Guidelines for band gap opening in graphene superlattices with periodic π-vacancy distribution

Abstract: Periodic $π$-vacancies in graphene superlattices (GSLs) provide a symmetry-based route to band-gap opening in graphene by modifying the $π$-band dispersion. However, the symmetry conditions that determine whether a vacancy motif can open a band gap remain unclear. Here, we investigate periodic $π$-vacancy GSLs using a nearest-neighbor tight-binding model with one $p_z$ orbital per carbon site to identify the symmetry requirements for gap opening. $π$-vacancies, representing functionalized, substituted, or missing carbon sites, are modeled as site deletions in the $π$ basis, with all hopping matrix elements to and from the deleted sites set to zero. We focus on $π$-vacancy motifs with $C_2$ and $C_3$ point-group symmetry. A $3n \times 3n$ GSL, where $n=1,2,3,\ldots$ is the integer scaling factor multiplying the honeycomb primitive-cell vectors, folds $K$ and $K'$ to $Γ$ and can therefore open a band gap. For $C_3$-type vacancies, the Dirac cones remain pinned at high-symmetry points and thus stay at $Γ$ in folded $3n$ GSLs. In contrast, $C_2$-type vacancies that reduce the global point group of the GSL to $D_{2h}$ by preserving a pair of perpendicular mirror symmetries, $σ_v \perp σ_d$, can also constrain the Dirac cones to $Γ$. When the $σ_v$ and $σ_d$ mirror planes are absent, the cones are allowed to shift away from $Γ$ to $(\pm Δq,\pm Δq)$ in the $3n$ superlattice.