Calculating the Space-Charge-Limited Current using the Vacuum Electric Field

The space-charge limited current (SCLC) represents the maximum steady state current for a diode¹. Extending SCLC theories to realistic systems requires particle-in-cell simulations², which can be computationally expensive. Recent studies derived the SCLC as a function of the cathode electric field or the diode capacitance without any particles². In this work, we use COMSOL to calculate the diode capacitance and the cathode electric field in vacuum to determine the SCLC of realistic systems. We recover prior 1D², 2D¹, and 3D¹ analytic results for planar geometries, concentric cylinders, and emission from a single tip. Using the cathode electric field permits us to examine complicated geometries, such as arrays of tips, and determine both local and global SCLC. Changes in the SCLC for different tip arrangements and determination of the canonical gap distance to apply to other emission mechanisms² will be discussed.

[1] N. R. S. Harsha, M. Pearlman, J. Browning, A. L. Garner, “A multi-dimensional Child–Langmuir law for any diode geometry,” Phys. Plasmas 28, 122103 (2021).

[2] A. G. Sinelli, L. I. Breen, N. R. Sree Harsha, A. M. Darr, A. M. Komrska, and A. L. Garner, IEEE Trans. Electron Devices 72, 397-403 (2025).