Objective Recently, mimicking the quantum electromagnetically induced transparency (EIT) effect using metamaterials in a classical way has attracted continuous attention. Achieving an active EIT effect is one of the important research directions owing to its great potential in many practical applications, such as active light switching and high-speed slow light modulation. So far, a variety of new working schemes have been proposed by integrating functional materials into the metamaterial structures, such as nonlinear media and photoactive and electroactive semiconductors. Graphene, composed of single-layer carbon atoms, exhibits excellent electrical and optical properties and the dynamic tuning of its optical conductivity is achieved by tuning its Fermi level (E_F) and carrier scattering time (τ). Based on this, graphene-based active metamaterials have successfully exhibited their potentials in light modulation in which high modulation speed is shown owing to the picosecond-level relaxation time of graphene. However, the previous studies mostly rely on tuning E_F to achieve the active control and the studies by tuning τ are relatively scarce. In this work, we theoretically proposed an active EIT device in the terahertz regime using graphene-metal hybrid metamaterials, in which we tune τ using the nonlinear effect of graphene under strong-field terahertz incidence. Owing to the ultrafast relaxation time of the carriers in graphene, such a nonlinear modulation route paves the way towards ultra-fast active devices.

Methods The proposed active EIT metamaterial is composed of graphene-metal hybrid structures on the silicon substrate. The metal structure part is composed of meanderline resonator (MLR) and split ring resonators (SRRs), as shown in Fig. 1. The two SRRs are placed vertically and symmetrically inside the MLR. The geometric parameters of the metal structure are: L₁=85 μm, L₂=75 μm, d =12.5 μm, l₁=29 μm, l₂=25 μm, w=6 μm, s=7 μm, g =0.5 μm, and D =53.5 μm, respectively. The period is P=100 μm, and the thicknesses of the metal and silicon substrate layers are 200 nm and 640 μm, respectively. The graphene structures here are designed to locate only in the gaps of the two SRRs that connect the gap end. In order to study the active EIT effect, the finite-difference time-domain (FDTD) method is applied to simulate the transmission spectra. In the simulation, the excitation source is a plane wave propagating along the z direction, the boundary conditions along the x and y directions are periodic while those along the z direction are open boundary conditions, the substrate is set as lossless silicon (ε =11.78), and the metal is set as aluminum with a conductivity of 3.72×10 ⁷S·m ^-1.

Results and Discussions When the graphene structures are not presented, the metal structure can exhibit a strong EIT effect under y-polarized incidence, as shown in Fig. 2, where the MLR and SRRs function as bright mode and dark mode, respectively. In order to study the nonlinear EIT modulation effect when the graphene structures are presented, the condition of strong-field terahertz incidence (~300 kV·cm ^-1) is mimicked by changing the relevant graphene parameter τ in simulation according to the previously reported nonlinear behavior of graphene. The E_F of graphene is fixed to be 0.15 eV, and the τ is increased from 1 fs to 13 fs (corresponding to gradually decreased terahertz field). Figure 3(a) shows the corresponding simulated transmission spectra. It can be seen that as τ increases, the overall resonance behavior gradually changes to the situation when there is only the MLR. In order to reveal the physical mechanism of the active EIT modulation, a coupled-mode theory is used to quantitatively describe the changing behavior. Figure 3(b) shows the fitted transmission spectra, which are in good agreement with the simulated results. Figure 5 shows the corresponding fitting parameters as a function of τ. It can be seen that the parameters γ₁, δ, and κ basically remain unchanged, while the damping rate γ₂ of the dark mode resonator obviously increases. This can be attributed to the enhanced shorting effect of the graphene structures and to the resonance of the SRRs due to the increased graphene conductivity. In addition, the influence of the gap size g of SRRs on the active EIT effect is also studied, as shown in Figs. 6(a) and (b). Figure 6(c) shows the field enhancement factor at the center of the SRR gap and the corresponding nonlinear modulation depth at different g, which both decrease as g increases. When g is small, under strong terahertz field incidence, the large field enhancement effect can greatly reduce the graphene conductivity and contributes to a strong EIT effect, while under weak terahertz field incidence, the opposite carriers at the two gap ends can easily recombine and contribute to the disappearance of the EIT effect. Thus, the modulation depth becomes larger as g decreases. Here, at g=0.5 μm, the field enhancement factor reaches 360.7 and the nonlinear modulation depth reaches 49.3%.

Conclusions In summary, a nonlinear EIT modulation effect in a composite metamaterial composed of graphene-metal hybrid structures has been studied in the terahertz regime. The inner mechanism lies in the combination of the field enhancement effect and the nonlinear effect of the graphene conductivity under a strong terahertz field. Owing to the ultrafast carrier relaxation time of graphene, the nonlinear modulation speed here is thus determined by the relaxation time of the structure resonances which is in the dozens of picoseconds level. The proposed metamaterials may have potential applications in high-speed slow light modulation and optical switching. And the proposed nonlinear modulation method provides a new way towards high-speed active devices.