Consistent solution for space-charge-limited current in the relativistic regime for monoenergetic initial velocities

Abstract

The solution for space-charge-limited (SCL) currents in electron vacuum diodes with monoenergetic initial velocity is extended to the relativistic regime. Two types of solutions are found: Type I corresponds to zero steady state surface electric field (field emission with high enhancement factor), and Type II corresponds to a finite steady state surface electric field (other emission mechanisms). Our solution compares well to the classical space-charge-limited currents with initial energy and relativistic space-charge limited currents without initial energy in the appropriate limits. The scaling law between the true SCL and the applied voltage is discussed and the two solution types are verified by particle-in-cell simulation.