Surface conductivity σ(ω) is often used to describe the optical properties of graphene, because its value can be measured over a wide frequency range. It is known that surface conductivity of graphene is determined by two processes: interband and intraband transitions of carriers. The gapless band structure of graphene leads to an unusual behavior of its conductivity. For visible and infrared ranges interband transitions dominate. In that frequency range surface conductivity is almost independent of the optical frequency ω and the chemical potential μc(Fermi level Ef). For THz frequency range the intraband transitions contribute greatly to the surface conductivity. In that frequency range there is a strong dependence of the surface conductivity with the Fermi level. Therefore, by changing the position of the Fermi level it is possible to control effectively the conductivity of graphene. This effect forms the basis of all tunable optoelectronic devices based on graphene.
The value of the surface conductivity of graphene can be theoretically calculated by the Kubo formula. It gives the expression of the complex conductivity of graphene monolayer, which takes into account both intraband
and interband transitions: (1) (2) (3)
where i is an imaginary unit; kB is the Boltzmann constant; e is the charge of an electron; ħ=h/2π is the reduced Planck′s constant. As seen from the expressions (2) and (3) the surface conductivity is related to the environmental temperature T; the relaxation time τ=1 ps; the optical frequency ω and the chemical potential μc.
Under optical pumping of graphene with the light of visible and infrared range, photogeneration of electron-hole pairs occurs with an efficiency of 2.3%. Immediately after the optical pumping the electron-hole pairs tend to move into a state with lower energy. The optical generation of electron-hole pairs in graphene is described by the chemical potential μc, where unexcited graphene is located at the intersection of the valence and conduction bands, and its energy is equal to zero. The value of the chemical potential under optical pumping is determined by:
where νF is the Fermi velocity(~106 m/s); α=1/137 is the fine-structure constant; τR=1 ns is the characteristic recombination time; Ω is the frequency of pumping source corresponding to the wavelength of λ=1 550 nm and Ipump is the intensity of a photo-doping pump source. The pumping source wavelength was chosen according to its prevalence in the communications.
The frequency dependent normalized conductivity of graphene depending on some selected values of the pumping intensity is shown in Fig. 1(a) and 1(b). Fig. 1(c) and 1(d) show the real and imaginary part of normalized surface conductivity of graphene as a function of the pump intensity and operation frequency.
Figure 1. Spectrum of real(a) and imaginary(b) parts of normalized conductivity for single layer graphene at various pumping intensities. Real(c) and imaginary(d) parts of normalized conductivity as function of pumping intensity at various frequencies
The variation of the real and imaginary parts of the graphene surface conductivity causes changes of the amplitude and phase correspondingly of the electromagnetic(EM) wave propagating through the graphene monolayer. According to this fact, Fig. 1(c) and 1(d) reveal that at low frequency there are significant losses and large possibilities of phase control of the EM wave, whereas at higher frequencies the reverse situation is observed. At this point high efficiency of graphene properties tuning can be achieved only by a compromise between low losses and sufficiently high phase changing. As seen from Fig. 1, the largest difference between the imaginary parts of surface conductivity with and without pumping is observed at the frequency of 0.18 THz, but at the same frequency the significant value of the real part of surface conductivity is present. For this reason, the resonant frequency of the switcher developed was chosen in the higher frequency region of the spectrum where the losses are minimum and the difference between the value of Im(σ) is still significant with or without optical pumping. The frequencies near 0.4 THz satisfy this condition. In addition, there is a sufficient number of radiation sources, operating at frequencies close to 0.4 THz, that can be useful for the experimental verification of the calculations.
Optically controlled narrowband terahertz switcher based on graphene
摘要: 本文提出了一种光控太赫兹开关，该开关采用覆盖单层石墨烯的十字金属谐振器超表面。利用石墨烯表面电导率模型和有限元法计算了这种复合结构的光谱特性。模拟结果表明，在0.2 W/mm2的光泵浦后，传输谱（调制深度为36.8%，Q-因子为250）出现了窄带共振衰减现象。另外，这种衰减的调制深度可以通过改变泵浦强度微调节。因此，光学可调谐太赫兹开关的设计将有助于太赫兹通信应用的功能组件开发。Abstract: This paper proposes an optically controlled terahertz switcher based on cross-shaped metal resonators metasurface covered by monolayer graphene. The spectral characteristics of proposed composite structure were calculated using the surface conductivity model of graphene and the finite element method. The modeling demonstrated the appearance of a narrowband resonant dip in the transmission spectrum with a modulation depth of 36.8% and a Q-factor of 250 after the optical pump intensity achieving to 0.2 W/mm2. In addition, the modulation depth of such a dip can be slightly tuned by varying value of the pump intensity. Thus, the design of the optically tunable terahertz switcher may contribute to the development of functional components for terahertz communication applications.
Figure 2. Schematic of the unit cell geometry under consideration: the periodic cross-shaped aluminum/graphene arrays with width K, length L and period G. The arrays are located on a PET substrate with thickness d. The incident EM wave is TE polarized with the electric fields along the y axis. The plane wave normally(along z) impinges on the switcher
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