STM Theory

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    Appl. Phys. A 51,281-288 (1990) Applied ,o,,,,,Physics A "n"urfaces Springer-Verlag 1990

    Theory of Scanning Tunneling MicroscopyG. D oye n a, E. Koetter 1, J.P. Vigneron 2, and M. Scheffler 11 Fritz-Hab er-Institut der M ax-Planck-Gesellschaft, Faraday weg 4-6 , D-t000 Berlin 332 Facultts Universitaires Notre-Dame de la Paix, Nam ur, BelgiumReceived 27 M arch 1990/Accepted 18 Jun e 1990

    Abstract. The present theoretical understanding of imaging clean and adsorb ate covered metalsurfaces in scanning tunneling microscopy is examined with special emphasis on a possibletheoretical foundation for the observed unexpectedly large corrugation on close-packed metalsurfaces. Several suggestions for explaining these experimental findings are investigated.Reson ance tunneling via t ip d-orbitals might b e a possible mechan ism of amplifying small la teralstructure o f electronic or elast ic or igin. Two com plem entary theo retical method s are applied. Thefirst one concentrates on a realist ic descr iption o f the potential and wave functions of the samplesurface whereas the seco nd one at te mpts to m odel a m ore realistic transition metal t ip. In the f irstapproach the t ip is represented by a Gaussian protrusion on an otherwise planar f ree-electronmetal surface. The sample surface is buil t f rom muff in- t in potentials accounting for the atomicstructure and the d-electrons. The spatial current distr ibution near the t ip region is obtained bysumming the cont r ibut ions of al l sca t tered waves. The metho d has been appl ied to s tudy thecurrent to AI(II1) and Pd(100) surfaces. The corrugation obtained is rather small and cannotexplain the experimental observations. The second approach studies two transit ion metal t ipsconsist ing of a single tungsten atom adso rbed on a f lat W(110) surface and o n a gro up of four otherW atoms. The cluster of four W atom s is coupled to a f lat W(110) surface by using an em bedd ingmetho d. The basis set on the W atom s includes 6s-, 6p-, and 5d-orbitals. The electronic structure ofthe tip exhibits a 5dz2-resonance near the Fe rmi level. The effects of tip d-orbitals and reson ancetunneling on the lateral contrast in STM are analyzed.

    PACS: 68.35, 73.20, 82.65

    Earlier theoretical studies of scanning tunn eling m icros-copy (STM) [1 ,21 were most ly based on Bardeen ' stransfer Hamiltonian approach [3] , which star ts f romperturbation theory and implies cer tain ambiguous as-sumpt ions abou t the behaviour of the decaying f ringingfields of the t ip and the sample surface. Recently var iousthree-dimensiona l approaches to the STM theory havebeen formulate d [4-10 ] , which do not rely on the transferHamiltonian approach, but st i l l apply one or anotherapproximation. Aspects which today appear relevant foran unde rstanding of the experimental technique o f scan-ning tunneling microscopy and therefore should be con-tained in any rel iable theory include:1. A prop er descr iption of the scatter ing behav iour ofthe tunneling electrons in the sample.2. A comp lete and co rrect set of basis wave functionsfor the outmost t ip a tom.

    3. A rel iable chemisorption theory to handle thebon ding of the t ip atom to the base o f the t ip mater ial .4. Inclusion of the electronic and elastic interactionbetween t ip and sample surface.5. An appro priate descr iption of the potential in thesample- t ip region, experienced by the tunn eling electrons.A theory enco mpassin g al l these aspects does not existat present . Different models of STM have to be con-strutted to deal separately with the var ious points.Two complemen tary approaches a re followed in th isarticle. The first one concen trates on a realistic descrip tionof the sam ple surface and yields the correct mult iplescattering be havio ur o f the electrons in the sample,whereas the second approach a t tempts to model a morerealist ic transit ion metal t ip in order to discuss theinf luence of the t ip w ave functions and the t ip electronicstructure on the tunnel current (points 2 and 3 above) .

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    282I t e m 4 w i ll b e i n v e s t ig a t e d b y m e a n s o f si m p l e m o d e la p p r o a c h e s . T h e e m p h a s i s t h r o u g h o u t t h i s a r ti c le l ie s o np r o v i d i n g a c l e a r p i c t u r e o f t h e p h y s i c s i n v o lv e d .A s t r o n g m o t i v a t i o n f o r c o n s t r u c t i n g a d e t a i le d t h e o r yo f S T M o r i g i n a t e s fr o m t h e f a c t t h a t s c a n n i n g t u n n e l i n gm i c r o s c o p y , h a s r e p e a t e d l y r e s o l v e d i n d i v i d u a l a t o m s o n an u m b e r o f m e t a l s u r fa c es [ 1 1 - 1 4 ] . A S T M i n v e s ti g a ti o n o ft h e A I ( l l l ) s u r f a c e b y W i n t t e r li n e t a l. [ 1 4 , 1 5 ] s h o w e ds u r p r i si n g l y a c o r r u g a t i o n o f u p t o 0 .8 A ( 40 n A , 5 0 m V ) .T h e c o r r u g a t i o n d e c r e a s e s e x p o n e n t i a l l y w i t h d i s t a n c ea n d i s o b s e r v a b l e o v e r a r e g i o n o f d i st a n c e s o f 2 A . T h i sm e a n s t h a t i t i s o b s e r v a b l e e v e n a t d i st a n c e s e q u a l o rl a r g e r t h a n t h o s e c o r r e s p o n d i n g t o t h e t u r n i n g p o i n t f o rH e s c a t t e r i n g . H e l i u m s c a t t e r i n g s h o w s a v e r y s m a l lc o r r u g a t i o n ( < 0 .0 2 A ) [ 1 6 ]. T h e S T M c o r r u g a t i o n a m p l i -t u d e d e p e n d s n o t o n l y o n t h e t i p - sa m p l e s e p a r a ti o n , b u ta l s o o n t h e a t o m i c s t r u c t u r e a n d c o m p o s i t i o n o f th e t ip ,w h i c h f r e q u e n t l y c h a n g e s i n a n u n c o n t r o l l e d m a n n e r .Be ing a s imple meta l , t he c lose -pack ed Al(111) sur faceh a s a p r a c t i c a l l y f l a t c h a r g e d e n s i t y w i t h o u t a n y s u r f a c es t a t e n e a r t h e F e r m i e n e r g y . T h i s h a s b e e n c o n f i r m e d b yvar ious f i r s t -pr inc ip les ca lcu la t ions [17 , 18] . The ca l -c u l a t e d c o r r u g a t i o n o f t h e c h a r g e d e n s i t y le v e ls o u tc o m p l e t e l y w i th i n o n e o r t w o a n g s t r o m s o f t h e f ir s t l a y e rof nuc le i.A c c o r d i n g t o t h e T e r s o f f - H a m a n n a p p r o a c h [ -1 ] t h et u n n e l c u r r e n t s h o u l d b e p r o p o r t i o n a l t o t h e c h a r g ed e n s i t y a t t h e c e n t e r o f t h e t i p a t o m . O b v i o u s l y t h eo b s e r v e d c o r r u g a t i o n o n A 1 (1 11 ) c a n n o t b e u n d e r s t o o dw i t h i n t h i s f r a m e w o r k a n d a m o r e g e n e r a l t h e o r y i srequi red , the ba s i s of wh ich i s ou t l ine in Sect . 1.V a r i o u s s u g g e s t i o n s h a v e b e e n f o r w a r d e d t o e x p l a i nt h e a n o m a l o u s c o r r u g a t i o n : T i p in d u c e d s t a te s i n t h e re g i o n b e t w e e n s a m p l e a n d t ip[19 , 20] d - o r b i t a l s c e n t e r e d o n th e ti p a t o m a s p r o b i n g w a v efunc t ions [ -21] E l e c t r o n i c r e s o n a n c e e ff e ct s [ 2 2 ] A n e l a st i c d e f o r m a t i o n o f t h e t ip i n d u c e d b y t h e t i p -s a m p l e i n t e r a c t io n [ 9 ] .T h e s e s u g g e s t i o n s w i ll b e c r i ti c a ll y e x a m i n e d w i t h t h et h e o r e t i c a l t o o l s a v a i l a b l e t o u s . A s w e c o n c e n t r a t e o nd i ff e r en t a s p e c t s o f S T M t h e t h e o r e t i c a l m o d e l s i m u l a t i n gt h e S T M i s c h a n g e d i n a n a p p r o p r i a t e w a y .

