We demonstrate that excited states in single-layer graphene quantum dots can be detected via direct transport experiments. Coulomb diamond measurements show distinct features of sequential tunneling through an excited state. Moreover, the onset of inelastic co-tunneling in the diamond region could be detected. For low magnetic fields, the position of the single-particle energy levels fluctuate on the scale of a flux quantum penetrating the dot area. For higher magnetic fields, the transition to the formation of Landau levels is observed. Estimates based on the linear energymomentum relation of graphene give carrier numbers of the order of 10 for our device.Graphene [1,2], the first real two-dimensional (2D) solid, consists of a hexagonal lattice of carbon atoms providing highly mobile electrons [3,4] for future applications in electronics, spintronics [5] and information processing [6]. However, confinement of charge carriers in graphene cannot be achieved as easily as in conventional two-dimensional electron gases by using electrostatic gates because of the gapless nature of graphene [2] and a relativistic phenomenon called Klein tunneling [7,8]. Cutting graphene into a desired geometry is an alternative to overcome this obstacle. Well-controlled nanostructures, such as nanoribbons [9, 10, 11], quantum interference devices [12,13,14], and single-electron transistors [15,16,17] have been created in several labs to date. Small spin-orbit and hyperfine interactions have been theoretically predicted [18], promising spin decoherence times superior to the GaAs material system in which solid-state spin qubits are most advanced today [19,20]. Therefore, the identification of individual orbital quantum states, well established in GaAs quantum dot devices, have so far remained on the wish list of physicists aiming at quantum information processing with graphene.An atomic force microscope image of our quantum dot (QD) is shown in Fig. 1a. It was fabricated with the standard procedure: Mechanical exfoliation of natural graphite led to single-layer graphene flakes. The desired structure was defined with electron beam lithography and subsequently cut using reactive ion etching based on Ar and O 2 . Contacts were also defined with electron beam lithography; then gold contacts were evaporated on top [15]. The single layer quality was experimentally verified with Raman spectroscopy [21]. The QD device consists of two about 60 nm and 70 nm wide graphene constrictions separating source (S) and drain (D) contacts from the graphene island (diameter 140 nm). The island can be tuned by a nearby plunger gate (PG), whereas the overall Fermi level is adjusted with a highly doped silicon back gate (BG). The sample was annealed for about 24 hours at 400 K directly before cool down. The experiments were carried out in a dilution refrigerator at a base temperature of 40 mK. Measuring the current I through the QD as a function of back gate voltage V bg allows us S D 200nmCurrent ( to identify a transport gap [15] extending roughly from V bg ...

We review transport experiments on graphene quantum dots and narrow graphene constrictions. In a quantum dot, electrons are confined in all lateral dimensions, offering the possibility for detailed investigation and controlled manipulation of individual quantum systems. The recently isolated two-dimensional carbon allotrope graphene is an interesting host to study quantum phenomena, due to its novel electronic properties and the expected weak interaction of the electron spin with the material. Graphene quantum dots are fabricated by etching mono-layer flakes into small islands (diameter 60-350 nm) with narrow connections to contacts (width 20-75 nm), serving as tunneling barriers for transport spectroscopy. Electron confinement in graphene quantum dots is observed by measuring Coulomb blockade and transport through excited states, a manifestation of quantum confinement. Measurements in a magnetic field perpendicular to the sample plane allowed to identify the regime with only a few charge carriers in the dot (electron-hole transition), and the crossover to the formation of the graphene specific zero-energy Landau level at high fields. After rotation of the sample into parallel magnetic field orientation, Zeeman spin splitting with a g-factor of g ≈ 2 is measured. The filling sequence of subsequent spin states is similar to what was found in GaAs and related to the non-negligible influence of exchange interactions among the electrons.

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Probing techniques with spatial resolution have the potential to lead to a better understanding of the microscopic physical processes and to novel routes for manipulating nanostructures.We present scanning-gate images of a graphene quantum dot which is coupled to source and drain via two constrictions. We image and locate conductance resonances of the quantum dot in the Coulomb-blockade regime as well as resonances of localized states in the constrictions in real space.Graphene has sparked intense research among theorists and experimentalists 1,2 alike since its first successful fabrication in 2004. 3 This is mainly due to graphene's extraordinary band structure, a linear relationship between energy and momentum without a band gap. The gapless band structure, however, prohibits confining charge carriers by using electrostatic gates. Hence, lateral confinement in graphene relies on etched structures and the appearance of a transport gap in graphene * To whom correspondence should be addressed † constrictions. [4][5][6] Nevertheless, already the first experiment on graphene nanoribbons by Han et al. 7 showed a discrepancy between the measured transport gap and a simple confinement-induced band gap. Theoretical models explain the observed gap by Coulomb blockade, edge scattering, and/orAnderson-type localization due to edge disorder. [8][9][10][11] On the experimental side, there is increasing evidence for Coulomb-blockade effects in nanoribbons. 6,[12][13][14][15] Transport through graphene quantum dots in the Coulomb blockade regime is typically modulated by resonances arising from the constrictions. 16 However, for both, nanoribbons and quantum dots, the microscopic origin of the transport gap and the resonances in the constrictions needs to be understood in more detail. Atomic-force micrographs of the sample after etching under ambient conditions (a) and of the completed device at T ≈ 2.6 K (b) are shown in Fig. 1. If not stated otherwise, the temperature of all measurements shown in this paper is T = 2.6 K. Fabrication details are given in the supporting information. 24 We first show a backgate sweep in Fig A symmetric bias of V bias = 300 µV was applied across source and drain and the tip was scanned at a constant height of ∆z ≈ 120 nm above the sample. Coulomb resonances of the quantum dot show up as concentric rings denoted by arrow (QD). The center of the Coulomb resonances are offset from the topographic center of the dot by ca. 240 nm. Such a behavior, known from previous scanning-gate experiments, is understood and of minor importance here. 28 The outline of the quantum dot and its connection to source and drain via the two constrictions, depicted with dashed, black lines, is corrected for the offset, assuming that the Coulomb resonances are centered in the quantum dot (see also supporting information 24 ). Most striking, however, is the appearance of two more sets of concentric rings which are highlighted by arrows (A) and (B) and which are centered around points in the constrictions. Th...

We analytically calculate the energy spectrum of a circular graphene quantum dot with radius R subjected to a perpendicular magnetic field B by applying the infinite-mass boundary condition. We can retrieve well-known limits for the cases R, B → ∞ and B → 0. Our model is capable of capturing the essential details of recent experiments. Quantitative agreement between theory and experiment is limited due to the fact that a circular dot deviates from the actual experimental geometry, that disorder plays a significant role, and that interaction effects may be relevant.

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The use of the parameter m in Eq. (11) of our publication is incorrect. Rather, it should read E = ±v F 2e B(m + 1 + p), (1) where m is the previously defined quantum number and p is an integer with p > −(m + 1). This follows from the fact that Eq. (6) in our publication can be simplified to [ (α) (−α)] −1 = 0 in the limit R/ l B → ∞ with α := k 2 l 2 B /2 − m − 1. This is fulfilled for α = ±p and p being an integer. The later restriction of p > −(m+) is then required to make the radicand non-negative. The mathematical derivation based on our ansatz as described in our original publication is correct.

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