The present application teaches, among other innovations, power semiconductor devices in which breakdown initiation regions, on BOTH sides of a die, are located inside the emitter/collector regions, but laterally spaced away from insulated trenches which surround the emitter/collector regions. Preferably this is part of a symmetrically-bidirectional power device of the “B-TRAN” type. In one advantageous group of embodiments (but not all), the breakdown initiation regions are defined by dopant introduction through the bottom of trench portions which lie within the emitter/collector region. In one group of embodiments (but not all), these can advantageously be separated trench portions which are not continuous with the trench(es) surrounding the emitter/collector region(s).