Scanning-gate-induced effects and spatial mapping of a cavity

Tailored electrostatic potentials are the foundation of scanning gate microscopy. We present several aspects of the tip-induced potential on the two-dimensional electron gas. First, we give methods on how to estimate the size of the tip-induced potential. Then, a ballistic cavity is formed and studied as a function of the bias-voltage of the metallic top gates and probed with the tip-induced potential. It is shown how the potential of the cavity changes by tuning the system to a regime where conductance quantization in the constrictions formed by the tip and the top gates occurs. This conductance quantization leads to a unprecedented rich fringe pattern over the entire structure. Finally, the effect of electrostatic screening of the metallic top gates is discussed.


Introduction
Top gated structures for nanoelectronic devices formed with two-dimensional electron gases (2DEG) are prospective candidates for future computing. They offer the possibility to build the basic elements, like quantum point contacts (QPCs) [1,2] and quantum dots [3], or cavities in general. Unlike etched structures, top gated devices can be tuned over larger ranges. To gather more information about the depletion of the 2DEG below gates, spatial information is required.
We investigated the formation of a cavity with scanning gate microscopy (SGM). The latter technique proved to be a powerful tool for the investigation of twodimensional systems [4,5]. It gives spatially resolved information on non-local transport measurements. Previous studies using SGM have explored quantum wires [6], graphenebased devices [7,8], nanotube quantum-dots [9], integer and fractional quantum Hall effect in narrow constrictions [10,11,12], interference effects in electron backscattering into QPCs [13], and in tip-formed quantum rings [14].
Those measurements showed that the tip-potential is usually well approximated by a Lorentzian. The properties of electrostatically defined QPCs were investigated by means of standard transport measurements [17]. In this paper we perform scanning gate experiments on a narrow constriction and a circular cavity defined electrostatically by applying a voltage to metallic top gates to determine the size of the tip-induced potential. We examine how the depletion border of a ballistic cavity changes as a function of applied gate voltage. It is also demonstrated that electrostatic screening of the tip-induced potential by the top gates causes a shift of QPC pinch-off regions due to the tip-induced potential away from the expected positions in the center of the top gate defined constrictions by several 100 nm.

Experimental setup
The investigated 2DEG is formed in a molecular-beam-epitaxy-grown GaAs/AlGaAs heterostructure with a density of 1.5 × 10 11 cm −2 , and a mobility of 3.8 × 10 6 cm 2 /Vs at a temperature of 300 mK. It is buried 120 nm below the surface. The electrons have a Fermi wavelength of 65 nm and an elastic mean free path of about 50 µm.
The sample under study is fabricated by etching a conventional Hall bar. On top Au/Ti gates [see figure 1a)] for the two cavities used in the following measurements are placed using electron beam lithography. The segmented design is intended to give flexibility in forming cavities with different diameters (d 1 = 1.0 µm for cavity I with gates g 1 − g 5 , d 2 = 1.5 µm for cavity II g 8 − g 12 ). The lithographic width of the constrictions used as openings of cavity I (gates g 1 and g 4 , g 3 and g 4 , as seen in figure 1) is 0.62 µm. The constrictions used for cavity II (g 8 and g 9 , and g 11 and g 12 ) are 0.4 µm wide.
The experimental setup is a home-built AFM operated in a 3 He cryostat [18] at a base temperature of 300 mK. A Pt/Ir wire, sharpened with chemical wet-etching and consecutive milling with a focussed ion beam is used as the tip. It is glued to a tuning fork sensor, which is controlled by a phase locked loop [19,20]. The structure in the 2DEG is formed by applying negative voltages [V g in figure 1b)] to the top gates thereby decreasing the charge carrier density below the gates. The gate pinch-off is around −0.35 V. Biasing the tip (V tip ≈ −3 .. − 8 V) 60 nm above the GaAs surface depletes the 2DEG underneath, and hence forms a movable gate. The transport measurements are carried out in a two-terminal configuration with a sourcedrain voltage (V SD ) of 100 µV modulated at 27 Hz [ figure 1b)]. The source-drain current (I SD ) is measured by standard lock-in techniques.

