1

we review the four available

models of the STM and the estimated resolution predicted by them. In Section 111 we

describe the wKB-type analysis of the multidimensional non-separable barrier and

its implication for the resolution of the STM.

11. REVIEW OF THE THEORETICAL MODELS FOR THE STM 11.1. The p l a n a r , q u a s i one-dimensional model

The f i r s t , and s i m p l e s t t h e o r y of t h e STM was p r e s e n t e d by Binnig e t a l . / l a / . They r e p r e s e n t e d t h e p o i n t c o n t a c t j u n c t i o n by an i n f i n i t e l y extended p l a n a r metal-vacuum- metal ( M V M ) j u n c t i o n w i t h a s e p a r a b l e p o t e n t i a l b a r r i e r depending only on t h e z-co- o r d i n a t e (normal t o t h e p l a n a r i n t e r f a c e s ) . T h i s e f f e c t i v e l y one-dimensional model was t r e a t e d i n t h e mean b a r r i e r approximation /11/, n e g l e c t i n g e n t i r e l y t h e image po- t e n t i a l . I n t h e presence of an a p p l i e d b i a s t h i s (one-dimensional) t u n n e l i n g b a r r i e r becomes t r a p e z o i d a l i n shape and was t r e a t e d i n t h e t h i c k - b a r r i e r l i m i t . Thus i n t h i s t h e o r y only t h r e e parameters c h a r a c t e r i z e t h e b a r r i e r : t h e mean work f u n c t i o n of t h e two e l e c t r o d e s , t h e i n t e r p l a n a r spacing, and t h e a p p l i e d b i a s . A t a c o n s t a n t b i a s , any v a r i a t i o n i n t h e observed c u r r e n t with t h e scanned c o o r d i n a t e s i n t h e p l a n e of t h e j u n c t i o n h a s t o be explained i n terms of a p o s i t i o n dependent b a r r i e r width and/or a p o s i t i o n dependence of t h e work f u n c t i o n of t h e sample being probed. The former was l i t e r a l l y i n t e r p r e t e d i n terms of s u r f a c e topography (e.g., s t e p s ) . The l a t t e r can presumably b e determined by a modulation technique i n which t h e l o g a r i t h m i c increment of t h e t u n n e l i n g c u r r e n t with r e s p e c t t o an increment i n t h e v e r t i c a l ( z ) p o s i t i o n of t h e t i p ( i . e . , t h e b a r r i e r width) i s measured. Here we emphasize t h a t t h i s model is t o t a l l y inadequate, because none of i t s premises a r e even remotely s a t i s f i e d : t h e non-planar geometry and m u l t i p l e image i n t e r a c t i o n determine t h e b a r r i e r shape /4/, and t h e mean b a r r i e r approximation i s g e n e r a l l y poor /4b/. Furthermore t h e t h i c k bar- r i e r l i m i t i s probably n o t g e n e r a l l y a p p l i c a b l e over t h e range of b a r r i e r widths of i n t e r e s t . The preceding h a s obvious and s e r i o u s i m p l i c a t i o n s . Furthermore, it a l s o i n v a l i d a t e s t h e model used by Binnig e t a l . t o i n f e r t h e a b s o l u t e v e r t i c a l h e i g h t of t h e t i p above t h e " s u r f a c e plane" from o s c i l l a t i o n i n t h e d i s t a n c e - v o l t a g e t u n n e l i n g c h a r a c t e r i s t i c / l a , l l b / . T h i s d i f f i c u l t y m a n i f e s t s i t s e l f i n t h e i n c o n s i s t e n c y be- tween t h e t h e o r e t i c a l e x p r e s s i o n s f o r t h e low b i a s t u n n e l i n g c u r r e n t quoted i n / l a / and / l e / , n e i t h e r of which a g r e e s w i t h t h e experimental r e s u l t s quoted i n / l e / . A

"very naive" c a l c u l a t i o n suggested t o Binnig e t a l . a l a t t e r a l r e s o l u t i o n 3&

where "R" i s t h e r a d i u s of c u r v a t u r e of t h e t i p . S i n c e t h i s c a l c u l a t i o n i s n o t r e produced, it i s d i f f i c u l t t o judge i t s m e r i t s . However, t h e s t a n d a r d e s t i m a t e of t h e p o i n t r e s o l u t i o n f o r scanning microscopes i s about f i d , where "dm i s t h e e f f e c t i v e beam diameter /7c/. I f we t e n t a t i v e l y i d e n t i f y 'Id" and "2R" t h e n t h i s s u g g e s t s a s e r i o u s flaw i n Binnig e t a l . ' s c a l c u l a t i o n . We s h a l l r e t u r n t o t h i s q u e s t i o n i n our g e n e r a l d i s c u s s i o n of r e s o l u t i o n , i n S e c t i o n 111.

