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Marius Eich, František Herman, Riccardo Pisoni, Hiske Overweg, Annika Kurzmann, Yongjin Lee, Peter Rickhaus, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Manfred Sigrist, Thomas Ihn, Klaus Ensslin

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[Spin and Valley States in Gate-Defined Bilayer Graphene Quantum Dots](https://mdr.nims.go.jp/datasets/fec9be13-dac9-4968-b331-959b6d817b90)

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Spin and Valley States in Gate-Defined Bilayer Graphene Quantum Dots Spin and Valley States in Gate-Defined Bilayer Graphene Quantum DotsMarius Eich,* Riccardo Pisoni, Hiske Overweg, Annika Kurzmann, Yongjin Lee, Peter Rickhaus,Thomas Ihn, and Klaus EnsslinSolid State Physics Laboratory, ETH Zurich, 8093 Zurich, SwitzerlandFrantišek Herman and Manfred SigristInstitute for Theoretical Physics, ETH Zurich, 8093 Zurich, SwitzerlandKenji Watanabe and Takashi TaniguchiAdvanced Materials Laboratory, NIMS, 1-1 Namiki, Tsukuba 305-0044, Japan(Received 28 March 2018; revised manuscript received 7 June 2018; published 24 July 2018)In bilayer graphene, electrostatic confinement can be realized by a suitable design of top and back gateelectrodes. We measure electronic transport through a bilayer graphene quantum dot, which is laterallyconfined by gapped regions and connected to the leads via p-n junctions. Single electron and holeoccupancy is realized and charge carriers n ¼ 1; 2;…50 can be filled successively into the quantum systemwith charging energies exceeding 10 meV. For the lowest quantum states, we can clearly observe valley andZeeman splittings with a spin g-factor of gs ≈ 2. In the low-field limit, the valley splitting depends linearlyon the perpendicular magnetic field and is in qualitative agreement with calculations.DOI: 10.1103/PhysRevX.8.031023 Subject Areas: Condensed Matter Physics,Quantum Physics,Semiconductor PhysicsI. INTRODUCTIONGraphene has been recognized early on as a primecandidate to host spin qubits [1]. With carbon being one ofthe lightest elements in the periodic table, spin-orbit inter-actions are expected to be weak. In addition, 99% of naturalcarbon consists of nuclear spin-free 12C. Therefore, the twomain spin decoherence mechanisms for spin qubits, namely,spin-orbit interactions and hyperfine coupling of nuclearand electronic spins, should be strongly suppressed in anycarbon-based solid state system. So far, these theoreticalconsiderations have not come to fruition in experiments.Until now, graphene quantum dots (QDs) have beenmostly realized by top-down lithography and etching ofsingle layer graphene [2–6]. While many of the basicquantum transport properties such as Coulomb blockade[2,3], charge detection [7], and electronic phase coherence[8,9] have been experimentally demonstrated, the under-standing of the orbital and spin character of specific stateshas remained elusive. In retrospect, we understand that theCoulomb blockade in these devices arises mostly fromlocalized states at the sample edges, which remain rough onthe atomic scale because of the limitations of top-downtechnology [6].More than a decade ago, experiments showed that a bandgap can be opened in bilayer graphene by vertical electricfields [10–12], and charge carrier confinement in bilayergraphene has been studied in theory [13–15]. Severalattempts to use split-gate electrodes to laterally confinecharge carriers in the absence of a magnetic field havesuffered from limited resistance values that can be exper-imentally obtained upon pinch-off and did not reach the last-electron regime [16–18]. Recently, we realized quantumpoint contacts that display quantized conductance and showpinch-off resistances orders of magnitude larger than thequantum of resistance h=e2 [19], a necessary requirement toelectrically isolate charge carriers from their environment.Here, the same fabrication technique has been adapted usingsuitable gate geometries to prepare QDs with an electronicquality thatmatcheswhat has been achieved in the traditionalsemiconductors Si and GaAs [20,21].In the first experiment, we demonstrate charging of abilayer graphene QD with a single and a few holes whencoupling the QD to n-type source and drain leads throughp-n tunnel barriers. Charging energies in excess of 10 meVare observed. We reverse the gate voltages and investigatesingle or few electron QDs connected to p-type leads,demonstrating the ambipolar operation of these QDs onthe same graphene flake in close vicinity to each other.