Recent advances in the Child-Langmuir law in one, two, and three dimensions

Non-uniform electron emission from a cathode surface is notoriously difficult to characterize since it is highly dependent on the emission mechanisms and cathode properties including cathode age, composition, roughness, and temperature, etc. None of these operating conditions is included in the classical one-dimensional (1D) Child-Langmuir law (CLL), which gives the maximum current density only in terms of the diode voltage, V, and gap spacing, D. This paper summarizes recent findings from simulation and analytic studies of the anode current versus temperature characteristic (also known as Miram or “rollover” curve) on a thermionic cathode, based on realistic work function distributions [1,2], and extensions of CLL to 2D and 3D [3,4]. Nonuniformly emitting patches on the cathode, with different work functions, do not emit independently. The edge effects are very important, especially when the emitting patch size is small compared with D. With significant non-emitting regions, such as 80 percent of the cathode surface, the average anode current density is still governed by the 1D CLL as if the entire cathode were emitting, at a sufficiently high cathode temperature, T [1]. The edge effects of the heavily emitting regions, which may constitute less than 20 percent of the cathode area, can make up a large fraction of anode current as their neighboring non-emitting regions do not provide space charge shielding. Transverse electron motion is relatively unimportant to the shape and smoothness of the Miram curves. Smaller emission (grain) size is correlated with more abrupt transition to CLL as T increases. The effects of local emission hot spots are examined [2]. The 1D Child-Langmuir current density appears to be the hard limit for the average current density over the entire cathode [4], regardless of the emission nonuniformity in space and in time [5].

 

[1] D. Chernin, et al., IEEE Trans. Plasma Sci. 48, 146 (2020).

[2] A. Jassem, et al., IEEE Trans. Plasma Sci. 49, 749 (2021).

[3] Y. Y. Lau, et al., Phys. Plasmas 30, 093104 (2023).

[4] D. Chernin, et al., Phys. Plasmas 31, 022103 (2024).

[5] P. Zhang, et al., Appl. Phys. Rev. 4, 011304 (2017).

 

The contributions from David Chernin, Dion Li, and Abhijit Jassem are gratefully acknowledged. This work was supported by AFOSR Grants Nos. FA9550-20-1-0409 and FA9550-21-1-0184.