    1 . F u n d a m e n t a l T h e o r yT h e t a s k o f a t h e o r y o f s c a n n i n g t u n n e l i n g m i c r o s c o p y ist o c a l c u l a t e t h e c u r r e n t f l o w i n g f r o m t h e t i p t o t h e s a m p l es u r f a c e o r v ic e v e r sa d e p e n d i n g o n t h e a p p l i e d v o l t a g e . I na s ingle -par t i c le p ic ture th i s can b e v iewed as a s ca t t e r ingp r o c e s s w h e r e e .g . a n e l e c t r o n i n c i d e n t f r o m t h e i n t e r i o r o ft h e t i p m e t a l s c a t t e r s f r o m t h e b a r r i e r a n d h a s a c e r t a i np r o b a b i l i t y o f p e n e t r a t i n g i n t o t h e s a m p l e s u r fa c e . T h es c a t t e r i n g p r o c e s s i s d e s c r i b e d b y a w a v e f u n c t i o n [i + )w h i c h i s a n e i g e n f u n c t i o n o f t h e t o t a l H a m i l t o n i a ninc luding the t ip potent i a l V ,p. i i nd ica tes the incom ingm o m e n t u m o f t h e e le c tr o n . W h e n m e a s u r i n g t h e c u r re n t ,t h e s c a t t e re d e l e c t r o n w i t h fi n al m o m e n t u m f i s d e t e c t e d i nt h e w a v e f u n c t i o n I f ) , w h i c h d o e s n o t e x p e r i e n c e t h e

    G. Do yen et al.p o t e n t i a l ~ t i p i n the ba r r i e r reg ion . Thi s mean s tha t I f ) hast o b e a n e i g e n s t at e o f t h e s a m p l e H a m i l t o n i a n .T h e c u r r e n t i t se l f i s th e n u m b e r o f e l e c tr o n s p e r u n i tt ime be ing de tec ted in the s t a t e I f ) t imes the e l ec t ronc h a r g e e. A c c o r d i n g t o L i p p m a n n ' s g e n e r a l i z a t io n [ 2 3 ] o fEhren fes t ' s theorem 1-24] the tunne l cur ren t can b e wr i t t eni n t h e f o r m :

    Ir eJ ----- ~ - ~ i [ (tq Vtipli+ ) [ 2 ~ ( E f - - E l ) ,f ," ( 1 )T h e s u m r u n s o v e r a ll s ta t e s b e t w e e n t h e t w o F e r m i l ev e ls .Thi s exp res s ion i s exac t . I t reduces to F erm i ' s go lden ru le,i f t he exa c t s ca t t e r ing s t a t e li + ) i s rep laced b y a s t a t e ] i )w h i c h d o e s n o t i n c l u d e t h e t i p - s a m p l e i n t e r a c t i o n . A ne x a c t e x p r e s s io n f o r t h e t r a n s i t io n m a t r i x e l e m e n t i s:(tqV, pli + ) = (flV~ipli) + Y~ V i a G a n V n i .A,B ( 2 )T h e l a b e ls A a n d B r e f er t o b a s i s f u n c t i o n s w h i c h f o r m ac o m p l e t e s e t f o r t h e e l e c tr o n s t a t e s n e a r t h e t i p a t o m i n t h ebar r i e r reg ion . VyA a n d Vgi a r e m a t r i x e l e m e n t s o f t h e t ippotent i a l V t ip . G A I~ i n d i c a te s a m a t r i x e l e m e n t o f th eG r e e n ' s o p e r a t o r . T h e i m p o r t a n t t h e o r e t i c a l t a s k i s t oe v a l u a t e t h e G r e e n ' s f u n c t i o n s G A g i n the t ip reg ion .I n t h e f o l l o w i n g w e a s s u m e a n a p p l i e d b i a s v o l t a g es u c h t h a t t h e c u r r e n t f l o w s f r o m t h e t i p t o t h e s a m p l e .P e r f o r m i n g t h e s u m m a t i o n o v e r a l l s t a t e s b e t w e e n t h et w o F e r m i l ev el s, w e e n d u p w i t h t h e f o l l o w i n g f o r m u l af o r t h e c u r r e n t :

