Abstract
We present transport measurements on a strongly coupled graphene quantum dot in a perpendicular magnetic field. The device consists of an etched singlelayer graphene flake with two narrow constrictions separating a 140 nm diameter island from source and drain graphene contacts. Lateral graphene gates are used to electrostatically tune the device. Measurements of Coulomb resonances, including constriction resonances and Coulomb diamonds prove the functionality of the graphene quantum dot with a charging energy of approximately 4.5 meV. We show the evolution of Coulomb resonances as a function of perpendicular magnetic field, which provides indications of the formation of the graphene specific 0th Landau level. Finally, we demonstrate that the complex pattern superimposing the quantum dot energy spectra is due to the formation of additional localized states with increasing magnetic field.
Introduction
Graphene [1,2], a twodimensional solid consisting of carbon atoms arranged in a honeycomb lattice has a number of unique electronic properties [3], such as the gapless linear dispersion, and the unique Landau level (LL) spectrum [4,5]. The low atomic weight of carbon and the low nuclear spin concentration, arising from the ≈99% natural abundance of ^{12}C, promises weak spin orbit and hyperfine coupling. This makes graphene a promising material for spintronic devices [6,7] and spinqubit based quantum computation [811]. Additionaly, the strong suppression of electron backscattering [4,5] makes it interesting for future high mobility nanoelectronic applications in general [12,13]. Advances in fabricating graphene nanostructures have helped to overcome intrinsic difficulties in (i) creating tunneling barriers and (ii) confining electrons in bulk graphene, where transport is dominated by Klein tunnelingrelated phenomena [14,15]. Along this route, graphene nanoribbons [1622] and quantum dots [2330] have been fabricated. Coulomb blockade [2325], quantum confinement effects [2628] and charge detection [29] have been reported. Moreover, graphene nanostructures may allow to investigate phenomena related to massless Dirac Fermions in confined dimensions [24,3136]. In general, the investigation of signatures of graphenespecific properties in quantum dots is of interest to understand the addition spectra, the spin states and dynamics of confined graphene quasiparticles.
Here, we report on tunneling spectroscopy (i.e. transport) measurements on a 140nm graphene quantum dot with open barriers, which can be tuned by a number of lateral graphene gates [37]. In contrast to the measurements reported in Ref. [27] the more open dot in the present investigation enables us to observe Coulomb peaks with higher conductance and the larger dot size reduces the magnetic field required to see graphene specific signatures in the spectra. We characterize the graphene quantum dot device focusing on the quantum dot Coulomb resonances which can be distinguished from additional resonances present in the graphene tunneling barriers. We discuss the evolution of a number of Coulomb resonances in the vicinity of the charge neutrality point in a perpendicular magnetic field from the lowfield regime to the regime where Landau levels are expected to form. In particular, we investigate the device characteristics at elevated perpendicular magnetic fields, where we observe the formation of multipledots giving rise to (highly reproducible) complex patterns in the addition spectra.
Device fabrication
The fabrication process of the presented graphene nanodevice is based on the mechanical exfoliation of (natural) graphite by adhesive tapes [24,25,28]. The substrate material consists of highly doped silicon (Si^{++}) bulk material covered with 295 nm of silicon oxide (SiO_{2}), where thickness (and roughness) of the SiO_{2 }top layer is crucial for the Raman [38] and scanning force microscope based identification of singlelayer graphene flakes. Standard photolithography followed by metallization and liftoff is used to pattern arrays of reference alignment markers on the substrate which are later used to reidentify the locations of individual graphene flakes on the chip and to align further processing patterns. The graphene flakes are structured to submicron dimensions by electron beam lithography (EBL) and reactive ion etching based techniques to fulfill the nanodevice design requirement. After etching and removing the residual resist, the graphene nanostructures are contacted by an additional EBL step, followed by metallization and liftoff.
