Topological phases in triangular lattices of Ru adsorbed on graphene: Ab initio calculations

C. Mera Acosta, Matheus P. Lima, R. H. Miwa, Antônio J. R. da Silva, and A. Fazzio

Phys. Rev. B 89, 155438 – Published 30 April 2014

Abstract

We have performed an ab initio investigation of the electronic properties of the graphene sheet adsorbed by Ru adatoms (Ru/graphene). For a particular set of triangular arrays of Ru adatoms, we find the formation of four (spin-polarized) Dirac cones attributed to a suitable overlap between two hexagonal lattices: one composed by the C sites of the graphene sheet, and the other formed by the surface potential induced by the Ru adatoms. Upon the presence of spin-orbit coupling (SOC) nontrivial band gaps take place at the Dirac cones promoting several topological phases. Depending on the Ru concentration, the system can be topologically characterized among the phases (i) quantum spin Hall (QSH), (ii) quantum anomalous Hall (QAH), (iii) metal, (iv) or trivial insulator. For each concentration, the topological phase is characterized by the ab initio calculation of the Chern number.

Received 22 January 2014

DOI:https://doi.org/10.1103/PhysRevB.89.155438

Authors & Affiliations

C. Mera Acosta^1,*, Matheus P. Lima^1,†, R. H. Miwa^2,‡, Antônio J. R. da Silva^1,3,§, and A. Fazzio^1,∥

¹Instituto de Física, Universidade de São Paulo, CP 66318, 05315-970, São Paulo, SP, Brazil
²Instituto de Física, Universidade Federal de Uberlandia, Uberlandia, MG, Brazil
³Laboratório Nacional de Luz Síncrotron, CP 6192, 13083-970, Campinas, SP, Brazil

^*acosta@if.usp.br
^†mplima@if.usp.br
^‡hiroki@infis.ufu.br
^§jose.roque@lnls.br
^∥fazzio@if.usp.br

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Vol. 89, Iss. 15 — 15 April 2014

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Images

Figure 1
Schematic representations of Ru/graphene systems with (a) ( $4 \times 4$ ), (b) ( $5 \times 5$ ), and (c) ( $6 \times 6$ ) periodicities, respectively. These are examples of the $[(3 n + 1) \times (3 n + 1)]$ , $[(3 n + 2) \times (3 n + 2)]$ , and ( $3 n \times 3 n$ ) Ru/graphene systems, respectively. Here $n$ is an integer. The geometric centers of the triangles formed by the Ru adatoms (barycenters) form a honeycomb lattice, represented in the figure.
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Figure 2
Evolution of the electronic band structure of Ru/graphene with a ( $5 \times 5$ ) periodicity, considering successively the contributions of (a) electrostatic potential generated by Ru adatoms and (b) exchange field. In (c) is shown the PDOS of (b). Dark gray (blue) and black lines are associated with the down and up spin, respectively. (d) Variation of the energetic separation $Δ_{C 2 - C 3}$ of the Dirac cones closer to the Fermi level ( $C 2$ and $C 3$ ) relative to the concentration of Ru adatoms. (e) Band structure of (10 $\times$ 10) Ru/grapehene system. (f) Expected picture of the band structure of ( $n \times n$ ) Ru/graphene systems with $n \to \infty$ .
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Figure 3
Local effect of Ru adatom. (a) Molecular spectrum of Ru adsorbed on the coronene molecule, as illustrated in the structure at the bottom left of the box. (b) Electronic band structure of (4 $\times$ 4) Ru/graphene system without spin-orbit coupling. The (red) boxes around C2 and C3 indicate the energy intervals used to calculate the LDOS. In the bottom left is represented the structure of the unit cell used. We indicate the barycenters with (red) balls. (c) LDOS around the LUMO (left) and HOMO (right) of molecular spectrum in arbitrary units. (d) LDOS with down and up spins around the $C 3$ and $C 2$ cones, respectively. We point out a C site associated with a barycenter, which is indicated in the structure in (b).
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Figure 4
LDOS for up and down spins around the energy level at which the (a) $C 2$ and (b) $C 3$ cones are formed for the (5 $\times$ 5) Ru/graphene system.
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Figure 5
(a) Electronic band structure and (b) LDOS for the (6 $\times$ 6) Ru/graphene system. The region of integration used to make the LDOS is represented by the (red) box in (a).
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Figure 6
The SOC effects in Ru/graphene systems. (a) Electronic band structure for the (5 $\times$ 5) periodicity with SOC. (b) Band structure of (a) near the Fermi level. (c) Energy band gaps for the $[(3 n + 1) \times (3 n + 1)]$ and $[(3 n + 2) \times (3 n + 2)]$ families with periodicities $5 \times 5$ or larger. Band structures for the (d) (4 $\times$ 4), (e) (6 $\times$ 6), and (f) (2 $\times$ 2) periodicities. (g) Band gap variation of (f) with the SOC strength $λ_{SOC}$ . The dashed (red) lines indicate the Fermi levels, whereas the black dotted lines is just to better visualize the energy gap.
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