Injection into insulators in the presence of space charge

Equations are derived that describe the effect of space charge on current injection into insulators under the approximations that, in the absence of space charge, lead to either the Fowler-Nordheim equation or the Schottky equation. The effect of the space charge is that the field in the insulator at the electrode interface is different from the field with no space charge, and that the triangular shape of the potential barrier is distorted. If only the alteration of the cathode field is taken into account, as is often the case, the effect of the space charge is overestimated. Explicit solutions are found for the case of constant space charge, where the effect of the shape distortion can be described in terms of a single parameter, which is different for the two injection mechanisms considered—zero temperature tunneling and thermionic emission over the potential barrier. For either mechanism, however, the parameter is a measure of the ratio of the charge on the electrode to the charge contained within a characteristic distance—the charge-free tunneling distance for tunneling injection or the position of the charge-free potential maximum for thermionic injection.