*meich@phys.ethz.chPublished by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.PHYSICAL REVIEW X 8, 031023 (2018)2160-3308=18=8(3)=031023(11) 031023-1 Published by the American Physical Societyhttps://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevX.8.031023&domain=pdf&date_stamp=2018-07-24https://doi.org/10.1103/PhysRevX.8.031023https://doi.org/10.1103/PhysRevX.8.031023https://doi.org/10.1103/PhysRevX.8.031023https://doi.org/10.1103/PhysRevX.8.031023https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/Applyingperpendicularmagnetic fields in the second experi-ment, we extract the single particle level spectrum, showingshell filling and orbital degeneracy. The pronounced valleysplitting is in agreement with calculations, which predict thatthe splitting depends on the dot size. In the third experiment,we carefully align thegraphene sheet hosting theQD to an in-plane magnetic field and find a Zeeman splitting with ag-factor gs ¼ 2.08� 0.22, agreeing with the expected valuefor carbon-based devices [22,23].Our bilayer QDs display high-quality electronic proper-ties comparable to standard semiconductor structures thathave been optimized for the last 30 years. While excellentdevices have also been reported for carbon nanotube QDs[24–28], graphene offers the advantage of a planar tech-nology [21] and the possible combination with other 2Dmaterials [29–31]. Our demonstration of excellent controland reproducibility opens up a wide field of possibilities forcarbon-based quantum electronics.II. CHARACTERIZATIONWe investigated the bilayer graphene device encapsu-lated in hexagonal boron nitride [32–34] shown in Fig. 1.The individual layers of the van der Waals heterostructurewere stacked and processed as in Ref. [19], protecting thenatural edges of the bilayer flake [white dashed lines inFig. 1(a)]. Opposite voltages applied to the split gates(green in Fig. 1) and the graphite back gate [red solidlines in Fig. 1(a)] lead to the formation of a band gap inthe bilayer regions underneath the split gates. For appro-priate voltages applied to theses gates, the Fermi level istuned to be in the band gap (for details, see Ref. [19]),rendering these regions insulating and defining approx-imately 100-nm-wide channels between the source anddrain contacts (contacts shown in yellow in Fig. 1).Finger gates [blue in Fig. 1, numbered 1 through 11 inthe x-direction; see Fig. 1(a)] crossing the channel on topof the two split gate pairs (insulated from them by 25 nmof Al2O3) are biased to control the charge carrier densitylocally in the channel.First, we investigate the conductance of the device,biasing only the uniform top gate crossing the entirewidth of the bilayer region [white asterisk in Fig. 1(a)].By applying large opposite voltages to the graphite backgate and this top gate, the strong displacement field opensa band gap in the bilayer region underneath the topgate. The two-terminal resistance measured between thesource and drain contacts reaches values on the order ofGΩ when tuning the Fermi level into the gap (seeAppendix B), demonstrating the high electronic qualityof our sample and the excellent insulating behavior ofthe gapped region. Biasing either pair of split gates inthis regime of high displacement field, charge carriers arelaterally confined and forced to flow through the narrowchannel between the split gates. This is the regime inwhich we form and operate our QDs.III. RESULTSA. Gate-defined quantum dotsIn the first experiment at 1.7 K, we investigate charging aQD with single holes. We measured nine QDs in total, allshowing qualitatively the same behavior. By recordingconductance maps as a function of finger gate and splitgate voltage, a particular QD can be tuned to an optimaloperation point (see Appendix B). Figure 2(a) shows theconductance of the device as a function of the finger gatevoltage VFG. Charge carriers can only flow through thenarrow channel, because the regions underneath the splitgates are insulating. The positive back gate voltage VBGinduces a finite excess electron density in the channel. Withdecreasing finger gate voltage VFG, the electron density isFIG. 