Electrons imitating light: Frustrated supercritical collapse in charged arrays on graphene

The photon-like electronic dispersion of graphene bestows its charge carriers with unusual confinement properties that depend strongly on the geometry and strength of the surrounding potential. Here we report bottom-up synthesis of atomically-precise one-dimensional (1D) arrays of point charges aimed at exploring supercritical confinement of carriers in graphene for new geometries. The arrays were synthesized by arranging F4TCNQ molecules into a 1D lattice on back-gated graphene devices, allowing precise tuning of both the molecular charge state and the array periodicity. Dilute arrays of ionized F4TCNQ molecules are seen to behave like isolated subcritical charges but dense arrays show emergent supercriticality. In contrast to compact supercritical clusters, extended 1D charge arrays exhibit both supercritical and subcritical characteristics 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 have their fall to the center frustrated by subcritical potentials in the near-field, similar to the trapping of light by a dense cluster of stars in general relativity. Graphene’s photon-like carrier dispersion provides fertile ground for testing exotic predictions of quantum electrodynamics, as well as for developing novel quantum electron optics1. Due to this relativistic behavior, electrostatic confinement of charge carriers in graphene is very different than that seen in more conventional materials2,3. Indeed, trapping electrons by placing point charges on graphene is formally analogous to trapping light by a gravitational field: something only possible near extremely dense matter4. Such localization, however, is possible for graphene around very strong Coulomb centers in the so-called supercritical regime5-9 which allows a degree of localization otherwise impossible to achieve in pristine graphene. This behavior is formally equivalent to the supercritical collapse of atoms having ultra-heavy nuclei in quantum electrodynamics (QED)10-14. This atomic analogy, however, is only useful for charge distributions that can be approximated as a single point charge. Here we demonstrate a new supercritical regime, “frustrated supercriticality”, that is accessible through careful arrangement of point charge distributions on a graphene surface. Frustrated supercriticality reflects an interplay between near-field and far-field electronic behavior for charge distributions that are globally supercritical but locally subcritical. Electronic behavior here is analogous to photons gravitationally trapped within a star cluster that has no black holes. The ability to charge and discharge such states via local electrodes raises the prospect of designing localized electronic states without compromising graphene crystallinity, and hence integrating them into extremely high-mobility nanoscale devices. Demonstrating frustrated supercriticality in graphene requires the ability to position static charges with a level of precision currently unobtainable by conventional top-down lithography. We achieved the necessary precision via a bottom-up synthesis technique that yields charge-tunable, periodic, self-assembled 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 fermions and induce no bound states in the nearby pristine graphene. For denser charged arrays with d ≤ 10 nm, however, STS shows the emergence of a new quasibound state with an energy near the Dirac point. This state extends into the pristine graphene and is able to trap charge, as observed through spatially-resolved charging maps. We are able to explain this behavior by modeling the combined array/graphene system via tight-binding calculations that take screening into account. Our simulations reveal that when intermolecular distance in a 1D array is greater than the graphene screening length then each molecule behaves like an isolated subcritical Coulomb center. For intermolecular separations less than the screening length, however, our simulations reveal the emergence of a new type of collective supercritical state with energy near the Dirac point. This frustrated supercritical state is seen theoretically even for systems composed of only two point charges and the wavefunction spread scales with intercharge separation. In the semiclassical limit this behavior is shown to be nearly equivalent to a general relativistic treatment of trapped light. 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 because their charge state can be reliably switched on (negative) and off (neutral) via the back-gate, as demonstrated previously15. 