Tip depletion size in the 2DEG
Information on the tip-induced potential is needed in order to interpret SGM results. In the following we show four methods which allow us to estimate the radius R tip of the tip-depleted region in the plane of the electron gas.
In figure 2a) the conductance G of cavity II is shown as a function of tip position. The black lines correspond to the edges of the biased top gates (V g8−12 = −0.4 V) which form the structure. The conductance decreases from approximately seven conductance quanta (7 × 2e 2 /h) for the tip at a position where it does not influence the cavity transmittance and zero in the vicinity of the two QPCs. The result are lens-shaped regions close to the two QPCs , labeled A and B in figure 2a) and b), similar as it has been observed in [10].
In order to understand how we can read the approximate size of the tip-depleted region from this image, we first concentrate on the solid green sectional line in figure 2a) and b). Figure 2c) shows the conductance and its derivative along this line together with schematic drawings of the tip position relative to the constriction. It is evident from the data and the schematics that 2R tip ≈ 0.6 µm. At the same time the electronic width of the constriction is seen to be W el ≈ 0.3 µm in agreement with the lithographic size and the depletion width caused by the applied voltage. This crude estimate, which we call method I in the following, regards the tip-depleted region to be hard-wall, simplifies the detailed geometry, and neglects all screening and stray-capacitance effects caused by the surface gates. It should therefore be taken as an order of magnitude estimate. The oscillations in the derivative of the conductance, also seen as fringes in figure 2b), reflect quantized conductance plateaux in the constriction formed between the tip and one of the QPC gates. Figure 2d) illustrates another geometric consideration for estimating R tip from the extent of the lens-shaped region along the green dashed line (method II). One finds where the width of the QPC gate is taken to be a ≈ 0.15 µm, the extent of the lensshaped region l ≈ 0.75 µm, and the electronic width of the constriction W el ≈ 0.3 µm. This result is in agreement with the previous estimate.
In figure 2b) we observe that the last fringe before depletion in the lens-shaped region can be followed into the interior as indicated by the blue dotted lines. These lines run at approximately constant distance from the edge of the gate directly indicating R tip ≈ 0.6 µm (method III) as illustrated in figure 2e). This estimate is again in reasonable agreement with the previous ones, given the fact that the density in the cavity may be enhanced compared to the constrictions (although possibly reduced compared to the bulk), and given the distinct electrostatic environment formed by the surface gates.
All previous estimates of R tip neglected the long-range tails of the tip-induced potential. The long-range capacitive coupling of the tip to a QPC [13] can be used to determine this tail quantitatively. To this end the tip, kept at constant voltage, is placed at several positions along the transport axis of the QPC. At each point the QPC depletion gate-voltage is determined. Using finite-bias spectroscopy this gate-voltage can be calibrated to an energy scale [17]. The resulting data is shown in figure 3a), where the horizontal axis represents the distance from the tip to the center of the QPC.
We fit these data with a lorentzian line shape where E 0 , A, x 0 , and γ are fitting parameters describing an energy offset, the peak amplitude, a position offset, and the line-width, respectively. The particular data shown in figure 3a) lead to E 0 = (0.24 ± 0.03) meV, A = (0.372 ± 0.005) meVnm 2 , x 0 = (−0.085 ± 0.003) nm, and γ = (0.160 ± 0.005) nm. The intersection point of this reconstructed tip-induced potential with the Fermi energy of the electron gas gives another estimate of R tip ≈ (0.17 ± 0.08) µm (method IV). The largest contribution of the uncertainty of this estimate stems from the energy offset E 0 , because this quantity results from the QPC gate-voltage to energy conversion. In figure 3b) R tip of four different tips is shown as a function of tip-voltage. The tip-surface separation and the depth of the 2DEG are the same for all measurements shown (60 nm and 120 nm). The plot confirms that the different methods are consistent for a given tip. At a given tip-voltage different values of R tip are brought about by unintentional differences in tip fabrication and by modifications of the tip shape during topography scans. The radius of the tip-depleted region is found to increase linearly with the tip voltage. The change of R tip with V tip is approximately 80 nm/V for tips of any radius.

