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; LE'ITERS EUROPHYSICS LETTERS

Europhys. Lett., 3 (1), pp. 101':106 (1987)

1 January 1987

ag, Ber;lin)

1985)16\0;, 54 (1985)

107;JETP

207.

A Barrier Potential Calculation for Tunneling Electrons at aMetal-Metal Interface.

:Nf:~VLOS

P. DE ANDRÉS(*),F. FLORES(*),P. M. ECHENIQUE(**)and R. H. RITCHIE(***)

(*) Departamento de Física del Estado SólÜÜJ,Universidad Autónoma de Madrid,Cantoblanco, 28049 Madrid, Spain(**) Cavendish Laboratory, Madingley Road, Cambridge, CB30HE, U.K.Euskal HerriJco Unibersitatea, Quimicas, Donostia - Euzkadi, Spain(***) Health and Safety Research Division, Oak Ridge National LaboratoryOak Ridge, TN-37830, USA

)NEY J. V.

(received 12 May 1986; accepted in final fonn 28 August 1986)

PACS. 73.20. - Electronic surface states.PACS. 73.40G ;...Tunnelling: general.PACS. 73.4OJ - Metal-to-metal contacts.

~

Abstract. - The interface potential for tunneling electrons between two interface metals hasbeen obtained as the sum of a local and a nonlocal contribution, the last one introducing imageforce effects. Our results show that the image potential is the dominant effect for interfacedistances between metal surfaces greater than 10 Á. For smaller distances, local effects controlthe barrier height. We have calculated tms barrier height for distances between 5 and 10 Á andfound its value ranging between 1.2 and 2.4 eV in good agreement with other independentevidence.

A complete theory of the scanning tunneling microscope [1] needs a full description oí Olebarrier that electrons feel when they tunnel between the tipand the sample [2]. Although insome theoretical approaches [3] that barrier has been assumed to be independent oí thedistance between the two surfaces of the microscope, theoretical and experimentalwork [4,5] have shown that the image force plays an important role lowering substantiallythe interface barrier height for intennediate distances between two metals(=(10.0+15.0)Á). A theoretical analysis oítbis problem is stilllacking: this is a cumbersometask, since a simple local formalism cannot be applied to it. Indeed, the image force is wellknown to be a nonlocal effect associated with the correlation energy oí an electron leaving asurface. Altbougb different approacbes [6,7] are available to calculate tbis image potential,local effects are not negligible for tbe distances of interest between tbe' tip and tbe sample

. (= 10 Á) in tbe microscope~ The difficulty of calculating the barrier beigbt is, tben,associated witb tbe necessity of including in tbe calculation, at the same time, local andnonlocal correlation effects. At present, tbere is a possibility oí performing tbis calculationby using a local density. fonnalism, supplemented witb a prescription to describe tbe

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102 EUROPHYSICS LETTERS

nonlocal correlation energy for electrons moving away from a surface [8, 9]. This is acumbersome self-consistent calculation, and in this paper we have followed a differentapproach by introducing s()me reasonable approximations in the general expression for theself-energy of an electron moving at the interface.

Figure 1 shows OUTmodel for a Au-W interface. We have sirnulated the two metals byme~s of a jellium model (for Au we take Tg=3.02 and for W Ts=2:5). The electronic charge

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Fig. 1. - Jellium model for the Au-Winterface showing the electronic density, the image planes (i.p.),the jellium edges and the infinite barriers (Lb.) for each surface.

\ at the interface can be calculated by using a localdensity formalism; the results of Smith andFerrante [10] show that the electron density of that case is practically the superposition ofthe charges of the two independent surfaces. With thrs assumption, andusing now thevalues calculated by LANGand KOHN[11] we can obtain the local charge and the barrierpotential, Vxc(z) for the interface. In fig. 2, 3 and 4 we show these potentials for threedifferent distances between the two crystals; we take d =5.4 A, 7.6 A and 9.7 A, d beingthe distance between the irnage planes of both crystals [11]. Two comments are relevant atthis point: i) first, notice that the barrier is substantially decreased by the overlapping of themetal electronic charges; this creates a deeper' potential due to the local exchange andcorrelation energy. This result shows the irnportance of local effects in lowering the barrierheight for tunneling electrons. ü) On the other hand, OUTcalculationsshow that the distancebetween the points defined by the crossing of Vxc(z) and the Fermi l~vel is close to d asdefined above. Indeed, for the three cases, we find the effective length of the barrier to be5.7 A, 8.1 A and 10.1 A, respectively.

