## 3 Results and Discussions
It is worth noting that in our hybrid metamaterial, the graphene layer is placed right between the U-shaped ring and the substrate, and this design causes strong coupling between the graphene strip and the radiative resonator, which accounts for the complete deformation of the EIT window. Figure 3: (**a–e**) The electric field distribution at the resonant frequency corresponding to Figure 2 labelled A, B, C, D, E, respectively. The Fermi energy of the graphene layer is 0.05 eV. Figure 2: Transmission spectra of the sole U-shaped ring, the sole cut-wire pair, the combined structure, and the hybrid structure with a graphene layer. To interpret the distinctive amplitude variations in different resonances, the surface current distribution at the transparency peak (1.71 THz) and the transmission dips (1.49 THz and 1.75 THz) with \(E_{F}\) = 0.05 eV and \(E_{F}\) = 0.20 eV are calculated and plotted in Figure 5. It is obvious from Figure 5a,d that the surface current intensity on the U-shaped resonator is weakened as \(E_{F}\) rises from 0.05 to 0.20 eV. As \(E_{F}\) = 0.20 eV, more electron-hole pair recombination occurs in the graphene layer, which renders the surface currents no longer concentrated on the U-shaped ring. As a consequence, the radiative mode is suppressed and the amplitude of the first transmission dip drastically increases. A similar phenomenon is observed at the second transmission dip, as shown in Figure 5c,f, despite the relatively slight changes in the surface current intensity. The attenuation of the current intensity is also observed, as shown in Figure 5b,e. Nevertheless, the intensity variation is rather small at the transmission peak and the amplitude exhibits a slight decrease because of the enhanced radiative loss caused by the more conductive graphene strip with higher Fermi energy. The EIT effect with dispersive curves can lead to the slow light behavior. The slow light capability can be characterized by the group delay of the transmitted wave, which can be defined as \(\mathrm{t_{g}}=-\mathrm{d}\varphi/\mathrm{d}\omega\), where \(\varphi\) is the transmission phase and \(\omega\) is the operation frequency. Figure 6a depicts the group delay spectra of the metamaterial with and without the graphene layer. The positive and negative values denote the slow and fast light effects, respectively. Apparently, without the graphene strip, a broadband slow-light region and a group delay up to 2.2 ps are realized. As the graphene layer is  
Figure 4: Transmission spectra of the hybrid metamaterial with different Fermi energy. Figure 5: The calculated surface current distributions of the proposed metamaterial at transmission peak and dips with different Fermi energy. (**a–c**) f = 1.49, 1.71, 1.75 THz as \(E_{F}\) = 0.05 eV; (**d–f**) f = 1.49, 1.71, 1.75 THz as \(E_{F}\) = 0.20 eV. placed between the U-shaped ring and the substrate, the slow light region is drastically narrowed but the maximum value of group delay is increased to 3.3 ps. Moreover, by applying gate voltages, the dynamical control of the group delay can be achieved based on the hybrid structure. With the Fermi energy increasing from 0.05 to 0.20 eV, the group delay is reduced from to 3.3 to 0.5 ps and the metamaterial gradually loses the slow light capability. It is noted that the adjustment to the Fermi energy required for group delay modulation is much smaller than that in ref. [21; 22; 23; 24], which may provide more feasibility in practical operation. This feature is due to the strong coupling caused by the direct contact between the graphene layer and the U-shaped ring in our design. The external voltages required for tuning the Fermi energy based on our structure can approximately be evaluated according to Equation (2). Assuming the Dirac point voltage \(V_{d}\) = 0 V, the relation between the gate voltages and the tuning range of Fermi energy is calculated and plotted.