Scanning tunneling microscopy and spectroscopy

We employed scanning tunneling spectroscopy (STS)23,24 to identify Kondo screening of the vacancy magnetic moment by the distinctive zero-bias resonance it produces in the dI/dV curves (I is the tunneling current and V the junction bias), hereafter called Kondo peak. We first discuss samples consisting of two stacked single layer graphene sheets on a SiO 2 substrate (G/G/SiO 2 ) capping a doped Si gate electrode (Fig. 1b). A large twist angle between the two layers ensures electronic decoupling, and preserves the electronic structure of single layer graphene while reducing substrate induced random potential fluctuations24,25,26. A further check of the Landau-level spectra in a magnetic field revealed the characteristic sequence expected for massless Dirac fermions2,23, confirming the electronic decoupling of the two layers (Supplementary Note 1).Vacancies were created by low energy (100 eV) He+ ion sputtering followed by in situ annealing22,27,28. In STM topography of a typical irradiated sample (Fig. 1c) the vacancies appear as small protrusions on top of large background corrugations. To establish the nature of a vacancy we zoom in to obtain atomic resolution topography and spectroscopy. Single atom vacancies are recognized by their distinctive triangular \(\sqrt 3 \times \sqrt 3 \;R30^\circ\) topographic fingerprint (Fig. 1c inset)27,28,29 which is accompanied by a pronounced peak in the dI/dV spectra at the DP reflecting the presence of the ZM. If both these features are present we identify the vacancy as a single atom vacancy (Supplementary Note 2) and proceed to study it further. In order to separate the physics at the DP and the Kondo screening which produces a peak near Fermi energy, E F ≡ 0, the spectrum of the vacancy in Fig. 1d is taken at finite doping corresponding to a chemical potential, μ ≡ E F −E D = −54 meV. Far from the vacancy (lower curve), we observe the V shaped spectrum characteristic of pristine graphene, with the minimum identifying the DP energy. In contrast, at the center of the vacancy (Fig. 1d upper curve), the spectrum features two peaks, one at the DP identifying the ZM and the other at zero bias coincides with the position of the expected Kondo peak3. (In STS the zero-bias is identified with E F .) From the line shape of the zero-bias peak (Fig. 1f inset), we extract T K = (67 ± 2)K by fitting to the Fano line shape30,31 characteristic of Kondo resonances (Supplementary Note 3). As a further independent check we compare in Fig. 1f the temperature dependence of the linewidth to that expected for a Kondo-screened impurity30,32 (Supplementary Note 4), \(\Gamma _{\mathrm{{LW}}} = \sqrt {\left( {\alpha k_{\mathrm{B}}}T \right)^2 + \left( {2k_{\mathrm{B}}T_{\mathrm{K}}} \right)^2}\) from which we obtain T K = (68 ± 2) K, consistent with the above value, and α = 6.0 ± 0.3 in agreement with measurements and numerical simulations on ad-atoms30,33. Importantly, as we show below, this resonance is pinned to E F over the entire range of chemical potential values studied, as expected for the Kondo peak34,35.

The gate dependence of the spectra corresponding to the hundreds of vacancies studied here falls into two clearly defined categories, which we label type I, and type II. In Fig. 2a we show the evolution with chemical potential of the spectra at the center of a type I vacancy. Deep in the p-doped regime, we observe a peak which is tightly pinned to, E F , consistent with Kondo-screening. Upon approaching charge neutrality the Kondo peak disappears for μ ≥ −58 meV and reenters asymmetrically in the n-doped sector, for μ ≥ 10 meV. As we discuss below, the absence of screening close to the charge neutrality point and its reentrance in the n-doped regime for type I vacancies is indicative of pseudogap Kondo physics for subcritical coupling strengths8,36. For type II vacancies, the evolution of the spectra with chemical potential, shown in Fig. 2b, is qualitatively different. The Kondo peak is observed in the p-doped regime and disappears close to charge neutrality, but does not reappear on the n-doped side. We show below that this behavior is characteristic of pseudogap Kondo physics for vacancies whose coupling to the conduction band is supercritical8,36.

Fig. 2 Evolution of Kondo screening with chemical potential. a dI/dV curves for a subcritical Kondo vacancy (type I in text) with reduced coupling strength Γ 0 /Γ C = 0.90 at the indicated values of chemical potential. Red (blue) shade indicates the presence (absence) of the Kondo peak (V b = −200mV, I = 20pA). The chemical potential is tuned by the backgate voltage28. b dI/dV curves for a supercritical Kondo vacancy (type II in text) with Γ 0 /Γ C = 1.83 Full size image

