2According to the band structure of graphene, the density of state (DOS) diverges at the M points in the first Brillouin zone (BZ). These points are known as van Hove Singularities (VHSs) 1 .When the Fermi energy EF is approaching to the VHSs, the DOS divergence may lead to system instabilities and phase transitions 2, 3 . Low-energy states such as charge density waves (CDW) may occur. The VHS features in Ca-intercalated few-layer graphene 4, 5 , artificially twisted bilayer graphene 2 and high-quality single layer graphene under a high magnetic field 6 have been studied theoretically and/or experimentally. However, the possibility of CDW phase transitions that involve VHSs in electrostatically doped graphene has been a subject of considerable debate since the discovery of graphene because extremely large doping concentrations are difficult to achieve to tune the Fermi level of graphene to VHSs.To effectively tune the Fermi level of graphene, we fabricate multilayer graphene field-effect double-layer transistors (EDLTs) biased through ionic liquid gating. A high density of charge carriers are effectively introduced into graphene via electric double layers formed at the interface between the ionic liquid and the topmost graphene surface 7 . Once the Fermi level of graphene is tuned to the VHSs by the ionic liquid gating, a sudden increase of the graphene channel resistance is repeatedly detected at about 100K, similar to the features of CDW phase transitions 8 . Evidently, the splitting of the Raman G peak further demonstrates the lattice reconstructions, which is associated with the CDW phase transitions when the sample temperature is below the critical point (Tm=100K) while the Fermi energy is close to VHSs 8 , 9 .
Graphene electric double-layer transistors (EDLTs)EDLT devices are chosen to achieve the CDW phase transition through effective tuning of the Fermi energy of graphene to the VHSs electrostatically (Supplementary Materials). Hall devices 3 with side gate electrodes are fabricated using standard electron beam lithography (see the inset in Fig.1(b)). A droplet of ionic liquid (DEME-TFSI) is applied onto the surface of graphene channel and the side gate electrodes (Section 1, Supplementary Materials). Under a positive (or negative) bias voltage at room temperature, cations (or anions) from the ionic liquid are derived onto the surface of graphene channel ( Fig.1(a)). The inducing carrier efficiency (estimated based on the equivalent capacitance) is ~5.23 10 13 cm -2 V -1 by evaluating the Hall effect of the devices ( Fig. 1(c)). The gating efficiency of the ionic liquid is approximately two orders of magnitude higher than that of widely used 300 nm-thick SiO2 (~10 11 cm -2 V -1 ). Importantly, the carrier density linearly depends on the gate voltage ( Fig.1(c)) and does not depend on the thickness of graphene samples. Therefore, the gate voltage dependence of carrier density can be estimated expediently and accurately. denotes the dielectric constant 13 . For quantitative estimation, we take DOS=0...