Diode Physics: From Child-Langmuir to Paschen’s Law

Abstract

While extensive studies have characterized diode physics since the seminal derivation of the Child-Langmuir law (CLL) for space-charge limited current (SCLC) over a century ago, much of this work has focused on planar diodes at vacuum 1 . However, practical devices, even in vacuum electronics, rarely operate under this narrow range of conditions. This paper extends this traditional view of diode physics by performing matched asymptotic analyses to directly link electron emission and gas breakdown mechanisms for planar geometries, or “nexus theory 2 ,” and extending the CLL to include non-planar geometries 3 . Nexus theory permits simultaneously comparing multiple emission mechanisms, including quantum SCLC, thermionic emission, field emission, vacuum SCLC, SCLC with collisions, the coupled field emission/Townsend avalanche (FE/TA) regime, and Paschen’s law (PL) 2 . Nexus theory elucidates the suitability of applying a simple emission theory or the necessity of linking multiple mechanisms. We also applied variational calculus and conformal mapping to derive closed form solutions for SCLC for concentric cylinders and spheres and multi-dimensional diodes. Incorporating the appropriate electric field profile permits linking field emission to PL for non-planar geometries. Finally, we summarize ongoing extensions of these efforts, including incorporation of an orthogonal magnetic field in SCLC calculations, incorporation of an RF field in the FE/TA regime, improved characterization of the ionization coefficient at the high reduced electric fields typical for microscale and smaller gaps, and experimental assessments of emission for sub-microscale diodes at atmospheric pressure.