- Email: [email protected]
Field-emission resonance measurements with mechanically controlled break junctions
Physica B 291 (2000) 246}255
Field-emission resonance measurements with mechanically controlled break junctions O.Yu. Kolesnychenko , Yu.A. Kolesnichenko, O.I. Shklyarevskii , H. van Kempen * Research Institute for Materials, University of Nijmegen, Toernooiveld 1, NL-6525 ED Nijmegen, The Netherlands B. Verkin Institute for Low Temperature Physics & Engineering, National Academy of Science of Ukraine, 47 Lenin Av., 310164, Kharkov, Ukraine Received 1 August 1999; received in revised form 20 December 1999; accepted 20 December 1999
Abstract The additional information provided by "eld emission resonance spectra signi"cantly increases the potential of the mechanically controllable break junction technique. We have found that the pronounced three-dimensional nature of the electrodes results in an extremely high sensitivity of the "eld emission resonance spectra to the "ne details of the surface geometry, electronic structure and electric "eld distribution at small electrode separations. At larger distances a quasiclassical approximation can be used for the determination of the metal work function, precise calibration of the electrode relative displacement and discrimination between `blunta and `sharpa electrodes on the basis of the distance}voltage z(<) dependence. 2000 Elsevier Science B.V. All rights reserved. PACS: 73.40.Gk; 73.90.#f; 73.30.#j; 79.70.#q Keywords: Break junctions; Vacuum tunneling; Field emission; Gundlach oscillations; Work function
1. Introduction In a rather short time since the mechanically controllable break junction (MCBJ) technique [1] (based on the early work of Moreland and Ekin [2]) was "rst reported it has become not only an excellent complementary method to both pointcontact spectroscopy and scanning tunneling microscopy (STM) but can also be regarded as an independent and extremely powerful tool in itself. * Corresponding author. Tel.: #31-24-3653499; fax: #3124-3652190. E-mail address: [email protected] (H. van Kempen).
This technique is the most promising for point contacts of submesoscopic size with which the interaction of electrons with an individual magnetic impurity [3] or single two-level #uctuator [4}6] (see also comment by Ref. [5] and reply by Ref. [6]) has been observed and conductance quantization in contacts of only a few atoms has been reported [7]. In the tunneling regime the exceptionally high stability of MCBJ has been exploited to investigate electrode interaction at ultrashort ((3 As ) distances [8]. The deviation of the current}distance dependences from exponential behavior due to the shape of the electrodes [9] or to adsorbed He atoms [10]
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 8 8 4 - 7
O.Yu. Kolesnychenko et al. / Physica B 291 (2000) 246}255
has been studied. In addition, with only a minor loss of junction stability MCBJ can be used in quasi-STM mode for scanning very small ((1000 As ) areas of the electrode surface [8,11]. Up to now all experiments with MCBJ in the tunneling regime were performed at relatively low bias voltages < . Here we present measurements at < much larger than the work function of the metal under investigation, emphasing the junction conductance oscillation e!ect, also known as Gundlach oscillations [12] or "eld emission resonances (FER). For vacuum gap tunneling this e!ect was observed for the "rst time by Binnig and Rohrer [13]. The ensuing publications [14,15] gave rise to a considerable theoretical interest [16}20]. The most detailed and thorough measurements of Coombs and Gimzewski [21], extending to large quantum numbers and gap spacing, revealed that the "ne structure of the oscillations is extremely sensitive to the electronic and physical structure of the emitting surface. This fact stimulated our interest in that the properties of the MCBJ electrode surfaces are not well characterized in general. Recent MCBJ experiments in the "eld emission range were aimed at precise calibration of the distance between the electrodes, accurate determination of the absolute vacuum gap and the metal work function [22] and investigation of the in#uence of adsorbed helium on the latter [23]. In the present paper we have concentrated mostly on deviations from standard behavior in FER spectra and voltage}distance dependences. These are related to the essential three-dimensional nature of the problem.
2. Theory In the Fowler}Nordheim regime [24] when the applied voltage < is larger than the work function
of the metal, part of the vacuum gap (Fig. 1a) becomes classically accessible. The wave function of an electron in this positive kinetic-energy region is determined by superposition of the transmitted wave and the wave re#ected from the potential step at the collector interface. The transmission coe$cient has a maximum if the electron energy coincides
247
Fig. 1. (a) Energy diagram for "eld emission. d-separation between electrodes, z -classical turning point, -work functions M of the electrodes. (b) Typical sample mount consisting of a bending beam with the sample wire attached to it. The dimensions indicated are referred to in the text. For clarity, all distances have been exaggerated. A bending force causes a vertical displacement y, which leads to a distance z between the electrodes.
with one of the virtual energy levels in the potential well formed by the barrier potential ;(z, <) and the potential step at d. In a quasiclassical approximation it is equivalent to the Bohr}Sommerfeld quantization rule for the shadowed area in Fig. 1a.
(2m B dz (e !;(z, < )"pn. X L
X
(1)
Here e is the electron energy in the direction X normal to the metal surface, ;(z, < ) is the potenL tial energy of an electron inside the barrier, m is the electron mass, n is a positive integer and is the Planck constant. It must be emphasized that there is no barrier at the point d and the electron energy spectrum is continuous. The ful"llment of Eq. (1) means that at <"< standing waves arise beL tween the surface of the collector and the classical turning point z resulting in an oscillating depend ence of tunneling probability and therefore of tunnel current as a function of the applied voltage <.
248
O.Yu. Kolesnychenko et al. / Physica B 291 (2000) 246}255
However, the quasiclassical approximation is inapplicable for calculation of oscillation amplitudes since it calls for the determination of the transmission coe$cient D(e ), including the pre-exponential X factor. Our further calculations are aimed at investigating the analytical expression for the tunnel current in the "eld emission region and its dependence on the main parameters involved. Let us consider the situation when e is relatively far from its limiting X values e "k# and e "k# !e< where k is the Fermi energy and are the work func tions of the left and right electrodes. The transmission coe$cient can be written in the form [10] D(e )"8 exp(!f ) X (e (e #e<)( #k!e )(e #e
Here
4 k# !e X f (e )" <1, X 3 eFl
4 e
I"I
$,
k#e< e
1 ; 1! ; 1#(e
f (k)
4 f (k)" , 3 eFl
e
where I is the Fowler}Nordheim tunnel current $, [24]: (k eF " exp+!f (k),. (7) $, k# (2p)
It should be noted that the oscillation amplitude (Eq. (5)) is rather large:
I
k#
<1. /I K
e
At high voltages e
I
(3)
F"(e<#* )/de is an e!ective electrical "eld and l\"(2m/ )(e<#* )/d. At low temperatures and e<'k the tunnel current is determined by the following expression:
em I I" de (k!e )D(e ). (4) X X X 2p The tunneling probability has a maximum for electrons with energy e Kk moving perpendicular X to the boundaries. If the characteristic energy range *e& /f to an exponential decay exp(!f ) in the tunneling probability is much smaller than k we can present Eq. (4) in the form
exp(!f ). Taking it outside of the integral sign and expanding f (k!g) in series around g"0, one can obtain after integration
em IK dg gD(g), f <1, (5) 2p where g"k!e . At relatively small voltages X (e
I"I
$,
k#e< 1 (k# ) (k#e< 1! e
2 (e
;sin(f (k)#U) , B
(8)
k#
<1; and e
2( (e
f (k)
This result shows that at e
O.Yu. Kolesnychenko et al. / Physica B 291 (2000) 246}255
the electrode separation. While the monotonic part decreases by a factor of e as the distance between the electrodes increases by *d +d/f , the phase of the oscillations changes its sign when the barrier width changes by *d +d/f ;*d . This means B that the interference of contributions to the tunnel current from the di!erent regions of the real 3D junction with only slightly di!erent gaps may result in a considerable suppression of the FER amplitude. This result provides us an important criterion for analysis of the in#uence of the three-dimensional electrode geometry on the FER spectra. Eq. (8) also permits the calculation of junction conductance at high voltages. Taking into account that the main contribution in dI/d< provides differentiation of rapidly changing functions of voltage exp(!f ) and sin(f ) one "nds that B
dI 3 f (k) 1 k#
cos f (k) " I 1! B 4 ( e< d< 2 < $, f (k)
e<<
(9)
, k.