    2roe 1OABtJCO rBCVDA ( 3 )= h A-E A, n,~C3~, etlp K , a m p l e V l /

    w i t hti pS t ~ = z ( A l k + ) ( k + [ B ) , (4)k

    s a m p l es s a m p l e _ _CD - - ~ ( C l k ) ( k l D ) . (5)kA E i s the e nergy be tw een the two Ferm i l eve ls . VBc i s ama t r ix e l em ent o f the po tent i a l V tip. S ~ i s a mat r ixe l e m e n t o f t h e p r o j e c t i o n o p e r a t o r o n t h e e i g e n f u n c ti o n so f t h e t o ta l H a m i l t o n i a n w i th b o u n d a r y c o n d i t io n sc o r r e s p o n d i n g t o i n c i d e n t w a v e s f r o m t h e t i p sid e . ~"*ABKvsarnplei s t h e c o r r e s p o n d i n g p r o j e c t i o n o n t h e s a m p l e w a v ef u n c t io n s . T h e d i a g o n a l m a t r i x e l e m e n t s m i g h t b e in t e r -p r e t e d a s t h e l o c a l d e n s i t y o f s t a te s i n t h e t i p r e g i o ni n t e g ra t e d o v e r t h e e n e rg y r a n g e b e t w e e n t h e t w o F e r m ileve ls . S~ i s the pro je c t ion o f the loca l s t a t e [A) in theb a r r i e r o n t h e c o m p l e t e e ig e n s t a t e s a n d w i l l b e t e r m e d t ipp r o j e c t e d l o c a l d e n s i t y ( ( T IP L O D ) . S ~mp~e s the p ro je cti ono f th e l o c a l s t a t e o n t h e e i g e n f u n c t io n s o f t h e s a m p l esur face and i s ca l l ed s a m p l e p r o j e c t e d l o c a l d e n s i t y( S A P L O D ) .I n p e r t u r b a t i o n t h e o r y , w h e r e [i + ) i s r e p l a c e d b y [ i) ,S~P doe s not v a ry wi th l a t e ra l t ip pos i t ion . In th i s l imi t the

    q ' s a m p l e which i s e s sen-u r r e n t w o u l d b e p r o p o r t i o n a l t o OAA ,t i a l ly the loca l charge dens i ty a t the Fermi l eve l averagedo v e r t h e t i p o r b i t a l [ A ) . F o r a n s - t y p e t i p o r b i t a l ] A ) a t

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    Theory o f Scanning Tunneling M icroscopysuf f ic i en t ly l a rge d i s t ance f rom the sam ple sur face th i sr e p r e s e n ts t h e T e r s o f f - H a m a n n a p p r o x i m a t i o n [ 1 ] .I t c a n b e s h o w n t h a t a n e q u i v a l e n t p r o c e d u r e o fe v a l u a t i n g t h e t u n n e l c u r r e n t i s t o c a l c u l a t e t h e c u r r e n td e n s i t y a s a f u n c t i o n o f p o s i t i o n f r o m t h e s c a t t e ri n g w a v efunc t ion ~vi( r)= , o n e o b t a i n s :< f l i + > = < f l i p > + ~ < lq Gp lB >A,B

    x < B IV .p lA > < A l l + > . ( 10 )

    2 . S i m p l e T i p P l u s M u f f i n T i n S u r f a c eF r o m t h i s e x p r e s s i o n t h e c u r r e n t d e n s i t y i s e v a l u a t e daccording to (6) .

    2.1 . De f in i t ion o f the Model and Formal i smA t h e o r y o f S T M h a s t o e v a l u a t e t h e e x a c t s c a tt e r in g w a v efunc t ion ~Pi which con ta ins the w ave sca t t e r ing t ak ingp l a c e i n t h e t i p a n d t h e f i rs t a t o m i c l a y e r s o f t h e s a m p l esur face . Obvious ly th i s i s a three-dimensional sca t t e r ingp r o b l e m .T h e f o l l o w i n g m o d e l s y s t e m i s u s e d : T h e s a m p l esur face i s bu i l t f rom muff in- t in potent i a l s [2 5] acc oun t ingfor the a tomic s t ruc ture and the d-e lec t rons . The t ip i sr e p r e s e n t e d b y a G a u s s i a n p r o t r u s i o n o n a n o t h e r w i s ep l a n a r f r e e - e l e c t r o n m e t a l s u r f a c e . T h e p o t e n t i a l i n t h ein te r ior of the t ip m eta l , lib . . . is cons ta n t for a f ree e l ec t ronmeta l .A G r e e n ' s f u n c t i o n a p p r o a c h i s u s e d t o e v a l u a t e t h es c a t t e r i n g w a v e f u n c t i o n a n d t o c a l c u l a t e t h e c u r r e n tb e y o n d f i r s t - o r d e r p e r t u r b a t i o n t h e o r y . T h e G r e e n ' sf u n c t i o n s f o r t h e s y s t e m w i t h o u t t i p a r e c a l c u l a t e d b y al a y e r- b y - la y e r K K R s c h em e d e v l o p e d b y K a m b e ,Schef fi er , and M aca [26 , 27] de f in ing a ze ro -orde rs y s t e m c o n s i s t in g o f t w o e l e c tr o d e s . B l o c h ' s t h e o r e ma l l o w s o n e to e x p a n d t h e w a v e fu n c t i o n a n d t h e G r e e n ' so p e r a t o r i n r e c i p r o c a l s p a c e .T h e t i p b r e a k s t h e p a r a l le l t r a n s l a ti o n a l s y m m e t r y b yin t rod uc ing a loca l pe r tur ba t i on Vtip represent ing the t ipp o t e n t i a l . T h e L i p p m a n n - S c h w i n g e r e q u a t i o n i s u s e d t oca lcu la te the exac t s ca t t e r ing wave func t ion:l i + > = lip) + GpV~i i + ) , (7)w h e r e l i p ) a n d G a r e t h e e i g e n f u n c t io n a n d t h e G r e e n ' so p e r a t o r f o r th e t r a n s l a t io n a l l y i n v a r i a n t s y s t e m w i t h o u tt ip . V tip i s res t r i c t ed to the reg ion a ro un d the t ip a tomw h i c h i s s p a n n e d b y t h e l o c a l iz e d b a s i s s e t { IA > } :l i + > = li ,> + GplB> t h e G r e e n ' s o p e r a t o r f o r th eH a m i l t o n i a n i n c l u d i n g t h e t i p i s n e e d e d . T h e G r e e n ' so p e r a t o r s f o r t h e t w o s y s t e m s ( w i t h o u t a n d w i t h t h e t ip )a r e c o n n e c t e d b y D y s o n ' s e q u a t i o n . T h i s e q u a t i o n i s

