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We performed a comprehensive scaling study of Schottky-barrier (SB) carbon nanotube transistors using self-consistent, atomistic scale simulations. We restrict our attention to SB carbon nanotube field-effect transistors (FETs) whose metal source-drain is attached to an intrinsic carbon nanotube channel. Ambipolar conduction is found to be an important factor that must be carefully considered in device design, especially when the gate oxide is thin. The channel length scaling limit imposed by source-drain tunneling is found to be between 5 nm and 10 nm, depending on the off-current specification. Using a large diameter tube increases the on-current, but it also increases the leakage current. Our study of gate dielectric scaling shows that the charge on the nanotube can play an important role above threshold.