Spin coupling around a carbon atom vacancy in graphene

We investigate the details of the electronic structure in the neighborhoods of a carbon atom vacancy in graphene by employing magnetization-constrained density-functional theory on periodic slabs, and spin-exact, multireference, second-order perturbation theory on a finite cluster. The picture that emerges is that of two local magnetic moments (one $\pi$-like and one $\sigma$-like) decoupled from the $\pi$ band and coupled to each other. We find that the ground state is a triplet with a planar equilibrium geometry where an apical C atom opposes a pentagonal ring. This state lies $\sim$0.2 eV lower in energy than the open-shell singlet with one spin flipped, which is a bistable system with two equivalent equilibrium lattice configurations (for the apical C atom above or below the lattice plane) and a barrier $\sim$0.1 eV high separating them. Accordingly, a bare carbon atom vacancy is predicted to be a spin-1 paramagnetic species, but spin-$\frac{1}{2}$ paramagnetism can be accommodated if binding to foreign species, ripples, coupling to a substrate, or doping are taken into account.

Figure 1

Optimized structure of a carbon vacancy in a $6\times 6$ unit cell.

Figure 2

The molecular model adopted for the wave function calculations, along with isosurfaces of singly occupied $\sigma$ (left) and $\pi$ midgap (right) orbitals for $\left|\varphi \right|=0.015$ Å${}^{-3/2}$, as obtained at the restricted open-shell Hartree-Fock level ($S=1$) for the minimum structure.

Figure 3

Magnetization-constrained energies as functions of the height $h_C$ of the apical carbon atom above the surface. Filled and empty symbols for $M=2,0{\mu }_B$, referenced to the minimum of the $M=2{\mu }_B$ case. Also reported as a thick line the results of magnetization-unconstrained calculations, referenced to their minimum, and the corresponding magnetization (dashed line, right scale).

Figure 4

Variation of the open-shell character of the singlet state as the apical carbon atom moves out of plane. The symbols give the weight $(c_1^2+c_2^2)\times 100$ of the coefficients in the minimal CAS(2,2) wave function with diabatized orbitals described in the main text. The insets show isosurfaces at $\left|\varphi \right|=0.015$ Å${}^{-3/2}$ of the corresponding $\sigma$-like CASSCF orbitals for representative values of $h_C=0.0,0.32,0.68$ Å, from left to right, respectively.

Figure 5

CASPT2 energies as function of the displacement ($h_C$) of the apical carbon atom out of the surface plane, for the singlet (open symbols) and the triplet (filled symbols) states. Also reported for comparison the magnetization-constrained DFT results of Fig. 3 (dashed lines). Energies are referenced to the triplet minimum at the corresponding theory level. The inset shows a blow-up of the $h_C\approx 0$ region for the singlet.

Figure 6

Variation of the Hund coupling constant as a function of the angle subtended by the $\sigma$ dangling bond and the graphene plane (filled symbols). Dashed line is a low-angle weighted fit of the data to $J_H=J_H^0-4At{g}^2\left(\theta \right)[1-2t{g}^2\left(\theta \right)]/3$, which gives $J_H^0=0.268$ eV and $A=7.59$ eV.

Figure 7

Optimized structure of a single hydrogenated vacancy. The CH bond is $1.080$ $\text{Å}$ long, and its carbon end lies $0.694$ $\text{Å}$ above the graphene plane.