Coulomb blockade phenomena observed in supported metallic nanoislands

1. Introduction

The study of Coulomb blockade (CB) phenomena in nanostructures is a very attractive area of condensed matter physics, because single electron electronics is considered to have a high potential for basic research and for technological applications (1–5). The field started more than 40 years ago with the investigation of small metal particles embedded in an oxide layer between two planar metal electrodes (6, 7). Subsequently, in order to avoid ensemble effects and to study single double-barrier tunnel junctions (DBTJ), CB phenomena have been investigated in small-area tunnel junctions prepared by lithographic methods developed for microelectronics (8, 9). More recently, junctions using a scanning tunneling microscope (STM) tip as one adjustable electrode came into the focus of research (10–13).

Consider single electron transport in the system shown in Figure 1: the two barriers constitute a DBTJ. For each junction the barriers are modeled by an ohmic resistor R_i in parallel with a capacitor C_i, i = 1 for the substrate-island junction and i = 2 for the island-tip junction, as shown in the equivalent electronic circuit in Figure 1B. In order to transfer a single electron through the DBTJ, the electron must overcome the Coulomb charging energy E_c = e²/C_∑, where C_∑ = ΣC_i. If the charging energy E_c is much larger than the thermal energy k_BT, the tunneling of an electron is blocked for bias voltages smaller than U_c = e/C_∑. This phenomenon has been termed CB (CB) (2, 3, 14–16). Moreover, in order to keep charge fluctuations sufficiently small, the coupling resistance of each junction has to be significantly larger than the quantum resistance: R_i ≫ R_Q = h/(2e²) ≈ 12.9 kΩ. These conditions imply that extremely small metal clusters have to be probed, or that the measurements have to be performed at low temperature.

FIGURE 1

Figure 1. (A) Schematic diagram of the Coulomb blockade system. The Pb nanoisland is embedded between the vacuum barrier and a few NaCl dielectric layers on Ag(111). (B) Equivalent double-barrier tunnel junction (DBTJ) circuit where each junction is represented by a set of capacitors and resistors, connected in parallel.

Previous studies of nanostructures in the CB regime by STM were performed on amorphous particles (11, 13, 17, 18), on particles covered by shells consisting of specific ligands and/or molecules (19–22), and to a lesser extent on supported clusters and metal islands (18, 23–26). The vast majority of the CB spectroscopy results have been successfully described within the orthodox theory of the DBTJ (2), which allowed the authors to determine quantitatively the tip-particle and the particle-substrate junction parameters (see Figure 1B). Within the context of single electron tunneling, the “orthodox theory,” a semi-classical theory, is based on the following three assumptions (3). (1) The electron energy quantization inside the conductors is ignored, i.e., the electron energy spectrum is continuous. (2) Coherent quantum processes consisting of several simultaneous tunneling events (“cotunneling”) are ignored. This assumption is valid if the resistance of all tunneling barriers of the system is much higher than the quantum unit of resistance. Only under this condition the quantum-mechanical uncertainty of electrons is suppressed because of the high tunneling resistance (i.e., less tunneling events). Thus one can neglect the concurrent tunneling events and treat one electron at a time, which makes controllable single-electron manipulation possible. (3) The tunneling time τ_t of an electron tunneling through the barrier is assumed to be negligible compared to other time scales (including the interval between subsequent tunneling events). For a typical tunneling event in a practical junction (3), τ_t is around 10⁻¹⁵ s. For a typical STM tunneling junction with 1 nA tunneling current, on average one electron tunnels per 10⁻¹⁰ s. Considering that the electronic lifetime of an electron in the states of the islands is around 10⁻¹³−10⁻¹⁵ s, electrons tunnel one at a time and consequently under these conditions cotunneling processes can be neglected.

Recently, in a low-temperature scanning tunneling spectroscopy (STS) study, we reported (27) on the observation of the dynamical Coulomb blockade (DCB) (28–30) in nanosized electrical contacts consisting of small Pb islands on various conducting, semiconducting, and partially insulating substrates. We observed a suppression of the differential tunnel conductance at small bias voltages due to DCB effects. The differential conductance spectra allowed us to determine the capacitances and resistances of the electrical contact which depend systematically on the island-substrate contact area. Calculations based on the theory of environmentally assisted tunneling agree well with the measurements.

