Abstract

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.

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