Edge channel confinement in a bilayer graphene n–p–n quantum dot

To deplete the region under the SG, we apply a positive voltage to the BG and a negative voltage to the SG (or vice versa). This leads to an asymmetry between the two graphene layers, which results in the opening of a band gap [22]. A detailed characterization of the band gap of this device at $B=\,0\,{\rm{T}}$ can be found in [26]. To find the strongest depletion of the region under the SG in a finite magnetic field, we measure the resistance as a function of SG voltage V_SG and BG voltage V_BG at $B=2\,{\rm{T}}$ (figure 2(a)) and identify the line of highest resistance (white dashed line). For this measurement the channel gate voltage was fixed at a large negative value (${V}_{\mathrm{CH}}=-12\,$V). With this channel gate voltage setting, the channel is depleted or p-doped for the entire range of the measurement and has a small coupling to the n-doped reservoirs (see also figure 2(b)). With a highly resistive channel and a highly n-doped bulk, the changes in resistance observed in this measurement stem primarily from the region under the SG (see insets). If multiple regions of the device had a considerable influence on the resistance, features with different slopes would be expected, because of different gate capacitances in different regions. The diagonal line of high resistance corresponds to the charge neutrality point under the SG. Along this line the displacement field increases in the direction of the arrow. We observe an increase of the resistance by two orders of magnitude along this line, ensuing from the increasing band gap under the SG. The high resistance achieved is important for the formation of a well defined quantum dot. In all following measurements the SG voltage is adjusted when the BG voltage V_BG is varied (see white dashed line in figure 2(a)), so as to keep the region under the SG depleted. We denote this by ${V}_{\mathrm{BG}}^{{\prime} }$.

Zoom In Zoom Out Reset image size

We now focus on the effect of the channel gate. The conductance at $B=2\,{\rm{T}}$ as a function of ${V}_{\mathrm{BG}}^{{\prime} }$ and V_CH is shown in figure 2(b). In the upper right corner, both the bulk and the channel are n-doped and therefore quantized conductance is observed, similar to [35, 36]. The observed lines with negative slope are defined by a constant filling factor in the channel. The transitions between the Landau levels are marked by their respective filling factors. They are used to determine the dependence of the charge carrier density on the gate voltages. The conductance inside the channel reaches $G=16$ e²/h, which shows that the edge channels of at least four Landau levels fit inside the channel. The extent of the wave function of the fourth Landau level is ${l}_{B}\sqrt{2N+1}=55\,\mathrm{nm}$, where l_B is the magnetic length. This extent is indeed smaller than the lithographic width of the channel. In the hatched area at the bottom of the figure the conductance is much lower. In this regime, where the channel is p-doped while the bulk is n-doped, a quantum dot as sketched in figure 1(c) forms.

Figure 3 shows the main result of this work. In figure 3(a) the conductance as a function of channel gate voltage and magnetic field is shown. For this measurement a dc bias of ${V}_{\mathrm{SD},\mathrm{DC}}=200\,\mu {\rm{V}}$ was applied to enhance the signal to noise ratio. Lines of higher conductance with a negative slope are observed. To enhance the visibility, we subtract a smoothened background (figure 3(b)). When moving in the direction of increasingly negative channel gate voltage, we can interpret each resonance as the addition of an individual charge carrier to the confined area. The negative slope implies that an increase of the magnetic field leads to the removal of charge carriers from the dot. This trend can be understood as follows: when the magnetic field increases, the hole-like Landau levels shift down in energy. Since the Fermi level in the dot is pinned by the reservoirs, this leads to a decrease of the number of occupied states in the dot. In the case of Aharonov–Bohm interferometry, which typically requires smaller dots [32], the opposite slope would be expected: an increase in magnetic field increases the flux through the interferometer, which has the same effect as increasing the area by applying a more negative gate voltage.

Zoom In Zoom Out Reset image size

Charging lines of quantum dots in a magnetic field often show a slope related to a certain filling factor [41, 42]. The slope of the lines in figure 3(b) is close to a filling factor of four inside the channel, but the error bar on the density inside the channel does not allow for a quantitative comparison. To calculate the density, we extracted the capacitance between the conducting channel and the channel gate C_CH from a Landau fan measured in a previous cool down (see [26]) using a plate capacitor model. We also extract the same capacitance from several maps of the conductance at fixed magnetic field, such as figure 2(b). The numbers we find are off by 30% and we therefore conclude that our estimates of charge carrier density and filling factor inside the channel have an error bar of 30%. Because of the vicinity of the SG to the channel, it could be the case that C_CH depends on the SG voltage. Moreover, C_CH might depend on the extent of the wave function inside the channel. These two factors are not accounted for in a simple capacitance model.

In figure 3(c) a line cut of figure 3(a) is shown. The spacing between the peaks is given by ${\rm{\Delta }}{V}_{\mathrm{CH}}=0.2$ V. For the addition of a single charge carrier, this corresponds to a capacitance between the dot and the channel gate of ${C}_{\mathrm{CH}}=e/{\rm{\Delta }}V=0.8\,$ aF, in rough agreement with the capacitance of ${C}_{\mathrm{CH}}=1.3\,$ aF extracted using the gate voltage spacing of the Landau levels in figure 2(b). For the latter estimate a dot area of $A=0.02\,\mu {{\rm{m}}}^{2}$ (the lithographic size of the dot) was assumed. The significant background signal indicates that, apart from the channel exhibiting Coulomb blockade, there is another conductive channel through the dot. The conductive background decreases with increasing magnetic field (see figure 3(a)), but above $B=3\,{\rm{T}}$ the tunnel coupling to the reservoirs also gets weaker, because the distance between edge channels increases. Therefore we do not obtain a clearer signal at higher magnetic fields. The line cut shows a series of alternating low and high current peaks. This is reminiscent of the work on GaAs dots by Baer et al [33], in which the pattern was explained by a different tunnel coupling to the reservoirs for the inner and outer edge channel present in the dot.

The conductance at $B=2\,{\rm{T}}$ as a function of density in the bulk n_BG and density in the channel n_CH (extracted from figure 2(b)) is shown in figure 3(d). In figure 3(e) the same measurement is shown with a smoothened background subtracted. From the data it is apparent that the resonances (marked by dashed lines) are independent of the density in the bulk of the device and only depend on the density inside the dot for the entire range of the measurement. This is in agreement with our expectation, since the density in the bulk is much higher than the density in the dot. It also shows that the dot is clearly located inside the channel, making a disorder induced dot as in [15] highly unlikely.

Finite bias measurements were performed to determine the energy spacing of the levels in the quantum dot. The differential conductance as a function of dc bias and channel gate voltage (see figure 4) at $B=2.33\,{\rm{T}}$ shows pairs of Coulomb diamonds (see dotted red lines), with a charging energy on the order of 1 meV. By approximating the dot as a disk in an infinite medium with a self capacitance of $C=8\epsilon {\epsilon }_{0}r$, we can extract a radius of $r=500$ nm. This is an upper bound for the dot radius, since the presence of closeby metal gates alters the capacitance of the dot significantly [43].

Zoom In Zoom Out Reset image size