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We study the electronic structure of graphene with a single substitutionalvacancy using a combination of the density-functional, tight-binding, andimpurity Green's function approaches. Density functional studies are performedwith the all-electron spin-polarized linear augmented plane wave (LAPW) method.The three $sp^2 \sigma$ dangling bonds adjacent to the vacancy introducelocalized states (V$\sigma$) in the mid-gap region, which split due to thecrystal field and a Jahn-Teller distortion, while the $p_z \pi$ statesintroduce a sharp resonance state (V$\pi$) in the band structure. For a planarstructure, symmetry strictly forbids hybridization between the $\sigma$ and the$\pi$ states, so that these bands are clearly identifiable in the calculatedband structure. As for the magnetic moment of the vacancy, the Hund's-rulecoupling aligns the spins of the four localized V$\sigma_1 \uparrow\downarrow$, V$\sigma_2 \uparrow $, and the V$\pi \uparrow$ electrons resultingin a S=1 state, with a magnetic moment of $2 \mu_B$, which is reduced by about$0.3 \mu_B$ due to the anti-ferromagnetic spin-polarization of the $\pi$ banditinerant states in the vicinity of the vacancy. This results in the netmagnetic moment of $1.7 \mu_B$. Using the Lippmann-Schwinger equation, wereproduce the well-known $\sim 1/r$ decay of the localized V$\pi$ wave functionwith distance and in addition find an interference term coming from the twoDirac points, previously unnoticed in the literature. The long-range nature ofthe V$\pi$ wave function is a unique feature of the graphene vacancy and wesuggest that this may be one of the reasons for the widely varying relaxedstructures and magnetic moments reported from the supercell band calculationsin the literature.

Electronic structure of the substitutional vacancy in graphene: Density-functional and Green's function studies