Properties of nano-graphite ribbons with zigzag edges -- Difference   between odd and even legs --
  Persistent currents and transport properties are investigated for the nano-graphite ribbons with zigzag shaped edges with paying attention to system length $L$ dependence. It is found that both the persistent current in the isolated ring and the conductance of the system connected to the perfect leads show the remarkable $L$ dependences. In addition, the dependences for the systems with odd legs and those with even legs are different from each other. On the persistent current, the amplitude for the cases with odd legs shows power-low behavior as $L^{-N}$ with $N$ being the number of legs, whereas the maximum of it decreases exponentially for the cases with even legs. The conductance per one spin normalized by $e^2/h$ behaves as follows. In the even legs cases, it decays as $L^{-2}$, whereas it reaches to unity for $L \to \infty$ in the odd legs cases. Thus, the material is shown to have a remarkable property that there is the qualitative difference between the systems with odd legs and those with even legs even in the absence of the electron-electron interaction. 
Hideo Yoshioka and Satomi Higashibata