Theory of third-harmonic generation in graphene: A diagrammatic approach

Abstract

We present a finite-temperature diagrammatic perturbation theory of third harmonic generation (THG) in doped graphene. We carry out calculations of the third-order conductivity in the scalar potential gauge, highlighting a subtle cancellation between a Fermi surface contribution, which contains only power laws, and power-law contributions of inter-band nature. Only logarithms survive in the final result. We conclude by presenting quantitative results for the up-conversion efficiency at zero and finite temperature. Our results shed light on the on-going dispute over the dependence of THG on carrier concentration in graphene.Introduction.-The non-linear optical properties of graphene 1 , the most studied two-dimensional (2D) material, are beginning to attract considerable interest. Using four-wave mixing, Hendry et al. 2 demonstrated experimentally that the third-order optical susceptibility of graphene is remarkably large (≈ 1.4 × 10 −15 m 2 /V 2 ) and only weakly dependent on wavelength in the nearinfrared frequency range. Third harmonic generation (THG) from mechanically exfoliated graphene sheets has been measured by Kumar et al. 3 who extracted a value of the third-order susceptibility on the order of 10 −16 m 2 /V 2 for an incident photon energy ω = 0.72 eV. Finally, Hong et al. 4 reported strong THG in graphene grown by chemical vapor deposition, in the situation in which the incident photon energy ω = 1.57 eV is in three-photon resonance with the exciton-shifted van Hove singularity.

2D Layered Materials: Synthesis, Nonlinear Optical Properties, and Device Applications

“…The third‐order susceptibility χ (3) is a complex number. Its imaginary part describes triple‐frequency phenomenon, including nonlinear absorption, third‐harmonic generation, parametric process, and Raman effect, while its real part describes the Kerr nonlinearity.…”

Strong enhancement of third-harmonic generation in a double layer graphene system caused by electron-hole pairing

“…The peaks predicted in [3,4,7] emerges at lower temperature (T 10 −2 µ). Under transition to the normal state the double peak at ω ≈ (2/3) µ 2 + ∆ 2 is transformed to the peak at ω = (2/3)µ, and the peaks at ω = (2/3)∆, ω = ∆, and ω = 2∆ disappear.…”

“…This equation yields the electric current that contains all odd Fourier harmonics with the amplitudes falling down very slowly with the harmonic number. The quantum approach [3][4][5][6][7] predicts resonant behavior for the third harmonic generation in graphene. The frequency dependence of the THG intensity has the main peak at ω = (2/3)ε F and two minor peaks at ω = ε F and ω = 2ε F , where ε F is the Fermi energy.…”

Nonlinear graphene plasmonics

“…The series of studies on graphene's numerous nonlinear optical processes, as shown in figure 7, can be traced back to contributions by Mikhailov since 2007 [58][59][60][61][62][63] [78][79][80] and more recently by other groups [81][82][83][84][85][86][87][88][89].…”

“…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.…”