Effective Current Enhancement vs. Aspect Ratio for Rectangular Ridge Cathodes

Summary form given only. In this paper we develop a theoretical model for surface field enhancement and the field emission current obtained from ridged, or "knife-edge" cathodes. As an illustrative example, we consider rectangular cross-section ridges. We use a conformal mapping approach to transform the electric field above a Hat surface to the one surrounding the rectangular ridge. From the exact expression for E(x,y) we derive an analytical scaling law for the local field enhancement factor beta_E = |E(x,y)|/E_avg vs. ridge height-to-width aspect ratio at the center and corner of the rectangular ridge. In addition, we calculate the effective enhancement factor, that one would infer from experimental measurements of field emission current versus cathode voltage. Using E(x,y) in the Fowler-Nordheim formula we obtain J(x,v) and numericallv integrate J(x.v) over the surface of the ridge. From J_total=A(beta_effE_gap)²e(-beta_effE_eff) we can find beta_eff. Again, we determine the scaling of this eff versus ridge height-to-width aspect ratio and compare it to the scaling of E. Our model can be extended to calculate the field enhancement factor of a composite geometry with multiple protrusions.