Current density scaling expressions for a bipolar space-charge-limited cylindrical diode

Exact analytical solutions for the currents in a bipolar space-charge-limited (SCL) diode in cylindrical geometry do not exist. Expressions for the electron and ion current densities as functions of the diode cathode and anode radii and the applied voltage are developed and fit to numerical solutions [1]. The results are provided for nonrelativistic and relativistic electrons. These provide scaling relations for the diode, which were not previously available. They also converge to known planar diode expressions in the appropriate limit. The relativistic scaling relations are then combined with nonrelativistic results using a fit function technique to provide a general expression for the current densities valid for all values of the diode voltage. The results are presented for both negative and positive polarities. For completeness, single-species SCL flow results in cylindrical geometry are also developed using the same techniques presented here for bipolar SCL flow and are also included.