Impurity states on the honeycomb lattice using the Green's function method

Using the Green's function method, we study the effect of an impurity potential on the electronic structure of the honeycomb lattice in the one-band tight-binding model that contains both the nearest neighbor ($t$) and the second neighbor ($t'$) interactions. The model is relevant to the case of the substitutional vacancy in graphene. If the second neighbor interaction is large enough ($t' > t /3$), then the linear Dirac bands no longer occur at the Fermi energy and the electronic structure is therefore fundamentally changed. With only the nearest neighbor interactions present, there is particle-hole symmetry, as a result of which the vacancy induces a "zero-mode" state at the band center with its wave function entirely on the majority sublattice, i. e., on the sublattice not containing the vacancy. With the introduction of the second neighbor interaction, the zero-mode state broadens into a resonance peak and its wave function spreads into both sublattices, as may be argued from the Lippmann-Schwinger equation. The zero-mode state disappears entirely for the triangular lattice and if $t'$ is large for the honeycomb lattice as well. In case of graphene, $t'$ is relatively small, so that a well-defined zero-mode state occurs in the vicinity of the band center.