Space charge limited current for bipolar flow with uniform initial velocity

Abstract

The characteristics of space charge limited (SCL) bipolar flow in a planar ion diode with uniform initial velocity are studied in terms of the potential distribution. As a negative potential minimum occurs, a general expression for the limiting current density as a function of the normalized initial kinetic energy of electron αe and ion αi, normalized potential minimum αm, and ratio of ion current density to electron current density q is shown to be given by Je=JCL[Γ(αe,αi,αm,q)]2 in the form of a definite integral, where the function Γ is independent of the gap voltage and gap spacing of the diode, and JCL is the classical Child–Langmuir law. According to the expression, the SCL current density can be found by traversing all the values of the potential minimum. We also proposed a separation method to numerically obtain the spatial profile of the electrostatic potential as a negative potential minimum develops in a steady state. The theoretical and numerical results are compared with some special cases that have been derived previously and also verified by the extrapolation method in the presence of ions.

2D Planar PIC Simulation of Space Charge Limited Current With Geometrical Parameters, Varying Temporal-Profile and Initial Velocities

“…Also, it is observed that when the injection current is larger, the average current density is even going to decrease under the stronger impact of the self-induced electric field which pulls the electrons back, as compared between the blue line and the black line shown in Fig. 2, and we term it as the over-SCL effect, which is also demonstrated by Zhu et al [46].…”

“…As shown in Fig. 2, larger injection current will decrease the transported current at some inflection point and the point is easier to obtain [46]. Thus, in this article, we choose the maximal transported current density as the value of interest to study.…”

Space-charge affected current flow: an analytical verification solution for kinetic and fluid simulation models

“…where η * = β 2 + y * . Similar to appendix A, equation (41) provides an expression allowing us to replace α in favor of η * . For y * = 0 (which gives η * = β), equation ( 41) reduces to α = α tr as expected since the location of the local potential minimum/maximum is exactly at the injection electrode.…”

“…Here the body of work is extensive [35], and includes the derivation of the simple SCL limit (i.e. the famous Child-Langmuir law) [22,36], the effect of a finite particle injection velocity [37][38][39][40][41], 2D and 3D geometries (including cylindrically and spherically symmetric systems) [23,[42][43][44][45][46], and even extensions to quantum and relativistic regimes [47][48][49].…”

A coordinate system invariant formulation for space-charge limited current with nonzero injection velocity

“…Since J Jaffé > J B , we define this as the SCLC. Other studies have looked at other additional velocity cases such as thermal distributions [55] and bipolar flows [56], but we will only be analyzing the case of monoenergetically emitted electrons.…”