Non-linear electromagnetic response of graphene

Abstract

It is shown that the massless energy spectrum of electrons and holes in graphene leads to the strongly non-linear electromagnetic response of this system. We predict that the graphene layer, irradiated by electromagnetic waves, emits radiation at higher frequency harmonics and can work as a frequency multiplier. The operating frequency of the graphene frequency multiplier can lie in a broad range from microwaves to the infrared.In the past two years a great deal of attention has been attracted by a recently discovered, new two-dimensional (2D) electronic system -graphene, built out of a single monolayer of carbon atoms with a honeycomb 2D crystal structure [1,2]. The band structure of the charge carriers in this system consists of six Dirac cones at the corners of the hexagonshaped Brillouin zone [3,4], with the massless, linear electron/hole dispersion. The massless electron spectrum leads to unusual transport and electrodynamic properties, which have been intensively studied in the literature, see e.g. [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31] and for review [32,33].Electrodynamic properties of graphene have been theoretically studied in Refs. [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. The frequency dependent conductivity [16,17,20,21,22], as well as plasmon [23,25,27,29,30], plasmon-polariton [24], and transverse electromagnetic wave spectra [31] have been investigated. In all these papers electrodynamic response of the system has been studied within the linear response theory (for instance, using the Kubo formalism, or the random phase approximation, or the self-consistent-field approach). In this Letter we show that, apart from all the fascinating and non-trivial properties of graphene predicted and observed so far, this material should also demonstrate strongly non-linear electrodynamic behavior. In particular, irradiation of the graphene sheet by a harmonic electromagnetic wave with the frequency Ω should lead to the emission of the higher harmonics with the frequencies mΩ, m = 3, 5, . . ., from the system. The operating frequency of such a frequency multiplier can vary from microwaves up to infrared, and the required ac electric field is rather low, especially at low carrier densities and low temperatures. The predicted non-linear electrodynamic properties of graphene may open up new exciting opportunities for building electronic and optoelectronic devices based on this material.To qualitatively demonstrate the non-linear behavior of graphene electrons consider a classical 2D particle with the charge −e and the energy spectrum ǫ p = V p = V p 2 x + p 2 y in the external electric field E x (t) = E 0 cos Ωt. Here V is the velocity of 2D electrons in the energy band (in graphene V ≈ 10 8 cm/s [1, 2]). According to the classical equations of motion dp x /dt = −eE x (t) the momentum p x will then be equal to p x (t) = −(eE 0 /Ω) sin Ωt, and the velocity v x = ∂ǫ p /∂p x is then v x (t) = −V sgn(sin Ωt). If there are n s particles per unit area, the corresponding ac ...

Optical bistability of graphene in the terahertz range

“…We also note that result (19) differs by a factor of 3 from the result for the same quantity computed by Mikhailov [19]. This difference exists, because Mikhailov treatment does not permit one to study the regime of ωτ 1, since by construction it assumes that the observation time is much smaller than τ .…”

Graphene photonics and optoelectronics

“…Calculations suggest that nonlinear frequency generation in graphene (e.g. harmonics of input light) should be possible for sufficiently high external electric field (> 100V /cm) 139 . Second harmonic generation (SHG) from a 150fs laser at 800nm was reported for a graphene film 140 .…”

Terahertz and Infrared Spectroscopy of Gated Large-Area Graphene

“…12-17 For example, it has been predicted that the response of Dirac fermions to an applied AC electric field of frequency ω would automatically contain all odd harmonics of (2n+1)ω, where n is an integer, implying extremely high nonlinearity. 13,14 Furthermore, creation of electrons and holes through interband optical pumping is expected to lead to population inversion near the Dirac point, resulting in negative σ(ω), or gain, in the THz to MIR range. 12,17 While initial experimental investigations on graphene have concentrated on DC characteristics, these recent theoretical studies have instigated a flurry of new experimental activities to uncover unusual AC properties.…”