Calculating the Volume-Averaged Space-Charge-Limited Current Density by Simulating Vacuum Capacitance

While analytic results for the space-charge-limited current density (SCLCD) exist for relatively simple geometries, more realistic geometries often require computational approaches, such as particle-in-cell (PIC) simulations. We applied a recent theory that relates the volume-averaged SCLCD (VASCLCD) to the vacuum capacitance to calculate the VASCLCD for 1-D and multidimensional geometries by performing electrostatic simulations using the commercial code COMSOL Multiphysics without requiring particles. We recovered the Child–Langmuir law (CLL) for a 1-D planar diode by choosing the computational domain to eliminate fringing field contributions. We also obtained the VASCLCD for a 1-D cylindrical geometry in agreement with the analytic solution. For 2-D geometries, the VASCLCD using the capacitance from COMSOL agreed well with the analytic results for computational domains sufficiently large to account for fringing fields. For 3-D geometries, the VASCLCD obtained analytically without corner fringing fields (CFFs) agreed well with COMSOL results without CFFs. COMSOL calculations incorporating CFFs disagreed strongly with theory for 3-D geometries where the electrode width$W$and length$L$were both smaller than the gap distance$D$. The difference between the VASCLCD obtained with or without CFFs diminishes for${L} / {D} \gt {1}$and/or${W} / {D} \gt {1}$, which mitigates the contributions of CFFs since the geometry becomes effectively 2-D. These results demonstrate the feasibility of using electrostatic simulations to rapidly approximate the SCLCD without requiring particles or time-domain simulations.