11.2. The p e r i o d i c , c o r r u g a t e d s u r f a c e ( p o t e n t i a l ) model.

G a r c i a e t a l . /3/ have suggested a model i n which t h e s i n g l e p o i n t c o n t a c t j u n c t i o n i s r e p l a c e d by a p e r i o d i c a r r a y of such junctions. The period i s chosen t o be s u f f i - c i e n t l y l a r g e t o permit t h e n e g l e c t of any i n t e r a c t i o n between neighboring t i p s . T h i s j u s t i f i e s t h e a p p l i c a t i o n of p e r i o d i c boundary c o n d i t i o n s i n t h e c o o r d i n a t e s i n t h e p l a n e of t h e j u n c t i o n s with presumably n e g l i g i b l e e f f e c t on t h e c a l c u l a t e d c u r r e n t pe t i p . The model a l s o i n c l u d e s s e v e r a l q u e s t i o n a b l e assumptions concerning t h e d e t a i l e d s t r u c t u r e of t h e t u n n e l i n g b a r r i e r . S p e c i f i c a l l y , t h e image p o t e n t i a l "was taken t o b e f l a t i n t h e r e g i o n between t h e two m e t a l s , " t h u s t h e z-dependence of t h e b a r r i e r was r e p r e s e n t e d by a t r a p e z o i d a l p o t e n t i a l . The dependence of t h e b a r r i e r on t h e co- o r d i n a t e s ( x y ) i n t h e p l a n e of t h e j u n c t i o n was assumed t o s e p a r a t e i n x and y , and was r e p r e s e n t e d by a c o r r u g a t i o n of t h e s u r f a c e . Thus t h i s model i s i n e f f e c t an ex- t e n s i o n of Binnig e t a l . ' s model. Again, t h e b a r r i e r i s parameterized by i t s width, t h e mean work f u n c t i o n and t h e b i a s . Furthermore, t h e amplitude and p e r i o d of t h e s u r f a c e c o r r u g a t i o n e n t e r a s new parameters, i n a d d i t i o n t o an e f f e c t i v e c u r v a t u r e parameter s t a t e d t o account f o r t h e t i p and sample geometry. Garcia e t a l . d e f i n e

K i m 112

t h e r e s o l u t i o n by t h e e f f e c t i v e beam diameter, deff = 2IT]

.

^{Here jm,} i s t h e peak c u r r e n t d e n s i t y a t t h e s u r f a c e and I i s t h e t o t a l c u r r e n t p e r t i p . A s we s h a l l s e e t h i s may b e a p e s s i m i s t i c e s t i m a t e .

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11.3. The t r a n s f e r Hamiltonian models.

T e r s o f f and Hamann /2/ (TH) suggested a more fundamental model f o r t h e STM, based on t h e transfer-Hamiltonian formulation of t u n n e l i n g /6,12/. T h i s formalism h a s an im- p o r t a n t i n t u i t i v e advantage, s i n c e it e x p r e s s e s t h e c u r r e n t i n t h e j u n c t i o n i n terms of c h a r a c t e r i s t i c s of t h e two uncoupled e l e c t r o d e s . , S p e c i f i c a l l y t h e t u n n e l i n g cur- r e n t i s g i v e n a s an i n t e g r a l o v e r energy of a weighted product of an a p p r o p r i a t e l y g e n e r a l i z e d e l e c t r o n i c d e n s i t y of s t a t e s ,

I = 2 e K

"i I M ( E ) ~ 2L-

f ( E )

-

f ( ~ + e V )

I

Pl(E)P2(E+eV) dE

.