The junction conductance peaks when
4 (2m (e< ! ) 1 L f (k)" "2p n! ; B 3
eF 4 n"1, 2, 32 ,
(10)
where < is the bias voltage at which the nth L maximum occurs. It is obvious from Eqs. (6) and (8) that by keeping I constant in the course of measurements we are also able to maintain the e!ective electrical "eld strength nearly constant at large voltages. Assuming FKconst and (n#) +n at n<1, one can "nd from Eq. (10):
3p e e< " # F n . L 2(2m
(11)
Not surprisingly, under the assumption made above we arrive at the same result as in Ref. [19] where it was obtained from the simple quasiclassical approximation. It should be realized that all calculations presented above were made using a simpli"ed model of a one-dimensional tunnel junction with a trap-
249
ezoidal barrier. In reality some additional factors are of considerable importance. (i) Image forces change the overall shape of the tunneling barrier and thus the amplitude of the tunnel current and position and period of dI/d< oscillations. Even including an approximate expression for the image potential into the SchroK dinger equation makes analytical solution of the problem impossible. Numerical calculations of FER spectra [17,19,20] demonstrated that inclusion of an image force in#uences results in an appreciable (10}20%) shift in the position of the "rst maxima towards the lower energies. Note that distortion of the barrier by image forces occurs mainly in the region close to the boundaries, a!ecting small-n oscillations. As n increases the relative contribution of this area in integral (1) decreases and the in#uence of image forces on Gundlach oscillations corresponding to larger n drops noticeably. As calculations [17,19,20] have been made for speci"c material parameters they have little relevance for improving the determination of d and
for an arbitrary junction. A more attractive suggestion [25] takes into account the image forces by introducing e!ective work functions H and elec trical "eld EH in the framework of a trapezoidal barrier model. (ii) In contrast to planar M}I}M junctions, the electric "eld distribution for both STM and MCBJ con"gurations is essentially three dimensional. However, because of the exponential dependence of the tunneling probability on the barrier width, the main part of the tunnel current #ows through the electrode region with the smallest gap. This makes the problem quasi-one-dimensional and the monotonic part of the tunnel current can be described by an expression analogous to the Fowler}Nordheim formula [24] with an additional coe$cient depending on the exact shape of the junction and proportional to the e!ective tunneling area [16,17,19].
3. Experiment In our experiments we used the traditional sample design presented in Fig. 1b and described
250
O.Yu. Kolesnychenko et al. / Physica B 291 (2000) 246}255
elsewhere (see e.g., Refs. [1,9]). It consists of a 50}250 lm "lament glued using two small drops of hard epoxy (Stycast 2850 FT) onto a phosphor}bronze bending beam which is covered with a 70 lm insulating layer of capton foil. The wire (Pt, Au, Cu 99.999% purity or Ni, Co, Fe, Be, Pb and Mg 99.99% purity) is then notched at the midpoint of the glued section for about 80}90% of its diameter. A narrow (300}400 lm) strip of Dy 50 lm foil was treated the same way. After mounting in a special setup [9] and cooling, the sample is broken at 1.2}4.2 K in a high vacuum environment by bending the substrate against the counter supports C, thus creating two atomically clean electrodes. The relation between the vertical displacement of the piezodriver D and the electrode separation D is W X determined by the length ¸, width and thickness of the bending beam, distance between anchoring points of the wire s and slope a at the "xation point. Additional corrections arise if the break does not occur exactly midway along the glued section (for detailed calculations see Ref. [9]). The approximate nature of this relation and uncertainty in determination of a allows only a rough estimation of D /D . This ratio usually ranges between 300 and W X 500 and can reach 3000 for lithographically fabricated and thin "lm break junctions [26}28]. Typically, we try to keep the distance between the anchoring points as small as possible (0.2}0.3 mm) to ensure high stability of the device. For FER measurements we deliberately increased s up to 0.5}0.7 mm (reducing D /D below 200) in W X order to cover a gap spacing of about 100 As with the piezodriver. The oscillatory behavior of the tunnel current for e<' can already be seen in a Fowler}Nordheim plot ln(I/<) versus 1/< and even more clearly observed in a logarithmic plot of dI/d<(<) taken at a "xed electrode separation. However, only the "rst two or three peaks can be observed at constant distance or at constant bias (by varying the vacuum gap alone) mode because of the limited dynamic range of the measuring equipment. In addition the rapid decrease of amplitude makes measurements of oscillations with large quantum numbers practically impossible. Following [14] the oscillating part of the junction conductance dI/d<(<) was measured in a con-
stant current (I"0.5}1 nA) mode by adding a feedback ramp with a relatively slow response for adjusting the gap distance between the MCBJ electrodes. A modulation voltage of frequency 1000 Hz and 200}300 mV amplitude was superimposed on the bias voltage signal and the conventional lock-in technique was used for the determination of dI/d<(<). The sweep time for the scan from $1 V to $25}35 V was 1000}2000 s which gives the possibility to use a lock-in time constant of 0.3}1 s. The piezovoltage signal < (<) proportional to the gap spacing was measured simultaneously. To change the bias polarity the feedback circuit was switched o! at an electrode separation of 20}50 As . The dI/d< curves can be reproduced extremely accurately for 1 h, which demonstrate the high junction stability and testi"es that the lateral drift of the electrodes during this time is negligibly small. In order to `changea the junction we modi"ed the surface relief of the electrodes by controlled `crashinga using either a piezodriver to create a point contact with resistance in the 10}10 ) range (estimated indenting of electrodes into each other 5}10 As ) or a di!erential screw (indenting up to 100}200 As ). We observed oscillation of the tunnel current for all materials investigated. The best results were obtained for platinum, cobalt, beryllium and gold MCBJ. Experimental data for more than 600 junctions from about 50 di!erent samples were analyzed.
