David P. Chernin | Searcharxiv

arXiv2026

Solution to an unsolved problem in diodes: Limiting current for small emission area and low emission energy

The maximum current that can be extracted from a diode is a central question in electronic devices, especially for the generation of radiation from microwaves to x-rays. The challenge in its prediction increases significantly when the electron emission is restricted to a small area for which the classical one-dimensional Child-Langmuir law is no longer applicable. We address this unsolved problem using a model that simultaneously includes a small emission area and a small electron emission velocity. New scaling laws are presented for this difficult regime. These results are obtained from three vastly different approaches: a differential equation formulation which provides extremely high resolution, an integral equation formulation which leads to the scaling laws, and a particle-in-cell simulation which shows the temporal-spatial evolution. Comparisons of the predicted maximum current among these three approaches are performed over a large range of parameters, paying special attention to the resolution of the potential minimum in the immediate vicinity of the cathode surface. Corroborations of the scaling laws with experiments on thermionic cathodes and on photoinjectors are indicated. The model consists of a periodic array of electron sheets of finite width in a planar diode, all emitted from a cathode with the same energy. Electron motion is restricted to the direction normal to the cathode.

physics.plasm-ph↗

arXiv2023

On the Child-Langmuir Law in One, Two, and Three Dimensions

We consider the limiting current from an emitting patch whose size is much smaller than the anode-cathode spacing. The limiting current is formulated in terms of an integral equation. It is solved iteratively, first to numerically recover the classical one-dimensional Child-Langmuir law, including Jaffe's extension to a constant, nonzero electron emission velocity. We extend to 2-dimensions in which electron emission is restricted to an infinitely long stripe with infinitesimally narrow stripe width, so that the emitted electrons form an electron sheet. We next extend to 3-dimensions in which electron emission is restricted to a square tile (or a circular patch) with an infinitesimally small tile size (or patch radius), so that the emitted electrons form a needle-like line charge. Surprisingly, for the electron needle problem, we only find the null solution for the total line charge current, regardless of the assumed initial electron velocity. For the electron sheet problem, we also find only the null solution for the total sheet current if the electron emission velocity is assumed to be zero, and the total maximum sheet current becomes a finite, nonzero value if the electron emission velocity is assumed to be nonzero. These seemingly paradoxical results are shown to be consistent with the earlier works of the Child-Langmuir law of higher dimensions. They are also consistent with, or perhaps even anticipated by, the more recent theories and simulations on thermionic cathodes that used realistic work function distributions to account for patchy, nonuniform electron emission. The mathematical subtleties are discussed.

physics.plasm-ph↗