Space charge limited field emission from a single sharp field emitter

Space-charge-limited (SCL) electron flow describes the maximum current density allowed for steady-state electron beam transport across a planar gap with gap spacing D and gap voltage Vg. It has been an area of active research in the development of non-neutral plasma physics, high current diodes, high power microwave sources, vacuum microelectronics, and sheath physics in plasma processing. Although the transition from field emission or Fowler Norheim (FN) law [1] to Child-Langmuir law [2] has been studied before, these models are one-dimensional and the effect of protrusive surface of the field emitters is not included. Some prior models of 2D or 3D CL law [3, 4] may be able access the importance of multidimensional space charge effects of field emission, but they still do not include the sharpness of the emitters. In this paper, we are interested to developed a protrusive (or higher dimensional) CL law to account for the sharpness of the emitters. We propose a method of calculating the maximum applied electric field Ec, which occurs when the emitter surface electric field Es is equal to the space-charge electric field Esp.