Theoretical studies for scanning tunneling microscopy

Physiea 127B (191t4) 137-142 North-Holland, Amsterdam

THEORET|CAL STUD~'ES F O R SCANNING T U N N E L I N G M I C R O S C O P Y N. G A R C L ~ at~d F. FLORES Dioisi6n de F~slea$. Uhlver~dad Autdnoma de Madrid, Cantoblanco, Madrid.34. Spain A theoretical approach, exactlysolvable,based on the seattering of electrons by tough surfaces is presented to explain tunnel-vacuum experiments. The electrodes, the tip as well as the crystal surface are dc.!~edbedby a jetEu~ model. A simple fotrmtla that 'we;ksex~l:ellentlyis givenfor the tunnel conductivitysb.owingthat this varies linearl7with the radius of curvature o¢ the til>.-sufface:l~stem.It is also shown that the classical image potential pla'.l,sa very important role in the cc,rreet [nterprctatlon elf the vacuum tunneling experiments. The imal;e potentials saturates at short distances (,=3.0 between jell~umedges) into ,theexchange and correlation potential as described by dansiw functionaltheory. I. in~ducfion and the mo~e'l

In a recent work Gareia, Ocal and Fl6res [1] have presented an exactly solvable theory for the tunnel current between two electrodes of general shape; described by a jeUium model. This is directly applicable to interpret and understand the scanning tunneling microscopy (STtd) experiments performed by Binnig, Rohrer, Gerber and Weibel [2] that gives informaRon of the surface structure.

Our model is graphi,:ally described in fig. 1. The left side is a scheme of the one-electron potential profile between the tip and the sample. There are three important contributions to that potential: (i) first, we have a narrow region of widl~:h around 1 2", near tI~e two metals ~3... where the potential ehange~; quicklty from the b,-Ik to the vacuum. (ii) On 1the other hand, at~ electrostatic potential between the two metals mast be included to equalise bcJth Fermi levels; for W and Au, the drop voltage is of 0.7 eV, i.e., the difference in work furmtions between the tip and the sample [4]. (iii) Finally, there must appear corrections introduced b~ the image potential [5]. ARhou~;h this effect is important, the image potential is very fiat in tht~ region between both metals, pre~enting ir~,po~a,~t variations only near both surfaces. According to this ,discussion, !in our model we simplify the interface potential and substitute it by th~" abrupt potent~.al shc,wn i~1 fig. in. we have

chosen the parameters ¢,f this model in order to simulate the tunneling associated with the s-wave funcJrioas e l both raetals [6], whgch are the ones gi*ing d:,e important contribution to the tipcurrent, ffhe following values of the Fermi energies of both media have been used: EF(Au)~ 5.5 eV and EFt'W)~ 8 eV [7] (see fig. 1). In the right side of fig. 1, we allfo show the scheme used in the real space for the silape and potential of both surfaces. Once defined the model, our problem is to solve for the current density, the transmitivity and tel[activity of one electron approaching the tip's surface from +~ and tunneling to the sample. The whole intensity is obtained from all the electrons conta£ned in a sphere shell around the tip Fermi surface with an energy width equal to Vc, the applied voltage, which in the experiments [2] is taken around 0.01 eV. This problem has been solved by repeating periodically the tips on tl~e surface (fig. Ib), at distanc~ L long enough to have them decoupled. We have used 'the same numerical technique, as developed in atora-[8] and light surface scattering ~9, 10] by using GG' or G R methods. Recently, Stoll et aL [11] have also used the same model and techniques. Vee have analyzed differen: systems with a corrugated surface (either W or A,u) having a spherical, cosinelike, parabolic and ~aw-tooth profiles, placed at d from a (either Au or W). By using 50 reciprocal vectors, we have obtained good convergent results,

0378-4363/84/$03.00 I~) Elsevier Science Publishers B.V. (North-Holland Physics Publishing D~vision)

N. Gurcta and F. Fl6res / Scanning t~nneli*zg microscopy

t238

/ ,Metal

sample

la)

,: (b) Pig. I, On the I hs the encr:~ctie scheme for the tip--sample sy.~lem is ~hown. On the rhs the tlp.-sampltr lateral profiles are described. The tip is period~cMly rcpeuled with a period L large enough to decouple the ti!~s.

