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Analytical and numerical methods are used to solve Poisson's equation for carbon nanotube field-effect transistors (FETs) with a cylindrical surrounding gate and Schottky-barrier contacts to the source and drain. The effect on the nanotube potential profile of varying the work functions of all the electrodes, and the thickness and permittivity of the gate dielectric, is investigated. From these results, the general trends to be expected in the above-threshold drain current-voltage characteristics of Schottky-barrier nanotube FETs are predicted. The unusual possibility of simultaneous electron and hole contributions to the drain current is revealed. The subthreshold characteristics are computed from a solution to Laplace's equation, and the subthreshold slope is found to depend on gate dielectric thickness in a different manner from that in other FETs.