1. (a) False color scanning force micrograph of the device. (b) 3D sketch of part of the device showing the different layers of gatesand dielectrics. White dashed lines and solid red lines in (a) outline the bilayer flake and graphite back gate, respectively. Edge contactsto the bilayer are colored in yellow, split gates are shown in green, and the finger gates are shown in blue. The top gate spanning theentire width of the bilayer flake is marked with an asterisk.MARIUS EICH et al. PHYS. REV. X 8, 031023 (2018)031023-2locally reduced until complete pinch-off is reached at thecharge neutrality point (CNP) at about −7.5 V.For VFG < −7.5 V, the region underneath the finger gateis tuned into the hole regime as shown schematically inFig. 2(b): We sketch at the top the n-type channel (red) withthe locally induced p-type region (blue). Below, we showthe dispersion relation near theK-point of the first Brillouinzone in the three spatial regions and at the p-n junctionsbetween them (dashed). At the p-n junctions, the Fermilevel EF lies in the gap leading to a region with zero chargecarrier density. These regions provide natural tunnel bar-riers separating the p-type dot from the n-type leads. Bylowering the finger gate voltage in this regime, the p-typeQD can be charged one by one with individual holes.FIG. 2. (a) Conductance trace for p-type QD 1. (b) Schematic of the band structure at different positions along the current direction inthe channel. Black dashed lines schematically show the band alignment between the dot region and the leads. (c) Coulomb diamonds inthe hole regime for QD 1, with an asterisk indicating regularly spaced resonances parallel to the diamond boundaries. (d) Conductancetrace and (e) Coulomb diamonds for the n-type QD 2. Numbers in (a) and (c), as well as (d) and (e) indicate the occupation of the QDswith holes or electrons, respectively.SPIN AND VALLEY STATES IN GATE-DEFINED … PHYS. REV. X 8, 031023 (2018)031023-3This is seen in Fig. 2(a) at VFG < −7.5 V, where sharpconductance resonances appear (see also inset).Finite DC bias spectroscopy of the QD tuned to thisregime yields the Coulomb diamonds shown in Fig. 2(c).For VFG > −8 V, we do not see additional states contrib-uting to transport through the QD, indicating a completelydepleted dot. Therefore, we label each diamond with theoccupation number of the QD [cf. inset of Fig. 2(a)].The regularly spaced features running parallel to theedge of the Coulomb diamonds [indicated by an asteriskin Fig. 2(c)] appear for all measured QDs, are stable over atemperature range from 50 mK to 1.7 K, and are currentlystill under investigation.To form an electron QD connected to p-type leads, wereverse all applied gate voltages with respect to the overallcharge neutrality point. The conductance trace in Fig. 2(d)mirrors the situation of the hole QD in Fig. 2(a). The n-typeQD can also be charged one by one with individualelectrons, proving the ambipolar operation of our bilayerQDs. The corresponding Coulomb diamonds for theelectron dot [Fig. 2(e)] again mirror the situation ofFig. 2(c). In the electron as well as in the hole regime,charging energies on the order of 10 meV are observed. Incontrast to the p-type QD presented in Fig. 2(c), the n-typeQD exhibits additional features in the region of zero chargecarrier occupation of the QD. These features depend on theprecise setting of the split gate voltage VSG and correspondto localized states in the leads close to the QD (see Fig. 6).In total, eight different QDs were measured both in theelectron and hole regime, all showing qualitatively thesame results.B. Level structureIn the second experiment at 1.7 K, we measure Coulombresonances of the electron QD 9 as a function of aperpendicular magnetic field B⊥ [conductance map inFig. 3(a)]. The shifts of the resonances in VFG as a functionof B⊥ correspond to shifts of energy levels of the QDevolving with B⊥. To extract the energy level spectrum ofour QD from the resonance spacings, we subtract thecharging energy [21,35,36] by shifting neighboring reso-nances such that they touch in a single point and convert thevoltage to an energy axis (see Appendix A). The extractedmagnetic field dependence of the energy levels is shownin Fig. 3(b), where the color scale indicates the peakFIG. 