Multi-Dimensional Space-Charge Limited Current Using Variational Calculus and Vacuum Capacitance

Abstract

Space-charge limited current (SCLC) is the maximum current that can flow in the steady-state operation of a diode. Analytical formulations of SCLC are extremely important for various practical applications, including nano vacuum transistors, electric thrusters, multipactor, and time-resolved electron microscopy 1 . The analytical solution for a 1-D planar diode was initially derived by Child and Langmuir over a century ago 1 , 2 . Lau extended this result to derive the SCLC density from a 2-D patch of width W on an infinite electrode for a planar diode when W/D >> 1, where D is the plate separation 3 . In this presentation, we apply variational calculus (VC) 4 and conformal mapping 5 , which have recently been used to derive SCLC in 1-D diodes with non-planar geometries, to derive a mathematical relationship between space-charge limited potential and vacuum electric potential for a diode of any geometry. We further extend this derivation to obtain a mathematical relationship between SCLC and vacuum capacitance, which we use to derive exact analytical solutions for 2-D SCLC for a practical geometry with the emission area on the same order of magnitude as the emitter dimensions and recover Lau’s result when W / D >>1. We also derive 3-D SCLC density for a rectangular parallel plate and disk electrodes. Implications of this approach for crossed-field diodes and emitter arrays, will be discussed.