In order to calculate the full potential seen by a tunneling electron, we start byintroducing the nonlocal self-energy [12], I(r, r'; w). We are interested in calculating thisquantity at the Fermi energy, w = EF. Now, we notice that, at the interface, the electronicdensity, n(z), is rather low, and defines a local plasma frequency, wp(z), and a local Fermimomentum, kF(z). We find that the following condition is well satisfied: wp(z)>>k¡'(z)l2m.This suggests to use Hedin's approximation [12] for I(r, r'; EF), and write

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I(r, r'; EF)=Isx(r, r') + !ICh(r= r'; w= O)O'(r- r') ,

where ~sx(r, r') is the screened exchange interaction, independent of w, andICh(r = r'; w= O) the Coulomb-hole potential, given by the self-interaction of a staticcharge. Equation (1) holds if the time of response associated with the interface dielectric

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P. DE ANDRÉS et al.: A BARRIER POTENTIAL CALCULATION FOR TUNNELlNG ELECTRONS ETC. 103

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Fig. 2. - Potential barrier in the Au-W interface for a distance between the irnage planes equal to5.4 A. Operídots: localexchange and correlation potential. FuII dots: total potential including localandnonlocal contribution (dot:; correspond to the calcuIated points).

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Fig. 3. - Potential barrier in the Au-W interface for a distance between the irnage planes equal to7.6 A. As fig. 2 for the differents curves.

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104 EUROPHYSICS LETTERS

o

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Fig. 4. - Potential barrier in the Au-W interface for a distance between the image planes equal 109.7 Á. As fig. 2 for the differents curves.'.

function is short compared with the time an electr(m takes to travel an inter-electronicdistance. Two typical frequencies appear in the interface dielectric function: the surfaceplasma frequency, ws, and the local plasma frequency, wp(z); on the other hand, in thetunneling region an electron crosses the interface With a velocity given by [13] -y2mrp,where rpis the tunneling barrier height. Conditions for using eq. (1) can be written asfolIows: .

wp(z) or ws» y(2mrp)/rs or ki-(z)/2m .

-These equations are equivalent to the condition rs(z)« 40 for rp== (1+ 2) eV, which is welIsatisfied if d~ (10+ 15) Á. At larger distances, however, local effects start to be negligible, .

the only significant frequency is ws, and eq. (1) is a good approximation to the self-energy.- Notice that the screened exchange self-energy is a very localized interaction. This

suggests to write eq. (1) as folIows:

2(r, r'; EF)==2sx(r,r')+ll'ch(r=r'; w=O, n(z»G(r-r')+

+ UXCh(r= r'; w= O)'-!XCh(r= r'; w=O, n(z»} G(r- r') , (2)

k;¡.;....~ ''1

where we have introduced the Coulomb-holeinteraction associated with the local densityn(z). The first two terms of eq. (2) are tbe local expression of the self-energy for an electrongas having the homogeneousdensity n(z). This \ocal contribution is a good approximation 10the local potential calculated by using the local density formalismoAccordingly, the last twoterms given by eq. (2) represent the spéci:ficcontribution associated with the nonlocal self-energy. This means' that the results calculated above for tbe local potential have to becorrected by the folIo"wg nonlocal form: .

vru(z) = 1{XCh(Z'= z; w = O)- l:Ch(ZI= z; w = O, n(z»} . (3)

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; LETIERS P. DE ANDRÉS et at.: A BARRIER POTENTIAL CALCULATION FORTUNNELING ELECTRONSETC. 105

With eq. (3), the problem oí calculating the nonlocal potential at the interface is reduced toobtaining the self-interaction oí a static charge. Obviously, in the limit d~ 00, the localterms are negligible and lEch(Z' = z; W = O) goes to the classical image potential.}JCh(Z' = Z; w = O) has been calculated using an infinite barrier model [14, 15] íor each metal,and by introducing a small uniform charge at the interface, trying to simulate the screeningeffect associated with the tails oí the electronic states (see fig. 5). The infinite barrier íoreach metal has been located imposing the condition that its iInage plane coincides with thelrnown image plane oí the metal [11]. On the other hand, the small uniform electronicchargehas been chosen to be the same as the one locally seen by the external charge. We have usedthe model shown in fig. 5 to calculate the self-energy, }JCh(Z = z'; w = O),oí a static charge at

Au

ll(K)

2

-d Z=O d

Fig. 5. - Model used to calculate the dielectric response in the Au-W interface. Different media arecharacterised by the bulk dielectric functions, t¡(k), t2(k) and t3(k).

,s equal to

(3)

a given positionz; this is equivalentto calculatingthe electrostaticpotential inducedby aunit charge at Z on itself. This has been performed by using, for metal! and 2, the infiniie-barrier model oí reí. [14], and íor medium 3 a semi-classicalinfinitebarrier modelwith theLindhard dielectric function corresponding to the electronic density at the point z. Weremark that, in this calculation, we extend the different region 1, 2 and 3 oífig. 5 to infiniteuniform media, each one having fictitious charges adjusted to give the adequate electrostaticboundary conditions at each interface (details will be published elsewhere; see also reí. [7],chapter 5).

We mention that we can only expect the simple model used here to be a goodapproximation íor points not to close to the metal surfaces. In our calculation, we have.onlyobtained the nonlocal contribution at the points oí the central region as shown in fig. 2, 3 and4; then, we have joined smoothly the calculated total potential to the local potential near thetwo surfaces (note that íor z approaching a surface, .ECh(z'= z; w = O) goes to}JCh(Z'= Z;w = O,n(z») and the nonlocal contribution goes to zero).