3. (a) Conductance map in a perpendicular magnetic field for QD 9 in the electron regime. (b) Single particle energy leveldispersion for QD 9 with perpendicular magnetic field B⊥ extracted from Fig. 3(a). Blue and red solid lines are the result of theoreticalcalculations of the lowest energy levels. (c) Addition (red), charging (blue), and single particle energy (green) as a function of the QDoccupation extracted from Fig. 3(a).MARIUS EICH et al. PHYS. REV. X 8, 031023 (2018)031023-4conductance of each level, which is proportional to thecoupling of the corresponding state to the leads [36].We see that the levels bunch in groups of four at zeromagnetic field, as expected from the twofold valley andtwofold spin degeneracy in bilayer graphene, similar tocarbon nanotubes [24,26]. Each shell of four states splitsinto 2þ 2 as the magnetic field is increased, one pairshifting up, and the other down in energy. The splittingbetween these states is linear in the accessible magneticfield range, and is 40 times stronger than the Zeemanslitting for a free electron.To compare the data with theory, we used the bilayer QDmodel presented in Ref. [14] and adapted it to our system.The allowed energy levels depend on the valley index(labeled by τ ¼ �1 in the theory corresponding to the Kand K’ valley), the angular momentum number m, theinterlayer asymmetry V, the confinement potential U, andthe radius R of the QD. The levels have to be calculatednumerically by matching four-component spinor states atthe QD boundary. From the displacement field applied inthe experiment, the interlayer asymmetry was estimated tobe V ¼ 60 meV [37]. Since electrons are confined electro-statically, the confinement potential U should be on theorder of the interlayer asymmetry V and we fixU ¼ V. Theremaining parameter R determines both the orbital energylevel difference at B⊥ ¼ 0 and the valley splitting asfunction of B⊥. To reproduce the observed orbital energylevel difference on the order of 5 meV, we obtainR ¼ 20 nm, which agrees well with the lithographic designof the device. The calculated energy levels (spin-degeneratein the theory) are shown in blue and red in Fig. 3(b) for theK and K’ states, respectively. To improve the theoreticalmodel, it should be adapted to the nonradial symmetryof the dot and the nonhomogeneous confinement laterallyand in the transport direction (see Appendix C). Theexperimentally observed valley splitting varies by 20%between different QDs, which could be a result of themicroscopic differences in the size of the individual QDscaused in fabrication.Figure 3(c) shows the experimental spacing of neighbor-ing states as a function of occupation number. The singleparticle level spacing at zero field and the charging energyare shown in green and blue, respectively. The chargingenergy decreases with an increasing number of electronsoccupying the QD, because the effective electronic dotbecomes larger [21,36]. The addition energy, the sum of thetwo former, is directly proportional to the spacing ofCoulomb resonances in finger gate voltage at zero field.We observe a clear fourfold level bunching [Figs. 3(b)and 3(c)], originating from the twofold spin and valleydegeneracy of bilayer graphene [13–15], which can alreadybe seen in Figs. 2(d) and 2(e). Until now, this intrinsicproperty specific to graphene QDs has not been observedexperimentally. The same level bunching was also observedfor the hole QDs at 35 mK (see Appendix B).C. g-factorIn the third experiment at 80 mK, we align the deviceparallel to the magnetic field in a revolving sample holder.The energy levels in Fig. 4(a) as a function of the parallelmagnetic field are extracted in the same way as forFig. 3(b), but vertically shifted by 1 meV for clarity.Blue lines are guides to the eye for purely Zeeman splitenergy levels with a spin g-factor of gs ¼ 2. Repeating themeasurement of Fig. 4(a) for different rotation angles ϕenables us to extract the spin Zeeman and orbital contri-bution to the splitting of energy levels as a function of theangle. The orbital contribution adds to the Zeeman splittingand is proportional to the perpendicular component of themagnetic field. With respect to horizontal lines in Fig. 4(a),the splitting is enhanced for levels K’↓ and K↑ and isreduced for levels K↓ and K’↑. Averaging over these pairsof levels leads to the data shown in Fig. 4(b), where solidlines represent fits to the data.In theory, the red and blue points should touch at gs ¼ 2for perfect alignment of the sample parallel to the magneticFIG. 