4. Results and discussion All STM measurements of Gundlach oscillations were carried out using a `blunta tip with a rather large radius against an atomically #at (over 10 As ) sample surface [14,21] in order to make the situation as close as possible to that in planar junctions. In spite of the fact that the surfaces of the MCBJ electrodes are very irregular [8,11] and essentially three dimensional, a surprisingly high percentage of dI/d<(<) curves (more than 90%) demonstrate oscillatory behavior in a voltage range up to 30}35 V with up to n"35}45 distinct peaks (Fig. 2). However, only a small part of FER spectra (:10%) display the theoretically predicted gradual
O.Yu. Kolesnychenko et al. / Physica B 291 (2000) 246}255
decrease in oscillation amplitude (curve 1 at Fig. 2). The < (<) (distance}voltage) dependences for such spectra show marked steps around the conductance maxima up to 12}15 V (see inset in Fig. 2) and nearly linear increase at higher voltages. The vast majority of spectra shows variations of the peak amplitudes (curves 2}4, Fig. 2) which is reminiscent of `beatinga or `wave packetsa. This e!ect was probably "rst observed in Ref. [14] as a disruption in the periodic structure of the dI/d<(<) dependence and attributed to the re#ection associated with atomic planes below the surface. Numerical calculations [19] for tunneling from four contributing areas with slightly di!erent gaps revealed not only a more rapid decay of the oscillation amplitude than was predicted for a single area, but also its non-monotonic behavior. Since the numerical analysis in Ref. [19] was restricted to the "rst 7}8 peaks authors did not pay due attention to this fact. The "ne structure of FER spectra was studied in great detail by Coombs and Gimzewski [21]. They suggested that in their ex-
Fig. 2. An example of "eld emission resonance spectra for Pt MCBJ showing continuous decay (1) and strong variation of the oscillation amplitude (2}4). The inset shows a distance}voltage dependence z(<) with distinct steps, related to the conductance maxima.
251
periments with a blunt tip, `beatinga arises as a result of electron waves interfering from the patches of the surface with di!erent work functions. In extreme situations (3}10% of all curves depending on the MCBJ material) the destructive interference results in a completely irregular spectrum with a majority of the peaks being suppressed. The processing of FER spectra with a heavy `beatinga is more complicated as some of the peaks can be missed and proper indexing must be restored to eliminate steps in the < (n ) dependence (for L a more comprehensive description see Ref. [22]). The problem of adequate peak numbering for the low index peaks, however, is far more complicated. Fig. 3a shows two FER spectra for a Co MCBJ
Fig. 3. (a) Two FER spectra for a Co MCBJ taken with a 3 h interval for the same junction. An extremely small lateral shift of electrodes (0.2}0.3 As ) results in the appearance of an extra maximum on curve 2 (dashed line) between 5 and 11 V whereas above this interval the curves remain unchanged. (b) Position of the maximum in the di!erential conductance versus the order of the resonance to the power two-thirds for curves 1 and 2 of Fig. 3a. Incorrect peak indexing for curve 1 (open circles) results in an unrealistically high value for the Co work function (5.55 eV).
252
O.Yu. Kolesnychenko et al. / Physica B 291 (2000) 246}255
measured sequentially for 3 h. The estimated lateral shift of the electrodes during this time does not exceed 0.3}0.5 As but its impact on the d
"4.5$0.1 eV which is very close to the conventional value of 4.41 eV [29]. The deviation of the peaks with small quantum numbers from the linear dependence (Fig. 3b) in MCBJ experiments is much larger than for the STM `blunt tip-#at samplea con"guration [21]. It can be explained by the fact that the 3D nature of the break-junction at relatively small distances is especially pronounced and "rst oscillations are more responsive to the geometry of a junction. First of all, at these separations the distortion of a trapezoidal barrier by image forces is much more signi"cant. Secondly, for this vacuum gap range the condition F"const most probably does not hold anymore in constant current mode. As can be seen from Eq. (6), at bias voltages comparable to /e the Gundlach oscillation is essentially nonharmonic. Finally, the phase shift between electron waves from tunneling areas with di!erent gaps (or di!erent work functions) decreases in inverse proportion to the electrode separation and is largest for the "rst peaks. All these factors make the analysis of this part of the FER spectra extremely di$cult. We made an attempt to detect missing or extra peaks using results obtained by numerical solution of the SchroK dinger equation. According to Ref. [19] the position of the "rst oscillations is linear in coordinates < versus n . But the "t to the experiL mental data (Fig. 4a) seems equally good for both Co FER spectra and the absent peak does not manifest itself in any manner. To avoid complications with peak indexing it was suggested [21] to use a *<\(<) plot (where *< is the voltage separation between the two ad-
Fig. 4. (a) position of the "rst maxima versus the square root from the order of resonance for Co FER spectra presented in Fig. 3a. The open circles correspond to curve 1 and the closed circles to curve 2. (b) *<\(<) dependencies for the "rst 12 peaks of the same FER spectra. In both cases there is no clear indication for the `misseda peak in curve 1.
jacent peaks) for analyses of FER spectra. Sharp spikes occur in this plot at voltages where the normal sequence of peaks is disrupted by interference. Unfortunately, this method is very unsuitable for analysis of small-n oscillations as the scattering of points in this range is too large to draw a de"nite conclusion if one of the peaks is really lost (Fig. 4b). The linear "t interception with the x-axis gives the approximate value of the work function but its accurate determination is highly conjectural. It is necessary to stress that the position of the "rst peak in the FER spectra < remains remark ably stable during the small changes of the surface relief or a local work function (caused e.g. by bringing electrodes together) whereas the overall shape of the spectrum can be drastically a!ected. This can be explained by the fact that at I"0.1}1 nA the "rst peak occurs for the majority of metals at separations of +7}10 As . At these distances almost all
O.Yu. Kolesnychenko et al. / Physica B 291 (2000) 246}255
the tunnel current is determined by the foremost atom or clusters of atoms and contributions from areas with larger gaps or di!erent are negligible. Using data for the best FER spectra we were able to draw an empirical relationships between the "rst peak position and the work function for some metals. For example, "(0.88$0.04) e< for Pt and "(0.84$0.04) e< for Be. The scattering of the data can be related to the dependence of the "rst peak position on the e!ective tunneling area as pointed out by Bono and Goods [19]. Nevertheless, these relationships can be used as a very reasonable estimation for the local work function. The absolute vacuum gap (an o!set of z(<) dependence) can be found by the integration of the one-dimensional SchroK dinger equation [14] to yield the constant transmission probability for the experimentally observed relative displacement of the tip corresponding to a "xed tunnel current. An alternative way of determining the absolute vacuum gap was suggested in Ref. [16] but has never been tested experimentally. Using z(<) dependencies taken in both polarities for two signi"cantly di!erent values of tunnel current (0.5 and 0.03 nA in our case), one can "nd the absolute vacuum gap as an interception point of the linear part of the curves for di!erent I (Fig. 5). The distance between these points in `#a and `!a polarity on the <-axis gives us 2* . We have found, however, from our data that the value of absolute vacuum gap (+16}18 As ) is larger than expected by a factor of 2 and * +1.1 eV di!ers from that determined using FER spectra by a factor of 5. It should be noted that the accuracy of the suggested method is not very high as it requires linear z(<) dependencies over a large voltage range. The greatly exaggerated value for the gap and * can be explained by the fact that for smaller current (at the same voltage) the e!ective tunneling area is larger and the corresponding z(<) curve rise more steeply [18], pushing the interception point further than expected. These considerations are also accountable for much larger * . Most probably this method yields a vacuum gap `averageda over all tunneling areas and the result depends on the exact shape of the electrodes. In the "eld-emission experiments with an arti"cially blunted tip in the STM con"guration the main
253
Fig. 5. Distance}voltage characteristics for a Pt MCBJ in positive and negative polarity taken in constant current mode at 500 and 30 pA. According to Ref. [16] the interception of linear extrapolations from high voltages (dashed lines) in each polarity gives the position of an absolute vacuum gap, and the voltage di!erence between intercepts at d"0 for `#a and `!a polarity determines 2* . The three dimensional character of the MCBJ leads to the largely exaggerated values.
reason for distance}voltage curve z (<) asym! metry is the di!erence of the electrode work functions. However, in MCBJ experiments we have found that in many instances the distance}voltage characteristics are very asymmetric in spite of * &0. (Fig. 6a) and thus the junction geometry accounts for the observed e!ect. For example in the case of a very sharp emitter the electrical "eld distribution inside the vacuum gap is not linear any more and the barrier can be considered as more transparent in comparison with the nearly trapezoidal barrier between planar electrodes. Consequently, a more steep distance}voltage dependence corresponds to negative polarity on a `sharpa electrode. Under an inhomogeneous "eld distribution the change of sign of the applied voltage can result in a substantial change of the electrode area involved in tunneling. Theoretical calculations [20] demonstrated that in the three-dimensional case the electric "eld cannot be assumed to be constant even at high voltages. They also revealed a dramatic dependence of the distance}voltage curves bending on the tip
254
O.Yu. Kolesnychenko et al. / Physica B 291 (2000) 246}255
total tunneling area as well as an increase of the contribution from the patches with a lower situated behind the foremost atoms.
