Robustness of the optical conductivity sum rule in bilayer graphene

We calculate the optical sum associated with the in-plane conductivity of a graphene bilayer. A bilayer asymmetry gap generated in a field-effect device can split apart valence and conduction bands, which otherwise would meet at two $\mathbf{K}$ points in the Brillouin zone. In this way, one can go from a compensated semimetal to a semiconductor with a tunable gap. However, the sum rule turns out to be “protected” against the opening of this semiconducting gap, in contrast to the large variations observed in other systems where the gap is induced by strong correlation effects.

Figure 1

(Color online) Top: bilayer band dispersion for (a) $\Delta =0$ and (b) $\Delta =0.4\mathrm{eV}$ as a function of $p\equiv \hslash v_{F}k$. Bottom: general scheme of the experimental setup for a tunable-gap bilayer device (notation defined in the text).

Figure 2

(Color online) Doping dependence of the $T=0$ sum rule for a bilayer system with a doping-dependent gap $\Delta \left(n\right)$, as shown in the inset. We show the results for different values of $n_0$, given in units of ${10}^{12}{\mathrm{cm}}^{-2}$. For comparison, we show the equivalent doping dependence of the sum rule when no gap is present between the two layers (dashed-dotted line).

Figure 3

(Color online) (a) Doping dependence of the $T=0$ kinetic energy for a bilayer system with a doping-dependent gap $\Delta \left(n\right)$ and without it. Observe the small variations of $K$ in the considered doping range. Insets: doping dependence of the chemical potential when a gap is (b) present or (c) not for parameters as in the main frame.