2. Calculations From our program we can obtain all the relevant q~tantities as for example, density current profiles, lateral resolution L~,~ and surface-tip distance giving a current I for an applied V.. A very interesting result is that the tunnel conductivity can be written as

cr = KoG( R~lr)exp[-2.14hd],

{la)

where I~ = (2m~b/f~z) It'', g~ = (ehl2m)N(E~)~1.846× 1W'~A ' ~ ' - ' eV "'* and N(E~).~ 0.02 A:~/eV the typical value for the density of states at the Fermi level for metals [7]. G(R~t~) i., a function of the effective radius of curvatnr: involved in the tip-surface tunnel current defined as

t

/1

t\(1

lX

+N)tN+ N].

(2)

The values R',, RI, R~ and R~ are the curvalure of the lip and surface at the shorter distance d between lhe two crystals (see figs 1 and 4). It is clear that for a plane surface and cilidrical symmetry for the tip R ~ r = R I ~ R ~ and d is the distance between the tip minima and the surf~tcc. As an example we present fig. 2a for a spheric

tip of R = 5 . ~ , L = 1 0 A and 4 ~ = 2 e V and 4..5 eV for a flat surface. As it can be seen both curves cut at the same point for d = 0 a~.~d the slope number 2.14±0.04 has validity for 2 / ~ R~a-~ 11 ~ . This value rather [1.2] than 2,3 appears because the maximum current density is that corresponding to the Fermi electrons that have a wave vector at angle 0 ~ 2 2 ° with the surface normal. The 2.14 changes slowly ,when R~jr changes; for example for R~tr--~ oo; the val~te is ~2,30. In fig, 2b we also present the plot of G(R¢,r) for differe~lt walues of R~tr; it can be seen that G(R¢,) changes linearly, in the range 2 11/~, with R~jr; i.e. croeR~tr. We should srres:~ that [ormula (1) and the pa:rame~ers given in it

have been obtained after litt~ng aJ~d checking for many diflerent ~oalues of R¢~t, d, 4~, etc ~nd using several geometries for the tip and surface: ~'pherical, coslne-tike, 17arabolie, p&tne, etc. The resuhs we believe to be good to a 95%. which wouh~ predict d tmlues for the Irip-surface distanc,¢ within ~0.05 ,~, ,error for the model presented ir~ #g 1.

3, Comoar~son wiq~lpertm'b~,~on ~eory Te~;o.t~ and Hamann [13] have also presented a theory for STM based in the perturbative approach of Bardeen [14] giving quite interesting

N, G~:rrclaand F. FlcSres/ Scanning tunneling microscopy

139

i) Flat jellium surfaces. The advantage of this case is that o a r formula ,7 gives an exact result and can be checked against O'-r-H for which we can also calculate #(r0, EF) for the jellium model. In fig. 3 we have plotted in ordinates the values of p(z, EF) for LEv= 8 eV (in thick Iines'~ versus the distance ,:o the jetlium edge z. The verb' interesting thing is that

O(z, E~) = e - 2 ' ' ~ 2

l,

6

8

b}

,< -"-" 2 N

1

~.

~,

s

8 Reff(~, )

(4)

as one sheuld expect but with 2.14 in the exponential. Nodce that for a spherical tip r,r= .R +d. In fig. 3 we have also plotted cr versus d(A) lot R~¢I = 2.5, 5 and 10 A (thin lines) and we see that in all cases the lines are straight and para|tel, in the insert of fig. 3 we show the same plot in an augmented scale and it is quite clear that ~r :~ R~,, and not quadratic as Tersoff and Hamann [t3] claim. This result is also confirmed by recent work using exact and approximate formulas [ t l ] . From the above considerations we can nevertheless correct formula (3) to give the right answer as

Fig. 2. a) Conductivity in .O-I versus distance d for the schcnte of lig, t with L~ 10A and a spheric tio of 5.& radius. F,~,rmuta (la, b) are applicable ~rom d>0.5A.: b) Values of G(R,,fd versus R~tr. Notice that for 2.5 ,ik
~rc ~- 0.086R~,re-"4kR"po(ra, EF).

features. In this theory the conductivity reads as

ii) Corrugated jellium surfctces, This case can also be solved exactly for the tunnel conductivity and the values of p(x, y, z, E~) using the same numerical scattering technique [8-11]- As an example we have taken a simulation of the Au(110) ( t × 2 ) in which this surface is considered to be reconstructed ray a missing row model according to experimental r~alt'.~ [13]. The shape of the surface is tai~:en te be a jelliam of shape

trT_~, ~ 0..1R 2e-:*'~p(r,~, EF),

(3)

where t~(t0, ;EF) is tl-.e 'charge density' at the center of curvature c~f a spherical tip, for the electrons, at the Ferrr, i level per unit of energy and R is the tip radiu~ of curvature. In the above formula ,Or.r_. is given :n O -x, distances in atomic units and energy in eteetroe volts, in order to compare with our results we write our equation (la) in the same units as tr ~ 3.75 X t0-'~ R=f,,e-z'4k~.