    2.2. Tunnel Curre nt to an AI(111) and a P d ( 1 0 0 ) Sur face fora T ip o f Gauss ian ShapeT h e c a l c u l a ti o n s i n c l u d e t h e a t o m i c s t r u c t u r e o f th esample and the re fore e l ec t ron d i f f rac t ion in the s ample .F o r t h e z e r o - o r d e r s y s t e m t h e r e g i o n o f fi a t p o t e n t i a lins ide the t ip mate r i a l i s s epara ted f rom the muff in- t inr e g i o n b y a s q u a r e b a r r i e r. F o r A I ( 11 1 ) th e d i s p l a c e m e n to f t h e b a r r i e r f r o m t h e f ir s t l a y e r o f a l u m i n u m a t o m s i s1 . 1 / k c o r r e s p o n d i n g t o h a l f t h e s e p a r a t i o n b e t w e e n t w oadjace nt (111) p lanes . The k ine t i c energy o f the tunn e l inge lec t ron i s 11 eV (ze ro appl i ed vol t age) .

    W i t h i n t h e b a r r i e r r e g i o n , a lo c a l a t t ra c t i v e p o t e n t i a li s i n t r o d u c e d . T h i s d e f o r m a t i o n i s o f G a u s s i a n s h a p epara l l e l to the sur face . Perp end icula r to the sur face i t i s aG a u s s i a n c u t o f f a t t h e b o r d e r s o f t h e r e c t a n g u l a r b a r r i e r .I t i s c e n t e r e d i n t h e m i d d l e o f t h e b a r r i e r a n d h a s a l a t er a lrad ia l range of ca . 3 /~ para l l e l to the sur face . The averag ed e c r e a s e o f t h e b a r r i e r p o t e n t i a l w i t h i n t h i s d e f o r m a t i o nis 2.7 eV.F i g u r e 1 s h o w s a s l i g h t p e r i o d i c c h a n g e i n t h e t o t a lt u n n e l c u r r e n t w h e n t h e t i p i s m o v e d p a r a l l e l t o t h es u r f a ce a t c o n s t a n t p e r p e n d i c u l a r s e p a r a t io n . T h e r e l a ti v ev a r i a t i o n o f t h e t o t a l t i p - i n d u c e d c u r r e n t i s < 1 0 - 3. T h ec u r r e n t i s e n h a n c e d i f t h e t i p i s a b o v e a h o l l o w s i t e w i th n oA 1 a t o m i n t h e s e c o n d l a y e r , w h i c h is n o t i n a c c o r d a n c ew i t h t h e e x p e r i m e n t a l f i nd in g s. H o w e v e r , t h e a b s o l u t em a g n i t u d e o f t h e t h e o r e t i c a ll y p r e d i c t e d v a r i a t i o n i s s os m a l l t h a t w e d o n o t i n t e n d t o g i v e a q u a l i t a t i v ein te rpre ta t ion .P a l l a d i u m h a s d - e l e c t r o n d e n s i t y a t t h e F e r m i l e v e la n d t h e r e f o r e e x h ib i t s a s t r o n g e r v a r i a t i o n o f th e c h a r g edens i ty pa ra l l e l to the sur face . Thi s i s expec ted to bere f l ec ted in the tunn e l cur rent . W e f ind a re la t ive va r i a t iono f th e t u n n e l c u r r e n t o f s e v e ra l p e r c e n t f o r t h e P d ( 1 0 0 )s u r f ac e w i t h m a x i m a a b o v e t h e a t o m s ( Fi g. 2 ). T h eb a r r i e r w i d t h a n d t h e t i p p o t e n t i a l a r e t h e s a m e a s f o r th eA l ( l l l ) - s u r f a c e . T h e k i n e t i c e n e r g y o f t h e t u n n e l i n ge lec t ron i s 7 .6 eV.

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    284 G. Doyen et al.

    Fig. 1. Layer-KKR tunnel theory: Variation of the total tunnelcurrent parallel o an AI(111) surface n the region of contact betweentip and sampleFig. 3. Layer-KKR tunnel theory:(TIPLOD) for Pd(100) Tip projected local density

    qFig. 2. Layer-KKR tunnel theory: Variation of the total tunnelcurrent parallel to a Pd(100) surface n the region of contact betweentip and sample

    J )

    F i g . 4 . Layer-KKR tunnel theory: Sample projected local density(SAPLOD) for Pd(100)

    2.3. Discuss ionThe tip potential we choose simulates roughly the tip-induced localized states (TILS), discussed by Ciraci et al.in order to explain the large corrugation found whenimaging sp-band metal surfaces [19, 20]. Ciraci et al. donot calculate the tunnel current bu t base their conclusionson the variation of the charge density which builds up inthe conductance channel.The variations found in the present calculations aresurprisingly small and cannot explain the experimentalfindings, which show variations o f the tunnel current ofthe order of 10%. The effects are small in the presenttheory, because th~ atomic A1 potential has a weakscattering power. Only at very small distances does itbecome attractive enough t o increase the electron flow.For Pd(100) we find a stronger tip-induced modifica-tion of the charge density for the top position comparedto the hollow position. Density of states is removed fromthe Fermi level because of the formation of bonding and

    anti -bonding states (Fig. 3). This is in agreement with thequalitative argument given by Ciraci et al. It leads to anenhanced corrugation in the tunnel current. The reasonfor this can be seen by looking at (4), which, for a single tipstate, reads:2~e 1 ~ t i p ~ s a m p l e l ,V 2 I , d ~ ( 1 1 )

    J = T A - -E ~ A A L ' A A ' " ~ x ' t "The tip-sample interaction leads to a variation of thequantity g]~ with lateral tip position (Fig. 3). As theresults show, the corrugation of this quantity followsquali tative ly that of S] ~ p~e (displayed in Fig. 4). Hence thelateral variation of the current has the same shape as thecharge density at the Fermi level, but the amplitude islarger by a fac tor of 10. However, this enhancement is stillnot large enough to explain the experimental results.The conclusion obtained from this KK R investigationis that the multiple scattering of the tunneling electronsinside the sample does not explain the observed corruga-tion. Even near contact, where this theory contains

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    Theory of Scanning Tunneling M icroscopyef fec ts of t ip - induced loca l i zed s t a t es , t he pred ic ted l a t e ra lST M con t ras t on AI( l l 1) i s negl ig ib le , Phys ica l e f fec t s ofd i f fe r e n t q u a l i t a ti v e n a t u r e h a v e t o b e i n v o k e d . O n e o f t h em a j o r u n r e s o l v e d q u e s t i o n s i n th e o p e r a t i o n o f t h e S T Mc o n c e r n s t h e r o l e p l a y e d b y t h e a t o m i c a n d e l e c t r o n i cs t r u c t u r e o f t h e t ip . T h e n e x t s e c t i o n c o n c e n t r a t e s o n t h e s easpec t s .

