Third harmonic generation of undoped graphene in Hartree-Fock approximation

TL;DR

This paper theoretically studies how Coulomb interactions influence third harmonic generation in undoped graphene using Hartree-Fock approximation, revealing significant enhancement at low photon energies and dependence on dielectric environment.

Contribution

It introduces a Hartree-Fock based theoretical framework to analyze Coulomb effects on third harmonic generation in graphene, highlighting the interaction's role in modifying optical responses.

Findings

Coulomb interaction increases third harmonic generation at low photon energies.

The conductivity's frequency dependence follows a power law with exponent 5.5 when Coulomb effects are included.

Background dielectric constant influences the third harmonic response.

Abstract

We theoretically investigate the effects of Coulomb interaction, at the level of unscreened Hartree-Fock approximation, on third harmonic generation of undoped graphene in an equation of motion framework. The unperturbed electronic states are described by a widely used two-band tight binding model, and the Coulomb interaction is described by the Ohno potential. The ground state is renormalized by taking into account the Hartree-Fock term, and the optical conductivities are obtained by numerically solving the equations of motion. The absolute values of conductivity for third harmonic generation depend on the photon frequency $\Omega$ as $\Omega^{-n}$ for $\hbar\Omega<1$, and then show a peak as $3\hbar\Omega$ approaches the renormalized energy of the $M$ point. Taking into account the Coulomb interaction, $n$ is found to be $5.5$, which is significantly greater than the value of $4$ found with the neglect of the Coulomb interaction. Therefore the Coulomb interaction enhances third harmonic generation at low photon energies -- for our parameters $\hbar\Omega<0.8$~eV -- and then reduces it until the photon energy reaches about $2.1$~eV. The effect of the background dielectric constant is also considered.