A photodetector heterostructure based on AlInSb/InSb was grown by molecular-beam epitaxy. Mesostructures of different diameters were fabricated. The temperature dependence of the dark current was measured. It was demonstrated that the built-in barrier blocks the flow of majority charge carriers, thus reducing the dark-current density (relative to that for a pin structure based on InSb). The measured dependence of the dark current on the mesostructure size revealed that the bulk current component prevails over the surface one.