Nonuniform Space Charge Limited Current for 2-D Bipolar Flow in Vacuum Diode

This article presents a self-consistent model to study the space charge limited (SCL) bipolar flow for the 2-D nonuniform and uniform currents injection. This model can simultaneously determine the unknown current densities and the potential distribution. By considering a planar diode with a gap spacing
${D}$
and a finite-width beam
${W}$
, the enhancement over the 2-D unipolar flow model is investigated as a function of
${W}/{D}$
. Due to the edge effect, the nonuniform current density profiles for both the electron and ion are observed. We also found that the 2-D total SCL electron current is universally given by
${F} \times {I}_{\mathrm {2D\_{}unipolar}}$
, where
${F}$
and
${I}_{\mathrm {2D\_{}unipolar}} $
, respectively, denote the 2-D enhancement factor due to the influence of the geometry and ions, and 2-D total SCL electron current in the absence of ions. The enhancement factor
${F}$
follows a
${W}/{D}$
scaling and gradually converges to the 1-D enhancement factor
${\Gamma }$
with the increasing of the ratio
${W}/{D}$
.