Two-dimensional space charge limited current in regime between accelerating diode and drift space for sheet and circular beam

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 eKV=mv02/2) 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≡KV/Vg≥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.