A graphene plasmonic structure consists of three graphene layers mingled with a silicon–air grating is proposed. We theoretically predict and numerically simulate the plasmon-induced transparency effect in this system at terahertz wavelengths, and a dual plasmon-induced transparency peaks can be successfully tuned by virtually shifting the desired Fermi energy on graphene layers. We investigate the surface plasmon dispersion relation by means of analytic calculations, and we can achieve the numerical solution of propagation constant got by the dispersion relation. A suitable theoretical model is established to study spectral features in the plasmonic graphene system, and the theoretical results agree well with the simulations. The proposed model and findings may provide guidance for fundamental research of highly tunable optoelectronic devices.
© 2017 Optical Society of America
Plasmon-induced transparency (PIT) is a typical destructive interference effect resulting from the strong coupling between two excitation states in meta-atoms of metamaterials, and it is a kind of analogues of electromagnetically induced transparency (EIT) [1, 2]. In general, the PIT effect can remarkably slow down photons velocities and enhance nonlinear properties . Besides, surface plasmon polaritons (SPPs) which are polariton modes of photon and electron density waves along a conductor and dielectric interface have undisputed advantages like strong enhancement of the local electric field and much better adaptability to nano architectures [4, 5]. So of course it has many underlying practical applications. For example, a large number of applications in optical sensors , optical switch , plasmonic waveguide filters  and slow light effect  have been proposed up to now. The PIT effect can be realized in the proposed applications, but PIT peaks are all modulated only by carefully changing geometric parameters of structures.
Graphene, a two-dimensional (2D) material composed of single-layer carbon atom in a honeycomb lattice, has attracted much attention in the past decade because it exhibits almost all the electrical properties and functions required for integrated photonic circuits [10, 11]. Considering its unique properties and highly reactive electric response, graphene can also support propagation of SPPs and it results in strongly localized plasmons residing within from near-infrared to terahertz region. In contrast to noble metals, graphene plasmonic resonances can be dynamically tuned through electrostatic biasing and enable a new generation of reconfigurable plasmonic devices. Therefore, it is proved as a promising material for multifarious plasmonic systems due to its remarkable characteristics, such as strong mode confinement, low loss, and active tunability. Compared with traditional bulk semiconductors, 2D graphene materials also provide additional values, such as mechanical flexibility, easy fabrication and integration. With those advantages, graphene provides a great opportunity in many plasmonic devices, such as modulators [12, 13], photodetectors , sensors  and many other practical applications [16–19].
In our paper, a novel graphene based on silicon–air grating structure is proposed to realize PIT phenomenon. We can achieve a dual-PIT peak by changing Fermi energy of the graphene layers at infrared and terahertz (THz) wavelengths in the designed structure. In contrast to the metal plasmonic waveguide structure , our graphene system has competitive advantages that the PIT can be tuned by extra gate voltage not geometric parameters. Furthermore, compared with the patterned or separating graphene devices , the graphene in our structure keeps continuous form. It has the benefit of preserving the high mobility of graphene and also simplifies these fabrication processes. Moreover, the modulation of PIT peaks with three graphene layers is investigated in detail. The designed structure exhibits a prominent PIT resonance peak in the finite-difference time-domain (FDTD)  simulated transmission spectrum, and its resonance mechanism is further discussed by the coupled mode theory (CMT). Furthermore, numerical simulation results for the PIT effects show a good agreement with theoretical expectations. Thus it can be achieved for building high performance active plasmonic devices, and the presented theoretical model and the pronounced features of this simple graphene plasmonic structure, such as the tunable PIT phenomenon and convenient integration, may have potential applications in the sensors, tunable switches, and slow light devices.
2. Structure and theoretical model
The schematic of dual-PIT graphene device, which is composed of three graphene layers separated with the dielectric silicon and air by a grating shape, is illustrated in Fig. 1(a). Most structural information of the system is also introduced in Fig. 1. The three graphene layers are separated by a dielectric spacer of thickness d1 and d2, respectively.
Numerical simulation method in this letter is selected as the finite-difference time-domain (FDTD). For simplicity, the spacer is assumed to be air with a permittivity of 1.00 in the simulation. A TM-polarized wave is injected along the negative direction of z-axis. For the sake of the clarity of presentation, the Kubo formula has governed the surface conductivity of graphene including the intraband and interband transition contributions, and then the conductivity of graphene σ obtains a Drude-like expression , as follows:10, 24]). The simulations are performed with the two-dimensional FDTD method with mesh grid size Δx = 1 nm and Δz = 0.1nm, respectively. The y-axis can be regarded as infinite. The calculated domain is surrounded by perfectly matched layer absorbing boundary at z-directions and periodic boundary at x-directions, respectively.
