Analytical expressions for the polarizability of the honeycomb lattice

TL;DR

This paper derives analytical formulas for the polarizability of graphene within a tight-binding model, capturing intra- and inter-band effects for small wave vectors and various chemical potentials, including near van Hove singularities.

Contribution

It provides explicit analytical expressions for the polarizability of graphene, extending previous Dirac cone approximations to more general chemical potentials and wave vectors.

Findings

Derived analytical expression for the imaginary part of polarizability.

Identified a square-root singularity at the Dirac energy.

Described behavior of polarizability near van Hove singularities.

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 $P_\mu(q_x,\omega)=P_\mu^<(\omega/\omega_q)+q_x^2P_\mu^>(\omega)$ with $\omega_q=v_Fq_x$ 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 $\mu\rightarrow0$. For $\mu<\sqrt{3}t$, there is a square-root singularity at $\omega_q=v_Fq_x$ independent of $\mu$. For doping levels close to the van Hove singularity, $\mu=t\pm\delta\mu$, $ImP_\mu(q_x,\omega)$ is constant for $\delta\mu/t<\omega/\omega_q\ll1$.