Band-gap switching and scaling of nanoperforated graphene
  In this paper, a framework of {w_1, w_2, R} classification for constructing the graphene nanomesh (GNM) of zigzag-edged hexagonal nanohole is systematically built. The three integer indexes w_1, w_2, 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_1+w_2-R=3n+1,n is integer, 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 w_1 respectively. The periodic nanoperforation induced gap sizes agreewith a linear fitting with a smaller sqrt(N_rem )/N_tot ratio, while deviates from that when (w_1+w_2)<R+1. Particularly, the {p, 1, p} and {1, q, q} 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 {1, q, q} and {1, 1, m} structures, which is sensitive to the atomic patters 
Haiyuan Chen, Xiaobin Niu
cond-mat.mtrl-sci