Now, we present the derivation of the relationship between thickness and wave vector of electromagnetic mode guided by graphene. TM wave is characterized by the existence of a Hy component of the magnetic field together with Ex and Ez components of the electric field. Here, x is the direction of propagation, z is the direction normal to the graphene, and y is the direction parallel to the graphene and perpendicular to z. The structure is surrounded with dielectrics of constants ε1 = 1.0 (air, on top of structure) and ε4 = 3.9 (silica, the substrate of structure). For definiteness we use ε5 = ε6 = 1.0 for air dielectric grating, and ε2 = ε3 = 11.9 corresponding to silicon dielectric grating, which corresponds to a typical experimental setup . According to Maxwell equations of , and the component of the wave vector perpendicular to the interface in the three layer graphene ki≡kz,i (i = 1, 2, 3), the expression of electric and magnetic fields is given as follows:
Finally, with the combination of the above equations, continuity of the tangential electric field(Ex1 = Ex2), boundary condition for the tangential magnetic field(H2-H1 = σEx) [25, 26], , and (β is propagation constant and k0 is the wave vector of the propagating wave in free space, respectively), the dispersion relation for the TM SPP is implicitly given as
From this equation, the propagation constant β of the graphene SPPs can be obtained. As a result, we can readily get the effective mode index of SPP, defined as neff = β/k0. As a proof of this concept, the real part of propagation constant β and effective index neff are numerically plotted in the Fig. 2(a) and 2(b), respectively. Obviously, from the Fig. 2(b), Re(neff) decreases for a fixed wavelength as the Fermi energy EF increases, which intends the graphene SPPs are better confined at lower Fermi energy. Importantly, with a slight change in Fermi energy, the Re(neff) varies greatly, which leads to the design of dynamically tunable peak modulation devices.
As frequency increases into the far-infrared, the surface wave becomes more tightly confined to the graphene layer, but becomes slow as energy is concentrated on the graphene surface. In this three layers graphene system, Fig. 2(c) shows the transmission spectra of the hybrid system with three layers graphene (blue), only upper-layer graphene (red), only middle-layer graphene (green), and only lower-layer graphene (purple) as Fermi energy EF = 1.0eV. From the spectra, we can see that the top layer graphene at z = d1 and the middle layer graphene at z = 0 couple efficiently to the incident wave, thus, the graphene SPPs are strongly excited which act as two excitation states and two wide continuum spectra are formed. However, the bottom layers graphene at z = -d2 couples weakly to the incident wave and leads to a narrow discrete one which play another excitation state. In other words, the top and middle layers graphene act as two bright elements, and the bottom layer graphene act as a dark elements. Therefore, a dual-PIT resonance results from the destructive interference between the bright and dark plasmon elements as showed in Fig. 2(c) and Fig. 3. More specific analysis to the activated modes will be described in detail at in later content.
With the incident waves pass through z-direction, the energy can be coupled into the three layers graphene and the dynamic transmittance characteristics of our proposed structure can be investigated by the CMT [27–29]. As shown in Fig. 1(c), the three equivalent resonators are named as A1, A2 and A3 take the place of the excitation state modes, respectively. The incoming and outgoing waves in the resonators are depicted by and (n = 1, 2, 3). The subscript ± represent two propagating directions of waveguide modes, as shown in Fig. 1(c). Thus, the complex amplitude an of the nth resonator (n = 1, 2, 3) can be expressed as
According to Eqs. (15)-(17) and the condition that the wave is only injected from the upper layer ( = 0), we can achieve the complex transfer coefficient of this system
3. Simulation and discussion
Next, we study the optical characteristics of our proposed structure. By properly setting separation distance d, Fermi energy EF and mobility μ of the three graphene layers, the resonance characteristics of plasmonic modes supported by the graphene could be accurately controlled. And then, a desired PIT effect can be obtained. The tuning scheme is outlined in Fig. 1(a) and is based on applying a voltage difference across the dielectric layer to inject or remove electrons from the graphene layers and modify their Fermi energy. Throughout this letter, the geometric parameters are fixed, like as L = 400nm, l1 = 250nm, l2 = 150nm, d1 = 250nm and d2 = 250nm. Based on the above analysis, we numerically calculated the transmission spectra of the grapheme plasmonic system with different Fermi energy EF = 1.05eV, 1.00eV, 0.95eV, 0.90eV, respectively, and electrical tuning of the transparency window has a close relation with the Fermi energy level as shown in Fig. 3. The Fermi energy of graphene could be experimentally modified from 0.2eV to 1.2eV after applying a high bias voltage . Thus in this system, we reasonably assume that Fermi energy EF can be dynamically tuned from 0.90eV to 1.05eV. The decay rates and the coupling coefficients obtained from theoretical calculation and the FDTD simulations are fitting parameters and their values are showed in the caption of Fig. 3. Then we put these parameters into Eq. (18)-(25) and get theoretical transmission curves, which agree well with the simulation results. This consistence also reveals that the formation of plasmon induced transparency can be described as the coupling between cavities in our system. In Fig. 3, the blue solid lines are simulated transmittance and the red cycle lines are theoretical fitting, respectively. And we can see that the FDTD simulations are in excellent agreement with the theoretical fittings, from which we can conclude that Eq. (18) is a qualified theoretical description of plasmon-induced transparency in the plasmonic graphene system. So, the theoretical analysis allows us to understand the response of the plasmonic graphene system as a function of their microscopic parameters. As we can see from the blue lines, the transmission spectrum exhibits an obvious dual-PIT peak for each value of Fermi energy. In this stacked graph, we can clearly see that the resonant wavelength blue shift with Fermi energy EF increases.