    3. Inf luence o f the Electronicand Geometrical Tip Structure on the Tunnel Current3.1. Tunn el Current fo r N oninteracting 7Tp Orbitals ofDifferent ShapeN e g l e c t i n g t h e t i p - s a m p l e i n t e r a ct i o n , t h e l a t e ra l c o n t r a s ti n im a g i n g a s u r f a c e i s d e t e r m i n e d b y t h e v a r i a t i o n o f t h eq u a n t i t y ~'aaesample"F igu re 5 d i sp lays the resu l t s for threea t o m i c t u n g s t e n o r b i t a l s ( 6 s , 6pz , 5d~2) with Pd(10 0) a ss a m p l e s u r f a c e . T h e w a v e f u n c t i o n s a r e t a k e n f r o m t h el i te r a t u r e a n d a r e o f m u l t i p l e - z e ta q u a l i t y ( P d : [ 2 8 ] ; W :[29 , 30]) . The d i s t ance o f the t ip cen te r f rom the f i r s t l ayero f P d a t o m s i s 2. 5 A .T h e i m a g e s o b t a i n e d d i ff e r q u a l i t a ti v e l y f o r t h e t h r e et ip orb i t a l s . Only for the 5d~2-orb i t a l i s an enhancemento n t o p o f a s u b s t r a t e a t o m f o u n d . T h i s i s i n t e r p r e t e d a s a ni n d i c a t i o n t h a t t i p d - o r b i t a l s p l a y a n i m p o r t a n t r o l e f o rt h e o b s e r v e d a t o m i c c o r r u g a t i o n o f m e t a l s u r f ac e s,b e c a u s e i n e x p e r im e n t a l S T M i n v e s t i g a t i o n s t h e m e t a la t o m s i n v a r i a b l y s h o w u p a s p r o t r u s i o n s . T h e r e l a t i v econ t ras t in ou r ca lcu la t ions i s 3 x 10 -z for 5d~2 and 10 -2for 6s. Thi s co nt ra s t w ou ld be va l id i f t he respec t ive t ipo r b i t a l w e r e t h e o n l y o n e r e l e v a n t f o r t u n n e l in g . I n r e a l i tys e v e r a l t i p o r b i t a l s c o n t r i b u t e t o t h e t u n n e l c u r r e n t a n dthe cont ras t i s then l ike ly to be d imini shed .T h e i d e a t h a t t i p d - o r b i ta l s l e a d t o a l a r g e r c o r r u g a -t i o n h a s a l r e a d y b e e n s u g g e s t e d b y C h e n 1 -2 1] . H o w e v e r ,i n c o n t r a s t t o C h e n ' s c o n c l u s i o n , t h is d o e s n o t e x p l a in t h eexper imen ta l f ind ings in the t rans fe r -H am i l tonian appro x-i m a t i o n , b e c a u s e t h e d - o r b i t a l s h a v e s m a l l e r w e i g h tc o m p a r e d t o t h e 6 s - a n d 6 p z - o r b i t a l s a n d b e c a u s e , a trea l i s ti c tunne l ing d i s t ances ( > 3 A) , the d-c or rug a t ion i sa l r e a d y t o o s m a l l p e r s e .T h e t i p o r b i ta l I A ) u s e d in t h e K K R t u n n e l t h e o r y o fS ec t. 2 d o e s n o t h a v e t h e s h a p e o f a n a t o m i c W o r b i t a l. I t sl a t e ra l ex tens ion i s smal l e r than tha t of the W 6s -orb i t a l .P r o j e c t i n g t h e P d ( 1 0 0 ) w a v e f u n c t i o n s a t t h e F e r m i l e v e lo n t o t h i s t i p o r b i t a l g i v e s a n e n h a n c e m e n t o v e r t h e P da t o m s ( F ig . 4 ) a l t h o u g h w i t h c o n s i d e r a b l y s m a l le r c o r r u -g a t i o n a m p l i t u d e t h a n f o r t h e 5 d ~ : t i p o r b i t a l.T h e t w o t h e o r e t ic a l a p p r o a c h e s f o l l o w e d u p t o h e re( K K R t h e o r y w i t h s i m p l e t i p s t r u c t u r e , r e a l i s t i c b u tn o n i n t e r a c t i n g t i p o r b i t a l s ) d o n o t e x p l a i n t h e o b s e r v e dc o r r u g a t i o n o n m e t a l s u r fa c e s. T h e c l u e h a s t o b e f o u n d i nf e a t u re s w h i c h a r e n o t i n c o r p o r a t e d i n th e s e a p p r o a c h e s ,e .g ., rea l is t i c e l ec t ronic s t ruc ture of the t ip , e l as t i c de for -m a t i o n o f t h e t ip .Fig. 5a-c. N oninteracting tip o rbitals: Sample projected local den-sity for Pd(100) n th e cas e of a 6s- (a), 6p,-(b), and 5dz2-tip e) orbital.The c enter of the tip is 2.5 A from the first layer of substrate atom s

    b

    G

    285

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    286 G. Doyen et al.- 2 . 25

    5 a to m i pn e u t~ -3 .7 5 - - k ~ r a f ~

    ~ ' ~ a t o m t i pc -O- 5 .2 5 - -

    I I3 .0 4 . 0 s . o 6 . 0distance tip - b a s e [ b o h r ]

    F i g . 6. Potential energy of the topm ost W tip atom for various tipmodels as d escribed in the text

