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Jiong Lu, Hsin-Zon Tsai, Alpin N. Tatan, Sebastian Wickenburg, Arash A. Omrani, Dillon Wong, Alexander Riss, Erik Piatti, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Alex Zettl, Vitor M. Pereira, Michael F. Crommie

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[Frustrated supercritical collapse in tunable charge arrays on graphene](https://mdr.nims.go.jp/datasets/9980abff-01ff-41d1-ac39-2e4427b8813c)

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Frustrated supercritical collapse in tunable charge arrays on grapheneARTICLEFrustrated supercritical collapse in tunable chargearrays on grapheneJiong Lu1,2,3, Hsin-Zon Tsai1,4,5, Alpin N. Tatan3,6, Sebastian Wickenburg1,5, Arash A. Omrani1, Dillon Wong1,5,Alexander Riss1,10, Erik Piatti1,7, Kenji Watanabe8, Takashi Taniguchi8, Alex Zettl1,5,9,Vitor M. Pereira 3,6 & Michael F. Crommie 1,5,9The photon-like behavior of electrons in graphene causes unusual confinement propertiesthat depend strongly on the geometry and strength of the surrounding potential. We reportbottom-up synthesis of atomically-precise one-dimensional (1D) arrays of point charges ongraphene that allow exploration of a new type of supercritical confinement of graphenecarriers. The arrays were synthesized by arranging F4TCNQ molecules into a 1D lattice onback-gated graphene, allowing precise tuning of both the molecular charge and the arrayperiodicity. While dilute arrays of ionized F4TCNQ molecules are seen to behave like isolatedsubcritical charges, dense arrays show emergent supercriticality. In contrast to compactsupercritical clusters, these extended arrays display both supercritical and subcritical char-acteristics and belong to a new physical regime termed “frustrated supercritical collapse”.Here carriers in the far-field are attracted by a supercritical charge distribution, but their fallto the center is frustrated by subcritical potentials in the near-field, similar to trapping of lightby a dense cluster of stars in general relativity.https://doi.org/10.1038/s41467-019-08371-2 OPEN1 Department of Physics, University of California at Berkeley, Berkeley, CA 94720, USA. 2Department of Chemistry, National University of Singapore, 3Science Drive 3, Singapore 117543, Singapore. 3 Centre for Advanced 2D Materials, National University of Singapore, 6 Science Drive 2, Singapore 117546,Singapore. 4 International Collaborative Laboratory of 2D Materials for Optoelectronic Science & Technology of Ministry of Education, EngineeringTechnology Research Center for 2D Material Information Function Devices and Systems of Guangdong Province, Shenzhen University, Shenzhen 518060,China. 5Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA. 6Department of Physics, National University ofSingapore, 2 Science Drive 3, Singapore 117542, Singapore. 7 Department of Applied Science and Technology, Politecnico di Torino, Torino 10129 TO, Italy.8 National Institute for Materials, Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 9 Kavli Energy NanoSciences Institute at the University of California atBerkeley, Berkeley, CA 94720, USA. 10Present address: Physics Department E20, Technical University of Munich, James-Franck-Straße 1, D-85748 Garching,Germany. These authors contributed equally: Jiong Lu, Hsin-Zon Tsai, Alpin N. Tatan. Correspondence and requests for materials should be addressed toJ.L. (email: chmluj@nus.edu.sg) or to V.M.P. (email: vpereira@nus.edu.sg) or to M.F.C. (email: crommie@berkeley.edu)NATURE COMMUNICATIONS |          (2019) 10:477 | https://doi.org/10.1038/s41467-019-08371-2 | www.nature.com/naturecommunications 11234567890():,;http://orcid.org/0000-0002-3462-524Xhttp://orcid.org/0000-0002-3462-524Xhttp://orcid.org/0000-0002-3462-524Xhttp://orcid.org/0000-0002-3462-524Xhttp://orcid.org/0000-0002-3462-524Xhttp://orcid.org/0000-0001-8246-3444http://orcid.org/0000-0001-8246-3444http://orcid.org/0000-0001-8246-3444http://orcid.org/0000-0001-8246-3444http://orcid.org/0000-0001-8246-3444mailto:chmluj@nus.edu.sgmailto:vpereira@nus.edu.sgmailto:crommie@berkeley.eduwww.nature.com/naturecommunicationswww.nature.com/naturecommunicationsGraphene’s photon-like carrier dispersion provides fertileground for testing exotic predictions of quantum elec-trodynamics, as well as for developing novel quantumelectron optics1. Due