Fig. 1. (a) Schematic structure of graphene nanomesh with periodically patterned holes; (b) the supperlattice cells (left) and Y-junction connection areas (right) for three types of vacant holes with different magnetic distributions, black beads represent carbon atoms distributed with net spin moment at the edge of G₄₂ (MA) and G₈₄ (MC) holes in graphene nanomeshes.

Fig. 2. Calculated electronic structures of G₆₀ patterned graphene nanomeshes: (a) Energy band structures of the G₆₀ nanomeshes with supplattice cell (N, N) (N = 10 – 15); (b) the electron density distribution of the σ* state at K point in the energy ～0.2 eV for N = 12; (c) the projected density of states on the carbon atoms of hole edge for N = 12. The reference energy zero is set as Fermi energy level indicated with horizontal dot line.

Fig. 3. The energy band structures of pristine graphene supperlattices with lattice vector extending from (1, 1) to (7, 7). Fermi energy level is referenced as energy zero indicated by horizontal dot line.

Fig. 4. Carbon-carbon atomic distance d at hole edge (a) and energy level of the σ^* state at K point (b) as a function of supperlattice cell size N for the G₆₀ (N, N) graphene nanomeshes, with Fermi energy level referenced as energy zero.

Fig. 5. Energy band structures of the graphene nanomeshes with (a) MA (G₄₂) and (b) MC (G₈₄) patterned holes, respectively. The up and down panels represent nanomeshes without and with hydrogen passivation at hole edge, respectively.

Fig. 6. Bandgap width varying with cell size (N, N) of (a) G₆₀, (b) G₄₂ and (c) G₈₄ graphene nanomeshes with NM, MC and MA vacancy holes respectively.