Optical and transport gaps in gated bilayer graphene

We discuss the effect of disorder on the band gap measured in bilayer graphene in optical and transport experiments. By calculating the optical conductivity and density of states using a microscopic model in the presence of disorder, we demonstrate that the gap associated with transport experiments is smaller than that associated with optical experiments. Intrinsic bilayer graphene has an optical conductivity in which the energy of the peaks associated with the interband transition are very robust against disorder and thus provide an estimate of the band gap. In contrast, extraction of the band gap from the optical conductivity of extrinsic bilayer graphene is almost impossible for significant levels of disorder due to the ambiguity of the transition peaks. The density of states contains an upper bound on the gap measured in transport experiments, and disorder has the effect of reducing this gap, which explains why these experiments have so far been unable to replicate the large band gaps seen in optical measurements.

Figure 1

Density of states (DOS) of intrinsic biased BLG with $u=0.2$ eV on SiO${}_2$ substrate. The case with no charged impurities is shown by the black line. The other lines show the DOS for increasing charged impurity concentration, with $n_{\mathrm{imp}}=1,5,10,20\times {10}^{12}{\mathrm{cm}}^{-2}$.

Figure 2

(a) Interband optical conductivity of biased intrinsic BLG with $u=0.2$ eV in the absence of disorder. The inset shows the band structure with the interband transitions indicated by arrows. (b) Optical conductivity of disordered biased BLG for impurity densities $n_{\mathrm{imp}}=0,1,5,10,20\times {10}^{12}{\mathrm{cm}}^{-2}$.

Figure 3

DOS gap and optical gap in intrinsic BLG as a function of impurity density. Parameters in (a) correspond to an SiO${}_2$ substrate and in (b) to a $h$-BN substrate (Ref. 31). To aid comparison, the charged impurity concentration of a SiO${}_2$ substrate is of the order of ${10}^{12}{\mathrm{cm}}^{-2}$ whereas for a $h$-BN substrate it is of the order of ${10}^{11}{\mathrm{cm}}^{-2}$.

Figure 4

(a) Interband optical conductivity of extrinsic biased BLG with carrier density $n=5\times {10}^{12}{\mathrm{cm}}^{-2}$ and $u=0.2$ eV in the absence of disorder. The inset shows the band structure with the interband transitions indicated by arrows and the dotted line shows the Fermi energy. (b) Optical conductivity of disordered biased BLG for several values of the impurity density.

Figure 5

Optical conductivity of biased BLG with $u=0.2$ eV on different substrates. (a) $n_{\mathrm{imp}}=5\times {10}^{12}{\mathrm{cm}}^{-2}$ on a SiO${}_2$ substrate and (b) $n_{\mathrm{imp}}={10}^{11}{\mathrm{cm}}^{-2}$ on a $h$-BN substrate. In each case, the black line shows the intrinsic case, and the other lines show different carrier densities for extrinsic graphene, measured in ${10}^{12}{\mathrm{cm}}^{-2}$.