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

• Published in: Physics of plasmas 2024-06, Vol.31 (6)
• Main Authors: Liu, Yao-Li, Ang, L. K.
• Format: Article
• Language: English
• Subjects: Convergence; Curved beams; Drift; Electron beams; Emission; Kinetic energy; Space charge; Steady state models; Two dimensional analysis
• ISSN: 1070-664X; 1089-7674
• DOI: 10.1063/5.0208823

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.