The relevance of electrostatics for scanning-gate microscopy

Scanning-probe techniques have been developed to extract local information from a given physical system. In particular, conductance maps obtained by means of scanning-gate microscopy (SGM), where a conducting tip of an atomic-force microscope is used as a local and movable gate, seem to present an intuitive picture of the underlying physical processes. Here, we argue that the interpretation of such images is complex and not very intuitive under certain circumstances: scanning a graphene quantum dot (QD) in the Coulomb-blockaded regime, we observe an apparent shift of features in scanning-gate images as a function of gate voltages, which cannot be a real shift of the physical system. Furthermore, we demonstrate the appearance of more than one set of Coulomb rings arising from the graphene QD. We attribute these effects to screening between the metallic tip and the gates. Our results are relevant for SGM on any kind of nanostructure, but are of particular importance for nanostructures that are not covered with a dielectric, e.g. graphene or carbon nanotube structures.


Introduction
Scanning-gate microscopy (SGM) is a local probing technique that can unveil microscopic processes governing the conductance of nanostructures. The conducting tip of an atomicforce microscope is used to induce a potential perturbance in the nanostructure, thus altering the conductance of a particular sample as a function of tip position. Conductance maps are then obtained by recording the conductance while scanning the tip over the nanostructure. With this technique, various material systems and different types of nanostructures have been explored in the past, e.g. carbon nanotube-based quantum dots (QDs) [1], GaAsbased quantum-point contacts [2]- [4], quantum wires [5], InGaAs-bases quantum rings [6], GaAs-based QDs [7,8], superconducting single-electron transistors [9] and recently graphene QDs [10].
Scanning a tip over nanostructures causes an unwanted effect: the electrostatic potential experienced by charge carriers in the nanostructure changes during a scan because the moving metallic tip partially screens the electric field emanating from gate electrodes. In other words, the mere presence of the tip alters the transport properties of a nanostructure, even if contact potential differences between tip and gates are compensated. Although this is an obvious effect, its importance has not been discussed in detail in the literature to our knowledge. Only in [1], where the authors vaguely attribute a conspicuous symmetry of the scanning-gate images to the electrostatic influence. However, they do not analyze their findings further.
With the advent of graphene, two aspects may increase the importance of these screening effects. Firstly, graphene samples are typically not covered with a dielectric and secondly, voltages up to several tens of volts are applied to lateral gates. In this paper, we will present SGM on a graphene QD sample. We will demonstrate that these screening effects can lead to apparent shifts of the QD resonances that are bigger than the lithographic extent of the sample. Moreover, under certain experimental conditions more than one set of Coulomb rings appears, which we also attribute to the strong screening effect. Our analysis will be corroborated with the constant-interaction model where we introduce the screening terms phenomenologically. We want to point out that our results are also of importance for other scanning-probe techniques, such as scanning-tunneling microscopy, although our findings are analyzed in the context of SGM.

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LG PG RG sou rce d r a i n

Measurements
We performed SGM on the same graphene QD that has already been discussed in [10]. The sample is shown in figure 1(a), where the bright areas are metallic gates contacting the graphene flake. The latter has been cut into the desired shape by reactive ion etching. The faint edges of the QD are retraced with continuous white lines between source and drain for better visibility of the structure. In-plane graphene gates (edges retraced with white dashed lines) used for tuning the device are denoted by 'LG', 'PG' and 'RG', standing for left, plunger and right gate, respectively. All other gates including the graphene nanoribbon (white, dashed lines to the lower left of the QD) were grounded throughout the experiment. The QD has a lithographic radius of 110 nm and a charging energy of E C = 3.5 meV. The charge neutrality point is around 30 V in back-gate voltage [10]. For a coherent set of data, we present only scanning-gate images obtained in the hole regime for back-gate voltages V BG 25 V. The temperature for the scanning-gate images presented here was between 2 and 2.6 K.
In figure 2(a), we present a scanning-gate image of the QD taken at V BG = 12 V. Coulomb resonances of the QD show up as ring-shaped conductance resonances (in the following called Coulomb rings) and are centered around the (white) cross that may be interpreted as the location of the QD. The conductance is given in color code, where higher conductance is given by brighter, up to yellowish, colors. The strong, ring-like modulation of the Coulomb rings is due to localized states that are located in the constrictions. Their behavior has been analyzed previously [10]. Figure 2   Two sets of Coulomb rings, which sense the same QD, are visible in this regime. By changing side-gate voltages, the centers of the two sets shift in opposite directions by in total I = 430 nm for the lower and II 450 nm for the upper set. The center of the Coulomb rings in (d) for set II cannot be determined. Hence, just a lower bound for the total shift can be given. In all scanning-gate images, conductance is given in color code with For a different back-gate voltage V BG = 16 V, we obtain qualitatively different images: figures 2(c) and (d) present two scanning-gate images where again in-plane gate voltages were changed as denoted in the images. We recognize two sets of Coulomb rings whose respective centers shift by 430 nm and more. This shift is significantly larger than the extent of the QD (diameter about 220 nm) and can therefore not be a real shift of the QD charge distribution.
In summary, we observe two unexpected effects, namely the appearance of more than one set of Coulomb rings and apparent shifts of their centers bigger than the extent of our device. For a correct interpretation of scanning-gate images, especially when we want to extract length scales, it is therefore crucial to understand where these two effects come from and how they can possibly be avoided in future experiments.
For a further analysis, we determined the centers of Coulomb rings of the QD and resonances of localized states in the constrictions for many scanning-gate images taken with this sample in the hole regime. The locations of all the centers are plotted in figure 3 with their corresponding error bars 3 . We distinguish the different resonances-Coulomb resonances of the QD or resonances of localized states in the constrictions-by using different symbols. The outline of the lithographic QD is shown as a black, dashed line to depict a reference for the geometry of the structure. Again, we see that the shifts observed in figure 3 cannot be accounted for by a shift in real space alone because they are much larger than the extent of the structure.

