Applying Conformal Mapping to Derive Analytical Solutions of Space-Charge-Limited Current Density for Various Geometries

While exact analytic solutions for space-charge-limited currents (SCLCs) are well-established for parallel plate geometries, they have only recently been derived for concentric cylindrical and spherical geometries using variational calculus (VC). However, actual diode systems and slow-wave structures are usually more complicated, making SCLC calculations more difficult. In this article, we apply conformal mapping to derive the analytical solutions for SCLC for various complicated geometries exhibiting curvilinear flow. We first replicate the exact solution of SCLC for concentric cylindrical electrodes from VC using conformal mapping to transform from the Child-Langmuir (CL) law for a planar geometry. We then derive SCLC in other geometries using conformal transformations to either the planar or the concentric cylinder solution. Because the SCLC calculated using such conformal mappings depends only on the CL law, this may permit future incorporation of relativistic or quantum corrections to determine the appropriate relationships for more complicated geometries.