Finite conductivity minimum in bilayer graphene without charge inhomogeneities

Boltzmann transport theory fails near the linear band crossing of single-layer graphene and near the quadratic band crossing of bilayer graphene. We report on a numerical study which assesses the role of interband coherence in transport when the Fermi level lies near the band-crossing energy of bilayer graphene. We find that interband coherence enhances conduction, and that it plays an essential role in bilayer graphene’s minimum conductivity phenomena. This behavior is qualitatively captured by an approximate theory which treats interband coherence in a relaxation-time approximation. On the basis of this short-range-disorder model study, we conclude that electron-hole puddle formation is not a necessary condition for finite conductivity in bilayer graphene at zero average carrier density.

Figure 1

The dotted curves depict the electrical conductivity of bilayer graphene (per spin/valley) as a function of carrier concentration computed according to the Kubo formula (2) for the series of model parameters specified in Table . The solid lines correspond to the analytical approximation which is the sum of the Drude conductivity ${\sigma }_{\text{D}}$ and an interband coherent correction $\Delta \sigma$ given by Eq. (5). The inset illustrates the decomposition of the conductivity for disorder model A into intraband and interband coherent contributions proportional, respectively, to the intraband and interband terms in the velocity operator in Eq. (2).

Figure 2

Comparison between Kubo conductivities, Eq. (2), of (a) the decoupled band model and (b) bilayer graphene. These results were obtained for a series of models with identical golden-rule relaxation times $\tau =0.3\times {10}^{-13}\text{s}$, and sample sizes $L=1.8\times {10}^{-5}\text{cm}$. (The concentration of scatterers $n_s$ was adjusted appropriately in each case.) One can see that the conductivity minimum for the decoupled band model vanishes whereas for the bilayer model it is finite and insensitive to the scattering potential strength. The thick solid lines show the naive prediction of (a) Drude theory and (b) our interband coherent Boltzmann model with golden-rule relaxation times.