On the high voltage behaviour of STM distance-voltage characteristics

Solid State Communications, Printed in Great Britain.

~01.60,No.5,

ON THE HIGH VOLTAGE

pp.427-430,

BEHAVIOUR

0038-1098/86 $3.00 + .OO Pergamon Journals Ltd.

1986.

OF STM DISTANCE-VOLTAGE

CHARACTERISTICS

G.C. Aers and C.R. Leavens National

Research

Council

(Received

of Canada, Ottawa,

17 June,

Canada KlA OR6

1986 by R. Barrie)

We investigate the recent claim that a linear extrapolation from high characteristic bias voltages of a "blunt tip" STM distance-voltage We find both analytically and should pass through the origin. numerically that this is not true in general, even in the absence of three dimensional effects. This is unfortunate because it would have provided a simple and direct method for determining absolute vacuum However, our calculations indicate that within one dimensional gaps. tunnelling theory linear extrapolation from high voltages gives a voltage intercept at zero vacuum gap that differs by twice the difference in electrode work functions from the corresponding intercept obtained upon bias reversal. Hence, if the distance-voltage characteristics for both positive and negative bias have well defined linear regions at high voltages, free of significant contributions from Gundlach oscillations and three dimensional effects, it is still possible to obtain a good estimate of the absolute vacuum gap by extrapolation, provided the work functions are known and the bias can be reversed without the STM tip losing station.

detailed analysis of references 3, 7, and 8. It would not be so simple in practice because of an upward curvature in the characteristics at large voltages (i.e. large vacuum ga s) due to the finite curvature of the ti# *l% . Fortunately, the Gundlach oscillations in the d-V characteristic are much less pronounced than in the conductance and, at least for W-Au, die out rapidly with increasing voltage. Hence, for sufficiently blunt tips there should be an extended linear region at high, but not too high, voltages allowing reliable extrapolation to zero voltage. When the characteristic that we had already calculated for tunnelling from W to Au* was extrapolated linearly from high voltages it passed very close to the origin. However, the calculated characteristic for tunnelling from Au to W extrapolated to a voltage intercept at d = 0 of -1.6 volts, which we soon realized is very close to 2($W-@Au)/e where $w and oAu are the work functions used in the calculation and -e is the charge on the electron. This observation led us to take a close look at the asymptotic dependence of distance-voltage characteristics predicted by one dimensional tunnelling theory. Garcfa et al based their statement that extrapolation of distance-voltage characteristics from high voltages should pass through the origin on an approximate expression for the tunnelling current due to Simmonsll. However, Simmons used an average work function in its derivation so that situations in which the electrodes have significantly different work functions are not accurately described. For this reason we rely instead on the work of Gundlachl who presented an exact treatment of

Two decades ago Gundlach' predicted strong oscillations of the tunnelling conductance in the Fowler-Nordheim regime. These oscillations have recently been observed2P3'4'5 with the scanning tunnelling microscope (sTM)~. Using one dimensional tunnelling theory Becker, Golovchenko, and Swartzentrubes demonstrated the sensitivity of the oscillatory conductance to the detailed shape of the barrier potential. They then exploited this sensitivity to estimate absolute vacuum gap distances (the STM only measures changes in vacuum gaps) by fitting theory to experiment using the unknown distance offset as an adjustable arameter. Garcia, S6enz, Soler, and Garcia s and Leavens and Aerss improved their procedure, obtaining absolute vacuum gaps that are larger by a few angstroms. The agreement with experiment (for tunnelling from W to Au) is encouraging but discrepancies remain. These have been attributed to three dimensional effects7 *BP9 and the particular choice of image potential used in the calculations? (the dominant first peak in the conductance is particularly sensitive to the behaviour of the image potential near the vacuum-Au interface). Because of these discrepancies it is not clear just how reliable the estimated vacuum gaps are. Hence, we were intrigued by the claim of Garcia et al7 that the constant current distance-voltage (d-V) characteristic of a "blunt tip" STM at high bias voltages should extrapolate linearly through the origin. If this were indeed the case then one could immediately obtain the absolute vacuum gap d(V) simply be extrapolating the measured characteristic Ad(V) = d(V) - doffset at high bias to zero voltage,

completely

bypassing

the 427

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428

BEHAVIOUR

OF STM DISTANCE-VOLTAGE

the tunnelling of electrons through a trapezoidal barrier allowing for different work functions in the two plane and parallel, freeThe parameters electron-metal electrodes. describing the trapezoidal barrier are defined tunnelling in Fig. 1. For the Fowler-Nordheim regime (eV>eR+sZ) Gundlach presented an approximate expression (equation (24) of reference 1) for the ratio of transmitted to incident current density for an electron incident on the barrier from the left electrode. Substituting the large bias limit of this expression in the standard formula (equation (1) of reference 1) for the zero temperature tunnelling current density J(V) we obtain

where A$ : QL - $R and m is the free electron Gundlach's equation (24) is accurate mass. provided that

