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

摘要

Using the Green's function method, we study the effect of an impuritypotential on the electronic structure of the honeycomb lattice in the one-bandtight-binding model that contains both the nearest neighbor ($t$) and thesecond neighbor ($t'$) interactions. The model is relevant to the case of thesubstitutional vacancy in graphene. If the second neighbor interaction is largeenough ($t' > t /3$), then the linear Dirac bands no longer occur at the Fermienergy and the electronic structure is therefore fundamentally changed. Withonly the nearest neighbor interactions present, there is particle-holesymmetry, as a result of which the vacancy induces a "zero-mode" state at theband center with its wave function entirely on the majority sublattice, i. e.,on the sublattice not containing the vacancy. With the introduction of thesecond neighbor interaction, the zero-mode state broadens into a resonance peakand its wave function spreads into both sublattices, as may be argued from theLippmann-Schwinger equation. The zero-mode state disappears entirely for thetriangular lattice and if $t'$ is large for the honeycomb lattice as well. Incase of graphene, $t'$ is relatively small, so that a well-defined zero-modestate occurs in the vicinity of the band center.