A scanning force microscope image of the final device studied here is shown in Figure 1a. The approximately 140 nm diameter graphene quantum dot is connected to source (S) and drain (D) via two graphene constrictions with a width of ≈75 nm and a length of ≈25 nm, both acting as tunneling barriers. The dot and the leads can be further tuned by the highly doped silicon substrate used as a back gate (BG) and three inplane graphene gates: the left side gate (LG), the plunger gate (PG) and the right side gate (RG). Apart from the geometry, the main difference of this sample compared to the device presented in Ref. [27] is the higher root mean square variation of the height (r_{h}) on the island. While there are no visible resist residues on the island of the sample in Ref. [27] with r_{h }≈ 0.35 nm, there are many dotlike residues on the sample presented here giving r_{h }≈ 1.1 nm.
Figure 1. Device characterization. (a) Scanning force microscopy of the graphene quantum dot device. The overall chemical potential of the device is tuned by a global back gate, where as the right side gate (RG) is used for local asymmetric tuning. The extension of the dot is around 140 nm with 75 nm wide and 25 nm long constrictions. The white dashed lines delineating the quantum dot perimeter are added for clarity. (b) Measurement of the source (S)drain (D) conductance for varying back gate voltage showing a transport gap from around 5 to 3 V (V_{b }= 200 μV). (c) Coulomb diamond measurements in the gap showing a charging energy of around 4.5 meV. This energy is lower than what has been measured in an other dot of similar size (Ref. [26]), most likely because of the increased coupling to the leads. The arrows point to faint lines outside the diamonds. The extracted energy difference of around 1 meV is a reasonable addition energy for excited states. Note that for the measurement in (c), in addition to the BG the right side gate was changed according to V_{rg }= 0.57·V_{bg }1.59 V.
Measurements
All measurements have been performed at a base temperature of T = 1.8 K in a variable temperature cryostat. We have measured the twoterminal conductance through the graphene quantum dot device by applying a symmetric DC bias voltage V_{b }while measuring the sourcedrain current through the quantum dot with a noise level below 10 fA. For differential conductance measurements a small AC bias, V_{b,ac }= 100 μV has been superimposed on V_{b }and the differential conductance has been measured with lockin techniques at a frequency of 76 Hz.
In Figure 1b we show the conductance G_{qd }as a function of back gate voltage at low bias (V_{b }= 200 μV) highlighting the strong suppression of the conductance around the charge neutrality point (5 < V_{bg }< 3 V) due to the socalled transport gap [1922]. Here we tune transport from the hole to the electron regime, as illustrated by the left and the right inset in Figure 1b. The large number of resonances with amplitudes in the range of up to 0.1 e^{2}/h inside the gap region may be due to both, (i) resonances in the graphene constrictions acting as tunneling barriers [4] (and thus being mainly responsible for the large extension of this transport gap) and (ii) Coulomb resonances of the quantum dot itself (see also examples of Coulomb diamonds in Figure 1c). At room temperature these resonances disappear and a conductance value of 0.76 e^{2}/h is measured at V_{bg }= 0 V.
Coulomb blockade measurements at B = 0 T
By focusing on a smaller back gate voltage range within the transport gap (indicated by the dashed lines in Figure 1b) and measuring the conductance as a function of V_{bg }and the right side gate V_{rg }much more finestructure appears, as shown in Figure 2. A large number of resonances is observed with sequences of diagonal lines (see white lines in Figure 2) with different slopes, corresponding to different lever arms (α's). By sweeping the right side gate (V_{rg}) we break the leftright symmetry of the transport response (see also Figure 1a). This allows us to distinguish between resonances located either near the quantum dot or the left and right constriction. The steeper the slope in Figure 2 the less this resonance can be electrostatically tuned by the right side gate and, consequently, the larger the distance between the corresponding localized state and the right side gate. Subsequently, the steepest slope (II, corresponding to ) can be attributed to resonances in the left constriction and the least steepest slope (III, ) belongs to resonances in the right constriction. Both are highlighted as white dashed lines in Figure 2. The Coulomb resonances of the quantum dot appear with an intermediate slope (I, ) and exhibit clearly the smallest spacing in back gate voltage, ΔV_{bg }≈ 0.1 V. This is a good indication that they belong to the largest charged island in the system, which obviously is the 140 nm large graphene quantum dot, which is much larger than the localized states inside the graphene constrictions acting as tunneling barriers.
