Kondo effect of a cobalt adatom on a zigzag graphene nanoribbon

Based on ab initio calculations we discuss the Kondo effect due to a Co adatom on a graphene zigzag nanoribbon. A Co atom located at a hollow site behaves as a spin $\mathcal{S}=1/2$ impurity with $d_{xz}$ and $d_{yz}$ orbitals contributing to the magnetic moment. Dynamical correlations are analyzed with the use of complementary approximations: the mean field slave boson approach, noncrossing approximation, and equation-of-motion method. The impact of interplay between spin and orbital degrees of freedom together with the effect of peculiarities of electronic and magnetic structure of the nanoribbon on many-body resonances is examined.

Figure 1

Structure of H-passivated 4ZGNR, where the dark gray and purple/light gray spheres represent the C and H atoms, respectively. Labeling of carbon atoms across the ribbon with division into sublattices is introduced. The possible adatom locations $h_1$, $h_2$, $h_3$ (hollow positions, most favorable); $t_1$, $t_2$: top sites; $b_1$, $b_2$: bridge positions; and perturbed hollow-like site of the edge $h_0$ are depicted.

Figure 2

Spin density plots of 4ZGNR showing up (blue/dark gray) and down (green/light gray) spin densities together with the corresponding values of local magnetic moments calculated with VASP code using (a) GGA-PBE exchange correlation potential and (b) nonlocal GGA-HSE potential.

Figure 3

(a), (b) VASP spin-degenerate bands of 4ZGNR with fat bands showing the amplitudes of the projection of each band on $p_z$ orbital of edge atom ${\mathcal{A}}_1$. Broken lines represent dispersions calculated by OPENMX code. ${\Delta }_0$ and ${\Delta }_1$ are direct band gap and gap of the zone boundary. (c) Width dependence of the nanoribbon gaps. Inset shows the width dependence of edge magnetic moments (solid line, VASP; broken line, OPENMX).

Figure 4

VASP energy dispersion curves of 4ZGNR (a) compared with the bands of 4ZGNR with Co impurity in $h_1$ position (b). The blue/dark gray fat bands highlight carbon $p_z$ contribution and green/light gray the Co $d$ contribution.

Figure 5

Spin- and orbital-resolved densities of states of Co adatom at $h_1$ position in 4ZGNR. Black denotes spin-up contributions and green/light gray the spin-down contributions. Insets of (b) and (c) present $yz$ and $xy$ partial DOS of Co adatom at $h_0$ site and insets of (d) compare $z^2$ partial DOS in $h_1$ and $h_0$ positions.

Figure 6

Spin density plots of Co adatom and 4ZGNR matrix ($8\times 1$ supercell). The blue/dark gray densities correspond to spin-up and green/light gray to spin-down components. (a) Co impurity in $h_1$ position. (b) Co in $h_2$ position.

Figure 7

Low-energy band structure of 4ZGNR (a) together with orbitally resolved hybridization functions of Co adatom in $h_1$ position [(b), (c), (d)]. Black color denotes $yz$ contribution and green/light gray $xz$. In (b) labeling of Van Hove singularities used further in the text is introduced. Panels (c) and (d) present spin-up and spin-down parts of hybridization, respectively. Insets of (c) and (d) are zoom views of lowest dispersion curves decorated with $xz$ and $yz$ contributions.

Figure 8

Orbital- and spin-resolved hybridization functions for Co at $h_1$ position in 4ZGNR. Solid black lines represent real parts ($\mathrm{Re}\left[{\Sigma }_{m\sigma }^{h_1}\right]$) and imaginary parts ($\mathrm{Im}\left[{\Sigma }_{m\sigma }^{h_1}\right]$) are shown by green/light gray filled curves.

Figure 9

Orbital-resolved hybridization function of Co in $h_2$ position of 4ZGNR. Solid black lines represent real part and green/light gray filled curves show the imaginary parts of hybridization.

