Analytical study of edge states in a semi-infinite graphene nanoribbon

By means of the transfer-matrix technique, we present an analytical solution of the edge states localized at the lateral zigzag edge of a semi-infinite graphene nanoribbon. The electric field tuning on the energy level, the localized length, and the local electron probability distribution of an edge state is then studied in detail. The dependence of the edge state on the size of the ribbon, the presence of impurities, and the structural variation in the lateral edge is discussed. The physical natures of some previous numerical conclusions about the edge state are clarified. For example, it was previously expected that any edge state cannot survive while the width of the graphene nanoribbon becomes smaller than three times of the lattice constant and whenever such a width increases by triple lattice constants, one more edge state is added. The physical reasons of these issues can be intuitively seen in our analytical treatment on the edge states.