Modeling of the gate-controlled Kondo effect at carbon point defects in graphene

We study the magnetic properties in the vicinity of a single carbon defect in a monolayer of graphene. We include the unbound $\sigma$ orbital and the vacancy-induced bound $\pi$ state in an effective two-orbital single-impurity model. The local magnetic moments are stabilized by the Coulomb interaction as well as a significant ferromagnetic Hund's rule coupling between the orbitals predicted by a density functional theory calculation. A hybridization between the orbitals and the Dirac fermions is generated by the curvature of the graphene sheet in the vicinity of the vacancy. We present results for the local spectral function calculated using Wilson's numerical renormalization group approach for a realistic graphene band structure and find three different regimes depending on the filling, the controlling chemical potential, and the hybridization strength. These different regions are characterized by different magnetic properties. The calculated spectral functions qualitatively agree with recent scanning tunneling spectra on graphene vacancies.

Figure 1

Left: Schematic picture of remaining electronic states for a missing carbon atom. The free $\sigma$ orbital hybridizes with both adjacent $\pi$ orbitals. The resulting impurity has the characteristic triangle shape. Right: Schematic two-orbital model. The $d$ orbital represents the free $\sigma$ orbital and is coupled directly to the conduction band. The $\pi$ orbital represents the vacancy-induced zero-mode bound state.

Figure 2

$\mathrm{\Gamma }\left(\omega \right)$ for flat graphene using a $t,t^{\prime }$ tight-binding model with $t=2.91\mathrm{eV}$ and $t^{\prime }=-0.16\mathrm{eV}$ (solid) in accordance with ab initio calculations [29, 30]. We performed a summation over the first Brillouin zone and evaluated the well-known single-particle dispersion Eq. (4). The dashed curve shows a simplified approximate $\mathrm{\Gamma }\left(\omega \right)$. The frequency is shifted in both cases such that the Dirac point lies at $\omega =0$ for $\mu =0$.

Figure 3

Scenarios for $n$ doping depending on the local curvature, i.e., varying ${\mathrm{\Gamma }}_0$ in our model. Stronger rippling of the graphene sheet increases the overlap between $\sigma$ and $\pi$ orbitals, resulting in a level repulsion. Depending on the strength of ${\mathrm{\Gamma }}_0$, we expect three different scenarios as a result of the orbital occupation. The ground-state energy of the corresponding isolated impurity is given by Eqs. (15), (14), and (13) (left to right). Kondo screening is only possible for small and medium hybridizations.

Figure 4

${\rho }_d\left(\omega \right)$ for the weak hybridization regime for (a) $p$ doping and (b) $n$ doping. The numbers in the plot refer to the chemical potential. The results for the realistic $t,t^{\prime }$ hybridization function combined with simple linear interpolation (dashed) and rediagonalization Eq. (16) (dotted) are shown. For the approximate hybridization, the Kondo effect sets in at higher $\mu$ and is not shown for visibility. All used parameters are listed in Table 1. The calculations were done for $T=4.2\mathrm{K}$. All curves are divided by ${\rho }_0$ being the maximum of all ${\rho }_d\left(\omega \right)$.

Figure 5

Occupation of the $d$ orbital, $n_{d,\sigma }$ (solid line), as well as the ZM represented by the local $\pi$ orbital, $n_{\pi ,\sigma }$ (dashed line), per spin $\sigma$ at $T=4.2\mathrm{K}$ for (a) weak, (b) intermediate, and (c) strong hybridization regimes as calculated by the NRG. We used (i) the approximated $\mathrm{\Gamma }\left(\omega \right)$, (ii) the realistic $\mathrm{\Gamma }\left(\omega \right)$ obtained from a $t,t^{\prime }$ TB model with a linear interpolation for the orbital evolution, and (iii) the realistic $\mathrm{\Gamma }\left(\omega \right)$ in combination with the $Z$-factor-based orbital splitting according to Eq. (16).

Figure 6

${\rho }_d\left(\omega \right)$ for the intermediate hybridization regime for (a) $p$ doping and (b) $n$ doping. The numbers in the plot refer to the chemical potential. Solid curves, data obtained with the approximate $\mathrm{\Gamma }\left(\omega \right)$; dashed curves, the realistic $t,t^{\prime }$; and dotted curves, the realistic $\mathrm{\Gamma }\left(\omega \right)$ and $Z$ factor. The calculations were done for $T=4.2\mathrm{K}$ and the parameter sets are listed in Table 1. All curves are divided by the same constant ${\rho }_0$ which is the maximum of all ${\rho }_d\left(\omega \right)$.