4. (a) Lowest single particle energy levels of QD 2 in theelectron regime as a function of parallel magnetic field. Energylevels are shifted by 1 meV for clarity. (b) Sum of the spin Zeeman(gs) and orbital (gm) contribution to the energy level splitting as afunction of the sample tilt angleϕwith respect to themagnetic field.Blue (red) points are averaged over levels K’↓ and K↑ (levels K↓and K’↑), and solid lines represent fits to the data.SPIN AND VALLEY STATES IN GATE-DEFINED … PHYS. REV. X 8, 031023 (2018)031023-5field, and the slope of the fits should correspond to theslopes of the lowest four levels in Fig. 3(b). The extractedorbital splitting for ϕ ¼ 90° is 35 (blue) to 40 (red) timesstronger than the Zeeman effect for a free electron, matchingthe experimental data from Fig. 3(b). The extracted sping-factor of gs ¼ 2.08� 0.22 also agrees well with thepredicted value for carbon-based devices [22,23].With the experiments in parallel and perpendicularmagnetic fields, we can show that the observed fourfoldlevel bunching originates from the twofold spin degeneracy(split in Bk) and the twofold valley degeneracy (split in B⊥)of bilayer graphene.Follow-up manuscripts reporting on double- and multi-QD systems in bilayer graphene have, meanwhile, appearedin the literature [38,39].IV. CONCLUSIONThe presented experimental results and the qualitativeagreement with theoretical calculations prove the quality andunderstanding of our bilayer QDs. The electrostatic confine-ment of single charge carriers in a planar technology is animportant step toward the promising implementation of spinand valley qubits in graphene-based devices. With the p-njunctions serving as natural tunnel barriers, QDs can becoupled in the future to create a series ofQDswith alternatingpolarity. We expect that the implementation of high-frequency read-out will enable the determination of spinand valley coherence in graphene quantum dots and open upnew horizons for spin and valley qubit research.ACKNOWLEDGMENTSWe thank Jelena Klinovaja and Francois Peeters forfruitful discussions, and Peter Märki, Erwin Studer, aswell as the FIRST staff for their technical support. We alsoacknowledge financial support from the European GrapheneFlagship, the Swiss National Science Foundation via NCCRQuantum Science and Technology, the EU Spin-Nano RTNnetwork and ETH Zurich via the ETH fellowship program.Growth of hexagonal boron nitride crystals was supported bythe Elemental Strategy Initiative conducted by the MEXT,Japan, and JSPS KAKENHI Grant No. JP15K21722.Author contributions: M. E. and R. P. fabricated thedevice. M. E. performed the measurements. F. H. per-formed the theoretical calculations. H. O., A. K., Y. L.,and P. R. supported device fabrication and data analysis.K.W. and T. T. provided high-quality boron nitride crys-tals. K. E., T. I., and M. S. supervised the work. The authorsdeclare that they have no competing interests.APPENDIX A: MATERIALS AND METHODS1. FabricationThe device was fabricated as described in Ref. [19].The van der Waals heterostructure was built up using thepick-up technique described in Ref. [33]. The stackedheterostructure was then deposited on a phosphorus-dopedSi chip with a dielectric layer of 285 nm SiO2. From thebottom to the top [see Fig. 1(b)], the stack contains a thingraphite flake (graphite back gate), the approximately27-nm-thick bottom boron nitride flake, the approximately20-μm-long and 1-μm-wide bilayer flake, and the approx-imately 24-nm-thick top boron nitride flake. After encap-sulating the bilayer graphene in boron nitride [32], thebilayer flake is protected from the following processingsteps. We work with the natural shape of the exfoliatedbilayer flake to protect its natural edges. In a firstprocessing step, side contacts [33] were patterned usingelectron-beam lithography (EBL) and etched by reactive-ion etching (40 sccm of CHF3, 4 sccm of O2, 132 W RFpower, 306 V DC bias), followed by evaporation of 10 nmchromium and 60 nm gold. The two top gate layers [greenand blue gates in Fig. 1(a)] are both produced in twoconsecutive EBL steps. The inner gate structure on top ofthe bilayer flake is written in a separate EBL step to achievehigh resolution with a thin EBL resist. The thinner