^(11.1)

Here f ( ~ ) i s t h e Fermi f u n c t i o n , M ( E ) i s an e f f e c t i v e m a t r i x element, and P i ( € ) i s t h e l o c a l d e n s i t y of s t a t e s of e l e c t r o d e i , averaged over a r e f e r e n c e i n t e r f a c e l o c a t e d i n t h e b a r r i e r . A l l e n e r g i e s a r e r e f e r e d t o t h e Fermi energy of e l e c t r o d e . I n t h e s t a n d a r d f o r m u l a t i o n of e l a s t i c ( i . e . , energy conserving) t u n n e l i n g t h e d e n s i t i e s of s t a t e s do n o t occur e x p l i c i t l y , s i n c e t h e y a r e formally included i n t h e squared m a t r i x element, M ' ( E 1, i . e . ,

where M ' ( E ) i s t h e i n t e g r a l o v e r t h e r e f e r e n c e s u r f a c e of t h e t r a n s i t i o n c u r r e n t ,

from s t a t e nl i n e l e c t r o d e 1 t o s t a t e m2 i n e l e c t r o d e 2.

The p r e c i s e l o c a t i o n and shape of t h e i n t e r f a c e ( o r r e f e r e n c e s u r f a c e ) i n t h e b a r r i e r a r e d i s p o s a b l e parameters t o b e chosen f o r mathematical convenience and i n t u i t i v e , i . e . , p h y s i c a l , a p p e a l . The t r a n s f e r Hamiltonian and s i m i l a r f o r m u l a t i o n s of tunnel- i n g have t h e i n t u i t i v e advantage of c l e a r l y s u g g e s t i n g t h e d e s c r i p t i o n of t h e S!l!M i n terms of an i n v a r i a n t " i n s t r u m e n t f u n c t i o n n c h a r a c t e r i s t i c of t h e t i p and a corre- sponding "sample c h a r a c t e r i s t i c " which i s probed by t h e STM, and h a s t o be e x t r a c t e d from t h e d a t a /6a/. A f u r t h e r advantage of t h e t r a n s f e r Hamiltonian approach i s t h a t i t emphasizes t h e r e l a t i o n between t u n n e l i n g and t h e l o c a l e l e c t r o n i c d e n s i t i e s of s t a t e . Thus it p r o p e r l y i d e n t i f i e s t h e s e f u n c t i o n s a s t h e fundamental sample charac- t e r i s t i c being probed by t h e STM. I n o r d e r t o o b t a i n more r e a d i l y q u a n t i t a t i v e re- s u l t s TH assume t h e s u r f a c e t o be p e r i o d i c i n t h e c o o r d i n a t e s x and y ( i n t h e p l a n e of t h e j u n c t i o n ) . The b a r r i e r i s t a k e n t o b e only a f u n c t i o n of z ( t h e c o o r d i n a t e normal t o t h e p l a n e of t h e j u n c t i o n ) and t o be square i n shape, i . e . , a l l image e f f e c t s a r e neglected. These assumptions a l l o w a n expansion of t h e corresponding wave f u n c t i o n i n a two-dimensional sum over t h e r e c i p r o c a l l a t t i c e , which i s t r u n c a t e d a f t e r t h e f i r s t few terms. The t i p i s r e p r e s e n t e d by a sphere. The t i p wave f u n c t i o n s a r e assumed t o be s p h e r i c a l l y symmetric, and t h e tip-vacuum p o t e n t i a l b a r r i e r i s s t r i c t l y a r a d i a l s q u a r e b a r r i e r , i . e . , once a g a i n a l l image c o n t r i b u t i o n s a r e ne- g l e c t e d , a s w e l l a s a n i s o t r o p i e s due t o symmetry favored t u n n e l i n g of h i g h e r a n g u l a r momentum s t a t e s /6b/. A l l of t h e s e s i m p l i f y i n g assumptions amount t o a c h o i c e of t h e r e f e r e n c e s u r f a c e t o b e t h e c e n t e r of t h e s p h e r i c a l t i p . The instrument f u n c t i o n r e duces t o ,

where

P t

i s t h e e l e c t r o n i c d e n s i t y of t h e t i p a t t h e Fermi energy, R i s t h e r a d i u s of t h e s p h e r i c a l t i p and k = (2m/rn24)ll2 is t h e i n v e r s e decay l e n g t h of t h e wave f u n c t i o n i n t h e vacuum ( i . e . , s q u a r e ) b a r r i e r . The corresponding "sample f u n c t i o n "

i s t h e l o c a l d e n s i t y of s t a t e s of t h e sample evaluated a t t h e f i d u c i a l reference-- t h e c e n t e r of t h e s p h e r i c a l t i p . Thus t h e experimental d a t a c o n s i s t i n g of c o n t o u r s of z a s a f u n c t i o n of x and y a r e t o b e i n t e r p r e t e d a s c o n t o u r s of c o n s t a n t l o c a l e l e c t r o n i c d e n s i t y of s t a t e s i n t h e s u r f a c e b a r r i e r of t h e sample.

