Analytical expressions for the polarizability of the honeycomb lattice

Abstract

We present analytical expressions for the polarizability $P_{\mu }(q_x,\omega )$ of graphene modeled by the hexagonal tight-binding model for small wave number $q_x$, but arbitrary chemical potential $\mu$. Generally, we find $P_{\mu }(q_x,\omega )=P_{\mu }^{<}(\omega /{\omega }_q)+q_x^2{P}_{\mu }^{>}(\omega )$ with ${\omega }_q=v_F{q}_x$ the Dirac energy, where the first term is due to intra-band and the second due to inter-band transitions. Explicitly, we derive the analytical expression for the imaginary part of the polarizability including intra-band contributions and recover the result obtained from the Dirac cone approximation for $\mu \to 0$. For $\mu <\sqrt{3}t$, there is a square-root singularity at ${\omega }_q=v_F{q}_x$ independent of $\mu$. For doping levels close to the van Hove singularity, $\mu =t\pm \delta \mu$, $Im{P}_{\mu }(q_x,\omega )$ is constant for $\delta \mu /t<\omega /{\omega }_q\ll 1$.

5 pages, 3 figures, 1 table

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