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B. Terrés, L. A. Chizhova, F. Libisch, J. Peiro, D. Jörger, S. Engels, A. Girschik, [K. Watanabe](https://orcid.org/0000-0003-3701-8119), [T. Taniguchi](https://orcid.org/0000-0002-1467-3105), S. V. Rotkin, J. Burgdörfer, C. Stampfer

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[Size quantization of Dirac fermions in graphene constrictions](https://mdr.nims.go.jp/datasets/4f9cffce-62e1-4a22-9b60-2b622e8e6d48)

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Size quantization of Dirac fermions in graphene constrictionsARTICLEReceived 28 Sep 2015 | Accepted 5 Apr 2016 | Published 20 May 2016Size quantization of Dirac fermions in grapheneconstrictionsB. Terrés1,2, L.A. Chizhova3, F. Libisch3, J. Peiro1, D. Jörger1, S. Engels1,2, A. Girschik3, K. Watanabe4, T. Taniguchi4,S.V. Rotkin1,5,6, J. Burgdörfer3,7 & C. Stampfer1,2Quantum point contacts are cornerstones of mesoscopic physics and central building blocksfor quantum electronics. Although the Fermi wavelength in high-quality bulk graphene can betuned up to hundreds of nanometres, the observation of quantum confinement of Diracelectrons in nanostructured graphene has proven surprisingly challenging. Here weshow ballistic transport and quantized conductance of size-confined Dirac fermions inlithographically defined graphene constrictions. At high carrier densities, the observedconductance agrees excellently with the Landauer theory of ballistic transport without anyadjustable parameter. Experimental data and simulations for the evolution of the conductancewith magnetic field unambiguously confirm the identification of size quantization in theconstriction. Close to the charge neutrality point, bias voltage spectroscopy reveals arenormalized Fermi velocity of B1.5� 106 m s� 1 in our constrictions. Moreover, at low carrierdensity transport measurements allow probing the density of localized states at edges, thusoffering a unique handle on edge physics in graphene devices.DOI: 10.1038/ncomms11528 OPEN1 JARA-FIT and 2nd Institute of Physics, RWTH Aachen University, 52056 Aachen, Germany. 2 Peter Grünberg Institute (PGI-9), Forschungszentrum Jülich,52425 Jülich, Germany. 3 Institute for Theoretical Physics, Vienna University of Technology, 1040 Vienna, Austria. 4 National Institute for Materials Science,1-1 Namiki, Tsukuba 305-0044, Japan. 5 Department of Physics, Lehigh University, Bethlehem, Pennsylvania 18015, USA. 6 Center for Advanced Materials andNanotechnology, Lehigh University, Bethlehem, Pennsylvania 18015, USA. 7 Institute of Nuclear Research of the Hungarian Academy of Sciences (ATOMKI),4001 Debrecen, Hungary. Correspondence and requests for materials should be addressed to F.L. (email: florian.libisch@tuwien.ac.at) or to C.S.(email: stampfer@physik.rwth-aachen.de).NATURE COMMUNICATIONS | 7:11528 | DOI: 10.1038/ncomms11528 | www.nature.com/naturecommunications 1mailto:florian.libisch@tuwien.ac.atmailto:stampfer@physik.rwth-aachen.dehttp://www.nature.com/naturecommunicationsThe observation of unique transport phenomena ingraphene, such as Klein tunnelling1, evanescent wavetransport2, or the half-integer3,4 and fractional5,6 quantumHall effect are directly related to the material quality, as well asthe relativistic dispersion of the charge carriers. As the quality ofbulk graphene has been impressively improved in the last years7,8,the understanding of the role and limitations of edges ontransport properties of graphene is becoming increasinglyimportant. This is particularly true for nanoscale graphenesystems where edges can dominate device properties. Indeed, therough edges of graphene nanodevices are most probablyresponsible for the difficulties in observing clear confinement-induced quantization effects such as quantized conductance9 andshell filling10. So far signatures of quantized conductance haveonly been observed in suspended graphene, however with limitedcontrol and information on geometry and constriction width11.More generally, with further progress in fabrication technology,graphene nanoribbons and constrictions are expected to evolvefrom a disorder-dominated12–15 transport behaviour to aquasi-ballistic regime where boundary effects, crystal alignmentand edge defects16,17 govern the transport characteristics. Thiswill open the door to investigate interesting phenomena arisingfrom edge states, including