    ~ ' 3 . 5 - -" ~ 2 . 7 - -

    1 . 9 - -o 1 . 1 - -

    0 3 0 . 3 - -- 0 . 5

    \ - - 5 a t o m ip" . , ~ i o n i z e d~ - s in g le a to m t ip

    n e u t r a l

    r I I r J3 . 5 4 . 2 4 . 9 5 . 6 6 . 3 7 . 0d i s t a n c e t i p - s a m p l e [ A ]

    Fig. 7. Tunnel current to a free electron surface for the tip m odelsof Fig. 6

    3.2. A More Realistic Electronic Tip StructureI n c a l c u l a t i n g t h e S T M c u r r e n t w e w i l l h a v e t o d e t e r m i n ethe e l ec t ro n i c s t ruc tu re o f t he t ip , i. e. , t he w ave fun c t ionsf o r t h e W s u r fa c e to g e t h e r w i t h t h e e m b e d d e d c l u st e r a n dt h e a d s o r b e d t i p a t o m o n t o p o f i t.W e c a n n o t a t p r e s e n t u s e f i r s t - p r i n c i p l e s m e t h o d s t os o l v e t h i s c o m p l i c a t e d p r o b l e m . R a t h e r w e u s e a m o d e lH a m i l t o n i a n w h e r e t h e o n e - a n d t w o - e l e c t r o n i n t e g r a l sa r e c h o s e n i n a s e m i - e m p i r i c a l w a y s u c h a s t o r e p r o d u c ee x p e r i m e n t a l a n d t h e o r e ti c a l d a t a f o r t h e n o n i n t e r a c t i n gsys t em [9 , 31 , 32 ] .F i g u r e 6 d e m o n s t r a t e s t h a t t h e a d s o r p t i o n o f a Wa t o m o n a n e m b e d d e d c l u s te r o f f o u r o t h e r W a t o m s ( a 5 -a t o m t i p ) i s q u a l i t a t i v e l y d i f f e r e n t t o t h e a d s o r p t i o nb e h a v i o u r o n a f l a t W ( 1 1 0 ) s u rf a c e , b e c a u s e t h e e l e c t r o nd e n s i t y i n t h e e n v i r o n m e n t o f t h is c l u s t e r d e v i a t e s s i g n if i-can t ly f rom tha t o f the f l a t su r face (F ig . 6 ). In f ac t , t hee l e c t r o n d e n s i t y t e n d s t o b e i n c r e a s e d n e a r t h e c l u s t e r s ot h a t t h e t i p o r b i t a l s h a v e a l a r g e r o v e r l a p w i t h o c c u p i e dm e t a l w a v e f u n c t i o n s . T h i s m a k e s t h e a d s o r p t i o n l e s ss t a b l e a n d a t s h o r t e r d i s t a n c e s p u s h e s t h e t i p - d e r i v e dl e ve l s u p i n e n e r g y . A t s o m e d i s t a n c e a 5 d - d e r i v e d le v e lw i ll c r o s s t h e F e r m i l e v e l a n d t h e e l e c t r o n i c c o n f i g u r a t i o no f l o w e s t e n e r g y c o r r e s p o n d s t o a p o s i t i v e l y c h a r g e d t i pa t o m .I n t h e c a s e o f t h e 5 - a t o m t i p t h e p o t e n t i a l e n e r g y c u r v ef o r t h e t i p a t o m s h o w s t w o m i n i m a c o r r e s p o n d i n g t od i f f e re n t e l e c t r o n i c s ta t e s. T r a n s i t i o n s b e t w e e n t h e s e t w om i n i m a m i g h t a c c o u n t f o r s u d d e n c h a n g e s in t h e t u n n e lc u r r e n t a s s o m e t i m e s o b s e r v e d i n e x p e r i m e n t .

    E x p e r i m e n t a l l y it is o b s e r v e d t h a t i n s t a b i l i t y o f t h e t i pi s a g e n e r a l c h a r a c t e r i s t i c fo r h i g h r e s o l u t i o n i n S T M . T h i sm a n i f e s t s i t se l f j n f r e q u e n t p e r t u r b a t i o n s o f th e t u n n e l i n gs i g n a l a n d i n a s u d d e n c h a n g e o f r e s o l u ti o n . T h e t i p r e a c t si n a p a r t i c u l a r l y s e n s i ti v e w a y i f t h e t u n n e l c u r r e n t i si n c r e a s e d a b o v e 1 0 0 h A .U s i n g t h e t i p m o d e l s d e s c r ib e d , t h e t u n n e l i n g t o a f r ee -e l e c t r o n m e t a l s u r f a c e i s i n v e s t ig a t e d . I n c l u s i o n o f t h ed - e l e c t r o n s a f f e c t s t h e l o c a l e l e c t r o n i c s t r u c t u r e a n dm o d i f i e s th e c u r r e n t . F o r s h o r t d i s t a n c e s a s i g n i f ic a n t p a r to f t h e t o t a l c u r r e n t f l o w s d i r e c t ly v i a t h e t i p d - o r b i t a ls .I n F i g . 7 t h e l o g a r i t h m o f t h e t u n n e l c u r r e n t i s p l o t t e dv e r s u s t h e t i p - s a m p l e s e p a r a t i o n f o r t h e t h r e e k i n d s o f t i p su n d e r i n v e s t i g a t i o n .