to this relativistic behavior, electrostaticconfinement of charge carriers in graphene is very different thanthat seen in more conventional materials2,3. Indeed, trappingelectrons by placing point charges on graphene is formally ana-logous to trapping light by a gravitational field: something onlypossible near extremely dense matter4. Such localization, how-ever, is possible for graphene around very strong Coulomb cen-ters in the so-called supercritical regime5–10, which allows adegree of localization otherwise impossible to achieve in pristinegraphene. This behavior is formally equivalent to the supercriticalcollapse of atoms having ultra-heavy nuclei in quantum electro-dynamics (QED)11–15. This atomic analogy, however, is onlyuseful for charge distributions that can be approximated as asingle-point charge. Here, we demonstrate a new supercriticalregime, “frustrated supercriticality”, that is accessible throughcareful arrangement of point charge distributions on a graphenesurface. Frustrated supercriticality reflects an interplay betweennear-field and far-field electronic behavior for charge distribu-tions that are globally supercritical but locally subcritical. Elec-tronic behavior here is analogous to photons gravitationallytrapped within a star cluster that has no black holes. The ability tocharge and discharge such states via local electrodes raises theprospect of designing localized electronic states without com-promising graphene crystallinity, and hence integrating them intoextremely high-mobility nanoscale devices.Demonstrating frustrated supercriticality in graphene requiresthe ability to position static charges with a level of precisioncurrently unobtainable by conventional top-down lithography.We achieved the necessary precision via a bottom-up synthesistechnique that yields charge-tunable, periodic, self-assembledone-dimensional (1D) arrays of F4TCNQ molecules on clean,back-gated graphene FET devices. STM spectroscopy (STS)measurements reveal that dilute charged arrays with large inter-molecule spacings d ≥ 10 nm scatter surrounding Dirac fermionsand induce no bound states in the nearby pristine graphene. Fordenser charged arrays with d ≤ 10 nm, however, STS shows theemergence of a new quasi-bound state with an energy near theDirac point. This state extends into the pristine graphene and isable to trap charge, as observed through spatially resolved char-ging maps. We are able to explain this behavior by modeling thecombined array/graphene system via tight-binding calculationsthat take screening into account. Our simulations reveal thatwhen intermolecular distance in a 1D array is greater than thegraphene screening length then each molecule behaves like anisolated subcritical Coulomb center. For intermolecular separa-tions less than the screening length, however, our simulationsreveal the emergence of a new type of collective supercritical statewith energy near the Dirac point. This frustrated supercriticalstate is seen theoretically even for systems composed of only two-point charges and the wavefunction spread scales with inter-charge separation. In the semiclassical limit, this behavior isshown to be nearly equivalent to a general relativistic treatment oftrapped light.ResultsStructural characterization of F4TCNQ molecular arrays ongraphene. Our FET devices were fabricated by placing a CVD-grown graphene monolayer on top of a hexagonal boron nitride(h-BN) flake resting on an SiO2 layer covering a doped Si wafer,the latter providing an electrostatic back-gate. F4TCNQ molecules(Fig. 1a) were used as the charge elements in this study becausetheir charge state can be reliably switched on (negative) and off(neutral) via the back-gate, as demonstrated previously16. One-dimensional lattices of F4TCNQ were created using an edge-templated self-assembly protocol that allows highly precisealignment of individual molecules. The template consists ofelectronically inert 10,12-pentacosadiynoic acid (PCDA), a linearchain molecule that self-assembles into monolayer-high islandson graphene with perfectly straight edges17 (Fig. 1a). As seen inthe STM image of Fig. 1b, these islands display a regular moirépattern with a period of a= 1.92 nm due to the lattice mismatchbetween graphene and the PCDA layer. When F4TCNQ isdeposited at room temperature onto PCDA-decorated graphene/h-BN, we observe the preferential