resistallows only for evaporating 5 nm chromium and 25 nmgold. In a second step with thicker EBL resist, the outergate structure connecting to the bond pads is written and10 nm chromium and 60 nm gold are evaporated. For thefirst top gate layer, an additional gentle reactive-ion etchingstep (80 sccm of Ar, 5 sccm of O2, 70 W RF power, 222 VDC bias) is performed to ensure that the gates are stickingon the boron nitride surface. To separate the two top gatelayers, we grow an approximately 25-nm-thick layer ofAl2O3 by atomic layer deposition at 150 °C (with precursorgases of trimethylaluminum and water).The device was cooled down four times in differentsetups: a dipstick 4He system at 4.2 K, a variable temper-ature insert reaching 1.7 K, a dry dilution refrigeratorreaching 35 mK, and a wet dilution refrigerator with arotatable sample holder reaching 80 mK.2. Extracting energy levelsFigure 3(a) shows Coulomb resonances as a function ofB⊥. The finger gate axis can be converted into an energyaxis, using the Coulomb diamond measurement and deter-mining the lever arm α of the finger gate [36]. To add anadditional electron to the QD, the electron has to paycharging energy (due to Coulomb interaction) in addition tothe single particle energy level difference. Thus, when twosingle particle levels of the QD are degenerate in energy, theadditional electron only pays charging energy, which isassumed to be independent of the magnetic field. In turn,this means that the minimal distance of Coulomb reso-nances in αVFG corresponds exactly to the charging energy.Subtracting the charging energy means shifting neighbor-ing resonances in αVFG such that they touch at one point.The result is shown in Fig. 3(b), yielding the single particleenergy level spectrum of the QD. Figure 3(c) shows theMARIUS EICH et al. PHYS. REV. X 8, 031023 (2018)031023-6subtracted charging energy (blue curve), which decreasesas expected with increasing electron occupation number[21]. The single particle energy (green curve) shows clearpeaks whenever the occupation number is an integermultiple of 4. Therefore, also, the addition energy (redcurve) shows maxima at the same position, which is whythe fourfold level bunching can already be observed in theCoulomb resonances of the conductance traces (see Fig. 6).APPENDIX B: SUPPORTING DATA1. Dirac pointsIn order to demonstrate the high electronic quality of oursample, Fig. 5(a) shows a resistance map as a function ofgraphite back gate voltage VBG and top gate voltage VTGapplied to a uniform top gate crossing the entire bilayerregion [gate marked by a white asterisk in Fig. 1(a)]. Allother gates (split and finger gates) are grounded for thismeasurement.The overall Dirac point [white triangle in Fig. 5(a)] isreached when VBG ¼ −0.34 V and VTG ¼ −0.62 V. Thehorizontal line of high resistance at VBG ¼ −0.34 V [whitediamond in Fig. 5(a)] corresponds to the Dirac point of thebilayer regions above the graphite back gate [red solid linein Fig. 1(a)], not being covered by any top gates [greengates in Fig. 1(a)]. The second horizontal line of highresistance [white square in Fig. 5(a)] corresponds to theDirac peak underneath the two pairs of split gates kept atVSG ¼ 0. To induce the same density in these regionscompared to the regions not covered by any gates on top,the split gates would have to be set to VSG ¼ −0.62 V.With the slope of the diagonal resistance peak, we cancalculate the offset in VBG that would compensate for theoffset in VSG. We find ΔVBG ¼ 0.95 V, which agrees withthe distance of the two horizontal resistance peaks in VBG.To recapitulate, the high resistance along the two horizontallines corresponds to the Dirac points in the regions abovethe graphite back gate covered and not covered bygrounded split gates.2. Pinch-off resistanceThe high resistance along the D-axis in Fig. 5(a)corresponds to the Dirac peak underneath the biased topgate [white asterisk in Fig. 1(a)], spanning the whole widthof the bilayer flake. Along this line, away from the overallDirac point [white triangle in Fig. 5(a)], the displacementfield in the bilayer increases, leading to the opening of aband gap in the region below the top gate [10]. At highdisplacement field [red circle in Fig. 5(a)], resistances in theGΩ regime can be reached [see Fig. 5(b)]. This is theregime in which we operate our QDs.The measurement shown in Fig. 5(a) was performed witha constant source-drain bias excitation of 500 μV. Wemeasure the current with a precision of 10 pA using an IVconverter. This means that we can only measure resistancesup to 50 MΩ. The extracted resistance values as a functionof VBG at VTG ¼ −4.5 V are plotted as a solid line inFig. 