Here, we extend the previous observations from the DCB to the orthodox CB and Coulomb staircase (CS) regime. We report on the observation of CB phenomena in Pb nanoislands, ranging from a lateral size of a few nanometers to tens of nanometers and a thickness of a few monolayers, grown on ultrathin dielectric films of NaCl deposited on Ag(111). For selected Pb islands of well-defined area, the local tunneling spectra reflect CB and CS phenomena in the DBTJ (see Figure 1) characteristic for single electron tunneling: Every additional electron tunneling to or from the Pb nanoisland is counted one by one. Using a model based on the orthodox theory of single electron tunneling (12, 16), we are able to simulate the experimental tunneling spectra and to extract the relevant parameters of the DBTJ, i.e., the resistances of island-substrate and tip-island junctions (R₁ and R₂), the capacitances of island-substrate and tip-island junctions (C₁ and C₂), and the residual charge (Q₀) on the island. The extracted C₁ capacitances fit well with the values corresponding to a planar capacitor, C₁ = ε₀ ε_rA_island/d, the island area A_island being measured from the STM topography and the insulator thickness determined by field emission resonances (FERs) analysis. Manipulation of Q₀ is achieved by controlled application of voltage pulses on the Pb islands. Moreover, under specific tunneling conditions, the influence of the adjustable tip-island junction on Q₀ is revealed in topographic images of the Pb islands.

In an STM measurement of a DBTJ, in general the tip-island junction resistance R₂ is always much larger than the quantum resistance R_Q. Then the electron transport through such a DBTJ can be separated into three different regimes (16): if the island-substrate resistance R₁ ≪ R_Q, the whole system corresponds to a single junction; if R₁ is of the order of R_Q, the dynamical Coulomb blockade (DCB) regime is reached; if R₁ ≫ R_Q, the CB and CS regime is entered. These different conditions can be selectively and controllably fulfilled in our experimental setup by choosing the appropriate size of the islands (by changing the growth parameters) and the pertinent tunnel parameters, including an adjustable tip-island distance (by changing the tip-island resistance R₂). In this way island-size-dependent CB gaps and CS have been unambiguously identified in the tunneling spectra obtained on the Pb quantum dots on NaCl/Ag(111).

2. Experimental

The Ag(111) single crystal substrate was cleaned by Ar⁺ sputtering and subsequent annealing cycles. NaCl powder was thermally evaporated from a crucible at a temperature of T = 620 °C onto the Ag substrate held at room temperature or heated to ≈ 420 K. Subsequently, Pb islands were grown by evaporation of Pb from a W filament onto the NaCl-covered Ag substrate cooled to 130 K to obtain Pb islands of the desired size.

The experiments were conducted in a homebuilt STM operated at 4.6 K (31). Differential conductance (dI/dV) measurements were performed with an open feedback loop using a lock-in technique with modulation voltage from 2 to 10 mV_pp at 300–400 Hz frequency with tunneling current ranging from 100 pA to a few nA.

3. Results and Discussion

3.1. Growth of Pb Nanoislands on NaCl/Ag(111)

A bulk NaCl crystal is a dielectric material with a wide band gap in the range of 8.5 − 9 eV (32–34). For thin NaCl films, the band gap is reported to be close to the bulk value (35) For 1 − 3 ML of NaCl on Ag(100), recent STS measurements of the energy positions of field emission resonances propose a work function of 3.2 eV (36), while ultraviolet photoemission spectroscopy (UPS) yields 3.5 eV (36, 37). From dI/dV spectra measured on top of a 2 ML NaCl film, we deduce that the gap lies within the voltage range of ± 2 eV around E_F, which is the energy range selected for the measurement of the dI/dV spectra of the Pb nanoislands on NaCl.

Studies of NaCl layers on other substrates abound, e.g., on Al(111) (35, 38), on Cu(111) (39), on Ag(100) (36, 40, 41), and on Au(111) (42–44). NaCl(100) layers are very versatile as a dielectric ultrathin film for the electronic decoupling of supported molecules (39, 42, 43) or other nanostructures (27) from a metallic substrate.

