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.