title: Electronic states of graphene nanoribbons and analytical solutions.
abstract: Graphene is a one-atom-thick layer of graphite, where low-energy electronic states are described by the massless Dirac fermion. The orientation of the graphene edge determines the energy spectrum of π-electrons. For example, zigzag edges possess localized edge states with energies close to the Fermi level. In this review, we investigate nanoscale effects on the physical properties of graphene nanoribbons and clarify the role of edge boundaries. We also provide analytical solutions for electronic dispersion and the corresponding wavefunction in graphene nanoribbons with their detailed derivation using wave mechanics based on the tight-binding model. The energy band structures of armchair nanoribbons can be obtained by making the transverse wavenumber discrete, in accordance with the edge boundary condition, as in the case of carbon nanotubes. However, zigzag nanoribbons are not analogous to carbon nanotubes, because in zigzag nanoribbons the transverse wavenumber depends not only on the ribbon width but also on the longitudinal wavenumber. The quantization rule of electronic conductance as well as the magnetic instability of edge states due to the electron-electron interaction are briefly discussed.
authors.affiliation: International Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Namiki 1-1, Tsukuba 305-0044, Japan; PRESTO, Japan Science and Technology Agency (JST), Kawaguchi 332-0012, Japan.
authors.forename: Katsunori
authors.initials: K
authors.lastname: Wakabayashi
authors.affiliation: International Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Namiki 1-1, Tsukuba 305-0044, Japan.
authors.forename: Ken-Ichi
authors.initials: KI
authors.lastname: Sasaki
authors.affiliation: Nanotube Research Center, AIST, Higashi 1-1-1, Tsukuba 305-8565, Japan.
authors.forename: Takeshi
authors.initials: T
authors.lastname: Nakanishi
authors.affiliation: Department of Chemistry, Tokyo Institute of Technology, Tokyo 152-8551, Japan.
authors.forename: Toshiaki
authors.initials: T
authors.lastname: Enoki
keywords: analytical solution; chirality; edge orientation; edge state; graphene; graphene nanoribbon; tight-binding model; wave mechanics