5. Conclusion
Fig. 6. Asymmetric distance}voltage dependences for: (a) Pt MCBJ with * &0; (b,c) Au MCBJ with FER spectra which are heavily distorted due to surface defects.
radius r though this e!ect is reduced by the slower increase in the e!ective tunneling area with voltage for tips with smaller r. Using results [20] we have evaluated the typical radius of electrodes for Pt MCBJ as 500}1000 As and for Co MCBJ (Fig. 3a) r+200}300 As . The most illustrative examples of the z(<) heavy bending and asymmetric behavior are presented in Fig. 6b,c for Au MCBJ. It is well known that the noble metals experience severe deformations during the low-temperature break which can result in a very irregular shape of the electrodes on an atomic scale and the emergence of small patches with di!erent values associated with non-equilibrium surface defects. We have found that for freshly broken samples FER spectra are heavily distorted at low voltages. The "rst peak and work function values are scattered over an interval of 4}5.5 eV and the z(<) curves are very nonlinear and asymmetric. After 20}30 h the quality of spectra improves noticeably (though intensity of the oscillations with large quantum numbers remain very low) and values tend towards 5 eV. This means that the relaxation of defects results in a smoothing of the electrode surface and restores . It also means that the nonlinear behavior of z(<) curves is related to electrodes with relatively small ((100 As ) radius and that the steep rise in z at high voltages can be attributed both to the rapid increase of the
The measurements of "eld emission resonance spectra performed with MCBJ demonstrated that in spite of the essentially three-dimensional nature of electrodes the oscillating structure of the spectra can be clearly observed and in most cases experimental data can be used for an accurate calibration of the relative electrode displacement. The theoretical treatment of the problem in the framework of a one-dimensional model revealed that, whereas for the "rst maximum in FER spectra the junction conductance changes by an order of magnitude, at high voltages the oscillation amplitude drops rapidly. It was also shown that the oscillation phase is extremely sensitive to the gap distance which makes the overall spectrum and especially the "rst peaks very responsive to small details of the surface relief. We have also found that the asymmetric behavior of the distance}voltage characteristics with respect to polarity and their shape can be used for discrimination between MCBJ electrodes with di!erent geometries.
Acknowledgements The authors are grateful to J. Hermsen and J. Gerritsen for invaluable technical assistance, Aidan Quinn and Antony Keen for carefully reading the manuscript. Part of this work was supported by the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is "nancially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). O.I.S. wishes to acknowledge the NWO for a visitor's grant.
References [1] C.J. Muller, J.M. van Ruitenbeek, L.J. de Jongh, Physica C 191 (1992) 485. [2] J. Moreland, J.W. Ekin, J. Appl. Phys. 58 (1985) 3888.
O.Yu. Kolesnychenko et al. / Physica B 291 (2000) 246}255 [3] I.K. Yanson, V.V. Fisun, R. Hesper, A.V. Khotkevich, J.M. Krans, J.A. Mydosh, J.M. van Ruitenbeek, Phys. Rev. Lett. 74 (1995) 302. [4] R.J.P. Keijsers, O.I. Shklyarevskii, H. van Kempen, Phys. Rev. Lett. 77 (1996) 3411. [5] G. ZaraH nd, Jan von Delft, A. Zawadowski, Phys. Rev. Lett. 80 (1998) 1353. [6] R.J.P. Keijsers, O.I. Shklyarevskii, H. van Kempen, Phys. Rev. Lett. 80 (1998) 1354. [7] J.M. Krans, J.M. van Ruitenbeek, V.V. Fisun, I.K. Yanson, L.J. de Jongh, Nature 375 (1995) 767. [8] J. Voets, R.J.P. Keijsers, O.I. Shklyarevskii, H. van Kempen, Phys. Rev. B 53 (1996) 1072. [9] R.J.P. Keijsers, J. Voets, O.I. Shklyarevskii, H. van Kempen, Low. Temp. Phys. 24 (1998) 730. [10] R.J.P. Keijsers, J. Voets, O.I. Shklyarevskii, H. van Kempen, Phys. Rev. Lett. 76 (1996) 1138. [11] N. van der Post, J.M. van Ruitenbeek, Proceedings of the 21st International Conference on Low Temperature Physics, Chech. J. Phys. 46 (Suppl.) (1996) 2853. [12] K.H. Gundlach, Solid State Electron. 9 (1966) 946. [13] G. Binnig, H.R. Helv, Phys. Acta 55 (1982) 726. [14] R.S. Becker, J.A. Golovchenko, B.S. Swartzentruber, Phys. Rev. Lett. 55 (1985) 987. [15] G. Binnig, K.H. Frank, H. Fuchs, N. Garsia, B. Reihl, H. Rohrer, F. Salvan, A.R. Williams, Phys. Rev. Lett. 55 (1985) 991.
255
[16] N. Garcia, C. Ocal, F. Flores, Phys. Rev. Lett. 50 (1983) 2002. [17] R. Garcia, J.J. Saenz, J.M. Soler, N. Garcia, J. Phys. C: Solid State Phys. 19 (1986) L131. [18] G.C. Aers, C.R. Leavens, Solid State Commun. 60 (1986) 427. [19] J. Bono, R.H. Good, Surf. Sci. 188 (1988) 153. [20] J.M. Pitarke, F. Flores, P.M. Echenique, Surf. Sci. 234 (1990) 1. [21] J.H. Coombs, J.K. Gimzewski, J. Microscopy 152 (1988) 841. [22] O.Yu. Kolesnychenko, O.I. Shklyarevskii, H. van Kempen, Rev. Sci. Instrum. 70 (1999) 1442. [23] O.Yu. Kolesnychenko, O.I. Shklyarevskii, H. van Kempen, Phys. Rev. Lett. 83 (1999) 2242. [24] R.H. Fowler, L. Nordheim, Proc. Roy. Soc. A 119 (1928) 173. [25] A.M. Gabovich, V.M. Rozenbaum, A.I. Voitenko, Physica B 218 (1996) 280. [26] J.M. van Ruitenbeek, A. Alvarez, I. Pin eyro, C. Grahmann, P. Joyez, M.H. Devoret, D. Esteve, C. Urbina, Rev. Sci. Instrum. 67 (1996) 108. [27] C. Zhou, C.J. Muller, M.R. Deshpande, J.W. Sleight, M.A. Reed, Appl. Phys. Lett. 67 (1995) 1160. [28] R.J.P. Keijsers, O.I. Shklyarevskii, J.G.H. Hermsen, H.van Kempen, Rev. Sci. Instrum. 67 (1996) 2863. [29] I. Grigoriev, E. Meilikhov (Eds.), Handbook of Physical Quantities, CRC Press, Boca Raton, 1997 (Chapter 25).