Figure 2. Conductance of the quantum dot with varying right gate and back gate voltage measured at bias voltage V_{b }= 200 μV. Coulomb resonances and modulations of their amplitude with different slopes are observed (dashed white lines). The extracted relative side gate back gate lever arms are , and . Lever arm (III) is attributed to resonances in the right constriction which are strongly tuned by the right side gate. In contrast resonances with lever arm (II) are only weakly affected by the right side gate and therefore attributed to states in the left constriction. The periodic resonances marked with (I) are attributed to resonances in the dot in agreement with the intermediate slope.
Corresponding Coulomb diamond measurements [39], that is, measurements of the differential conductance as a function of bias voltage V_{b }and V_{bg }(i.e. V_{rg }= 0.57·V_{bg } 1.59 V) have been performed along the (diagonal) solid gray line in Figure 2 and are shown in Figure 1c. From the extent of these diamonds in bias direction we estimate the average charging energy of the graphene quantum dot to be E_{c }= 4.5 meV, which is in reasonable agreement with the size of the graphene quantum dot [23,25,26]. Moreover, we observe faint strongly broadened lines outside the diamonds running parallel to their edges, as indicated by arrows in Figure 1c. The extracted energy difference of roughly 1 meV is reasonable for electronic excited states in this system [26].
Coulomb resonances as a function of a perpendicular magnetic field
In Figure 3 we show a large number of Coulomb resonances as function of a magnetic field perpendicular to the graphene sample plane. The measurement shown in Figure 3a has been taken in the back gate voltage range from V_{bg }= 5 to 3.5 V, at V_{rg }= 0 V (highlighted by the horizontal line (A) in Figure 1b). Thus we are in a regime where transport is dominated by holes (i.e. we are at the left hand side of the charge neutrality point in Figure 1b), which is also confirmed by the evolution of the Coulomb resonances in the perpendicular magnetic field as shown in Figure 3a. There is a common trend of the resonances to bend towards higher energies (higher V_{bg}) for increasing magnetic field, in good agreement with Refs. [27,28,3234]. The finite magnetic field introduces an additional length scale which competes with the diameter d of the dot. Therefore, the ratio d/ℓ_{B }is a relevant parameter for the observation of Landau levels in graphene quantum dot devices. Here, the comparatively large size (d ≈ 140 nm) of the dot promises an increased spectroscopy window for studying the onset and the formation of Landau levels in graphene quantum dots in contrast to earlier work [27,28] (where d ≈ 50 nm). Moreover, we expect that in larger graphene quantum dots, where the surfacetoboundary ratio increases edge effects should be less relevant. In Figure 3a, c, d we indeed observe some characteristics of the FockDarwinlike spectrum [3234] of hole states in a graphene quantum dot in the near vicinity of the charge neutrality point: (i) the levels stay more or less at constant energy (gate voltage) up to a certain Bfield, where (ii) the levels feature a kink, whose Bfield onset increases for increasing number of particles, and (iii) we observe that the levels convergence towards higher energies (see white dashed lines in Figure 3a). The pronounced kink feature (see arrows in Figure 3c, d) indicate filling factor ν = 2 in the quantum dot. However, this overall pattern is heavily disturbed by additional resonances caused by localized states, regions of multidot behavior, strong amplitude modulations due to constriction resonances and a large number of additional crossings, which are not yet fully understood. This becomes even worse when investigating the electron regime (see horizontal line (B) in Figure 1b), as shown in Figure 3b. Individual Coulomb resonances can (only) be identified for low magnetic fields B < 2 T and a slight tendency for their bending towards lower energies might be identified (please see white dashed lines in Figure 3b). For magnetic fields larger than 3 T it becomes very hard to identify individual Coulomb resonances in the complex and reproducible conductance pattern.
Figure 3. Evolution of Coulomb peaks under the influence of a magnetic field in different gate voltage regimes (V_{b }= 200 μV). (a) More on the hole side. (b) More on the electron side. In contrast to (a) V_{rg }= 2.15 V is applied to the right gate in (b). The effect of the right gate to the dot is taken into account in the back gate scale to allow comparison with Figure 1b. (c, d) Reproducibility of the measurement for different magnetic field sweep directions (07 T in (c), 70 T in (d)). The right side gate is changed according to V_{rg }= 0.57·V_{bg } 1.59 V (see Figure 2), with an applied bias of V_{b }= 200 μV.
