Sum rules for the optical and Hall conductivity in graphene

Abstract

Graphene has two atoms per unit cell with quasiparticles exhibiting Dirac-like behavior. These properties lead to interband in addition to intraband optical transitions and modify the $f$-sum rule on the longitudinal conductivity. The expected dependence of the corresponding spectral weight on the applied gate voltage ${V}_{g}$ in a field-effect graphene transistor is $\ensuremath{\sim}\mathrm{const}\ensuremath{-}{\ensuremath{\mid}{V}_{g}\ensuremath{\mid}}^{3∕2}$. For ${V}_{g}=0$, its temperature dependence is ${T}^{3}$ rather than the usual ${T}^{2}$. For the Hall conductivity, the corresponding spectral weight is determined by the Hall frequency ${\ensuremath{\omega}}_{H}$ which is linear in the carrier imbalance density $\ensuremath{\rho}$, and hence proportional to ${V}_{g}$, and is different from the cyclotron frequency for Dirac quasiparticles.