(57

This formula and formula (lb) give the same answer and are an exact solution to the jellium flat sin*face and constant tunnel barrier &

h(x) =

2-/]" a

O.ia cos --- x

'

(,6)

O.b)

Now we proceed to compare both formulas for cr and ~-_~,.

with no corrugation in y direction. The surface periodicity is a = 8 . 1 4 A . The metal i~ at Z < h ( x , y ) and the vacuum at Z > h ( x , y ) and the

N. Galena and F. Fl6re,i. I Scanning tunneling microscopy

140

to2,

10" b

0

2

3

~

zG,)

..~~o

k

8

z

12

15

2o

zl~j

d(,~,:, Fig, 3. Value.~ of O(Z. Ev) (thick line) in a,u./eV or tr(D ~l) ~thin lines'~ verzus Z and d for a platte ~ufface of jdlium, Notice that all lines arc parallel. The insert is the sa1:0e in an augmented scale au~d ~ R~,v

interphase is described by an a b r u p t tunnel barrier 6 = 4 , 5 . Fig. 4 present by continuous thick lines the value of p(x -- 0, Z, E~) versus Z and by d a s h e d thicl.: lines O(x = a/2, Z, E F) versus Z. W e see that both fines are straight with small devia--

1(~

k

-\\ 2

z

6

8

z(~,}

I d-O.lal (2i3 Filz. 4. Values ,:ffp(x = O.Z. E v) and O(x '~ af2, Z, E~r}(thick couLinllotls arid dashed lines) a~d o'(O-l), for tile case A of Ihc in.~¢rl, in lhe thirl line. For ~:he case B the value of ¢r is Ill.it ;I S~.I'~liglll llne and now the Cl.tryglalflows from two Sil:esat the su':face. Nflticc tha~ lot a COl'ruRated surface We have actually Iwo 9il:.s checking each other and then R~ is the I'O[OVlllllparanlcter.

tions at large Z. The tunnel eonductivities for a spheric tip of R ] - - R ~ = 5 ~ w h e n the center of the tip is at x = 0 (case A in the insert) are given by the line indicated by G(R¢~r = 2.7 A). By using the corrected Tersoff and H a m a n n formula (5) for o,~, we again obtain curve G. Notice that n o w it is not clear h o w to use eq. (3) because we can see the p r o b l e m as two tips checking each o t h e r and it is quite reasonable that the relevant p a r a m e t e r is t h e effective radius of ~urvature given in eq. (2). In the case B of the in:~ert w h e n the tip c e n t e r of curvature is at x = a / 2 the situation is m o r e complicated because n o w the contribution to the current comes from two sites at the same tip as it is ~llustrated in t~g. 4. Nevertheless applying formulas ( l a , b) or (.5) we again obtain a good re:;ult taking into account that now for a given dislrance d (see fig. 4) t~erc: a p p e a r s two equal intensities I, so that o n e has to find the value of d such that the intensity at each site is half of the desired tunnel current.

4. Image p o t e n ~ A s we said in o u r introduction the electron will also 'see' il~ its tunnel path the image potential at

N. Uarcta and F. Fl6res / Scan)~ing lunnding microxcopy

long distance [ 1.5] and gaturates to the local density exchange and con'ela~l:~on potential near ~he surface. This implies ¢~at the tunnel barrier is a function of the distance d between the electrodes as well zs of tb.e (x, y) surface components (ib(x, y, d). Recent exl?edmental and theoretical work by Bhmig et al. ~5] have shown clearly that for a correct L,~erpretatiou of the vacuum tunholing experiment8 corrections due to the image potential have to be introduced. Because of the lack of space we cannot give here a complete account of ref. 5, but ir can be said that the ~/ dependence of the image potential (neglecting i~ first order x, y) reads

4~(d)=6a

(7)

d-l.~'

where &o(eV) is the work function of the metal, and ,~ = 9 . 9 7 e V A -~. The value 1.5 appears because the relevant parameter is the distance between the image planes [3] that ar,~ 0.75 ,~ from each jetlium edge. Experiments skew that ln(tr) versus d is a straigh~ line. This is in agreement with the theoretical view; by introducing <~(d) in (1) we find d(ln tr) = 2.14tb~ l: dd

x[l+

141

taiued are 0.45 ,~ and 1.4 ~ for the (1 x 2) and (1×3) reconstruction of A u ( l l 0 ) respectively. The theoretieal corrugations are in excellent agreement with theory.

