A new double gate graphene nanoribbon field effect transistor with dual material for gate namely DMG-GNRFET is proposed. DMG-GNRFET includes a gate which is divided to the left and right side materials with different work functions. The left side metal (the source side) has lower work function than the right side one. This difference creates a step in potential profile which decreases band to band tunneling. Due to this step, the proposed structure has lower subthreshold swing, higher saturation current, lower leakage current and consequently higher current ratio than conventional structure. These advantages make DMG-GNRFET more suitable for digital applications. Transport properties of the proposed structure are compared to conventional GNRFET. The devices have been simulated based on self-consistent solution of Poisson and Schrodinger equations within non-equilibrium Green׳s function formalism. The effects of third nearest neighbor, dangling bands, and edge roughness have been included in the procedure to obtain more reliable simulations.