Multiorbital electronic correlation effects of Co adatoms on graphene: An ionic Hamiltonian approach

In the present work, we propose an ionic Hamiltonian for describing the interaction of graphene with an adsorbed Co atom. In this approach, the electronic correlation effects, related to the many $d$ orbitals involved in the interaction, are taken into account by selecting appropriate electronic configurations of the adsorbed atom. The Hamiltonian parameters are calculated considering the localized and extended features of the atom-surface interacting system. The physical quantities of interest are calculated by using a Green functions formalism, solved by means of the equations of motion method closed up to a second order in the atom-band coupling term. The charge and spin fluctuations in the adsorbed Co atom are inferred from density functional theory calculations and assuming that the lower energy configurations obey Hund's rules. The calculated spectral densities and the occurrence probabilities of the different atomic configurations are analyzed as a function of the Co energy level positions and the surface temperature. In addition, the conductance spectra are calculated by using the Keldysh formalism and compared with existing measurements. We analyze the behavior, under variable bias and gate potentials, of resonancelike features in the conductance spectra which can be related to transitions between atomic configurations of low occurrence probability.

Figure 1

DOS and band structure of pure graphene (upper left) and square modulus of the atom-band couplings ${\left|V_{n\mathbf{k}{d}_i}\right|}^2$ for the five $d$ orbitals of Co at the equilibrium distance on the hollow position. The colors show the correspondence between the graphene bands $n$ and the ${\left|V_{n\mathbf{k}{d}_i}\right|}^2$. The contour plots show the ${\left|V_{n\mathbf{k}{d}_i}\right|}^2$ for each orbital for the $\pi$ and ${\pi }^{*}$ bands. The $\mathbf{k}$ path followed in the plots ($\mathrm{K}\mathrm{\Gamma }\mathrm{MK}$) is indicated in the contour plots. The jumps observed at the $K$ point for the $\pi$ and ${\pi }^{*}$ bands and at the $\mathrm{\Gamma }$ point for some $\sigma$ bands are not numerical artifacts and they correspond to an accumulation of contour lines in the contour plots.

Figure 2

Our calculated ${\mathrm{\Gamma }}_d^0$ compared with those from Refs. [11, 12] for (a) $E1$ and (b) $E2$ groups. (c) Presents comparatively our calculated ${\mathrm{\Gamma }}_d^0$ for the three symmetry groups, (d) is a zoom and (e) corresponds to the real part of the noninteracting self-energy, ${\mathrm{\Lambda }}_d^0$.

Figure 3

(a) Energy levels as a function of the Co-graphene distance $z$ for each group, shown relative to the vacuum. The asymptotic value $E_I^{*}$ and the work function of graphene $\mathrm{\Phi }$ are indicated. The detail at the Co equilibrium position (1.52 Å) shows the level positions with respect to the graphene Fermi level ($E_F$). (b) Co LDOS obtained with DFT compared with the results of the simplified model (SM) described in the text.

Figure 4

Occurrence probabilities for the configurations with (a) three and (b) two holes. (c) Hole occupation per symmetry group. (d) Shifted energy levels ${\epsilon }_D+{\mathrm{\Lambda }}_D^0\left({\epsilon }_D\right)$ for the noninteracting case, broadened by ${\mathrm{\Gamma }}_D^0\left({\epsilon }_D\right)$. The results are shown as a function of the energy level position ${\epsilon }_{E1}$ after shifting rigidly the reference levels ${\epsilon }_{E1}=-0.55$ eV (indicated by a dotted line), ${\epsilon }_{E2}=-0.50$ eV, and ${\epsilon }_{A1}=-0.65$ eV.

Figure 5

Spectral densities ${\rho }_D$ calculated at several values of the energy levels, keeping constant their splitting and indicating their positions with the value of ${\epsilon }_{E1}$. The main resonance in each case is associated to the fluctuation between the three and two holes configurations labeled with $P$ (upper line) and $Q$ (lower line). The inset in (a) shows a zoom near zero, where other fluctuations of lower probability can be observed. The insets in (c) show the spectral densities for the levels well below and well above the Fermi level.

Figure 6

Spectral densities ${\rho }_D$ calculated at several Fermi level shifts. The energy levels are maintained at the position corresponding to ${\epsilon }_{E1}=2.45$ eV from the Fermi level in all cases.

Figure 7

Occurrence probabilities for the configurations with (a) $S=3/2$ and (b) $S=1$, calculated by including 11 NEF (full line), 4 NEF (dotted line), and 2 NEF (dashed line) as a function of the energy level positions. Spectral densities obtained from each approximation for (c) $E1$ and (d) $E2$ orbitals, at the reference energy levels (shown with a doted line in the upper panels). The insets show a zoom near zero, where structures appearing in the 11 NEF case can be appreciated.

Figure 8

[(a)–(c)] Spectral densities at several temperatures, for the three symmetry groups and (d) total spectral density. [(e) and (f)] Occurrence probabilities for the configurations with three and two holes as a function of the temperature. (g) Variation of the hole occupation as a function of the temperature.

Figure 9

(a) Co-tip noninteracting width ${\mathrm{\Gamma }}_{tD}^0$ per symmetry group. (b) Geometry of the system. (c) Scheme indicating the position of the graphene Dirac point (DP) relative to the chemical potential of the surface ${\mu }_s$, the Co energy levels ${\epsilon }_D$ and their shift $\mathrm{\Delta }{\epsilon }_D$, the position of the tip chemical potential ${\mu }_t$ when a bias voltage $V_s$ is applied and the interaction between the Co atom and each lead represented by ${\mathrm{\Gamma }}_{sD}^0$ and ${\mathrm{\Gamma }}_{tD}^0$.

Figure 10

Differential conductance for several $E_G$ shifts. The Dirac point position with respect to ${\mu }_s=0$ eV corresponds to $E_G$ and is indicated with an arrow. The calculations are made for different rigid shifts of the Co energy levels $\mathrm{\Delta }{\epsilon }_D$.