Nonlinear optical conductivity of a generic two band systems, with application to doped and gapped graphene

TL;DR

This paper develops a general nonlinear optical conductivity model for two-band systems, applying it to doped and gapped graphene to analyze nonlinear effects beyond linear response.

Contribution

It introduces a unified analytical framework based on optical Bloch equations for nonlinear optical conductivity in two-band systems, including doping and temperature effects.

Findings

Derived the universal ac conductivity of graphene in the linear regime.

Identified a key parameter controlling optical nonlinearities.

Analyzed high-intensity nonlinear deviations in doped and gapped graphene.

Abstract

We present a general formulation to calculate the dynamic optical conductivity, beyond the linear response regime, of any electronic system whose quasiparticle dispersion is described by a two band model. Our phenomenological model is based on the optical Bloch equations. In the steady state regime it yields an analytic solution for the population inversion and the interband coherence, which are nonlinear in the optical field intensity, including finite doping and temperature effects. We explicitly show that the optical nonlinearities are controlled by a single dimensionless parameter which is directly proportional to the incident field strength and inversely proportional to the optical frequency. This identification leads to a unified way to study the dynamical conductivity and the differential transmission spectrum across a wide range of optical frequencies, and optical field strength. We use our formalism to analytically calculate the nonlinear optical conductivity of doped and gapped graphene, deriving the well known universal ac conductivity of $\sigma_0={e^2}/4\hbar$ in the linear response regime of low optical intensities (or equivalently high frequencies) and non-linear deviations from it which appear at high laser intensities (or low frequencies) including the impact of finite doping and band-gap opening.