2D Child-Langmuir law for planar diode with finite-radius emitter

A planar diode with finite-radius emitter is studied with particle-in-cell simulations to extend the classical one-dimensional (1D) Child-Langmuir law to two dimensions. It is found that the 2D space-charge-limited current density in units of the classical 1D value follows a monotonically deceasing function of the dimensionless ratio of emitter radius (R) to gap separation (D), and the function can be extended to relativistic regime without significant modification. It is also verified that Lau's analytical prediction of the 2D Child-Langmuir law is in good agreement with the simulation results in comparison with Rokhlenko's.