adsorption of individualF4TCNQ molecules at the PCDA island edge sites that corre-spond to a maximum in the moiré pattern (Fig. 1b, c). The precisemoiré periodicity facilitates the assembly of 1D molecular arraysthat remain strictly periodic over hundreds of nanometers, asshown in Fig. 1c. By controlling the dosage of F4TCNQ onto thesurface, this edge-templating process results in tunable arrays thatcan exhibit periodicities (d) with unit cells having multiples of themoiré period a. F4TCNQ arrays with d= 2a, 3a, 4a, and 5a canbe seen in Fig. 2a–d. Gate voltage control allows the moleculeswithin an array to be toggled between negative and neutral chargestates (Supplementary Fig. 1)15. The molecular charge state, forexample, is negative when the gate voltage is 30 V for all mole-cular arrays down to (and including) a periodicity of 2a.Probing electronic structure of graphene near charged mole-cular arrays. We investigated how charged 1D molecular arraysa5 nmba2aF4TCNQF4TCNQPCDA1 nmcCFNPCDAF4TCNQ VSVgG/BNSiO2Doped SiFig. 1 STM images of one-dimensional F4TCNQ molecular arrays. a Schematic illustration of edge-templated synthesis of F4TCNQ molecular arrays on agated graphene FET device. b A close-up view of the PCDA edge-anchored F4TCNQ molecular array having a period of 2a (a= 1.92 nm is the moiré latticeconstant of the PCDA monolayer on graphene). c STM image of an 80-nm long section of an atomically precise F4TCNQ molecular array having the 2astructure and anchored to the edge of a PCDA island. All STM images were acquired at T= 4.5 KARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-08371-22 NATURE COMMUNICATIONS |          (2019) 10:477 | https://doi.org/10.1038/s41467-019-08371-2 | www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsaffect graphene’s Dirac fermions by probing the energy-dependent local density of states (LDOS) in the vicinity ofarrays having different periodicity. This was done by performingdI/dV point spectroscopy on pristine graphene at different dis-tances from the center of an F4TCNQ molecule along a lineperpendicular to the charged array (Fig. 2e–h). All dI/dV spectraexhibit a gap feature (~130 meV) pinned at EF (arising fromphonon-assisted inelastic tunneling18) and another local mini-mum at Vs ≈−0.18 V for Vg= 30 V that indicates the Dirac pointenergy (ED). ED is seen to lie 115 meV below the Fermi energyafter accounting for the inelastic gap, corresponding to a carrierdensity of ne ≈ 9.5 × 1011 cm−2 for Vg= 30 V. In arrays with alarge intermolecular spacing of 5a, the spectra at points adjacentto F4TCNQ molecules (Fig. 2e) exhibit the characteristicparticle–hole asymmetry expected for an isolated subcriticalnegative charge (here, Z < ZC, where Ze is the charge on amolecule and ZCe is the supercritical charge threshold; ZC= 1/2α0 and α0 is the fine structure constant for graphene, see Sup-plementary Fig. 4)5,6,16,19–22. The graphene LDOS, however,changes substantially when the array period is decreased. As seenin Fig. 2f–h, the hole-side of the dI/dV traces (i.e., E < ED)develops a systematically higher spectral weight and a clearresonant structure near ED as the array period is reduced to 2a(Fig. 2h). The resonance decays rapidly with distance from thearray and fades beyond 10 nm (Supplementary Fig. 3). This newfeature cannot be attributed to a localized molecular orbital sinceF4TCNQ molecular states are more tightly bound and vanish atdistances s > 1.25 nm from an F4TCNQ center (SupplementaryFig. 2), whereas the new resonance is observed over the range 1.8nm < s < 10 nm.Since isolated charged F4TCNQ molecules generate only asubcritical Coulomb potential16, the development of a resonancenear ED in more closely packed arrays suggests a collective effectwhereby the array somehow surpasses the supercritical thresholdand induces new quasi-bound states7. This hypothesis issupported by charging behavior observed near dense d= 2aarrays, as seen in Fig. 3. Figure 3a shows a continuous region ofthe surface where the left side is imaged via an STM topograph(showing the 2a array) and the right side is imaged via a dI/dVmap that shows electronic structure in the pristine graphene tothe right of the array for VS=−0.12 V and Vg= 20 V. Sharprings are seen on the right that are indicative of charging behavior(similar rings have been seen previously by STM due to thecharging