    C o m p a r i n g t h e s in g le a t o m t i p t o t h e n e u t r a l 5 - a t o mt i p t h e p r o j e c t i o n s o f t h e t ip o r b i t a l s o n t h e F e r m i l ev e ls o ft h e re s p ec ti v e m e t a l su r fa c e s ( T I P L O D a n d S A P L O D ) d on o t c h a n g e s i g n i f ic a n t ly , b u t t h e p o t e n t i a l i n t h e t i p r e g i o nh a s b e e n m o d i f i e d c o n s i d e r a b l y . T h i s y i e ld s a r e d u c t i o n o ft h e t u n n e l c u r r e n t b y a f a c t o r o f t w o t o t h r e e .C o m p a r i n g t h e i o n i z e d 5 - a t o m t i p t o t h e n e u t r a l 5 -a t o m t ip t h e p o t e n t i a l h a s c h a n g e d a g a i n i n s u c h a w a y a st o a c c i d e n t a l l y g iv e th e s a m e t u n n e l c u r r e n t a s t h e s i n g lea t o m t i p a t l a r g e s e p a r a t i o n s .A n i m p o r t a n t f e a t u r e o f t h e i o n i z e d t ip i s t h a t a5 d - le v e l h a s j u s t p a s s e d t h r o u g h t h e F e r m i l e v el s o t h a t as ign i f i can t pa r t o f t he t a i l o f t h i s 5d - re son ance s t i l l r e achesbe low the Fe rmi l eve l . Th i s g ive s t he 5d- t i p o rb i t a l ap a r t i c u l a r l y l a r g e s p e c t r a l w e i g h t a t t h e t u n g s t e n F e r m il ev e l. I t m e a n s t h a t t h e i m p o r t a n c e o f t h e 5 d - o r b i t a l f o rt u n n e l i n g h a s b e e n i n c r e a s e d o v e r - p r o p o r t i o n a t e l y c o m -p a r e d t o t h e o t h e r t i p o r b i t a l s . T h i s is s e e n i n t h e t u n n e lc u r r e n t a t s h o r t e r d i s t a n c e s , w h e n t h e o v e r l a p o f t h e 5 d -o r b i t a l w i t h t h e A 1 w a v e f u n c t i o n s b e c o m e s i m p o r t a n t a n dl e a d s t o a d e v i a t i o n f r o m t h e s t r a i g h t l i n e f o r t h el o g a r i t h m o f t h e t u n n e l c u r r e n t .T h i s i s r e m i n i s c e n t o f r e s o n a n c e s c a t t e ri n g . I f a t i po r b i t a l w e r e s i t u a t e d f i g h t a t t h e F e r m i l e v e l , t h e t r a n s -m i t t i v i ty w o u l d b e u n i t y c o r r e s p o n d i n g t o t h e q u a n t u ml i m i t o f r e s i s ta n c e o f 1 2 9 06 .4 O h m s . T h i s r e s o n a n c e h a s af i n i t e wid th , i .e ., i f t he cen t e r o f t he r e so nan ce i s pos i t i on eds l i g h t l y a w a y f r o m t h e F e r m i l e v e l w e s t i l l h a v e ac o n s i d e r a b le e n h a n c e m e n t o f t h e t u n n e l c u r r e n t .I n o r d e r t o q u a n t i f y t h e id e a w e w r it e th e T I P L O Da c c o r d i n g t o (2 ) a n d ( 4) i n t h e f o r m :

    ti p~ t~p = ( A[ G - [A ) ( A[G+ [A) W2 ( A) ( A l k ) ( k [ A ) (12)k

    w i t h( A [ G - [ A ) ( A [G + ] A ) = [(Everm -- ca) 2 + (Im {qa} ) 2] - 1.(13)I m { q a } i s t h e i m a g i n a r y p a r t o f th e s e l f - e n e rg y o f t h e t i po r b i t a l, w h i c h d e t e r m i n e s t h e w i d t h o f t h e r e s o n a n c e , e a i st h e c e n t e r o f t h e r e s o n a n c e i n c l u d i n g t h e i n t e r a c t i o n w i t hthe sam ple su r face . I f i t is nea r t h e Fe rm i l eve l, a sm a l lv a r i a t i o n o f e a m i g h t o b v i o u s l y le a d t o a l a r g e v a r i a t i o n o ft h e T I P L O D , i f t h e w i d t h o f th e r e s o n a n c e i s s u ff ic ie n tl ysma l l .

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    Theory of Scanning Tunneling MicroscopyThe results displayed in Fig. 7 have been obtained fora free electron sample surface where no corrugation ispresent. If the surface were to exhibit only a slightcorrugation in its electronic structure as discussed abovefor the AI(111) and Pd(100) surfaces, this might affect theenergetic position of the 5d-resonance via electronic tip -sample interaction. A minor modification of this reso-

    nance can alter the tunnel current to an appreciableextent and hence lead to an amplification of the samplecorrugation.Definite conclusions must be based on a full theory. Atthis point we merely want to illust rate what k ind of effectresonance tunneling might have on the STM image. Forthis qualita tive discussion we restrict ourselves to a singletip orbital IA). As an example we might think of imagingan oxygen atom adsorbed on an AI(lll) surface. Atheoretical investigation [33] demonstrated that thecharge density at the Fermi level exhibits oscillationswhich are not visible in the tunnel current. The contoursof constant tunnel current were predicted to representsimple protrusions for all tunneling distances.These theoretical investigations need to be re-examined in view of the additional insight into the STMprocess as described above. Points of concern are theneglected d-wave functions on the tip atom and the factthat the tip charge distribution was frozen in thesecalculation. Hence there was no influence of the adsorbatecharge distribu tion on the tip electronic structure, which,according to the arguments just presented, might have aconsiderable effect on the appearance of the STM image.Assuming that the adsorbate induced change of theSAP LOD has a simple Gaussian shape, we investigate theinfluence of a modulation of the resonance position. Thiswill be reflected in the T IP LO D according to (12) and (13).Results are displayed in Fig. 8. If there are oscillationsin the adsorbate-induced charge density, these lead tooscillations in the TIP LOD and will therefore appear in

    adsorbate induced cha rge densi ty at Fermi leve l(O/AI(I 1 )

    modulus squared of ip Green funct ion

    S A P L OD

    -10 0 10distance paral le l to sur face [bo hr ]

    F i g . 8 . A d s o r b a t e - i n d u c e d s t r u c t u r e p a r a l l el t o t h e s u r fa c e fo r o x y g e non a freeelectronsurface:charge densityat the Fermi evel;modulussquared of the tip Green's function,determining he variation of theTIPLOD; SAPLOD determining he tunnel current for constantTIPLOD; tunnel current for resonance tunneling

    287the tunnel current contours for the case of resonancetunneling.The slightly confusing theoretical situation is hencethe following: From the simplest available tunnel theory(the Tersoff-Hamann approximation) one expects to seeoscillations in the adsorbate-induced tunnel current whenimaging O/AI(11). A more refined theory which improvesover the transfer-Hamiltonian concept [33] predicts tha tthis detailed structure should n o t appear in the tunnelcurrent. If finally the concept of resonance tunneling isintroduced it is conceivable that adsorbate-inducedFriedel oscillations superimposed on the atomic substratestructure are detectable in STM.A similar effect can occur i f the intrinsic corruga tion ofthe sample electronic structure leads to a small elasticdeformation of the tip, which will also affect the energeticposition of the 5d-resonance.

