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Register a new userElectronic structure of the substitutional vacancy in graphene: Density-functional and Greens function studies
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We study the electronic structure of graphene with a single substitutional vacancy using a combination of the density-functional, tight-binding, and impurity Greens function approaches. Density functional studies are performed with the all-electron spin-polarized linear augmented plane wave (LAPW) method. The three $sp^2 sigma$ dangling bonds adjacent to the vacancy introduce localized states (V$sigma$) in the mid-gap region, which split due to the crystal field and a Jahn-Teller distortion, while the $p_z pi$ states introduce a sharp resonance state (V$pi$) in the band structure. For a planar structure, symmetry strictly forbids hybridization between the $sigma$ and the $pi$ states, so that these bands are clearly identifiable in the calculated band structure. As for the magnetic moment of the vacancy, the Hunds-rule coupling aligns the spins of the four localized V$sigma_1 uparrow downarrow$, V$sigma_2 uparrow $, and the V$pi uparrow$ electrons resulting in 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 $pi$ band itinerant states in the vicinity of the vacancy. This results in the net magnetic moment of $1.7 mu_B$. Using the Lippmann-Schwinger equation, we reproduce the well-known $sim 1/r$ decay of the localized V$pi$ wave function with distance and in addition find an interference term coming from the two Dirac points, previously unnoticed in the literature. The long-range nature of the V$pi$ wave function is a unique feature of the graphene vacancy and we suggest that this may be one of the reasons for the widely varying relaxed structures and magnetic moments reported from the supercell band calculations in the literature.
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We present a numerically efficient technique to evaluate the Greens function for extended two dimensional systems without relying on periodic boundary conditions. Different regions of interest, or `patches, are connected using self energy terms which encode the information of the extended parts of the system. The calculation scheme uses a combination of analytic expressions for the Greens function of infinite pristine systems and an adaptive recursive Greens function technique for the patches. The method allows for an efficient calculation of both local electronic and transport properties, as well as the inclusion of multiple probes in arbitrary geometries embedded in extended samples. We apply the Patched Greens function method to evaluate the local densities of states and transmission properties of graphene systems with two kinds of deviations from the pristine structure: bubbles and perforations with characteristic dimensions of the order of 10-25 nm, i.e. including hundreds of thousands of atoms. The strain field induced by a bubble is treated beyond an effective Dirac model, and we demonstrate the existence of both Friedel-type oscillations arising from the edges of the bubble, as well as pseudo-Landau levels related to the pseudomagnetic field induced by the nonuniform strain. Secondly, we compute the transport properties of a large perforation with atomic positions extracted from a TEM image, and show that current vortices may form near the zigzag segments of the perforation.
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