Influence of average electron injection velocity on space-charge-limited current in vacuum devices

Space-charge-limited current is a fundamental constraint in electron emission systems, traditionally described by the Child–Langmuir law for one-dimensional parallel plate geometry assuming zero electron injection velocity from the cathode. First, a simplified one-dimensional model of space charge is introduced in which emitted electrons are treated as a collinear sequence of point charges with a fixed average electron injection velocity, each contributing to a reduction in the emission barrier. A semiclassical approach is then developed to self-consistently solve the charge transport and Poisson equations for three canonical geometries (parallel plates, concentric cylinders, and concentric spheres), based on a modification of a recent fully quantum-mechanical algorithm developed by Hernandez et al. [J. Appl. Phys. 139, 174503 (2026)] to calculate the average electron injection velocity for non-planar metallic cathodes. The approach is used to calculate the field emission characteristics and Miram curves of all-metallic vacuum diodes with three different geometries using a physically accurate average injection velocity distribution and compared to those obtained using a conventional fixed average injection velocity to stress the importance of the former. For the case of a vacuum diode with a barium-on-tungsten cathode with parallel plate configuration, our simulations are in good agreement with the field emission measurements of Barbour et al. [Phys. Rev. 92, 45 (1953)], stressing the importance of accurately modeling injection velocity distributions in field emission theory.