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In a current density versus temperature (   $\textit{J}$   &#x2013;   $\textit{T}\text{)}$    (Miram) curve in thermionic electron emission, experimental measurements often yield a smooth transition between the exponential region and the saturated emission regions, which is sometimes referred to as the &#x201C;roll-off&#x201D; or &#x201C;Miram curve knee.&#x201D; 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 the patch field effect are key pieces of physics, which set the Miram curve shape. 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. For our analyses, we use a periodic, equal-width striped (&#x201C;zebra crossing&#x201D;) 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. Our results indicate that there are three main physical parameters that significantly impact the shape of the Miram curve: 1) the normalized work function difference; 2) the patch-diode size ratio; and 3) the patch-diode field ratio. These parameters relate the patch size, work function values, anode&#x2013;cathode voltage, and anode&#x2013;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.

Physical Factors Governing the Shape of the Miram Curve Knee in Thermionic Emission