CHARACTERISTICS

Vol. 60, No. 5

istics considered here. It follows immediately from (1) that the high voltage constant current distance-voltage characteristic for a trapezoidal barrier is given by d = constant

(eV+A$)

(3)

when the conditions (2) are satisfied. Extrapolating this expression linearly to d=O we obtain the voltage intercept V. = -A$/e which is zero only if the two electrodes have the same work function. To check this simple result we computed distance-voltage characteristics for tunnelling through a trapezoidal barrier by accurate numerical solution of the SchrEdinger equation as was done in references 7 and 8. The

results for tunnelling from W into Au and Au into W are shown in Fig. 2 (to compress the figure the electrodes have been switched instead of reversing the bias). These characteristics extrapolate linearly from high voltages to give voltage intercepts very close to (4,-$,)/e and (OW-$,)/e respectively, as predicted. Even if one questions our objectivity in extrapolating these characteristics, it is obvious that one cannot consistently extrapolate both curves through the origin. Unfortunately, the calculations with a trapezoidal barrier are primarily of academic interest because the image potential has a very smficant effect on both the conductance and the d-V characteristic. Since the image potential lowers the barrier a larger vacuum gap is required to maintain the same constant current. Hence it is most unlikely that the voltage intercepts will not shift when the image The potential is included in the calculation. results of such a calculation are shown in Fig. 3. The extrapolation from high voltages is much

These inequalities are satisfied at high voltages for the distance-voltage character-

r d = 15 Angstroms

d

No Image Potential 0

5

10

15

20

z (Angstroms) Fig. 1. Barrier potential U(z) with (solid curve) and without (broken curve) the multiple image potential 1.15 e*d In 2/4z(d-z) suggested by Simmonsll but with the singularities at the vacuum-electrode interfaces at a=0 and z=d removed by joining it smoothly and linearly to the inner potentials at these points. @L and @R are the work functions of the left and right hand electrodes; EF and EF are the R L In the following corresponding Fermi energies. figures eV is regarded as a positive quantity so that electrons tunnel from left to right. Instead of reversing the bias we interchange the The parameters used in the two electrodes. figures are $W = 4.55 eV, @Au = 5.37 eV and EF = 5.51 for both W and Au. In this figure the left electrode is W and the right one Au. ~a is the energy associated with motion perpendicular to the plane of the barrier measured downwards from the Fermi level of the left hand electrode.

0 -5

.* 0

5

Bias Voltage

10

15

20

(Volts)

Fig. 2. Constant current distance-voltage characteristics for tunnelling from tungsten into gold (W-Au) and gold into tungsten (Au-W) The dashed lines through a trapezoidal barrier. are linear extrapolations from high voltages. The vertical arrows mark the predicted The tunnelling current is 1 intercepts at d=O. nA for an assumed effective tunnelling area of 200 82.

Vol. 60, No. 5

HIGH VOLTAGE

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Image Potential 40 1 30 20 1

Fig. 3. Constant current distance-voltage characteristics for tunnelling between tungsten and gold. The trapezoidal barrier has been reduced by a multiple image potential. The dashed lines are linear extrapolations from high voltages. The horizontal two headed arrow is of The tunnelling current is 1 length 2(0Au-@W). n.4 for an assumed effective tunnelling area of 200 82.

easier because the image potential has significantly damped the Gundlach oscillations. In addition, the voltage range has been extended by an additional 5 volts to make the extrapolation even more certain. The characteristic for tunnelling from W into Au does have a voltage intercept very near the origin but that for Au into W is displaced by an amount very close to 2(@,*,). We repeated the calculation assuming a much larger difference, 2eV, in work functions and found that the voltage intercepts differed by very close to 4 volts. The predicted voltage intercept is independent of current density for the trapezoidal barrier (provided (2) holds). Figure 4 shows that this remains approximately true when a multiple image potential is introduced. Multiplying the current by factors of 10 and l/10 leads to at most only small changes in the voltage

60 , 50

-I Image Potential

I /i

Fig. 4. Distance-voltage characteristics for tunnelling from gold into tungsten at constant currents of 0.1, 1, and 10 nA (upper, middle, and lower solid curves respectively) for an assumed effective tunnelling area of 200 a*. The dashed lines are linear extrapolations from high voltages.

