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The scaling behaviors of graphene nanoribbon (GNR) Schottky barrier field-effect transistors (SBFETs) are studied by self-consistently solving the nonequilibrium Green's function transport equation in an atomistic basis set with a 3-D Poisson equation. The armchair edge GNR channel shares similarities with a zigzag carbon nanotube; however, it has a different geometry and quantum confinement boundary condition in the transverse direction. The results indicate that the I-V characteristics are ambipolar and strongly depend on the GNR width because the bandgap of the GNR is approximately inversely proportional to its width, which agrees with recent experiments. A multiple gate geometry improves immunity to short channel effects; however, it offers smaller improvement than it does for Si MOSFETs in terms of the on-current and transconductance. Reducing the oxide thickness is more useful for improving transistor performance than using a high-k gate insulator. Significant increase of the minimal leakage current is observed when the channel length is scaled below 10 nm because the small effective mass facilitates strong source-drain tunneling. The GNRFET, therefore, does not promise to extend the ultimate scaling limit of Si MOSFETs. The intrinsic switching speed of a GNR SBFET, however, is several times faster than that of Si MOSFETs, which could lead to promising high-speed electronics applications, where the large leakage of GNR SBFETs is of less concern.