magnetic order at zig-zag edges18,an unusual Josephson effect19, unconventional edge states20,magnetic edge-state excitons21 or topologically protectedquantum spin Hall states22.In this work we report on the observation of size quantizationand localized trap states in ballistic transport through grapheneconstrictions approximating quantum point contacts. Away fromthe Dirac point, the current features evenly spaced, reproduciblekinks superposed on a linear background, in agreement withtransport simulations. Scattering at the rough constrictionedges reduces quantization steps to kinks in both experimentand theory. The kink spacing, and their evolution with magneticfield, allows us to unambiguously identify them as signatures ofsize quantization. Close to the Dirac point, deviations fromballistic behaviour allow for probing the density of localizedtrap states.ResultsBallistic transport. We prepared four-probe devices based onhigh-mobility graphene–hexagonal boron nitride (hBN) sand-wiches on SiO2/Si substrates and use reactive ion etching topattern narrow constrictions (see Methods) with widths rangingfrom WE230 to 850 nm, connecting wide leads (Fig. 1a–c). Thegraphene leads are side-contacted8 by 80-nm-thick chrome/goldelectrodes. A back-gate voltage is applied on the highly doped Sisubstrate to tune the carrier density in the graphene layer,n ¼ aðVg�V0g Þ ¼ aDVg, where a is the so-called lever arm andV0g is the gate voltage of the minimum conductance, that is, thecharge neutrality point. To demonstrate the high electronicquality of our graphene–hBN sandwich structures we show thegate characteristic of a reference Hall bar device (Fig. 1d andSupplementary Fig. 1). From this data we extract a carriermobility in the range of around 150.000 cm2 V� 1 s� 1(Supplementary Note 1), resulting in a mean free pathexceeding 1 mm at around DVg¼ 4.6 V. Thus, the mean freepath is expected to clearly exceed all relevant length scales in ourconstriction devices giving rise to ballistic transport.Back gateGraphenehBN0 200 400 600 8000200400600800bWVsd+–Isd–Vb/2+ –cafElectronsHolese0 50 100 150 250050100150200200kF (106 m–1)310 nm 440 nm 590 nm 850 nm590850 nm440310280230250d2001601208040–40 –30 –20 –10 0 10 20 30 400Hall barWidth, W (nm)G (e2 /h)G (e2 /h)c 0W (nm)ΔVg (V)Vb/2Figure 1 | Width-dependent ballistic transport in etched graphene nanoconstrictions encapsulated in hBN. (a) Schematic illustration of a hBN–graphenesandwich device with the bottom and top layers of hBN appearing in green, the gold contacts in yellow, the SiO2 in dark blue and the Si back gate in purple.(b) SEM images of four investigated graphene constrictions patterned using reactive ion etching. Black scale bar, 500 nm. (c) False coloured atomic forcemicroscope (AFM) image of a fabricated device. Transport is measured in a four-probe configuration to eliminate any unwanted resistance of theone-dimensional contacts8. The yellow colour denotes the gold contacts, green the top layer of hBN and brown the SiO2 substrate. White scale bar,500 nm. (d) Low-bias back-gate characteristics of a Hall bar device (see arrow) and of five constriction devices with different widths ranging from 850 to230 nm (see e for colour code). The dashed grey lines are fits to the data. (e) Low-bias four-terminal conductance of graphene quantum point contacts asfunction of kF extracted in the high carrier density limit for seven different samples. The colour encodes the different samples with different constrictionwidths (see labels). Grey lines represent a linear fit at high values of kF, inserted as guide to the eye. Conductance deviates from the expected linear slopefor small kF. Electron (hole) conductance is plotted as solid (dashed) line. Data are taken at temperatures below 2 K. (f) Comparison of c0W fromconductance traces (e) with the width W (extracted from SEM images).ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms115282 NATURE COMMUNICATIONS | 7:11528 | DOI: 10.1038/ncomms11528 | www.nature.com/naturecommunicationshttp://www.nature.com/naturecommunicationsWe measure the conductance as function of gate voltage for anumber of constrictions with different widths W (Fig. 1d;see labels in Fig. 1e). The observed square root dependenceG /ffiffiffiffiffiffiffiffiffiDVgp/ffiffiffinp(see dashed lines in Fig. 1d) is a first indicationof highly ballistic transport in our devices. Indeed, according tothe Landauer theory for ballistic transport, the conductancethrough a perfect constriction increases by an additionalconductance quantum e2/h whenever WkF reaches a multiple of pG ¼ 4e2hX1m¼1yWkFp�m� �; ð1Þwhere kF ¼ffiffiffiffiffiffipnpis the Fermi wave number, the factor fouraccounts for the valley and spin degeneracies, y is the stepfunction and we have neglected minor phase contributions due todetails of the graphene edge23 for simplicity. Fourier expansion ofequation (1) yieldsG ¼ 4e2hc0WkFpþ 4e2hX1j¼1cj sin 2jWkF�fj� �� c02" #: ð2ÞFor an ideal constriction c0¼ 1, fj¼ 0 and cj¼ 1/(jp), j40. In thepresence of edge roughness, c0 is reduced to a value below 1 dueto limited average transmission, and the higher Fouriercomponents are expected to decay in magnitude and acquirerandom scattering phases fja0. Consequently, the sharpquantization steps turn into periodic modulations as will beshown below. Averaged over these modulations only the zeroth-order term in the expansion (equation (2)) survives. This meanconductance G(0) of a constriction of width W thus features alinear dependenc on kF, or, equivalently, a square-rootdependence as a function of back-gate voltage assuming anenergy-independent transmission c0 of all modes, in accord withFig. 1d.By measuring the carrier-density-dependent quantum Halleffect at high magnetic fields4,24, we can independently determinethe gate coupling a for each device (Supplementary Fig. 2,Supplementary Table 1 and Supplementary Note 2). We can thusunfold the dependence on Vg and study both the electron andhole conductance as function of kF (Fig. 1e). From the linearslopes of G(kF), the product c0W can be extracted for each deviceand compared with its width W (Fig. 1f) determined fromscanning electron microscopy (SEM) images (see, for example,Fig. 1b). The estimates for c0W extracted from G(0) lie onlyslightly below the width W, where c0 decreases for decreasingwidth. This suggests that for the narrower devices reflections,most likely due to device geometry and edge roughness, areplaying a more important role. From the data in Fig. 1f we canextract c0E0.56 for our smallest constriction. Below we will showthat, indeed, reflections at the rough edges of the constriction andnot a reduction in active channel width is responsible for thedeviation of the experimentally extracted c0W from the SEMwidth W.Localized states. For small kFo50� 106 m� 1 (that is, low carrierconcentrations) the measured conductances systematically devi-ate from the expected linear behaviour (Fig. 1e). This deviationfrom the square-root relation between G and n (that is, DVg)becomes more apparent when focusing on G around the charge0510152025d00.81.20.4ExperimentaG (e2 /h)G (e2 /h)05101520250 0.5 1 1.5 0 100 200–200 –100–1.5 –1 –0.5� (1017 eV–1)� (1017 eV–1)0 100 200kF (106 m–1) kF (106 m–1)kF (106 m–1)n (1012 cm–2)–200 –100e00.81.20.4Theory0 100 200–200 –100 0 1|�|2 (a.u.)1 2 3123230 nm–100 meV–30 meV250 meV230 nmb cFigure 2 | Conductance through graphene quantum point contacts. (a) Conductance traces of two different cool-downs (black and green curve) of thesame constriction (WE230 nm) as a function of charge carrier density. For the black (green) cool-down, shaded grey (light grey) regions denote deviationsfrom the ideal Landauer model G /ffiffiffinpshown in red. At higher conductance values we observe well-reproduced ‘kinks’ with spacings on the order of 2e2/h(see arrows and horizontal lines). (b) Experimental conductance trace as a function of kF after correction for the density of trap states (black and greencurves) and theoretical simulations of the graphene quantum point contact (blue curve). Theoretical results are rescaled to experimental device size asdetermined from a. Ideal transmission pkF is shown in red as guide to the eye. Curves are offset horizontally for clarity. The inset gives an example for theprobability distribution of a simulated scattering state. (c) Local density of states of the graphene quantum point contact from tight-binding simulations, atthree different energies (� 100, � 30 and 250 meV; see also arrows in e). (d) Graphene density of states extracted from experiment (fit to a Gaussian)and e from simulation. Both experiment and theory find a substantial contribution from trap states around the Dirac point.NATURE COMMUNICATIONS | DOI: 10.1038/ncomms11528 ARTICLENATURE COMMUNICATIONS | 7:11528 | DOI: 10.1038/ncomms11528 | www.nature.com/naturecommunications 3http://www.nature.com/naturecommunicationsneutrality point (CNP). The conductance as function of n for twodifferent cool-downs of the same graphene constriction(WE230 nm, Fig. 2a), shows marked cool-down-dependent low-carrier-density regions with substantial deviations from G /ffiffiffinp.Far away from the CNP, the conductance as function of n forboth cool-downs shows (i) an identicalffiffiffinpbehaviour leading tothe very same c0W and (ii) almost identical, regularly spaced kinkstructures (see arrows in Fig. 2a), which are, however, slightlyshifted relative to another on the carrier density axis n(Supplementary Fig. 8). These observations suggest that thesquare-root relation between the Fermi wave vector kF andthe gate voltage Vg, that is, n needs to be modified. While thequantum capacitance of ideal graphene can be neglected25–27,a small additional contribution nT(DVg) from, for example,localized trap states modifies the relation between n and kF toaDVg ¼ n ¼ k2Fp� 1þ nT DVg� �: ð3ÞFar away from the Dirac point (k2F � pnT), we recover theexpected square-root relation. Close to the Dirac point, however,aDVg will be strongly modified by deviations nT from the lineardensity of states of ideal Dirac fermions and approaches nT(DVg)near the CNP. The trap states do not contribute to transport, yetthey contribute to the charging characteristics28. Such trap statescan for instance be found at the rough edges of patternedgraphene devices, which feature a significant number of localizedstates. A tight-binding simulation of the local density of states ofthe experimental geometry yields a strong clustering of localizedstates at the device edges (Fig. 2c), which energetically lie close tothe CNP (Fig. 2e). The deviation of G from theffiffiffinpscaling alsoopens up the opportunity to extract nT from experimentalconductance data (for example, Fig. 2d), and thus a new pathwayfor device characterization. Inspired by the tight-bindingsimulation, we approximate the distribution of trap states asfunction of Fermi wave vector by a Gaussian distribution. We fitthe position, height and width of the Gaussian by minimizing thedifference between the measured G(kF) and the correspondinglinear extrapolation to very low values of kF (Fig. 2b, Supplemen-tary Fig. 3 and Supplementary Note 3). We find good qualitativeagreement between simulation and experiment (compareFig. 2d,e). Quantitative correspondence would require adetailed, microscopic model for the trap state density nT. Notethat the only difference between different traces in Fig. 2a,b,d isthe exposition of the device to air for several days leading to awider carrier density region of substantial deviations (greentrace). The number of trap states (that is, the deviations aroundthe CNP) is significantly enhanced (compare also green and black0204060200 400 600 8000204060Length (nm)230 nmElectronsHolesexthexth150 200 250 300 350 400f40035030025020015026101418ExperimentTheoryG (e2 /h)G (e2 /h)G (e2 /h)0 40 80 100kF (106 m–1) kF (106 m–1) kF (106 m–1)230 nm0 50 100 150412243240c801620283604812160 50 100 1504122480162028280 nm 310 nmWFT≈ 230 nmelho42elhoElectronsHolesWFT (nm)Width, W (nm)F [δG(kF)]F [�G(kF)]a bdeFigure 3 | Size quantization signatures. (a) Comparison of the low-energy conductance between theory (blue) and experiment (black). (b,c) Measuredelectron (el; black trace) and hole (ho; red trace) conductance including kink or step-like structure (see arrows) as a function of kF for two differentconstriction widths (see insets). The hole conductance traces are horizontally offset for clarity. (d) Fourier transform of the G�G(0) electron conductanceF½dGðkFÞ� through the 230-nm graphene constriction, for experiment (ex; black trace) and theory (th; blue trace). The first peak of theFourier transform clearly corresponds to the width W of the quantum point contact (marked by arrows). (e) Same as d for the hole conductance. The sizeof the first peak is substantially reduced for both experiment and theory due to the presence of localized states that lead to additional scattering.(f) Comparison of width WFT extracted from the Fourier transform of the conductance traces (as shown in d,e) to geometric constriction width W from fourdifferent devices (extracted from SEM images).ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms115284 NATURE COMMUNICATIONS | 7:11528 | DOI: 10.1038/ncomms11528 | www.nature.com/naturecommunicationshttp://www.nature.com/naturecommunicationstrace in Fig. 2d). As the active graphene layer is completelysandwiched in hBN, only the graphene edges are exposed to airand, very likely, experience chemical modifications. In line withour numerical results, we thus conjecture that localized states atthe edges substantially contribute to nT, leading to the strongcool-down dependence we observe in our measurements. Whilethis interpretation seems plausible and is consistent with our data,alternative explanations such as electron–hole puddles29 orcharged impurities13 cannot be ruled out.Away from the CNP our data agrees remarkably well withballistic transport simulations through the device geometry usinga modular Green’s function approach30 (see blue trace in Fig. 2b):we simulate the four-probe constriction geometry taken from aSEM image, scaled down by a factor of four to obtain anumerically feasible problem size31. To account for the etchededges in the devices, we include an edge roughness amplitude ofDW¼ 0.2W for the constriction. This comparatively large edgeroughness (which is consistent with the systematic reduction oftransmission through the constriction when using the averageconductance) is probably due to microcracks at the edges of thedevice.Quantized conductance. Superimposed on the overall linearbehaviour of G(kF), we find reproducible modulations (kinks) inthe conductance (Fig. 3a–c and Supplementary Fig. 4). The kinksare well reproduced for several cool-downs (see arrows in Fig. 2a,Supplementary Figs 5 and 6 and Supplementary Note 4), as wellas for different devices (Supplementary Fig. 7), generally showinga spacing DG varying in the range of (2� 4)e2/h (see arrows inFig. 3b,c). The ‘step height’ and its sharpness depend on thecarrier density (that is, kF), as well as on the constriction widthand is strongly influenced by the overall transmission c0 (Fig. 1f).Remarkably, we observe a spacing DG of the steps close to 4e2/hfor one of our wide samples (WE310 nm) at elevatedconductance values on both the electron and hole sides (seearrows and horizontal lines in Fig. 3c and Supplementary Fig. 4b)Our assignment of the conductance ‘kinks’ as signaturesof quantized flow through the constriction is supported byour theoretical results. Theory and experimental data fromthe smallest constriction show similar smoothed, irregularmodulations (Fig. 3a), instead of sharp size quantization steps32.The replacement of sharp quantization steps by kinks reflects thestrong scattering at the rough edges of the device33,34, resulting inthe accumulation of random phases in the Fourier componentsof G (equation (2)). We note that calculations with smalleredge disorder show a larger average conductance, yet verysimilar ‘kink’ structures. As the present calculation includesonly edge-disorder-induced scattering while neglecting otherscattering channels such as electron–electron or electron–phononscattering, the good agreement with the data suggestsedge scattering to be the dominant contribution to theformation of the ‘kinks’. By contrast, both experimental andtheoretical investigations of, for example, semiconducting GaAsheterostructures show very clear, pronounced quantizationplateaus35. In these heterostructures, the electron wavelengthnear the G point is very long, and cannot resolve edge disorder onthe nanometre scale. By contrast, K�K0 scattering in grapheneallows conduction electrons to probe disorder on a muchshorter length scale. Consequently, edge roughness substantiallyimpacts transport. The comparison between experimental andtheoretical data (Fig. 3a) unambiguously establishes the observedmodulations to be consistent with the smoothed size quantizationeffects predicted by theory.By subtracting the zeroth-order Fourier componentpkF(orffiffiffinp), the superimposed modulations of the conductancedG(kF)¼G�G(0) provide direct information on the quantizedconductance through the constriction (equation (2)). One keyobservation is that the Fourier transform of dG(kF) offers analternative route towards the determination of the constrictionwidth complementary to that from the mean conductance G(0).For example, the pronounced peak of the first harmonic at230 nm (red arrows in Fig. 3d,e) is consistent with theconstriction width W derived from the SEM image. Oursimulation