Algorithm to calculate the field and temperature dependence of average electron injection velocities from non-planar metallic cathodes

An algorithm is presented to first derive the average kinetic energy of the injected electrons during emission from flat metallic cathodes as a function of temperature and external electrostatic field. The average injection velocity is then calculated at a de Broglie wavelength beyond the classical turning point of the potential energy barrier in front of the cathode. This approach is then used to obtain the average injection velocity distribution across all emission regimes, including thermionic emission, thermal-field emission, and pure field emission for the case of non-planar metallic cathodes. An analytical expression for the average kinetic energy on injected electrons is proposed which is in good agreement with numerical simulations with a 3% accuracy over a wide range of temperature and external electrostatic field. We illustrate the strong spatial and electric field dependence of the average electron injection velocity distribution for the case of a nanoscale vacuum diode with a spheroidal tungsten cathode facing a planar anode.