Diffusion on edges of insulating graphene with intravalley and   intervalley scattering
  Band gap engineering in graphene may open the routes towards transistor devices in which electric current can be switched off and on at will. One may, however, ask if a semiconducting band gap alone is sufficient to quench the current in graphene. In this paper we demonstrate that despite a bulk band gap graphene can still have metallic conductance along the sample edges (provided that they are shorter than the localization length). We find this for single-layer graphene with a zigzag-type boundary which hosts gapless propagating edge states even in the presence of a bulk band gap. By generating inter-valley scattering, sample disorder reduces the edge conductance. However, for weak scattering a metallic regime emerges with the diffusive conductance G = (e^2/h)(l_KK' / L) per spin, where l_KK' is the transport mean-free path due to the inter-valley scattering and L >> l_KK' is the edge length. We also take intra-valley scattering by smooth disorder (e.g. by remote ionized impurities in the substrate) into account. Albeit contributing to the elastic quasiparticle life-time, the intra-valley scattering has no effect on G. 
G. Tkachov and Martina Hentschel
cond-mat.mes-hall