also correctly reproduces the experimental observationthat the peak in the Fourier spectrum of dG(kF) is morepronounced on the electron side (Fig. 3d) than on the hole side.This results from the slightly asymmetric energy distribution ofthe trap states relative to the CNP, which is accounted for in ourtight-binding calculation.Performing such a Fourier analysis for several devices(Supplementary Fig. 9 and Supplementary Note 5) yields muchcloser agreement with the geometric width W (Fig. 3f andhorizontal axis of Fig. 1f) than an estimate based only on thezeroth-order Fourier component c0W (first term in equation (2);see vertical axis of Fig. 1f). Fourier spectroscopy of conductancemodulations thus allows to disentangle reduced transmission dueto scattering at the edges (c0W) from the effective width of theconstriction, and proves the relation between the observedFourier periodicity and the device geometry.Bias voltage spectroscopy measurements yield an estimate forthe energy scale of the size quantization steps11,36. For example,cb7654321–20 2001020–10–2000.4 0.8 1.230262218202428G (e2 /h)g (e2 /h)2012 168Vg (V)T (K)21.71021230 nm–0.5 V–16Vb (mV)Vg (V)Vb (mV)aFigure 4 | Finite bias and temperature dependence of the quantizedconductance. (a) Zero B-field differential conductance g as a functionof bias voltage Vb, measured at T¼6 K, taken at fixed values ofback-gate voltage Vg from �0.5 to 3.0 V in steps of 30 mV (see lowerright label). The dense regions correspond to plateaus in conductance.(b) Transconductance qg/qVg in units of e2/hV (see colour scale) as afunction of bias and back-gate voltage for a different cool-down of the samedevice (Supplementary Note 6). At Vb¼0, the transitions betweenconductance plateaus appear as red spots. At finite bias voltage, weobserve a diamond-like shape, which provides an energy scale for thesub-band energy spacing DEE13.5±2 meV (see dashed black lines andwhite arrow), which is also in good agreement with the energy scaleobserved in a (Supplementary Note 6). (c) Conductance traces as afunction of temperature and back-gate voltage. We observe features withdifferent temperature dependencies. Above around 10 K only kinks relatedto quantized conductance survive (see arrows).NATURE COMMUNICATIONS | DOI: 10.1038/ncomms11528 ARTICLENATURE COMMUNICATIONS | 7:11528 | DOI: 10.1038/ncomms11528 | www.nature.com/naturecommunications 5http://www.nature.com/naturecommunicationsby analysing finite bias measurements from our smallestconstriction device we extract a sub-band energy spacing ofDE¼ 13.5±2 meV near the CNP (Fig. 4a,b, SupplementaryFigs 10–12 and Supplementary Note 6). With the geometricwidth of 230 nm also confirmed by the Fourier spectroscopy(Fig. 3c) we can estimate the Fermi velocity near the CNP asvF¼ 2WDE/h¼ (1.5±0.2)� 106 m s� 1. This is a clear signatureof a substantially renormalized Fermi velocity in nanostructuredgraphene, possibly enhanced by electron–electron interaction37.Moreover, the extracted energy scales are consistent with theweak temperature dependence of the quantized conductance(Fig. 4c, Supplementary Figs 13 and 14 and SupplementaryNote 7).Transition from quantized conductance to quantum Hall.Additional clear fingerprints of size quantization appear in theparametric evolution of the conductance steps38 with magneticfield B. The transition from size quantization at zero B-field toLandau quantization at high magnetic fields occurs when thecyclotron radius lC becomes smaller than half the constrictionwidth W. For the Landau level m the transition should occur at2lC ¼ 2ffiffiffiffiffiffiffi2mplB � W with lB the magnetic length. This transitionline in the B� n plane (see black dashed curve in Fig. 5a) agreeswell with the onset of Landau level formation in our data (seeSupplementary Fig. 15 and Supplementary Note 8 for similar datafrom a 280-nm constriction device). The evolution of the lowestquantized steps (at B¼ 0 T) to the corresponding lowest Landaulevels at low temperatures (T¼ 1.7 K) can be easily tracked(Fig. 5b,c). At higher temperatures (T¼ 6 K) the evolution ofquantized sub-bands to Landau levels is observed even for higherconductance plateaus (Fig. 5d,e). For a comparison, we calculatethe evolution of size quantization of an infinitely long ribbon ofwidth W as function of magnetic field. We take WE230 nm fromthe SEM data, which leaves no adjustable parameters. Our