Field-emission resonance measurements with mechanically controlled break junctions O.Yu. Kolesnychenko , Yu.A. Kolesnichenko, O.I. Shklyarevskii , H. van Kempen * Research Institute for Materials, University of Nijmegen, Toernooiveld 1, NL-6525 ED Nijmegen, The Netherlands B. Verkin Institute for Low Temperature Physics & Engineering, National Academy of Science of Ukraine, 47 Lenin Av., 310164, Kharkov, Ukraine Received 1 August 1999; received in revised form 20 December 1999; accepted 20 December 1999
Abstract The additional information provided by "eld emission resonance spectra signi"cantly increases the potential of the mechanically controllable break junction technique. We have found that the pronounced three-dimensional nature of the electrodes results in an extremely high sensitivity of the "eld emission resonance spectra to the "ne details of the surface geometry, electronic structure and electric "eld distribution at small electrode separations. At larger distances a quasiclassical approximation can be used for the determination of the metal work function, precise calibration of the electrode relative displacement and discrimination between `blunta and `sharpa electrodes on the basis of the distance}voltage z(<) dependence. 2000 Elsevier Science B.V. All rights reserved. PACS: 73.40.Gk; 73.90.#f; 73.30.#j; 79.70.#q Keywords: Break junctions; Vacuum tunneling; Field emission; Gundlach oscillations; Work function
1. Introduction In a rather short time since the mechanically controllable break junction (MCBJ) technique [1] (based on the early work of Moreland and Ekin [2]) was "rst reported it has become not only an excellent complementary method to both pointcontact spectroscopy and scanning tunneling microscopy (STM) but can also be regarded as an independent and extremely powerful tool in itself. * Corresponding author. Tel.: #31-24-3653499; fax: #3124-3652190. E-mail address: [email protected] (H. van Kempen).
This technique is the most promising for point contacts of submesoscopic size with which the interaction of electrons with an individual magnetic impurity [3] or single two-level #uctuator [4}6] (see also comment by Ref. [5] and reply by Ref. [6]) has been observed and conductance quantization in contacts of only a few atoms has been reported [7]. In the tunneling regime the exceptionally high stability of MCBJ has been exploited to investigate electrode interaction at ultrashort ((3 As ) distances [8]. The deviation of the current}distance dependences from exponential behavior due to the shape of the electrodes [9] or to adsorbed He atoms [10]
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 8 8 4 - 7
O.Yu. Kolesnychenko et al. / Physica B 291 (2000) 246}255
has been studied. In addition, with only a minor loss of junction stability MCBJ can be used in quasi-STM mode for scanning very small ((1000 As ) areas of the electrode surface [8,11]. Up to now all experiments with MCBJ in the tunneling regime were performed at relatively low bias voltages < . Here we present measurements at < much larger than the work function of the metal under investigation, emphasing the junction conductance oscillation e!ect, also known as Gundlach oscillations [12] or "eld emission resonances (FER). For vacuum gap tunneling this e!ect was observed for the "rst time by Binnig and Rohrer [13]. The ensuing publications [14,15] gave rise to a considerable theoretical interest [16}20]. The most detailed and thorough measurements of Coombs and Gimzewski [21], extending to large quantum numbers and gap spacing, revealed that the "ne structure of the oscillations is extremely sensitive to the electronic and physical structure of the emitting surface. This fact stimulated our interest in that the properties of the MCBJ electrode surfaces are not well characterized in general. Recent MCBJ experiments in the "eld emission range were aimed at precise calibration of the distance between the electrodes, accurate determination of the absolute vacuum gap and the metal work function [22] and investigation of the in#uence of adsorbed helium on the latter [23]. In the present paper we have concentrated mostly on deviations from standard behavior in FER spectra and voltage}distance dependences. These are related to the essential three-dimensional nature of the problem.
2. Theory In the Fowler}Nordheim regime [24] when the applied voltage < is larger than the work function
of the metal, part of the vacuum gap (Fig. 1a) becomes classically accessible. The wave function of an electron in this positive kinetic-energy region is determined by superposition of the transmitted wave and the wave re#ected from the potential step at the collector interface. The transmission coe$cient has a maximum if the electron energy coincides
247
Fig. 1. (a) Energy diagram for "eld emission. d-separation between electrodes, z -classical turning point, -work functions M of the electrodes. (b) Typical sample mount consisting of a bending beam with the sample wire attached to it. The dimensions indicated are referred to in the text. For clarity, all distances have been exaggerated. A bending force causes a vertical displacement y, which leads to a distance z between the electrodes.
with one of the virtual energy levels in the potential well formed by the barrier potential ;(z, <) and the potential step at d. In a quasiclassical approximation it is equivalent to the Bohr}Sommerfeld quantization rule for the shadowed area in Fig. 1a.
(2m B dz (e !;(z, < )"pn. X L
X
(1)
Here e is the electron energy in the direction X normal to the metal surface, ;(z, < ) is the potenL tial energy of an electron inside the barrier, m is the electron mass, n is a positive integer and is the Planck constant. It must be emphasized that there is no barrier at the point d and the electron energy spectrum is continuous. The ful"llment of Eq. (1) means that at <"< standing waves arise beL tween the surface of the collector and the classical turning point z resulting in an oscillating depend ence of tunneling probability and therefore of tunnel current as a function of the applied voltage <.
248
O.Yu. Kolesnychenko et al. / Physica B 291 (2000) 246}255
However, the quasiclassical approximation is inapplicable for calculation of oscillation amplitudes since it calls for the determination of the transmission coe$cient D(e ), including the pre-exponential X factor. Our further calculations are aimed at investigating the analytical expression for the tunnel current in the "eld emission region and its dependence on the main parameters involved. Let us consider the situation when e is relatively far from its limiting X values e "k# and e "k# !e< where k is the Fermi energy and are the work func tions of the left and right electrodes. The transmission coe$cient can be written in the form [10] D(e )"8 exp(!f ) X (e (e #e<)( #k!e )(e #e
Here
4 k# !e X f (e )" <1, X 3 eFl
4 e
I"I
$,
k#e< e
1 ; 1! ; 1#(e
f (k)
4 f (k)" , 3 eFl
e
where I is the Fowler}Nordheim tunnel current $, [24]: (k eF " exp+!f (k),. (7) $, k# (2p)
It should be noted that the oscillation amplitude (Eq. (5)) is rather large:
I
k#
<1. /I K
e
At high voltages e
I
(3)
F"(e<#* )/de is an e!ective electrical "eld and l\"(2m/ )(e<#* )/d. At low temperatures and e<'k the tunnel current is determined by the following expression:
em I I" de (k!e )D(e ). (4) X X X 2p The tunneling probability has a maximum for electrons with energy e Kk moving perpendicular X to the boundaries. If the characteristic energy range *e& /f to an exponential decay exp(!f ) in the tunneling probability is much smaller than k we can present Eq. (4) in the form
exp(!f ). Taking it outside of the integral sign and expanding f (k!g) in series around g"0, one can obtain after integration
em IK dg gD(g), f <1, (5) 2p where g"k!e . At relatively small voltages X (e
I"I
$,
k#e< 1 (k# ) (k#e< 1! e
2 (e
;sin(f (k)#U) , B
(8)
k#
<1; and e
2( (e
f (k)
This result shows that at e
O.Yu. Kolesnychenko et al. / Physica B 291 (2000) 246}255
the electrode separation. While the monotonic part decreases by a factor of e as the distance between the electrodes increases by *d +d/f , the phase of the oscillations changes its sign when the barrier width changes by *d +d/f ;*d . This means B that the interference of contributions to the tunnel current from the di!erent regions of the real 3D junction with only slightly di!erent gaps may result in a considerable suppression of the FER amplitude. This result provides us an important criterion for analysis of the in#uence of the three-dimensional electrode geometry on the FER spectra. Eq. (8) also permits the calculation of junction conductance at high voltages. Taking into account that the main contribution in dI/d< provides differentiation of rapidly changing functions of voltage exp(!f ) and sin(f ) one "nds that B
dI 3 f (k) 1 k#
cos f (k) " I 1! B 4 ( e< d< 2 < $, f (k)
e<<
(9)
, k.
