Simplified Theory of Space-Charge-Limited Currents in an Insulator with Traps

An ohmic contact between a metal and an insulator facilitates the injection of electrons into the insulator. Subsequent flow of the electrons is space-charge limited. In real insulators the trapping of electrons in localized states in the forbidden gap profoundly influences the current flow. The interesting features of the current density-voltage ($J-V$) characteristic are confined within a "triangle" in the $logJ-logV$ plane bounded by three limiting curves: Ohm's law, Child's law for solids ($J\propto V^2$) and a traps-filled-limit curve which has a voltage threshold and an enormously steep current rise. Simple inequalities relating the true field at the anode to the ohmic field facilitate qualitative discussion of the $J-V$ characteristic. Exact solutions have been obtained for an insulator with a single, discrete trap level in a simplified theory which idealizes the ohmic contact and neglects the diffusive contribution to the current. The discrete trap level produces the same type of nonlinearity discovered by Smith and Rose and attributed by them to traps distributed in energy.