of adsorbed molecules and defects on varioussurfaces23–27). The rings of Fig. 3a, however, are centered awayfrom the molecules on the pristine graphene, indicating that theyarise from states localized in the pristine graphene rather than inthe molecular orbitals.This charging behavior can be better seen in the gate-dependent dI/dV point spectra of Fig. 3b, acquired with the2.2 nm3.2 nm4.5 nm6.1 nm2.2 nm3.2 nm4.5 nm6.1 nm1.8 nm3.2 nm4.5 nm6.1 nm2.6 nm3.2 nm4.5 nm6.1 nm2.6 nm3.2 nm4.5 nm6.1 nm2.2 nm3.2 nm4.5 nm6.1 nm2.2 nm3.2 nm4.5 nm6.1 nm–0.4 –0.2 0.0 0.2 0.4–0.4 –0.2 0.0 0.2 0.40123401234012345012341.8 nm3.2 nm4.5 nm6.1 nm5a5aExperimental Theoretical1 nmefa b c dghijkl4a4a3a3a2a2a5a4a 3a2aSample bias (V)dI/dV (a.u.)Vg = 30 VVg = 30 VVg = 30 VVg = 30 V12340123401234012340Fig. 2 Emergence of supercritical features in 1D charged molecular arrays. a–d STM images of 1D F4TCNQ molecular arrays with tunable periodicity from5a to 2a (the molecular arrays are anchored to PCDA islands at the surface of a graphene FET, and a= 1.92 nm is the PCDA/graphene moiré latticeconstant). e–h dI/dV spectra measured at different distances from the center of an F4TCNQ molecule along a line normal to the 1D array axis for chargedarrays having different periods as shown in (a–d). All spectra were taken at the same back-gate voltage (Vg= 30 V) and tip height. i–l Theoreticallysimulated dI/dV spectra for equivalent arrays of point charges on graphene at the same probing distances as in the experimental traces shown in(e–h). The calculation used an effective valence per molecule of Z= 0.86 and an effective Coulomb screening length λS= 10 nm, as described in the maintext. All experimental data were obtained at T= 4.5 KNATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-08371-2 ARTICLENATURE COMMUNICATIONS |          (2019) 10:477 | https://doi.org/10.1038/s41467-019-08371-2 | www.nature.com/naturecommunications 3www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsSTM tip held at the edge of the ring marked in Fig. 3a. The peakmarked “A” shows the new graphene resonance induced by thecharged molecular array as seen in Fig. 2h. As Vg is lowered fromVg= 30 V to 24 V, this feature moves up in energy, as expectedfor a density-of-states feature when EF is lowered by reduction ofVg. An additional peak marked “B” can also be seen that movesopposite to A as Vg is lowered, indicating that it is a charging peakrather than a density-of-states feature16,27. For Vg > 20 V, peak Bis caused by the discharging of state A (i.e., by loss of an electron)as it is pulled above EF by STM tip-induced local gating. For Vg <20 V, peak B jumps across EF and continues to move down inenergy, as expected for a charging peak since state A has nowcrossed to the other side of EF (the empty state side) and must bepulled below EF to become charged (i.e., by gain of an electron).The charging behavior observed for this new graphene stateconfirms its localized nature (see Supplementary note 6 andSupplementary Figs 5 and 6 for additional details).Modeling the electronic structure of graphene near one-dimensional charge pattern. To understand the microscopicorigin of this new state, we set out to answer the question of howsuch a localized state might arise in pristine graphene from theeffect of subcritical molecular Coulomb potentials. We started bycalculating the LDOS for electronic states in the vicinity of asimulated 1D array of point charges on graphene. The simulationwas performed by locating point charges at positions coincidingwith the center of each molecule in the experiment and thencalculating the LDOS via a recursive method28 (see Supplemen-tary Notes 7 to 11 and Supplementary Figs 7–17). Electrons ingraphene were modeled using a single-orbital, nearest-neighbortight-binding approximation29,30 (Supplementary Note 7), andelectrostatic screening was incorporated through the use of anappropriate dielectric function15,31 (Supplementary Note 8). Theresulting theoretically predicted dI/dV traces are shown inFig. 