The junction conductance peaks when
4 (2m (e< ! ) 1 L f (k)" "2p n! ; B 3
eF 4 n"1, 2, 32 ,
(10)
where < is the bias voltage at which the nth L maximum occurs. It is obvious from Eqs. (6) and (8) that by keeping I constant in the course of measurements we are also able to maintain the e!ective electrical "eld strength nearly constant at large voltages. Assuming FKconst and (n#) +n at n<1, one can "nd from Eq. (10):
3p e e< " # F n . L 2(2m
(11)
Not surprisingly, under the assumption made above we arrive at the same result as in Ref. [19] where it was obtained from the simple quasiclassical approximation. It should be realized that all calculations presented above were made using a simpli"ed model of a one-dimensional tunnel junction with a trap-
249
ezoidal barrier. In reality some additional factors are of considerable importance. (i) Image forces change the overall shape of the tunneling barrier and thus the amplitude of the tunnel current and position and period of dI/d< oscillations. Even including an approximate expression for the image potential into the SchroK dinger equation makes analytical solution of the problem impossible. Numerical calculations of FER spectra [17,19,20] demonstrated that inclusion of an image force in#uences results in an appreciable (10}20%) shift in the position of the "rst maxima towards the lower energies. Note that distortion of the barrier by image forces occurs mainly in the region close to the boundaries, a!ecting small-n oscillations. As n increases the relative contribution of this area in integral (1) decreases and the in#uence of image forces on Gundlach oscillations corresponding to larger n drops noticeably. As calculations [17,19,20] have been made for speci"c material parameters they have little relevance for improving the determination of d and
for an arbitrary junction. A more attractive suggestion [25] takes into account the image forces by introducing e!ective work functions H and elec trical "eld EH in the framework of a trapezoidal barrier model. (ii) In contrast to planar M}I}M junctions, the electric "eld distribution for both STM and MCBJ con"gurations is essentially three dimensional. However, because of the exponential dependence of the tunneling probability on the barrier width, the main part of the tunnel current #ows through the electrode region with the smallest gap. This makes the problem quasi-one-dimensional and the monotonic part of the tunnel current can be described by an expression analogous to the Fowler}Nordheim formula [24] with an additional coe$cient depending on the exact shape of the junction and proportional to the e!ective tunneling area [16,17,19].
3. Experiment In our experiments we used the traditional sample design presented in Fig. 1b and described
250
O.Yu. Kolesnychenko et al. / Physica B 291 (2000) 246}255
elsewhere (see e.g., Refs. [1,9]). It consists of a 50}250 lm "lament glued using two small drops of hard epoxy (Stycast 2850 FT) onto a phosphor}bronze bending beam which is covered with a 70 lm insulating layer of capton foil. The wire (Pt, Au, Cu 99.999% purity or Ni, Co, Fe, Be, Pb and Mg 99.99% purity) is then notched at the midpoint of the glued section for about 80}90% of its diameter. A narrow (300}400 lm) strip of Dy 50 lm foil was treated the same way. After mounting in a special setup [9] and cooling, the sample is broken at 1.2}4.2 K in a high vacuum environment by bending the substrate against the counter supports C, thus creating two atomically clean electrodes. The relation between the vertical displacement of the piezodriver D and the electrode separation D is W X determined by the length ¸, width and thickness of the bending beam, distance between anchoring points of the wire s and slope a at the "xation point. Additional corrections arise if the break does not occur exactly midway along the glued section (for detailed calculations see Ref. [9]). The approximate nature of this relation and uncertainty in determination of a allows only a rough estimation of D /D . This ratio usually ranges between 300 and W X 500 and can reach 3000 for lithographically fabricated and thin "lm break junctions [26}28]. Typically, we try to keep the distance between the anchoring points as small as possible (0.2}0.3 mm) to ensure high stability of the device. For FER measurements we deliberately increased s up to 0.5}0.7 mm (reducing D /D below 200) in W X order to cover a gap spacing of about 100 As with the piezodriver. The oscillatory behavior of the tunnel current for e<' can already be seen in a Fowler}Nordheim plot ln(I/<) versus 1/< and even more clearly observed in a logarithmic plot of dI/d<(<) taken at a "xed electrode separation. However, only the "rst two or three peaks can be observed at constant distance or at constant bias (by varying the vacuum gap alone) mode because of the limited dynamic range of the measuring equipment. In addition the rapid decrease of amplitude makes measurements of oscillations with large quantum numbers practically impossible. Following [14] the oscillating part of the junction conductance dI/d<(<) was measured in a con-
stant current (I"0.5}1 nA) mode by adding a feedback ramp with a relatively slow response for adjusting the gap distance between the MCBJ electrodes. A modulation voltage of frequency 1000 Hz and 200}300 mV amplitude was superimposed on the bias voltage signal and the conventional lock-in technique was used for the determination of dI/d<(<). The sweep time for the scan from $1 V to $25}35 V was 1000}2000 s which gives the possibility to use a lock-in time constant of 0.3}1 s. The piezovoltage signal < (<) proportional to the gap spacing was measured simultaneously. To change the bias polarity the feedback circuit was switched o! at an electrode separation of 20}50 As . The dI/d< curves can be reproduced extremely accurately for 1 h, which demonstrate the high junction stability and testi"es that the lateral drift of the electrodes during this time is negligibly small. In order to `changea the junction we modi"ed the surface relief of the electrodes by controlled `crashinga using either a piezodriver to create a point contact with resistance in the 10}10 ) range (estimated indenting of electrodes into each other 5}10 As ) or a di!erential screw (indenting up to 100}200 As ). We observed oscillation of the tunnel current for all materials investigated. The best results were obtained for platinum, cobalt, beryllium and gold MCBJ. Experimental data for more than 600 junctions from about 50 di!erent samples were analyzed.
4. Results and discussion All STM measurements of Gundlach oscillations were carried out using a `blunta tip with a rather large radius against an atomically #at (over 10 As ) sample surface [14,21] in order to make the situation as close as possible to that in planar junctions. In spite of the fact that the surfaces of the MCBJ electrodes are very irregular [8,11] and essentially three dimensional, a surprisingly high percentage of dI/d<(<) curves (more than 90%) demonstrate oscillatory behavior in a voltage range up to 30}35 V with up to n"35}45 distinct peaks (Fig. 2). However, only a small part of FER spectra (:10%) display the theoretically predicted gradual
O.Yu. Kolesnychenko et al. / Physica B 291 (2000) 246}255
decrease in oscillation amplitude (curve 1 at Fig. 2). The < (<) (distance}voltage) dependences for such spectra show marked steps around the conductance maxima up to 12}15 V (see inset in Fig. 2) and nearly linear increase at higher voltages. The vast majority of spectra shows variations of the peak amplitudes (curves 2}4, Fig. 2) which is reminiscent of `beatinga or `wave packetsa. This e!ect was probably "rst observed in Ref. [14] as a disruption in the periodic structure of the dI/d<(<) dependence and attributed to the re#ection associated with atomic planes below the surface. Numerical calculations [19] for tunneling from four contributing areas with slightly di!erent gaps revealed not only a more rapid decay of the oscillation amplitude than was predicted for a single area, but also its non-monotonic behavior. Since the numerical analysis in Ref. [19] was restricted to the "rst 7}8 peaks authors did not pay due attention to this fact. The "ne structure of FER spectra was studied in great detail by Coombs and Gimzewski [21]. They suggested that in their ex-
Fig. 2. An example of "eld emission resonance spectra for Pt MCBJ showing continuous decay (1) and strong variation of the oscillation amplitude (2}4). The inset shows a distance}voltage dependence z(<) with distinct steps, related to the conductance maxima.