To get more insight into the physical mechanism of this observed dual-PIT effects, Fig. 4 shows the peaks and dips of the spectral transmittance with Fermi energy EF varying from 0.90eV to 1.05eV. To illustrate a difference, the dip at the shortest wavelength is called as dip1, the dip at the longest wavelength is called as dip3, and the middle one is called as dip2. The peak which lies between dip1 and dip2 is called as peak1 and which lies between dip2 and dip3 is called as peak2, respectively. We can see the approximately linear relationship of the peak/dip wavelengths versus the Fermi energy EF. In order to elaborate this phenomenon, we have theoretically modeled the effects of dual-PIT on the Fermi energy EF from 0.90eV to 1.05eV, as showed in Fig. 5(b).
FDTD simulated electric field intensities of THz metamaterial with three graphene layers are showed in the Figs. 4(c)-4(f) with Fermi energy EF = 0.9eV at dip1, dip2, dip3 and EF = 0.95eV at dip, dip2, dip3, respectively. Here, these field intensities provide a basic understanding of the mechanisms involved. From the field intensity diagrams, we can get that the SPPs mode is excited at the interface of graphene–silicon boundary. At transmission dip 1 and transmission dip2, only the top graphene layer is excited, and the other two layers are not excited. However, at dip3, the two upper layers are excited at the same time and the bottom layer is still not directly excited. Through above simulation and analysis, we can draw a conclusion that the two upper graphene layers can be regarded as two bright elements and the bottom graphene layer acts as a dark element. The two modes at lower wavelength are together excited by the top and bottom graphene layers and the mode at maximum wavelength is excited by the middle and bottom graphene layers, respectively. So, the three resonance modes are together controlled by the three graphene layers. The bottom graphene layer cannot be excited by the incident wave functioning as the dark element, and the destructive interference between the bright and dark elements give rise to a dual-PIT effect.
The dispersion characteristics of plasmonic modes in the three graphene layers device can be obtained by solving the Eq. (14). From the equations, we can see that the dispersion characteristics are dependent on the surface conductivity of graphene, which can be controlled by the Fermi energy. What is more, we can tune the Fermi energy by the applied bias voltage Vg , like following formula
where, dsub is the thickness of insulated substrate material (silica). Using above formula, we can plot an evolution of the Fermi energy EF versus bias voltage Vg, as showed in the Fig. 5(a). Because the graphene layers in our plasmonic system exist in a continuous whole block, it is much easier to structurally realize the tunability compared with other discrete graphene tunable devices. Furthermore, to investigate spectral characteristics more specifically, the evolution of the transmission spectra versus Fermi energy EF and wavelength λ is displayed in Fig. 5(b). As expected, Fig. 5(b) shows that there are three transmission dips which present blue shift with the increase of Fermi energy. Moreover, the transparent resonance dips and peaks exhibit a linear relationship. Figure 5(b) clearly shows that the resonance characteristics of the proposed graphene plasmonic system are efficiently tuned by altering the Fermi energy EF from 0.90eV to 1.05eV.
In summary, by using the temporal CMT and FDTD method, we have proposed and demonstrated a tunable dual-PIT phenomenon by means of three layers graphene nanostructures based on silicon–air grating structure, and the metamaterial is designed at terahertz frequencies. At first, we analyze the dispersion relations of our proposed structure. Re(β) and Re(neff) decrease for a fixed wavelength as the Fermi energy EF increases. Then, the numerical results are calculated by FDTD method for different Fermi energy EF and the theoretical results are calculated by CMT for different Fermi energy EF. The consistency between the theoretical and numerical results validates the correctness of the theoretical description. Compared with the devices based on patterned and separating graphene, our structure keeps graphene in the continuous form. It has the benefit of preserving the high mobility of graphene and also simplifies the fabrication processes. With these advantages and the impressive dual-PIT resonance characteristics, our proposed graphene metamaterial structure may open up avenues for terahertz devices and sensing technology, and may provide meaningful guidance and potential applications for designing graphene metamaterials.
This work is supported by the Hunan Provincial Innovation Foundation for Postgraduate (Grant No. CX2017B042) and the National Natural Science Foundation of China (Grant No. 61275174).
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