Space charge limited field emission in a plane parallel electrode geometry

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Abstract
Poisson's equation in parallel geometry is solved analytically with the inclusion of the electron emission velocity spectrum predicted by quantum mechanics. The mathematical difficulty of classically defining the cathode surface is overcome, to a first approximation, by postulating a virtual cathode situated in front of the real cathode such that the majority of the emitted electrons have penetrated the potential barrier at that position. The treatment is nonrelativistic, assumes that the emission is exclusively by barrier penetration, and is expressed in parameters that enable the results to be applied to most situations of interest. It is shown that the effect of nonzero emission velocity is negligible except for the cases of cathodes of low work function operated at very high current densities. Particular attention is paid to the departure of the experimentally measured current from that predicted by the Fowler-Nordheim theory and the subsequent approach to the Langmuir-Child fully space charge limited current. It is shown that the point of departure can quantitatively be defined in terms of the fully limited current and the bearing of this on the design of field emission electron guns is considered.