A Tutorial on Calculating Space-Charge-Limited Current Density for General Geometries and Multiple Dimensions

Space-charge-limited current (SCLC) remains a critical issue across material phases (vacuum, gas, liquids, and solids) and timescales. The solution for one-dimensional (1D), planar SCLC density (SCLCD), referred to as the Child–Langmuir (CL) law, remains the common benchmark for these calculations; however, practical devices are neither 1D nor planar. This article reviews recent progress on developing approaches to derive SCLCD for nonplanar and multidimensional diodes. First, we summarize the application of variational calculus (VC) to derive 1D SCLCD based on minimizing the current in the gap. We also describe the nuances of applying Poisson’s equation to 1D nonplanar diodes compared to three-dimensional (3D) geometries. We next discuss the application of conformal mapping (CM) to extend 1D, planar SCLCD to more complicated 1D geometries, including a detailed justification of the application of CM for solving Poisson’s equation. We then discuss the application of vacuum capacitance and a generalized relationship between the electric potential in vacuum and space-charge-limited (SCL) diodes to derive exact analytic solutions for SCLCD in two-dimensional (2D) and 3D diodes with emission from the full cathode. We further relate the SCLCD for 3D planar square and disk electrodes across a wide range of electrode sizes. We conclude by discussing future theoretical and practical extensions of these approaches.