Here, NaCl was deposited on the substrate held at two different temperatures, 300 K, and ≈ 420 K, the latter favoring the formation of more extended, flatter, and defect-free layers. Figures 2A,B show STM images corresponding to these two different sample preparations, yielding Pb islands of different average size. This behavior is mainly due to the higher density of nucleation points on the NaCl layer in (b) with respect to (a). The NaCl layer has different characteristics in the two cases: the layer is flat and presents very few defect in (a), while it shows the coexistence of various thicknesses and a higher density of edges in (b). In (a) the side length of the equilateral triangular Pb islands is in the range of 25 to 60 nm, while in (b) it is ≤ 10 nm. For the STS measurements, triangular islands of various sizes were selected. The Pb islands expose (111) crystal faces. A determination of the thickness of NaCl layers at a given location is not straightforward. Moreover, the apparent height of NaCl layers varies depending on tunneling conditions, such as bias voltage and tip conditions.

FIGURE 2

Figure 2. STM topographies acquired on two different samples with Pb islands grown on NaCl/Ag(111). Pb islands are visible in both images, most of them with triangular shape and a few with rounded shape. The smaller island size in (B) as compared to (A) is obtained on a NaCl film exhibiting more defects. (A): +1 V, 20 pA; (B): +4 V, 20 pA.

3.2. Coulomb Blockade and Coulomb Staircase on Pb Nanoislands

In a STM double junction geometry, there are some intrinsic limitations in the relations between resistances and capacitances of the two junctions. As discussed in the introduction, in order to observe CB phenomena, the condition R_i ≫ R_Q has to be satisfied. In the tunneling regime, this condition is easily satisfied for junction 2 (tip-vacuum-island), for which resistances R₂ of 1 MΩ to 1 GΩ are usual. For junction 1, this condition requires the presence of an insulating layer of sufficient thickness, leading in our case to resistances R₁ of the order of 1 MΩ to 100 MΩ. Concerning the capacitance, although the tip-island distance is of the same order as the insulating layer thickness (0.5 to 1 nm), the NaCl dielectric constant plays an important role, leading to capacitances C₁ ten times larger than typical values for C₂. By varying the set-point current, the tip-island distance is modified and consequently also R₂ and C₂. The resistance, however, decreases exponentially with decreasing distance, while the capacitance increases only weakly.

Figure 3 shows dI/dV spectra acquired on four different Pb islands. The spectra present the typical signature of CB, i.e., a zero-conductance gap around the Fermi energy (E_F). Notice that bulk Pb is superconducting at the measurement temperature of T = 4.6 K. We neglect here the possible existence of the superconducting gap in the Pb islands because the energy scale of the Coulomb gap is much larger than the one of the superconducting gap of ≤ 1 meV (45). Consequently, we treat the tunneling junctions as if they are in the normal state. The width of the gap is inversely proportional to the island area A_island: the gap is ≈ 120 mV wide for the smallest island (A_island ≈ 20 nm²), and ≈ 18 mV for the largest one (A_island ≈ 220 nm²). This is the expected behavior, since for C₂ < C₁ the gap width is governed by the tip-substrate capacitance C₁. Therefore, as C₁ increases for increasing island area, it leads to narrower gaps. The 220 nm²-island (a triangle with 22 nm side) represents the upper limit for an unambiguous observation of a zero-conductance gap at this temperature. From the measured gap, we estimate C₁ to approximately 10 aF. For the 20 nm²-island, C₁ is ten times smaller. These values are in agreement with values calculated using the expression for a parallel plate capacitor C₁ = ε₀ ε_rA_island/d_NaCl, with the island supported on a 3 ML NaCl film of thickness d_NaCl = 0.8 nm and dielectric constant of bulk NaCl ε_r = 5.5 (46) The small features in Figure 3A just below ± 0.1 eV are related to the capacitance C₂ (see discussion of Figure 5).

FIGURE 3

Figure 3. dI/dV spectra labeled (A to D) acquired on four Pb islands with different area A_island. A Coulomb gap of decreasing size is visible in going from spectrum (A to D) reflecting a Coulomb blockade process.

Figure 4 displays the typical evolution of the tunneling spectra from the CB to the CS regime for a selected Pb triangular island with a side length of 14 nm. The dI/dV spectrum A shows a simple Coulomb gap while spectra B to E display a series of equidistant peaks (steps in the I − V characteristics, a CS) with increasing intensity. This progression is achieved by measuring the tunnel spectra with increasing initial current, corresponding to decreasing tip sample separation and decreasing resistance R₂, in going from spectrum A to E. Our simulation within the orthodox theory reproduces the general features of the measured spectra revealing CB and CS characteristics. The simulation of these spectra within orthodox theory allows us to extract the pertinent parameters of the double barrier system, i.e., the capacitances and resistances of both junctions, as well as the residual non-integer charge Q₀ on the Pb island, see figure caption. Notice that, with decreasing tip-sample distance, R₂ decays exponentially, whereas C₂ increases linearly (13, 18). As this latter variation is small, C₂ is assumed to be constant throughout this series of simulations. To summarize, tunneling spectra on such nanoislands may display a simple CB gap or a CS depending on the parameters of the DBTJ. Well-developed CS's are observed with an asymmetric resistance ratio R₂ < R₁.