2i–l, beside the experimental ones of correspondinggeometry. To capture the inelastic phonon gap seen experimen-tally, we convolved the theoretical LDOS as described in ref. 32(Supplementary Note 9).Comparison of theory and experiment shows good agreementin all the key features: the overall particle–hole asymmetry, themarked increase of spectral weight for energies below ED as thearray density is increased, the emergence of a clear resonance inthe vicinity of ED, and the rate of decay of the resonance withperpendicular distance from the array (see also SupplementaryNote 10). Since our calculation included no perturbation to thegraphene other than point charges, this confirms that the newstructure in the dI/dV curves is due to the collective Coulombfield of the charged F4TCNQ array. Our best theory/experimentagreement is obtained for an effective valence per molecule ofZ= 0.86 and a Coulomb screening length of λS= 10 nm (thesevalues agree with previous estimates of λS and Z for isolatedmolecules adsorbed to graphene16,33, see Supplementary Note 11).The estimated value of λS is consistent with the experimentalspatial extent of the resonant state, which is seen to disappear atdistances s > 10 nm from an array (Supplementary Fig. 3).In order to better understand the spatial distribution of thisresonance state, we computed representative wave functions atenergies within the resonance via exact diagonalization of thetight-binding model. As shown in Fig. 4a, a supercriticalwavefunction is found that is confined to within a few nm ofthe array centerline and can thus be characterized as a quasi-localized state. This explains the strong, spatially decayingresonant state imprinted in the dI/dV spectra of Fig. 2 as wellas the fact that the resonance can be charged/discharged throughlocal tip-gating (Fig. 3). The experimentally observed spatial offsetof the charging circle to the side of the molecule (seen in Fig. 3a)can be explained by decreased tip-gating efficiency over themolecule’s center due to the presence of highly concentratednegative charge on the F4TCNQ molecules (see SupplementaryNote 6 for details).ba dI/dV1 nmVS = –0.12 VVg = 20 Vx dI/dV (a.u.)Sample bias (V)–0.4024681012–0.2 0.0 0.2 0.422 V20 V19 V18 V17 V15 VB30 V26 V24 VBBB BBBBBAAAAAAAAAA12 VVgBFig. 3 Gate-dependent charging behavior of supercritical quasi-bound state. a Left side: STM image of a portion of a charged 2a F4TCNQ array. Right side:dI/dVmap of the pristine graphene region adjacent to the array shows charging rings in the near-field region (Vg= 20 V, VS=−0.12 V). b Gate-dependentdI/dV spectra acquired at the position marked “x” in panel (a). The supercritical resonance is labeled “A”, and the corresponding tip-induced charging/discharging feature is labeled as “B”ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-08371-24 NATURE COMMUNICATIONS |          (2019) 10:477 | https://doi.org/10.1038/s41467-019-08371-2 | www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsPoint chargeabc2 nm0 nm2 nm4 nm0.4 nm1.3 nmd = 2.1 nm 3.0 nm(no state)Evolution of unscreened supercritical charge pair, λs = ∞ Evolution of screened supercritical charge pair at fixed separation, d = 1.3 nm–6 –4 –2 0 2Distance x (mm)4 6Min Maxddλs = 14.2 nm λs = 7.1 nm λs = 4.7 nm λs = 3.6 nm |�(r )|2|�(r )|2|�(x)|2Fig. 4 Theoretical wave functions for frustrated supercritical states. a Density plot of the wavefunction associated with a supercritical resonant state in graphenenear the Dirac point obtained from exact diagonalization of the Hamiltonian discussed in the text (same parameters as in Fig. 2). Black dots mark the positionsof the Coulomb centers used in the calculation and the colored disks reflect the state’s local probability density, both through size and color. The charges areseparated by d= 3.8 nm as in the experimental 2a array and the total system has 16,000 carbon atoms spanning 19 × 21 nm2 (the image shown is cropped).The top inset shows a close-up near the central charge, where rapid decay is visible against the underlying honeycomb lattice. The bottom inset shows thewavefunction cross-section along a line perpendicular to the array (boxed region, cf. Supplementary Note 9). b Wavefunction of the most bound supercriticalstate for a pair of unscreened charges at the following charge separations: d=0 nm, d=0.4 nm, d= 1.3 nm, d= 2.1 nm, and d= 3.0 nm (Z=0.8 ZC). Eachwavefunction is shown in the region where its value is at least 1% of its maximum. The characteristic wavefunction extension is ~d. c The same as (b) but with afixed charge separation (d= 1.3 nm) and a varying screening length λS as indicated. Supercritical states disappear for λS≤ 3.6 nm (cf. Supplementary Fig. 