Dynamically controllable plasmon induced transparency based on hybrid metal-graphene metamaterials

Novel hybrid metal-graphene metamaterials featuring dynamically controllable single, double and multiple plasmon induced transparency (PIT) windows are numerically explored in the terahertz (THz) regime. The designed plasmonic metamaterials composed of a strip and a ring with graphene integration generate a novel PIT window. Once the ring is divided into pairs of asymmetrical arcs, double PIT windows both with the spectral contrast ratio 100% are obtained, where one originates from the destructive interference between bright-dark modes, and the other is based on the interaction of bright-bright modes. Just because the double PIT windows are induced by two different mechanisms, the continuously controllable conductivity and damping of graphene are employed to appropriately interpret the high tunability in double transparency peaks at the resonant frequency, respectively. Moreover, multiple PIT windows can be achieved by introducing an additional bright mode to form the other bright-bright modes coupling. At the PIT transparent windows, the dispersions undergo tremendous modifications and the group delays reach up to 43 ps, 22 ps, and 25 ps, correspondingly. Our results suggest the existence of strong interaction between the monolayer graphene layer and metal-based resonant plasmonic metamaterials, which may hold widely applications in filters, modulators, switching, sensors and optical buffers.

tunable dual-band PIT based on hybrid metal-graphene 25 . Both of them extensively investigated the shift of the resonant frequency by changing the Fermi energy of the graphene. However, the intensity modulation of PIT has been rarely reported and the key role of graphene in the hybrid metal-graphene metamaterials has not been fully understood. Hence, the intensity modulation of PIT and how graphene works remain to be studied in the hybrid metal-graphene metamaterials.
Motivated by the previous observations, we investigate the dynamically controllable single, double and multiple PIT effects based on hybrid metal-graphene metamaterials by varying the Fermi energy in graphene, which consist of a strip and a ring with graphene integration structure. Both the strip and the ring are served as bright modes, which are induced by electric dipole oscillations. A novel PIT window caused by the hybridization of bright-bright modes can be controlled because of the oscillations of the metallic constructions damped by the dissipation of the graphene layer. By dividing ring into pairs of asymmetrical arcs, the other PIT transparency window with the spectral contrast ratio 100% emerges due to the excitation of magnetic dipole resonance. The continuously controllable conductivity of graphene can also explain the dynamically controllable amplitude of this PIT. Then, by introducing an additional bright mode to form the other bright-bright modes coupling, multiple PIT windows can be achieved and can be effectively controlled with different modulation depth approximately 39%, 86%, and 89% when the Fermi energy of graphene increases from 0.0 eV to 0.8 eV. At the transparent windows, the tremendous changes in the dispersions and the group delays reach up to 43 ps, 22 ps, and 25 ps, correspondingly. Compared with previously PIT effects, our structured plasmonic metamaterials presented have high feasibility and easy modulation. Figure 1(a) schematically shows our proposed plasmonic metamaterials, which consist of a strip and an asymmetric split ring (ASR) separated with a semi-infinite silicon substrate by a monolayer of graphene. All the graphene finger sets are connected to the bilateral gold strips as the top electrode and the lower gold square is used as the bottom electrode. And in general, similar high values of the Fermi energy in graphene can be observed with relatively low electric voltage applied 27 . Therefore, by applying the different gate voltages between the top electrode and the bottom electrode, we can achieve the flexibly controllable Fermi energy of graphene finger sets 28 . The unit cell is arranged in a two-dimensional array with a lattice constant P x = P y = 80 μm, which is illustrated in Fig. 1(b). The inner radius of the ASR is r = 18 μm with a ring width of 6 μm and spans different angles corresponding to θ′ = 155° and 180°. The strip placed 30 μm away from the center of the ASR has length L = 60 μm, width w = 6 μm. All metallic elements are chosen to be aluminum (Al) and have the same thickness t = 200 nm.

Results and Discussions
To our knowledge, the ASR supports electric dipole with currents flow symmetrically and magnetic dipole with currents flow antisymmetrically 29,30 . In the ASR, electric dipole, performed as the bright mode, is accessible from the free-space excitation with a broad transmission dip caused by the strong radiative losses, while magnetic dipole, regarded as the dark mode, cannot be directly coupled to the incident field because the magnetic arm of the incident plane wave is perpendicular to the array plane. However, the magnetic dipole can interact through near-field coupling with the bright mode, resulting in collective resonances of the ASR and the suppression of radiation damping 31,32 , forming the spectrally narrow dark mode.