251
periments with a blunt tip, `beatinga arises as a result of electron waves interfering from the patches of the surface with di!erent work functions. In extreme situations (3}10% of all curves depending on the MCBJ material) the destructive interference results in a completely irregular spectrum with a majority of the peaks being suppressed. The processing of FER spectra with a heavy `beatinga is more complicated as some of the peaks can be missed and proper indexing must be restored to eliminate steps in the < (n ) dependence (for L a more comprehensive description see Ref. [22]). The problem of adequate peak numbering for the low index peaks, however, is far more complicated. Fig. 3a shows two FER spectra for a Co MCBJ
Fig. 3. (a) Two FER spectra for a Co MCBJ taken with a 3 h interval for the same junction. An extremely small lateral shift of electrodes (0.2}0.3 As ) results in the appearance of an extra maximum on curve 2 (dashed line) between 5 and 11 V whereas above this interval the curves remain unchanged. (b) Position of the maximum in the di!erential conductance versus the order of the resonance to the power two-thirds for curves 1 and 2 of Fig. 3a. Incorrect peak indexing for curve 1 (open circles) results in an unrealistically high value for the Co work function (5.55 eV).
252
O.Yu. Kolesnychenko et al. / Physica B 291 (2000) 246}255
measured sequentially for 3 h. The estimated lateral shift of the electrodes during this time does not exceed 0.3}0.5 As but its impact on the d
"4.5$0.1 eV which is very close to the conventional value of 4.41 eV [29]. The deviation of the peaks with small quantum numbers from the linear dependence (Fig. 3b) in MCBJ experiments is much larger than for the STM `blunt tip-#at samplea con"guration [21]. It can be explained by the fact that the 3D nature of the break-junction at relatively small distances is especially pronounced and "rst oscillations are more responsive to the geometry of a junction. First of all, at these separations the distortion of a trapezoidal barrier by image forces is much more signi"cant. Secondly, for this vacuum gap range the condition F"const most probably does not hold anymore in constant current mode. As can be seen from Eq. (6), at bias voltages comparable to /e the Gundlach oscillation is essentially nonharmonic. Finally, the phase shift between electron waves from tunneling areas with di!erent gaps (or di!erent work functions) decreases in inverse proportion to the electrode separation and is largest for the "rst peaks. All these factors make the analysis of this part of the FER spectra extremely di$cult. We made an attempt to detect missing or extra peaks using results obtained by numerical solution of the SchroK dinger equation. According to Ref. [19] the position of the "rst oscillations is linear in coordinates < versus n . But the "t to the experiL mental data (Fig. 4a) seems equally good for both Co FER spectra and the absent peak does not manifest itself in any manner. To avoid complications with peak indexing it was suggested [21] to use a *<\(<) plot (where *< is the voltage separation between the two ad-
Fig. 4. (a) position of the "rst maxima versus the square root from the order of resonance for Co FER spectra presented in Fig. 3a. The open circles correspond to curve 1 and the closed circles to curve 2. (b) *<\(<) dependencies for the "rst 12 peaks of the same FER spectra. In both cases there is no clear indication for the `misseda peak in curve 1.
jacent peaks) for analyses of FER spectra. Sharp spikes occur in this plot at voltages where the normal sequence of peaks is disrupted by interference. Unfortunately, this method is very unsuitable for analysis of small-n oscillations as the scattering of points in this range is too large to draw a de"nite conclusion if one of the peaks is really lost (Fig. 4b). The linear "t interception with the x-axis gives the approximate value of the work function but its accurate determination is highly conjectural. It is necessary to stress that the position of the "rst peak in the FER spectra < remains remark ably stable during the small changes of the surface relief or a local work function (caused e.g. by bringing electrodes together) whereas the overall shape of the spectrum can be drastically a!ected. This can be explained by the fact that at I"0.1}1 nA the "rst peak occurs for the majority of metals at separations of +7}10 As . At these distances almost all
O.Yu. Kolesnychenko et al. / Physica B 291 (2000) 246}255
the tunnel current is determined by the foremost atom or clusters of atoms and contributions from areas with larger gaps or di!erent are negligible. Using data for the best FER spectra we were able to draw an empirical relationships between the "rst peak position and the work function for some metals. For example, "(0.88$0.04) e< for Pt and "(0.84$0.04) e< for Be. The scattering of the data can be related to the dependence of the "rst peak position on the e!ective tunneling area as pointed out by Bono and Goods [19]. Nevertheless, these relationships can be used as a very reasonable estimation for the local work function. The absolute vacuum gap (an o!set of z(<) dependence) can be found by the integration of the one-dimensional SchroK dinger equation [14] to yield the constant transmission probability for the experimentally observed relative displacement of the tip corresponding to a "xed tunnel current. An alternative way of determining the absolute vacuum gap was suggested in Ref. [16] but has never been tested experimentally. Using z(<) dependencies taken in both polarities for two signi"cantly di!erent values of tunnel current (0.5 and 0.03 nA in our case), one can "nd the absolute vacuum gap as an interception point of the linear part of the curves for di!erent I (Fig. 5). The distance between these points in `#a and `!a polarity on the <-axis gives us 2* . We have found, however, from our data that the value of absolute vacuum gap (+16}18 As ) is larger than expected by a factor of 2 and * +1.1 eV di!ers from that determined using FER spectra by a factor of 5. It should be noted that the accuracy of the suggested method is not very high as it requires linear z(<) dependencies over a large voltage range. The greatly exaggerated value for the gap and * can be explained by the fact that for smaller current (at the same voltage) the e!ective tunneling area is larger and the corresponding z(<) curve rise more steeply [18], pushing the interception point further than expected. These considerations are also accountable for much larger * . Most probably this method yields a vacuum gap `averageda over all tunneling areas and the result depends on the exact shape of the electrodes. In the "eld-emission experiments with an arti"cially blunted tip in the STM con"guration the main
253
Fig. 5. Distance}voltage characteristics for a Pt MCBJ in positive and negative polarity taken in constant current mode at 500 and 30 pA. According to Ref. [16] the interception of linear extrapolations from high voltages (dashed lines) in each polarity gives the position of an absolute vacuum gap, and the voltage di!erence between intercepts at d"0 for `#a and `!a polarity determines 2* . The three dimensional character of the MCBJ leads to the largely exaggerated values.
reason for distance}voltage curve z (<) asym! metry is the di!erence of the electrode work functions. However, in MCBJ experiments we have found that in many instances the distance}voltage characteristics are very asymmetric in spite of * &0. (Fig. 6a) and thus the junction geometry accounts for the observed e!ect. For example in the case of a very sharp emitter the electrical "eld distribution inside the vacuum gap is not linear any more and the barrier can be considered as more transparent in comparison with the nearly trapezoidal barrier between planar electrodes. Consequently, a more steep distance}voltage dependence corresponds to negative polarity on a `sharpa electrode. Under an inhomogeneous "eld distribution the change of sign of the applied voltage can result in a substantial change of the electrode area involved in tunneling. Theoretical calculations [20] demonstrated that in the three-dimensional case the electric "eld cannot be assumed to be constant even at high voltages. They also revealed a dramatic dependence of the distance}voltage curves bending on the tip
254
O.Yu. Kolesnychenko et al. / Physica B 291 (2000) 246}255
total tunneling area as well as an increase of the contribution from the patches with a lower situated behind the foremost atoms.