15d)NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-08371-2 ARTICLENATURE COMMUNICATIONS |          (2019) 10:477 | https://doi.org/10.1038/s41467-019-08371-2 | www.nature.com/naturecommunications 5www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsThe full 1D array simulations of Figs 2i–l and 4a reproduce ourexperimental data quite well, but they do not give us deep insightinto the inner workings of frustrated supercriticality, includingthe interplay between near- and far-field behavior for Diracquasiparticles interacting with distributed point charges. In orderto gain a better intuition into this behavior, we analyzed quasi-localized states formed near globally supercritical charge distribu-tions containing just two identical subcritical charges as afunction of their separation34,35 and screening length (eachcharge was given a valence Z= 0.8 ZC). Figure 4b shows theresults of exact diagonalization of the tight-binding model for thispair of charges with different separations, d, at the energy of thequasi-bound resonance (see also Supplementary Note 12).Localization of the quasi-bound state cannot be seen aroundany one charge center, because the near-field regions reflect thesubcritical valence of the individual charges. Localization is seenrather in the far-field at distances r > d, where the aggregatecharge of the interior can be seen as supercritical. As the twocharges are pulled apart, the size of the quasi-bound state is seento monotonically increase and push the far-field region outwardfrom the origin. For unscreened systems, this process scaleswithout limit as the subcritical charges are pushed apart toinfinity.The effect of screening on this process can be seen in Fig. 4cwhich shows the same two charges as in Fig. 4b, but for fixedseparation d= 1.3 nm and different values of the screening lengthλS. The bulk of the wavefunction is seen to localize within r ≤ 4nm, and so the state is essentially unchanged so long as λS > 4 nm.As λS is reduced below 4 nm, however, the supercritical staterapidly quenches and the charge distribution reverts tosubcriticality. The rapid quenching arises from two simultaneouseffects. First, the two Coulomb potentials become physicallyseparate as λS approaches d and, second, the supercriticalwavefunction (which extends out a distance d) becomesconstricted when the reduced screening length cuts into thepotential that supports it. This explains why no signs ofsupercriticality are seen experimentally for our d= 5a arrays,since the inter-charge separation in this case is on the order of λS.Supercriticality develops for denser arrays as the inter-chargespacing falls below the screening length (d < λS).DiscussionThe contrasting behavior we observe here for the near- and far-field of a pair of subcritical charges each with Zc/2 < Z < Zc can besummed up in a semi-classical description of graphene carriersunder the effective potential, Vtot(r), of a point charge distributionthat is supercritical in the far-field but subcritical in the near-field.The supercritical regime is generally characterized semiclassicallyby the existence of a finite potential barrier that traps carriers onthe charge distribution side of the barrier (details in Supple-mentary Note 13.1). For a carrier in the far-field, the potentialappears supercritical, as schematically represented in Fig. 5 (left),and the relativistic nature of graphene renders the potential sin-gularly attractive, namely Vtot ~− 1/r2. The centrifugal barrier isunable to counterbalance this singularity and the orbits becomecollapsing spirals (see Supplementary Fig. 18)19,36,37. The far-fieldsingularity, however, is removed at short distances from indivi-dual charge centers since Z < ZC. The about-to-collapse far-fieldorbit is thus modified when it reaches the near-field of the cluster,where collapsing orbits can’t exist due to the centrifugal barrier.The “collapse to the center” that seemed inevitable in the far-fieldis thus frustrated, as sketched in Fig. 5 (right), by the regular near-field behavior. Instead of collapsing, the particle becomes trappedwithin a region that extends out to ~d, the distance betweencharges.A useful analogy for this electronic behavior is the propagationof light near cosmic mass distributions accroding to generalrelativity. If a single, continuous mass distribution is compactenough that its spatial