Discussion
There are three possible explanations for the observed shifts of Coulomb-ring centers, namely (i) a drift of the sample holder with respect to the tip over time, (ii) bending of the tip because of electrostatic forces between the tip and the surrounding and (iii) a change in the electrostatic fields and potentials shifting the electrochemical potential of the QD.
In order to make sure that the QD structure was at the same position throughout, topographic scans of the sample were taken in certain intervals. A drift of the sample holder was not detectable and can therefore be excluded. We give a rough estimate of the ratio of the electric force acting on the tip and the mechanical shear force necessary to lead to a deflection of the tip, We approximate the tip-sample capacitance C(z) with a plate capacitor with area A which will lead to a drastic overestimation of the energy and hence of the electric force. The tip-sample separation is assumed to be z = 100 nm and the tip deflection 400 nm (about the largest shift observed in our measurements) for a rather long tip of 500 µm. This yields a deflection angle of γ = 0.4 µm/500 µm. The shear modulus for platinum, which is a good approximation for our PtIr-tip, is G Pt = 6 × 10 10 Pa, and we assume a rather large tip-sample voltage of U = 10 V. The small ratio of 10 −3 that is an upper bound clearly indicates that electrostatic forces cannot lead to significant bending of the tip.
Hence, we are left with the third explanation, namely that the shift of Coulomb rings is due to a change in the electrostatic potentials when tuning gates. In fact, when we take a closer look at figure 2, we see that strong voltage differences between gates (especially in (c) and (d)) lead to a significant distortion of the Coulomb rings. This distortion cannot be explained with a drift of the sample holder or bending of the tip. It rather corroborates the importance of electrostatics for the interpretation of SGM images.
The appearance of two sets of Coulomb rings was previously assigned to a double tip [11]. Double-tip behavior can show up in topographic images as double features since a topographic feature, e.g. an edge of a metallic contact, is imaged by each tip individually. The excellent quality of the topographic images (cf figure 1(a)) indicates that we use a sharp, single tip.
In the following, we will present a different explanation that will lead to double features in SGM as observed in figure 2. For a first qualitative understanding, we only consider the tipdot capacitance C tip (x tip ) and the plunger gate-dot capacitance C PG (x tip ) for tip positions x tip along the (red) arrow in figure 1(a). A plot is shown in figure 1(b). For x tip → ±∞, C tip = 0 and C PG = const. C tip has a maximum somewhere at the dot position x dot , whereas C PG has a minimum when the tip is located somewhere between the dot position x dot and the gate x PG due to screening by the tip.
Following the constant-interaction model, the electrochemical potential of the QD with N -trapped charge carriers is described by where the total capacitance C (x tip ) = C 0 − C tip (x tip ) − C PG (x tip ) depends on the tip position x tip and C tip , C PG < 0 are capacitance coefficients. A constant background capacitance C 0 was incorporated into the total capacitance C . We neglect the single-particle energy since the QD under investigation did not show any signatures of excited states. Coulomb rings are centered at the extrema of µ N (x tip ) in an SGM image. We calculate the derivative ∂µ N /∂ x tip of equation (1) and set it to zero. The total capacitance C depends only weakly on x tip because its value is dominated by the constant background capacitance C 0 . Experimental findings [11] and our numerical model below confirm this approximation. Using ∂C /∂ x tip ≈ 0, we then find This is an implicit equation for the value of x tip where the Coulomb rings are centered. The interpretation of this expression is straightforward with the help of figure 1(b): if the extrema of the two individual capacitances are sufficiently sharp and displaced in space, the equation can be fulfilled for two distinct values of x tip and two sets of Coulomb rings will be observed. Their centers can be shifted by applying different voltages to the tip and the gate. In the following, we want to extend this qualitative model to a more realistic one in two dimensions based on the real QD structure. The details of the model can be found in the appendix. The spatial dependence of the tip capacitance C tip is assumed to have a Lorentzian shape with an amplitude of 2.4 aF and a width of 600 nm deduced from experimental values. We consider three gates corresponding to the left, plunger and right gate of the real device; equation (1) is extended accordingly. Their capacitances to the dot approach a constant value of 4 aF if the tip is placed far away but are screened by the tip for tip positions in the dot-gate region. The precise function is discussed in the appendix. The total capacitance C also includes a constant background capacitance of 46 aF. In figures 2(e) and (f), we show the results of a numerical evaluation of the model for two different gate-voltage settings. We plot the contour lines of the electrochemical potential for different integer values N . This is in analogy to the SGM measurement. Our model clearly resembles the measurements presented in panels (c) and (d): we observe at least two sets of Coulomb rings, and their centers can be shifted in real space by applying different gate voltages. If we apply a higher voltage to a particular gate, the corresponding peak in µ N induced by that gate becomes more pronounced and more Coulomb rings appear. This is also observed in both the measurements and the numerical results.
We see that our model can explain our experimental findings; however, it does not exclude the possibility of the existence of a double tip. Additional support for our model is provided by comparing the scanned resonances with the sample topography: If our argumentation is relevant for our measurements, then we should see a spatial correlation between the gateelectrode arrangement and the appearance of Coulomb rings. In the case that the different sets of Coulomb rings were created by a double or multiple tip, such a correlation would be accidental and is therefore very unlikely. The overlays of the arrangement of the gate electrodes with scanning-gate images in figures 4(a) and (b) show that the pattern of the measured Coulomb rings correlates with the geometry of the device. This is strong evidence that our explanation for the formation of multiple sets of Coulomb rings is relevant for our sample.

