Turning graphene into nodal-line semimetals by vacancy engineering

We elaborate that single-layer graphene can be turned into a nodal-line or nodal-loop semimetal by introducing periodic vacancies, opening the possibility of fabricating graphene-based electronic or spintronic devices with novel functionalities. The principle is that by removing carbon atoms such that the lattice becomes nonsymmorphic, every two sublattices in the unit cell will map to each other under glide plane operation. This mapping yields degenerate eigenvalues for the glide-plane operation, which guarantees that the energy bands must stick together pairwise at a boundary of the Brillouin zone. Moving away from the Brillouin zone boundary causes the symmetry-enforced nodal lines to split, resulting in accidental nodal lines caused by the crossings of the split bands. Moreover, the vacancy-engineered graphene has a dramatically enlarged density of states at the Fermi level. This mechanism is applicable to a variety of crystalline structures, irrespective of the details of the systems, and valid even in the presence of strong spin-orbit coupling.