extent lies within the Schwarzschild radius,RSC (i.e., a black hole, see Supplementary Note 13.2), then lightwill be gravitationally trapped and inexorably fall through theevent horizon toward the center4, precisely the analogue ofelectronic supercritical collapse in the presence of a singlesupercritical impurity (i.e., graphene carriers here are mappedonto photons and the supercritical charge onto a black hole). Onthe other hand, if a mass distribution consists of isolated massesthat each have no event horizon (e.g., a star cluster) but thatextend close to RSC of the aggregate, then photons incident fromoutside of RSC can be trapped gravitationally in an extreme case ofgravitational lensing. Unlike near a black hole, the photon’s orbitwill not end with a fall onto one of the stars, but will rathermeander endlessly within the cluster, permanently bound by itsgravitational field. This is completely analogous to the frustratedsupercritical orbits of graphene charge carriers that remaintrapped in the near-field of a cluster of subcritical charges whosetotal charge > ZC (cf. Supplementary Fig. 19 and SupplementaryNote 13.3).In conclusion, we have discovered a new physical regime offrustrated supercriticality that is accessible experimentally due toFar fieldSupercritical cluster Subcritical chargesVtot(r)Vn(r)Zeff = N Z ZZdr rNear fieldFig. 5 Far-field vs. near-field semiclassical trajectories for frustrated supercriticality: The far-field potential Vtot(r) of a supercritical cluster (left) inducescollapse because N Z > ZC. Orbits here describe a collapsing spiral toward the charge cluster. In the near-field, on the other hand, each individual potentialVn(r) is subcritical (right) and the orbits approach the charges without falling into them. The supercritical collapse is thus frustrated by the subcriticalindividual charges in the near-field. This is analogous to light rays gravitationally trapped by a dense cluster of stars (Supplementary Note 13.2)ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-08371-26 NATURE COMMUNICATIONS |          (2019) 10:477 | https://doi.org/10.1038/s41467-019-08371-2 | www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsadvances in our ability to create atomically precise mesoscopicarrangements of Coulomb potentials on graphene. This createsnew opportunities for manipulating charge states in high-mobility graphene devices and provides new insight into theirbehavior by analogy to astrophysical gravitational lensing ofphotons.MethodsGraphene device fabrication. A back-gated graphene/h-BN/SiO2 device wasprepared by overlaying CVD-grown graphene onto hexagonal boron nitride (h-BN) flakes exfoliated onto a SiO2/Si substrate. h-BN flakes were exfoliated ontoheavily doped silicon wafers and annealed at 500 °C for several hours in air prior tographene transfer. The graphene was grown on copper foil by the CVD methodand transferred to the h-BN/SiO2 substrate via a poly methyl methacrylate stamp38.Electrical contact was made to the graphene by depositing Ti (10-nm thick)/Au(30-nm thick) electrodes using the stencil mask technique.STM/STS measurements. STM/STS measurements were performed under UHVconditions at T= 5 K using a commercial Omicron LT STM with tungsten tips.STM topography was obtained in constant-current mode. STM tips were calibratedon a Au(111) surface by measuring the Au(111) Shockley surface state before allSTS measurements. STS was performed under open feedback conditions by lock-indetection of an alternating tunnel current with a bias modulation of 6–16 mV (r.m.s.) at 400 Hz added to the tunneling bias. WSxM software was used to process allSTM images39.Theoretical modeling. The theoretical calculations are described in detail in thefollowing sections of the supplementary information: Tight-binding model of thecharged arrays in graphene (Supplementary Note 7), Simulated dI/dV curves fromthe bare LDOS calculations (Supplementary Note 8), Decay of the computed LDOSwith distance (Supplementary Note 9), Screened Coulomb potential (Supplemen-tary Note 10), Estimation of the effective potential parameters (SupplementaryNote 11), Supercritical threshold of an array of subcritical charges (SupplementaryNote 12), Effective radial potentials (Supplementary Note 13).Data availabilityThe data that support the findings of this study are available from the corre-sponding author on reasonable request.Received: 13 July 2018 Accepted: 2 January 2019References1. Chen, S. et al. Electron optics with p-n junctions in ballistic graphene. Science353, 1522 (2016).2. Katsnelson, M. I. et al. Chiral tunnelling and the Klein paradox in graphene.Nat. Phys. 2, 620 (2006).3. 