The field enhancement factor at the emitter tip and its variation in a close neighbourhood determines the emitter current in a Fowler-Nordheim like formulation. For an axially symmetric emitter with a smooth tip, it is shown that the variation can be accounted by a cosθ˜ factor in appropriately defined normalized co-ordinates. This is shown analytically for a hemiellipsoidal emitter and confirmed numerically for other emitter shapes with locally quadratic tips.

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Multi-stage cathodes are promising candidates for field emission due to the multiplicative effect in local field predicted by the Schottky conjecture and its recently corrected counterpart [Biswas, J. Vac. Sci. Technol. B 38, 023208 (2020)]. Due to the large variation in length scales even in a 2-stage compound structure consisting of a macroscopic base and a microscopic protrusion, the simulation methodology of a gated field emitting compound diode needs to be revisited. As part of this strategy, the authors investigate the variation of local field on the surface of a compound emitter near its apex and find that the generalized cosine law continues to hold locally near the tip of a multi-scale gated cathode. This is used to emit electrons with appropriate distributions in position and velocity components with a knowledge of only the electric field at the apex. The distributions are consistent with contemporary free-electron field emission model and follow from the joint distribution of launch angle, total energy, and normal energy. For a compound geometry with local field enhancement by a factor of around 1000, a hybrid model is used where the vacuum field calculated using COMSOL is imported into the Particle-In-Cell code PASUPAT, where the emission module is implemented. Space charge effects are incorporated in a multi-scale adaptation of PASUPAT using a truncated geometry with “open electrostatic boundary” condition. The space charge field, combined with the vacuum field, is used for particle-emission and tracking.

The apex field enhancement factor (AFEF) γa of a cylindrical emitter depends sensitively on its end-cap geometry. The hemispherical end-cap is well studied due to its simplicity, but, in general, a cylindrical emitter may terminate in a variety of end-cap shapes. It is well known that the AFEF depends on the ratio h/Ra, where h is the total height of the emitter and Ra is the apex radius of curvature. The authors show here that there can be a large variation in γa as the end-cap geometry is altered while keeping h/Ra fixed. They carry out a systematic numerical study and determine an approximate formula for γa in terms of measurable end-cap geometry parameters such as its height H, the radius of the cylinder R, and the apex radius of curvature Ra. They show that the formula is robust and can predict the net field emission current with errors generally less than 40%.

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