In order to demonstrate the reproducibility of these complex patterns we show an up (Figure 3c) and a down (Figure 3d) sweep of the very same B  V_{bg }parameter space. These two measurements, have different resolution and thus different sweep rates in both the B and V_{bg }direction. However, all the individual features are highly reproducible (but hard to understand) despite the fact that we find some small hysteresis in magnetic field for B < 3 T (see white arrows in Figure 3c, d). The origin of the complex patterns shown in Figure 3 can be understood when having a closer look at charge stability diagrams (such as Figure 2) for different magnetic fields.
In Figure 4a we show an example of a sequence of dot Coulomb resonances in the V_{rg}V_{bg }plane. The slope corresponding to and the spacing of ΔV_{bg }≈ 0.1 V are in good agreement with Figure 2, and lead to the conclusion that we observe single quantum dot behaviour over a large parameter range. However, if we measure the current in the very same V_{rg } V_{bg }parameter space at B = 7 T the pattern changes significantly and the diagonal lines are substituted by a strong hexagonal pattern (see dashed lines) typical for two coupled quantum dots [40]. The two states forming the hexagon pattern show relative lever arms of and . While the resonances with are attributed to the original dot, corresponds to a new and strongly coupled localization formed close to the right constriction. Additional resonances from the right constriction with (see above) are still visible.
Figure 4. Dot conductance as a function of right gate and back gate voltage at a magnetic field of (a) 0 T and (b) 7 T. The spectrum is dominated by dot resonances marked with the solid line in (a) with a relative lever arm of (see also Figure 2). (b) At a magnetic field of 7 T a hexagon pattern with two characteristic slopes is observed. Their corresponding lever arms are attributed to the dot and origin around the right constriction.
We interpret the magnetic field dependence in the following way. At low but increasing magnetic field we see in almost all measurements an increase of the conductance through the dot (see, e.g. Figure 3). Assuming diffusive boundary scattering such a conductance onset in magnetic field occurs due to reduced backscattering [41] and has been observed in other measurements on graphene nanoribbons [42,43]. The maximum conductance is reached around B ≈ 1.5 T corresponding to a magnetic length nm in rough agreement with the size of the constrictions. As the magnetic field is further increased the complex pattern with many crossings starts to emerge, attributed to the formation of additional quantum dots around the right constriction with strong coupling to the original dot. The formation of such localized puddles is understood as a consequence of the increased magnetic confinement where ℓ_{B }is getting smaller than the extension of potential valleys induced by disorder.
Conclusion
In summary, we have presented detailed studies of transport through an open and larger graphene quantum dot (compared to Ref. [27]) in the vicinity of the charge neutrality point as a function of perpendicular magnetic field. The evolution of Coulomb resonances in a magnetic field showed the signatures of Landau level formation in the quantum dot. Indications for the crossing of filling factor ν = 2 are obtained by the observation of kinks in spectral lines before bending towards the charge neutrality point. However, the observation is disturbed by the formation of a pronounced additional localized state at high magnetic fields in the vicinity of the right constriction. Although the use of open constrictions enhances the visibility of the Coulomb peaks and reduces the transportgap region, emerging pronounced parasitic localized states make the analysis very difficult. For a further indepth analysis of the addition spectra around the electronhole crossover, it is hence beneficial to minimize the amount of disorder and to use clearly defined constrictions. These should be thin compared to the dot diameter to get different energy scales for quantum dot resonances and constriction resonances, which are easy to distinguish. However, the constrictions need to be wide enough to enable conductance measurements around the electronhole crossover without a charge detector.
Abbreviations
BG: back gate; EBL: electron beam lithography; LL: Landau level; LG: left side gate; PG: plunger gate; RG: right side gate; SiO_{2}: silicon dioxide.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
KE, TI, CS and JG designed the experiment. JG fabricated the sample. TF and JG carried out the transport measurements. All authors analyzed the measurements. JG and CS wrote the paper. All authors read and approved the final manuscript.
Acknowledgements
The authors wish to thank F. Libisch, P. Studerus, C. Barengo, F. Molitor and S. Schnez for help and discussions. Support by the ETH FIRST Lab, the Swiss National Science Foundation and NCCR nanoscience are gratefully acknowledged.
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