Figure 10

Expectation values of slave boson operators, polarizations, and expectation values of orbital pseudospin of Co impurity at $h_2$ position of 4ZGNR: (a) triple occupation slave boson operators $|t_m{|}^2=|t_{m+}{|}^2={\left|t_{m-}\right|}^2$; (b) double and full occupation slave boson operators $|d_{\mathcal{T}0}{|}^2$, $|d_{\mathcal{T}1}{|}^2$, $|d_{\mathcal{T}\overline{1}}{|}^2$, $|d_{\mathcal{S}1}{|}^2={\left|d_{\mathcal{S}\overline{1}}\right|}^2$, $|d_{\mathcal{S}0}{|}^2={\left|d_{\mathcal{T}0}\right|}^2$, and ${\left|f\right|}^2$; (c) orbital polarization of Co at $h_2$ site [${\mathcal{P}}_{\mathcal{O}}=\frac{{\varrho }_{xz}\left({\mathcal{E}}_{\mathcal{F}}\right)-{\varrho }_{yz}\left({\mathcal{E}}_{\mathcal{F}}\right)}{{\varrho }_{xz}\left({\mathcal{E}}_{\mathcal{F}}\right)+{\varrho }_{yz}\left({\mathcal{E}}_{\mathcal{F}}\right)}$, where ${\varrho }_m\left(\mathcal{E}\right)={\sum }_{\sigma }{\varrho }_{m\sigma }\left(\mathcal{E}\right)$ and ${\varrho }_{m\sigma }$ denotes spin-orbital partial density of states] and expectation value of Co orbital pseudospin (${\mathcal{T}}_z$). Inset of (a) shows total, spin, and orbital $d_{xz},d_{yz}$ occupancies.

Figure 11

Selected partial orbital DOS of Co at $h_2$ site in 4ZGNR. The vertical red/thin lines indicate positions of Fermi levels.

Figure 12

Expectation values of slave boson operators, polarizations, and expectation values of orbital pseudospin and magnetic moment of Co impurity at $h_1$ position of 4ZGNR: (a) triple-occupation slave boson operators $|t_{m\sigma }{|}^2$; (b) double- and full-occupation SB operators $|d_{\mathcal{T}0}{|}^2$, $|d_{\mathcal{T}1}{|}^2$, $|d_{\mathcal{T}\overline{1}}{|}^2$, $|d_{\mathcal{S}1}{|}^2$, $|d_{\mathcal{S}\overline{1}}{|}^2$, $|d_{\mathcal{S}0}{|}^2={\left|d_{\mathcal{T}0}\right|}^2$, and ${\left|f\right|}^2$; (c) local spin polarization of the nanoribbon around $h_1$ (${\mathcal{P}}_{{\mathcal{S}}_{\text{ZGNR}}}={\sum }_i\frac{{\varrho }_{i+}\left({\mathcal{E}}_{\mathcal{F}}\right)-{\varrho }_{i-}\left({\mathcal{E}}_{\mathcal{F}}\right)}{{\varrho }_{i+}\left({\mathcal{E}}_{\mathcal{F}}\right)+{\varrho }_{i-}\left({\mathcal{E}}_{\mathcal{F}}\right)}$, where ${\varrho }_{i\sigma }$ denotes local ZGNR density of states at the n.n. carbon sites around hollow position), orbital (${\mathcal{P}}_{\mathcal{O}}$) and spin polarization of Co adatom (${\mathcal{P}}_{\mathcal{S}}$); (d) magnetic moment and orbital pseudospin of Co impurity. Inset of (a) presents spin- and orbital-resolved contributions to the occupancies of Co.

Figure 13

(a) Density of states map of Co impurity in $h_1$ position of 4ZGNR. (b) $\Delta \varrho$, difference of spin-up and spin-down densities of states of Co at $h_1$ site of 4ZGNR. (c), (d), (e) Partial orbital densities of states for the selected values of the Fermi levels. Insets are the zoom views of the spin- and orbital-resolved DOS.

Figure 14

Characteristic quasiparticle temperatures $T_{\mathcal{K}}$ of Co in 4ZGNR for $xz$ and $yz$ channels vs position of the Fermi level. Dash-dotted line represents Kondo temperature of Co at hollow position in graphene. Panel (b) shows details of dependencies close to $\text{VH}2\nu$ singularity.

