Methods of forming a transistor include providing a semiconductor epitaxial structure including a channel layer and barrier layer on the channel layer, forming a gate electrode on the barrier layer, etching the semiconductor epitaxial structure using the gate electrode as an etch mask to form a trench in the semiconductor epitaxial structure, and depositing a source metal in the trench. The trench extends at least to the channel layer, and the source metal forms a Schottky junction with the channel layer. Related semiconductor device structures are also disclosed.