Simulations of the Pressure Dependence of the Space-Charge Limited Electric Potential

Several studies have assessed the implications of collisions on the space-charge limited current (SCLC) [1,2], which is the maximum steady-state current in a diode [3]. For no collisions, the SCLC is given by the Child-Langmuir law (CLL); for many collisions, it is given by the Mott-Gurney law (MGL) [3]. For the CLL, the electric potential Φ varies with position x as Φ ∝ x^4/3; for the MGL, Φ ∝ x^3/2 [2]. In between, Φ ∝ x^ξ with 4/3 ≤ ξ ≤ 3/2 [2]. Here, we use the particle-in-cell code XPDP1 to assess the spatial dependence of Φ at the SCLC to determine ξ for various gap distances and pressures by using the relationship between the current density and the electric field at the anode. For sufficiently small gap size and low pressure, ξ ≈ 4/3, as expected for vacuum. With increasing gap size and pressure, more collisions occur, and ξ increases. Incorporating these results into the closed form equation for general collisional SCLC [2] will be discussed.

[1] L. I. Breen and A. L. Garner, Phys. Plasmas 31, 032102 (2024).

[2] A. L. Garner and N. R. S. Harsha, IEEE Trans. on Electron Devices 72, 1419-1425 (2025).

[3] P. Zhang, Y. S. Ang, A. L. Garner, Á. Valfells, J. W. Luginsland, and L. K. Ang, J. Appl. Phys. 129, 100902 (2021).

This work was supported by SCALE: U.S. DoD under Contract W52P1J-22-9-3009