Enhanced space charge limited current for curved electron emitters

TL;DR

This paper analytically and numerically demonstrates that curved electron emitters can achieve higher space charge limited currents than flat ones, with the current scaling influenced by the emitter's apex field enhancement.

Contribution

It introduces a nonlinear line charge model showing increased limiting current for curved emitters, extending the Child-Langmuir law to include shape effects.

Findings

Curved emitters have higher SCL current than planar emitters.

The limiting current scales with the apex field enhancement factor.

Numerical simulations confirm the analytical predictions.

Abstract

The maximum current that can be transported across a vacuum diode is limited by forces arising due to space charge. In a planar diode configuration, the space charge limited (SCL) current density from a planar emitting patch is given by the Child-Langmuir (CL) law $J_{CL} \sim V_g^{3/2}/D^2$ where $V_g$ is the potential difference across the diode and $D$ is the separation between the anode and cathode. We show here analytically using the nonlinear line charge model that for a curved emitter in a planar diode configuration, the limiting current obeys the scaling relationship $J_{SCL} \sim \gamma_a V_g^{3/2}/D^2$ where $\gamma_a$ is the apex field enhancement factor of the curved emitter. For an emitter with large height ($h$) to apex radius of curvature ($R_a$) ratio, the limiting current far exceeds the planar value. The result is verified using the particle-in-cell code PASUPAT for two curved emitters shapes.