Engineering quantum anomalous/valley Hall states in graphene via metal-atom adsorption: An ab-initio study

We systematically investigate the magnetic and electronic properties of graphene adsorbed with diluted 3$d$ transition and noble-metal atoms using first-principles calculation methods. We find that most transition-metal atoms (i.e., Sc, Ti, V, Mn, Fe) favor the hollow adsorption site, and the interaction between magnetic adatoms and the $\pi$ orbital of graphene induces sizable exchange-field and Rashba spin-orbit coupling, which together open a nontrivial bulk gap near the Dirac $K/K^{\prime }$($\Gamma$) points in the $4\times 4$ ($3\times 3$) supercell of graphene leading to the quantum anomalous Hall effect. We also find that the noble-metal atoms (i.e., Cu, Ag, Au) prefer the top adsorption site, and the dominant inequality of the AB sublattice potential opens another kind of nontrivial bulk gap exhibiting the quantum-valley Hall effect in the $4\times 4$ supercell of graphene.

Figure 1

(a) A $4\times 4$ supercell of graphene. ${\vec{a}}_{1,2}$ (${\vec{a}}_{1,2}^4$) indicates the primitive vectors for the $1\times 1$ ($4\times 4$) supercell. Three possible adsorption sites of single adatom on graphene are labeled as follows: hollow (H), top (T), and bridge (B). (b) Corresponding reciprocal momentum space structures: ${\vec{b}}_{1,2}$ (${\vec{b}}_{1,2}^4$) denotes the reciprocal primitive vectors for the $1\times 1$ ($4\times 4$) supercell. $K$ ($K^4$), $K^{\prime }$ ($K^{\prime 4}$), and $\Gamma$ (${\Gamma }^4$) are the high symmetry points for the $1\times 1$ ($4\times 4$) supercell. Note that $K$ ($K^4$) and $K^{\prime }$ ($K^{\prime 4}$) are separated and thus distinguishable.

Figure 2

(a) A $3\times 3$ supercell of graphene. ${\vec{a}}_{1,2}$ (${\vec{a}}_{1,2}^3$) indicates the primitive vectors for the $1\times 1$ ($3\times 3$) supercell. Three possible adsorption sites of single adatom on graphene are labeled as follows: hollow (H), top (T), and bridge (B). (b) Corresponding reciprocal momentum space structures: ${\vec{b}}_{1,2}$ and ${\vec{b}}_{1,2}^3$ are for reciprocal vectors for the the $1\times 1$ and $3\times 3$ supercells, respectively. $K$ and $K^{\prime }$ points for the $3\times 3$ supercell of graphene are folded into the $\Gamma$ point.

Figure 3

Left column (${\mathrm{a}}_1$)–(${\mathrm{e}}_1$): Full bulk band structures based on GGA for Sc (${\mathrm{a}}_1$), Mn (${\mathrm{b}}_1$), Fe (${\mathrm{c}}_1$), Cu (${\mathrm{d}}_1$), and Ni (${\mathrm{e}}_1$) adsorbed at the hollow site on a $4\times 4$ supercell of graphene along high symmetry lines without involving spin degrees of freedom. (${\mathrm{f}}_1$) Full bulk band structure based on GGA for Cu adsorbed at top site on a $4\times 4$ supercell of graphene. Right column (${\mathrm{a}}_2$)–(${\mathrm{e}}_2$): Magnification of the bands of panels (${\mathrm{a}}_1$)–(${\mathrm{e}}_1$) around $K$ and $K^{\prime }$ Dirac points showing linear dispersion. (${\mathrm{f}}_2$) Magnification of the bands panel (${\mathrm{f}}_1$) with an obvious bulk band gap opening.

Figure 4

Left column (${\mathrm{a}}_1$)–(${\mathrm{e}}_1$): Full bulk band structures based on GGA for Sc (${\mathrm{a}}_1$), Mn (${\mathrm{b}}_1$), Fe (${\mathrm{c}}_1$), Cu (${\mathrm{d}}_1$), and Ni (${\mathrm{e}}_1$) adsorbed at the hollow site on a $4\times 4$ supercell of graphene along high symmetry lines by including the magnetization. (${\mathrm{f}}_1$) Full bulk band structure based on GGA for Cu adsorbed at top site on a $4\times 4$ supercell of graphene. Right column (${\mathrm{a}}_2$)–(${\mathrm{f}}_2$): Magnification of the bands around $K$ and $K^{\prime }$ Dirac points. Red and black are used to denote spin-up and spin-down bands.