Figure 15

Dependence of the Kondo state characteristics on the value of exchange parameter: (a), (b) Slave boson expectation values $|t_{m\sigma }{|}^2$ for $\mathcal{N}\approx 3$ (${\varepsilon }_0=-7.5$ eV) and $\mathcal{N}\approx 2.7$ (${\varepsilon }_0=-7$ eV), ${\mathcal{E}}_{\mathcal{F}}=-0.8$. (c) Magnetic moments of Co impurity. Inset shows orbital occupancies: for ${\varepsilon }_0=-7.5$ dashed line (${\mathcal{N}}_{yz}$) and dash-dotted line (${\mathcal{N}}_{xz}$), ${\varepsilon }_0=-7$ solid line (${\mathcal{N}}_{yz}$) and dotted line (${\mathcal{N}}_{xz}$). (d) Relative quasiparticle temperatures.

Figure 16

Comparison of $xz$ (dotted line) and $yz$ (solid line) contributions to the DOS of Co adatom in $h_1$ position of 4ZGNR (${\mathcal{E}}_{\mathcal{F}}=-0.8$) (NCA). The vertical dashed lines in (a) determine the positions of bare Coulomb peaks and in (b) the positions of singularities in the interacting self-energies.

Figure 17

Density of states of Co impurity at $h_1$ position (${\mathcal{E}}_{\mathcal{F}}=-0.8$). (a), (b) Temperature evolution of orbital partial densities of states calculated with the use of NCA approximation (insets are the extended views for $T=0.6$ K and $55.7$ K). (c), (d) Comparison of NCA DOS with EOM spectra and SBMFA densities of states.

Figure 18

Partial $xz$ DOS of Co adatom calculated by EOM with realistic (DFT) hybridization function of 4ZGNR with Lacroix's decoupling (solid blue line), without dynamical correlations (red dotted line). Dashed black curve presents the corresponding DOS calculated with Lacroix's decoupling, but using energy-independent, constant hybridization function.

Figure 19

(a) LDA total energy curves for different positions of Co adatom on 4ZGNR. Inset is the zoom view of spin-resolved energy curves for $h_1$ position presented close to minimum. (b) GGA+$U$ total energy curves for $h_1$ position for selected values of $U$. (c) Comparison of GGA+$U$ energy curves for LS ($h_1$) and HS ($t_2$) states for $\mathcal{U}=4$ eV and in the inset for $\mathcal{U}=3$ eV.

Figure 20

GGA+$U$ estimations of total magnetic moment of Co impurity in 4ZGNR for different values of Coulomb interaction parameter $\mathcal{U}$. Inset shows magnetic moment of Co placed at $h_1$ position of 4ZGNR at a distance $h$ from the nanoribbon plane. The dots represent additional effect of exchange $\mathcal{J}=0.9$ eV for $h_0$ (circle), $h_1$ (square), and $h_2$ (triangle).

Figure 21

(a) Density of states of NZGNR: $N=4$ (blue line/gray line), $N=8$ (black), $N=14$ (green/light gray line). (b) Highest valence bands (HV) of NZGNR (${\varepsilon }_{k\sigma }^{HV}$) (solid lines, right axis of ordinates) and the corresponding edge state contributions $|\langle p_{z\sigma }^{{\mathcal{A}}_1}|k{\sigma }^{HV}\rangle {|}^2$ (dashed lines for spin up and dotted lines for spin down, left axis of ordinates). Assignment of the line colors to the ribbon widths is the same as in (a).

Figure 22

Characteristic quasiparticle temperatures $T_{\mathcal{K}}$ of Co in NZGNR [(a), (b)] and NAGNR [(c), (d)] for $xz$ [(a), (c)] and $yz$ [(b), (d)] channels. Dots represent the results for central hollow location ($h_c$) and squares for hollow positions next to the edge ($h_1$). Dotted lines correspond to ${\mathcal{E}}_{\mathcal{F}}=-0.25$ eV case and solid lines to ${\mathcal{E}}_{\mathcal{F}}=-0.8$ eV. Inset of (a) is a zoom picture of the band structure of 14ZGNR and illustrates a crossing of the Fermi level (${\mathcal{E}}_{\mathcal{F}}=-0.8$ eV) by the next valence band. The upper horizontal axes show the nanoribbon widths expressed in graphene lattice constant units a ($a=1.42$ Å).