Abstract
We develop a general electrostatic steady-state model to calculate the two-dimensional (2D) space-charge-limited current (SCLC) with an initial velocity v0 (i.e., kinetic energy
e
K
V
=
m
v
0
2
/
2) injected from a cathode into a vacuum diode with a spacing D and a potential difference of Vg. We consider two types of beam cross section: (a) a sheet beam with a width W and (b) a circular beam with a radius R, where both W and R are larger than D in the calculation. By introducing a parameter
ε
≡
K
V
/
V
g
≥
0, we can tune the operating regime to be in between the accelerating diode (
ε
≪
1) and drift space (
ε
≫
1). Hence, we verify the 2D SCLC model by converging to some prior analytical findings such as the 2D Child–Langmuir law at ε = 0, 2D SCLC at drift space at
ε
≫
1, and their 1D limits at
W
/
D
≫
1 or
R
/
D
≫
1. Our 2D SCLC model indicates that there is a maximal value at
ε
∼
1
/
60, which is independent of W/D or R/D. Furthermore, the hysteresis behavior is also studied when the injected current density is beyond the SCLC threshold and its dynamics are mainly influenced by ε instead of geometrical effect. This model is useful in high current electron beam emission with a finite emission area and with a non-negligible arbitrary injection kinetic energy.