Analytical expressions for the polarizability of the honeycomb lattice

We present analytical expressions for the polarizability $P_μ(q_x,ω)$ of graphene modeled by the hexagonal tight-binding model for small wave number $q_x$, but arbitrary chemical potential $μ$. Generally, we find $P_μ(q_x,ω)=P_μ^<(ω/ω_q)+q_x^2P_μ^>(ω)$ with $ω_q=v_Fq_x$ 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 $μ\rightarrow0$. For $μ<\sqrt{3}t$, there is a square-root singularity at $ω_q=v_Fq_x$ independent of $μ$. For doping levels close to the van Hove singularity, $μ=t\pmδμ$, $ImP_μ(q_x,ω)$ is constant for $δμ/t<ω/ω_q\ll1$.

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