Vacancy mediated magnetization and healing of a graphene monolayer

Vacancy-induced magnetization of a graphene layer is investigated by means of a first-principles DFT method. Calculations of the formation energy and the magnetization by creating the different number of vacancies in a supercell show that a clustering with a big number of vacancies in the cluster is rather favorable to that of isolated vacancies, homogeneously distributed in the layer. The magnetic moment of a cluster with a big number of vacancies is shown to not be proportional with the vacancy concentration, which is in good agreement with the recent experimental results. Our studies support the idea that, although the vacancies in graphene create a magnetic moment, they do not produce a magnetic ordering. It is shown that, although the Lieb's rule for the magnetization in a hexagonal structure violates, two vacancies, including a di-vacancy, in the supercell generates a quasilocalized state when they belong to the different sublattices and, instead, two vacancies generate an extended state when they belong to the same sublattices. Analytical investigation of the dynamics of carbon atom and vacancy concentrations according to the nonlinear continuity equations shows that the vacancies, produced by irradiation at the middle of a graphene layer, migrate to the edge of the sample, resulting in a specific “segregation” of the vacancy concentration and self-healing of the graphene.

Figure 1

The relaxed structure and the deformation potentials around the vacancies, the spin-polarized DOS, and the band structures for (a) mono-, (b) di-, and (c) trivacancy, when one carbon atom or two and three nearest-neighboring carbon atoms are removed in the unit cell. The DOS, corresponding to spin-up and spin-down states, are plotted, respectively, by red and blue curves in opposite directions for clarity.

Figure 2

(a) Band structure of a pristine graphene monolayer and a supercell of two-vacancy states, produced by removing a reference (red circle) and second C atoms 1, 2, 3, and 4 (green circles) located, correspondingly, in the $1d$, $2d$, $3d$, and $4d$ interatomic distances with $d$ being equal to C—C bond length. The relaxed supercell, DOS, and the band structure of two vacancies located at (b) one atomic distance $\left(1d\right)$, (c) $2d$, (d) $3d$, and (e) $4d$ interatomic distances.

Figure 3

Two other possible structures for three-vacancy defect, where all three vacancies are not bounded to each other, which is different from the trivacancy structure in Fig. 1 with three bounded removal C atoms.

Figure 4

Different topologies of four vacancies in the supercell, their band structures and the relaxed supercell. The removal atoms in the supercell are shown by the red circles.

Figure 5

Distribution of the vacancy concentration with time (or in space) in a sample with dimensionless length $L=7$ (a) at fixed value of $\alpha =1.2$ and for $N_{0V}=1.0$ solid (yellow) curve, $N_{0V}=0.9$ dot-dashed (green) curve, $N_{0V}=0.8$ dashed (blue) curve, $N_{0V}=0.7$ dotted (violet) curve, and $N_{0V}=0.6$ double dot-dashed (red) curve; and (b) at fixed value of $N_{0V}=0.9$ and for $\alpha =1.1$ solid (yellow) curve, $\alpha =1.2$ dot-dashed (green) curve, $\alpha =1.5$ dashed (blue) curve, $\alpha =1.6$ dotted (violet) curve, and $\alpha =1.8$ double dot-dashed (red) curve.