5(b). When the Fermi energy lies within the gapopened by the strong displacement field, the resistancereaches higher values and the measured current approacheszero. Therefore, we record additional IV characteristics ofthe device in this regime. The extracted resistance valuesare plotted as solid circles in Fig. 5(b), reaching resistancesin the GΩ regime.3. Tuning into the QD regimeThe Coulomb resonances (white star in Fig. 6) of QD1 inthe hole regime can be clearly identified in the logarithmicconductance map as a function of channel and split gatevoltage. The middle region of low conductance correspondsto charge neutrality underneath the split gates such thattransport outside of the channel is strongly suppressed. Atmore negative (positive) split gate voltage, p-type (n-type)FIG. 5. (a) Logarithmic two-terminal resistance of the device asa function of the graphite back gate voltage VBG and the top gatevoltage VTG [asterisk in Fig. 1(a)]. (b) Two-terminal resistance asa function of the graphite back gate voltage at fixed top gatevoltage VTG ¼ −4.5 V. The solid line was measured in a constantsource-drain bias setup, while the circles correspond to resistancevalues extracted from IV characteristics recorded at the given gatevoltage setting.SPIN AND VALLEY STATES IN GATE-DEFINED … PHYS. REV. X 8, 031023 (2018)031023-7conductance underneath the split gates leads to increasedconductance parallel to the channel. In addition to theCoulomb resonances, localized states close to the QD canbe identified (white circle in Fig. 6). However, they do notinfluence the QD at the operation point indicated by thedashed line [cut along the dashed line corresponds toFig. 2(a)].The cross capacitance of the QD and the split gates leadsto the finite slope of the Coulomb resonances in Fig. 6.Additionally, increasing the voltage VSG applied to all splitgates squeezes the hole wave function of the QD, leading toan increase in confinement energy and, thus, Coulombresonance spacing.4. Level bunchingAs mentioned in the main text of the manuscript, the holeQDs also showed a fourfold level bunching in the cool-down to 35 mK. The corresponding conductance curve as afunction of the finger gate voltage for QD 8 in the holeregime is shown in Fig. 7(a). Numbers indicate theoccupation of the QD with holes, and we observe anincreased addition energy whenever the occupationnumber is an integer multiple of 4 (with the exceptionof 16). This agrees with shell filling of our QDs [36] andthe fourfold degeneracy of bilayer graphene.Figure 7(b) shows the same measurement for QD 4 in theelectron regime, mirroring the result of Fig. 7(a). Overall, 8of our 11 QDs show qualitatively the same behavior, bothin the electron and in the hole regime. All QDs show asequence of groups of 4 levels, sometimes with groups of 8or 12 showing up, depending on the specific QD.APPENDIX C: THEORETICALCONSIDERATIONSFor comparison of the measured data with theory, weadopted the solution for the energy levels of a bilayer QD ina perpendicular magnetic field [14]. The allowed energylevels depend on the valley number τ (þ1 for K and −1 forK’), the total angular momentumm, the interlayer potentialasymmetry V (potential difference between upper andlower layer), the boundary potential U, and the radius Rof the circular dot. The resulting energy levels are deter-mined numerically.1. Ground stateThe result of fitting the ground state is shown inFig. 