Mathematical Modeling of Field Emission From a Microscale-Size Cathode to a Vacuum

In this article, we propose a numerical method for calculating solutions of the PDE (the heat equation for the cathode temperature and the continuity equation for the current density inside and outside the cathode) with discontinuous flux at the emission surface of a cathode placed in a vacuum. The discontinuity of the flux appears because of the Nottingham effect and tunneling of emitting electrons. In the present article, this approach is applied to mathematical modeling of field emission from a conic silicon cathode of small size (about ten micrometers) to a vacuum, but can be used in other problems with similar types of flux jumps.