## 3 Results and Discussions
Through the fitting parameters, as displayed in Figure 4a, one can see \(\gamma_{2}\), \(\delta\) and \(\kappa\) are roughly constant, whereas \(\gamma_{1}\) increases significantly with increasing \(E_{F}\). Hence, the characteristic of modulation of the PIT resonance can be dominated by the variation of the damping rate in the bright resonator through the graphene layer. This implies that graphene layer with high conductivity in the gap tend to shorten the SRR1 circuit and reinforce the damping of the bright mode resonator. As the graphene Fermi energy increases, rising damping leads to reduce the LC resonance in SRR1 and suppresses the destructive interference between the bright mode resonator SRR1 and dark mode resonator SRR2. Finally, the damping rate \(\gamma_{1}\) becomes too large to sustain the bright mode excitation by the incident light, giving rise to the disappearance of the PIT effect. The coupled two-particle model and the coupled circuit model describe the same physical essence [17, 37]. With the increase of the damping rate of the bright mode, graphene induced loss resistor, \(R_{Dmmp}\), describing the resistive loss of the graphene in the bright resonator, will increase gradually, which suppresses bright mode resonance. Despite the simplicity of the analytic model, the calculation results reflect the function of the graphene layer in the metamaterial structure and can be used to design hybrid metal-graphene devices. In order to further reveal the function of active two-dimensional graphene in the PIT device, the absorption, \(A(\omega)\), is derived by the formula \(A(\omega)=1-R(\omega)-T(\omega)\) under different values for the graphene Fermi energies, where \(R(\omega)\) and \(T(\omega)\) denote the reflection and transmission, respectively. The absorbance spectra of PIT structure, as presented in Figure 4b, are normalized to that of the silicon  
Figure 4: (**a**) The fitting parameters for different Fermi energies. (**b**) The simulated spectra of the absorption in the PIT structure with different Fermi energies. substrate. It can be clearly seen that the absorption of proposed PIT structure gradually increases with the increase of Fermi energy. In the terahertz band, the thickness of the monoatomic layer of graphene is much smaller than the skin depth of the terahertz wave, which can be regarded as a zero-thickness conductive film. Thus, varying the conductivity of graphene introduces tunable energy loss to the metamaterial modulating PIT resonance. From the perspective of energy location, the electric field distribution at the resonance frequency can further reflect the modulation mechanism of PIT resonance, as shown in Figure 5. For the only metal-based PIT structure, the electric field is mainly focused around the gaps of bright mode SRR1 and dark mode SRR2, but the electric field strength SRR2 is stronger than that of SRR1, which is a typical PIT resonance distribution. However, for the structure with a graphene layer under the SRR1 gap, the electric field of the bright and dark resonators gaps are weakened. The former indicates that the coupling between the bright mode SRR1 and the incident electric field is reduced, and the latter means that the near-field coupling between dark mode SRR2 and the bright mode SRR1 is weakened. The introduction of the conductive graphene layer attenuates the capacitance effect of the SRR1 gap, resulting in the suppression of the bright mode resonance excited by the incident field. When \(E_{F}=0.8\,\mathrm{eV}\), the electric field enhancement around the gap of dark resonator are completely suppressed since the destructive interference between the bright and dark modes disappears. Therefore, the modulation of PIT effect essentially results from the tunable conductivity of the graphene via changing the Fermi energy. Figure 5: The distributions of electric field at the PIT resonance frequency for the cases of (**a**) without graphene; (**b**) \(E_{F}=0.2\,\mathrm{eV}\); (**c**) \(E_{F}=0.8\,\mathrm{eV}\).