1D lattices of F4TCNQ were created using an edge-templated self-assembly protocol that allows highly precise alignment of individual molecules. The template consists of electronically inert 10,12pentacosadiynoic acid (PCDA), a linear chain molecule that self-assembles into monolayerhigh islands on graphene with perfectly straight edges16 (Fig. 1a). As seen in the STM image of Fig. 1b, these islands display a regular moiré pattern with a period of a = 1.92 nm due to the lattice mismatch between graphene and the PCDA layer. When F4TCNQ is deposited at room temperature onto PCDA-decorated graphene/h-BN, we observe the preferential adsorption of individual F4TCNQ molecules at the PCDA island edge sites that correspond to a maximum in the moiré pattern (Fig. 1b, c). The precise moiré periodicity facilitates the assembly of 1D molecular arrays that remain strictly periodic over hundreds of nanometers, as shown in Fig. 1c. By controlling the dosage of F4TCNQ onto the surface, this edgetemplating process results in tunable arrays that can exhibit periodicities (d) with unit cells having multiples of the moiré period a. F4TCNQ arrays with d = 2a, 3a, 4a, and 5a can be seen in Figs. 2a-d. Gate voltage control allows the molecules within an array to be toggled between negative and neutral charge states (supplementary Fig. S1)15. The molecular charge state, for example, is negative when the gate voltage is 30 V for all molecular arrays down to (and including) a periodicity of 2a. We investigated how charged 1D molecular arrays affect graphene’s Dirac fermions by probing the energy-dependent local density of states (LDOS) in the vicinity of arrays having different periodicity. This was done by performing dI/dV point spectroscopy on pristine graphene at different distances from the center of an F4TCNQ molecule along a line perpendicular to the charged array (Figs. 2e-h). All dI/dV spectra exhibit a gap feature (~130 meV) pinned at EF (arising from phonon-assisted inelastic tunneling17) and another local minimum at Vs ≈ -0.18 V for Vg = 30 V that indicates the Dirac point energy (ED). ED is seen to lie 115 meV below the Fermi energy after accounting for the inelastic gap, corresponding to a carrier density of ne ≈ 9.5×10 cm for Vg = 30 V. In arrays with a large intermolecular spacing of 5a, the spectra at points adjacent to F4TCNQ molecules (Fig. 2e) exhibit the characteristic particle-hole asymmetry expected for an isolated subcritical negative charge (here Z < ZC where Ze is the charge on a molecule and ZCe is the supercritical charge threshold; ZC= 1/2α0 and α0 is the fine structure constant for graphene, see supplementary Fig. S4)5,6,15,18-21. The graphene LDOS, however, changes substantially when the array period is decreased. As seen in Figs. 2f-h, the hole-side of the dI/dV traces (i.e., E < ED) develops a systematically higher spectral weight and a clear resonant structure near ED as the array period is reduced to 2a (Fig. 2h). The resonance decays rapidly with distance from the array and fades beyond 10 nm (supplementary Fig. S3). This new feature cannot be attributed to a localized molecular orbital since F4TCNQ molecular states are more tightly bound and vanish at distances s > 1.25 nm from an F4TCNQ center (supplementary Fig. S2), whereas the new resonance is observed over the range 1.8 nm < s < 10 nm. Since isolated charged F4TCNQ molecules generate only a subcritical Coulomb potential,15 the development of a resonance near ED in more closely packed arrays suggests a collective effect whereby the array somehow surpasses the supercritical threshold and induces new quasibound states7. This hypothesis is supported by charging behavior observed near dense d = 2a arrays, as seen in Fig. 3. Fig. 3a shows a continuous region of the surface where the left side is imaged via an STM topograph (showing the 2a array) and the right side is imaged via a dI/dV map that shows electronic structure in the pristine graphene to the right of the array for VS = -0.12 V and Vg = 20 V. Sharp rings are seen on the right that are indicative of charging behavior (similar rings have been seen previously by STM due to the charging of adsorbed molecules and defects on various surfaces22-26). The rings of Fig. 3a, however, are centered away from the molecules on the pristine graphene, indicating that they arise from states localized in the pristine graphene rather than in the molecular orbitals. This charging behavior can be better seen in the gate-dependent dI/dV point spectra of Fig. 3b, acquired with the STM tip held at the edge of the ring marked in Fig. 3a. The peak marked “A” shows the new graphene resonance induced by the charged molecular array as seen in Fig. 2h. As Vg is lowered from Vg = 30 V to 24 V this feature moves up in energy, as expected for a density-of-states feature when EF is lowered by reduction of Vg. An additional peak marked “B” can also be seen which mov