Exact analysis of surface field reduction due to field-emitted vacuum space charge, in parallel-plane geometry, using simple dimensionless equations
Abstract
This paper reports (a) a simple dimensionless equation relating to field-emitted vacuum space charge (FEVSC) in parallel-plane geometry, namely 9ζ2θ2-3θ-4ζ+3=0, where ζ is the FEVSC "strength" and θ is the reduction in emitter surface field (θ =field-with/field-without FEVSC), and (b) the formula j =9θ2ζ/4, where j is the ratio of emitted current density JP to that predicted by Child's law. These equations apply to any charged particle, positive or negative, emitted with near-zero kinetic energy. They yield existing and additional basic formulas in planar FEVSC theory. The first equation also yields the well-known cubic equation describing the relationship between JP and applied voltage; a method of analytical solution is described. Illustrative FEVSC effects in a liquid metal ion source and in field electron emission are discussed. For Fowler-Nordheim plots, a "turn-over" effect is predicted in the high FEVSC limit. The higher the voltage-to-local-field conversion factor for the emitter concerned, then the higher is the field at which turn over occurs. Past experiments have not found complete turn over; possible reasons are noted. For real field emitters, planar theory is a worst-case limit; however, adjusting ζ on the basis of Monte Carlo calculations might yield formulae adequate for real situations.
- Publication:
-
Journal of Applied Physics
- Pub Date:
- October 2008
- DOI:
- 10.1063/1.2996005
- Bibcode:
- 2008JAP...104h4303F
- Keywords:
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- 79.70.+q;
- 77.22.Jp;
- 52.80.-s;
- Field emission ionization evaporation and desorption;
- Dielectric breakdown and space-charge effects;
- Electric discharges