Empirically extending 1D Child-Langmuir theory to a finite temperature beam

TL;DR

This paper develops an empirical formula based on numerical solutions to extend the classical Child-Langmuir law to finite temperature beams, enabling quick estimates of transmitted current in vacuum diodes with realistic particle velocity spreads.

Contribution

The paper introduces a new empirical formula derived from PIC simulations that generalizes the Child-Langmuir law to finite temperature beams, filling a gap in analytical solutions.

Findings

The empirical formula accurately characterizes space charge effects for finite temperature beams.

It provides a practical tool for estimating current in diode devices with non-monoenergetic particles.

The formula is validated against particle-in-cell simulation results.

Abstract

Numerical solutions to the 1D steady-state Vlasov-Poisson system are used to develop a straightforward empirical formula for the electric current density transmitted through a vacuum diode (voltage gap) as a function of gap distance, gap voltage, the injected current density, and the average velocity and temperature of injected particles, as well as their charge and mass. This formula generalizes the 1D cold beam Child-Langmuir law (which predicts the maximum transmitted current for mono-energetic particles in a planar diode as a function of gap voltage and distance) to the case where particles are injected with a finite velocity spread. Though this case is of practical importance, no analytical solution is known. Found by a best-fit to results from particle-in-cell (PIC) simulations, the empirical formula characterizes the current transmitted across the diode for an injected velocity distribution of a drifting Maxwellian. It is not meant to yield a precise answer, but approximately characterizes the effect of space charge on transmitted current density over a large input space. The formula allows quick quantitative estimation of the effect of space charge in diode-like devices, such as gate-anode gaps in nanoscale vacuum channel transistors.