3(b). The fit was performed with fixed U ¼V ¼ 60 meV, calculated from the voltages applied in theexperiment. We find R ¼ 20 nm, where the ground statehas the total angular momentum number m ¼ þ1. Weobserve that the parameters U and R have the strongestinfluence on the orbital energy level spacing (approxi-mately 5–10 meV in the experiment) and their magneticfield dependence for small magnetic fields (B⊥ ∼ 0–2.5 T).FIG. 6. Logarithmic two-terminal conductance as a function ofthe split gate voltage VSG and the finger gate voltage VFG for QD1 measured at VBG ¼ 5.55 V. The cut along the dashed linecorresponds to Fig. 2(a).FIG. 7. Conductance traces for (a) QD 4 in the hole regime and(b) QD 8 in the electron regime measured at 35 mK and 1.7 K,respectively. Numbers indicate the occupation number of therespective QD.MARIUS EICH et al. PHYS. REV. X 8, 031023 (2018)031023-82. Linear magnetic field regimeSince the experimental results in Fig. 3(b) show anessentially linear magnetic-field dependence of the energylevels in the covered field range, we use a perturbativeapproach starting from the zero-field lowest energy states.We analyze the linear correction to the energy of the statesin lowest-order perturbation theory in B⊥ and obtain anestimate for the orbital g-factor gorb. Based on the theo-retical approach in Ref. [14], we write the eigenstates asΨðρ;ϕÞ ¼ eimϕffiffiffiρp0BBB@1 0 0 00 e−iϕ 0 00 0 1 00 0 0 eþiϕ1CCCAψmðρÞ; ðC1Þwhere ψmðρÞ is the radial part of the Dirac spinor depend-ing on m and the radius ρ ¼ r=R. We then use the solutionof the ground state to calculate the correction to the energylevel due to the magnetic field in first-order perturbation asEB ¼ τRlF����R∞0 dρρψmðρÞ†γ5ψmðρÞR∞0 dρψmðρÞ†ψmðρÞ����|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}MμBB⊥¼ gorbμBB⊥; ðC2Þwhere we introduced the Fermi length lF ¼ ℏ=ðvFmeÞ,which has, considering vF ¼ 106 m=s, the valuelF ¼ 1.16 Å. The 4 × 4 matrix γ5 can be expressed bythe Pauli matrices asγ5 ¼�σ2 00 −σ2�: ðC3ÞNote that the transformation ðm; τÞ → ð−m;−τÞ simplyleads to a change of sign in EB.Let us elaborate briefly on this approximation of EB. Theperturbative result only applies in the field range wherethe correction to the zero-field energy is smaller than theenergy level spacing at B⊥ ¼ 0. Through the spinor ψmðρÞ,gorb depends on both m and U. However, the matrixelement M in Eq. (C2) varies only between 0.05 and0.35, as can be seen in Table I, such that the strongesteffect on gorb actually originates from the ratio R=lF. Thedifference between the exact and the perturbative result issmall in the low-field limit.3. Noncircular dotsWe briefly want to discuss the effect of a modifiedboundary potential U, for which we have so far taken ahard wall potential, Uðx; yÞ ¼ 0 for r ≤ R and U for r > R(r2 ¼ x2 þ y2). We study the effect of a more generalpotential, which may be noncircular and layer dependent.This modification is responsible for two effects. Thelayer dependence of the potential lifts the degeneracy atB⊥ ¼ 0. In addition, the noncircular structure yields anenergy shift, which again can be calculated by means ofperturbation theory to lowest order,EU ¼RdxdyΨðx; yÞ†Uðx; yÞΨðx; yÞRdxdyΨðx; yÞ†Ψðx; yÞ : ðC4ÞIn order to examine the effect of B⊥ on this shift, weconsider the shape of the radial wave function, jΨðρÞj2 ¼ψmðρÞ†ψmðρÞ (including both valleys, τ ¼ �1), plotted inFig. 8 for fields B⊥ ¼ 0, 2.5 T. In the legend, the states areTABLE I. Matrix element M introduced in Eq. (C2), calculatedfor the first four lowest energy levels of the bilayer quantum dotmodel presented in Ref. [14], considering U ¼ V ¼ 60 meV andR ¼ 20 nm.m 1 0 2 −1M 0.17 0.18 0.33 0.06FIG. 8. jΨðrÞj2 for the four lowest energy levels in both valleys considering R ¼ 20 nm and U ¼ V ¼ 60 meV.SPIN AND VALLEY STATES IN GATE-DEFINED … PHYS. REV. X 8, 031023 (2018)031023-9ordered according to their energy, with m ¼ 1 being theground state. It becomes obvious that the magnetic fieldonly slightly influences jΨðρÞj2 for all displayed m.Therefore, it is clear that the energy shift due to themodified boundary potential leads to an essentially field-independent shift of the energy, i.e., gorb remains practicallyunchanged, if the mean radius R is kept constant.To demonstrate this effect, we useUðx;yÞ¼UL½ðx2=R2ÞþχLðy2=R2Þ� for x2 þ χLy2 ≤ R2 and Uðx; yÞ ¼ U forx2 þ χLy2 ≥ R2, where L ¼ 1, 2 is a layer index and χL agenerally layer-dependent anisotropy parameter. In Fig. 9,we compare the result of the standard hardwall potential (reddashed) with the layer-dependent potential above (blacksolid) using the parameters U1 ¼ U2 ¼ 6 meV, χ1 ¼ 0.9,and χ2 ¼ 0.95, while keeping U ¼ 60 meV.The small potential difference between the two layersleads to a splitting of the degeneracy of the pair ðm; τÞ andð−m;−τÞ at B⊥ ¼ 0, which might be one of the reasonsfor the observed splitting of energy levels at B⊥ ¼ 0 in theexperiment.[1] B. 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