  In a current density versus temperature (J-T) (Miram) curve in thermionic
electron emission, experimental measurements demonstrate there is a smooth
transition between the exponential region and the saturated emission regions,
which is sometimes referred to as the "roll-off" or "Miram curve knee". The
shape of the Miram curve knee is an important figure of merit for thermionic
vacuum cathodes. Specifically, cathodes with a sharp Miram curve knee at low
temperature with a flat saturated emission current are typically preferred. Our
previous work on modeling nonuniform thermionic emission revealed that the
space charge effect and patch field effect are key pieces of physics which
impact the shape of the Miram curve knee. This work provides a more complete
understanding of the physical factors connecting these physical effects and
their relative impact on the shape of the knee, including the smoothness, the
temperature, and the flatness of the saturated emission current density. For
our analyses, we use a periodic, equal-width striped ("zebra crossing") work
function distribution as a model system and illustrate how the space charge and
patch field effects restrict the emission current density near the Miram curve
knee. The results indicate there are three main physical parameters which
significantly impact the shape of the Miram curve. Such physical knowledge
directly connects the patch size, work function values, anode-cathode voltage,
and anode-cathode gap distance to the shape of the Miram curve, providing new
understanding and a guide to the design of thermionic cathodes used as electron
sources in vacuum electronic devices (VEDs).
