Structures for a high-electron-mobility transistor and methods of forming a structure for a high-electron-mobility transistor. The high-electron-mobility transistor has a first semiconductor layer, a second semiconductor layer adjoining the first semiconductor layer along an interface, a gate electrode, and a source/drain region. An insulator region is provided in the first semiconductor layer and the second semiconductor layer. The insulator region extends through the interface at a location laterally between the gate electrode and the source/drain region.