5. Condasions From the above results we can conclude the following points: 1) Two simple precise formula ((la, b) and (5)) are given for the tunnel conductivity which show an exponential dependence for t~e shorter tip--surface distance. Those formulas m'e precise to a 95 A, of the conductivity and gi.ve a ~.etermination of the tib'-sudace distance within 0.05 error. 2) The conductivity varies linearly with the effective radius of curvature of the tip-surface system; i.e. g :~ R~# for a given d and ~b3) The classical image potential plays an important role in the correct interpretation of the vacuum tunnel experiments [5]. Tiffs image potentia~ changes into the local densiE¢ potential when the dista.,'~ce oetween the jellium edges is =~3.0 A.

References

~2

1

8 4 ~ ( d - 1.5)2 ~ - ~ ( ~ ) ] .

(8)

So in a first apprr,ximafion to 1/d the slope of the conductivity I,?gadthm is ff~Jz, At shorter distances formula '-.8) predicts an increase of the observed slope a,l~.:l thi:~ has been verified experimentaly in rzf. 5. We have ~nalysed the A u ( l i 0 ) (1. x2) and (1 × 3) surface by using the image potenti~d effect ~1] and found that for h~ving agreement with experimental d~ta [2] we need a cosine corrugation profile of ~.I ~. maxir.u;m to minimum and ~. tip radius of :ik5-4 i~ and the shorter distance between tip and suffaoe (minimum tip zo maxima s urtace x = 0) in the notation of section 3) d~,,-4 ~ between jell~um edge,~ that correspon ~ to - 5 ,~. between ~Lhetip jellium edge and the surface A u ions. The th~,'oreti¢:al corrugations ob-

[1] N. Garcfa, C. Ocal and F. Fl6rcs, Phys. I.,ctt, 50 (1983) 2002. [2.~ G. Biarfig. H. Rohrer. Ch. Gerber and E. Weiber. Appl. Phys. Lett. 40 (1982) 178; Phys. Rev. Len. e~9 (t982) 57; 50 (1983) 120; and recent resutts for Au(110): Sad. Sci. Lett. t26 {1983) 236; Holy. Phys. Acta 51; (I983)

726.

[3] N~D, Lang and W, Kohn, Phys. Rev, B 1 (1970) 4555; N.D. Lang. Solid State Phys. 28 225 (Academic Press. New York. 1973), 124] J ~6~ and F.K. Schultn, in: Solid Stlrface fhysics (Springer Tracts in Modern Physics, VOl. 85, ,979) Table 4-.3, lx 98. (5] G, Binnig, N Gate/a+ H. Rohrer, J,M. Soler and F. F[6res, Phys. Rev. B. (15 September,,1984, issue),, ~6] N. Garcla. J.A. Basket and I.P. Batra, J. Electron Spectroscopy 30 (1983) 137; Solid State Comm. d7 (I983) 485. [7] C. Kittel, Introduction to Solid State Physic, ~th ed. (Wiley, New York, 1971}Table 1, p, 248. [8] N, Garcia. J. Chvn,, Phys. 67 (1977) 897; N, GaWfam~d N, Cabrera. Phys. Roy, B 18 (1978) 576.

142

N. Garcfa and F. FIc~re~ / Scanning tunneling microscopy

[9] N. Garc~a, Optics Commun. 45 (1983} 1037. [101 It should be said that convergent results arc obtained for this p~oblem as well ~L~fOr llghz scatte~ng for much lar/~cr corrugation than for hard wall in atom-surface scatterin~ be~luse of the softness bound~'y conditions at the matching points. [11] E, StolL A, B=lratoff, A. Selloni and P. Camevali, J. of Phys. C 17 (1984) 3073.

~12] C.B. Duke, Tunneling in Solids (Solid State Physics, Suppl. 10) (Academle P~ss, New York, 1969)o [13] J. Tezsoff and D.R. Hamann, Phys. Ray. Lctt. 50 {1983) 1998. [14] J. Bardeen, Pl~ys. Rev. Lett. 6 (1961) 57. [15] M. Biittilccr and R. Landauelr, Phys. Ray. Lellt. 49 (1982) 1739. A. Puri and W.L. Schaich, Phys. ]?~cv.B 28 (1983) 1781.