Effect of space charge on vacuum pre-breakdown voltage and electron emission current

The recent developments in nanosecond pulsed power supplies facilitate the emission of high density electron bursts but their safe operation demands avoiding breakdowns. Using the theoretical and numerical modeling of the electron emission phenomena from a tip (micro-protrusion), the breakdown threshold (pre-breakdown) is analyzed considering it as the highest value of the voltage preserving the system out of the thermo-emission instability regime. However, the space charge that builds up in front of the tip limits the performance of these electron sources by decreasing the local electric field and consequently the thermo-field emission as well as the temperature of the emissive surface. Hence, it is found that the system can safely hold higher voltages (without breakdown) in the presence of dense space charge. In direct current, for a titanium elliptic tip, the highest operation voltage increases by about 15%, whereas for a tungsten hyperbolic tip, it increases by 70%. Remarkably, the emitted current close to the pre-breakdown voltage stays unchanged with or without taking into account the space charge. Surprisingly, when very short pulses (3ns) are applied to a tungsten hyperbolic tip, the pre-breakdown voltage additionally increases by 30%, and the Coulomb screening, very effective in front of the tip apex, enlarges the electron emission area by 60%, releasing about 1.3 times more electrons compared to vacuum emission (without the space charge). Moreover, the ring effect, experimentally discovered by Dyke and Trolan [Phys. Rev. 89, 799 (1953)] on the radial electron density distribution, can be microscopically observed and understood with your model.