FIGURE 4

Figure 4. Evolution from Coulomb blockade to Coulomb staircase: experimental STS data acquired on a Pb island with 14 nm side (A_island ≈ 85 nm²) (dots) and simulation (black line) based on the orthodox theory. The spectra are offset for clarity. Parameters of the simulation: R₁ = 39 MΩ, C₁ = 7.46 aF, and C₂ = 0.7 aF; (A): R₂ = 300 MΩ and Q₀ = −0.15 e, (B): R₂ = 105 MΩ and Q₀ = −0.2 e, (C): R₂ = 26 MΩ and Q₀ = −0.26 e, (D): R₂ = 13 MΩ and Q₀ = −0.26 e, (E): R₂ = 4.5 MΩ and Q₀ = −0.28 e.

Concerning the robustness of the simulations and the precision of the derived parameters we note that Q₀ is always determined with an error of < 10%, as it corresponds to an energy shift of the tunneling spectra with respect to zero bias voltage. The capacitances C₁ and C₂ are also reasonably well determined, especially in situations like those depicted in Figure 5 or Figure 6 where two well separated energies are well-defined. The errors of the determined values are thus also < 10%. The sum of R₁ and R₂ is fixed by the experimentally determined I(V) characteristics. Their ratio is most favorably evaluated in the CS regime, as the height and the width of the various peaks are linked directly to their ratio. In this situation a precision of the order of 10% is obtained (see Figure 4). In the simple CB case, the precision of the resistance ratio is ≥ 10%.

FIGURE 5

Figure 5. Spectra (A to D): increase of the tip-island capacitance C₂ with decreasing tip-island distance (increasing tunnel current). The spectra, offset for clarity as dI/dV(0V) = 0, have been acquired on a Pb island of 14 nm side. Parameters of the simulation, performed for spectra (A) and (D) curve: C₁ = 5.0 aF, C₂ = 0.77 aF, R₁ = 5 MΩ, R₂ = 1700 MΩ, Q₀ = 0.062 e; (D) curve: C₁ = 5.0 aF, C₂ = 0.86 aF, R₁ = 5 MΩ, R₂ = 145 MΩ, Q₀ = 0.07 e.

FIGURE 6

Figure 6. Manipulation of the residual charge Q₀. dI/dV spectra measured on the same island with 7.9 nm side, showing features related to both, C₁ and C₂, with varying Q₀, and corresponding simulations. The residual charge (Q₀) of the island is changed by a bias voltage pulse (duration: 50 μs, current: 1 nA). Parameters of the simulation: C₁ = 3.8 aF, C₂ = 0.335 aF, R₁ = 25 MΩ, R₂ = 2300 MΩ; (A): Q₀ = 0.03 e, (B): Q₀ = 0.33 e, (C): Q₀ = −0.41 e.

A closer comparison of calculations and data in Figure 4 shows that there are some deviations in the intensity of the structures. For the present island of a thickness of 11 ML, a quantum well state, reflecting the vertical confinement, located at +0.2 eV above E_F contributes a considerable background to the spectra (47). Therefore, a quantitative fitting of the tunnel spectra within the orthodox theory should consider the density of states (DOS) of the sample (48).

If the parameters of the DBTJ are appropriately chosen, spectral features reflecting both, tip-island (C₂) and island-substrate (C₁) capacitance can be observed simultaneously. Note that the electrostatic energy of the island is given by the Coulomb charging energy E_c = e²/C_Σ, where C_Σ = ∑ C_i. However, the threshold voltage necessary for the transfer of an electron across the tip-island (island-substrate) junction is inversely proportional to the capacitance C₂ (C₁) (12, 13). Spectra of Figure 5 were acquired on a Pb island of ≈ 14 nm side length. Spectrum A shows a Coulomb gap around E_F as well as two peaks at ≈ ±0.1 V, which are related to the C₂ capacitance. For decreasing tip-island distance (increasing tunnel current), the tip-island capacitance C₂ is expected to increase slightly. This behavior is observed in Figure 5B: by approaching the tip to the island from A to D, the value of C₂ increases leading to a decreased separation between the two C₂ peaks, as indicated by the vertical lines. In addition, if the island area A_island becomes comparable to the effective tip area A_tip, there will be also a contribution to C₂ due to the tip-substrate capacitance. In the present measurement A_island turns out to be smaller than A_tip. C₂ is then evaluated using the simple parallel-plate capacitance formula. In addition to the capacitance between the metallic island and the tip, the capacitance between the tip and the substrate, including the vacuum and the NaCl layers, is considered in the model.

