Nonlinear optical conductivity of a generic two-band system with application to doped and gapped graphene

摘要

We present a general formulation to calculate the dynamic interband optical conductivity, beyond the linear response regime, of any electronic system whose quasiparticle dispersion is described by a two-band model. Our formulation is based on the optical Bloch equations with phenomenological damping constants. In the nonlinear 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 strengths. We use our formalism to analytically calculate the nonlinear interband optical conductivity of doped and gapped graphene, deriving the well known universal ac conductivity of σ_0 = e~2/4h in the linear response regime of low optical intensities and nonlinear deviations from it which appear at high laser intensities including the impact of finite doping and band-gap opening.