Band-gap switching and scaling of nanoperforated graphene.

A framework of \textbraceleft w1, w2, R\textbraceright classification for constructing the graphene nanomesh (GNM) of zigzag-edged hexagonal nanohole is systematically built. Three integer indexes w1, w2, and R indicate the distances between two neighboring sides of nanoholes in two directions and the nanohole size respectively, which leading to a straightforward gap opening criteria, i.e., w\textunderscore 1$+$w\textunderscore 2-R$=$3n$+$1,n$\in $Z, steered via DFT band structure calculations. The guiding rule indicates that the semimetallic and semiconducting variation is consistent with a peculiar sequence “010” and “100” (“0”/“1” represent gap closure/opening) with a period of 3 for odd and even w1 respectively. The periodic nanoperforation induced gap sizes agree with a linear fitting with a smaller $\surd $(N\textunderscore rem )$/$N\textunderscore tot ratio, while deviates from that when (w\textunderscore 1$+$w\textunderscore 2)\textless R$+$1. Particularly, the \textbraceleft p, 1, p\textbraceright and \textbraceleft 1, q, q\textbraceright structures demonstrate each unique scaling rule pertaining to the nanohole size only when n is set to zero. Furthermore, the coexistence of Dirac and flat bands is observed for \textbraceleft 1, q, q\textbraceright and \textbraceleft 1, 1, m\textbraceright structures, which is sensitive to the atomic patters.