T e r s o f f and Hamann s u g g e s t t h a t a measure of t h e r e s o l u t i o n can be obtained by a re- i n t e r p r e t a t i o n of t h e t r u n c a t i o n of t h e r e c i p r o c a l l a t t i c e expansion of t h e sample wave f u n c t i o n . Namely, by assuming t h i s t r u n c a t i o n t o be due t o t h e f i n i t e s p a t i a l band width of t h e instrument which i s t a k e n t o have a G a u s s i a n - t r a n s f e r f u n c t i o n -exp (-lc2/a). Taking G o , t h e l a r g e s t r e c i p r o c a l l a t t i c e v e c t o r k e p t i n t h e expan- s i o n , t o b e of t h e o r d e r of k, g i v e s a s p a t i a l r e s o l u t i o n

where d i s t h e p l a n e - t i p d i s t a n c e ( b a r r i e r width.) I t i s not c l e a r why t h e mathe- m a t i c a l approximation of t r u n c a t i n g an expansion should r e f l e c t t h e r e s o l u t i o n of t h e STM. Nor, whether and why a-1/2 i s a measure of t h e diameter of t h e t u n n e l i n g beam, o r a l t e r n a t i v e l y of an independent a d d i t i o n a l e f f e c t l i m i t i n g t h e r e s o l u t i o n .

Feuchtwang e t a l . /6a/ have a p p l i e d t h e t r a n s f e r Hamiltonian formalism i n a more r i g - o r o u s t h e o r e t i c a l a n a l y s i s of t h e STM. They f i n d t h a t i n g e n e r a l t h e t u n n e l i n g cur- r e n t depends on t h e more g e n e r a l s p e c t r a l d e n s i t y f u n c t i o n s

which reduce t o t h e l o c a l d e n s i t i e s of s t a t e o n l y i f r l = r 2 .

A s might be expected, t h e more r i g o r o u s t h e o r y a l s o i n d i c a t e s t h a t t h e d e t e r m i n a t i o n of t h e s p e c t r a l d e n s i t y of t h e sample from t h e experimental d a t a , while p o s s i b l e i n p r i n c i p l e , r e q u i r e s a r a t h e r involved mathematical procedure. However, it a l s o in- d i c a t e s t h a t i n t h e l i m i t of small beam d i a m e t e r s ( i . e . , high r e s o l u t i o n ) a r e s u l t analogous t o t h a t of TH o b t a i n s , namely, a t low b i a s and temperature,

Hence t h e c o n s t a n t c u r r e n t z ( x , y ) d a t a determine c o n t o u r s of c o n s t a n t , ps, i . e . , con- s t a n t l o c a l d e n s i t y of s t a t e s a t EF. T h i s r e s u l t d i f f e r s from t h a t of TH o n l y i n t h e c h o i c e of t h e f i d u c i a l r e f e r e n c e s u r f a c e .

111. THE TUNNELING BARRIER AND RESOLUTION OF THE STM 111.1. General comments on r e s o l u t i o n

I n d i s c u s s i n g r e s o l u t i o n of microscopes it i s n e c e s s a r y t o d i s t i n g u i s h between two r e s o l u t i o n limits. The f i r s t i s t h e f a m i l i a r p o i n t r e s o l u t i o n which r e f e r s t o t h e d i s t a n c e between t e s t o b j e c t s ( e . g . , p o i n t s ) a t which t h e i r 'images' can no l o n g e r be unambiguously s e p a r a t e d without image processing techniques. The second i s t h e 'in- formation r e s o l u t i o n l i m i t . " T h i s g i v e s t h e u l t i m a t e s p a t i a l band of t h e i n s t r u m e n t and i n d i c a t e s t h e h i g h e s t r e s o l u t i o n of d e t a i l s t h a t could i n p r i n c i p l e be e x t r a c t e d by image p r o c e s s i n g t e c h n i q u e s o r by comparison with computed images /7d/. We s h a l l r e s t r i c t o u r s e l v e s t o t h e d i s c u s s i o n of p o i n t r e s o l u t i o n , while bearing i n mind t h a t t h e use of r e f i n e d deconvolution t e c h n i q u e s could conceivably e n a b l e us t o achieve a b e t t e r r e s o l u t i o n of d e t a i l s i n t h e s p e c t r a l d e n s i t y f u n c t i o n of t h e t r a n s f e r Hamiltonian model, above.

