Incorporating Resistance Into the Transition From Field Emission to Space Charge Limited Emission With Collisions

Field emitters and microplasmas often use series resistors to mitigate the rapid increase in current density that reduces device stability. This paper investigates the impact of external resistance on the transition of electron emission mechanism as a function of applied voltage Vapp, gap distance D, and electron mobility μ. For low μ (high gas pressure), the circuit transitions from Fowler-Nordheim (FN) to space charge limited emission by Mott-Gurney (MG) and Child-Langmuir (CL) before reaching Ohm's law (OL). At higher μ, a triple point arises where the asymptotic solutions for FN, MG, and CL intersect. This triple point is uniquely defined by D, μ, or gap voltage V
_g
while also defining a specific gap impedance Z
_tp
. When R ≤ Z
_tp
, the electron emission transitions from FN to MG to CL to OL with increasing V
_app
while MG and CL are bypassed at higher R. For a given R, increasing the applied voltage or emission current causes the gap to appear as a short.