Guidelines for band gap opening in graphene superlattices with periodic {\pi}-vacancy distribution

Abstract

Periodic $\pi$-vacancies in graphene superlattices (GSLs) provide a symmetry-based route to band-gap opening in graphene by modifying the $\pi$-band dispersion. However, the symmetry conditions that determine whether a vacancy motif can open a band gap remain unclear. Here, we investigate periodic $\pi$-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. $\pi$-vacancies, representing functionalized, substituted, or missing carbon sites, are modeled as site deletions in the $\pi$ basis, with all hopping matrix elements to and from the deleted sites set to zero. We focus on $\pi$-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 $\Gamma$ 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 $\Gamma$ 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, $\sigma_v \perp \sigma_d$, can also constrain the Dirac cones to $\Gamma$. When the $\sigma_v$ and $\sigma_d$ mirror planes are absent, the cones are allowed to shift away from $\Gamma$ to $(\pm \Delta q,\pm \Delta q)$ in the $3n$ superlattice.