Abnormal Electronic Transport in Disordered Graphene Nanoribbon

We investigate the conductivity $\sigma$ of graphene nanoribbons with zigzag edges as a function of Fermi energy $E_F$ in the presence of the impurities with different potential range. The dependence of $\sigma(E_F)$ displays four different types of behavior, classified to different regimes of length scales decided by the impurity potential range and its density. Particularly, low density of long range impurities results in an extremely low conductance compared to the ballistic value, a linear dependence of $\sigma(E_F)$ and a wide dip near the Dirac point, due to the special properties of long range potential and edge states. These behaviors agree well with the results from a recent experiment by Miao \emph{et al.} (to appear in Science).