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A vertical partial-gate carbon-nanotube (CNT) field-effect transistor (FET), which is amenable to the vertical CNT growth process and offers the potential for a parallel CNT array channel, is simulated using a self-consistent atomistic approach. We show that the underlap between the gate and the bottom electrode (required for isolation between electrodes) is advantageous for transistor operation because it suppresses ambipolar conduction. A vertical CNTFET with a gate length that covers only 1/6 of the channel length has a much smaller minimum leakage current than the one without underlap while maintaining comparable on current. Both n-type and p-type transistor operations with balanced performance metrics can be achieved on a single partial-gate FET by using proper bias schemes. Even with a gate underlap, it is demonstrated that increasing the CNT diameter still leads to a simultaneous increase of on current and minimum leakage current. Along with a partial gate, the simulated transistor features a significant amount of air between the surface of the channel CNT and the gate insulator, as is caused by the vertical CNT growth process. Filling this pore with a high-kappa insulator is shown to have the potential to decrease the on current due to electrostatic phenomena at the source-channel contact.