Next we n o t e t h a t t h e STM does not e x h i b i t most of t h e e f f e c t s which normally l i m i t p o i n t r e s o l u t i o n /7/: (1) The geometric m a g n i f i c a t i o n i s e s s e n t i a l l y d i t y .

( 2 ) The e l e c t r o n s a r e e s s e n t i a l l y monochromatic.

( 3 ) There i s no l e n s e system f o r focusing.

( 4 ) No " t r a n s v e r s e ' v e l o c i t y d i s t r i b u t i o n .

( 5 ) There i s no r e s o l u t i o n l o s s due t o sample penetra- t i o n e f f e c t s which plague t h e o r d i n a r y SEM's /7c/.

On t h e o t h e r hand, t h e s t a n d a r d r e s o l u t i o n c r i t e r i o n f o r scanning microscopes /7c/,

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where "dm i s t h e i n c i d e n t beam d i a m e t e r , a p p l i e s t o t h e STM. However, a s we s h a l l s e e , "dn i s determined by quantum f l u c t u a t i o n s r a t h e r t h a n e l e c t r o n gun d e s i g n l i m i t - a t i o n s . I t i s c l e a r t h a t depending on t h e c o n t r a s t of t h e s u r f a c e f e a t u r e s probed, and t h e sharpness-of t h e beam edge t h e a c t u a l r e s o l u t i o n may b e i n c r e a s e d by a s much a s a f a c t o r of 242, l e a d i n g t o

R

= d/2. F i n a l l y , t o conclude t h i s somewhat academic a n a l y s i s we should n o t e t h a t i n p r a c t i c e much p a i n s t a k i n g work goes i n t o t h e experi- mental d e t e r m i n a t i o n and/or v e r i f i c a t i o n of t h e o r e t i c a l l y p r e d i c t e d r e s o l u t i o n /7c,d/.

Thus f o r t h e STM we a l r e a d y a r e aware of t h e p o s s i b i l i t y t h a t , f o r a g i v e n t i p g e m - e t r y , t h e r e s o l u t i o n i s not n e c e s s a r i l y t h e same f o r a d s o r b a t e s , i s o l a t e d s t e p s and

" s h o r t wave l e n g t h c o r r u g a t i o n s " / l c / . T h i s probably r e f l e c t s a c o n t r a s t dependence of t h e r e s o l u t i o n . Furthermore, one has t o r e s o l v e t h e i n h e r e n t d i f f i c u l t y with t h e unambiguous d e t e r m i n a t i o n of t h e s c a l e of t h e f e a t u r e s being probed. S p e c i f i c a l l y one has t o determine t h e constancy of t h e t i p s t r u c t u r e and t h e minimal displacement of t h e t i p . F i n a l l y one has t o be c a r e f u l not t o m i s i n t e r p r e t t h e a c t u a l minimal d i s - placements due t o a p o s s i b l e erroneous i d e n t i f i c a t i o n of t h e observed f e a t u r e . These remarks a r e p a r t i c u l a r l y r e l e v a n t i n o v i e w of t h e published v a l u e of t h e vacuum gap width ( i . e . , p o t e n t i a l b a r r i e r ) of 4A / l a , b / .

111.2. The vacuum t u n n e l i n g b a r r i e r

I t may b e i n f e r r e d from our d i s c u s s i o n i n S e c t i o n I1 t h a t t h e vacuum p o t e n t i a l b a r r i e r p l a y s a major r o l e i n t h e o p e r a t i o n of t h e STM. This s u g g e s t s t h e i n t e r e s t i n t h e s t u d y of t u n n e l i n g a c r o s s r e a l i s t i c , non-separable multidimensional b a r r i e r s . Un- f o r t u n a t e l y t h i s r a t h e r d i f f i c u l t problem i s o n l y p a r t i a l l y understood. However, due t o i t s r e l e v a n c e t o f i e l d t h e o r y /9,10/, multidimensional t u n n e l i n g h a s r e c e n t l y re- c e i v e d much a t t e n t i o n . I n p a r t i c u l a r , p r o g r e s s has been achieved i n t h e so-called s e m i c l a s s i c a l , o r WKB t y p e , approximation. For our purposes a q u a l i t a t i v e summary of t h e more p e r t i n e n t c o n c l u s i o n s of t h i s work w i l l s u f f i c e .