Figure 5

Left column (${\mathrm{a}}_1$)–(${\mathrm{e}}_1$): Full bulk band structures based on GGA for Sc (${\mathrm{a}}_1$), Mn (${\mathrm{b}}_1$), Fe (${\mathrm{c}}_1$), Cu (${\mathrm{d}}_1$), Ni (${\mathrm{e}}_1$) adsorbed at the hollow site on a $4\times 4$ supercell of graphene along high symmetry lines by including the spin-orbit coupling. (${\mathrm{f}}_1$) Full bulk band structure based on GGA for Cu adsorbed at top site on a $4\times 4$ supercell of graphene. Right column (${\mathrm{a}}_2$)–(${\mathrm{f}}_2$): Magnification of the bands around $K$ and $K^{\prime }$ Dirac points (in black), and the Berry curvatures $\Omega \left(\mathit{k}\right)$ for the whole valance bands (in red).

Figure 6

Left column (${\mathrm{a}}_1$)–(${\mathrm{f}}_1$): Full band structures of adatom-$3\times 3$ supercell of graphene for Sc, Mn, Fe, Cu, Ni, and Co adsorbed on top of the hollow position without including magnetization and spin-orbit coupling along the high symmetry line. Right column (${\mathrm{a}}_2$)–(${\mathrm{f}}_2$): Magnification of the circled bands around $\Gamma$ point in panels (${\mathrm{a}}_1$)–(${\mathrm{f}}_1$). Bulk band gaps are opened at the $\Gamma$ point due to the intervalley scattering effect. Note that there are two equivalent band gaps at $\Gamma$ point except in the band structure of Cu adsorption.

Figure 7

Projected bands analysis for the bands near the two bulk band gaps in Fig. 6(${\mathrm{b}}_1$). We can see that the bands near the two bulk gaps possess the same orbital components.

Figure 8

Left column (${\mathrm{a}}_1$)–(${\mathrm{f}}_1$): Full band structures of adatom-$3\times 3$ supercell of graphene for Sc, Mn, Fe, Cu, Ni, and Co adsorbed on top of the hollow position by including the magnetization along the high symmetry line. Right column (${\mathrm{a}}_2$)–(${\mathrm{f}}_2$): Magnification of the circled bands around $\Gamma$ point in panels (${\mathrm{a}}_1$)–(${\mathrm{f}}_1$). Colors are used to distinguish the spin-up (black) and spin-down (red) bands. In the right column all the spin-up and spin-down bands are crossing due to the large magnetization except that for Cu and Ni adsorption. In panel (${\mathrm{d}}_2$) for Cu adsorption the magnetization is less than the original band gap amplitude; therefore, the relative shift between the spin-up and spin-down bands does not cross to close the band gap. In panel (${\mathrm{e}}_2$) for Ni adsorption the band structure is exactly the same as that in Fig. 6e because of the vanishing magnetic moment of Ni adatom.

Figure 9

Left column (${\mathrm{a}}_1$)–(${\mathrm{f}}_1$): Full band structures of adatom-$3\times 3$ supercell of graphene for Sc, Mn, Fe, Cu, Ni, and Co adsorbed on top of the hollow position by including both the magnetization and spin-orbit coupling along the high symmetry line. Right column (${\mathrm{a}}_2$)–(${\mathrm{f}}_2$): Magnification of the circled bands around $\Gamma$ point in panels (${\mathrm{a}}_1$)–(${\mathrm{f}}_1$). In panels (${\mathrm{a}}_2$)–(${\mathrm{c}}_2$) and (${\mathrm{f}}_2$), the nontrivial bulk band gap is open at the crossing points. However, only bulk band gaps are globally opened in panels (${\mathrm{a}}_2$)–(${\mathrm{b}}_2$) for Sc and Mn adsorption, while the gaps in panels (${\mathrm{c}}_2$) and (${\mathrm{f}}_2$) are just locally opened. Therefore, only the Sc/Mn adsorbed $3\times 3$ graphene structure can exhibit the nontrivial insulating properties when the Fermi level is artificially adjusted to the bulk band gaps. In addition, we also plot the total Berry curvatures $\Omega \left(k\right)$ in panels (${\mathrm{a}}_2$)–(${\mathrm{b}}_2$) for all the valence bands below the bulk band gaps.