A general formulation of multi-dimensional space-charge-limited current density with nonzero monoenergetic initial velocity ^*

Abstract Thermionic energy converters (TECs) offer a direct method for converting heat into electricity, providing potential applications in waste heat recovery and renewable energy generation. While extensive research has focused on developing low work function materials, the performance of vacuum-based converters remains fundamentally hindered by space-charge effects, necessitating improved theoretical characterization of the space-charge limited current (SCLC). In this study, we derive exact theories for SCLC for multidimensional diodes with nonzero initial velocity (approximating high cathode temperatures) to demonstrate how to design TECs with higher steady-state currents to enhance the conversion of heat to electricity. We show that injected currents exceeding the SCLC limit result in a sharp decrease in the collector current, defining the bifurcation condition. Simulations using the two-dimensional particle-in-cell codes XOOPIC and VSim for both the SCLC and bifurcation regimes yield SCLC within 7% of the theoretical predictions. Finally, the sharp knee characteristic of the transition from one-dimensional thermionic emission to SCLC, known as the Miram curve, is extended to multidimensional diodes characterized by a single sharpness parameter σ , and its implications are discussed.