Current filaments formed in the parameter regime of the low‐temperature impurity impact ionization breakdown ofp‐germanium were investigated by measuring tunneling and point contact characteristics of a scanning tunneling microscope tip fixed at certain sample positions under variation of the sample voltage. The results clearly indicate that the tip current is determined by local potential variations which can be interpreted by a model for the filament taking into account the finite electric resistivity of the sample contacts. ©1996 American Institute of Physics.