model(black lines in Fig. 5e,f) reproduces the evolution from the kinksat small fields (lBcW) to the Landau levels for large fields(lBoW) remarkably well, further supporting the notion that theyare, indeed, a signature of size quantization.DiscussionWe have shown ballistic conductance of confined Dirac fermionsin high-mobility graphene nanoconstrictions sandwiched byhBN. Away from the Dirac point, we observe a linear increasein conductance as function of Fermi wave vector with a slopeproportional to constriction width. Close to the Dirac point,the charging of localized edge states distorts this linearrelation. Superimposed on the linear conductance, we observereproducible, evenly spaced modulations (kinks). Tight-bindingsimulations for the device reproduce these structures related tosize quantization at the constriction. We can unambiguouslyidentify these ‘kinks’ as size quantization signatures by bothFourier spectroscopy at zero magnetic field and their evolutionwith magnetic field, finding good agreement between theory andexperiment.0G (e2 /h)n (1012 cm–2)n (1012 cm–2) n (1012 cm–2)n (1012 cm–2)n (1012 cm–2)n (1012 cm–2)2468021B (T)B (T)0B (T)12300.210.40.60.804843210–1–2–3 1.51 2 10.7 1.3–0.3–0.6–0.9 0.80.4 0.350 0.7012B (T)2–3∂G/∂B (e2/h T–1)∂2G/∂B∂n(109 e2/h T –1cm2)∂2G/∂B∂n(109 e2/h T –1 cm2)∂2G/∂B∂n(109 e2/h T –1cm2)3–2.30 0–1.11.57.53fc d0.6 1�= –14�= –10 –6 +6 +10–10 –6 –2 +2 +6 +10 +14 +10 +22+18B (T)a ebFigure 5 | Magnetic-field dependence of the size quantization. (a) Landau level fan of the graphene quantum point contact of width WE230 nm,measured at T¼ 1.7 K. Landau levels emerge at high magnetic fields. The magnetic-field quantization of Landau level m dominates over size quantizationas soon as 2ffiffiffiffiffiffiffi2mplB (where the magnetic length lB � 25=ffiffiffiffiffiffiffiffiB½T�pnm) is smaller than the constriction width (B-field values above dashed black line).(b,c) Double-derivative plots of the regions delimited by thin dashed lines in a showing the evolution of the lowest quantization plateaus with magneticfield: we observe the full transition from quantized sub-bands (B¼0 T) to Landau levels at large B-field. (d) The same magnetic-field evolution is visible inthe conductance as a function of magnetic field and charge carrier density for a different cool-down of the same device, also measured at 1.7 K. The bluearrows highlight the expected quantum Hall conductance plateaus at 2, 6 and 10 e2/h. (e) Double-derivative plot of the conductance as a function ofmagnetic field and charge carrier density measured at T¼ 6 K. The solid black lines denote the theoretical expectations for the evolution of the sizequantization with magnetic field. The thick dashed black line corresponds to the boundary of the Landau level regime, also appearing in a. (f) Zoom-inof e for small magnetic fields Br1 T.ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms115286 NATURE COMMUNICATIONS | 7:11528 | DOI: 10.1038/ncomms11528 | www.nature.com/naturecommunicationshttp://www.nature.com/naturecommunicationsMethodsExperimental methods and details. The hBN–graphene–hBN sandwichstructures8 have been etched by reactive ion etching in an SF6 atmosphere, priordeposition of a B10-nm-thick Cr etching mask. Residues of Cr oxide are removedby immersing the samples in a tetramethylammonium hydroxide solution forabout 30–35 s. All transport measurements are performed in a four-probeconfiguration using standard lock-in techniques. Since the distances between thecontacted current-carrying electrodes and the voltage probes are small comparedwith the other length scales of the system, we have an effective two-probeconfiguration. Importantly, this way we exclude the one-dimensional contactresistances.Electrostatic simulations and transport calculations. We simulate theexperimental device geometry using a third-nearest neighbour tight-bindingansatz. We rescale our device by a factor of four compared with experiment,to arrive at a numerically feasible geometry. We determine the Green’sfunction using the modular recursive Green’s function method30,39. The localdensity of states and transport properties can then be extracted by suitableprojections on the Green’s function. For more technical details seeSupplementary Note 9.References1. Young, A. F. & Kim, P. Quantum interference and Klein tunneling in grapheneheterojunctions. Nat. Phys. 5, 222–226 (2009).2. Tworzydlo, J. et al. Sub-Poissonian shot noise in graphene. 