Ovonic threshold switching (OTS) in chalcogenide materials has attracted considerable interest for application in electronic devices to suppress leakage current in cross-point array structures. Although OTS appears to originate from an electronic process, the exact mechanism of OTS remains unclear with respect to sub-threshold conduction and threshold switching. In this study, we demonstrated that the sub-threshold conduction characteristics is affected by the structures of the exponentially distributed trap states in amorphous chalcogenide materials. Using binary AsxTe1−x thin films, we investigated the effects of trap distribution on OTS characteristics, analyzed the band structures, and developed a trap-controlled space-charge-limited conduction model.