Magnetic Response of Graphene Flakes

The orbital diamagnetism and spin paramagnetism of graphene flakes is computed as a function of size, shape, boundary type, and temperature, using a nearest--neighbor tight binding Hamiltonian. In general the orbital magnetization per atom grows as the size of the flake increases, but not necessarily monotonically. Flakes with "open" boundaries, where open means there are atoms on the edges of the flake with only one nearest neighbor, exhibit a smaller magnetization than flakes with "closed" boundaries, where closed means every atom has at least two nearest neighbors. The orbital susceptibility increases with temperature for open flakes, whereas the closed flakes show little change at low temperatures.
The paramagnetic spin response also depends on the boundaries. In particular, for certain types of edge defects states near zero energy are created. The wave functions of these states are not localized to the boundary. They produce a paramagnetic response that looks like an isolated spin so that the net magnetization of the flake is paramagnetic at low fields and diamagnetic at higher fields. Such magnetization vs. B plots have been observed.