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Hassler, F. Haupt and B.J. van Wees.Support by the HNF, the DFG (SPP-1459), the ERC (GA-Nr. 280140), the EU projectGraphene Flagship (Contract No. NECT-ICT-604391) and Spinograph, and the AustrianScience Fund (SFB-041 VICOM, SFB-049 NextLite and DK-W1243 Solids4Fun) isgratefully acknowledged. Calculations were performed on the Vienna Scientific Clusters.Author contributionsB.T. and C.S. conceived the project; B.T. and J.P. fabricated the samples, performed theexperiments and interpreted the data; S.E. assisted during measurements; B.T., D.J.and J.P. analysed the data; L.A.C. and F.L. performed the numerical calculations andtheoretical analysis; A.G. and F.L. developed the numerical code; T.T. and K.W.synthesized the hBN crystals; J.B., S.V.R. and C.S. advised on theory and experiments;B.T., L.A.C., F.L., J.B. and C.S. prepared the manuscript; all authors contributed indiscussions and writing of the manuscript.Additional informationSupplementary Information accompanies this paper at http://www.nature.com/naturecommunicationsCompeting financial interests: The authors declare no competing financial interests.Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/How to cite this article: Terrés, B. et al. Size quantization of Dirac fermions in grapheneconstrictions. Nat. Commun. 7:11528 doi: 10.1038/ncomms11528 (2016).This work is licensed under a Creative Commons Attribution 4.0International License. The images or other third party material in thisarticle are included in the article’s Creative Commons license, unless indicated otherwisein the credit line; if the material is not included under the Creative Commons license,users will need to obtain permission from the license holder to reproduce the material.To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/NATURE COMMUNICATIONS | DOI: 10.1038/ncomms11528 ARTICLENATURE COMMUNICATIONS | 7:11528 | DOI: 10.1038/ncomms11528 | www.nature.com/naturecommunications 7http://www.nature.com/naturecommunicationshttp://www.nature.com/naturecommunicationshttp://npg.nature.com/reprintsandpermissions/http://npg.nature.com/reprintsandpermissions/http://creativecommons.org/licenses/by/4.0/http://www.nature.com/naturecommunications title_link Results Ballistic transport Figure™1Width-dependent ballistic transport in etched graphene nanoconstrictions encapsulated in hBN.(a) Schematic illustration of a hBN-graphene sandwich device with the bottom and top layers of hBN appearing in green, the gold contacts in yellow, the Si Localized states Figure™2Conductance through graphene quantum point contacts.(a) Conductance traces of two different cool-downs (black and green curve) of the same constriction (Wap230thinspnm) as a function of charge carrier density. For the black (green) cool-down, shad Figure™3Size quantization signatures.(a) Comparison of the low-energy conductance between theory (blue) and experiment (black). (b,c) Measured electron (el; black trace) and hole (ho; red trace) conductance including kink or step-like structure (see arrow Quantized conductance Figure™4Finite bias and temperature dependence of the quantized conductance.(a) Zero B-field differential conductance g as a function of bias voltage Vb, measured at T=6thinspK, taken at fixed values of back-gate voltage Vg from -0.5 to 3.0thinspV in step Transition from quantized conductance to quantum Hall Discussion Figure™5Magnetic-field dependence of the size quantization.(a) Landau level fan of the graphene quantum point contact of width Wap230thinspnm, measured at T=1.7thinspK. Landau levels emerge at high magnetic fields. The magnetic-field quantization of Landa Methods Experimental methods and details Electrostatic simulations and transport calculations YoungA. F.KimP.Quantum interference and Klein tunneling in graphene heterojunctionsNat. Phys.52222262009TworzydloJ.Sub-Poissonian shot noise in graphenePhys. Rev. Lett.962468022006NovoselovK. S.Two-dimensional gas of massless Dirac fermions in grapheneNat We acknowledge stimulating discussions with F. Hassler, F. Haupt and B.J. van Wees. Support by the HNF, the DFG (SPP-1459), the ERC (GA-Nr. 280140), the EU project Graphene Flagship (Contract No. NECT-ICT-604391) and Spinograph, and the Austrian Science F ACKNOWLEDGEMENTS Author contributions Additional information