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 intervalley scattering, sample disorder reduces the edge conductance. However, for weak scattering a metallic regime emerges with the diffusive conductance $\mathcal{G}=(e^2/h)({\ell }_{{}_{K{K}^{\prime }}}/L)$ per spin, where ${\ell }_{{}_{K{K}^{\prime }}}$ is the transport mean-free path due to the intervalley scattering and $L\gg {\ell }_{{}_{K{K}^{\prime }}}$ is the edge length. We also take intravalley scattering by smooth disorder (e.g., by remote ionized impurities in the substrate) into account. Albeit contributing to the elastic quasiparticle lifetime, the intravalley scattering has no effect on $\mathcal{G}$.

Figure 1

Edge states ${\varepsilon }_{+}$(k) and ${\varepsilon }_{-}\left(k\right)$ within bulk band gap $[-\Delta ,\Delta ]$ in valleys $K$ and $K^{\prime }$, respectively: (a) zigzag boundary [see Eqs. (24) and (25) with $n_z=0.99$] and (b) mass confinement [see, e.g., Eq. (29)]. Edge momentum $k$ is measured in units of $\Delta /\hslash v_0$.

Figure 2

Diagrammatic representations of (a) Kubo formula (32) for the conductance, (b) Dyson Eq. (36) for disorder-averaged Green's functions (thick lines) and (c) Eq. (45) for renormalized current vertex in ladder approximation. Thin and dashed lines correspond to the unperturbed Green's function and disorder correlator, respectively.