Experimental observations have long-established that there exists a smooth roll-off or knee transition between the temperature-limited (TL) and full-space-charge-limited (FSCL) emission regions of the emission current density-temperature J-T (Miram) curve, or the emission current density-voltage J-V curve for a thermionic emission cathode. In this paper, we demonstrate that this experimentally observed smooth transition does not require frequently used a priori assumptions of a continuous distribution of work functions on the cathode surface. Instead, we find the smooth transition arises as a natural consequence of the physics of nonuniform thermionic emission from a spatially heterogeneous cathode surface. We obtain this smooth transition for both J-T and J-V curves using a predictive nonuniform thermionic emission model that includes 3-D space charge, patch fields (electrostatic potential nonuniformity on the cathode surface based on local work function values), and Schottky barrier lowering physics and illustrate that a smooth knee can arise from a thermionic cathode surface with as few as two discrete work function values. Importantly, we find that the inclusion of patch field effects is crucial for obtaining accurate J-T and J-V curves, and the further inclusion of Schottky barrier lowering is needed for accurate J-V curves. This finding, and the emission model provided in this paper have important implications for modeling electron emission from realistic, heterogeneous surfaces. Such modeling is important for improved understanding of the interplay of emission physics, cathode materials engineering, and design of numerous devices employing electron emission cathodes.
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