Simulating multi-scale gated field emitters—A hybrid approach

Abstract

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.

Predicting space-charge affected field emission current from curved tips

“…We shall first outline the model in section II and state the various approximations that may be used to improve the prediction. This will be followed by a comparison with a published MD simulation for planar geometry and our own PIC simulations using PASUPAT 21,26,33 .…”

“…Planar PIC simulations also display oscillations 12 but these have not been observed yet in 3-D simulations using curved emitters 39 which are considerably more resource intensive. We shall henceforth compare the model predictions with PIC simulations performed using PASUPAT 21,26,33 with a field emission module based on the cosine law 26,33,40 . Our focus will be on the steady state values of the apex field and emitted current and these will be compared with the predictions of the model for various geometric diode parameters.…”

Gamow factors and current densities in cold field emission theory: a comparative study

“…It also allows for an approximate analytical expression for the net emission current using (in several cases) a knowledge of the local field variation around the emitter apex 15,16 . Finally, an analytical expression for the distribution of emitted electrons 10 also leads to a better and faster modelling of the emission process in Particle-In-Cell codes [17][18][19] .…”

Approximate universality in the electric field variation on a field-emitter tip in the presence of space charge

“…The cosine law greatly simplifies the job of calculating total emitted current from the emitter. It has a special significance for emission modeling in Particle-In-Cell 24 (PIC) simulation codes 25 . Note that the locally parabolic assumption includes all generic emitters with a smooth endcap, although the extent upto which parabolicity holds depends on the nature of the endcap.…”

“…An alternate method 25 is to use the generalized cosine law of local field variation on the surface of the emitter 22,23 . The net emitted current for a given apex field is known analytically for locally parabolic tips where the cosine law holds 11 .…”