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 ( Γ-M direction) but arbitrary chemical potential μ . Generally, we find P_μ(q_x,ω)=P_μ^<(ω/ω_q)+q_x²P_μ^>(ω) with ω_q=v_Fq_x the Dirac energy, where the first term is due to intraband and the second due to interband transitions. Explicitly, we derive the analytical expression for the imaginary part of the polarizability including intraband contributions and recover the result obtained from the Dirac cone approximation for μ→0 . For μ<3t , 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±δμ , ImP_μ(q_x,ω) is constant for δμ/t<ω/ω_q≪1 .