A Numerical Approach for Space Charge Limited Bipolar Flow in Cylindrical diodes

In this paper, a numerical iteration approach to resolve the space charge limited (SCL) bipolar flow problem in cylindrical geometries has been developed. Such an approach is basis for the simultaneous determination of the unknown current densities and the potential distribution. We employed this method to study the characteristics of the SCL bipolar flow. By considering a cylindrical geometry with a cathode radius
$R_{\mathrm{ c}}$
and an anode radius
$R_{\mathrm{ a}}$
, the enhancement over the classical Langmuir-Blodgett (LB) law is investigated as a function of
$R_{\mathrm{ c}}/R_{\mathrm{ a}}$
. It is found that for the bipolar flow model, the SCL current density can be given by
$F\mathbf {\times }J_{\mathrm{ LB}}$
, where F and
$J_{\mathrm{ LB}}$
represent the enhancement factor on account of the influence of ions and the LB law, respectively. The enhancement factor F follows a
$R_{\mathrm{ c}}/R_{\mathrm{ a}}$
scaling and gradually converges to a constant with increasing
$R_{\mathrm{ c}}/R_{\mathrm{ a}}$
. The planar bipolar flow solution is recovered in the condition where the values of
$R_{\mathrm{ c}}$
and
$R_{\mathrm{ a}}$
are much greater than that of the gap spacing.