In the initial setup, we study the metamaterials composed of a strip and a ring without graphene integration. In all the simulations, a normally THz plane wave with x-polarization propagates along the z-axis and the geometric parameters are consistent with Fig. 1 except for θ′. As shown in Fig. 2(a), a PIT window with over 83% transmission at 0.962 THz located between two plasmon resonance dips at 0.945 THz and 1.073 THz. In order to fully understand the underlying physical mechanism behind the PIT effect, we present the simulated z-component of electric field (Ez) distributions in Fig. 2(c-e) corresponding to f 1 = 0.945 THz, f 2 = 0.962 THz, and f 3 = 1.073 THz. From the Ez distributions at f 1 = 0.945 THz (Fig. 2(c)), it is obvious that the strip is strongly excited by normal incidence light while the ring is scarcely excited. As for the other resonance dip at 1.073 THz (Fig. 2(e)), the strong interaction with exterior electric field primarily concentrates on the ring. Both plasmon resonance frequency at f 1 and f 3 serve as bright mode resonances. Notably, as a bright mode, the Ez distributions are not always observable at the plasmon resonance frequency f 1 and f 3 on account of the existence of the coupling field. In fact, it is electric dipole oscillations that form these bright modes. As shown in Fig. 2(d), we can see that both two parts are excited simultaneously due to the resonance detuning at f 2 = 0.962 THz. The hybridization of the two parts leads to two almost identical but a antiphase quadrupole resonance, resulting in the PIT effect. It is worth reminding that the quadrupolar resonance can't be excited by normal incidence light due to its vanishing dipole moment. Two different implementations are employed to activate the quadrupolar resonance: one is based on the highly angled illumination, but impractical, the other is based on the coupling with a bright dipolar resonance 6 . Thus, the quadrupole resonance mentioned above is exactly based on the latter in our proposed metamaterials.
Moreover, the most significant characteristic of the PIT effect is to achieve large positive group delay. Figure 2(b) provides the group delay, calculated by: Here, ϕ is the phase shift introduced by the PIT effect. At the PIT window, there occurs a strong dispersion, giving rise to a large group delay, which means trapping photons for a long time inside our structure, and reaches up to 22 ps. The large positive group delay of our proposed metamaterials will have a better performance in the field of routing optical information 33 and enhancing light-matter interactions 9 .
We now add a monolayer graphene to the bottom of the ring to achieve dynamically controllable single PIT effect. In our current-in-plane device geometry, the work function of graphene is assumed to be equal everywhere by the use of a gate voltage to balance the effect of metal on the Fermi level [34][35][36][37] . The transmittance of the hybrid metal-graphene metamaterials with different Fermi energy of graphene is shown in Fig. 3(a). As the Fermi energy of graphene increases from 0.0 eV to 0.3 eV with a step of 0.1 eV, the transmission intensity of PIT window undergoes strong modulation, with transmission peak gradually shrinks at the fixed frequency, in the end, the PIT window drops distinctly. The change in this trend, is articulately illustrated in the inset of Fig. 3(a). In order to quantitatively describe the change of the transmission intensity with the graphene, the modulation depth in transmittance is defined as: here T and T g represent the transmission intensity without and with the graphene layer, respectively. When the Fermi energy of graphene is equal to 0.8 eV, the modulation depth M depth of 83% at the frequency of 0.962 THz is realized. Therefore, the proposed metamaterials exhibit a high-performance optical switching property. During the modulation of PIT peak, we observe weakly dispersion, which will result in a significant decrease in group delay. We also calculate the group delay as shown in Fig. 3(b), the group delay declines from 22 ps to 8 ps when the Fermi energy of graphene increases from 0.0 eV to 0.3 eV. Our results may be useful for slow-light applications.
To explore the physical mechanism behind this novel phenomenon, the surface charge density distributions at the resonant frequency without (E f = 0.0 eV) and with graphene at different Fermi levels are compared in Fig. 3(c-f). We observe that the surface charge distributions get more indistinctive when the Fermi energy of graphene increases from 0.0 eV to 0.3 eV. The physical mechanism lies in the additional dissipation caused by the resonant fields in the monolayer graphene, namely, the monolayer graphene can significantly damp the oscillation of plasmonic elements when it overlaps with the strong oscillation electric fields driven by the surface charge densities in the metallic elements. This phenomenon can also be quantitatively described as a dissipative power density 34 : g res 2 σ = here σ g and E res represent the conductivity of graphene and the resonant electric field, respectively. Due to the surface conductivity of graphene is relatively small, a big enough dissipative power density also demands that the resonant electric field is large enough. In the entire frequency range, only the frequency correspond to the PIT Peak has a large enough resonant electric field, thus, the transmission intensity of PIT peak undergoes strong modulation at the fixed frequency. When the resonant electric field is large enough at the resonant frequency, the enhancement of graphene conductivity via increasing its Fermi energy can strengthen the dissipative power density, as a result, the transmission intensity of PIT peak is substantially reduced and even turned off, which is in agreement with our trend presented above. The dynamically controllable PIT effect together with the physical mechanism can be strategically crucial in designing active hybrid metal-graphene metamaterials.