5. Conclusion
Fig. 6. Asymmetric distance}voltage dependences for: (a) Pt MCBJ with * &0; (b,c) Au MCBJ with FER spectra which are heavily distorted due to surface defects.
radius r though this e!ect is reduced by the slower increase in the e!ective tunneling area with voltage for tips with smaller r. Using results [20] we have evaluated the typical radius of electrodes for Pt MCBJ as 500}1000 As and for Co MCBJ (Fig. 3a) r+200}300 As . The most illustrative examples of the z(<) heavy bending and asymmetric behavior are presented in Fig. 6b,c for Au MCBJ. It is well known that the noble metals experience severe deformations during the low-temperature break which can result in a very irregular shape of the electrodes on an atomic scale and the emergence of small patches with di!erent values associated with non-equilibrium surface defects. We have found that for freshly broken samples FER spectra are heavily distorted at low voltages. The "rst peak and work function values are scattered over an interval of 4}5.5 eV and the z(<) curves are very nonlinear and asymmetric. After 20}30 h the quality of spectra improves noticeably (though intensity of the oscillations with large quantum numbers remain very low) and values tend towards 5 eV. This means that the relaxation of defects results in a smoothing of the electrode surface and restores . It also means that the nonlinear behavior of z(<) curves is related to electrodes with relatively small ((100 As ) radius and that the steep rise in z at high voltages can be attributed both to the rapid increase of the
The measurements of "eld emission resonance spectra performed with MCBJ demonstrated that in spite of the essentially three-dimensional nature of electrodes the oscillating structure of the spectra can be clearly observed and in most cases experimental data can be used for an accurate calibration of the relative electrode displacement. The theoretical treatment of the problem in the framework of a one-dimensional model revealed that, whereas for the "rst maximum in FER spectra the junction conductance changes by an order of magnitude, at high voltages the oscillation amplitude drops rapidly. It was also shown that the oscillation phase is extremely sensitive to the gap distance which makes the overall spectrum and especially the "rst peaks very responsive to small details of the surface relief. We have also found that the asymmetric behavior of the distance}voltage characteristics with respect to polarity and their shape can be used for discrimination between MCBJ electrodes with di!erent geometries.
Acknowledgements The authors are grateful to J. Hermsen and J. Gerritsen for invaluable technical assistance, Aidan Quinn and Antony Keen for carefully reading the manuscript. Part of this work was supported by the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is "nancially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). O.I.S. wishes to acknowledge the NWO for a visitor's grant.
References [1] C.J. Muller, J.M. van Ruitenbeek, L.J. de Jongh, Physica C 191 (1992) 485. [2] J. Moreland, J.W. Ekin, J. Appl. Phys. 58 (1985) 3888.
O.Yu. Kolesnychenko et al. / Physica B 291 (2000) 246}255 [3] I.K. Yanson, V.V. Fisun, R. Hesper, A.V. Khotkevich, J.M. Krans, J.A. Mydosh, J.M. van Ruitenbeek, Phys. Rev. Lett. 74 (1995) 302. [4] R.J.P. Keijsers, O.I. Shklyarevskii, H. van Kempen, Phys. Rev. Lett. 77 (1996) 3411. [5] G. ZaraH nd, Jan von Delft, A. Zawadowski, Phys. Rev. Lett. 80 (1998) 1353. [6] R.J.P. Keijsers, O.I. Shklyarevskii, H. van Kempen, Phys. Rev. Lett. 80 (1998) 1354. [7] J.M. Krans, J.M. van Ruitenbeek, V.V. Fisun, I.K. Yanson, L.J. de Jongh, Nature 375 (1995) 767. [8] J. Voets, R.J.P. Keijsers, O.I. Shklyarevskii, H. van Kempen, Phys. Rev. B 53 (1996) 1072. [9] R.J.P. Keijsers, J. Voets, O.I. Shklyarevskii, H. van Kempen, Low. Temp. Phys. 24 (1998) 730. [10] R.J.P. Keijsers, J. Voets, O.I. Shklyarevskii, H. van Kempen, Phys. Rev. Lett. 76 (1996) 1138. [11] N. van der Post, J.M. van Ruitenbeek, Proceedings of the 21st International Conference on Low Temperature Physics, Chech. J. Phys. 46 (Suppl.) (1996) 2853. [12] K.H. Gundlach, Solid State Electron. 9 (1966) 946. [13] G. Binnig, H.R. Helv, Phys. Acta 55 (1982) 726. [14] R.S. Becker, J.A. Golovchenko, B.S. Swartzentruber, Phys. Rev. Lett. 55 (1985) 987. [15] G. Binnig, K.H. Frank, H. Fuchs, N. Garsia, B. Reihl, H. Rohrer, F. Salvan, A.R. Williams, Phys. Rev. Lett. 55 (1985) 991.
255
[16] N. Garcia, C. Ocal, F. Flores, Phys. Rev. Lett. 50 (1983) 2002. [17] R. Garcia, J.J. Saenz, J.M. Soler, N. Garcia, J. Phys. C: Solid State Phys. 19 (1986) L131. [18] G.C. Aers, C.R. Leavens, Solid State Commun. 60 (1986) 427. [19] J. Bono, R.H. Good, Surf. Sci. 188 (1988) 153. [20] J.M. Pitarke, F. Flores, P.M. Echenique, Surf. Sci. 234 (1990) 1. [21] J.H. Coombs, J.K. Gimzewski, J. Microscopy 152 (1988) 841. [22] O.Yu. Kolesnychenko, O.I. Shklyarevskii, H. van Kempen, Rev. Sci. Instrum. 70 (1999) 1442. [23] O.Yu. Kolesnychenko, O.I. Shklyarevskii, H. van Kempen, Phys. Rev. Lett. 83 (1999) 2242. [24] R.H. Fowler, L. Nordheim, Proc. Roy. Soc. A 119 (1928) 173. [25] A.M. Gabovich, V.M. Rozenbaum, A.I. Voitenko, Physica B 218 (1996) 280. [26] J.M. van Ruitenbeek, A. Alvarez, I. Pin eyro, C. Grahmann, P. Joyez, M.H. Devoret, D. Esteve, C. Urbina, Rev. Sci. Instrum. 67 (1996) 108. [27] C. Zhou, C.J. Muller, M.R. Deshpande, J.W. Sleight, M.A. Reed, Appl. Phys. Lett. 67 (1995) 1160. [28] R.J.P. Keijsers, O.I. Shklyarevskii, J.G.H. Hermsen, H.van Kempen, Rev. Sci. Instrum. 67 (1996) 2863. [29] I. Grigoriev, E. Meilikhov (Eds.), Handbook of Physical Quantities, CRC Press, Boca Raton, 1997 (Chapter 25).