Single vacancy defect in graphene: Insights into its magnetic properties from theoretical modeling

Magnetic properties of a single vacancy in graphene is a relevant and still much discussed problem. The experimental results point to a clearly detectable magnetic defect state at the Fermi energy, while calculations based on density functional theory (DFT) yield widely varying results for the magnetic moment, in the range of $\mu =1.04–2.0{\mu }_B$. We present a multitool ab initio theoretical study of the same defect, using two simulation protocols for a defect in a crystal (cluster and periodic boundary conditions) and different DFT functionals—bare and hybrid DFT, mixing a fraction of the Hartree-Fock (HF) exchange. We find that due to the $\pi$ character of the Fermi-energy states of graphene, delocalized in the in-plane and localized in the out-of-plane direction, the inclusion of the HF exchange is crucial, and moreover, that defect-defect interactions are long-range and have to be carefully taken into account. Our main conclusions are two-fold. First, for a single isolated vacancy we can predict an integer magnetic moment $\mu =2{\mu }_B$. Second, we find that due to the specific symmetry of the graphene lattice, periodic arrays of single vacancies may provide interesting diffuse spin-spin interactions.

Figure 1

Cluster models adopted here for graphene, hexagonal $D_{6h}$ symmetry (a) HAC $C_{222}{H}_{42}$ arm-chair and (b) HZZ $C_{216}{H}_{36}$ zig-zag edges.

Figure 2

Electronic energy levels for hexagonal H clusters in the region near the Fermi level, results from PBE and PBE0. Solid (dotted) lines indicate occupied (unoccupied) states. Upper panel HAC-$C_{222}{H}_{42}$ and lower panel HZZ-$C_{216}{H}_{36}$. Energies aligned to the Fermi energy of the perfect crystal by the C-$1s^2$ average energy. Isosurfaces for the molecular orbitals (HOMO and LUMO) at the frontier energies from the PBE0 calculations.

Figure 3

Vacancy in graphene (a) visualization of the atoms in the close vicinity. Model clusters: fully optimized structures with spin-polarized PBE (b) VHAC-$C_{222}{H}_{42}$, (c) VHZZ-$C_{216}{H}_{36}$. All structures remain flat after the structural relaxation.

Figure 4

Band structure for the vacancy defect in the region near the Fermi energy, results from spin-polarized PBE, for the supercells $\left(6\times 6\right)$ at left, $\left(7\times 7\right)$ at center, and $\left(8\times 8\right)$ at right. Solid (dotted) lines indicate occupied (unoccupied) states. Energies aligned to the Fermi energy of the perfect crystal by the C-$1s^2$ average energy.

Figure 5

Isosurfaces for the $d$'$\pi ,l\pi$, and $V\pi$ defect states (indicated in Figs. 4 and 6) obtained with the spin-polarized PBE functional; isosurfaces for periodic structures at the $\mathrm{\Gamma }$ point. The $V\sigma$ state shown here for the $\left(7\times 7\right)$ SC and the HZZ cluster presents very similar character in all simulations, and the $l\pi$ state is similar in the $\left(8\times 8\right)$ SC.

Figure 6

Electronic energy levels for the vacancy in the clusters HAC-$C_{222}{H}_{42}$ (left) and HZZ-$C_{216}{H}_{36}$ (right), in the region near the Fermi energy. Results from spin-polarized PBE (top) and PBE0 (down) functionals. Solid (dotted) lines indicate occupied (unoccupied) states. Energies aligned to the Fermi energy of the perfect crystal by the C-$1s^2$ average energy.

Figure 7

Band structure for the vacancy defect in the region near the Fermi energy, results from spin-polarized PBE0 for the supercells $\left(6\times 6\right)$ at left, $\left(7\times 7\right)$ at center, and $\left(8\times 8\right)$ at right. Solid (dotted) lines indicate occupied (unoccupied) states. Energies aligned to the Fermi energy of the perfect crystal by the C-$1s^2$ average energy.

Figure 8

Isosurfaces for the spin density $\left(0.05\AA {}^{-3}\right)$ produced by the array of vacancies in graphene obtained through PBE0 for (a) $6\times 6$, (b) $7\times 7$, and (c) $6\times 9$ SCs.