Forming a cavity with the top gates
The aim of this section is to show how much the depletion width at the borders of the gate-defined cavity I [see figure 1 a)] changes with gate voltage. In order to get such spatial information, a set of 2d scans with the biased AFM tip and varying gate voltages is taken. For a first set of five scans the voltage on g 4 is varied while the voltage on g 1 and g 3 is kept constant at −0.55 V. For the second set the roles of g 4 and g 1 , g 3 are interchanged. The first set, shown in figure 4b)-f), leads to a fringe pattern in dG/dx filling the whole cavity. In figure 4a) the conductance G(x, y) corresponding to figure 4b) is given. The origin of the fringes is the same as in figure 2b): a quantized constriction forms between the tip-depleted region and one of the cavity gates. There are two groups of fringes, group I/II related to gate g 1 and g 3 , and group III/IV related to g 4 [see labeling in figure 4 c)]. The sequence of images in figure 4b)-f) shows that the group III/IV fringes shift in space with changing V g4 , whereas group I/II stays in place. This shift contains the desired quantitative information about the change of the depletion width.
The exact positions of the fringes can be extracted from the cuts [ figure 5a)] along the green dashed line shown in figure 4b) for the five gate voltages applied to g 4 . The fringes are labeled starting from the center of the cavity. These positions are indicated by filled circles in figure 5a). In figure 5b) we plot these points and fit them linearly as a function of V g using where ∆V g = V pinch−off − V g is the difference of gate voltage from the gate pinch-off (determined to be -0.35 V), and l 0,i is an arbitrary length offset irrelevant for the determination of the α i , with i as the fringe number.
In figure 5c) we show the α i determined from all scans. In addition, with the cavity divided into four regions I-IV [see figure 4c)], one characteristic cross-section is analyzed in each region for each scan. Points connected by solid lines refer to the situation where the constriction forms between the tip and the gate that is varied (case 1). Points connected by dashed lines refer to the situation where the gate is varied whose action on the constriction is screened by the tip (case 2). In the latter case the values of α i are smaller, and they increase with fringe number (tip position) due to reduced screening of the gate voltage by the tip. The α 1 parameter, which indicates the change of the depletion width, is of the same order of magnitude for all regions. At the same time the α i vary only very little within each region in case 1.

The origin of the fringe pattern shape
The shape of the fringe pattern does not reflect the cavity gate outline, instead it can be divided into four regions indicated in figure 4c). The fringes in these regions surround the lens-shaped regions A' and B' in figure 4a). This suggests that the constrictions involving the tip form mainly with the openings of the cavity, similar as discussed for figure 2. Additionally, the lens-shaped regions are shifted into the cavity from the geometric center of the constriction. A striking observation is made when the tip moves along the dotted line from point α to β in figure 4b). While we would naively expect the conductance to increase we observe a decrease. Tentatively we ascribe this effect to enhanced screening of the tip-induced potential by the surface gates. By moving the tip closer to the constriction, the distance to the surface gates decreases, the tip-induced Figure 5. a) Cut of the dG/dx maps of the set of varying g 4 . The cut direction is the same as in figure 4a), but only for region IV, starting from the center. b) Position of the conductance plateaux (fringes in dG/dx) along the cut of the different gate voltages. c) All shift parameters for the fringes of the different areas of both sets of measurements. Figure 6. Simulations of the electrostatic potential on the 2DEG induced by the biased tip. The gate configuration is the same as in the measurements above. The gate g 2 (grey) is grounded, g 1 , g 3 , and g 4 (black) are biased . The insets show a 2d map at the Fermi energy, thus the depletion of the 2DEG. a) The tip is placed in the center of the entrance of the cavity. b) The tip is moved towards the cavity center until it blocks the constriction. potential gets increasingly screened, and the 2DEG is no longer depleted below the tip. Figure 6 shows electrostatic simulations supporting this interpretation. Calculations were carried out with COMSOL treating the 2DEG as a grounded metallic plane 100 nm below the metallic top gates. The GaAs material was modeled as a dielectric with = 13. The tip, implemented as a metallic cone with a hemisphere with radius 50 nm at its end, is placed 70 nm above the surface. We determine the induced density in the 2DEG and consider regions to be depleted if the induced density exceeds the sheet density of the electron gas.
In figure 6a) the tip is placed in the opening of the cavity [position α in figure 6a)]. The screening of the induced potential by the gate g 2 leads to an open conductance channel, as indicated in the inset showing the depletion area below the gates and the tip. In figure 6b) the tip is moved along the dotted line towards the cavity center until the constriction is closed (position β ). These simulations show that the zero conductance regions A' and B' are shifted relative to the constriction center into the cavity in the presence of grounded gates close to the constriction.

Conclusion
We have presented methods for estimating the size of the tip-induced depletion region in the 2DEG using a biased AFM tip and the investigation of the shape of a ballistic cavity. Even though most of the methods use simplified geometric assumptions their errors may play a minor role compared to electrostatic screening effects encountered in the experiments. But even with such limitations, fully quantized transport resulting in an unprecedented clear fringe pattern covering the entire stadium is observed. The findings are pointing towards the accessibility of the local density of the electronic states in ballistic cavities for optimized structures regarding tip potential screening by the top gates.