The Fowler-Nordheim equation predicts the electron current density resulting from quantum tunneling through the surface barrier of metals. The resulting values are extremely sensitive to the work function of the material and the electric field on the surface. The effective use of Trak and OmniTrak (or any numerical code) for this application requires care on the part of the user.
In the ideal situation, the emitter is a clean, homogeneous material with known and you know its exact shape. Furthermore, you have set up a mesh with high resolution near the tip for an accurate calculation of applied electric field. (In this circumstance, it may be useful to create a microscopic electrostatic solution with the BOUNDARY command.) Trak and OmniTrak calculate electric field values along each facet of the surface (with self-consistent space-charge effects), apply the Fowler-Nordheim equation, and emit model electrons to represent the predicted current density. The model gives the spatial distribution of current density over the emitter (i.e., source size) and the angular divergence of trajectories. In principle, the code also predicts the absolute emitted current for a given set of parameters.
What can go wrong with this scenario?
- The mesh is too coarse near the tip. Check the listing file. If all the current comes from one facet, then the spatial resolution is inadequate. Furthermore, the electric field calculation is probably inaccurate.
- The current predicted by the Fowler-Nordheim equation is extremely sensitive to shape imperfections or variations in the work function from surface contamination. In this case, the absolute current predicted by the codes may differ from experimental measurements.
- The F-N current density will shoot up to astronomical levels for unrealistic electric fields (i.e., a field strong enough to pull atoms from the tip). In this case, the codes will crash. A careful user will check the field levels in the equations documented in the Trak and OmniTrak manuals.
- With considerable reservations, we have included a knob in the field emission model that you can turn to match experimental conditions. The parameter ß is a field enhancement factor. The surface electric field multiplied by ß is used in the current density calculation. This feature is supposed to account for field enhancement through surface irregularities (e.g., carbon nanotubes). If you find yourself using high values of ß, it is questionable that Fowler-Nordheim emission is applicable. High observed values of current density could be the result of explosive plasma emission.
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