Numerical renormalization group calculations

To better understand the experimental results we performed numerical-renormalization-group (NRG) calculations for a minimal model based on the pseudogap asymmetric Anderson impurity model (AIM)6,16,37 comprising the free local σ-orbital coupled to the itinerant π-band (Supplementary Note 6). This model gives an accurate description of the experiment in the p-doped regime where the ZM is sufficiently far from the Kondo peak so that their overlap is negligible. Upon approaching charge neutrality, interactions between the two orbitals through Hund’s coupling and level repulsion become relevant. As described in Supplementary Note 6 we introduced an effective Coulomb interaction term to take into account this additional repulsion. The single orbital model together with this phenomenological correction captures the main features of the Kondo physics reported here (Fig. 3). Results from a comprehensive NRG calculation using a two-orbital pseudogap AIM to model the problem38 similarly indicate that this simplified one-orbital approach qualitatively describes the experimental results. The single orbital AIM is characterized by three energy scales, ε d , U, and Γ 0 , corresponding to the energy of the impurity state, the onsite Coulomb repulsion, and by the scattering rate or exchange between the impurity and the conduction electrons, respectively (Supplementary Note 7). In the asymmetric AIM, which is relevant to screening of vacancy magnetic moments in graphene, the particle-hole symmetry is broken by next-nearest neighbor hopping and by \(U

e 2\left| \varepsilon_{\mathrm d} \right|\). The NRG phase diagram for this model is controlled by the valence fluctuation (VF) critical point, Γ C 6,7,8,39,40. At charge neutrality (μ = 0), Γ C separates the NRG flow into two sectors: supercritical, Γ 0 > Γ C , which flows to the asymmetric strong-coupling (ASC) fixed point where charge fluctuations give rise to a frozen impurity (FI) ground state41, and subcritical, Γ 0 < Γ C , which flows to the local moment (LM) fixed point where the impurity moment is unscreened. At the FI fixed point, the correlated ground state acquires one additional charge due to the enhancement of the particle-hole asymmetry in the RG flow. In a simplified picture, the fixed point spectrum can be understood by the flow of \(\varepsilon _{\mathrm d} \to - \infty\)

Fig. 3 Chemical-potential dependence of the Kondo temperature. a Chemical potential dependence of T K obtained from the Fano lineshape fit of the Kondo peak. In the regions where the peak is absent we designated T K = 0. b NRG result for the vacancies in panel a. T K is estimated by fitting the numerically simulated Kondo peak (Supplementary Note 3) Full size image

, leading to an effective doubly occupied singlet impurity state that decouples from the remaining conduction band6,8,41. In terms of the real physical orbitals, however, the NRG reveals a distribution of this additional charge between the conduction band and the local orbital with a small enhancement of n σ = 1.2–1.3. For Γ 0 < Γ C and μ ≠ 0, the appearance of relevant spin fluctuations gives rise to a cloud of spin-polarized electrons that screen the local moment below a characteristic temperature T K which is exponentially suppressed8 (\({\mathrm{ln}}T_{\mathrm{K}} \propto - 1/\left| \mu \right|\)). As a result, at sufficiently low doping, T K must fall below any experimentally accessible temperature, so that for all practical purposes its value can be set to zero (Fig. 4a). Using NRG to simulate the experimental spectra (Supplementary Note 7) we found ε d = −1.6 eV for the bare σ-orbital energy11,36, U = 2 eV11,42,43 and a critical coupling Γ C = 1.15 eV that separates the LM and the FI phases at μ = 0. From the NRG fits of the STS spectra we obtained the value of the reduced coupling Γ 0 /Γ C for each vacancy shown in Fig. 3 (Supplementary Note 8). The values, Γ 0 /Γ C = 0.90, and 1.83 obtained for the spectra in Fig. 2a, b place these two vacancies in the sub-critical and super-critical regimes, respectively.

Fig. 4 Quantum phase transition and Kondo screening. a μ−Γ 0 phase diagram at 4.2 K. The critical coupling Γ C (circle at Γ 0 /Γ C = 1.0) designates the boundary between Frozen-Impurity and the Local-Magnetic-Moment phases at μ = 0. Dotted lines represent boundaries between the phases (Supplementary Note 8). b STM topography for the G/G/SiO 2 (top) and G/G/BN (bottom) samples with the same scale bar (V b = −300mV, I = 20pA). The arrows point to the vacancies. c Typical line profile of the STM topographies of graphene on different substrates with the same scanning parameters as in b. d The evolution of the hybridization strength with the curvature. Error bars represent the uncertainty in obtaining the angle between the σ-orbital and the local graphene plane orientation from the local topography measurements. Inset: sketch of the curvature effect on the orbital hybridization Full size image

In Fig. 3 we compare the chemical-potential dependence of the measured T K , with the NRG results. The T K values are obtained from Fano-fits of the Kondo peaks leading to the T K (μ) curves, shown in Fig. 3a. The corresponding values of Γ 0 /Γ C and the T K (μ) curves obtained by using NRG to simulate the spectra are shown in Fig. 3b. The close agreement between experiment and simulations confirms the validity of the asymmetric AIM for describing screening of vacancy spins in graphene. In Fig. 4a we summarize the numerical results in a μ–Γ 0 phase diagram. At charge neutrality (defined by the μ = 0 line), the critical point Γ 0 /Γ C = 1 signals a quantum phase transition between the LM phase and the FI phase36. The Kondo-screened phase appears at finite doping (μ ≠ 0) and is marked by the appearance of the Kondo-peak,8,9. The phase diagram clearly shows the strong electron-hole asymmetry consistent with the asymmetric screening expected in in this system4.

Dependence of Kondo screening on corrugation amplitude