Third-order nonlinearity of graphene: Effects of phenomenological relaxation and finite temperature

Abstract

We investigate the effect of phenomenological relaxation parameters on the third-order optical nonlinearity of doped graphene by perturbatively solving the semiconductor Bloch equation around the Dirac points. An analytic expression for the nonlinear conductivity at zero temperature is obtained under the linear dispersion approximation. With this analytic formula as a starting point, we construct the conductivity at finite temperature and study the optical response to a laser pulse of finite duration. We illustrate the dependence of several nonlinear optical effects, such as third harmonic generation, Kerr effects and two photon absorption, parametric frequency conversion, and two-color coherent current injection, on the relaxation parameters, temperature, and pulse duration. In the special case where one of the electric fields is taken as a dc field, we investigate the dc-currentand dc-field-induced second-order nonlinearities, including dc-current-induced second harmonic generation and difference frequency generation.

Transient nonlinear plasmonics in nanostructured graphene

“…Currents associated with these intraband transitions undergo a strongly anharmonic motion that reflects the linear electronic dispersion in graphene and leads to a highly nonlinear response to external electromagnetic fields [9][10][11][12][13][14][15][16][17]. However, most experimental studies on graphene nonlinear optics have dealt with interband effects, including reports of large third-order susceptibilities linked to wave mixing [18], harmonic generation [19][20][21][22][23], and the Kerr effect [24][25][26].…”

Identification of photocurrents in topological insulators

“…Two-color photocurrents have been observed [7], and the expressions for the nonlinear optical response that describes them have been worked out in detail [8] for frequencies where states near the K points are important, with scattering included phenomenologically.Unlike most semiconductors, where two-color injection currents have been studied forhω < E g < 2hω, in undoped graphene one-photon absorption is possible at both 2ω and ω, and so complexities in the injected currents arise [9]. Yet since the lattice structure has inversion symmetry there are no one-color effects.…”

Nonlinear graphene plasmonics

“…Ooi et al [111] substituted the full expression of the linear susceptibility, which includes the electronic scattering frequencies, and found that graphene may exhibit negative nonlinear refraction for certain E F values and wavelengths. Subsequent Bloch derivations by Cheng et al [75] also yield negative nonlinear conductivities for both the real and imaginary parts, when appropriate scattering frequencies and room temperature are applied, as shown in figure 8. Similarly, while early experimental investigations did not explicitly distinguish between the signs of graphene's nonlinear coefficient, very recent works have unanimously found negative nonlinear refractive indices in the visible and telecommunication wavelengths.…”

“…However, to compare with experimental measurements, the recasting of the 2D nonlinear conductivity into the 3D nonlinear refractive index via graphene's thickness is applied, as in the case of the modelling of graphene's linear refractive index. This is prominently practised by Cheng et al [73,75,76] in several of their landmark papers on the theoretical derivations of graphene's nonlinear optical response. Quantitatively, the theoretical models predict a Kerr-type nonlinear coefficient of the order of n 2 ∼ 10 −11 -10 −13 m 2 W −1 around the visible and telecommunication wavelengths.…”

“…Perturbative models are developed to supplement the interband physics, for example the quantum mechanical (QM) Dirac equation model [61,62,64,66,67,89], and also the semiconductor Bloch equations [73,75,76,85]. The QM model replaces v g in equation (3.1) with v g = ψ|∂ H/∂k|ψ , where ψ is the wave function of graphene and H is the Dirac Hamiltonian, which replaces N(p) with N(p) = f (−ε k ) − f (ε k ), the Fermi distribution difference between the conduction and valence bands.…”