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Scattering theory and ground-state energy of Dirac fermionsin graphene with two Coulomb impurities. Eur. Phys. J. B 87, 187 (2014).36. Gamayun, O. V. et al. Supercritical Coulomb center and excitonic instabilityin graphene. Phys. Rev. B 80, 165429 (2009).37. Boyer, T. H. Unfamiliar trajectories for a relativistic particle in a Kepler orCoulomb potential. Am. J. Phys. 72, 992 (2004).38. Jung, H. S. et al. Fabrication of gate-tunable graphene devices for scanningtunneling microscopy studies with Coulomb impurities. J. Vis. Exp. 101,e52711 (2015).39. Horcas, I. et al. WSXM: a software for scanning probe microscopy and a toolfor nanotechnology. Rev. Sci. Instrum. 78, 013705 (2007).AcknowledgementsThis research was supported by the Director, Office of Science, Office of Basic EnergySciences, Materials Sciences and Engineering Division, of the US Department of Energyunder contract no. DE-AC02-05CH11231 (Nanomachine program-KC1203) (STMimaging and spectroscopy), by the Molecular Foundry (graphene growth, growth char-acterization), and by the National Science Foundation grant DMR-1807233 (samplefabrication). J.L. acknowledges support from the Singapore Ministry of Education grantunder R-143-000-A06-112 (data analysis). H.-Z.T. acknowledges fellowship supportfrom the Shenzhen Peacock Plan (Grant no. 827-000113, KQJSCX20170727100802505,KQTD2016053112042971). V.M.P. acknowledges support from Singapore NationalResearch Foundation under its Medium-Sized Centre Programme (theory formalismdevelopment), and by the Singapore National Research Foundation award “Novel 2Dmaterials with tailored properties: beyond graphene” NRF-CRP6-2010-05 (charged arraysimulation).Author contributionsJ.L., H.Z.T. and S.W. designed and performed the experiments and analyzed the data.A.N.T. and V.M.P. performed the theoretical modeling and analysis, with A.N.T. com-puting the simulated LDOS and dI/dV curves, and V.M.P. performing the exact diag-onalization and analytical calculations. A.A.O., D.W. and A.R. helped with theexperiments and gave technical support and conceptual advice. A.Z. and E.P. facilitatedthe sample fabrication. K.W. and T.T. gave technical support and grew h-BN for thedevice. M.F.C. supervised the experiments and data analysis. J.L., H.Z.T, V.M.P. andM.F.C. wrote the paper.NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-08371-2 ARTICLENATURE COMMUNICATIONS |          (2019) 10:477 | https://doi.org/10.1038/s41467-019-08371-2 | www.nature.com/naturecommunications 7www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsAdditional informationSupplementary Information accompanies this paper at https://doi.org/10.1038/s41467-019-08371-2.Competing interests: The authors declare no competing interests.Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/Journal peer review information: Nature Communications thanks the anonymousreviewers for their contribution to the peer review of this work. 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To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2019ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-08371-28 NATURE COMMUNICATIONS |          (2019) 10:477 | https://doi.org/10.1038/s41467-019-08371-2 | www.nature.com/naturecommunicationshttps://doi.org/10.1038/s41467-019-08371-2https://doi.org/10.1038/s41467-019-08371-2http://npg.nature.com/reprintsandpermissions/http://npg.nature.com/reprintsandpermissions/http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/www.nature.com/naturecommunications Frustrated supercritical collapse in tunable charge arrays on graphene Results Structural characterization of F4TCNQ molecular arrays on graphene Probing electronic structure of graphene near charged molecular arrays Modeling the electronic structure of graphene near one-dimensional charge pattern Discussion Methods Graphene device fabrication STM/STS measurements Theoretical modeling References References Acknowledgements Author contributions ACKNOWLEDGEMENTS Competing interests Supplementary information ACKNOWLEDGEMENTS