Microwave (2ω1 − ω2) frequency mixing by mobile carriers in electron-doped and gapped graphene

Abstract

We theoretically study the generation of microwave radiation at frequency 2ω1−ω2, produced by two microwaves, of frequencies ω1 and ω2, normally incident on monolayer graphene grown epitaxially on a SiC substrate and doped by electrostatic gating. Two mechanisms responsible for this third-order nonlinear effect are considered: (i) the graphene’s conduction band nonparabolicity arising from the substrate-induced bandgap opening at the Dirac points of the graphene’s Brillouin zone and (ii) the energy dependence of the electron-momentum relaxation time. Both mechanisms are incorporated in our Boltzmann-equation-based treatment of the graphene’s third-order nonlinear response to the applied microwave radiation field. Within the framework of the approach, we evaluate the output power P2ω1−ω2 of the (2ω1−ω2) microwave radiation as a function of the graphene’s carrier density nS controlled by the applied gate voltage. Our formulation predicts an unexpected nonmonotonous behavior of this function: there is a pronounced minimum in the P2ω1−ω2(nS), which is followed by a maximum. This finding may serve to provide a rational explanation for this type of oscillation as observed by Dragoman et al. [Appl. Phys. Lett. 97, 093101 (2010)] in the output power of the microwave harmonic generation in gated graphene, which are not properly construed so far.

Unusual anisotropic behaviour of free-carrier-induced third-harmonic generation in phosphorene at microwave frequencies

“…As we shall see further, we need to know both the linear surface conductivity σ (1) (ω) and the third-order nonlinear surface conductivity σ (3) (ω) ≡ σ (3) (−3ω; ω, ω, ω) in order to calculate the intensity of the third harmonic radiation generated in the reflected direction when the phosphorene layer is deposited on a dielectric substrate, such as SiO 2 . In deriving explicit expressions for those conductivities, we rely on the perturbative approach that we have used in our previous papers [45][46][47][48][49] dealing with graphene. Within the framework of the approach, equation ( 10) is solved iteratively in powers of the electric field amplitude E 0 up to the third order in E 0 .…”

“…[13]). In this case, as in the case of an electron-doped and gapped graphene monolayer [45][46][47][48][49], free mobile carriers can cause a substantial nonlinear microwave response of phosphorene due to its non-parabolic conduction band [50].…”