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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).

Experimental observations of thermionic electron emission demonstrate a smooth transition between TL and FSCL regions of the emitted-current-density-versus-temperature (J-T) (Miram) curve and the emitted-current-density-versus-voltage (J-V) curve. Knowledge of the temperature and shape of the TL-FSCL transition is important in evaluating the thermionic electron emission performance of cathodes, including predicting the lifetime. However, there have been no first-principles physics-based models that can predict the smooth TL-FSCL transition region for real thermionic cathodes without applying physically difficult to justify a priori assumptions or empirical phenomenological equations. Previous work detailing the nonuniform thermionic emission found that the effects of 3-D space charge, patch fields, and Schottky barrier lowering can lead to a smooth TL-FSCL transition region from a model thermionic cathode surface with a checkerboard spatial distribution of work function values. In this work, we construct a physics-based nonuniform emission model for commercial dispenser cathodes for the first time. This emission model is obtained by incorporating the cathode surface grain orientation via electron backscatter diffraction (EBSD) and the facet-orientation-specific work function values from density functional theory (DFT) calculations. The model enables construction of two-dimensional emitted current density maps of the cathode surface and corresponding J-T and J-V curves. The predicted emission curves show excellent agreement with experiment, not only in TL and FSCL regions but, crucially, also in the TL-FSCL transition region. This model improves the understanding on the relationship between thermionic emission and cathode microstructure, which is beneficial to the design of vacuum electronic devices.

Analysis of temperature-limited flow, space-charge-limited flow, and the transition between them using a simple planar diode with a thermionic cathode, in which the cathode surface has spatially nonuniform emission properties, is presented. Our theoretical results, which are derived from a model based on solutions to the Vlasov and Poisson equations, compare well with the results of particle-in-cell simulations. We find that the location and the shape of the knee in the anode current versus temperature characteristic (Miram or “rollover” curve) are significantly affected by non-uniformities in the space-charge density in the A–K gap, but are relatively unaffected by the electron motion parallel to the electrode surfaces. In particular, emission from an actively emitting region is strongly affected by the forces (or lack thereof) exerted by the space-charge of the electrons emitted by their neighbors. Perhaps, most remarkably, we find that the limiting current reaching the anode is approximately given by the classical 1-D Child–Langmuir law, even if a significant fraction of the cathode surface is non-emitting.

Previously, we constructed a first-principles statistical model to predict the non-uniform emission from polycrystalline tungsten cathodes, which incorporated microstructure characterization results, crystallographic-orientation-specific work function values via density functional theory (DFT), and temperature-limited (TL) emission physics. This previous model could only predict the TL region of the Miram curve and not the transition between TL and full-space-charge-limited (FSCL) regions. In this work, we have expanded our model to predict emission along the entire Miram curve, including the transition from TL to FSCL regions, without any empirical assumptions on work function distribution or empirical emission equations. This more advanced model provides a pathway to understanding the complex physics of emission from heterogeneous cathode surfaces, which is a key issue for the commercial production and use of thermionic cathodes in vacuum electronic devices.

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., IEEE Transactions on Electron Devices (2021)

We have constructed a model capturing the statistical nature of non-uniform thermionic electron emission from polycrystalline W cathodes. This model incorporates the proportion of different crystallographic emitting surfaces from commercial cathode samples via electron backscatter diffraction (EBSD), the effects of surface roughness from optical interferometry measurements, and surface-specific work function values calculated using density functional theory (DFT). Using this model, we aim to calculate 2D emission maps and the corresponding Miram curves for real cathodes. This model provides a pathway to understanding the complex physics of emission from inhomogeneous cathode surfaces, which is a key issue for the commercial production and use of impregnated cathodes in vacuum electronic devices.

In this work, we have constructed a simplified statistical model of electron emission for polycrystalline W cathode surfaces. Construction of this emission model requires knowledge of which surfaces are present, the area fractions of each surface, and the work function values of each surface. To create our model, we have drawn upon previous work which used Density Functional Theory (DFT) to explore the effect of O, Ba and Ba-O adsorbates on the work function and stability of W surfaces, and have combined these DFT data with experimentally-derived surface grain orientations from polycrystalline W cathodes obtained using electron backscatter diffraction (EBSD). Using our model, we have constructed semiempirical cathode Miram curves and practical work function distribution (PWFD) curves based on DFT data, EBSD data of real cathode surfaces, and well-known empirical emission equations.