Solution for space charge limited field emission current densities with injection velocity and geometric effects corrections

When particles are injected according to the Fowler–Nordheim (FN) field emission equation, the transmitted current density will transition to the space charge limited (SCL) current density, with increasing applied diode voltage. The actual transmitted current density is the so-called SCL-FN current density. In this work, Barbour’s analytic solution for the SCL-FN current density is modified with consideration of injection velocity and also geometric effects, by solving the advanced FN equation with the effective field enhancement factor, the energy conservation equation with an initial velocity term, and Poisson’s equation simultaneously. The solution is also extended to the relativistic regime where similar transition process is found. This solution has been verified using particle-in-cell simulation with varying diode voltage, electron injection velocity, and field enhancement factor.