CHARACTERISTICS

429

intercepts. If the shift of the extrapolated voltage intercept of 2((1~-eR) upon reversal of bias is a general result, as appears to be the case, then it could be used to give a simple estimate of the distance offset for characteristics obtained with a sufficiently blunt STM tip. Of course, it would be necessary to know the work function difference and to ensure that-there was no change in the offset distance during bias reversal. The latter would involve reprogramming the constant current feedback mechanism while the tip remained stationary to allow for the change in direction of the current. In reference 8 the calculated and measured d-V characteristics were compared after choosing the offset distance for the experimental one3 so that the curves matched at 10 volts. There was then an extended region of good agreement where expected: poor representation at somewhat lower voltages could reasonably be attributed to an inadequate choice of image potential and the increasing discrepancy at higher voltages to the finite curvature of the blunt tipg. If this interpretation is correct and if the latter source of discrepancy can be considerably reduced by producing blunter tips then the method proposed here should give good estimates of absolute vacuum gaps. WyderX;cently van de Walle, van Kempen, and determined the structure of a tungsten STM tip by pushing it carefully into the surface of Ag single crystal and comparing two dimensional STM scans of the indented area taken before and after. In the context of the present paper, it would be interesting to repeat their experiment for a series of STM tips after measuring d-V characteristics for each to very high voltages. The object of the experiment would be to look for correlations between tip structure and discrepancies between measured and calculated characteristics, particularly at high voltages. If these discrepancies are in fact due to the three dimensional nature of the tip79**g they should be larger for the "sharper" tips. Three groups3 P7 Se have reported calculations of the Fowler-Nordheim conductance in promising agreement with experiment3 for tunnelling from W into Au. However, the work functions used are quite different: Becker et alg, Garcia et a17, and Leavens and Aers* used 4.8, 4.3, and 5.37 eV (the literature value) for @Au(llO) and 3'g, 5.0, and 4.55 eV for I$~(?), respectively. (Note in particular that I$~ -$I~ = -0.7 eV in reference 7 and +0.8 eV in reyerence 8.) The tungsten tip used in the experiment was poorly characterized (the tunnelling may even have been through a gold atom or cluster captured by the tip) so that it is not at all clear what work function was appropriate. On the other hand, the Au surface was well characterized and the one volt spread in the choice of work functions is particularly disturbing. Moreover, since the peak positions in the conductance shift as the value of the constant current is varied, the choice made for the effective tunnelling area influences the agreement between theory and experiment to some extent. Hence, to narrow the possibility that the present encouraging agreement is fortuitous it would be advisable to enlarge the data base

430

HIGH VOLTAGE BEHAVIOUR OF STM DISTANCE-VOLTAGECHARACTERISTICS

in future experiments by using several, widely different, constant current values and by reversing the bias for each, preferably in such a way that there is only one unknown distance

Vol. 60, No. 5

offset for all of the data. This would provide a much more stringent test of the theory without allowing the introduction of further adjustable parameters.

REFERENCES 1. K.H. Gundlach, Solid State Electron. 2, 949 (1966). 2. G. Binnia and H. Rohrer. Helvetica Phvsica Acta z,-726 (1982). . 3. R.S. Becker, J.A. Golovchenko and B.S. Swartzentruber,Phys. Rev. Lett z, 987 (1985). 4. G. Binnig, K.H. Frank, H. Fuchs, N. Garcfa, B. Reihl, H. Rohrer, F. Salvan, and A.R. Williams, Phys. Rev. Lett. z, 991 (1985). 5. R.S. Becker, J.A. Golovchenko, D.R. Hamann, and B.S. Swartzentruber,Phys. Rev. Lett. E, 2032 (1985). 6. G. Binnig, H. Rohrer, Ch. Gerber, and E. Weibel, App. Phys. Lett. s, 178 (1982);

7. a. 9. 10. 11. 12.

Phys. Rev. Lett. 2, 57 (1982); Phys. Rev. Lett. x, 120 (1983); Surf. Sci. 131, L379 (1983). R. Garcfa, J.J. Sahnz, J.M. Soler and N. Garci'a,J. Phys. C E, L131 (1986). C.R. Leavens and G.C Aers, Solid State Commun., to be published. R.S. Becker, private communication. R. Young, J. Ward, and F. Scire, Rev. Sci. Inst. 2, 999 (1972). J.G. Simmons, J. Appl. Phys. 34, 1793 (1963). G.F.A. van de Walle, H. van Kempen, and P. Wyder, Surface Science 167, L219 (1986).