We study the magnetic properties of nanometer-sized graphene structures with triangular and hexagonal shapes terminated by zigzag edges. We discuss how the shape of the island, the imbalance in the number of atoms belonging to the two graphene sublattices, the existence of zero-energy states, and the total and local magnetic moment are intimately related. We consider electronic interactions both in a mean-field approximation of the one-orbital Hubbard model and with density functional calculations. Both descriptions yield values for the ground state total spin $S$ consistent with Lieb’s theorem for bipartite lattices. Triangles have a finite $S$ for all sizes whereas hexagons have $S=0$ and develop local moments above a critical size of $\approx 1.5\mathrm{nm}$.

Hydrogen atom makes graphene magnetic: Graphene has many extraordinary mechanical and electronic properties, but it's not magnetic. To make it so, the simplest strategy is to modify its electronic structure to create unpaired electrons. Researchers can do that by, for example, removing individual carbon atoms or adsorbing hydrogen onto graphene. This has to be done in a very controlled way because of a peculiarity of the graphene's crystal lattice, which consists of two sublattices. Gonzales-Herrero et al. deposited a single hydrogen atom on top of graphene and used scanning tunneling microscopy to detect magnetism on the sublattice lacking the deposited atom (see the Perspective by Hollen and Gupta). Science , this issue p. 437 ; see also p. 415

We address the electronic structure and magnetic properties of vacancies and voids both in graphene and graphene ribbons. By using a mean-field Hubbard model, we study the appearance of magnetic textures associated with removing a single atom (vacancy) and multiple adjacent atoms (voids) as well as the magnetic interactions between them. A simple set of rules, based on the Lieb theorem, link the atomic structure and the spatial arrangement of the defects to the emerging magnetic order. The total spin $S$ of a given defect depends on its sublattice imbalance, but some defects with $S=0$ can still have local magnetic moments. The sublattice imbalance also determines whether the defects interact ferromagnetically or antiferromagnetically with one another and the range of these magnetic interactions is studied in some simple cases. We find that in semiconducting armchair ribbons and two-dimensional graphene without global sublattice imbalance, there is a maximum defect density above which local magnetization disappears. Interestingly, the electronic properties of semiconducting graphene ribbons with uncoupled local moments are very similar to those of diluted magnetic semiconductors, presenting giant Zeeman splitting.

By computing spin-polarized electronic transport across a finite zigzag graphene ribbon bridging two metallic graphene electrodes, we demonstrate, as a proof of principle, that devices featuring 100% magnetoresistance can be built entirely out of carbon. In the ground state a short zigzag ribbon is an antiferromagnetic insulator which, when connecting two metallic electrodes, acts as a tunnel barrier that suppresses the conductance. The application of a magnetic field makes the ribbon ferromagnetic and conductive, increasing dramatically the current between electrodes. We predict large magnetoresistance in this system at liquid nitrogen temperature and 10 T or at liquid helium temperature and 300 G.