We present theoretical and experimental studies of Schottky diodes that use aligned arrays of single-walled carbon nanotubes. A simple physical model, taking into account the basic physics of current rectification, can adequately describe the single-tube and array devices. We show that for as-grown array diodes, the rectification ratio, defined by the maximum-to-minimum-current-ratio, is low due to the presence of metallic-single-walled nanotube (SWNT) shunts. These tubes can be eliminated in a single voltage sweep resulting in a high rectification array device. Further analysis also shows that the channel resistance, and not the intrinsic nanotube diode properties, limits the rectification in devices with channel length up to 10 μm.