  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.
