Low work-function cathodes from Schottky to field-induced ballistic electron emission: Self-consistent numerical approach

A systematic study was done in order to relate the current density J-applied field F characteristic variation with three emission mechanisms: thermionic, tunneling, and ballistic. All three are now effective during the field emission from cathodes with work function Φ less than 2 eV. The current density is computed using the transmission probability for an electron to cross the barrier between the electron sea of the cathode and the vacuum. The corresponding Schrödinger equation is solved by means of the self-consistent Lippmann-Schwinger equation, with values of the effective potential corrected with the image potential between the cathode and the anode, and resolved by spatial discretization. This method allows computing the exact current within a zero emitted current approximation. It fills the gap left by the former analytical and numerical resolutions. The ln(J∕F²) vs (1∕F) plot shows, from the very beginning of the electron emission, three zones for the current variation. The first zone, corresponding to low applied electric fields, is a nonlinear variation of the current specific to thermionic-field emission; it is followed by a second zone having a linear variation with a slope proportional to Φ^3∕2 characteristic of the conventional Fowler-Nordheim field emission. The third zone, concerning high field values, indicated a current saturation behavior related to a field-induced ballistic emission.