Space charge effects on externally injected current in planar diodes: Existence of multiple stationary states

We study the effects of space charge on the properties of a system with a specified steady current. The problem is solved exactly for a planar one dimensional geometry, but we expect similar results for more realistic systems. We find that in many cases, the stationary current density cannot exceed some fixed value J max and in other cases be lower than a fixed J min depending on the method of injection. There are also intervals of intermediate current densities, which cannot exist in the stationary conditions. In general, there are values of J in some range where there are two different stationary regimes, which correspond to two different electric fields at the emitter surface. Connection with results obtained from numerical simulations and possible applications are also considered.