It is widely recognized that the asymmetry degree in our proposed metamaterials plays an important role in optimizing the PIT window. So, the asymmetry degree is investigated with the decrease of θ′ and is defined as θ = 180−θ′. Figure 4(a) illustrates that an additional PIT window appears distinctly as the asymmetry degree gets larger on reason that the otherwise forbidden dark mode emerges in our proposed metamaterials. The additional PIT window grows in strength and has a blueshift as the asymmetry degree increases from 10° to 30° due to the decrease of effective length. In order to analyze the reason of the additional PIT window, we plot the simulated transmission spectra of the proposed metamaterials composed of a strip and ASR without the graphene layer as shown in Fig. 4(b), here, we select θ as 25°. Contour profiles of field Ez corresponding to the PIT peaks represented by A and B in Fig. 4(b) are also presented in Fig. 4(f) and (g). Apparently, the formation of the PIT B and the initial PIT discussed above have exactly the same mechanism, a result of quadrupole resonance. However, the occurrence of the additional PIT A is related to the destructive interference between the direct excitation of the electric dipole oscillation by the incident field in the strip and the excitation by coupling with magnetic dipole oscillation in the entire ASR. It should be noticed that the resonance frequency of the PIT channel A could be shifted by changing the asymmetry degree while the resonance frequency of PIT channel B is not affected. Thus, the operating frequency of optoelectronic devices based on the PIT (A) can be designed by altering the asymmetry degree of the metamaterials.
To evaluate the performance of the metamaterials in optical applications, the spectral contrast ratio of the PIT effect is introduced to be a key parameter, which is defined as: where T peak and T dip represent the maximum transmission intensity and the minimum transmission intensity, respectively. As can be seen from Fig. 4(b), both the spectral contrast of the double PIT reach 100%. Thanks to the high spectral contrast ratio of the PIT, this proposed metamaterials authentically tend to exhibit as a promising candidate for the optical filtering, switch, and label-free sensing. Next let us consider the dynamically controllable double PIT windows in our proposed hybrid metal-graphene metamaterials. The simulated transmission spectra Fig. 4(c) and the calculated group delay Fig. 4(d) with different Fermi levels of graphene are presented. As the Fermi energy of graphene increases from 0.0 eV to 0.3 eV, the PIT peak A decreases from 95% to 45% with 53% modulation depth and the PIT peak B decreases from 85% to 33% with 61% modulation depth, corresponding to the group delay declines from 17 ps to 0 ps and 20 ps to 5 ps. Figure 4(e) is the evolution of the transmission spectrum versus Fermi energy and frequency. It is found that the double PIT windows undergo strong modulation and the modulation depths of 59% and 78% are achieved when the Fermi energy of graphene increases from 0.0 eV to 0.8 eV. According to the calculated surface charge density distributions in Fig. 5, we can syllabify the underlying physics. At the PIT A, a great quantity of the opposite kind of charges accumulate at each split gap, forming a circulation current along the entire ASR and generating a magnetic dipole along m as shown in Fig. 5(a). Therefore, the radiative losses of ASR are near completely suppressed. For the PIT B, a large amount of the same kind of charges locate at ends of each split gap, inferring a strictly symmetrical currents along the arcs and forming electric dipole along d as shown in Fig. 5(e). Once a monolayer graphene is placed on the bottom of the ASR, the opposite kind of charges are recombined and neutralized due to the shorting of the gaps by the conductive graphene. The magnetic dipole in the ASR gradually evolves into the electric dipole and the electric dipole in the strip was excited with the increase of the Fermi energy of graphene as shown in Fig. 5(a-d), which originates from the decline in the destructive interference between the electric dipole oscillation and magnetic dipole oscillation. However, the same kind of charges do not experience the charge redistribution and the surface charge distributions just get more indistinctive as the Fermi energy of graphene increases from 0.0 eV to 0.3 eV as shown in Fig. 5(e-h), which exhibits the same tendency with the previous discussion in the initial single PIT effet. Thus, this novel phenomenon is well explained by two different mechanisms. It is the continuously tunable conductivity and damping of graphene which lead to the dynamically controlled PIT A and B.