The present measurements and their analysis allow us to obtain the dielectric constant of the ultrathin NaCl film within a parallel plate capacitor model, if we take the island substrate capacitance C₁ from the simulation of the spectra, the area A_island of the capacitor plate from the STM topographies, and the thickness d_NaCl of the NaCl film as deduced from STS of FERs (36) With a typical island size of A_island = 100 nm² and d_NaCl = 0.8 nm (3 ML), we obtain ε_r = 4, a reasonable value compared to the one of ε = 5.5 for bulk NaCl (46), and the one extracted from FERs measurements, ε = 3.5 (36).

In contrast to the situation in a planar tunnel junction (49), our setup has no gate electrode. However, by externally applying a bias voltage pulse to the junction (12), a gate voltage change is mimicked. In Figure 6 we present three tunneling spectra, obtained on the same position on a triangular island of 7.9 nm side, where subsequently a bias voltage pulse of ±5 V has been applied to the junction. The spectra clearly show the result of this action, a pulse-bias-dependent shift of the central Coulomb gap away from the symmetric zero-voltage position, accompanied by a shift of the C₂-related spectral features. The background above +0.2 V originates from the tail of a quantum well state at higher energy (47). This background contributes also to the observed intensity variations of the spectral features. The asymmetric displacement of the central Coulomb gap and of the C₂-related peaks under the action of an external voltage pulse reflects the different values of the fractional residual charge Q₀, obtained from our simulation, on the Pb island.

It has been proposed that the residual charge Q₀ is related to the difference in work function of the various metals constituting the junctions (12):

$\begin{array}{cc}{Q}_0=\left[C_2(\Delta {\varphi }_2)-C_1(\Delta {\varphi }_1)\right]/e& (1)\end{array}$

where Δϕ₁ is the difference between substrate and island work functions, and Δϕ₂ the difference between tip and island work functions. As a result of voltage pulses, charge can be trapped in defects or impurities in the NaCl layer beneath the Pb nanoisland, leading to a permanent modification of the residual charge Q₀.

In the following section we demonstrate the influence of the tip-island junction on the residual charge Q₀ via the variation of C₂, variation due to the displacement of the tip with respect to the Pb island during scanning. In Figures 7A,B two different Pb islands are displayed. In the closeup-view topographic images of Figures 7C,D, very striking concentric lines following the contours of the island are visible. Figures 7E,F shows typical dI/dV spectra obtained on these islands. Our measurements show that, by scanning laterally across the surface, an energy shift of the spectrum is observed which corresponds to a maximum fluctuation of the residual charge Q₀ = ±e/2. We deduced the Q₀ values by simulating each local STS spectrum in the CS regime. For the Pb island shown in Figures 7A,C, we obtain a value of Q₀ ≈ +0.35e for the spectrum acquired at the center of the island, and Q₀ ≈ −0.15e for the one acquired in the surrounding region. For the system of Figures 7B,D, up to six regions of alternating negative and positive Q₀ are detected.

FIGURE 7

Figure 7. (A,B) Topographic images of Pb islands on NaCl; (A): −1 V, 20 pA, (B): 2.5 V, 20 pA; (C,D) Close-up view; (C): −0.15 V, 200 pA; (D): −0.01 V, 20 pA); (E,F) dI/dV spectra acquired on the locations represented by the corresponding symbol, showing different values of Q₀ (see text).

