## 3 Results and Discussions  
In order to explore the response of the hybrid metamaterial, the numerical calculations are performed based on the frequency domain method and employing the electromagnetic simulation software CST Microwave Studio based on the frequency domain Finite Element Method. The terahertz wave normally illuminates on the structure along \(z\) direction with linear polarization in \(x\) direction. The unit cell boundary conditions are utilized in both \(x\) and \(y\) directions and the open boundary conditions are utilized in z direction. The investigated frequency range is from 1.0 to 2.4 THz. Figure 2 shows the transmission spectra of the sole U-shaped ring, the sole cut-wire pair, the combined structure, and the hybrid structure with the graphene layer. It can be seen that the U-shaped ring manifests a single resonance with a transmission dip at 1.74 THz, which can serve as bright mode. By contrast, the cut-wire pairs display a near-unity transmission under the terahertz illumination in the investigated range, which can be considered as dark mode. When the two resonators are combined in the same unit cell, a noticeable transparency window appears while the transmission dip splits to two branches, indicating an evident EIT behavior. Moreover, for the hybrid structure with a graphene layer positioned between the substrate and the U-shaped resonator, the EIT effect is drastically damped and the transmission amplitude experiences an evident decline. The Fermi energy of graphene is 0.05 eV. To understand the physical mechanism of the EIT effect, the electric field distributions of the isolated resonators and the combined structure with/without graphene layer are calculated. Figure 3 shows the electric field distribution at the transmission dips and the transparency peaks, which are labeled in Figure 2. It is observed that the U-shaped ring can be directly excited by the incident wave. The field distribution of the isolated cut-wire pair is not shown as the field amplitude is much smaller compared with those of other circumstances, implying inactive behaviour under illumination. At transmission dips of the combined structure, the cut-wire pair is excited indirectly with the near-field interaction between two resonators, as shown in Figure 3b,c. At the transmission peak, the electric field  
Figure 1: **(a)** The geometrical schematic of the proposed metal-graphene metamaterial and **(b)** the unit cell of the structure. locates mainly on the edge of the cut-wire pairs and the initial radiative element, i.e., the U-shaped rings, are not induced, as can be seen in Figure 3d. This indicates a destructive interference between two resonators and thus the radiative mode is suppressed. Moreover, with the graphene layer placed under the U-shaped ring, there exists more electric field concentrated on the edges of the graphene strip and the U-shaped ring, which results in the noticeable attenuation at the transparency peak. As mentioned above, by applying external voltages, the Fermi energy and the surface conductivity of the graphene layer can be dynamically tuned, thus the modulation of the spectral EIT window can be realized. The transmission spectra with different Fermi energy in graphene are calculated and plotted in Figure 4. It can be seen that as the Fermi energy increases, the EIT window is gradually damped and the resonance frequencies manifest red shift. The first transmission dip increases drastically from 0.21 to 0.50 as the Fermi energy is tuned from 0.05 eV to 0.15 eV, meanwhile the other dip and the transmission peak experience slight fluctuations. When increasing the Fermi energy to a maximum value of 0.20 eV, the transmission dips gradually disappear and consequently the EIT window vanishes. The distinctive variations that occur in the transmission peak and dips lead to the deformed transparency window of the hybrid metamaterial.