Comparison with previous scanning-gate microscopy (SGM) experiments, summary
To our knowledge, a strong shift of the centers of Coulomb rings as observed here has not been reported in the literature yet. If more than one set of rings appeared, it was attributed to a double tip [11]. The question remains as to whether previous results have to be re-interpreted in the light of our findings or whether our findings were irrelevant for samples previously investigated. In [1], the authors describe SGM of carbon nanotubes on a back-gated Si-substrate. In fact, the observed Coulomb rings are strongly distorted and resemble the symmetry of the device. This indicates that the gate-electrode arrangement also plays a significant role. This fact is mentioned in the paper but not further discussed.
Most SGM-experiments have been performed on GaAs QDs [7,11,12]. The interpretation given there for two sets of Coulomb rings, namely a double tip, seems reasonable since no obvious spatial correlation between SGM images and the topography is observed. The authors do not report any significant shifts, either. In their structures, fabricated using AFM-oxidation techniques, gate voltages do not exceed a few hundred mV's-values which can be exceeded by more than a factor of ten in graphene nanostructures. Consequently, any shifts observed in GaAs will not be as pronounced as in graphene. Moreover, in the case of GaAs-dots, the in-plane gates are formed by the 2DEG. Metallic electrodes contacting the 2DEG are tens of micrometers away from the QD. The GaAs capping layer with a dielectric constant ε GaAs ≈ 13 may further reduce the relevance of mutual screening effects between the tip and the in-plane gates.
SGM was also performed on a superconducting SET [9]. It is interesting to compare this structure to graphene QDs because in both cases the sample is fabricated on a SiO 2 -substrate and the metallic gates are very close to the dot and the SET. The SGM images of the SET show a pronounced distortion along the axis connecting the source and drain contacts. This may indicate that mutual screening effects of the tip and the gate electrodes is important for this sample as well. The authors, on the other hand, ascribe the distortion to an asymmetric tip shape. However, the main findings reported in the paper are independent of the detailed interpretation of the distortion.
A fully tunable graphene QD consists of at least five electrodes-source and drain, two gates for tuning the tunnel barriers, and a plunger gate-plus a back gate. The effects described in this paper are intrinsic to a graphene sample and cannot be avoided using more advanced technology. Therefore, it is important to take these effects into account whenever local properties shall be deduced from an SGM image. In principle, it is possible to measure the influence of each individual gate as described in [12] for the tip. Since the contributions to the electrochemical potential are additive, each contribution can be subtracted once determined by measurement. However, this is a tedious task. Additional sets of Coulomb rings and strong shifts of their centers are therefore a phenomenon that probably has to be dealt with again in future scanninggate experiments, particularly on graphene devices.   [11]. All images were calculated for an area of 2 × 2 µm 2 .
even in the absence of the tip. The electrochemical potential is then given by LG + e C RG (r tip ) C (r tip ) V RG + const.
With these expressions at hand, it is now possible to calculate the electrostatic influence in our sample numerically and compare them to our experimental findings. In figure A.2, we present the left-gate capacitance C LG in (a), the resulting left-gate lever arm α LG in (b), the total capacitance C in (c), and the corresponding charging energy E C in (d). All quantities are plotted for tip positions within an area of 2 × 2 µm 2 . The calculated values are in good agreement with experimental findings; in particular the spatial fluctuations of the charging energy of about 10% of its absolute value are confirmed by earlier experiments [11]. This also justifies the approximation ∇ r tip C (r tip ) ≈ 0 in the main paper for the qualitative understanding.
Calculations of the electrochemical potential for a fixed number of charge carriers N = 1 are presented in figure A.3. The outline of the geometry, i.e. the parabolas of the gates where the screening by the tip is strongest, is also depicted there with a dashed line. If we vary the number of charge carriers N and determine contour lines of µ N , we obtain the simulated Coulomb rings shown in figures 2(e) and (f).