Multidimensional t u n n e l i n g b a r r i e r s a r e bounded by c a u s t i c s u r f a c e s . On t h e s e sur- f a c e s V ( r ) = E, i . e . , t h e y a r e t h e multidimensional g e n e r a l i z a t i o n of " t u r n i n g p o i n t s . " Consider a b a r r i e r bounded by two c a u s t i c s . From each p o i n t on one of t h e s e c a u s t i c s t h e r e emerge one o r more probably t u n n e l i n g p a t h s ending on t h e o t h e r c a u s t i c . These p a t h s minimize t h e i m a g ~ n a r y a c t i o n j[2m/Ch2(v-~)

l1l2

d s and hence can b e determined. Furthermore t h e r e e x i s t s one o r more most probable t u n n e l i n g p a t h s a- c r o s s t h e b a r r i e r , t h e s e g i v e r i s e t o " g l o b a l " a s opposed t o " l o c a l " minima of t h e a c t i o n . The r e l a t i v e p r o b a b i l i t y amplitude f o r t u n n e l i n g along any o t h e r p a t h de- c r e a s e s e x p o n e n t i a l l y . That i s t h e dominant c o n t r i b u t i o n t o t h e t u n n e l i n g proba- b i l i t y amplitude a r i s e s from a summation o v e r "tubes" surrounding t h e most probable t u n n e l i n g paths. The t u n n e l i n g c u r r e n t i s n e g l i g i b l e o u t s i d e t h e s e t u b e s and may b e v i s u a l i z e d a s flowing i n a narrow beam. I n b a r r i e r s which a r e not s p h e r i c a l l y symmet- r i c t h e number of " t u b e s " i s f i n i t e ( p o s s i b l y j u s t one) and a r e well spaced. The width of t h e t u b e s i s determined by quantum f l u c t u a t i o n s which a r e t h e f i r s t correc- t i o n s t o t h e z e r o o r d e r WKB-type t h e o r y , which i n t u r n determines t h e most probable path. We now can understand why G a r c i a e t a l . ' s estimated r e s o l u t i o n which i n e f f e c t assumes t h e t u n n e l i n g c u r r e n t d e n s i t y t o be uniform a c r o s s a "tube" i s probably a lower l i m i t .

It i s expected t h a t t h e most probable t u n n e l i n g p a t h should p a s s through a s a d d l e p o i n t of t h e b a r r i e r and t h e r e i s no simple r e l a t i o n between t h e s i z e of t h e "tube"

o r i g i n a t i n g a t some " e m i t t i n g " s t r u c t u r e ( s u c h a s a whisker, edge of a f a c e t , o r pro- t r u s i o n ) and t h e l i n e a r dimension of t h e , s t r u c t u r e . I n t h e model p o t e n t i a l computed by Lucas e t a l . /5b/, t h e a x i s of symmetry i s t h e most probable t u n n e l i n g path. T h i s p o t e n t i a l s u g g e s t s t h a t t h e t u n n e l i n g beam diameter may b e reduced by proper "design"

of t h e image p o t e n t i a l b a r r i e r . A s noted i n S e c t i o n 111.3. t h i s may s i g n i f i c a n t l y s i m p l i f y t h e r e d u c t i o n of t h e experimental d a t a , b e s i d e s improving t h e p o i n t r e s o l u - t ion.

I V . CONCLUSIONS

Recent p r o g r e s s i n t h e a p p l i c a t i o n of t h e s e m i c l a s s i c a l approximation t o multidimen- s i o n a l t u n n e l i n g t h e o r y may s i g n i f i c a n t l y enhance o u r u n d e r s t a n d i n g of t h e STM. I t w i l l probably p r o v i d e a more p r a c t i c a l a l g o r i t h m f o r t h e r e d u c t i o n of t h e d a t a w i t h i n t h e framework of t h e t r a n s f e r Hamiltonian. The d i s c u s s i o n of t h i s p i n t had un- f o r t u n a t e l y t o b e a n i t t e d . F i n a l l y t h e concept pf t u n n e l i n g t h r o u g h " t u b e s " may b e developed i n t o a p r a c t i c a l t o o l f o r t h e s y s t e m a t i c d e s i g n and e v a l u a t i o n of STM's.

h his

r e s e a r c h was s u p p o r t e d i n p a r t by t h e O f f i c e of Naval Research, A r l i n g t o n , V i r g i n i a , C o n t r a c t No. ~00014-82-K-0702.

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