Figure 7

${\rho }_d\left(\omega \right)$ for the strong hybridization regime for (a) $p$ doping and (b) $n$ doping. The numbers in the plot refer to the chemical potential. Solid curves, data obtained with the approximate $\mathrm{\Gamma }\left(\omega \right)$; dashed curves, the realistic $t,t^{\prime }$; and dotted curves, the realistic $\mathrm{\Gamma }\left(\omega \right)$ and $Z$ factor. The calculations were done for $T=4.2\mathrm{K}$ and the parameter sets are listed in Table 1. The spectral functions for $n$ and $p$ doping are divided by a different constant each for visibility.

Figure 8

${\rho }_{\pi }\left(\omega \right)$ for the intermediate hybridization regime for (a) a constant $\mu =-60\mathrm{meV},{\mathrm{\Gamma }}_0=1.21\mathrm{eV}$, and varying broadening parameter $b$ and (b) a constant $b=0.4$ and varying $\mu$. The approximated $\mathrm{\Gamma }\left(\omega \right)$ and the linear interpolation for the level energies were used. All parameters are listed in Table 1. The calculations were done for $T=4.2\mathrm{K}$.

Figure 9

Different regimes for finite temperature $T=4.2\mathrm{K}$. The parameters are listed in Table 1. We used the realistic $\mathrm{\Gamma }\left(\omega \right)$ and the $Z$ factor to determine the level positions for arbitrary hybridization strength ${\mathrm{\Gamma }}_0$. For the color coding, we used the residual entropy times the impurity occupation.

Figure 10

Different regimes for temperature $T=1.6\times {10}^{-8}\mathrm{K}$. The parameters are listed in Table 1. We used the realistic $\mathrm{\Gamma }\left(\omega \right)$ and the $Z$ factor to determine the level positions for arbitrary hybridization strength ${\mathrm{\Gamma }}_0$. For the color coding, we used the residual entropy times the impurity occupation.

Figure 11

(a) Spectral function and (b) orbital occupation for the intermediate regime ${\mathrm{\Gamma }}_0=1.7\mathrm{eV}$ obtained for the realistic $\mathrm{\Gamma }\left(\omega \right)$ and $Z$ factor. The spectral functions are calculated for finite $T=4.2\mathrm{K}$ while the occupation is shown for finite $T$ and $T\to 0$ as well. The parameters are listed in Table 1.

Figure 12

Kondo temperatures calculated via Fano fit extracted from the zero-energy peak of ${\rho }_d\left(\omega \right)$ (solid line) and Goldhaber-Gordon fit of the zero-bias conductivity (dashed line). The Fano fits were carried out at finite $T=4.2\mathrm{K}$ while the zero-bias conductivity was calculated at $T=1.6\times {10}^{-8}\mathrm{K}$. The first two graphs are from the strong regime, the next two from the intermediate regime, and the last one from the weak hybridization regime. We used the realistic hybridization and Eq. (16) for all calculations.

Figure 13

${\rho }_d\left(\omega \right)$ for smaller onsite $U=0.65\mathrm{eV}$ and relatively strong Hund's coupling $J_H=0.35\mathrm{eV}$. The solid (dashed) curve represents the intermediate (strong) hybridization regime and the impurity parameters are ${\epsilon }_d=-0.37(-0.40)\mathrm{eV},{\epsilon }_{\pi }=0.20\left(0.23\right)\mathrm{eV}$, and ${\mathrm{\Gamma }}_0=0.40\left(0.50\right)\mathrm{eV}$. The calculations are done for $T=1.6\times {10}^{-5}\mathrm{K}$. The Kondo peak is clearly pronounced but very narrow, indicating a small $T_K$.

Figure 14

${\mathrm{\Gamma }}_{\pm }\left(\omega \right)$ stemming from the even or odd linear combination in Eq. (25). We used a direct $\vec{k}$-space summation. For a direct comparison, we normalized both effective DOSs separately. Both hybridizations show the characteristic linear behavior close to the Dirac point but with different slopes.

Figure 15

Direct comparison of ${\rho }_d\left(\omega \right)$ for (i) $\mathrm{\Gamma }\left(\omega \right)$ calculated with a realistic $t,t^{\prime }$ DOS (solid), (ii) ${\mathrm{\Gamma }}_{-}\left(\omega \right)$ as defined in Eq. (25) (dashed), and (iii) ${\mathrm{\Gamma }}_{+}\left(\omega \right)$ (dotted). All hybridization functions are normalized in the same way according to Eq. (11). We adopted the $t,t^{\prime }$ parameters of the impurity for the even and odd hybridization functions.