Microscopic mechanisms of the supercritically charged single-atom vacancy in graphene: screening asymmetry, STM-pulse charge buildup, and stability limits

A single-atom vacancy in graphene can be positively charged by scanning-tunneling-microscope voltage pulses until it enters the supercritical regime, trapping conduction electrons in quasi-bound 'atomic-collapse' states that constitute a gate-tunable artificial atom (Mao et al., Nature Physics 12, 545, 2016). Three mechanism-level questions about this object remain open: (1) the microscopic origin of the pronounced electron-hole asymmetry in screening of the positive vacancy charge, whereby holes screen it poorly while electrons screen it efficiently; (2) the atomic-scale mechanism and energetics by which STM voltage pulses of either polarity irreversibly and stochastically drive a bare vacancy toward its fully charged ground state; and (3) the stability limits of the retained charge (in particular its temperature dependence beyond cryogenic conditions) and the feasibility of controlled arrays of charged vacancies. Solving these would establish how a supercritical Coulomb center is screened and written in a Dirac-fermion host and would constrain the use of charged vacancies as atomically precise quantum building blocks.

Background

In graphene the low-energy charge carriers behave as massless Dirac fermions, so the two-dimensional Coulomb problem reproduces the 'supercritical' physics of nuclei with charge beyond the critical value $Z_c$ at which the lowest bound state dives into the negative-energy (hole) continuum — a phenomenon called atomic collapse. The strength of the Coulomb coupling is measured by the dimensionless parameter $\beta = Z e^2/(\kappa \hbar v_F)$, where $Z$ is the charge number, $\kappa$ the effective dielectric constant of the environment, and $v_F$ the Fermi velocity; in the continuum Dirac description the critical value is $\beta_c = 0.5$, corresponding to an effective charge $Z_c \approx 1$ for typical device dielectrics. Theory established that above threshold the impurity is screened by vacuum polarization (the polarization of the Dirac sea, i.e. the filling of negative-energy states by electron-hole pairs) and that quasi-bound, quasi-Rydberg 'atomic-collapse' (AC) resonances appear (Pereira, Nilsson and Castro Neto, 2007; Shytov, Katsnelson and Levitov, 2007; Fogler, Novikov and Shklovskii, 2007), and the resonances were first observed experimentally in artificial nuclei made of charged calcium clusters on graphene (Wang et al., Science 2013). A different route to a supercritical center is the single carbon-atom vacancy: removing one atom breaks the sublattice symmetry and, according to density-functional theory (DFT), lattice relaxation after sputtering leaves a vacancy that stably hosts a positive charge of order $+1|e|$, with an associated magnetic moment (Ugeda et al., 2010). Screening of charged impurities by doped graphene is already known to be nonlinear and, in sign-dependent fashion, asymmetric between electron and hole doping (Ghaznavi, Mi\u0161kovi\u0107 and Goodman, 2010). In the experiment of Mao et al. (Mao, Jiang, Moldovan, Li, Watanabe, Taniguchi, Ramezani Masir, Peeters and Andrei, Nature Physics 12, 545, 2016; arXiv:1508.07667), vacancies created by He+ sputtering and annealing in graphene/hexagonal-boron-nitride stacks were probed by scanning tunneling microscopy and spectroscopy (STM/STS) together with Landau level (LL) spectroscopy, in which the on-site shift of the $N=0$ LL gives the local effective charge. As-made vacancies carried only $\beta \approx 0.1$; voltage pulses (2-3 V, 1-10 s) of either polarity irreversibly built the charge up step by step to $\beta \approx 1.29$, past the supercritical threshold. In the local density of states (DOS) the tightly localized vacancy peak (VP) is then joined by spatially extended AC resonances (R1, its sublattice-symmetry-breaking satellite R1', and R2), forming an electron disk surrounded by a halo of hole states. Gate-tuning the carrier density showed that the screening of the positive vacancy charge is strongly asymmetric: the R1 peak persists across the entire hole-doped (p-type) range, indicating weak screening by holes, while it is rapidly suppressed on the electron-doped (n-type) side, indicating efficient screening by electrons. The authors state that this asymmetry may reflect charge-induced electron-hole asymmetry in the local DOS or many-body correlations, 'but more work is needed to understand this phenomenon'. The observed charge is stable only 'for as long as the experiment is kept cold', and the authors propose, without demonstrating, that the charging technique 'could enable the fabrication of artificial atom arrays for electrostatic control and guidance of electrons in graphene'.

Problem Statement

Establish the microscopic physics of the supercritically charged single-atom vacancy ('artificial atom') in graphene, as reported by Mao et al. (Nature Physics 12, 545 (2016)), along the three directions the source work leaves open: (1) Determine the microscopic origin of the strong electron-hole asymmetry in the screening of the positively charged vacancy, in which hole doping leaves the atomic-collapse resonances (R1 and related quasi-bound states) essentially unscreened over the measured density range while electron doping rapidly screens them away; decide whether this asymmetry arises from the charge-induced electron-hole asymmetry of the local density of states or from many-body correlation effects, and give a quantitative account of the gate-dependent evolution of the resonances. (2) Identify the atomic-scale mechanism by which STM voltage pulses of either polarity incrementally, irreversibly and stochastically build up and stabilize the positive charge on a bare (as-made, subcritically charged) vacancy up to and beyond the supercritical threshold: the charge-transfer or structural-relaxation pathways involved, their energy barriers and kinetics, why both pulse polarities produce the same positively charged final state, and why pristine graphene and the substrate show no charging under identical pulses. (3) Establish the stability limits and scalability of the charged-vacancy state: the temperature dependence of charge retention beyond the cryogenic conditions under which stability has so far been demonstrated, and whether controlled arrays of individually charged vacancies can be fabricated and would behave as arrays of artificial atoms. These three questions share one scientific objective, a mechanism-level understanding and control of the supercritically charged vacancy; the proposed mechanisms must account for the measured Landau-level and gate-dependent tunneling spectra of the source experiments.

References

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