## 1 Introduction  
The past decade has witnessed great developments in the artificially engineered material metamaterial, which has attracted extensive interests due to its unique electromagnetic properties [1]. Tailored in diverse architectures, metamaterials provide opportunities to control the electromagnetic wave in a wide range, from microwave to near-infrared regions [2; 3; 4]. Potential applications are reported in different fields including absorbers, filters, antennas, and polarizers. Recently, researchers have demonstrated that the metamaterial can mimic the electromagnetically induced transparency (EIT) in an atomic system, which refers to a quantum interference that renders a narrow transparent window within an opaque medium [5; 6; 7]. The metamaterial analogue of the EIT effect effectively frees researchers from the cumbersome experimental conditions in an atom system, such as low temperatures and pumping lasers. Moreover, it overcomes the restriction of operation frequency range. The EIT effect in metamaterials mainly results from two interference pathways: bright-dark mode coupling and bright-bright mode coupling [8]. Since the earlier study was fulfilled by [PERSON] et al., various metamaterial designs are proposed and demonstrated to achieve the EIT effect with broadband,multiple transparency windows and polarization-insensitive properties [9; 10; 11; 12; 13]. The imitation of EIT is useful in applications such as slow light, modulating, and sensing purposes [14]. In practical occasions, it is desirable to actively control the EIT effect after the structures are fabricated. Various techniques such as thermal tuning, optical control, and micro-electromechanical (MEMS) systems have been employed to achieve the dynamic manipulation of the EIT metamaterials [15; 16; 17]. Among all approaches, electrical modulation is one of the most convenient methods in actual operations. Graphene, a two-dimensional material with carbon atoms arranged in a hexagonal lattice, promises a good candidate as its surface conductivity can be tuned by changing the Fermi energy via external voltages [18; 19]. Graphene can interact strongly with both the electromagnetic waves and the adjacent metal patterns when integrated in the metamaterial. In the past few years, a series of graphene-based metamaterial designs have been investigated to achieve the dynamic control of the EIT effect, exhibiting potential applications in controllable buffers and optical modulators [20; 21; 22; 23; 24]. Nevertheless, despite extensive studies on the dynamic response of the graphene-based structure, the deformation of the EIT window in the modulating process has not been fully investigated. On the other hand, a large adjustment in Fermi energy is required when tuning the group delay spectra. This may lead to practical difficulty since higher external energy is required. Thus, it is essential to get insight into the tuning mechanisms and improve the design of the dynamically tunable graphene-based EIT structure. In this work, we present a metal-graphene hybrid metamaterial composed of metal cut-wires and U-shaped rings as well as graphene strips. Numerical investigations of the transmission spectra are systematically conducted. The transparency peak is induced via the interactions between the U-shaped ring and the cut-wire pairs under terahertz illumination. As a graphene layer is inserted under the metal patterns, the transmission of the metamaterial decreases drastically and the EIT effect is seriously weakened. When increasing the Fermi energy, the transparency window manifests an evident deformation with distinctive variations in the transmission peak and dips. The surface current distribution on the metamaterial is analyzed to investigate the tuning mechanism of the deformed window. Furthermore, the slow light effect accompanied with the EIT phenomenon can also be modulated with different Fermi energy.