Semiconducting graphene nanomeshes

Symmetry arguments are used to describe all possible two-dimensional periodic corrugations of graphene (``nanomeshes'') capable of inducing the tangible semiconducting gaps. Such nanomeshes or superlattices break the initial graphene translational symmetry in a way that produces mixing and subsequent splitting of the Dirac $\mathbf{K}$ and ${\mathbf{K}}^{\ensuremath{'}}$ states. All of them have hexagonal Bravais lattice and are described by space groups that are subgroups of the graphene group. The first-principles calculations show that the gaps of about 0.5 eV can be induced at strains safely smaller than the graphene failure strain. Experimental realization with the use of lattice mismatching and tailored substrates is discussed.