Based on the previous structure comprising of a strip and ASR (θ′ = 25°) with the graphene layer, another strip regarded as bright mode is implemented to validate the multiple dynamically controllable PIT effects. The unit cell is shown in the inset of Fig. 6(a), which exhibits the simulated multiple PIT transmission responses with the spectral contrast ratio 95%, 100%, and 100%. When we change the double PIT metamaterials into the multiple PIT metamaterials, the transmission peak of PIT channel A decreases from 95% to 62% based on Fig. 4(c) and Fig. 6(a). The reason for this phenomenon is that the interaction between the incident plane wave and our designed structure becomes even greater. Compared with the results in Fig. 4(b), an extra PIT peak is obtained at the frequency of 1.016 THz, which is also caused by the hybridization of bright-bright modes. Obviously, when the Fermi energy of graphene increases from 0.0 eV to 0.3 eV, the transmission intensity of PIT channel A, channel B, and channel C varies from 62% to 38%, 78% to 11%, and 78% to 9%, correspondingly. Figure 6(c) shows the evolution of the observed multiple PIT effects versus the Fermi energy and frequency. It is found that as the Fermi energy increases the transmission intensity of PIT windows drops, and the modulation depth approximately 39% at 0.781 THz, 86% at 0.974 THz, and 89% at 1.016 THz is achieved when the Fermi energy of graphene increases from 0.0 eV to 0.8 eV. To understand the role of the Fermi energy in more detail, we present the modulation depth ( Fig. 6(d)) of peak A/B/C against the Fermi energy of graphene, which once again confirms our proposed physical mechanisms. Finally, at the transparent windows, the phases experience continuous steep variation, leading to strong dispersions and tremendous group delays. As shown in Fig. 6(b), the group delays of the PIT transparency peaks are suppressed with the increase of the Fermi energy because of greater conductivity and dissipative power density. By dynamically tuning the Fermi energy of graphene, the group delay is controlled between 0.2 ps and 43 ps, 7 ps and 22 ps, and 5 ps and 25 ps, corresponding to the delay of a 60 μm and 12900 μm, 2100 μm and 6600 μm, and 1500 μm and 7500 μm distance of free space propagation. Therefore, demonstrating group delay control and profoundly characterizing the group delay spectrum, is a significant step toward the field of optical information routing and optical pulse trapping in the quantum information processing and optical communication.

Conclusions
In conclusion, we propose dynamically controllable single, double and multiple PIT windows based on hybrid metal-graphene metamaterials consisting of a strip and a ring with graphene integration by tailoring the Fermi energy of graphene. The ring and strip are strongly excited by the incident field and are served as bright mode. The hybridization caused by the coupling of bright-bright modes generates a novel PIT window. When asymmetry is introduced to our structure, double PIT transparency windows originating from two different mechanisms are observed. One is based on the interaction between the electric dipole oscillation and magnetic dipole oscillation. And the other originates from the interaction between the electric dipole oscillation and the electric dipole oscillation. The continuously tunable conductivity and damping of graphene are employed to appropriately explain the controllable of double transparency amplitudes at the fixed frequency, respectively. Besides, multiple PIT windows are also predicted in our proposed plasmonic metamaterials, comprising a symmetric double-strip and a ring. More importantly, we can effectively control the intensities and group delays of the single, double and multiple transparency peaks with different modulation depth by tailoring the Fermi energy of graphene. These results will have widely applications in filters, modulators, switching, and sensors.

Methods
We have calculated the results with the finite-difference time-domain (FDTD) method, using perfectly matched layer (PML) absorbing conditions in the propagation directions of the normally incidence plane wave and the periodical boundary conditions in the x and y directions to model the periodic array corresponding to the saturated array size. In the THz regime, the complex relative permittivity of aluminum ε Al is given by the Drude model: where ε ∞ = 1 is the dielectric constant at the infinite frequency, γ = 81.8 meV = 1.22 10 14 × rad/s is the damping constant, and ω p = 14.75 eV = 2.24 × 10 16 rad/s is the plasmon frequency 38 . Also, we select semi-infinite silicon as substrate whose refractive index is n si = 3.42.
Compared with the noble metal, graphene's surface conductivity can be flexibly tailored by applying a gate voltage. In our simulations, graphene is modeled as a 2D flat plane whose conductivity presented by the Kubo formula 39   where e, k B , T, ћ, Γ, E f , and ω is the electron charge, the Boltzmann's constant, the temperature, the reduced Planck's constant, the phenomenological scattering rate, the Fermi energy of graphene, and the angular frequency, respectively. The relaxation time τ μ = − Γ = E ev 1/ (2 ) / ( ) f f depends on the carrier mobility μ, the Fermi energy of graphene E f , and the Fermi velocity v f .. Notably, a high relaxation time in the graphene also reduces the transmission attenuation and means the difficulty in fabrication. So, we select the relaxation time τ = 70 fs, which consistent with the refs 40,41 . Figure 7 shows the real and imaginary part frequency dependent surface conductivity of graphene corresponding to different Fermi energy. It is clear that the increase of Fermi energy remarkable leads to an enhancement of the surface conductivity of graphene.