Space-Charge Limited Current: Extensions to Non-planar, Multidimensional, and Crossed-field diodes*

Space-charge limited current (SCLC), first derived over a century ago by Child and Langmuir for one-dimensional (1-D) planar diodes, represents the maximum current that can flow in a diode and remains a critical limit for vacuum device operation [1]. We have used variational calculus (VC) [2] and conformal mapping [3] to derive analytic solutions for SCLC for more complicated geometries. We extend this approach to multidimensional non-planar diodes by deriving a general relationship dependent on only diode vacuum electric potential and capacitance. We apply this approach to extend SCLC for 1-D planar diodes with an external magnetic field perpendicular to the electric field [4] to non-planar and multidimensional diodes. This approach permits rapid screening of SCLC for more realistic diodes to guide further simulations and experimental studies.

[1] P. Zhang et al., J. Appl. Phys. 129, 100902 (2021).

[2] A. M. Darr and A. L. Garner, Appl. Phys. Lett. 114, 014103 (2019).

[3] N. R. Sree Harsha and A. L. Garner, IEEE Trans. Electron Devices 68, 264 (2021).

[4] Y. Y. Lau, P. J. Christenson, and D. Chernin, Phys. Fluids, 5, 4486 (1993).