We explore the electronic properties of finite-length graphene nanoribbons as well as graphene nanodisks with various sizes and shapes in quest of metallic ones. For this purpose, it is sufficient to search zero-energy states. We find that no zero-energy states exist in finite-length zigzag nanoribbons though all infinite-length zigzag nanoribbons have zero-energy states. The occurrence of zero-energy states is surprisingly rare. Among typical nanodisks, only trigonal zigzag nanodisks have degenerate zero-energy states and show metallic ferromagnetism, where the degeneracy can be controlled arbitrarily by designing the size. A remarkable property is that the relaxation time is quite large in spite of its small size in trigonal zigzag nanodisks.

• Received 2 July 2007

DOI:https://doi.org/10.1103/PhysRevB.76.245415