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The behavior of the leakage current in the semi‐insulating substrate of a GaAs device is more complicated than previously recognized. The voltage dependence of this current seen in a conventional voltage‐controlled experiment has hysteresis, which arises from an S‐type negative differential conductivity (S‐NDC). This is incompatible with the conventional trap‐fill‐limited model, and we propose an alternative explanation based on the impact ionization of deep‐level traps. We show how this simple model can account qualitatively for the S‐NDC and the associated current instability, and how it can be extended when the deep traps are photoexcitable.