We show that charge carrier transport in graphene exhibit sharp resonances in the presence of spatially and temporarily modulated scattering. Resonances occur when the period of an applied a-c field corresponds to the time taken by quasi-ballistic carriers to drift over a spatial scattering period, provided the latter is shorter than the distance taken by carriers to emit an optic phonon. We show that such system can be achieved with interdigitated gates energized with an a-c bias on graphene layers. Gate separation and fields to achieve ballistic transport would result in resonances in the terahertz range, with the generation of higher harmonics characterized by large Q-factors, which are tunable with gate spacing.As a one atom thick, two-dimensional (2D) material made up of carbon atoms arranged in a honeycomb structure, [1,2] graphene is characterized by linear dispersion relation between carrier energy and momentum E = v F |k| at K and K symmetry points of the 2D Brillouin zone. As a consequence, all charge carriers in graphene move with the constant Fermi velocity v F ∼ 10 8 cm/s. [3,4] The interaction of these carriers with the 2D environment such as lattice vibrations or oscillatory electro-magnetic fields, is of fundamental interest since it opens a new way to study quantum electrodynamics of charged particles in solid state physics.[5] It may also have important implications in semiconductor technology. [6][7][8] Owing to their large velocity, the weak interaction between charge carriers and acoustic phonons (APs) results in long mean free paths for the former that experience quasi-ballistic motion over distance of the order of several micrometers.[9, 10] At high energy however, carriers lose their momentum (and energy) to efficient optic phonons (OPs) that have much higher vibration frequency ( ω OP ∼ 0.2eV) than in conventional semiconductors.[11] Therefore, in the presence of a constant electric field F o , the carrier motion results in a succession of quasi-ballistic acceleration (when their energy E < ω OP ) and scattering (when their energy is E ≥ ω OP ).[12] This stop-and-go motion of carriers can occur over long distances during picosecond flight times.[13] An interesting effect could arise if the carriers are placed in periodic long range and time varying scattering to achieve transport resonance as well as possible frequency mixing. This kind of situation could be realized in free standing graphene sheets lying over periodically spaced narrow electric gates that would be regulated by an a-c field of appropriate frequency to modulate coulomb scattering of remote oxide impurities when carriers pass in front of the gates (Fig. 1). In the device, a DC field F o is set up between the source S and the drain D, and the a-c field F 1 is applied between successive gates, so that charge carriers experience periodic electric fields and scattering times varying in time and distance along the channel.

In graphene, after the electric field is turned on, the ballistic acceleration of charge carriers up to the monochromatic optic phonon energy generates a back-and-forth motion of the whole distribution function between the zero-point energy and the phonon energy. This effect is predicted to manifest in damped terahertz oscillations of the carrier drift velocity and average energy. It only takes place within a voltage and sample length window as the direct consequence of the interplay between the electric force and the randomizing nature of deformation potential optic phonons in the linear band structure of graphene.

We show that the onset of soft parametric resonance (SPR) at ω=ηωF, where ωF=eFovf/ℏωOP for hot carriers interacting with optic phonons ℏωOP in graphene is independent of temperature, whereas η ∼ 0.56 is practically independent of the DC field Fo, which makes SPR tunable over a sizeable range of Fo. Our model based on the linearization of the time-dependent Boltzmann transport equation (BTE) predicts a secondary resonance at ω∼ωF, emerging from a weak shoulder in the ac current at T = 300 K, when the temperature decreases to T = 77 K and lower. The two resonances behave differently as scattering at low energy strengthens.

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