    4 . E las t i c Deformat ion o f the T ipThe influence of an elastic deformation of the tip on thecorrugation has previously been investigated in a simplemodel calculation neglecting d-functions on the atomforming the apex of the tip [9]. The tip at om can move inthe direction perpendicular to the surface if a force isexerted on it.In this model the A1 sp-band charge density does notexhibit any corrugation at all. If the W tip atom ap-proaches the surface, it will eventually interact with thecore potential of the A1 atoms. This leads to a corrugationof the interaction potential in the repulsive region. As isobvious from physical intuition, the potential is morerepulsive for the top posi tion than for the center positionbetween three A1 atoms. Depending on the lateral po-sition the tip ato m recedes to a different extent from the Wsurface.For this simple calculation the total potential isobtained by superposing the W-AI and the W- W interac-tion potential. For a given electrode separation the W tipatom will find its equilibrium position between the twoelectrodes such that the net force is zero.How does this influence the tunnel current? Let us forsimplicitly assume that the tunnel current flows domi-nantly via the 6s tip orbital. The current is then approxi-mately given by the product o f the spectral weight of the6s-orbital at the tip Fermi level (6s-TIPLOD) times the 6s-spectral weight at the sample Fermi level (6s-SAPLOD)(times an effective value of the potential):

    2xe 1J = --h ~ E S~Ps,6s S~smg e W 2(6 s) . (i 4)The variation of both quantities with distance isdisplayed in Fig. 9. The 6s-TIPLOD varies much morestrongly than the 6s-SAPLOD. If the W tip atom movestowards the W(ll0) surface the 6s-TIPLOD increasesstrongly, whereas the SAPLOD decreases only slightly.The net effect is hence an increase of the tunnel currentwhich is larger in the top position than in the centerposition. The result is a corrugation in the tunnelcurrent. It means that in the constant current mode the

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    288

    0 . 0 2 3 p r o d ~ . . . _ I0 . 0 1 90 . 0 1 50 . 0 1 10 . 0 0 7 -distancei p a t om- A I ( 111 ) [ A ]

    oo

    0A1( 111) W ( 110)

    Fig. 9. TI PL O D and SA PLO D determining the influence of anelastic tip deformation on the tunnel current (see text)

    s e p a r a t i o n o f t h e e l e c t r o d e s h a s t o b e in c r e a s e d i n t h e t o pp o s i t i o n a s c o m p a r e d t o t h e c e n t e r p o s i t i o n .R e c e n t l y C i r a c i e t a l . a r g u e d t h a t e l a s t i c d e f o r m a t i o no f t h e t ip w o u l d l e a d t o a n a n t i c o r r u g a t i o n b e c a u s e t h et i p a t o m i s r e p e l l e d f r o m t h e s a m p l e s u r f a c e a n d t h e r e f o r et h e t i p a t o m - s a m p l e d i s t a n c e i s in c r e a s e d 1 -2 0 ]. T h i sc o n c l u s i o n w a s d e r i v e d w i t h o u t c a l c u l a t i n g t h e t u n n e lc u r r e n t a n d o v e r l o o k e d t h e r o l e p l a y e d b y t he q u a n t i t y~ ]ip I t is i n f a c t q u i t e p o s s i b l e t h a t t h i s q u a n t i t y v a r i e s-d r a s t i c a l l y i f r e s o n a n c e t u n n e l i n g v i a d - o r b i t a l s i s in -c l u d ed . T h i s m i g h t t h e n y i e ld t h e r ig h t o r d e r o f m a g n i t u d ef o r th e c o r r u g a t i o n , b u t a d e t a i le d a n d r e l ia b l e c a l c u l a t i o ni s n e e d e d t o c o n f i r m t h i s i d e a .

    5 . C o n c l u s i o n sA t h e o r e t ic a l i n v e s t i g a t i o n o f S T M a v o i d i n g t h e a p p r o x -i m a t i o n s i n v o l v e d i n th e t r a n s f e r - H a m i l t o n i a n a p p r o a c hh a s b e e n p r e s e n t e d . I t r e v e a l e d t h a t a c o m p l e t e t h e o r y i ss t i l l mi s s in g .

    A K K R t h e o r y c o m b i n e d w i th a s im p l e m o d e l f o r th et i p d o e s n o t r e p r o d u c e t h e e x p e r i m e n t a l l y o b s e r v e dl a t e r a l c o n t r a s t i n S T M . T h i s a p p r o a c h w i ll b e e x t e n d e d i nt h e f u t u r e t o i n c l u d e a b e t t e r d e s c r i p t i o n o f t h e t i p a t o ma n d t h e b a r r i e r . M o d e l c a l c u l a t i o n s p e r m i t t i n g t h e i n c l u -s i o n o f d - e l e c t r o n s o n t h e t ip d e m o n s t r a t e t h a t t i p d - w a v ef u n c t i o n s i n f lu e n c e t h e t u n n e l c u r r e n t b y m o d i f y i n g t h ee l e c t r o n i c s t r u c t u r e a n d , a t s h o r t d i s t an c e s , b y d i r e c t l yc o n t r i b u t i n g t o t h e c u r r e n t .

    W e h a v e m o d e l l e d a m o r e r e a l is t ic m u l t i - a t o m t ip a n df o u n d s e v e r a l p o t e n t i a l e n e r g y c u r v e s c o r r e s p o n d i n g t od i f f e r e n t e l e c t r o n i c c o n f i g u r a t i o n s o f t h e t i p. S t r u c t u r a lt r a n s i t i o n s b e t w e e n d i f f e re n t p o t e n t i a l e n e r g y c u r v e sm i g h t e x p l a i n i n s t a b i l i t i e s a s s e e n i n e x p e r i m e n t . T h ee l a s ti c d e f o r m a t i o n o f t h e t i p l e a d s t o c o r r u g a t i o n o ns p - b a n d m e t a l s a t s h o r t d i s t a n c e s .

    G. Doyen et al .R e s o n a n c e t u n n e l i n g m i g h t b e a k e y id e a f o r u n d e r -

    s t a n d i n g e x p e r i m e n t a l l y o b t a i n e d r e s o l u t i o n w h i c h i s n o te x p l a i n e d b y s i m p l e r a p p r o a c h e s .Acknowledgements. G.D. is grateful to D. Dra kova for valuablediscussions and for a critical reading o f the manusc ript. F inancialsup por t by the Deutsche Forschungsgem einschaft (SFB6) isacknowledged.R e f e r e n c e s

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