The results oí our calculations are shown in fig. 2, 3 and 4. In these figures we show thelocal interface potential, and the total potential, obtained by adding to the localpotential thenonlocal contribution. Notice that in the three cases presented in this paper, we find a smallwiggle in the total potential near the middle oí the interface; tbis is an effect which in ourmodel is due to the assymetry oí the interface, since W and Au have been modelled bydifferent electronic densities. In particular, we find the nonlocal correction to the totalpotential to be more important at regions oí low electronic densities: this appears not exactlyat the middle oí the interface, but a little shifted toward the Au surface. A word oí cautionshould be added: these wiggles will be considerably reduced and may disappear if a self-consistent calculation is performed. The main results coming out oí our calculation are thefollowing: i) the nonlocal contribution is always negative and reduces tbe barrier height; ü)this nonlocal potential is a small fraction oí the correction due to the local poteiltial íord = 5.4 Á. Nonlocal eííects do not change very much in the range oí values oí d taken in thispaper; however, as local effects decrease with d, nonlocal effects become more important,

~lectronice surface.d, in the-y2nl':?,Titten as

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U density1electronmation to~ last twolocal self-¡ve to be

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106 EUROPHYSICS LE'ITERS

dominating the interface potential for large d. In particular, for d = 9.7 A, nonlocal effectsovercome the localones. ili) Finally, our results show that the barrier height is very muchreduced for the distances considered in this paper.

We have also calculated the effective barrier by means of the following equation:b

J~j(z) dz = (b - a) ~!ff ,

~:.::\

where ~(z) is the interface potential, and b and a are defined by the crossing of ~(z) and theFermi leve!. We have obtained for ~effthe following values: 1.2 eV, 1.9 eV and 2.4 eV ford = 5.4 A, 7.6 A and 9.7 A, respectively. These barriers are in very good agreement withthe ones given in the fig. 2 ofref. [4]ifwe assume that the barrier collapses for do=3.5 A (inref. [4] it was found do=3.4 A).

In conclusion, we have given a simple method to calculate the interface potential fortunneling electrons between two surface metals. Our results show the importance ofintroducing local and nonlocal effects to obtain the interface barrier. Image potential is thedominant effect for distances between metal surfaces greater than 10 A. For smallerdistances local effects control the interface barrier.

***

\

Partial financialsupport of this work is gratefu1ly acknowledged to the USA-Spain JointCommittee for Scientific and Technological Cooperation and the C.A.I.C.Y.T. One of us(PME) gratefully acknowledges help and support from Iberduero SA and the MarchFoundation.

REFERENCES

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[1] BINNINGG., ROHRERH., GERBERCH. and WEIBELE., Appl. Phys. Lett., 40 (1982)178;Phys.Rev. Lett., 49 (1982)57.

[2] BINNIGG., FRANKK. H.) FUCHSB., GARCÍAN., REIHL B., ROHRERH., SALVANF. and. WILLIAMSA. R., Phys. Rev. Lett., 55 (1985)991; ECHENIQUEP. M., FLORESF. and SOLSF.,Phys. Rev. Lett., 55 (1985)2348.

[3] TERSOFFJ. and HAMMAND. R., Phys. Rev. Lett., 50 (1983) 1999.[4] BINl\'IGG., GARCÍAN., ROHRERH., SOLERJ. M. and FLORESF., Phys. Rev. B, 30 (1984)4816.[5] GARCÍAN., OCALC. and FLORESF., Phys. Rev. Lett., 50 (1983)2002.[6] RITCHIER. H., Phys. Lett. A, 38 (1972)189;ECHENIQUEP. M., RITCHIER. H., BARBERANN.

.and INKSONJ., Phys. Rev. B, 23 (1981)6468;MANSONJ. R. and RITCHIER. H., Phys. Rev. B, 24(1981)4867. .

[7] GARcÍA-MOLINERF. and FLOims F., Introduction to the Theory of Solid Surfaces (CambridgeUniversity Press, Cambridge) 1979.

[8] GUNNARSSONO. and JONESR. O., Phys. Ser., 21 (1980)394;ALVARELLOSJ. E., TARAZONAP.and CHACÓNE., Phys. Rev. B, to appear.

[9] OSSICTh'IS., BERTONIC. M. and GIES P., Europhys. Lett., to appear.[10] FERRANTEJ. and SMITHJ. R., Phys. Rev. B, 31 (1985)3427.[11] LANGN. D., Solid State Physics, Vol. 28 (Academic Press, New York, N. Y.) 1973.[12] HEDINL. and LUNDQVISTS. O., Solid State Physics, Vol.23 (AcademicPress, New York, N. Y.)

1969, pp. 1-43. .

[13] BÜTTIKERM. and LANDAUERR., Phys. Rev. Lett., 49 (1982) 1739.[14] GARRIDOL. M., FLORESF. and GARcÍA-MoLINERF., J. Phys. F, 9 (1979) 1097.[15]RITCHIE R. H. and MARUSAKA., Surf. Sci., 4 (1966)234.

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