Analysis of electric field screening by the proximity of two knife-edge field emitters

The electric field of two semi-infinitely wide knife-edge cathodes with arbitrary separation is calculated by using a Schwarz-Christoffel transformation. This geometry could also represent a trench (or scratch) on a flat surface. It is found that the magnitude of the electric field on the knife-edge cathodes depends strongly on the ratio h/a, where h is the height of the knife-edge cathodes and 2a is the distance between the cathodes. When h/a increases, the magnitude of the electric field on the cathode's surface decreases. This shows the screening of one cathode by another cathode; for example, keeping the height fixed and decreasing the distance between the cathodes, the field enhancement on the corner decreases. Analytic approximations for the divergent electric field in the immediate vicinity of the sharp edge are derived for the cases where h/a >> 1, and h/a << 1. These results lead to insight on the relationship of the density of field emitter in field emitting arrays and field emission from rough surfaces. [doi: 10.1063/1.3615846]