The ambipolar conduction which is one of the serious obstacles in the development of the graphene nanoribbon field-effect transistors (GNR-FETs) has been successfully controlled by the electrostatic modulation of the energy bands in this work. With targeted redistribution of the doping profile on the graphene surface inside the channel region near the source and drain and also extensions of them, symmetrical step-shaped potential energy band on the graphene surface is created to increase the effective potential barrier and band-to-band tunneling width, considerably. The most important result is the remarkable reduction in ambipolar current in the proposed structure. Also, the thermionic mechanism is amplified in our structure to reach a higher driving current. The ballistic transport has been assumed as conduction mechanism to quantify the current flow of the structures in this the article. For that reason, the Schrodinger equation is solved by the method of nonequilibrium Green function (NEGF) in a self-consistent manner with coupled Poisson electrostatic equation to extract the induced carrier density on the graphene surface. To accurately study the carrier transport on the graphene surface, edge bond relaxation has also been considered in the simulations. Not only the ambipolar conduction has been well-controlled but also the key parameters in terms of OFF and ON currents, ratio of ${I}_{ \mathrm{\scriptscriptstyle ON}}$ to ${I}_{ \mathrm{\scriptscriptstyle OFF}}$ , output conductance, transconductance, voltage gain, gate capacitance, and cutoff frequency have been improved in the proposed structure which is very desirable and proper in the digital and analog circuits.