In order to explain this surprising effect we have to consider that the parameters of junction 2 depend on the tip position with respect to the island. As already mentioned, the residual charge Q₀ is related to the difference in work function of the various metals constituting the junctions, see Equation 1. For a given set of measurements C₁ is fixed by the Pb island and the NaCl layer characteristics and the work functions are well defined. In these conditions, Equation 1 shows that a variation of Q₀ can only originate from a variation of C₂. As discussed above, the lateral extension of the tip can exceed the island area. Therefore, to evaluate C₂, not only the capacitance between tip and island, but also the one between tip and substrate has to be included. When scanning over a Pb island, the ratio between these two contributions changes continuously, leading to variations of the residual charge within the island. Now, considering the CS spectra shown in Figures 7E,F, we recall that with increasing bias voltage every additional CS peak in the differential conductance spectrum indicates the hopping of an additional electron to/from the island. In an STM scan at a bias voltage close to a CS peak, e.g., at 0.075 eV in Figure 7E, a small change in Q₀ (due to the changing tip-island capacitance) will lead to a situation where the measurement will be performed above or below the respective CS peak, corresponding to one more (above the peak) or one less electron (below the peak) tunneling to the island. This effect leads to the observed symmetric triangular step patterns in the STM topographies shown in Figures 7A,B. Evidently, the sharp apparent height steps visible in the STM images reflect the hopping of one additional electron on/off the Pb island, while the transition regions correspond to a continuous variation of Q₀. Notice that, contrary to the first impression gained from the images, at each instant the value of the charge Q₀ is the same over the entire island, as expected for a conducting object. These features are visible when C₁Δϕ₁ ≈ C₂Δϕ₂, a condition which can be fulfilled for Δϕ₁ ten times smaller than Δϕ₂, since C₁ is typically ten times larger than C₂. This implies that Δϕ₁ has to be small in order to observe these features. Moreover, Δϕ₂ strongly depends on the specific tip conditions, explaining the fact that these effects were not systematically observed. In this situation, a C₂ variation of the order of 10% can induce a modification of Q₀ as large as ±e/2.

Finally, we note that the observation of Coulomb charge rings with scanning probe methods has been made before for quantum dots inside carbon nanotubes (50), for quantum dots of a two-dimensional electron gas inside semiconductor heterostructures (51), and for quantum dots inside InAs nanowires (52). Controlled charge switching has been performed for single Au atoms on NaCl (53), for a single Si donor in Si doped GaAs (54), and for a small conductive grain on an InAs surface (55). The present measurements on Pb nanoislands show that these intriguing CB phenomena and the related charging effects are also observed in metallic quantum dots. While the observed phenomena are similar they are not identical. The essential difference between the present measurements and the SPM investigations on the quantum dots before is that in our case the STM tip is in the tunneling mode measuring the conductance through the Pb island without a gate electrode, while the former investigations use a standard conductance measurement of the nanowires (quantum dots) and there the SPM tip represents an external electrode the field of which shifts the CS spectra of the observed quantum dots.

4. Conclusions and Outlook

We investigated the transport properties of single crystalline metallic nanostructures in the CB regime. To this end, small Pb islands with well-defined geometries, ranging from a diameter of a few to tens of nanometers and a thickness of a few monolayers are grown on crystalline NaCl layers on Ag(111) substrate. The tunneling spectra obtained at 4.6 K on the metallic quantum dots show clearly the presence of CB phenomena. By reducing the tip-island distance, the tunneling spectra reflect the evolution from CB to CS. Our simulations employing the semi-classical orthodox theory describe well the observed spectra in both regimes allowing us to extract quantitatively the parameters of the DBTJ, i.e., the resistances, capacitances, and the residual charges.

The present STM/STS studies have shown that not only the size and shape of the metallic nanoislands can be well determined but that, importantly, also their local electronic structure and their transport properties can be quantitatively elucidated. In the past, e.g., the superconducting properties of small particles have been investigated (56) revealing a number parity property. STS experiments of isolated superconducting islands would allow us to observe simultaneously the pairing correlation on islands with different parity. Moreover, in small metallic particles containing magnetic impurities, the Kondo resonance was predicted to be strongly affected in a parity-dependent way (4), another exciting phenomenon to look at with local probes. We believe that STM/STS will continue to be a decisive player for the elucidation of the mesoscopic physics of nanostructures.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

We would like to thank K.-H. Müller and C. Flindt for stimulating discussions. We are also indebted to V. S. Stepanyuk, N. N. Negulyaev, P. Ignatiev, K. Horn and D. Roditchev for their very valuable input. Wolf-Dieter Schneider would like to thank H.-J. Freund for stimulating discussions and for his generous hospitality. We acknowledge financial support of the Swiss National Science Foundation.

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