Charge injection instability

We show that charge injection at a field-enhancing defect in a dielectric free of carriers but with a nonlinear carrier mobility corresponds to an instability of the insulating state. The formation of the charged state is governed by two different processes with clearly separated time scales. First, due to a fast growth of a charge-injection mode, a localized charge cloud forms near the injecting defect. Secondly, the charge slowly redistributes in the bulk. In the present paper we discuss the linear instability of the insulating state for cylindrical and spherical geometries. The theory explains an experimentally observed increase of the critical electric field with decreasing size of the injecting contact. Numerical results are presented for dc and ac biased insulators.