Competing magnetism in $\pi$-electrons in graphene with a single carbon vacancy

One intriguing finding in graphene is the vacancy-induced magnetism that highlights the interesting interaction between local magnetic moments and conduction electrons. Within density functional theory, the current understanding of the ground state is that a Stoner instability gives rise to ferromagnetism of $\pi$-electrons aligned with the localized moment of a $\sigma$ dangling bond and the induced $\pi$ magnetic moments vanish at low vacancy concentrations. However, the observed Kondo effect suggests that $\pi$-electrons around the vacancy should antiferromagnetically couple to the local moment and should carry nonvanishing moments. Here we propose that a phase possessing both significant out-of-plane displacements and $\pi$ bands with antiferromagnetic coupling to the localized $\sigma$ moment is the ground state. With the features we provide, it is possible for spin-resolved scanning tunneling microscopy, scanning tunneling spectroscopy, and angle-resolved photoelectron spectroscopy measurements to verify the proposed phase.

Figure 1

(a) In-plane structure with a single carbon vacancy. The lengths between C2 and C4 atoms are 1.86, 1.84, and $1.84\AA$ in the F, Q, and AF phases, respectively. Electron spin density of (b) F, (d) Q, and (f) AF phases. The majority (positive) and minority (negative) spin density at $0.0003e/{\mathrm{bohr}}^3$ are colored in red and blue, respectively. The side views of the out-of-plane structures are presented for (c) Q and (e) AF phases with the enlarged out-of-plane displacements ($d_z$). The height of the C1 atom measured from the C3 atom is $0.46\AA \left(0.68\AA \right)$ in the Q (AF) phase. The plots are generated by xcrysden [33].

Figure 2

The band structures of the F, Q, and AF phases in the representation of the Brillouin zone corresponding to the primitive unit cell without any vacancies. The spectral weight is represented by the diameter of a circle. The spin majority is colored in black, and the spin minority is colored in red (gray).

Figure 3

Local density of states of the (a) C1 and (b) C3 atoms in the F phase compared to the (c) C1 and (d) C3 atoms in the AF phase with a Gaussian broadening of 0.05 eV. The contributions of the C1 and C3 atoms in this energy range are mainly from the $s{p}^2$ and $p_z$ orbitals, respectively. The spin-majority density of states is presented by the positive value, whereas the negative value is chosen for the spin-minority density of states.